ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 17
This Page Intentiona lly Left Blank
Advances in
Electronics and Electron Physics EDITEDBY L. MARTON
.
Nqtiorial Bureau of Standards, Washington, D . C.
Assistant Editor CLAIRE MARTON EDITORIAL BOARD
W. B. Nottingham E. R. Piore M. Ponte A. Rose L. P. Smith
T. E. Allibone H. B. G. Casimir L. T. DeVore W. G. Dow A. 0. C. Nier
VOLUME 17
1962
ACADEMIC PRESS
New York and London
COPYRIGHT 01962, BY ACADEMICPRESSINC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS
ACADEMIC PRESS INC. 1 1 I FIFTHAVENUE
NEWYORK3, N. Y.
United Kingdom Edition
Published by ACADEMIC PRESS INC. (LONDON) LTD. BERKELEY SQUARE HOUSE,LONDON W. 1
Library of Congress Catalog Card Number 49-7504
PRINTED I N THE UNITED STATES OF AMERICA
CONTRIBUTORS TO VOLUME 17 W . G. DOW,Department of Electrical Engineering, College of Engineering, University of Michigan, Ann Arbor, Michigan J. P. HOBSON, Radio and Electrical Engineering Division, National Research Council, Ottawa, Canada
J . M . HOUSTON,General Electric Research Laboratory, Schenectady, New York FRANK E. JAUMOT, JR., Delco Radio Division, General Motors Corporation, Kokomo, Indiana
ERIC KAY, International Business Machines Corporation, Research Laboratory, San Jose, Calijornia E, V . KORNELSEN, Radio and Electrical Engineering Division, National Research Council, Ottawa, Canada
P. A. REDHEAD, Radio and Electrical Engineering Division, National Research Council, Ottawa, Canada
H . F. WEBSTER,General Electric Research Laboratory, Schenectady, New York
This Page Intentiona lly Left Blank
FOREWORD For many years we endeavored to issue at least one of our regular volumes per year, sometimes two. T h e present one was slated to appear some time in the fall of 1962 but the publication of the proceedings of the second international conference on photo-electronic image devices as our volume 16 interrupted somewhat our regular schedule and, at the same time, shortened the time lapse between publication dates of the next volumes. I sincerely hope that we will be able to bring to you in relatively rapid succession Volumes 18 and 19 together with our first supplementary volume on electroluminescence (by H. F. Ivey). At present we expect to publish the following reviews in forthcoming volumes: G. Broussaud and Simon F. P. Brooks, Jr. M. A. Biondi D. P. Kennedy G. Mollenstedt and F. Lenz F. E. Roach A. H. Schooley G. Birnbaum K. L. Bowles J. F. Dennisse M. Knoll J. L. Jackson and R. A. Piccirelli K. G. Emeleus R. G. Fawler L. S. Chernov S. H. Autler L. A. Russell J. W. Herbstreit J. Kistemaker and C. Snoek G. K. Wehner
Endfire Antennae Advanced Computer Systems Planning Atomic Collisions Semiconductor Device Evaluation Electron Emission Microscopy Night Air Glow Electronic Instrumentation for Oceanography Light Optical Masers Scattering in the Upper Atmosphere Radioastronomy Biological Effects of Atmospheric Ions Cooperative Phenomena Plasma Oscillations Electrons as a Hydrodynamical Fluid Microwave Applications of Plasma Cryogenic Magnets High Speed Magnetic Core Memory Technology Tropospheric Propagation Atoms Produced in Sputtering Experiments Cathode Sputtering
There are, however, two or three other possible contributions which, at this time, are not sufficiently well defined to be listed here. I t remains for me to express my heartfelt thanks to the authors of this volume for their excellent cooperation and to all those who helped to make it a good addition to our series.
Paris, Frunce December 1962
L. MARTON
This Page Intentiona lly Left Blank
CONTENTS
................... FOREWORD ........................... CONTRIBUTORS TO VOLUME 17
V
vii
The General Perturbational Theory of Space-Harmonic Traveling-Wave Electron Interaction W G Dow
. .
I. Introduction . . . . . . . . . . . . . . . . . . . . . . I1 Self and Mutual Beam-Circuit Capacitances. and Beam-Circuit
. Coupling Coefficients
.................. ........... . . . . . . . . . . .
I11. The Electrokinetic Bunching Equations IV The Circuit Equations . . . . . . .
. . . . .
V Complex Exponential Notation for Beam and Circuit Variable VI Voltage-Ratio Beam-Circuit Coupling Coefficient I, for a Spatially Periodic Structure . . . . . . . . . . . . . . . . . . . . VII Beam-Circuit Self-capacitance C , Governing the Space-Charge Voltage . . . . . . . . . . . . . . . . . . . . . . . . VIII Charge-Ratio and Voltage-Ratio Beam-Circuit Coupling Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . IX The Space-Harmonic Interaction Characteristic Equation . . . X The Gain. Space-Charge. and Circuit Loss Parameters . . . . XI Normalization Relative to the Gain Parameter C , . . . . . . . XI1 Expressions for Growing-Wave Hot-Circuit Gain and ColdCircuit Loss in Terms of Tube Length . . . . . . . . . . . XI11 The First-Order Quartics for Propagation-Constant Perturbation Due to Space-Harmonic Interaction . . . . . . . . . . . . XIV Graphical Study of the Perturbational Quartic Equation . . . . XV The Quartic for Drive of the Forward Total Wave . . . . . . XVI Frequency Offset Effects; Reduction to Cubic . . . . . . . . XVII The Nearly-Cubic Quartic. for Zero Values of Frequency Offset. Loss. and Space Charge . . . . . . . . . . . . . . . . . . XVIII The Electrokinetic Boundary Conditions at Electron Entrance into the If Field . . . . . . . . . . . . . . . . . . . . . . . XIX The Electromagnetic Conditions at Electron Entrance to and Exit from rf Field . . . . . . . . . . . . . . . . . . . . XX Relative Amplitudes of the Four Perturbed Waves; Voltage Gain XXI Phasor Representation of Backward-Wave Gain-Producing Interference . . . . . . . . . . . . . . . . . . . . . . . . . XXII Backward-Wave Gain Evaluation . . . . . . . . . . . . . XXIII The Start-Oscillation Conditions . . . . . . . . . . . . . . XXIV Start-Oscillation Current and Device Length; Voltage Tunability; Gain Bandwidth Product . . . . . . . . . . . . . XXV Concluding Comments . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
. . . . .
.
. . . . . . . .
. .
.
ix
1
3 5 9 14 17 29
42 45 53 60
66 68
70 76 79 84 87
93 95
105 109 112 120 121 123
X
CONTENTS
Thermionic Energy Conversion J. M HOUSTON AND H . F. WEBSTER
.
I. Introduction . . . . . . . . . . . . . . I1. Idealized Model of a Thermionic Converter I11. The Work Function of Various Surfaces .
........ ......... ......... IV. Vacuum Thermionic Energy Converters . . . . . . . . . . V. Cesium Thermionic Energy Converters . . . . . . . . . . . VI . Devices Using Auxiliary Discharges . . . . . . . . . . . . VII . Devices Using Fission Fragments for Ion Production . . . . . VIII . Applications of Thermionic Converters . . . . . . . . . . .
.
I X Summary and Future Trends References . . . . . . . .
............... ...............
125 130 142 154 169 193 195 195 199 201
Thermoelectricity FRANK E . JAUMOT. JR. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . I1. Basic Considerations . . . . . . . . . . . . . . . . . . . I11. Materials . . . . . . . . . . . . . . . . . . . . . . . . IV. Practical considerations . . . . . . . . . . . . . . . . . V Thermoelectric Applications . . . . . . . . . . . . . . . V I Theory and Problems . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
. .
207 208 215 220 233 235 242
Impact Evaporation and Thin Film Growth in a Glow Discharge ERICKAY I. Introduction . . . . . . . . . . . . . . . . . . . . . . 245 I1. Emission of Charged Particles from Metal Surfaces by Energetic Particles . . . . . . . . . . . . . . . . . . . . . . . . 247 I11. High Vacuum Impact Evaporation (Cathode Sputtering) . . . . 257 IV. Glow Discharge Characteristics . . . . . . . . . . . . . . 284 V. Nucleation and Film Growth in High Vacuum Environment . . 289 VI . Film Growth in Glow Discharge Environment . . . . . . . . 297 VII . Reactive Impact Evaporation . . . . . . . . . . . . . . . 311 References . . . . . . . . . . . . . . . . . . . . . . . 317 Ultrahigh Vacuum P. A. REDHEAD. J .P. HOBSON. E. V. KORNELSEN I. Introduction . . . . . . . . . . . . . . . . . . . . . . I1. Physical Processes . . . . . . . . . . . . . . . . . . . . I11. Technology of Ultrahigh Vacuum . . . . . . . . . . . . IV . Applications . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . AUTHORINDEX SUBJECT INDEX
......................... .........................
.
323 325 371 417 422 433 445
The General Perturbational Theory of Space-Harmonic Traveling-Wive Electron Interaction W . G . DOW Department of Electrical Engineering. College of Engineering. University of Michigan Ann Arbor. Michigan Page i Self and Mutual Beam-Circuit Capacitances. and Beam-Circuit Coupling Coefficients ...................................................... 3 The Electrokinetic Bunching Equations .............................. 5 The Circuit Equations ............................................ 9 Complex Exponential Notation for Beam and Circuit Variable .......... 14 Voltage-Ratio Beam-Circuit Coupling Coefficient 5, for a Spatially Periodic Structure ........................................................ 17 Beam-Circuit Self-capacitance C,. Governing the Space-Charge Voltage . . 29 Charge-Ratio and Voltage-Ratio Beam-Circuit Coupling Coefficients 42 The Space-Harmonic Interaction Characteristic Equation . . . . . . . . . . . . . . 45 The Gain. Space-Charge. and Circuit Loss Parameters . . . . . . . . . . . . . . . . . . 53 60 Normalization Relative to the Gain Parameter C, ...................... Expressions for Growing-Wave Hot-Circuit Gain and Cold-Circuit Loss in Terms of Tube Length ............................................ 66 The First-Order Quartics for Propagation-Constant Perturbation Due to Space-Harmonic Interaction ........................................ 68 Graphical Study of the Perturbational Quartic Equation . . . . . . . . . . . . . . . . 70 76 The Quartic for Drive of the Foward Total Wave .................... Frequency Offset Effects; Reduction to Cubic ........................ 79 T h e Nearly-Cubic Quartic. for Zero Values of Frequency Offset. Loss. and Space Charge .................................................... 84 The ElectrokineticBoundary Conditions at Electron Entrance into the rf Field 87 The Electromagnetic Conditions at Electron Entrance to and Exit from rf Field .......................................................... 93 Relative Amplitudes of the Four Perturbed Waves; Voltage Gain . . . . . . . . 95 Phasor Representation of Backward-Wave Gain-Producing Interference . . 105 Backward-Wave Gain Evaluation .................................... 109 The Start-Oscillation Conditions .................................... 112 Start-Oscillation Current and Device Length ; Voltage Tunability; Gain Bandwidth Product ................................................ 120 Concluding Comments ............................................ 121 References ........................................................ 123
I . Introduction
.
I1
. . .
111 IV V VI
.
VII. VIII IX X
. . .
XI .
.
XI1
. XIV . XV . XVI . XVII . XVIII . XIX . XI11
xx .
.
XXI XXII XXIII XXIV
. . .
XXV
.
....................................................
1
2
W. G. DOW
I. INTRODUCTION There will be developed in this paper the small-signal equations for traveling-wave interaction of an electron beam with an adjacent spaceharmonic circuit, including the effects of competition between the first forward space harmonics of the forward and backward total waves on the circuit. The basic equation for the perturbation of the propagation constant is found to be a quartic, having as its most important given quantity a frequency offset parameter, which for this study serves the same purpose that Pierce’s overdrive parameter b serves in the study of forward-wave interaction in a helix (I).Only if the slow wave on the space-harmonic structure is not very slow, or the gain parameter is very small, or the frequency offset parameter numerically substantial, does the quartic reduce to the familiar cubic equations used by Pierce (I) and Johnson (2). When the frequency offset is chosen to result in forward-wave amplifier action, the limiting cubic is that obtained by Pierce, whereas if the frequency offset calls for backward-wave amplification, the limiting cubic is that used by Johnson.If the frequency offset is numerically small, in general the complete quartic must be used. There are then four interacting waves, usually either three forward and one backward total waves, or three backward and one forward total waves. As the author has pointed out heretofore (3),the fact that all spaceharmonic components or none must be present in space-harmonic propagation, means that any reasonably complete analysis of spaceharmonic interaction must deal with the total wave on the structure. This total wave, comprising the assembly of all space-harmonic field configurations, has in general a phase velocity substantially greater than that of the interacting space harmonic, with a correspondingly greater wavelength. For a genuinely slow-wave structure, the first forward space harmonics respectively of the total forward and backward waves are so near together in wavelength and in /?-values that it is not legitimate to assume a priori interaction only with one or the other. This treatment therefore deals with both of the first forward space harmonics, and establishes the criteria to determine whether or not interaction with one total wave or the other dominates. A very simple circuit model is used for illustrative purposes, to permit clarification of concepts as to the relation between the space charge of a bunched beam and the detail geometric aspects of the spatially periodic circuit structure. T h e analytical model is an annular beam idealized into a bunched charged sheet. For the greater part of the study a quasistationary state is assumed, in which the dimensions are small enough
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
3
relative to the free-space wavelength to permit use of the Laplace equation in the region inside the beam, and between the beam and the adjacent circuit digits.
11. SELFAND MUTUALBEAM-CIRCUIT CAPACITANCES, AND BEAM-CIRCUIT COUPLING COEFFICIENTS When a longitudinally extensive bunched beam exists near and parallel to a circuit structure, the rf effects of the beam on the circuit are conveniently evaluated by determining the induced charge and the beam-location potential variation caused by the beam when the circuit structure is an equipotential at zero potential. The use of scalar potential concepts here implied corresponds to the quasi-stationary state assumption introduced above. There will be used the following capacitance definitions, as applying in the case of an axially sinusoidal variation of the beam density along the circuit direction. Amplitude of the beam charge density variation
c, = Amplitude of the voltage vari-
(this is the beam selfcapacitance in the presence of the circuit
ation along the beam, the circuit being an equipotential Amplitude of the induced cir-
ation along the beam, the circuit being an equipotential.
The first of these proves to be of major importance as to the effects of the beam's own space charge on traveling-wave interaction. In a space-harmonic circuit the numerator of Czgis the amplitude of a long-wavelength total-wave charge variation, whereas the denominator is the amplitude of a short-wavelength space-harmonic voltage variation. For C,, both numerator and denominator refer to properties of a spaceharmonic variation. In analyzing the closed-loop spatial feedback which occurs in various microwave devices employing traveling waves, it is convenient to use charge-ratio and voltage-ratio beam-circuit coupling coefficients 5, and I,, usually expected to be equal in value, and always opposite in sign.
W. G . DOW
4
They are fractional quantities, typically between 0.5 and 1.O, defined as follows (3): Amplitude of the voltage variation along the beam location, no beam present 5v
=
Amplitude of the voltage variation along the circuit Amplitude of the induced surface charge density variation, the circuit being an equipotential
I, = - Amplitude of the beam charge
The voltage-ratio beamcircuit coupling coefficient, numerically positive
The charge-ratio beamcircuit coupling coefficient, numerically negative
density variation,
The denominators of the two Eq. (1) expressions are the same, and their numerators are the charge amplitudes in Eq. (2b). Therefore
For any particular beam and circuit geometry, 6, is obtained by a study of the field due to the circuit in the absence of the beam, whereas Cia, C,,and from them ec, are obtained by a study of the fields in the presence of the beam with no wave on the circuit, the circuit electrode system being at a common zero potential. In a space-harmonic circuit, the numerator oft,, and the denominator of &, ate amplitudes for the short-wavelength space harmonic type of variation, whereas the denominator of I,, and the numerator of tC, are amplitudes for the long-wavelength total-wave type of variation. The spatial feedback behavior that will be analyzed in terms of these Eq. (l), Eq. (2) quantities is as follows: A propagating signal characterized by a long-wavelength voltage on the circuit makes its contribution to establishing a short-wavelength propagating rf voltage at the beam location, this contribution being smaller than the circuit voltage by the factor I,. If there is appropriate near-synchronism of advance between the beam-location propagating potential wave and the dc velocity of the electrons of the beam, this rf potential in the beam due to the circuit results in short-wavelength bunching of the beam. Because of the self-capacitance C,, the bunched beam makes its own contribution to the shortwavelength beam-location rf voltage. The short-wavelength bunches electrostatically induce long-wawelength rf surface charges in the circuit, opposite in sign to the inducing bunch charges in the beam. The amplitude of thk long-wavelength induced charge in the circuit is less by the factor IC than the amplitude of the short-wavelength rf inducing charges in the beam. The movement along the circuit, at t h e fast long-wavelength propagational velocity, of the self-induced and space-charge-induced charges together, constitutes an rf current
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
5
whose flow through the circuit’s characteristic impedance produces the advancing rf potential wave in the circuit, that this argument initially assumed to be present; thus the spatial feedback loop is closed. T h e presence of the long-wavelength space-charge-induced surface charges in the circuit may, if their phase relation to the circuit potential wave is appropriate, cause there to be growing and declining as well as uniform-amplitude waves. T h e space-charge voltage that exists due to the beam’s bunches, while modifying the over-all behavior, does not contribute directly to the transfer of energy from the beam to the circuit. Power gain can occur either because of the presence of a growing wave, or, as in the Crestatron (4), as the result of a standing-wave pattern due to interference between unchanging waves of differing phase velocities. In the Haeff electron wave amplifier each of two beams serves in effect as the circuit for the other, and there can be growing, declining, and unchanging waves, with power gain due either to a growing wave or to a standing-wave pattern (5, 6).
111. THEELECTROKINETIC BUNCHINGEQUATIONS Existing published derivations of the ballistic or electrokinetic equation for electron-beam rf motions are for the most part inadequate because of their failure to indicate clearly the very-small-signal nature of the small-signal limitation. A derivation that does indicate this is initiated by the following statement of Newton’s force law and the Lorentz equation in combination, for a confined-flow annular beam in which only x-directed electron motion is permitted:
Here and later, in mks units,
U,, u, up, will symbolize respectively the electrons’ dc velocity, their total local and momentary velocity, and its varying component ; V,, v, vq, symbolize the similar components of the beam-location potential, T,,7 , T ~ symbolize , the similar components of the circumferentially integrated beam charge, units being coulombs per meter ; the dc quantity symbol T, is to be interpreted as Greek capital tau, corresponding to the use of lower case tau for the corresponding variational quantity, I,, i, iq, are the similar components of the beam’s convection current, ih, T ~ vh, , symbolize local and momentary values respectively of the circuit’s total-wave current, total-wave self-induced charge, and total-wave voltage (the between-wire voltage for a balanced line),
6
W. G . DOW
including contributions from all space-harmonic components in combination ; T~ is the local and momentary value of the total-wave induced charge on the circuit, in coulombs per meter of axial length; me,qe, symbolize respectively the mass and absolute value of the charge of an electron; Y, z, t , symbolize respectively the independent radial and axial space variables, and the time variable; f,w, symbolize the signal frequency respectively in cycles and radians per second; V g m , Tqn’, I i t n are the anlplitudes of ih, vIL, 7iL, Ihm, v h m , Thnr, iq, wg, T ~ it, , and similarly for other variables; the T’s are to be interpreted as Greek capital tau’s. U , 8, A, symbolize respectively the phase velocity, the radian wave number, and the absolute value of the wavelength, of the total-wave rf propagation, involving all space harmonics in combination, for the entire beam and circuit configuration. From the above symbol definitions,
and in combination, since i = ru,
The important dc relations are
As yet, no small-signal limitation has been imposed. It will later be postulated that uq ub, rq Tb, and iq
<
<
7
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
relation employed, the T ~ term U ~ in Eq. (5f) drops out as being of secondorder magnitude; subtraction of Eq. (6a) from Eq. (5f) then gives: ja = TbUq
+
UbTqr
(the
small-signal
(7)
As to signs, note that and Tb are inherently numerically negative, and in the present study Ub is positive. U and 9 , are positive for a forward total wave, negative for a backward total wave. The usual expansion converts the left-side total derivatives of Eq. (4) to the sum of two partials; thus, u being a function of x and t,
Use from Eq. (5) of u =
u, + uq and
v =
vb
+ vq converts this
to
If the rf velocity component uq is small enough so that:
% the force law statement becomes
The left side of this equation has a form that occurs frequently enough to warrant the use of a special symbolism, that permits restatement of this as
Note that the limitation is not a severe enough restriction validate Eqs. (11) and (12))'
(13)
ub(to
because the two terms on the right of Eq. (10) have opposite signs in the end result, and are not greatly different in magnitude. The actual restriction necessary to satisfy Eq. (lo), thereby validating Eq. (1 l), is that uq U , UbJ where U , is the dc excess velocity, defined as
< <
ue= ub - u;
(this defines the excess velocity
u,).
(14)
8
W. G. DOW
Thus the small signal equations used here, and those used by Pierce ( I ) and Johnson (2), are valid only as long as UQ
< UP..
( 154
Also, because of the near-synchronism requirement for bunching,
<
ue
Thus in combination uq
(15b)
ub*
< < ue
(15c)
ub
is the over-all small-signal requirement. In the present symbolism, the one-dimensional equation for continuity
This expands and rearranges into the following:
aT
aT
u-+-=-7ax at
au ax
*
Use of the Eqs. (5a) and (5c) forms T = T, gives, after expansion and rearrangement, a% ax
-
LITb
aTP
Tb
ax
+--TbI
+
T~
and u = U,,
aTa +--+--"up aT, at
Tb
aZ
auq
Tb
ax
*
+ up (17)
The last two terms on the right are nonlinear; the expression is linearized by introducing the small-signal restriction that the sum of these two terms must be small relative to the sum of the remaining terms on the right-hand side. The inequality stating this restriction takes the following form after division by U,:
The earlier Eq. (10) restriction applying to Eq. (8) can be similarly
I n order to compare these, the limitation uq as
< U,< U, can be restated
It turns out that this satisfies both of the Eq. (18) expressions.
9
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
Use of the Eq. (18a) restriction permits writing Eq. (17) as
a@,
- __ = ub 82
aTq
Tb (82, Ubt)
a
If Eq. (19), and therefore also Eq. (ISa), are satisfied, this expresses the small-signal form of the equation of continuity of charge, for the present model. The next step in the analysis of the electron motion is to eliminate the velocity upbetween the Eq. (1 1) Newton's force law statement and the Eq. (20) statement of the continuity of charge. This is accomplished by applying the two-term Eq. (12) operator to Eq. (1 I), and the operator a/az to Eq. (20), then adding. The result can be expressed in the two following alternative forms:
and QeTb a2V,
me
822
'
overrun bunching equation (the electrokinetic for the stationary frame of reference )'
(2 1b)
Validity of these is subject to the Eq. (19) small-signal limitation. Employment of Eq. (21) in subsequent analytical steps is facilitated by eliminating T,, qe, and me from the bunching equation, by using the Eqs (6a) and (6b) relations I,, = UbTb and q e v b = (1/2) m,ui. The result is
IV. THECIRCUITEQUATIONS For the cold transmission circuit, no beam present, appropriate total-wave differential-equation relations are, in general (7)
10
W. G . DOW
I n these transmission-line forms, per meter of axial length, C,,, L,, R,, are the distributed capacitance, inductance, and effective series resistance, in farads per meter, henrys per meter, and ohms per meter.
As discussed later, L h and Rh particularly are derived rather than directlyspecified properties of the transmission structure. Note that c, is a circuit voltage, not a space potential; for the capacitively-loaded balancedpair model used in the present study, it is the between-wire voltage difference. More generally, it can describe the space-time variational swing of the modulation envelope the fast total wave imposes on the voltages between the spaced elements of the structure. Equations (23) are reduced to a single equation in v h by applying a/at to the first member, a/& to the second, then eliminating ai,,/az and a2il,,/az8tby substitution. T h e result is, for the cold circuit
which is expressible in the Eq. (12) notation as
Here, because L, is not readily determinable,
Il
/ a h I is now defined as being the measurable U,,
(244
where as appears shortly, +Uo, -?Yo, are the directly measurable forward and backward cold-circuit total-wave phase velocities, that is, these total-wave phase velocities in the absence of interaction with a beam. I n applying customary transmission-line principles it is desirable here, because of the central part played by electrostatic charge induction in the perturbations due to electron interaction, to introduce the characteristic impedance as a voltage-current amplitude ratio. First there is introduced the symbolism: =
&
v,,m/I,,m;(thisdefinesZ&);
(254
the f signs permit application individually to the forward and backward total waves. Now define for this study the cold-circuit characteristic impedance 2,; thus
zo
3
(the absolute value of the cold-circuitZ h ) .
(25b)
11
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
In the usual way it is found that for microwave circuit designs, nearly enough,
I n relation to the preceding equation, and because Lh is not easily measurable or calculable,
Idml
is now defined as being the measurable 2,.
(25d)
From this and Eq. (24c),
which permit evaluating C, and L,&from measured values of the totalwave 2, and U,. For either the hot or the cold circuit, and applying to either the forward or backward total wave described basically by Eqs. (24), let 'I = the complex propagation constant, = the radian wave number, 01 = the growth or decay exponent factor, defined for this paper as being numerically positive for a total wave whose amplitude is an increasing function of axial distance z. Thus P
= j/3
- CL.
Use of the usual transmission-line principles now shows that, for usual microwave transmission line design magnitudes,
For any propagating wave, the phase velocity U is w/@. Therefore from Eq. (26b), for the cold circuit, this shows that V,is the numerically positive phase velocity of the forward cold-circuit total wave
(27)
W.
12
G. D O W
With the nature of U, thus established, let this defines Is, as the (measurable) absolute value of the radian wave number of either the forward or backward cold-circuit total wave
To3 jP0; (a wholly imaginary quantity).
(28b)
Definitions and descriptions based on Eq. (26c) and Eq. (25d), and related to the fact that Rh is not directly measurable, are ”0
= (the measurable cold-circuit value of
-u),
(294
With a,, Uo, and Z, known, R, and the “time constant” 2L,IR, are determinable. The cold-circuit self-induced circuit charge depends on the circuit voltage as T h = chvh, (30) where c h is as used in Eq. (24). This can now be employed in an extension of the Eq. (23) circuit differential-equation pair, to express the electromagnetic effects of the over-running beam on the propagating wave. To extend the circuit differential equations as thus required, and presuming as a first approximation that the interaction with the beam causes no changes in c h j Lh, R,, so that U,, Z,, Po,Po, a,, have the meanings and values identified by earlier equations, the first step employs Eq. (30) to convert Eq. (23a) to:
Then note that in the presence of the beam the charge along the circuit includes the beam-induced charge T~ as well as T ~ so, that this becomes
a;, az
+
_-
Expansion and re-introduction of
at Th
Ti)
(32)
= c h v h gives this the useful form
(33)
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
13
to be employed in combination with the restated Eq. (23b):
-av, - - - i,R,
az
a;,
- L, __
at
(34)
*
T o solve this pair, apply ajat to the first, 3/33 to the second, then eliminate ai,,/az and a2ih/azat by substitution. T h e result can be stated as
It is useful to restate this in a form describing the second time derivative of a voltage, as follows:
Use now of Eq. (24c) and Eq. (25d) convert this to
This expresses the circuit behavior as modified by the charge electrostatically induced into the circuit by the bunched beam. T h e dependence of the magnitude of T~ relative to Th, in Eq. (32), can now be described in steps, as follows, neglecting the space-charge voltage: (a) Th depends linearly on v h ; thus: Th = w,C,; (b) w, depends linearly on v,; thus: vq = .$,,v,'; (c) T~ depends, linearly in the small-signal theory, on wu according to Eq. (22); (d) T~ depends linearly on r g ; thus: T~ = & T ~ .
For a spatially periodic structure the (b) and (d) relations are not literally correct as stated, in that vq, T~ have the space-harmonic periodicity and q, T ~ ,Th, the total-wave periodicity. They are essentially correct, however, as relationships between amplitudes. I n summary, then, as to logical dependence, neglecting the space-charge voltage,
''''
Ti -rh
Ch
X (the Eq. 22 electrokinetic dependence of
T~
on wa).
(38)
Note that Eq. (37) is not subject to a small-signal limitation; the electromagnetic behavior that it describes is linear for large as well as
14
W. G. D O W
small signals; it is only in regard to the electrokinetic behavior that nonlinearity appears for large signals. In using the Eq. (21) and Eq. (37) differential equations it is necessary to distinguish between the beam-location voltage due to the circuit and that due to the beam’s space charge. Thus where
= vd
+
vc,
(394
wd, we, symbolize the local and momentary contributions to the rf beam-location voltage vq, due respectively to the rf circuit voltage v, and to the beam’s own space charge 7,.
The amplitude V,, of v, is related to the amplitude T,, of the charge according to Eq. (la), that is
v,,
TI?,
=-
el ’
(39b)
and the amplitude V,,, of w, is related to the amplitude V,,, of the circuit voltage according to Eq. (2a), that is vd$n = tuVAnr.
(394
The Eq. (39a) sum converts to a complex-quantity sum rather than to an amplitude sum, because w, and w, may not have a common phase.
V. COMPLEX EXPONENTIAL NOTATION FOR BEAMAND CIRCUIT VARIABLES An essential aspect of the small-signal analysis is that all rf variables, including the displacement of an electron from the position the electron would have if the electron stream were not disturbed by an rf wave, behave as traveling waves, and are therefore expressible in complex exponential symbolism. The notation to be used will be illustrated with reference to the circuit voltage vh. For a growing wave, the physically real dependences of this voltage on distance x and on time t are, in terms of w , 8, and the exponential growth factor a,
also expressible as
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
15
Here, for this backward-wave amplifier study, p is normally negative, and Va, Vho,are the rms values of wk, respectively at any z, and at the rf output location z = 0,where the electrons enter the rf field, and V,, is the amplitude of the voltage of the growing wave at any value of z. 6,,,is the radian-measure phase angle by which the rf circuit voltage at z = 0 leads any rf quantity having a zero phase angle; the magnitude of ev0is governed by the choice of the moment for which t = 0. I n the exponential symbolism Eq. (40) becomes, in terms still of a physically real entrance amplitude: = the
real part of
2/z v,, exp (jut -jpz + az +jevo).
(43)
In terms of complex amplitudes this becomes v,, = the
Use of
real part of
fi P h o exp ( j u t -jpz
+ az),
(44)
I' = jp - a from Eq. (26a) converts this to: vh = the real part of
1/2 P,, exp ( j u t - I%).
(45)
Here and later Va, Vao, are the complex rms values of vh, respectively at any z, and at the z = 0 rf output location. In subsequent phasor equations the reader is expected to supply the phrase "the real part of"; thus the last equation for the total wave becomes: P)h = 4 2 P,, exp ( j u t - rz). (46)
In general, in a small-signal study only, several total waves are considered to exist simultaneously, the complete behavior being given by their linear algebraic sum. The various I"s have variously positive and negative values for their a's and P's. As a very simple illustration of this, for the cold rf circuit, no beam present, but with resistive loss, and using = ;Po from Eq. (28):
ro
1
The complex propagation constant for the forward total wave in a cold lossy spatially periodic structure, having a phase velocity U,, in the +z-direction, wavelength A,,
i i
The complex propagation constant for the backward total wave in a cold lossy spatially periodic structure, having a phase velocity -Uo,being in the -2-direction, wavelength A.
(474
=jPO =
r o + "0.
=
-iflo
-
~
(47b)
16
W. G . DOW
Here Po ao, U , are the inherently numerically positive quantities that, for the cold-circuit forward total wave, describe the respective values taken on by b, -a, and U. For the circuit voltage, with no beam present, but taking into account the decline in voltage due to the resistive circuit loss, Eq. (46) becomes, expressed for both waves,
vtL= 2/z ?,o,
exp [ j u t - (jso
+ a0)Z];
(the forward total wave);
(48a)
v,, = lh Ptfo,exp [jut - (-jp0 - a,,) z ] ; (the backward total wave); (48b)
+
thus r is Po a0 for the forward, and - To- a. for the backward, and 8,,, indicate cold-circuit total waves; the f and b subscripts in that these complex quantities may at z = 0 differ both as to magnitude and phase. T h e several total-wave quantities of interest are expressed as follows, for any one of the total backward waves that this small-signal analysis describes as being present simultaneously: v h = 2/z P,foexp (jut- r x ) ; (the circuit voltage), =d
T/&=
exp (jut - ' z ) ;
' h o
d 'hO
exp (.iut- rz);
(494
current in the circuit due to the (movement at velocity U of both T, and charge (self-induced on the circuit
7)
(49b) (494
) 9
The several first forward space harmonic quantities related to the particular I' total wave are expressed as follows:
axial electric field at the beam location
eQ =
2/z ' g o
rq =
~ T , ,exp o ( j u t - Tnz);(rf charge in the beam),
u, =
V'2 OQ0exp ( j u t - rn+
exp ( j u t - 'Tl');
(
1,
rf electron velocity in the beam
(
),
(49h)
(49i) (49d
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
17
Here r, is the complex hot-circuit propagation constant for all variables at the beam location, being that of the first forward space harmonic, T h e physically real amplitudes will sometimes be used, given for typical quantities as follows:
and similarly for other rf quantities.
VI. VOLTAGE-RATIO BEAM-CIRCUIT COUPLING COEFFICIENT 4, FOR A SPATIALLY PERIODIC STRUCTURE (3)
A backward-wave amplifier or oscillator, the latter given the name “Carcinotron” by the French (a),employs interaction between a forward electron beam and the first forward space-harmonic component of a backward rf wave in a spatially periodic structure ( I , 2). T h e model used in this paper’s backward-wave analysis employs a numerically positive dc electron velocity U,, the electrons moving in the +z direction, and a backward total rf wave for which the group velocity and the total-wave phase velocity, also their corresponding radian wave numbers, are numerically negative. T h e backward-wave amplifier rf input: is at the point of beam exit from the rf field, and the rf output at the point of beam entrance. Thus on first entering the field the electrons are exposed to a strong rf field, rather than to a weak rf field as in a forwardwave amplifier. With a given input frequency, the total-wave and space-harmonic phase velocities are determined. When the beam energy Va is such as to make the associated electron velocity U, approximately equal to the forward phase velocity of the backward wave’s first forward space harmonic, there will be electrokinetic interaction which results in bunching of the beam, and in the simultaneous appearance-for the smallsignal case-of several waves having different complex propagation constants. In a backward-wave device the presence of these waves can result in significant gain even if all of the waves are of the unchanging (i.e., constant-amplitude) type. Typically gain exists in a backward-wave device primarily because of a standing wave due to interference between the several waves having slightly different wavelengths. Ideally the input appears at or near a node, and the output at or near an antinode, of this standing-wave pattern. When the d-c current reaches what is
18 W. G. DOW
I
-1
I
0
-1
s
it
0
>
5:
nu
Ez
a
s
z
W
v)
z
a
called the “starting current” value, the voltage at the node vanishes, so that the gain becomes infinite; the device is then an oscillator. In the interest of clarity, the backward-wave analysis will be applied primarily to a circuit structure model (Figs. la, Ib), consisting of a balanced pair of wires capacitively loaded with thin vanes, the vanes having holes of appreciable size (3). T h e electron beam is idealized into a bunched cylindrical charge sheet of zero thickness.
w
0 0 a
-I J
a I (3
3
0 a T
iW
-1 0
r
FIG. 1. Simple space-harmonic structure, consisting of a balanced pair capacitiveiy loaded with interleaved vanes, with an annular beam of radius Y, passing through holes of radius Y, in the vanes. In (c) the vanes are shown in a waveguide rather than mounted on a balanced pair.
t
w
o va.
B
T F O R THE FORWARD \TOTAL WAVE
FOR THE BACKWARD TOTAL WAVE
w vs.
7
B
x FOR THE
SPACE HARMONICS
0
-Bvs. B diagram for a capacitively-loaded balanced pair, as for example Figs. la and b, with phase-velocity slope , all phase velocities of space harmonics of the buckward lines shown for a particular value of w . Note that U,,, Us*,U,,, U I 2are total wave. whose total-wave phase velocity is -Uo, this being also the group velocity for this frequency for this structure. All of the harmonics are present if any of them are.
FIG.2. The
w
20
W. G . DOW
I n relation to such a structure, let
rj = the radius of the holes in the vanes, rb = the radius of the bunched charge-sheet beam, A, = the fundamental spatial interval of the structure, & = h / h d = the radian wave number corresponding to this Spatial interval, U, = w/& = the wholly fictitious phase velocity corresponding to the wavelength Ad and any radian frequency w ; no propagation at this phase velocity occurs. Figure (1 c) suggests a circuit consisting of vanes supported in a waveguide structure rather than from wires. Figures 2 and 3 are w/P diagrams for
FIG.3. T h e w vs. j3 diagram for a spatially periodic structure having both a low-frequency and a high-frequency cutoff; as for example the Fig. lc vane-loaded waveguide. There has been chosen for representation a propagation for which the total wave having a forward total wave phase velocity has also a forward group velocity. For the next higher mode, the total wave having a forward total wave phase velocity would have a backward group velocity. Note that in this figure U i l ,Uie, Un, Urs,are all phase velocities, for a particular value of w , of space harmonics of the backward total euaoe, whose phase velocity is -Uo, group velocity U,, for this frequency; thus -Uo and U, are both numerically negative. All of the space harmonics are present if any are.
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
21
this vane type of spatially periodic circuit structure having respectively the balanced pair support of Figs. 1a and I b and the waveguide supports of Fig. 1c. T h e usual cylindrical-geometry differential equations for electromagnetic wave propagation apply in the hole through which the beam passes, and within each between-vane region individually out to where significant field modifications appear due to the supporting wires and attachments thereto. Electromagnetic wave propagation in a spatially periodic structure is characterized by two primary quantities, the total-wave phase velocity U,and the group velocity U,, which may or may not have a common direction (9, 10). Corresponding to the total-wave phase velocity is the
AXIAL DISTANCE, z a. D-C VOLTAGE VdSc BETWEEN THE WIRES
AXIAL DISTANCE.2 b. A PROPAGATING WAVE OF TOTAL-WAVE WAVELENGTH 10 ON THE STRUCTURE.
FIG.4. Axial variation of on-axis voltage for a pinhole version of the Figs. la and I b capacitively-loaded balanced pair, (a) with a dc voltage across the pair, (b) with an rf wave of total-wave wavelength A, propagating along the structure.
22
W. G . DOW
total-wave radian wave number 8, which is numerically positive if U is positive, and negative if U is negative. I n all cases U = w / F , and U, = dw/dP. T h e dashed-line curves in Figs. 2 and 3 illustrate the relations between these various total-wave quantities and the radian frequency w for circuits in which U and U, have a common direction. I n the cold circuit, U = f U,. Figure 4a illustrates, for a dc potential V,, between the wires, the potential distribution along the axis of a pinhole version of Figs. l a and Ib, in which the r j hole radius has become vanishingly small. This dc potential distribution, measured relative to a zero value at I = 03, has the Fourier series form:
If the between-vane spacings are precisely equal, all the even harmonics vanish, and the first three odd-harmonic coefficients are: A,
8
=
2;
A,
8
=
g;
A,
=
8
etc.
If the cold-circuit voltage between the wires has the form v h = V,, cos (wt poz), corresponding to the presence of a backward total wave propagation, the pinhole structure's on-axis voltage becomes, with the phase angle Os0 included,
+
This is illustrated for a particular moment by Fig. 4b. T h u s the entire Fourier-harmonic potential structure is subjected to a space and time modulation envelope of amplitude (1/2) Vhm. For a circuit like Figs. l a and Ib, it is easy to visualize the voltage v,-in this case the voltage between the supporting wires- that has the total-wave form vh = V,,m cos ( w t &z). For dispersive structures such as Fig. Ic this is more difficult, although it can be done by choosing an appropriate transverse-voltage line integral path. However, the Fig. 4b type of modulation envelope applies to all the electrical variables that are subject to the space-harmonic variation. For this reason the total-wave phase velocity U,, is always measurable. Expansion of Eq. (53) in the usual way gives the space-harmonic components of v. For the pinhole structure the expansion, through the
+
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
23
first forward and first backward space harmonics of the forward total wave only, gives
Thus the first forward and first backward space harmonics of the total wave have a common amplitude. This treatment assumes the vh circuitvoltage modulation to apply throughout the region between the vanes ; in fact it governs only the potentials of the vanes. For the true potential pattern the sawteeth of Fig. 4b are straight lines as in Fig. 4a, whereas Eq. (53) attributes to them a curvature obeying the modulation. This Pd, and will be used in this approximation is acceptable as long as /3 section. For a space-harmonic voltage, or field, or current, etc., both members of each space-harmonic pair are present at equal amplitudes if any member of any of the pairs is present. Each pair combines in a way illustrated as follows:
<
T h e similar combination of every other pair gives the same dependence on cos ( w t - Pz), which describes propagation with the phase velocity U . Thus this phase velocity is an attribute of every pair, and is therefore a primary property of the total wave; it is a somewhat more useful property in analytical processes than is the group velocity U,, which is also a property of the total wave. For backward-wave lossless cold-circuit propagation along this structure, the total-wave potential difference i v,,/2 between either supporting wire and infinity is,
For the simple circuit of Figs. l a and lb, the total-wave phase velocity for the low-frequency nondispersive behavior is determined by the inductance of the supporting balanced pair and the capacitive loading from the vanes. Each of the space-harmonic components has a phase velocity much slower than the velocity of light-the reason for using a structure of this kind is that its space harmonics can have phase velocities
24
W. G . DOW
approximately the same as electron velocities obtainable with moderate beam voltages. Even when the vanes are supported within a waveguide, as in Fig. lc, the phase velocities of the space harmonics are slow, partly because the total-wave phase velocity is reduced by the capacitive loading of the vanes, partly because the space-harmonic phase velocities are much slower than the total-wave value. For the large-hole version of Fig. I , the Fourier treatment parallels that for the pinhole version. Because the spacings from vane to vane are considered small relative to a quarter-wavelength of free-space propagation at the operating frequency, the Laplace equation rather than the wave equation will be used for the potential fields. In the open channel, r < Y ~ , the cylindrical-geometry form applying is
Because the axially periodic potential variation is at a minimum value along the axis, becoming greater near the vanes, the solution is an infinite series of terms having cosine-factor axial variance and I , Bessel function radial dependence, the spatial periodicity being dependent on vane spacing. T h e argument factors are successively &, 313,, 5&, etc., as symmetry removes the even-harmonic terms, For the region farther out than the open channel, so that r > r j , the field within each individual between-vane region satisfies either the cylindrical Laplace equation, or, if deep enough, the wave equation, but the solution for each region is not continuous with those in the adjacent regions on either side. Therefore an axial Fourier expansion of the total outer region potential from end to end of the structure gives harmonic components that are like those of Eq. (51) and Eq. (54)in not describing a continuous Laplacian or wave-equation field. For the Laplacian case, within each individual between-vane region the KOradially declining Bessel-function type of field solutions applies, describing the fringing field that provides the match from the discontinuous sawtooth voltage variation to the axially continuous field in the open channel. Figure 5 illustrates properties of this field for the non-propagating condition in which there is a dc voltage between the supporting wires. Comments are as follows: (a) This sawtooth voltage, given by the Eq. (51) series, with A,, A,, etc., as in Eq. (52), describes the Laplacian potential along a path for which r is enough larger than rj so that all of the KO fringing components have become negligibly small. (b) This curve describes the variation, at vane-edge radius r j , of
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
25
the total Laplacian field; the vane edges require the cusps. This potential variation is the algebraic sum of the (a) and (c) potentials, both being sections of Laplacian components of the total field.
FIG.5. Details of the match of the betwecn-vane potential to the open-channel field, for the Fig. l(a) space-harmonic circuit: (a) (b) (c) (d) (e)
The The The The The
potential at Y 9 ~ j . potential at Y = y j . fringing field contribution to (b). fundamental-interval Fourier component of (c). third harmonic component of (c).
This is the variation, at r = rj, of the fringing-field potential between the vanes, which is described within each between-vane region by an infinite series of terms each having a KO declining type of radial dependence and a sine-term (not cosine-term) axial dependence, with successive argument factors 2Pd, 4&,
26
W. G . DOW
6&, etc. These curves all have zero values at the vanes, and at the zero-potential planes midway between the vanes. Note that each between-vane section has its own axial dependence, with a .rr-radian discontinuity across each vane; this is a section of the Laplacian fringing-field component of the total field. (d) This is the fundamental-interval Fourier component of the (c) vane-edge potential section treated as an end-to-end whole; it is, as so obtained, not necessarily a section of a Laplacian field. However, it is the fringing-field contribution to the vane-edge potential that the fundamental-interval component of the openchannel field must match, and this open-channel component is a Laplacian field. (e) This is, qualitatively, the third-harmonic component of the (c) curve treated as an end-to-end whole; note the first appearance of the tendency toward the cusps of (c), which exist because of the sharpening effect due to inclusion of the similar fifth, seventh, etc. components, I n combination with (d), this illustrates the presence, in the matching vane-edge fringing potential, of only odd harmonics, just as for the primary (a) sawtooth distribution. This (e) curve in combination with the A, component of the (a) curve provides the total third-harmonic variation that must be matched by the open-channel I,, (3pdr) potential component. T h e axial potential variation at radius Y* is described by the algebraic sum of Eq. (51), which accounts for the Fig. 5a sawtooth behavior remote from the holes, and a similar series accounting for the effects of the KO fringing field between the vanes and near the holes. The amplitudes of the Figs. 5d and 5e curves are the first and third harmonic components of this fringing-field series. T h u s for this dc situation, with 2: = 0 at a vane, ~j
=
4 v,, [(A,- F,) cosfl&
f
In this and later equations,
(A3 -
Fs)cOS 38& f (A, - Fb) Cos 58dZ f
...I. (58)
Vj,,, are respectively the local and momentary value and the amplitude of the potential at r = ri; in Eq. (58) vj is a local dc value, because as yet no traveling wave has been introduced; F,,F,, etc., describe the relative amplitudes of the harmonic components.of the fringing-field contribution to vi in this dc situation; &F,V,, and &F3V,, are the amplitudes respectively of the Figs. 5c and 5d curves. Each F is usually a moderately small fraction of the associated A, depending on the hole aspect ratio rj&. vj,
SPACE-HARMONIC TRAVELING-WAVE
27
ELECTRON INTERACTION
The open-channel continuous Laplacian I , potential distribution must match to Eq. (58)at r = r j ; therefore the potential at any point in the open channel is, for this nonpropagating dc condition,
where I,, (pdr)and l0(,Bdri)are zero-order modified Bessel functions of the first kind, of the respective arguments. T h e F,,F3, etc., terms in the last two equations account mathematically for the intuitively evident weakening of the axial field between the vanes but near the holes caused by the nearby presence of the holes. Now consider the effects of the presence in this cold-circuit structure of a backward wave having the circuit voltage oh of Eq. (56). Much as for Eq. (53), this calls for the substitution of VhWcos ( o t - /3z) for VdCin Eq. (58). The useful aspect of the expansion of the result is the appearance of first backward and first forward space-harmonic components of the rf voltage wq at the radius rb where the annular beam will later appear. The forms resemble Eq. (54);for these two harmonics only
Note particularly that: (a) The V,,8,and V,, amplitudes are for traveling-wave voltages having greatly different wavelengths and radian wave numbers. (b) The phase angle el., carries through without change from the total-wave, Eq. (56), to the Eq. (60) expression; thus the complex voltages P, and P, have a common phase angle, in spite of their strikingly different phase velocities ; (E) These equations apply only when j3 &, for the reasons given just below Eq. (54).
<
The voltage-ratio beam-circuit coupling coefficient as defined by Eq. (2a) becomes, for either space harmonic, nearly enough if j3 &,
<
28
W. G. DOW
A solid round beam may be treated as an assembly of annular beams, each having the Eq. (62a) coupling. For a given bunching influence at an annulus, the resulting bunch density is in the small-signal theory proportional to the dc space charge available for bunching. Therefore a measure of the bunching effectiveness of the circuit voltage on the entire beam is obtained by weighting the contributions from the individual increments according to their dc space-kharge content. Note that the phase of the circuit’s influence is the same throughout the beam. T h e averaging process is straightforward for a beam of uniform density, the result being, for a beam of radius r, passing through vane holes of radius ri: A1
- Fi
211(8dYs)
lW =7 )Bdy,~o()Bdyi)
. ’
the average 6, for a solid round beam of uniform dc space-charge density
This employs the fact that Fl is a property of the size of the hole, not of the beam. Here II(&rs) is a first-order Bessel function of argument &s*
T h e physical-concept significance of the magnitude of tV as in Eq. (62a) is of interest. First note that Fl = 0 in the pinhole versionof the Fig. 1 structure; in this limiting case 5, is less by a factor of 4 than the nearly-unity quantity A , = 8/rr2. This contrasts with the helical-circuit traveling-wave amplifier concept that the beam-wave coupling approaches unity as the beam approaches indefinitely close to t h e circuit. T h i s contrasting factor of 4 arises as the product of two factors of 2 ; this situation can be summarized briefly as follows: (a) There is a semantically-based factor of 2, arising because the helix circuit voltage concept is that of a voltage measured longitudinally along the helix relative to a zero-potential plane perpendicular to the helix, whereas for a space-harmonic circuit the total-wave circuit voltage is conceived as existing transversely between the two members of a balanced pair, or between the two faces of a eoaaeguide enclosure-thus not in general being considered relative to a zero-potential plane ; (b) Another factor of 2 appears due to the fact that each of the two oppositely-traveling space-harmonic sets provides only half of the circuit voltage when in additive phases. T h e balanced pair may be converted into a completely unbalanced circuit by replacing one of the wires of the pair by an infinite “ground” plane, I n this case the one wire’s voltage amplitude relative to zero becomes V,,, rather than V,,,/2. There then appears a fast space-harmonic
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
29
component, with a phase velocity the same as for the total wave, whose amplitude is just V,,ll,/2.Thus there must be provided by the rest of the space harmonics, when in additive phases, just the remaining V,,/2 voltage relative to zero. This represents no change in the space-harmonic components from their balanced-pair values. Any partially-unbalanced condition incorporates a fast space harmonic component having whatever voltage amplitude to ground is necessary to maintain the total-wave between-wire voltage amplitude at V,,l,4,therefore maintaining the additive-phase sum of the other space harmonics at V,,,J2 relative to zero potential.
VII. BEAM-CIRCUIT SELF-CAPACITANCE C,, GOVERNING THE SPACE-CHARGE VOLTAGE T h e determination of space-charge voltage effects, governed by C,, is closely related to the boundary-value determination of the charge-
ratio coupling coefficient [,, because in studying both of these quantities the circuit structure is treated as a family of equipotential surfaces. T h e logical pattern for the boundary-value determination of C, will be described in this section, and approximate results of the treatment given, with the detail analysis omitted in order to avoid major interruption of the over-all chain of logic dealing with electron interaction, In the next section the extension of the boundary-matching logic to the determination of 5, and of will be described, and approximate results given, again with details omitted. Because the beam’s bunches advance at the slow-wave phase velocity of the space harmonic, the electric field set up by the beam’s charge obeys, nearly enough, the Laplace equation in the region between the beam and the circuit, and inside of the annular beam. For most useful combinations of frequency and hole aspect ratio rj/Ad,the between-vane fringing field’s radial decay distance is so short that only a small fraction of the cyclic time period is required for outward radial wave-type transmission of the time-varying signal through this decay region that lies just outward from the edge of the open channel. This makes it permissible to use the Laplace equation in studying the fringing-field match between the open channel field and that between the vanes. T h e useful boundary-value matching quantities at r = rj are the potential and the radial potential gradient of the Laplacian fields, respectively in the open channel, and between the vanes adjacent to the opening into the channel. It is common practice in evaluating the dispersion properties of a
30
W. G . DOW
propagating space-harmonic structure to use as matching quantities at r = rj an assumed axial field form and an average circumferential rf magnetic field intensity (ZZ), these being adequate because the true details of the match have only second-order effects on the dispersion and impedance properties of the total wave. However, in relation to beam-circuit coupling and the space-charge voltage for an annular beam close to the vane edges, the wave components due to the fringing field may have substantial effects, so that it is desirable to be able to determine with reasonable correctness the actual form of the potential and its radial gradient at the matching radius rj, rather than using an assumed form for one matching quantity and an average for another. Before presenting the matching logic there will be given a qualitative discussion of the nature of C, and the space-charge voltage it governs. Figure 6 illustrates schematically the varying spatial phase relationship
t-, -’ + + + - - - :,*++ --I-
---
1 1 1 . 1-I
-I
+ + +
-; + + ----
--I
An
FIG. 6. Contrasting phase relations between a space-harmonic bunched beam and the vanes or digits. At A there is no induced charge, and so a maximum in the envelope of the space-chargevoltage at the beam. At B there is a maximum induced charge, therefore a minimum in the envelope of the space-charge voltage; this is a nonzero minimum.
between the bunches and the vanes, for a bunched beam in a spatially periodic structure like that of Fig. 1. Because the beam’s radian wave +/3) is for backward-wave operation less than &, number pn(=& the bunch wavelength slightly exceeds the like-side vane spacing Ad. At location A in Fig. 6 the phase of the bunches resembles that in Fig. 7a, which illustrates a space-charge model for which bunch and vanestructure wavelengths are equal, and the vanes at equipotentials of the space-charge-produced field, so that no net flux terminates on the vanes. Therefore all the electric flux from each bunch terminates on its two
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
31
adjacent bunches. Thus the vanes have no effect on the space-charge field, which means that the space-charge voltage has at A in Fig. 6 its maximum value, and C, a minimum value, corresponding to removal of the circuit entirely. At B in Fig. 6 the resemblance is to Fig. 7b in that maximum flux terminates on the vanes, the space-charge voltage being therefore at a minimum and C, at a maximum; see the later Eqs. (74) and (90).
0 -
+ + - - ++ - -
+ + --
ELECTRON BUNCHES
++ ---
++ ELECTRON BUNCHES
FIG.7. Space-charge models with bunch spacing equal to like-side wave spacing A,: for (a) the phase of the beam is such as to produce zero induced charge; for (b) the phase is such as to cause a maximum induced charge.
T h e model for this section’s logical pattern is one in which the circuit structure is an equipotential at zero potential, and there exists at I = r,, a bunched annular beam of T~ coulombs per meter rf charge, having the space-harmonic propagational attributes. T o preserve clarity attention is directed to a short enough section so that rf growth or decay can be neglected. Thus, nearly enough T*
= Tqw, cos [wt - ( B d
+ B) z + &I.
Because the charge may have its own entrance phase, here 8, is the phase angle associated with Too;that is, Tq,,= T,,, expjo,.
(63)
32
W. G. DOW
Expansion of Eq. (63) into two traveling waves having a common phase velocity aids qualitative understanding ; the expansion gives
T h e first term here describes a wave that resembles Fig. 4b in having a fine structure of interval A,, and a spatial modulation envelope of the total-wave periodicity, but differing from Fig. 4b in the important respect that the underlying fine-structure wave form is not now the Fig. 4a sawtooth, but rather a pure cosine, being T,,,&cos &dz. This underlying wave form corresponds to the Fig. 7b bunch placement in that the maximum values of 7, occur in the planes of vanes. T h e second term in Eq. (64) has also an underlying sinusoidal fine structure of interval A,, being sin &z, with a total-wave modulation envelope 90" out of phase with that for the first term. T h e fine structure for the second term corresponds to the Fig. 7a bunch arrangement in that the bunches are at midplanes between vanes. To permit using Eq. (64) in a qualitative study of the beam selfcapacitance, let
C,, = the beam self-capacitance as determined for a field distribution that produces no charge on the vanes, as for example that for the second-term, sin &z Eq. (64) charge distribution illustrated by Fig. 7a, and let C,, = the beam self-capacitance as determined for a field distribution giving the maximum charge on the vanes, as for example that for the first-term, cos &z, Eq. (64) distribution, illustrated in Fig. 7b. In this qualitative study based on Figs, 7a and 7b, the space-charge voltage can be expressed by applying to each Eq. (64) term the appropriate self-capacitance. It is because of the contrasts in field pattern between locations A and B in Fig. 6 that different C,,,C,,,self-capacitances apply to the respective terms in Eq. (64). This in turn affects the relative magnitudes of the first forward and first backward space harmonics of the total backward propagating wave, this latter existing to aid in providing a transverse field match between the bunched beam and the zero-potential circuit of this section's model. Let wcla, vcbl,symbolize the local and momentary values of the rf potential at the beam location, respectively for the A,,-wavelength first forward space harmonic, and for the first backward space harmonic of the backward total wave, as existing due to the Eq. (64) bunched beam.
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
33
Since both harmonics are required for a match in the presence of unlike capacitances for the two terms, w,,
+ vCb,
T,,,cos (wt
= __
- /3z) cos P,z
c q c
T,,, sin ( w t .+C,
Pz) sin /3,/z. (65a)
Trigonometric rearrangement employing the definitions given for v,, and vCblshows that
T h e second member of this pair is not of direct interest, because as the backward space harmonic it is remote from synchronism with the bunched beam and therefore causes no interaction. T h e first member describes the space-charge voltage that affects the forward space-harmonic interaction for the backward total wave. Thus it becomes
where I/,,,& is the space-charge voltage amplitude, occurring at the positive crest phase moment and position of the first forward space harmonic of the backward total wave. Clearly then, since by definition
T,, = ~
, ~ C ? , l
1 -1 -
c,
2c,, +
z; 1
is describes the beam self-capaci( a c e C, in a space-harmonic structure).
(65e)
Note from the form of Eq. (65c) that it is only because CQBfCQc that the system requires a first backward space harmonic of the total backward wave. For the present model C,, is given later by Eq. (74). T h e first step in the boundary-value logic of the space-charge voltage effects is to express the solution of the Eq. (57) Lapiace equation for the open channel region, r < r j , as existing in the presence of the Eq. (64) advancing bunched beam, but with the vane edges replaced by a cylindrical equipotential-a drift tube model. In doing this there will be used the strictly Laplacian solution in which the Bessel function argument factor is the same as the sine-wave argument factor. This treatment is correct in the nondispersive range of frequencies. Adaptation to study of behavior in a dispersive frequency range requires use in general
34
W. G. DOW
of a yq for eaqh component in place of this treatment's /Iq, where 7: = 3/: - p&. Here /3,# is the radian wave number for free-space electromagnetic wave propagation occurring at the velocity of light. T h e hot-circuit wave, and so the advancing Eq. (64) bunches, have the hot space-harmonic phase velocity U,, therefore a radian wave number which is the value of 3/, for this driving component of the open-channel potential field. T h e Laplacian expressions for the open channel are:
x cos (wt
-Q ,Z
+ er).
(66b)
Here and later KO and K , refer as usual to modified Bessel functions /I. At Y = r b , both of of the second kind, and of course 3/, = /Id these reduce to the beam's potential due to its own charge, in the environment of a drift tube of radius r j , that is,
+
=
=
v,, cos iwt
-
(p,
+ p) a + e,].
(6W
At r = rj Eq. (66b) reduces to 2,
= vj = 0,
(6W
corresponding to the fact that the drift tube's surface must be an equipotential. T h e determination of the beam charge and the drift tube surface charge implies recognition that at any radius, with c0 symbolizing the permittivity of a vacuum, Radial electric flux density
=
-c
8V
O a r '
Application of this to the drift tube model proceeds as follows: O n the inner face of the charge sheet:
On the outer face of the charge sheet:
(67)
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
35
O n the inner face of the drift tube:
x cos (wt - /3,,z
+ Or).
(69a)
Use of the Wronskian Bessel-function identity converts this last equation
x cos (wt - , 4 2
+ s,).
(69q
T h e induced charge amplitude Ti,on the drift tube interior is obtained from the right-hand side of this equation by omitting the minus sign and the cosine factor. Solution for Ti,JV(,,,' then obtainable shows that, for this bunched annular beam at radius r, in an equipotentialsurface drift tube of radius yj, with bunch wavelength X, = 21118, = 27r/(15, p), the Eq. (1 b) mutual capacitance for the drift tube become:
+
c.
-Ko(pnyb)
2m0 for an annular bunched beam, spatial - lo(pnya) Ko(pnrj);(interval An, in a drift tube of radius r)'
lo(pnyj)
(704 T h e two Eq. (68) expressions that combine to give T~ are of unlike sign. However, their magnitudes combine additively to give a positive value of T~ when the cosine is positive. When the two Eq. (68) expressions are so combined, clearing of fractions carried out, the Wronskian identity employed, the cosine factor replaced by unity to give an amV,,,,, there plitude expression and the result solved for the ratio Tp,,r/ results for the Eq. ( 1 a) self-capacitance Cqd in a drift tube 2nc0
Io(Bnrj) '0,
=
0
nyb) KO(~ynb)zO(/%yj)
- zO(&yb)
for an annular bunched beam, spatial interval A,, in a drift tube of radius r ,
(70b)
For this model of an annular beam in a drift tube it is convenient to adopt the following definition: f,.d
is defined as being Ciqd/CQd.
(71)
Use here of the previous two equations shows that, for the drift tube,
W. G . DOW
36
Note that f s d is numerically positive. One may also employ a numerically negative f c d , where f r d = -fed, this being the ratio of the induced charge on the drift tube wall to the inducing rf charge in the bunched beam. For a solid round beam in a smooth-cylinder equipotential environment-this is the basis of the helical-circuit space-charge voltage study-the Cqclis obtained by a process which sums the effects of the charge over annular increments comprising the beam, each increment’s contribution being weighted according to its rf space-charge density andphnse. If the dc and rf space-charge densities and rf phase are uniform throughout the beam, the integration to obtain the total C,, is straightforward. T h e result is, for a solid round beam of radius rs passing through vane holes of radius ri, conveniently expressed as
)
for a solid round beam, spatial interval A,, in a drift tube of radius rj, if the dc and rf spacecharge densities, and the rf phase, are sectionally uniform; basis is a Laplacian model
’
(73)
For the sheet helix that has been widely used as a theoretical model, the annular-beam mutual and self capacitances C,, and C,, and the voltage coupling coefficient f,., are obtainable by using the axiallydirected helical transmission radian wave number ,B in place of B,, in Eqs. (70a), (70b), and (72). Of course f c = -fs in the helix. Equation (73) similarly gives C‘, for the solid round beam in the helix, with /3 used for &. This form is consistent with equations in various published discussions of the solid-beam “space-charge reduction factor” (IZ). Since C,,, as defined a little above Eq. (65), is unaffected by the circuit, it is evaluated by removing the drift tube to infinity for Eqs. (70b), and (73), making ri --t a,which gives
ca s --
c,, =
2rTf0 ZO(pnyb) KO(pnyb) ;
TTfOP3;4r2,
1 - 211(pnr,rs) K m
for the annular (beam of radius rb)’
(74)
for the uniform solid
; (round beam of radius Y .
These depend only on the beam, not at all on circuit geometry, except to the extent that circuit geometry influences the bunch wavelength. Evaluation of the dependence of C,, on geometry is found from the boundary-match study.
37
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
A model appropriate to the boundary-value analysis is the Fig. 1 circuit, with an advancing bunched beam phased as in Fig. 6 with respect * ~27r/(/3, p). to the circuit, the bunch wavelength being A, = 2 ~ / / 3= It should be clear here, with general correspondence as to concept with the fringing-voltage illustrations in portions of Fig. 5, that in the space-harmonic circuit-as distinct from the drift tube model of Eqs. (70a) and (70b)-there will exist at the edge of the open channel, Y = y i , a voltage component of wavelength A, having a nonzero amplitude. Thus although Eqs. (66a) and (66c) are still valid when the spatially periodic circuit replaces the drift-tube wall, Eqs. (66b) and (66d) are changed, the latter becoming, for the spatially periodic circuit, at r = ri
+
= v, =
vjnlcos
-
(pd + 8) z
+ e,].
(76)
Here and later the symbolism is that Vjnd= the amplitude along r = r j of the traveling component of rf voltage having the wavelength A, = 27-r/Pn = 2n/(& /I). One of the objectives of the boundary-value study is to establish the relation between V,,,, and the beam-location voltage amplitude V,,,,. T h e new form replacing Eq. (66b) is:
+
T h e use here of u', rather than 'u indicates that this expresses only the component of the advancing potential that has the space-harmonic wavelength .A, Treatment paralleling that given Eq. (66b) leads to an expression resembling but longer than Eq. (68b), containing a term in VjW,.Rearrangement and treatment paralleling that used in obtaining Eq. (70b) then gives T u m = V c i n C u d - Vjmc(q~, (78) where C,, and C,,, are the beam-circuit capacitances of Eqs. (70a) and gives (70b). Division by V,, and recognition that C, = Tpl,,/VC,,d,
Employment here of Eq. (71), with format adaptation for later use in relation to Eq. (65e), gives
38
W. G . DOW
Recall here that V,, and V,, are the amplitudes, at hole radius rj and beam radius rb respectively, of the advancing voltage component having the wavelength .,A Two limiting conditions are of interest in relation to Eq. (80),as follows: (a) Maximum coupling, with retention of a substantial space-charge voltage. There may be the extreme in “tight” coupling between the beam and the circuit, existing when the beam is very near the vane --t 1. edges. Then V,, obviously approaches identity with V,,, and tad I n this case the first Eq. (80) parenthesis approaches zero, as C,, of Eq. (70b) becomes indefinitely large. Their product remains finite for this maximum coupling, in contrast with the infinite value of C,-given by C,, with r b = rj-for maximum coupling in the sheet helix. Thus in the space-harmonic circuit there remains a substantial space-charge voltage with the beam adjacent to the circuit, in contrast to a zero space-charge voltage for maximum coupling in a sheet helix. It will be seen below that both C,, and C,, of Eq. (65e) contribute to the space-charge voltage that exists in the case of maximum coupling. (b) Weak coupling, the space-charge voltage approximating that in a drift tube or sheet helix. There may be relatively weak coupling between the bunched beam and the circuit, obtained by moving the annular bunched beam radially inward away from the beam edges. T h e V,,, and tvdbecome rapidly less than V , and unity respectively, and tvdVim/Vcmin Eq. (80) becomes much less than unity. As the coupling falls off, C, approaches the corresponding drift-tube (and sheet-helix) value C,, of Eq. (70b). I t should be clear from these comments that the space-charge voltage can be much more important in a space-harmonic structure than in a comparable sheet helix structure, and particularly so for the close coupling obtained by using an annular beam maintained close to the circuit. These comments apply for spatially periodic circuits generally, with the simple Fig. 1 circuit used as an illustration. A detail boundary-value study must employ: first, a statement of the infinite set of I,,(&r) space harmonics, each with an undetermined amplitude coefficient, that in combination provide a zero-circuit-potential Laplacian backward total wave in the open channel, as driven by the bunched beam; and, second, a statement of the infinite set of K,,(k&r) solutions of the radially-declining between-vane Laplacian potential. T h e q indices identify the several open-channel space-harmonic components, and the k indices identify the several between-vane potential
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
39
components. By requiring a match at r = rj of the potential and potential gradients of these two sets of field equations, there is obtained a doubly infinite determinant for the coefficients of the KO(k&r) terms of the between-vane structure. There is also obtained in the process an exin terms of these between-vane &coefficient, pression for [( Vc,,,L/(L,aVjm)-l] which thus permits using Eq. (80) for finding C,. T h e results are conveniently illustrated by specialization to satisfy the two following conditions:
<1;
(a) A genuinely slow-wave structure, for which (/3/pd)
(b) A hole-vane aspect ratio comparable with or greater than unity, for which rj is comparable with or greater than h d ; this turns out to be the requirement for making K,(k/3rj) approximate closely Ko(k/3rj),which greatly simplifies the expressions.
A rather good approximate solution to the doubly infinite determinant for this set of conditions shows illustratively that when /3/& is small and rj 2 A,, nearly enough,
T h e digital computer solution of the doubly infinite determinant converges rather slowly as the number of rows and columns used is increased; however, the first few terms of the solution, which dominate the results obtained in using it, are reasonably well established by using 70 rows and 70 columns. T h e solution as so obtained results in replacing [2 - (r2/8)], which is 0.78, by 0.82. Thus, from the computer solution, is small and rj => h,, nearly enough, when
Use of this in Eq. (80) shows that, for small enough, based on the computer solution:
/?/pa and
ri 2 A,, nearly
This permits evaluation of C, and therefore of the magnitude of the space-charge voltage, for these conditions. As an aid in interpretation of this and related results, and in connection with Eqs. (70a), (70b),
W. G. DOW
40
(72), and (74), note that, for any beam,
/?/&
and any rj/Ad, for the annular
Thus it is clear that quite generally, for the present model, and in anticipation of using Eq. (65e),
When
/3/&
is small, of course
p, +&;
then if in addition rj 2 A,,
Use of this, and of the corresponding p, + fld form of Eq. (72) for tCd, reduces Eqs. (83), (84), and (85) to the following, applying for small /3/& and rj 1 A,:
Use of this and of Eq. (82) in Eq. (65e) shows that, for small with rj 2 A,,
/?/&and
Thus as illustrated in Fig. 8, there applies the concept of two “fringing capacitances,” which appear in series in an equivalent circuit type of representation. The equations are, for small ,8//& and ri 2 A,, nearly enough,
SPACE-HARMONIC TRAVELING- WAVE ELECTRON INTERACTION
41
It is clear from this and Fig. 8 that: Cfs,Cfo are the contributions made by the fringing field to the beam
self-capacitance, due to flux patterns corresponding respectively to the Fig. 7a and Fig. 7b bunch phase positions.
(COS
s, z 1
(SIN P d z )
FIG. 8. Four different equivalent circuit representations of C,, indicating the series equivalence nature of the contributions to C, respectively from C,, and C,,,with subsequent separation into the effect due to C,, from the drift tube analysis and the two fringing-capacitance effects C,, for the Fig. 7(a) bunch phase and C,, for the Fig. 7(b) bunch phase.
42
W. G . DOW
Thus quite generally, for any space-harmonic circuit geometry 1 1 - 1 _ --+-+---,
c*
c,, that C ,,
Cod
1 Cf,
(93)
with of course the requirement must be defined relative to a drift-tube surface along the digits, whatever their gross geometry may be. When r, + y3, of course C,, + 0 3 ; thus, quite generally for any model, when the beam grazes the edges or faces of the digits, 1 1 -+-+-
c,
Cf.?
1 Cfc
(94)
VIII. CHARGE-RATIO AND VOLTAGE-RATIO BEAM-CIRCUIT COUPLING COEFFICIENTS There are adequate heuristic reasons for expecting the charge-ratio and voltage-ratio beam-circuit coupling coefficients 5, and 5, to be equal and opposite in sign; for a sheet helix circuit this is easily shown as given by Eq. (72). to be true, with 5, = -tC having the value tvd In this section tCwill be dealt with directly in terms of the preceding section’s equipotential circuit model. First note that Eq. (78) rearranges into the form
which with the bracket known from the preceding section’s study gives a form for T,, to use in finding 5, as -Ti,/Tgm. Next note that the infinite set of K,,(&r) Ressel-function solutions of the between-vane potential fields can be used to determine the flux density terminations on the lateral faces of the vanes. Integration of this flux density on both sides of the vanes from r = r3 to r = 00 then gives the digitalized time-varying charge on each vane. The amplitude of this is the product Ti, of induced total-wave charge per meter of length by +nearly enough in the condition of small p/pd.For reasons suggested by the shapes of the curves in Fig. 9, the Fig. 7b, cos /3,z bunch phase dominates the induced charge magnitude, and for small p/& the Fig. 7a, sin &z contribution is negligible. Use of the solution of the infinite determinant permits relating Ti, to Vt,,&.T h e result, based on an approximate solution to the determinant, appears as
5, = - l .4e d ~
(2 - 2 In 2),
(96)
SPACE-HARMONIC
TRAVELING-WAVE
43
ELECTRON INTERACTION
that is,
5,
=
+4 x 0.61. tud
(97)
A more complete study would show some dependence of
4,
on ,B/&.
N
N
-a
2
+
---
N
2f
---
w t =o
FIG.9. Illustrative diagrams of the variation with z of the voltage o, at channel radius I = r,; note that: in (a) the near-continuity of slopes at vanes indicates very little induced charge on the vanes; in (b) the cusps indicate a very large induced charge on the vanes.
The boundary-matching study to determine f,o directly employs a model in which there is no beam in the open channel, but there is a total-wave propagation along the circuit.
44
W. G. DOW
Thus the analysis employs: (a) A complete family of q-index open-channel space harmonic waves just as for the previous section, but of different relative magnitudes, (b) A family of k-index radially invariant and completely known between-vane standing waves that in combination provide at points well outward from vane edges the total-wave voltage variation at the vanes, and at the same time provide a uniform axial potential gradient between adjacent vanes-this potential structure represents the “drive” for the system, (c) A family of k-index radially declining between-vane fringingfield standing waves. By requiring at r = ri a continuity of both potential and radial potential gradient, as between the (a) item on one hand and the (b) and (c) items in combination on the other hand, the entire potential structure is determinable. T h e useful result of the matching process appears as follows, for the condition in which is small and ri 2 A,, and based on the approximate determinant solution, (98)
where A, is as in Eq. (52), and VlWL = the amplitude of the total-wave voltage between supporting wires; see Fig. 4 ; Vi,,, = the amplitude at r = r j of the first forward space harmonic component of the potential. T h e manner of use of the Bessel-function ratio in Eq. (61), and the Eq. (72) definition of trdas being this ratio, shows that in this model
Combination of this with the preceding equation gives: A, 7ra - 4 I, = I q - (7). Comparison with Eq. (62a) shows that, for these conditions, and based on the approximate determinant solution,
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
45
Introduction of A, = 8/7r2 now shows that for this Fig. I circuit model, when ,3/& is small and the open-channel radius is comparable with or greater than the spatial interval A,, nearly enough, and using an approximate determinant solution, ( 102a)
‘The difference between this and Eq. (97) is not fortuitous. It arises from the fact that the calculations employed in arriving at Eq. (96) and Eq. (98) employed approximate rather than exact solutions of the two somewhat different infinite determinants for the k-index coefficients. One of these determinants applies for the C,, &, model consisting of a bunched beam and an equipotential space-harmonic circuit, the other for the 5,. model consisting of a space-harmonic circuit propagating a total wave in the absence of a beam. In principle - t,.= t,,,at least when p/&l is small. For the digital computer solutions of the two infinite determinants, as the number of rows and columns used is increased, and the number of the terms used in finding - t,,and f u is also increased, the value leading to - 5,. slowly increases, and that to 4, slowly decreases, toward an apparent common limit that clearly lies between 0.61 and 0.63. Therefore, for application small in subsequent studies in this paper to the Fig. I model, with and the hole radius comparable with or greater than A,, and based on the digital computer solution, nearly enough, - f , == f
w
= 0.62f,d4,
( 102b)
where tpdis as given by Eq. (72). Correspondingly the computer-based counterpart of Eq. (101) is FJA1
=
1
R2
-
0.62 - = 0.235. 8
(102c)
IX. THESPACE-HARMONIC INTERACTION CHARACTERISTIC EQUATION T h e Eq. (21) and Eq. (37) differential equations describe respectively the electrokinetic and electromagnetic overrun bunching behavior in the backward-wave amplifier or oscillator. However, because the first of these equations applies to the first forward space-harmonic behavior, and the second to the total-wave circuit behavior, they have different complex exponential propagation factors. T h e analysis will deal with interaction of a forward beam, U, being numerically positive, with the
46
W. G . DOW
first forward space harmonic of a backward wave in which both group and total-wave phase velocities are backward, as for the circuit of Figs. la, and lb, whether for balanced-pair or waveguide support of the vanes. However, the detail treatment as given applies to this circuit only for frequencies below its first ‘‘stop band.” For the treatment so chosen, the total-wave phase velocity and the /3 of r = $3 are numerically negative, with the space-harmonic phase velocity U , numerically positive. With this in mind, note that For the electrokinetic equation (Eq. (21b) the exponential factor is):
exp [jut - (rd
For the electromagnetic equation (Eq. (37) the exponential factor is)’
exp ( j u t
-
+1‘ zl; (lo3) (104)
rz)’
Thus r is the hot-circuit total-wave propagation constant. Underlying propagating circuit properties appear as follows: U,, 4,, symbolize respectively the inherently numerically positive and wholly real phase velocity and radian wave number for forward-wave cold-circuit propagation in a lossless line ;
ro =jP
0
The wholly imaginary propagation constant for - lossless cold-circuit forward-wave propagation
(
r, =j p b = jwlu,, =
j
(1054
The wholly imaginary and fictitious propagation constant” at velocity U b
(((
Z,is the inherently numerically positive circuit characteristic impedance. T h e ensueing analysis will deal with and evaluate two alternative kinds of complex “perturbations” of propagation constants resulting from interaction between the electrons and the circuit as follows: (a) T h e complex pushing perturbation I‘9 = -( -ro)describes the small departure of the hot-circuit backward total wave propagation constant r from the -Po value applicable for the lossless accounts for the effects line; for mathematical convenience of the line’s own resistive loss as well as for electron interaction; has primarily conceptual value; (b) T h e complex slip perturbation = r b - describes negatively the departure of the hot-circuit forward space harmonic propagation constant r, from the wholly fictitious and wholly imaginary quantity r, -jSb = j w l u , ; thus in effect it describes the perturbation of the beam by the circuit; is more useful for engineering analysis purposes than F2,; it is analogous to the “slip” of an induction motor.
r
r,
r,
r,
r,
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
47
Immediately significant symbolism and interrelations are: p,, r,, U , symbolize respectively the radian wave number, the complex propagation constant, and the phase velocity for the space harmonic at which the beam is being bunched (in the present case the first forward space harmonic), for the hot circuit condition; thus these quantities will differ slightly but significantly from: p,, r,, U,, symbolizing respectively the same quantities as above, but for the cold-circuit backward wave. Pr and U , are inherently numerically positive; of course P, is wholly imaginary; useful interrelations are “f
= rd
- ro;
Pf = P d - P O ;
cold-circuit for(the ward space harmonic) ; the hot-circuit forward space harmonic
rf= r + ro;
-
PfUf= &U*
a;
(as defined earlier);
( 106d)
of the hot-circuit . (onespace harmonics ).
( 106e)
(106f)
= w.
(The complex pushing perturbation) =
(The complex “slip” perturbation) =
( 106b)
the “pushing” perturbation due to the bunched beam
r=jjl- a ;
r,,= jPn
.
( I06a)
rp= r
-
(-To) = r $. To = r, - rf; (107a)
r, = F, - (r,+ r)= r b r,. -
(107b)
T h e perturbations expand into equations dejning flB and Bs:
These result from using Eq. (106) in Eq. (107). T h e exponent growth factor cy appears with opposite signs in Eqs. (108a) and (108b), because one of the perturbations employs in its definition r; the other, Addition of Eqs. (107a) and (107b) eliminates F,giving the following for conversion between r, and P,, or between &, and Bs:
-r.
48
W. G. DOW
here and later: U,, r, symbolize respectively the “overdrive” velocity, and the measures of overdrive in terms of changes ,/3, and r, in radian wave number and propagation constant; F, is wholly imaginary as it is = jp,.; definitions are:
PI,
r,
ur = ub8,
=p b
this defines the overdrive velocity (U,, for space-harmonic operation
‘ f ‘
-Pf; r,.=
r b
-
(1 10a)
rr; (these define /3r, r,).
(1 lob)
Useful interrelations are: Pr _ -_ _ur- ___ Uf Pb
r, - the overdrive ratio;
(1 IOC)
.-
r b
(1 IOd) Bd
-Po
Ufrf= UJ,
= Pb =
(1 10e)
-Pr;
uorb = jw.
(1 10f)
Here, as in general for quantities measuring changes in a /3 or a U,P,., UfBf do not equal w .
r,
(1 log)
The left side of Eq. (109a) is the small algebraic sum of two quantities each of which is individually small; this sum is equal to the right-hand quantity which is the small difference between two large quantities. The operational multipliers applying for Eq. (21b) and Eq. (37) are: Application of the Is, for the usual Is, for perturbation operator in this nonperturbational study of the Eq. (21) space-harmonic column analysis the equivalent of multiplica- variation as exp tion by (jwt-r,,z), the equivalent of multiplication by the slip quantity
-r ju
-r,,-r ro- r rb- r
-r
b
+ re
rb u b
Is, for perturbation study of the Eq. (37) total-wave variation as exp (jut-rz) the equivalent of multiplication by the pushing quantity
r0-Tn roUo
(llla) (Illb)
- r, (iiic) - r, -1.- 2T‘o(111d (Ille)
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
49
With these operators employed, with 1/ UoC, expressed from Eq. (25e) as Z,, and, from Eq. (24c), I/L,C, expressed as U,Z, Eq. (21b) and Eq. (37) become, after cancelling each one's exponential factor and using definitions below Eq. (4),
+ r,) + R , l ~ , ~ , ~ , l
[rD(-2rO
VhO =
(RhUOrO
+ zouor,",Tie. (1 12b)
For present use the Eq. (2b) beam-circuit charge ratio coupling relation, and the Eq. (39) combination of the two beam-location voltages, can be expressed as follows:
TfO= & t o , and
to =
6 v 6 0
(1 12c)
+ (T*oIC*)-
(1 12d)
Here C,, f c , and 5, have meanings as in Eq. (la) and Eq. (2). All four of the electrical variables rfqo, Pgo, rfio, and ph0 can be eliminated between these four equations, leaving the following characteristic equation for the combination of Eq. (21b) and Eq. (37):
T h e numbering of the terms facilitates later discussion. This is the characteristic equation for forward space-harmonic excitation of a backward total wave in a space-harmonic structure.
ul
0
--I
\
*
--Po-
r;k T
% S
FIRST FORWARD
SPACE HARMONIC
-4
-
P
Pd
FIG. 10. An w-B diagram for a Fig. 1(c) type of space-harmonic structure having a low-frequency cutoff, the total-wave relationship as indicated by the dashed curve, and the first forward space harmonic relationship by the solid-line curve. P, at frequency wb, is a reference point identified by the space-harmonic curve and the beam-velocity line OP of slope ub. For an operating radian frequency w , chosen here to have backward-wave amplifier relation to P, the four cardinal points Q, R, S, T, are involved in the solution to determine U,, the phase velocity for one of the hot-circuit waves, identified by the point N at frequency w . For a frequency w', chosen to have a forward-wave amplifier relationship to P', the cardinal points are R', Q', S', T.
r Q
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
51
Figure 10 is an w / p diagram for the Fig. (lc) type of circuit that has a low-frequency cut-off because the vanes are supported in a waveguide. Operation in a slightly dispersive frequency range is presumed. Slopes of the various straight lines describe velocities as indicated. Thus ub, the slope of OQPQ’, is the dc electron velocity, being unaffected by the operating frequency, whether w , or w ’ , or some other value. T h e slope Ub determines the reference radian frequencies wb and w;, being those at reference points P, P’, the intersections of the beam-velocity straight line with the space-harmonic w / p curve. T h e phase velocity of the backward total wave is - U,,, which has, for the Fig. 10 w / p properties, a significant frequency dependence. Backward-wave amplifier operating frequencies are typically less than wb, hence the placement in the figure of the operating frequency w . For this frequency, the slopes U , and U , of OR and O T are the cold-circuit phase velocities for the first forward space harmonics of the backward and forward total waves respectively. For this w , U , > U,, the difference being measured in radian wave number terms by /Ir. If there is no coupling between the beam and the circuit, there can exist (a) two circuit-related total waves, one backward, one forward, and (b), two beam-related space-charge waves, both forward, and unaffected by rf properties of the circuit. When significant coupling exists, giving the Eq. ( I 13) form, the solution still provides for four waves, none being in principle purely circuit waves or purely space-charge waves. All four exhibit total-wave rf field properties. I n Fig. 10, one of these hot-circuit waves that is a backward total wave might have the numerically negative total-wave phase velocity U and a first forward space-harmonic phase velocity U,, where ub > U , > U,. For this wave pb < pn < PI, and both 8, and 18, are numerically negative, the “slip” and “pushing” both being numerically positive. Thus the electrons overrun (“slip” past) the hot-circuit space-harmonic wave crests, whose phase velocity has been “pushed” above its cold-circuit value by the electron interaction. For the space-harmonic circuit, just as for the helical circuit ( I ) , solutions for the “first-order” perturbations of the propagation constant result in a slip perturbation r, having a magnitude small relative to r b , so that Ip,I < p b , as illustrated for one wave in Fig. 10. Typical forward-wave amplifier operation would, for Fig. 10, take place at frequencies in the neighborhood of w i . For such forwardwave operation, the small-signal solutions permit three forward total waves, all affected by interaction, and one backward total wave not so affected, existing if at all as a cold-circuit wave. For the usual backwardwave amplifier operation, as at w in the figure, there can be three interact-
52
W. G. DOW
ing backward total waves, and one non-interacting forward total wave, exhibiting the cold-circuit total-wave and space-harmonic phase velocities +U, and U,. I n either typical case the characteristic equation for finding the propagation perturbation reduces to a cubic. However, there can often exist a range of frequencies, from a little below wb to a little above wt:, for which interaction, if occurring at all, will involve all four waves. I n backward-wave operation, the resulting excitation of the forward total wave may (a) convert second-order to first-order perturbation effects, (b) significantly modify the backward-wave amplifier equations, (c) appreciably alter the oscillator starting current, and (d) make possible a type of rf power generation that employs reflection of the forward wave from a mismatch in the input line. T h e cubic characteristic equation used in Johnson's backward-wave oscillator analysis (2) is obtained from Eq. (113) by dropping out the r, in the factors and the TI, in the 2F0-r,] factors. This latter step is not n priori justifiable (3),because in a spatially periodic structure 2p0 and /Il, may be of comparable magnitude (see Fig. 10) even when p, Therefore in the present study the r,, will be retained in the 2ro-P2>factors, and the first-order form of Eq. (1 13) will appear as a quartic. A criterion that must be satisfied for reduction to the cubic will be derived. A characteristic equation form that is convenient for conceptual study is obtained by a "pushing" type of normalization, accomplished Also, it is convenient for subsequent by dividing Eq. ( 1 13) through by study to divide Term 4 into two parts. T h e form so obtained becomes, after using Eqs. (IlOd), (110f), and (109a):
rb-r,9
<&
r;.
(5)
(1 14a)
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
53
T h e corresponding form employing slip normalization is obtained by dividing Eq. (1 13) through by Ft, then using = - P, from Eq. (109a), also using FJFb = - U,/U, from Eq. (1 lOc), and using r o U , = r b u b from Eq. (110f); this gives
r,, r,
(1 14b)
X. THEGAIN,SPACE-CHARGE, AND CIRCUITLoss PARAMETERS I n Eq. (113) and Eqs. (114a) and (114b), Term (1) establishes the existence of a propagating wave, with or without an electron beam; Term (2) describes the effects of space charge on the propagation; Terms (3) and (4) describe possible gain-producing mechanisms and Term ( 5 ) accounts for the resistive loss in the circuit. T h e statement of the exponential growth or decay constant OL of a growing or declining wave is contained in the imaginary parts of the “fractional complex slip” (-rS/Tb)or the “fractional complex pushing’’ (-FJr,), the real parts containing the physically measurable slip --p,s//f?b and fractional pushing -&Jt??j, that is, ( U , - U,)/ U,,, and (P, - &,)/p,. T h e Eq, (114a) and Eq. (114b) forms of the characteristic equation are respectively suitable for obtaining T l , / r for F,JTb. Any solution other than zero for these quantities describes some presumably realizable traveling-wave behavior, and such a solution is provided by Term ( I ) , which always exists even if all other terms are zero. T h e significance of the various terms will now be discussed in order. Term (I), establishing the travelinpwave behavior. This term is unique,
54
W. G . DOW
in that it always exists, with a value of other than zero, whether or not a beam is present. With a lossless circuit and a beam current essentially zero, so that, nearly enough, I,, = 0 and R , = 0, there remains only this one term in the differential equation. I n this case Eq. ( I 13) can by means of Eqs. (107a) and (107b) be expressed as
Obviously the solutions take the forms rn= I', and I'= f To. As these roots have no imaginary parts, 01 = 0; only constant-amplitude waves are described by Term (1) taken by itself. = 0 is the characteristic equation for Eq. (24a) T h e factor when R , = 0, and describes the forward and backward cold-circuit =0 waves in a lossless transmission circuit. T h e factor reduces to #In = co/Ub, describing a constant-amplitude traveling wave having as its phase velocity the dc beam velocity. T h e corresponding physical-model behavior is that the entering electrons are initially bunched, but there are so few of them (I,,--t 0), that the electrokinetic effects accounted for by Terms (2), (3), and (4) are ignorably small, so that the bunches initially present advance at the dc electron velocity, their advance not being appreciably affected either by space-charge forces or by the presence of the circuit. However, they do induce surface charges in the circuit and therefore a circuit current i,,, which obeys the same differential equations as vA. T h e present direction of attention to the backward total wave appeared in the Eq. (107) definition r,,= r - (-To), and in the Eqs. (1 10a) properties and ( I lob) definitions of U , and r,.in terms of the U,, of the first forward space-harmonics of the lossless backward coldcircuit total wave. Instead, attention might have been directed to the forward total wave, by using the properties of the forward cold-circuit tot a1 wave. Since in the study of interaction in a spatially periodic structure it I 2r0,Term (1) in is not permissible to assume, a priori, that I Eqs. (114a) and (114b) must remain a fourth-degree term in either r,,or r,. In contrast to this, when the principles here introduced are applied to a helical circuit device, it is always true that r, 2 r 0 , so that no terms higher than the third degree in TI, or remain. Thus there are two significant differences between the forward-wave helicalcircuit study and that for the general backward total wave study for a spatially periodic structure, as follows:
(r2 r:)
(r, rn)2
r,
r,
<
r,
<
(a) There is in Term ( I ) of Eqs (1 13), ( I 14a) and (1 14b), a change in
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
55
sign of the factor 2, due to change from the forward to the backward total wave ; (b) I n the helical-circuit study it is acceptable to drop out r, or I'k relative to 2r0,but this is not permissible in the general study for a spatially periodic structure. I n his excellent study of backward-wave oscillator behavior, Johnson
(2) recognized the change in sign of the factor 2, which Pierce ( I ) had not done; the author pointed out the need for the (b) change in an earlier paper (3). Term (2), the space-charge wave and space-charge voltage term. T h e combination of Terms (1) and (2) in Eq. ( I 13), with all other terms zero because the apparatus design makes -tC 6,. and Rl1 vanishingly small, reduces to the following characteristic equation:
T h e presence here of the factor (r2= 0 implies the presence of a transmission circuit, because the use of Po implies a PO and therefore a U , = l/dLlyCl,. But if L , and C,l have interesting values, so also does 2, = dLll/C,,.Thus Term (3) cannot be dropped by assuming 2, = 0, as that would destroy also the (r2 - Ti)= 0 factor; it cannot be dropped by assuming = 0, as that would remove also T e r m (2). T h u s Terms (1) and (2) can exist without Term (3) only because the coupling -4, Et. is vanishingly small. For such a condition the bracket in Eq. ( 1 16) must be thought of as the characteristic equation factor for the space-charge-wave voltage at the beam location, and the parenthesis as the characteristic equation factor for the transmission-line voltage at the circuit. Each kind of propagation then takes place independently of the other. T h e bracketed factor in Eq. (1 16) provides a quadratic equation for in which, because -Ib/2vbUbcq is numerically positive for the reason that I J U b is always negative, the two roots are wholly real when expressed in terms of r, treated as a simple algebraic. Therefore [Y = 0; Terms 1 and 2 in combination but without other terms, just as for Term (1) alone, can describe only unchanging (i.e., constantamplitude) waves. T h e bracketed factor in Eq. (1 16) describes spacecharge-wave propagation in a drift tube. I n a traveling-wave amplifier study, Term (2) accounts for the effects of the space-charge voltage in modifying the complex propagation constant that results from the combined effects of Terms (I), (2), and either
rt)
r,
56
W. G. DOW
(3) or (4). There appears, either for the drift tube or the interaction gain application, a space-charge voltage parameter C, defined as follows: cp
=
1 (*6&JP2 I* (
)
this defines a dimensionlessspacecharge-voltageparameter called Cp ‘
(1174
As to magnitudes, C, may be expected to lie between 0.01 and 0.2.
(117b)
Other dimensionless ratio parameter forms have been employed in incorporating space-charge voltage effects into the analysis. Pierce ( I ) chose to use an impedance-ratio parameter which he called Q; here it will be called Qp, defined for a particular hot-circuit space-harmonic of phase velocity U, as follows: Q =---21
1
2
u l/unc,. ub (,f,z,,’
The minus sign is used because 1 Qp
=
2(-tc5,)
T-Jn
).
this defines the space-charge-voltage parameter Q,,, called Q by Pierce
(
(118a)
ec is numerically negative. Conceptually,
An rf beam self-impedance The rf circuit characteristic impedance
.
(118b)
Comparison with Eq. (117a) shows that (118c) In Eq. (1 18a) the rf beam impedance is stated as 1/ UnCq rather than as 1/U,C, because there is an actual hot-circuit bunched beam whose phase velocity is U,. An important merit of the Q p parameter is that, in combination with the gain parameter defined later by Eq. (120a), it provides the numerically convenient coefficient QpCa;thus QpCaranges from 0 to perhaps 1.0 in useful devices;
(118d)
also Q,C,
=
C:/Ci.
(118e)
Other writers have incorporated the beam current dependence of the space-charge voltage by using the electron density radian frequency w e = d-qePa/rome, where P, (to be interpreted as Greek capital rho) is the dc space-charge density. Their treatment employs a “reduced plasma electron frequency” which is smaller than we by a geometrical
57
SPACE-WARMONIC TRAVELING-WAVE ELECTRON INTERACTION
“space-charge reduction factor.” For a solid uniform round beam of radius r , in a drift tube of radius ri, such treatment describes C, as follows, in terms of C,, as in Eq. (70b):
),
%&‘~(for a solid round beam
in a drift tube
).
(l
where w e is the true electron density frequency in the solid beam. This value of Cp” would also be that in an infinite planar beam in which P, and therefore we are less than their true round-beam values, the reduction factor for we being the square root of the parenthesis. Thus use of Eq. (73) indicates that (11)
This space-charge reduction factor and the corresponding wavelength for the “reduced plasma electron frequency” are frequently used in preference to Pierce’s Q,. However, for a beam idealized into a charge sheet the space-charge reduction concept has no meaning, because the space-charge density is infinite. Handling of the space-charge voltage in terms of C,, defined by Eq. (1 17a) in terms of the beam self-capacitance C,,applies equally well to annular and solid round beams. Also, the use of C, permits direct employment, in studies of space-harmonic interaction, of the two C,, and Cqpcontributions to C,, each with its appropriate magnitude. For analysis of space-charge-wave propagation along a beam in a drift tube, as for example in the drift space of a klystron, the reduced electron density frequency and its corresponding wavelength can be convenient working concepts. Term (3), providing gain through interaction with an rf transmission circuit. This term, in combination with Terms (1) and (2), accounts for the gain-producing interaction with a lossless transmission circuit having a substantial characteristic impedance 2, and a significant zdirected rf field at the beam location, as called €or by the coupling coefficient 8,. This becomes possible because the form of Term (3) permits the characteristic equation to have complex roots, so that TsJToand TJT, may have imaginary components containing growth or decline coefficients a. Because in Terms (I), (2), and (3), without (4) and (9,there are only even-ordered time derivatives, the correspond-
58
W. G. DOW
ing characteristic equation forms Eq. (1 14a) and Eq. (1 14b) for F8/Fb and FJFf have wholly real coefficients; therefore the complex roots occur in conjugate pairs in which the equal and opposite imaginary parts contain the equal and opposite growth coefficients a. Thus if there is a growing wave there is also a declining wave, and the two have a common phase velocity. There appears in Term (3), for the first time it has been defined in the analysis, the very important gain parameter C,,for electron interaction gain with a transmission-type circuit, as follows ( I , 2):
“=I(
E cE v M - 0
4vb
this defines, for the space-harmonic circuit, the traveling-wave amplifier transmissionline gain parameter, called C by Pierce
)li3i (
This transmission-line gain parameter dominates the interaction behavior when Rh fioZo,as this reduces Terms (4) and ( 5 ) to relative insignificance. The absolute-value bars are necessary to eliminate (from this definition) the complex cube roots of unity. For most useful engineering devices
<
c: =
(,fJbzo
4Vb
is of the order of lo-’, i.e., very much less than unity
(
)’
( 120b)
Therefore ( 120c)
C, ranges between perhaps 0.02 and 0.2.
Another form of the Eq. (120a) definition is ( 120d)
where Z, is a beam-location rf impedance parameter, given by 2,
=
I E c E J o I;
this defines the beam location impedance, for interaction
(with the space harmonic having the properties P,,
I,,
Term 4, permitting gain through interaction with a resistive circuit environment. I n its Eq. (113) form, Term (4), in combination with Terms (I), (2), and (9,accounts for the appearance of gain-producing interaction with an rf voltage at the beam that is due to passage of rf current through the ohmic resistance of an environmental envelope. Term (4),.like Term (3), permits gain under some conditions because it permits complex roots of the characteristic equation. However, because r, = j&, this equation has imaginary coefficients, so that the complex roots are not usually complex conjugate pairs.
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
59
There appears in Term (4) of Eq. (1 13) another dimensionless gain parameter, thus, generally paralleling Eq. ( I20a):
This defines the resistive-wall traveling-wave interaction gain parameter, called here C,. Because 5, is numerically negative, the two terms in the parenthesis combine additively. This resistive-wall gain parameter dominates interaction behavior if Rh W L h . A drift tube with a resistive wall has this property, asL, is extremely small; the charge involved in its C,, comprises the terminations of the electric flux interior to the drift tube between rf voltage maxima and minima along the surface. Before using C, in an interaction analysis, the other terms of Eq. (1 13) must be recast by a fresh start from Eqs. (23a) and (23b), in which entirely new r" and Zo" = Vh,,,/Ij,,,tare obtained. There appears in Terms (4a) and (4b) a quantity identifiable as a dimensionless parameter that measures the effect on the total wave of the resistive loss property of the circuit; thus in relation to Eq. (29b) and Terms (4a) and (4b) above, let
>
this defines the numerically positive resistive attenuation coefficient C,, which is equally useful in studying interaction with helical, drift-tube, or spatially periodic circuits. I n the shift to the helix from typical slowwave spatially periodic circuits, U,/Uo+ 1 in Eqs. (1 13), (1 14a), and (1 14b). In Eq. (1 14a), U,/Uooccurs twice in Term (4b), but only once in Term (4a); as a result, in the shift to the helical circuit Term (4a) drops by an order of magnitude relative to Term (4b), so that Term (4) reduces to Term (4b) in the "first-order" helical circuit perturbation study. This is the reason for splitting the term into two parts. Note in this connection that in Terms (4a) and (4b) of Eq. (1 14a), the first parenthesis can be restated as follows:
This quantity has comparable magnitudes in typical spatially periodic and helical circuits, and correctly indicates centering of design magnitudes around r,, which is governed by beam voltage which is usually comparable for the two classes of circuits if built for similar objectives.
W. G . DOW
60
It is the Eq. (124) quantity that is compared with Cu and C, for both classes of circuits. Term (9,accounting for resistive attenuation in the circuit. This term carries over directly from the characteristic equation for the Eq. (24) differential equation for a lossy transmission line, and correspondingly accounts for the effect of the ohmic resistance of the transmission-line circuit in reducing the gain or causing there to be a net attenuation rather than gain. The Eq. (123) dimensionless parameter is, of course, used in Term (5).
XI. NORMALIZATION RELATIVE TO
THE
GAINPARAMETER Cu
Normalization of Eq. (114a) with respect to C, from Eq. (120a) permits study of interaction details and terminal-plane boundary effects following very much the Pierce pattern for forward-wave helical operation. However, there is present now the new design parameter U,/U,. For the rather common types of slow-wave space-harmonic structures in which the first forward space harmonic of the backward wave has a phase velocity much smaller than that of the total wave, U,/Uo is small relative to unity, and is therefore appropriately normalized with respect to C,. Even for circuits in which the ratio U,/Uo is only moderately small, its description in normalized form aids in determining in what respects and to what extent this circuit property affects the interaction. T o study the pushing-normalized Eq. (1 14a), let this is the ratio of the phase velocity of the first forward space harmonic of the backward wave to that of the total wave, normalized relative to C,; its magnitude is generally of the order of unity
( 125a)
B = U ,
(used as a “frequency offset” parameter),
(125b)
CUUb
(a space-charge .voltage parameter) (125d)
Here C, is given by Eq. (120a); C, is defined by Eq. (1 17a), and
Q,,by
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
61
Eq. (118a); C,is defined by Eq. (123), being an attribute of the total wave. Also, as a normalized measure of pushing, let ( 126a)
This pushing measure
is complex, thus
A
= -
Aa -jA,,
(126b)
where (1 26c)
and
Aa
( 126d)
= 4Cai%
The slip-normalized equation, Eq. (1 14b), is generally found more useful than Eq. (114a). For use in Eq. (114b)' and largely following Johnson's modification of Pierce's treatment ( I , 2): (1 27a) b = - =u, --=-d = -ub 1 =
u~ ca
& = B - u, b
rr carb
tau,
u,
Ub Or, = D -
a frequency offset para(meter, being Johnson's C)'
(a circuit loss parameter),
Uf
c&b
(127b) (127c)
(127d)
Comparisons with Eq. (125)' with employment of Eq. (log), show that
Also, as a normalized measure of slip, let ja
=
-~ rs
'
(the 6 so defined is Johnson's 8).
( 129a)
The slip perturbation measure so defined is complex; thus again using Pierce's notation, ( 1 29b) 8 = x +jy,
W. G. DOW
62 where
(1 29d)
x = a/CJ$.
It is sometimes desirable to separate out the electron interaction growth or decay from the cold-circuit decay. Let a0, ai, 01 symbolize respectively: the exponent decay factor for the lossy
cold circuit, the exponent growth or decay factor resulting from the electron interaction, and the exponent growth or decay factor resulting from the combination of interaction with the loss due to the cold circuit; - a,, is always numerically negative; mi is numerically positive for interaction gain of a forward wave, negative for interaction loss of a forward wave. Obviously then exp ax
=
exp aizexp (-a0z)
so that (Y
=
exp (ai - a,,) z,
( 129e)
(129f)
= ai - a”.
Division by Capb or -Capr, followed by comparison with Eq. (126d) for a,, Eq. (129d) for x and with Eq. (12%) and Eq. (127c) for D and d, gives relationships interpretable as 2‘
=
’+
Lai =
La
this defines x, as a normalized measure of the interaction gain, in slip symbolism ) ?
(
= ag/capb’
+
=
ml/Capf’
(129g)
this defines A,, as a normalized measure of (the interaction gain, in pushing symbolism)’
(129h) Interrelations between the
a = - 1a-+CaB jB ’ =
and 6 quantities are as follows: and
B -in and 1 - CaB ’
’=l-CaB’
and
=
jn
-
=
A,=
a+$. 1
+ Cab ’
(130a)
b-js. 1 +Cab’
(130b)
Y + b . 1 +Cab ’
(130c) (130d)
(1
+ Cab)(1 - CaB)= 1.
(130e)
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
63
The “first-order” perturbation equations are those obtained by assuming that Cab, CaB, Can,C,S and all similar quantities are small relative to unity; thus, first-order relations include:
-in,
jS
B
y
A,-B,
x
G
and and
jn
b -jS;
( I30f)
A@r y + b ;
( 13%)
Am.
(130h)
The Eq. (114a) and (114b) forms of the characteristic equation are made use of by first dividing through by C:, next introducing either the various Eqs. ( 125), ( 1 26) or the Eq. ( I27), (1 29) perturbation or difference quantities, then using the methods outlined in succeeding sections for or solving the resulting equations for the real and imaginary parts of i? of A. The space-harmonic and total-wave complex propagation constants r, and r obey the following equations:
(131d)
Note in relation to this last equation that for the helical circuit, in which for the mode usually employed U,= Uo and 8, = Po, the Eq. (125a) definition of G reduces to G = l / C m .Also note that for the spatially periodic circuit,
The Eq. (1 3 1 e) and Eq. (1 31f ) pairs rearrange to show that P n = PA1
-YCJ
= PA1 -
AvCm).
(131h)
T h e illustrative Fig. 11 w / p diagram will be used to show why b and B are called “frequency offset’’ parameters in this study of space-harmonic
64
W. G . DOW
interaction. The designer of the device specifies the w / p diagram, including the structural spatial interval A, which determines pd = 27r/Ad, and both the total-wave and first forward space harmonic curves. T h e
FIG. 1 1 . Illustration of the identification of the reference radian frequency wb through point P, the intersection of the u-8 curve with the beam velocity line ub. Also identification of points Q, R, S, T at signal frequency o. This figure is used to show why b is called a frequency offset parameter.
operator of the device determines the beam voltage V,, and therefore also the beam velocity, I n the figure a straight line through the origin having as its slope the beam velocity U, intersects the space-harmonic w / p curve at the reference point P, thus specifying the following magnitudes: wb = the reference radian frequency, being the radian frequency at the P intersection of the beam-velocity straight line with the w / p curve for the first forward space harmonic of the backward wave ; &, j?,,, symbolizing the radian wave numbers at the reference frequency, respectively for the backward total wave and its first forward space harmonic. U,, = the numerically positive phase velocity of the forward total wave at the reference frequency w,; the backward total wave’s phase velocity at this frequency is -Urn;
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
65
U,, = the numerically negative group velocity of the backward wave, of course for all space harmonics and the total wave. Because
PR is short, nearly enough, U , at R equals Ugh. All of these quantities are determined before the appearance of any rf signal in the device, merely by the structural design of the circuit and the choice of the beam voltage. For a backward-wave amplifier, the operator is free to choose arbitrarily the operating frequency w , being the frequency of the input signal voltage. I n general, as illustrated in Fig. 11, he will choose an operating frequency near to but somewhat below the reference frequency, as being necessary for satisfactory voltage gain, for reasons that will be apparent later, Thus there is introduced the definition wb-w
=
radian frequency offset, for operation at signal fre(The quency when the reference frequency has the value w ) w,
Because the operating frequency is fairly close to the reference frequency the group velocity can be considered frequency invariant within the range of useful frequencies; that is, it is satisfactory within the short distance PR in Fig. 11 to identify the w//3 line with its tangent at P. In general it is not acceptable to treat the total-wave phase velocity similarly, as in a dispersive circuit the phase velocity may change substantially for small changes of frequency, especially as the cutoff point is approached. In the nondispersive frequency range for the total circuit, U, = -U0, and their common value is invariant within the RP type of frequency interval. It is apparent from the Fig. I 1 construction that
(I 32b) note that U, and 8, are both numerically negative, for the condition illustrated in the figure. This expression is now divided by /3* = w / U , and by C,; use of the Eq. (127b) definition of the parameter b then gives wb
b=
4
ub
-W 4
(1
this describes b in terms of ob - w
--);UQ (the frequency offset
).
(133a)
Because U , is numerically negative for a backward wave, the two terms in the parenthesis are additive; note that b is positive when the Eq. (132) frequency offset is positive. This relationship is the reason for calling b and B frequency offset parameters, in their initial introduction
W. G. D O W
66
in Eqs. (125b) and (127b). It is apparent that for the determination of b there are needed the w / p diagram, the dc beam velocity, the input signal frequency, and the gain parameter C,. For operation where the total-wave propagation is nondispersive, so that U, = -Uo, use of the definition of g in Eq. (127a) converts the last equation to b=
%
-
wca
(l
+gCa)$
the total-wave prop(when agation is nondispersive )’
(133b)
In some problems dealing with backward-wave start-oscillation conditions it is desirable to be able to express the frequency offset in terms of the reference frequency and the offset parameter b ; for this purpose Eqs. (133a) and (133b) can be rearranged as follows:
wb
-w wb
1
+ Ca(b + g) , K
Y
(the nondispersive case).
(133d)
XII. EXPRESSIONS FOR GROWING-WAVE HOT-CIRCUITGAINAND COLD-CIRCUIT Loss IN TERMS OF TUBE LENGTH Three quantities, the tube length, either x or A,, and either C, or G, can be used in combination to find the over-all power gain or loss due to any one of the several waves of a traveling-wave amplifier, whether of the spatially periodic or helical-circuit type. T h e tube length is conveniently described either in terms of the somewhat artificial beamvelocity wavelength A, (calling for use of x and C,), or of the total-wave cold-circuit wavelength A, (calling for use of A, and G). These wavelengths are defined as follows: /?b
= 2n/hb,
that is,
Po = 2n/h0, that is,
hb
=
Vb/(signalfrequency);
(this defines
13;
h, = Uo/(signalfrequency); (this defines a,,);
(134a) (134b)
Then let Nb, No symbolize respectively the number of fictional electronbeam wavelengths (A,) and of total-wave cold-circuit wavelengths (A,) for the length 1 of the interaction region ( z = 0 to z = 1); thus
Nb = I/&, and No = l/A,,
and Pol
= 2nNb;
(this defines NP,);
= 2nN0; (this defines No).
(134c) (134d)
67
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
As to voltage gain, from the definition of a, and using Eqs. (129d), (126), and (125a), for any one growing or declining wave: Voltage ratio change in the +z-direc(tion, per electron-beam wavelength A,)
= exp
= exp ( 2 v a / f l b )
= exp (2rrCv.x);
(1 35a) Voltage ratio change in the +z-direction (per total-wave cold-circuit wavelength A,)
-
2rra
- exp OLA0 = exp A = exp
2rAa
G ’ (135b)
Then as to power gain, use in connection with these of the definition of db power gain as 20 log,, (voltage gain) leads to the forms, useful for any one wave: DB power change in the +z-direction per beam-velocity wavelength A b
(
power change in the +z-direction, per (DBtotal-wave cold-circuit wavelength A, )-
)=
wO@f/pO)
(135c)
wbc,;
“O ;‘/
DB power change from z = 0 to z = 1, corresponding to numbers of wavelengths N r and No = WbNp,C, = W,N,/G;
)
where
(135d)
(135e)
(135f)
The factor 54.575 is, of course, the numerical value of 20 log,, (exp 277). Note that from Eq. (130h), for the first-order study x = A,, so that in such a study W, = W,. This means that in general, from Eq. (135e), for any one wave -No- b w GC, GC,. (135h) ~
Nb
W’O
The above power Eqs. (135c), (135d), (135e), are completely useless relative to typical backward-wave operation, because they refer only to any one wave, and backward-wave gain is caused by an interference pattern existing because of the “beating” against one another of several waves having different phase velocities. The requisite terminal boundary conditions governing backward-wave gain are discussed in a later section. Equations (135c), (1 35d), and (135e), can be useful in analysis of forward growing wave amplification. Frequently the distributed cold-circuit power loss is described in a manner similar to Eqs. (135f) and (135g). Paralleling Eq. (135a) and
W. G. DOW
68
(135b), the cold-circuit voltage ratio change in the direction of the group velocity, per beam-velocity wavelength is exp( -adb),and that per totalwave cold-circuit wavelength is exp (-a,,X,,). Then from Eq. (127c), a0 = CUpbd,and, from Eq. (125a) and (125c), 01, = P,D/G, so that, Cold-circuit voltage ratio change in the direction of the group velocity, due to ohmic resistance of the circuit, for Nabeam-velocity wavelengths
= exp (-27rC,Nbd);
(136a)
Cold-circuit voltage ratio change in. the direction of the group velocity, due to ohmic resistance of the circuit, for N o cold-circuit total-wave wavelengths
= exp
(1 36b)
1
(-27rD/G);
these resemble Eqs. (135a).and (135b) as to form. From these and the db-loss definition, similar to that for the db power gain, where
L,i = 54.575NbCud = 54.575NoD/G,
(1 36c)
La = the power loss, in decibels, due to the ohmic resistance of the circuit, being used as a measure of d and of D, for the interaction length I having N, and No numbers of wavelengths of the two kinds. Note also that, more simply, L,
=
54.575N,(a0//9,)= 54.575N,(ao//9").
(1 36d)
It is important to note here that as indicated by this last equation, Ld is a property of the cold circuit only. The significance is that La is easily measured experimentally ; the measured value is then used in Eq. (136c) to find d and D,which in combination with other parameters govern the hot-circuit behavior. For operation of a helical circuit in its usual mode, U,becomes U,,so that Pr = Po; in this case Eq. (125a) reduces to 1/G = C,; substitution of this into Eqs. (135b), (135d), (135e), (135h),(136b), and (136c) gives the appropriate complex pushing types of relation for the helical circuit.
XIII. THEFIRST-ORDER QUARTICS FOR PROPAGATION-CONSTANT PERTURBATION DUETO SPACE-HARMONIC INTERACTION In order to permit use of the several parameters defined in the previous rather section, the characteristic Eq. (1 13) is divided through by C,", than by Cz as customary in the forward-wave helical-circuit study. The results, expressed for the Eqs. (1 14a) and (1 14b) transmission-line form of Term(4), are as follows for the slip and pushing expressions respectively:
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
69
For the complex slip, obtained by using Eq. (1 14b): -
(jS)2 ( b -jS) (-2g = -2g( 1
-
+jS) ( 1 +jCaS)' +j4d( 1 +jCaS)2+j2a2gd(1 +jCaS)' -(js)2j2gd.
+jS) + u2(h -jS) (-2g
b
+
-
b
( 1 37a)
For the complex pushing, obtained by using Eq. (1 14a): - j n ( B -jA)"(-2G =
-in)+ j A 2 A ( - 2 G
-2G(1 - CaB) (1 -
-in) (1 -jC.A)2
+j4D(1 - CaB) (1 (1 37b)
+j2A2DG(1 -j C a n ) 2-j2DG(B -j A ) 2 .
The concise first-order equations are obtained by rearrangement after omitting all terms containing C,; this converts Eq. (137a) and Eq. (137b) to -
- u2]
[(in- B)'
[2g(-b - j d +jS)
- A2] [ 2 G ( - j n
-
(b -jS)2] = -2g + j 4 d ; (138a)
-jD) - ( j A ) 2 ] = 2G
- j4D.
(138b)
In straightforward first-order quartic form these become: - (is)*
+ ( j S ) a 2(b + g )
-
(jay [2g(b + j d )
+ b2 - a']
-jS2a2(b
+g )
+ [-2g +j4d + 2gu2(6 +j d ) + u2b2]= 0; -
(1 39a)
[2G(2B - J D ) - (B' - A')] (in)'+ 2(B - G ) + +jA2[G(B2- A2) -j2GBDI + [-2G +j4D -j2GD(B2 - A2)] = 0. (139b)
An important difference between Eq. (138) and its forward-wave counterpart is indicated by noting that when d, b, a2, and B, D, A 2 are all zero, but with 2g and 2G large enough to make the cubic term mask the fourth-degree term, the limiting cubic forms are S3 = j , and
=
-j.
(140)
This is opposite in sign to Pierce's familiar simplest-case cubic, but has the same sign as Johnson's simplest-case cubic (1, 2). The terms 2g and 2G on the right sides in Eqs. (138a) and (138b) are of the order of unity for a space harmonic whose phase vefocity is much less than that of the total wave. The appearance here, on the right, of the terms containing d and D is for most applications not of
70
W. G. DOW
great importance ; conceptually their presence indicates that the voltage at the beam due to ohmic circuit resistance may have a first-order effect. Primary engineering interest attaches to the no-loss forms of the first-order equations, obtained by letting d and D vanish in Eq. (138). For the complex slip dependence this gives
-(W
=
28 ( b -is) (2g + b
or [($)a
- a21 ( b -jS) (2g
-is)
+b -jS)
- a2,
(141a)
= -2g.
(141b)
For the complex pushing dependence the result is ( 142a)
or [(jn - B)2 - AqjA(2G
+in)+ 2G = 0.
(142b)
The corresponding straightforward quartic forms are obtained by letting d and D vanish in Eq. (139). Note that Eqs. (141a) and (141b) are obtainable from Eqs. (142a) and (142b) by the following first-order operations: Replace: jn by (b -$);
XIV. GRAPHICAL STUDYOF
THE
B by b ; G by g.
( I42c)
PERTURBATIONAL QUARTIC EQUATION
T h e relationship of the perturbations to space-harmonic interaction with the beam will be initially illustrated by a study of the first-order Eq. (141), for the perturbations, there being presumed no ohmic circuit loss, but an appreciable space-charge voltage; attention will be given to a considerable range of values of the frequency offset parameter b. Occurrence of the roots can be studied by means of graphical representations, as in Figs. 12 and 13, of the following quartic equation based on Eq. (141b):
p
=
[( is)2- 4 ( b - j s ) (2g
+ b -is) + 2g,
(143a)
where, of course, p = 0 at each of the four roots, and the independent variable is j S . Alternative interesting forms are
p
=
[ ( j V - a21 [(.is)' - 2(g 3-b) js
+ b(2g + 41 + 2g,
(143b)
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
71
As so stated for the variable j S , the coefficients of Eq. (143) are all wholly real, so that the roots in general occur in two sets of two each, each set Q
I
1
/
4
I
/
t2p+bI2-a2
FIG. 12. Illustrative chart of p vs. jS curve for a case having a growing, a declining, and an unchanging backward total wave, and a perturbed forward total wave, from the roots of the quartic for jS. Note points Q. R, S, T, in reference to points similarly marked in the Fig. 1 1 w - 8 diagram.
consisting either of a complex conjugate pair or of two real roots. T h e form of Eq. (143b) suggests expression as follows, for convenience in graphical representation: p = pap, 2g (the quartic), ( 144a) where pa = ( - a2, (the space-charge-wave parabola), (144b)
+
pb
=
( b - j 8 ) (2g
+b
-
j 8 ) = [ i s - (g
+ b)I2 - g2, (the heam-circuit parabola).
(144~)
72
W. G. DOW
T h e vertex of the space-charge-wave parabola occurs at j S = 0, pa = -a2, and of the beam-circuit parabola at j8 = g b, p b = -g2. Introduction of these values forjS into Eq. (144a) gives the two following points on the quartic:
+
At j 8
= 0,
At j 8 = g
p
= -u2(b’
+ b,
p
+ 2gb) + 2g;
= g[2 -#
(the point Q abscissa).
(145a)
+ 2bg + g2 - a”] ; (the point S abscissa) (145b)
The complete quartic is easily sketched by using these two points and the additional knowledge that p = 2g at each of the four points jS = f a, j S = b, and jS = 2g b, these being the four roots of the first term on the right of Eq. (144a), that is, the four points at WhiChp,pb = 0. For the condition g = 1, b = 1, the quartic and its two Eq. (144b) and (144c) contributing parabolas appear in Figs. 12 and 13 respectively, using a = 0 and a = 1, that is, QpCa= 0 and QpCa= 0.25. Note in Fig. 12 the existence of two wholly real roots jS, and jS4, corresponding to unchanging waves, and the pair of complex conjugate
+
FIG. 13. Chart of p vs. jS for a condition having no growing or declining waves, with three backward total waves from jS, jSe, jS,, and a perturbed forward total wave from jS,.
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
73
roots j61, jS,, for growing and declining waves, having real-part values that are negative but quite near to the j S = 0 nonzero minimum. T h e occurrence of this pair here suggests correctly that it is reasonable to expect a pair of complex conjugates near any nonzero extreme whose convex aspect points toward the horizontal axis of coordinates. An w / p diagram corresponding approximately to Fig. 12 appears in Fig. 14; here at the left there are straight lines from the origin having slopes U , = U,, U,, and U,, being the total-wave phase velocities for the roots. The primary effect of the interaction is to produce these four totaE waves, which can be thought of as perturbations from the cold-circuit waves on the undisturbed circuit. At the right in Fig. 14
---
0
ad
8-
FIG. 14. Chart of w vs. /? for a system like that for Fig. 12 having both a growing and a declining backward total wave of equal /?-values, at is,,j&,, a fast backward total wave at iss,and a perturbed forward total wave at is,.
are straight lines from the origin whose slopes describe the phase velocities of the first forward space harmonics of the four total waves, which are secondary but very useful aspects of the effects of the perturbations. There are four “cardinal points” of any w/p space-harmonic interaction diagram and its corresponding quartic equation, being the points Q, R, S, T, shown on Figs. 12 and 14 for one set of design and operating parameters, and on Fig. 13 for a different set. Note with respect to the Fig. 12, 14, w/P diagrams that: are definedas the two intersections, respectively of the straight slant
Q , S (U, line, and of the vertical &axis, with the horizontal line the operating frequency w ; at Q, An = /?,;atS, fin = &
at
74
W. G. DOW
are defined as the two intersections, respectively for the forward and backward space harmonics, of the w / s curve with the horizontal line at the operating frequency w ; at R, )In = fl,; at T, fin = 28,
+
By virtue of definitions in Eqs. (127a), (I27b), and (129a), of g, b, and jS, the real parts of j S at the cardinal points can be stated as follows: Cardinal point:
Q;
Real part ofj8: zero; g
S;
R;
T;
+ b;
b ; 2g
+ b.
(146c)
With respect to the quartic equation graphs of Figs. 12 and 13, applicable only when d = 0, note that
Q,+
are placed at the vertices of the two contributing Eqs. (144b) and (144c) paraolas, that is, Q occurs at jS = 0, pa = -as, and S at jS = g b, pa = -gz; as ; so placed, Q and S are in general not points on the quartic curve for p
+
)
(146d) are placed at the two roots of the p b parabola expressed by Eq. (144~): that is, R occurs at jS = b, p b = 0, and T occurs at jS = 2g 6, p b = 0; both R and T are always points on the quartic curve, at the ordinate p = 2g
+
)
(146e)
These definitions on the w//? diagram, and placements on the quartic no-loss curves, are used for the four cardinal points primarily because of the following relations that follow directly from Eqs. (127a) and (127b). B d - Bf
=
(147a)
cdb ’
(147b)
Also note that, in accordance with Eqs. (129b) and (129c) defining y , -J’
The real part of jS at /% -/% = (any root of the quartic) =
c,Bb
(147c)
For the lossy hot circuit, d f 0, procedures paralleling those just given but suggesting complex-plane graphical representation, use the complex-coefficient quartic for j S as follows, growing out of Eq. (138a):
Here p may be complex; just as in Eq. (143), p = 0 at each of the four roots of the quartic, which may all be expected to describe growing
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
75
or declining wave propagational properties. This complex quartic can also be expressed, paralleling Eq. (144), as
P
=p d b
+ 2g - - I 4 4
( 149a)
where
p,, = (
(149b)
- a',
and pb =
[(h
+j d -is) (2g + b - j d
=
[js - (b
- j & ) - d"],
(149c)
also pb
+ g)I2- gz +j2gd.
(1 49d)
The two parentheses in this equation become zero at the cold-circuit space-harmonic propagation, respectively for the forward and backward total waves, so that the complex-function placement of the four Fig. 14 cardinal points occurs as follows: ~t
R: i s
=b
+jd9
Pb
coincident with the backward cold-circuit propagation
(
=
The vertices of the pa and f i b functions, found where the respective derivatives relative to j8 vanish, appear as follows: At Q: jS
= 0,
At S: j 8
=g
p,
+ b,
= pb
- a2, = -g2
(1 50c)
(propagation with zero slip),
+j2gd, ).
the point of division as between forward (and backward total-wave propagation (150d)
Study will show that none of the four cardinal points as so placed are on the complex-function representation of the quartic; Q is on the p , representation, and R, S, T are on the pb representation. It is desirable for later use to state the values of the p quartic function for the values of jS at Q and S. From Eq. (149) and paralleling Eq. (145), At the abscissaja = 0 corresponding to Q:
+ 2bg) + 2~ j 2 4 2 + a'g); At the abscissa jS = g + b corresponding to S: p = g (2 - g[(b + g)' a"} - j2d (2- g[(b + g)2 - a"}. p
=
-a2(b2
-
-
(1 50e)
( 1 50f)
76
W. G. DOW
It is the presence of the j d terms in Eqs. (138a), (148), and (149) that causes all four roots of the complete lossy-circuit p quartic to be in general complex, but not consisting of complex conjugate pairs; unchanging waves are not to be expected. It will appear later that although R and T are never roots of the quartic, for various extreme conditions certain of the roots can be made to approach indefinitely close to the values of jS at R and T. XV. THEQUARTIC FOR DRIVEOF
THE
FORWARD TOTAL WAVE
Space-harmonic interaction can be used to drive the forward total wave as well as to drive the backward total wave. Figure 15 will be used to show how to adapt the backward-wave perturbational quartic to aid study of forward-wave interaction, and to illustrate a somewhat more
FIG. 15. Chart of velocity lines and w / P curves showing the reference frequency w i at P' for studying interaction with the forward total wave.
general form symmetrically related to forward-wave and backwardwave interaction. I n this figure the cardinal points Q, s, T, R have the same definitions and physical meanings as in Fig. 14 and Eq. (146). The significant changes are in the reference frequency concepts; thus let to; symbolize the new forward-wave reference radian frequency
defined as the frequency at the point P' of intersection of the straight slant beam-velocity line U, with the wJP curve of the first forward space harmonic of the forward total wave;
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
77
the symmetry reference radian frequency, defined as the frequency at the point H of intersection of the U, beam-velocity line with the vertical Pd axis of symmetry of the forward space harmonic w / P curve; also Pio,/?& symbolize the values of Pb and Po at point P’, reference frequency wh =
4
U;, /?; symbolize respectively the phase velocity and the radian wave number for the cold-circuit propagation at the operating radian frequency w, corresponding to the point T on the forward total wave branch of the curve for the first forward space harmonic. Then paralleling previous usage:
b ’ = U’L - -
this defines the forward-wave frequency offset parameter
--
‘a’;
this evaluates the forward-wave frequency offset parameter
A symmetrical frequency offset parameter is used in statingthesymmetrica1 perturbation quartic; let h=
pd
- fib
capb
h=
(frequency this defines the symmetryoffset parameter h
(1524
this evaluates the symmetryfrequency offset parameter h
(1 52b)
It is also easily shown geometrically that, (1 52c)
By using these definitions it is easy to show that
+ h, so that b + g = h ; b’ = g + h, so that b’ - g = h ;
b
=
b‘
-g
+ b = 2h;
b‘ - b
= 2g.
(1 53a) (153b) (1 53c)
78
W. G . DOW
I n the w/,3 type of diagrams of Figs. 14 and 15, and equally in the type of quartic graph of Figs. 12 and 13: Numerically positive or negative values of h correspond respectively to placements of Q to the left, or to the right, of S
(
T h u s in Fig. 15, h is numerically negative, and Ihl < g ; therefore b’ is positive, b markedly negative. Optimum forward-wave amplifier operation usually occurs at or near to zero value of the forward-wave frequency offset parameter b’, in sharp contrast to the occurrence of optimum backward-wave gain for substantial positive values of the backward-wave offset parameter b. T h e first-order quartic Eq. (148) is converted by means of the Eq. (153) relation b = b’ - 2g, to the following forward-wave form:
p
=
[(jS)z
-
a21
[(-2g
+ b‘ f j d -jS) (b’ -j d -jS) --
dZ]
+ 2g
-
j4d.
( I 54a) The
p
= 0 roots satisfy the following rearranged form:
- [(jS)” - us] [g(b’ -j d -jS)
- *(b‘
= -g
+j2d.
(154b)
This is the first-order quartic in a form convenient for a forward-wave study. To obtain the complete (i.e., not first-order equation), the substitution b = b’ - 2g may be made in Eq. (137a). T h e b’ here has the same significance as the b in Pierce’s study of the helical forward-wave amplifier (I). T h e first-order quartic Eq. (148) is converted to the more general h, the result symmetric form by the Eq. (153) substitution b = - g being
+
p
= [(jS)z - a2] [(-g
+jd
+ h -jS)
(g - j d
+h
-
jS) - d2]
+ 2g -j4d. (1 5 5 4
The
p = 0 roots satisfy the following rearranged form: - [(
- G ~ [g(g ] -j 2 d ) - (h -jS)2] = - 2g fj4d.
(1 55b)
This equation is at the same time the most general and the simplest form of the first-order quartic, and the Eq. (152) frequency offset parameter employed is the simplest to determine from the 0//3 diagram. T h e preferential use made in this text of b and b’ rather than h is based only on the grounds that there exists a large body of literature employing them, and that most individual applications are primarily oriented
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
79
either toward backward-wave or forward-wave operations. For a lossless circuit, d = 0, Eq. (155a) reduces to
p
=
[(jS)O - u2] [(jS - h)O - g2]
+ 2g,
(15%)
which is obtainable from Eq. (143) by the Eq. (153) substitution b+g=h.
XVI. FREQUENCY OFFSET EFFECTS; REDUCTIONTO
CUBIC
This section presents itemized comments having to do chiefly with the effects of the frequency offset, measured by b or b’, on space-harmonic interaction, based on the previous section’s figures and equations. Primary attention is devoted to the lossless circuit, and to criteria for reduction to the cubic.
A. Roots to the Right of S Represent drive of the forward Total Wave. I t is important to note regarding the backward-wave quartic Eq. (144a) that any root that occurs to the right of the point S in Figs. 12 and 13, this being the vertex of the P b parabola, represents a space harmonic of the forward total wave.
B . Mirror-Image Symmetry of the Quartic, for Positive and Negative Placements of the Cardinal Point S A shift of the vertex at S of the p b parabola from any positive value of jS to the same negative value produces a mirror image change in the quartic about the vertical jS = 0 axis in Fig. 12 and Figs. 13 because of the symmetries of the two p , and pa parabolas about their respective axes. Interpretation of this mirror image change in terms of the Eqs. (153) shows that it corresponds to an exchange of --h for h, and of -b’ for b. T h e two mirror images of course become mutually coincident when b = b’ = -g, that is, h = 0, corresponding to placement of Q in coincidence with S on the Fig. 14 w//3 diagram, and of S in coincidence with Q on the Fig. 12 type of quartic graph. This behavior extends to the Eq. (149) complex-p condition. C. Approach of Roots to Cardinal Points T or R Requires in the limit non-interacting Waves Any interaction between the beam and a wave on the circuit results in bunching of the beam; bunches induce current in the circuit, thereby perturbing the propagation constant. A negative corollary of this
80
W. G . DOW
statement is that if there is no perturbation, there is no interaction between the beam and the wave. Therefore any occurrence of a root of the quartic at either T or R, that in the Fig. 14 and 15 w//3 diagrams call respectively for cold-circuit forward and backward total-wave propagation, implies that propagation in the appropriate direction will take place at the operating frequency corresponding to the stated frequency offset parameter, without being affected by the presence of the beam. If there are two noninteracting circuit roots, one in the limit at T, the other at R, no beam-wave interaction occurs at the frequency described by the stated offset parameter. The other two roots then describe pure space-charge waves requiring electrokinetic rather than circuit voltage excitation. This condition is approached when both g2 and u2 are very large.
D. Reduction to the Backward-Wave Cubic, when the Slow Wave is not Very Slow, or the Gain Parameter is very Small Figures 12 and 13 illustrate conditions in the backward-wave amplifier range of interesting frequency offsets, employing numerically positive values of h. In these figures h = 2, g = 1, making b = - 1 2 = 1. Although for the general positive-h range of conditions depressing b, the S vertex by use of a large g2 moves the j S , root very close to 2g or to 2g b - j d for the complex quartic, the ji3, root remains reasonably remote from the values b or b j d of the abscissa of the cardinal point R, for useful engineering design magnitudes. Study of the quartic graph suggests correctly that when the S vertex is greatly depressed the left-hand portion of a positive-h quartic becomes essentially a cubic curve whose shape is unaffected by the now large value of g2. The cubic form is obtained from Eq. (143c) by omitting the terms that become small when g is large, thus obtaining, for the lossless circuit, - [(j8)2 - a21 (b -j 8 ) 1. (156)
+ +
+
+
' z-
+
Of course p / 2 g is the useful ordinate to use for the cubic graph. The same procedure, of using a large g , applies for reducing to cubic forms the quartics that describe the behavior in the presence of circuit loss, d # 0. The result, applying only for positive values of h, that is. with Q to the left of S of the quartic graph, is (
( b -j8) - d ( b -j8) (1
+juZd(l +jcaS)'
-
+jC,S)2 = - (1 -t jC,8)z the backward-wave cubic, applying when both g and C , are relatively large
(
). (157a)
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
81
This complete (i.e., not first-order) cubic form is likely to be necessary when g is made large by using a circuit in which the phase velocity of the space harmonic is a substantial fraction of the total-wave phase velocity. If g is made large by using a small gain parameter, as when the dc current is very small, the first-order form may be applicable, and is obtained by omitting interior terms containing jC,, to give the following form extensively studied by Johnson (2): (” -k
“1
(b + j d -j8)
=
”
the first-order backward-wave cubic, (applying when g is large and C , is small
). (157b)
This is consistent with Eq. (141b).
E. Reduction to the Forward- Wave Cubic, when g is Large T h e forward-wave amplifier’s range of frequency offset values employs numerically negative values of h, for which Q is to the right of S in an 0//3 diagram, and S to the left of Q in the quartic graph. T h e mirror image counterparts of Figs. 12 and 13 that might illustrate the negative4 1 - 2 = - 1, behavior would employ h = - 2, g = 1, making 6’ = from Eq. (153b). When, by depressing the S vertex, jS, in such a mirrorimage graph is moved very close to the cold-circuit backward-wave value - 2g b’ j d that applies at R (now at the extreme left of the graph), the jS, root remains reasonably remote from the b’ -j d value at T. There remain three S, S, and 6, interacting forward waves; with appropriate design magnitudes gain may appear as a result of one of these being a growing wave; the presence of gain as a result of an interference pattern is not expected to be the dominating feature of forward-wave amplification. T h e right-hand portion of the mirror-image quartic graph now resembles a cubic of shape independent of g2. T h e firstorder forward-wave cubic equation is obtainable by employing a large g in the Eq. (154) relation that is appropriate when h is negative. T h e form commonly used is (1)
+
+ +
(P+ a2)(b’ -j d -j 8 ) = - 1,
first-order forward-wave cubic, (Pierce’s applying when g is large and C , is small
This is identical in form with the forward-wave cubic obtained in Pierce’s study of the helical forward-wave amplifier. T h e corresponding complete (i. e., not first-order) forward-wave cubic equation can be obtained from Eq. (137a) by first making the Eq. (153) substitution b = b’ - 2g, then lettingg become large enough to mask all terms not containing it as a factor, then canceling it.
82
W. G . DOW
F. Design and Operational Criterion for “Largeness” of g , to Eliminate the fourth Perturbed Wave and Permit Use of the Cubic In relation to the adequacy of either the Eq. (157b) or the Eq. (158) cubic, the first step in finding a criterion for the “largeness” of g is to note that the right-hand minimum of the Fig. 12 and 13 positive-h type of quartic curve must be very much below the p = 0 axis, in order to insure bringing jS, very close to 2g b, or to 2g b - j d , thereby providing essentially cold-circuit forward-wave propagation for the 6, wave. Similarly, in the negative-h type of quartic that resembles the mirror image of Fig. 12 or Fig. 13, the left-hand minimum of the quartic must be very much below p = 0 to insure bringing jS, very close to -2g b’, or to -2g b’ j d , thereby providing essentially coldcircuit backward-wave propagation for the 6,-wave. For either type of representation a sufficient condition for the appropriate minimum to be low is to make the S vertex of the p, parabola low. Thus, a sufficient condition to eliminate perturbation of the forward wave, and so permit reduction to the appropriate cubic, is that, at the S value of jS, (2g -p ) 2g. Use of Eq. (145b) shows this to mean that
+
+
+
+ +
>>
+
g2[(b g)2 - a2]
> 2g
’is a sufficient condition for reduction of the cubic to a quartic, and elimination of the perturbation of the fourth wave
Note that this means, in somewhat more general symbolism, that (159b)
provides the sufficient condition; also that ( 159c)
provides the sufficient condition. A study of Eq. (150f) shows that these describe also sufficient conditions for elimination of the perturbation of the fourth wave, and therefore for reduction to the cubic, when the circuit is lossy. I n this case, the underlying requirement is that, at the S-value of j6, (2g -j 4 d - p ) (2g -j4d). Use in Eq. (159) of Eq. (127) for a and for g , and of Eq. (152b) for h, leads to the following statement of the sufficient condition for reduction to $he cubic, with or without circuit loss:
>
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
83
Note that as the point H of symmetry approaches the cutoff frequency, U,, on the right becomes indefinitely large. Then there are four perturbed waves, and the quartic must be used. A similar situation applies; based, however, on a somewhat different analysis, at a high-frequency cutoff point (12). Rearrangement and introduction of Eq. (120b) and Eq. (117a) for C: and C, leads to he following forms in terms of design and operational parameters, respectively in terms of the backward-wave and the symmetry-point frequency offsets:
e,,
and U, symbolize negative quantities, the latter because it is Here I,, the group velocity at P on the w / / l diagram for the first forward space harmonic of the backward total wave. All other symbols stand for numerically positive quantities, by virtue of their definitions in this text. Because of the Eq. (159b) dependence on g (rather than, for example, on g2) these sufficient condition criteria can be satisfied, making the cubic adequate, for only moderately large values of g. I t is probable that the inequality of Eqs. (161) and (162) will usually be adequately satisfied when the left-hand member becomes 5 to 8 times the righthand member, depending on the degree of freedom from forward-wave competition required in the engineering objective.
G . The Third Root and the Cardinal Point R, for Start-Oscillation Conditions I n the typical positive-6 start-oscillation condition for a lossless circuit, the wholly real root j6, is near to but appreciably to the right of the Fig. 12 or 13 abscissa-value of R, and the start-oscillation frequency offset is such as to give b a value somewhat greater than 1 . Oscillation will not occur at a frequency offset that gives b too small a value, nor will it occur with values of the other parameters that place the $3, root too remote from R, because under either condition the amplitude of the 6, wave is too small, relative to the 6 , and 6 , waves, to produce a zero-value node of the spatial interference pattern at the input. Neither can it occur with the j6, root too close to b, because then the 6 , and 6, waves have extremely small amplitudes, and cannot contribute to the interference pattern significantly. I n general between these extremes
84
W. G . DOW
there will be, for sufficient dc current in the beam, a particular value of b, corresponding to a particular frequency offset, for which a zero input node exists; oscillation occurs at this frequency offset. This vanishing of the need for an input signal occurs typically for relatively large positive values of b, usually in the range between 1.25 and 2. For such values of b, the Eqs. (159), (160), and (161) inequalities are likely to be satisfied, so that many devices have shown reasonably good agreement between the actual start-oscillation conditions and the prediction based on the limiting cubic. In a device for which the cubic gives satisfactory prediction of the start-oscillation behavior, the backward-wave amplifier performance at currents below the start-oscillation value may not be well predicted by use of the cubic, especially for frequencies near the backward-wave reference frequency. Maximum gain will in general occur for appreciably positive values of 6, that is, at frequencies somewhat below the backwardwave reference frequency. There evidently exists a possibility for oscillation of a type that employs the forward-wave excitation, by causing its reflection from a mismatch in the input line, thus providing input to the device as a backward-wave amplifier. T h e start-oscillation frequency offsets and currents for such operation can only be predicted by use of the quartic, and must be very different from those that presume no backward-wave signal in the input line beyond the input plane.
H . Appearance of a Moderate Space-Charge Voltage Eliminates the Growing and Declining Waves For the positive-h condition of Figs. 12 and 13, b being not greatly 1, the rather moderate space-charge voltage introduced different from in going from one figure to the other converts the complex conjugate pair of roots to a wholly real pair. Thus, the condition a2 = 1 ; Q,Ca = 0.25 has a space-charge voltage well in excess of that necessary to make all four roots wholly real, for b = 1. For large a2, approaches - a ; in the limit no fast space-charge-wave bunching is generated, and any initially existing propagates unaffected by the circuit.
+
QUARTIC,FOR ZEROVALUESOF XVII. THENEARLY-CUBIC FREQUENCY OFFSET,Loss, AND SPACE CHARGE This section deals illustratively with the quartic case for negligible frequency offset, circuit loss, and space-charge voltage, so that a2,
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
85
b, d, and A2,B, D, are all zero, making Eqs. (141) and (142) reduce to (j6)* - 2g( j6)3 = -2g,
+
( j ~ l ) 2G( ~ jil)3= -2 G.
(163) (164)
The first of these leads to the following special case of the Eq. (144a) quartic: ( 165) p = (j6)4 - 2g( j ~ 3 + ) ~2g. This is charted for g = 2 in Fig. 16 together with the contributing
P
-2
-
-3-4-
-5-6-
-7
-
-8-
FIG.16. Illustrations of a particular p vs. j8 quartic as being studied in terms of two parabolas pa, pb describing the two factors whose product pupa form part of p . For this figure a = b = d = 0, g = 2.
86
W. G. DOW
pa, p , ; parabolas. For any positive value of g such a curve of p vs. jS has a zero-slope point of inflection at jS = 0, and in general a pair of complex conjugate roots associated with the upward bend from this singular point. I n Fig. 16, for which 2g = 4, the fourth root is at jS = 3.9343, a departure of 0.0657 from 4.000. This indicates a small but perhaps significant perturbation of the forward total wave. For this g = 2, b = a = d = 0 case, Eq. (159c) becomes 8 2. This g = 2 case provides interesting comparisons with the Eq. (140) form (jS), = 1 taken by the limiting cubic when b = a2 = d = 0, that applies when g -+ m. Thus, for b = ae = d = 0 the following numbers are obtained from Eq. (138a):
>
6,
g = 2, as for example
g very large, as if in Eq. (114b)
when UJ U, = 0.05, and
2Ub/U, is a substantial fraction, or,
C, = 0.025.
C, is small,
= 0.7974
+j0.5247
6, = 0.866
+j0.500
(the declining backward wave), (166a)
6, = -0.7974
+j0.5247
6,
=
-0.866
+j0.500
(the growing backward wave),
6,
=
-jl.llSO
8, = -j 3.9343 (= nearly 4.00)
(166b)
the unchanging
8,
=
-j
( b ackward wave ),(I664
8, = - j m the unchanging but (modified forward wave).
(166d)
Note here that at g = 2 the appearance of the fourth root is associated with rather small differences of the other three roots from their values 00. Yet, if g is reduced from 2 to 1, as by changing the circuit for g so that U,/U,, = j3&, = 0.10, there appear very marked changes, including the conversion of jS, and jS, to a complex conjugate pair describing growing and declining perturbations of the forward total wave. Thus, a factor of 2 in g may have a very pronounced effect on the extent of the competitive interaction with the forward wave. At g = 1, Eq. (159c) becomes the obviously unsatisfied inequality 1 2, implying necessity for use of the quartic. --f
>
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
87
XVIII. THEELECTROKINETIC BOUNDARYCONDITIONS AT ELECTRON ENTRANCE INTO THE RF FIELD In any device employing traveling-wave interaction the relative amplitudes of the several small-signal theory over-run bunching waves are governed by boundary conditions at the plane of entrance of the electron beam into the rf field, Figure 17 is a flight-line diagram re-
FIG. 17. Flight-line diagram used for studying entrance boundary conditions at z
=
0.
presenting the electron entrance condition for some one wave. For all negative values of z the straight, parallel, equally-spaced flight lines of slope U, represent the undisturbed beam. For all positive values of z the flight-line slopes obey the small-signal velocity relation:
where uQis the varying component of the axial velocity. oqo is the complex rms value of the rf component of the electron velocity at the entrance plane. At any point such as Q in Fig. 17 the departure of an electron’s
88
W. G . DOW
actual flight line from the extrapolation of its undisturbed flight can be described by stating; x, = the axial displacement (positive at Q in the figure) of the electron’s actual position from the position along the extrapolated straight negative-x flight line at the moment QS, and t , = the lead time interval (positive at Q in the figure) by which its arrival at the location RQ leads arrival according to the extrapolated straight flight line. Just as the rf velocity uq is a function up(x, t ) of position and time, so the displacement and lead time are functions z, (x,t ) and t , (z, t ) of position and time. Toward determining zd(x,t ) , observe that
(1 68a) For the small-signal analysis this “total” derivative is handled just as in obtaining the left side of Eq. (1 1); thus
(168b) This two-term operator implies taking a derivative along a flight line, which according to Eq. (1 1 le) is the equivalent of multiplication by Correspondingly, integration along a flight line is the equivalent of division by r,. Thus, for the small-signal case,
r,.
xd = d2 Zdoexp ( j w t
where
z,
GO
= 7,.
-
mz),
( 169a)
(169b)
It is evident here that
zd0= the complex rms value, at the x = 0 entrance of the electrons into the rf field, of the electron displacement from the extrapolation of its own negative-z straight flight line.
This illustrates the familiar statement that all the rf attributes of any one wave, including its displacement as at S in Fig. 17 from the undisturbed position Q, have the space-harmonic variation exp ( j w t - F‘%z). Now define: xdo,tlO,as the entrance-plane (z = 0) values of the zd, t , , displacement and lead time respectively, as indicated near location N in Fig. 17, being functions of time but not of position; of course, uo = Ub
+
uqo.
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
89
As applied at N in Fig. 17, the tl,, concept implies that, with this one wave alone present, the BNAG electron departs from the zero plane L
a -
WAVE
-b
WAVE
c r3 WAVE
t
-t+
'd 03
d SUM
i/
OF 3 WAVES
Y
/I
2 =O -t+
/I FIG. 18. Flight-line diagram showing how the flight lines for one electron as subject to three distinct waves may show absurd abrupt jumps for each wave yet have continuity of position and slope for the sum of the three waves.
90
W. G . DOW
before its arrival there, or, alternatively, that on arrival at N it abruptly jumps to G-this latter is the xao concept. Either is obviously absurd. However, as indicated for a single flight line in Fig. 18, the combination of more than one wave, each having the abrupt time discontinuity at z = 0, can if magnitudes are right have at z = 0 continuity of both slope and time. Nature in fact imposes this condition-this states qualitatively the electrokinetic entrance boundary requirement. It is evident from the NAG triangle in Fig. 17 that uotl, = zdo, that is, ( U ,
+ u,,) t,,
= zdo.
( 170)
I n this small-signal study uPotl,is dropped as being a second-order rf term, so that this becomes states that the expression of the entrance discontinuity as a time (this advance is proportional to its expression as a distance displacement1
UbtrO = sdO;
'
(171) When combined with Eq. (169b) and expressed in complex terms this becomes T,,= -a 0 1 (172) u l l T.s' where plo is the complex rms value of the t l , entrance lead time. Expression of Eq. (7) in. terms of complex rms beam current beam velocity and beam charge Tq, gives
oP0,
1, = UbTq+ TBUq.
I*,
(173)
T h e complex rms symbolism, Eqs. ( l l l ) , and this last Eq. (173), will now be employed to restate i n turn Eq. (20) for continuity of charge, a derived dependence of current on velocity, the Eq. (11) Newton's force law expression, and the dependence e, = -&+Jax of axial electric field on potential, all in form to relate various rf quantities to the rf velocity as a common conceptual reference:
- -Tb r n oa0;(beam charge); u b
&,
= TbPb
p E' 00
-
qe
me'' 4e
(1744
Oqo; (convection current);
( 174b)
rsouo;(beam-location potential);
( 174c)
rsod0;(axial electric field in the beam).
(174d)
m'Ub
PO
r,
r n
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
91
For this treatment's one-dimensional problem there are two electrokinetic and two electromagnetic boundary conditions. For rather general conditions which apply not only to the ordinary traveling-wave amplifier, but also to the entrance from a drift tube into an rf transmission-line field of a beam bunched because it carries a space-charge wave, the electrokinetic conditions may be stated as follows: The entering rf velocity
( electron
1
=
The sum of the rf electron velocities at entrance, for the several waves
(
The sum of the rf convection currents The entering rf (convection current) = at entrance, for the several waves
(
Because from Eqs. (I 74b), (I 72), and ( I 69b) the rf convection current, the positional displacement, and the lead time relative to the undisturbed (as contained in r,)their conbeam, have the same dependence on tinuity requirements are redundant; mathematically any one of the three is as appropriate as any other for use in Eq. (175b). Continuity of the lead time is more straightforward than the others as to concept, as illustrated by Figs, 17 and 18. Thus with Eq. (175b) expressed for lead time rather than convection current, the two electrokinetic conditions in combination state the requirement, obvious because electrons have inertia, that each one must leave the entrance plane at the same moment as that at which it arrives, and with the same velocity. However, continuity of convection current jq is better adapted than continuity of either lead time or displacement to handling the transition from a space-charge wave in a drift tube to a circuit-coupled wave, and has been quite widely used by other authors; it is therefore chosen for use as stated in Eq. (175b). Because convection current is the product of velocity by space-charge content, if there is continuity of both velocity and convection current across the entrance plane in accordance with Eq. (175), there will be continuity of space-charge density. Thus up to this point in the argument there exists a requirement for continuity of space charge content 'pqo of Eq. (174a) not because of any direct electron inertia property, but rather as a consequence of Eq. (175). For use in mathematical formulation of the electrokinetic requirements, it is convenient to express the axial field, the velocity, convection current, and space-charge quantities in terms of the beam-location voltage This employs the obvious relation ego = I',vq0, and combinations and rearrangements of Eqs. (174a), (174b), and (174c); thus, at the beam,
r,
'
rqo.
Ego = r,,P,,, (the axial field;
( 176a)
W. G. DOW
92
oqo=
-6'
me.Ub
P,,,
n '
(electron velocity);
(1 76b)
(convection current);
(1 76c)
(space-charge content).
(1 76d)
r.s
r: P,
f0 0 -
a0
-
qeTb rbrn meUb
--2.--! Tb "P -
m, U ; r2
QU'
Here q,Tb/m,Ub can be replaced by IbI2vby also q,lm,Ub by ub/2vb; both T b and I b are numerically negative. For the simple traveling-wave amplifier, whether forward-wave or backward-wave, the electron exit counterpart of Eq. (175) is not required. There are, however, applications of traveling-wave interaction in which they are of interest, as for example to describe the bunching behavior at entrance to a drift-tube section through which a beam passes during transit from one circuit section to another. The I', and F, quantities are now eliminated from Eqs. (176) by using = I'b(1 +jSC,) and F, = - j & c a r b to give:
r,
ego= (1 +jSca)rbpQ(I, (the axial field);
==+ P,, zv, c; $('&):y
O~Q
ub
Ib
fm =
T,
=
+*'a'
jaca
1 jaCa p (j8)S
Po,,
(177a)
(electron velocity);
(177b)
(convection current);
(1 77c)
(space-charge content);
(177d)
The signs here can be verified by considering an unchanging first-order wave for which ub > U,, so that the electrons overrun the spaceharmonic wave. Note that to an observer advancing synchronously with this wave, the crests of the potential wave are locations of maximum electron velocity, therefore of electron scarcity, therefore of positive net charge. These three rf quantities transfer to the stationary frame of reference without change of relative phase; therefore oqo, ~ q , , , and Tqoin this first-order case all have a common phase. This is provided for in the equations by the facts that for an unchanging wave jS is wholly real and numerically positive when u b > u,, and that is numerically negative. In the stationary frame of reference fq of Eq. (173) is dominated by the ubTqterm, with Tboq a small subtractive term; therefore in the unchanging wave f,, appears to a stationary observer to have the same phase as Ppo,and this is provided for by Eq. (177c).
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
93
It is convenient to express the boundary conditions in terms of circuit voltage P,. I n general accordance with Eqs. (39a), (39b), and (39c), P,, and Tho are related as follows:
By using Eq. (176d) for rfqo, then employing Eq. (1 17), and the Eq. (127) definition of u2, 'QO
=
- (ja)s +
+jsc,)z
I, G o .
(1 79a)
T h e first-order form of this is
P*"= - (-$ 9 (is)' 2 +
,LAO'
(1 79b)
Use of the first of these to eliminate converts Eqs. (177b), and (177c), to give, for any one perturbed wave,
T h e corresponding first-order forms are (181a)
(181b)
T h e relationships of Eqs. (180), and (181) adapt well to expressing
Uqoand Iqo in terms of Vqofor the complete electron entrance boundary matching process.
CONDITIONS AT ELECTRON ENTRANCE XIX. THEELECTROMAGNETIC TO AND EXITFROM RF FIELD I n addition to the Eq. (175) electrokinetic boundary conditions, there are electromagnetic field continuity requirements at the electron entrance and exit locations. Continuity is in fact required for both of two complete
94
W. C. DOW
and distinct but superposed families of fields; one comprises the spacecharge-caused fields associated with the bunched beam, for which the circuit appears as an equipotential. The other comprises the spacecharge-free fields caused by the total-wave propagation on the circuit. For the present Laplacian model, there must be continuity at both z = 0 and z = 1 of both potential and potential gradient, for all members of the two families of fields. For the more general model of a dispersive circuit the “potential” requirement is replaced by an rf magnetic field continuity requirement. Continuity of the space-charge-caused family is provided for by means of the Eq. (175) electrokinetic requirements, in that with oqand fq both continuous, so also is 4‘, and the sets of space harmonics associated with it. For the present model the circuit structure is initially presumed continuous in every detail across both the z = 0 and z = I boundary planes. Therefore the field configurations and relative values of the space harmonics comprising total-wave propagation will be similarly continuous. This implies that provision of continuity at the bounding planes of both potential and potential gradient at any one radius in the open channel for the first forward space harmonics will provide continuity at all values of the radius for all space harmonics. I t is convenient analytically to require this continuity at the beam radius, because it has already been used in setting up the electrokinetic requirements. Thus there are required, as stated for the backward-wave amplifier, Continuity at x = 0 of: Continuity at z = 0 of:
Continuity at z = 1 of:
‘n6~9h;
the beam-location potential (due t o the circuit voltage)’
.
(1 82a)
the axial beam-location field , (due to the circuit voltage)’
(182b)
the axial beam-location field due to the circuit voltage
(183b)
These continuities are provided for by matching 4 perturbed waves between z = 0 and z = 1 to space-charge-free waves in z < 0 and z > 1 regions. For physically real models the concept may be that the beam geometry rather than the circuit geometry is continuous at z = 0 and at z = 1; the relationships of Eqs. (182), and (183) are still valid if there are perfect nonreflecting rf circuit matches at z = 0 and x = 1. Thus the true concept is that of a physical separation at these terminations of the
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
95
channels of transmission of the signal on the circuit and any signal on the beam. For example, z = 0 may be the juncture of a bunched beam in a drift tube with an rf circuit carrying toward this point a total backward wave signal.
XX. RELATIVEAMPLITUDES OF THE FOURPERTURBED WAVES; VOLTAGE GAIN Primary emphasis will here be given to backward-wave amplifiers and oscillators, for which the z = 0 electron entrance plane is the rf output plane, and the z = I electron exit plane the rf input plane. For a backward-wave amplifier the operator will in general prefer to have no forward total wave in the output line; to avoid such a wave there must be no reflective mismatch in the output line to the left of the z = 0 output plane. I n general then four boundary conditions-two electrokinetic at x = 0, and two electromagnetic, one each at z = 0 and at z = I respectively-permit finding the circuit voltage amplitudes of the four S,, S,, S,, and 6 , perturbed total waves in the z = 0 to z = 1 interaction region, regardless of which perturbations describe waves interacting with backward and with forward waves. The amplitudes of these four perturbed waves can then be used to find the voltages of the rf waves transmitted away from the interaction region, one being a backward wave at I = 0, the other a forward wave at z = I. In a backward-wave amplifier, the backward wave transmitted away from x = 0 is the rf output wave, whereas the forward wave transmitted away from z = 1 is due to a reflection back along the input line caused by the electrokinetic discontinuity existing because of the removal of the bunched beam from the circuit at x = 1. The complex amplitudes of the four perturbed waves can also be used to find the complex amplitude of the circuit voltage wh or of the beam-location voltage vq at any value of I between z = 0 and x = 1. A knowledge of these resultant voltages in the interaction region is essential to an understanding of the phase interference between the waves, that gives rise to a spatial interference pattern and can cause backward-wave gain, even if all waves are of unchanging amplitude. As discussed in earlier sections, if for positive-h, primarily backwardb - jd, wave, operation, the value ofg is large enough so thatjS, = 2g any forward wave transmitted toward the device along the x = 0 output line will pass through toward the z = I input with the coldcircuit propagation constant, unaffected by the beam. Similarly if for
+
96
W. G. DOW
negative-h, primarily forward-wave operation, the value of g is large b’ j d , any backward wave transmitted enough so that jS, = -2g toward the device along the z = I (now output) line will pass through to the z = 0 input with the cold-circuit propagation constant, unaffected by the beam. In either case these will be the only waves of their kind present in either output line, so that there will be no reflections from the electrokinetic mismatch at the point of beam removal from the line. As an aid in stating the boundary relations quantitatively, let
+ +
r,l, symbolize respectively the hot-circuit total-wave propagation constant, the hot-circuit propagation constant for the = first forward space harmonic, and the measures r,, = Fn1-Ff,of the complex slip and the complex pushing for the first one of the four perturbed waves, with subscripts 2, 3, and 4 applying to similar symbolism for the other waves; for usual backward-wave-amplifier values of g and b, the 1, 2, and 3 waves are forward total waves, and the 4 wave a backward total wave.
I‘,, F,,,
r,, r,-r,,,
the ph0 complex rms circuit voltages at the z = 0 output plane, for the four perturbed total waves; P,,, PbO, symbolize the complex rms circuit voltages, at the x = 0 output plane, of the respective forward and backward total waves in the space-charge-free portion of the circuit just ahead of the x = O electron entrance plane; of course p b o is the useful output signal voltage, and p,, usually exists, if at all, because of reflection from an imperfect match of the rf circuit to its output transmission line ;
Vfiol, Pfi02, P h o 3 , p,04 symbolize
P,,
p b d , symbolize the complex rms circuit voltages, at the z = I input location, of the respective forward and backward total waves in the input circuit line, just beyond the plane of electron exit from the rf field.
goo, fgW,symbolize the complex rms values, if
any, infinitesimally to the left of the z = 0 electron entrance plane, of the rf electron velocity Oqand the rf convection current fg respectively; these are of course the consequence of any prebunching that may have been established at some negative-z point; these two quantities are in principle under the control of the designer and operator, and are therefore “inputs” to the analytical problem ; for a simple backwardwave amplifier or oscillator both are zero.
Ogiz,Igdd,symbolize similarly the complex rms values of Og and fg existing infinitesimally to the right of the 2 = I input plane; these are of course the consequence of the perturbation of the beam
97
SPACE-HARMONIC TRAVELING- WAVE ELECTRON INTERACTION
between z = 0 and x = I, and must commonly have nonzero values. T h e meanings of P,, and P,,, are indicated by their manner of occurrence in the following expressions for the circuit voltage vh and the beamlocation first forward space-harmonic voltage [ , V , , stated for the combined forward and backward total waves in the space-charge-free region x 5 0, but assuming the same circuit loss properties as between I = 0 and x = I : vh
= The
real part of:
.\/z p,
+ a,,) z]
exp [ j u t - (r,
+ .\/z Lpb0exp [ j u t r,
-
(r,- ao)21.
(184b)
+
Here and (r, 2T0) are the imaginary parts of the propagation constants of the first forward space harmonics, respectively for the forward and backward cold-circuit waves, and correspond to the Fig. 15 phase velocities U , and U;.Of course = r, 2F0 is obtained by 243,, which is evident from the figure. applying the factor j to = pt T h e meanings of V , and Vbi are similarly indicated by the following equations for the combined forward and backward waves in the spacecharge-free region z 2 I:
+
v,, = The real part of:
fvwh = The real part of:
+
d2 f,, exp [ j u t - (r,+ %) (z - !)I
2/2 tvPfiexp [ j u t - (r,+ 2F0 +
+ .\/z Phi exp [jut - (rf- 06)
(2
01")
- z)].
( z - Z)]
( 184d)
Note in Eqs. (184b) and (1846) the use of tvvil rather than v Q ; there /CQterm of Eq. are two reasons for this omission of the effect of a ? (178). First, there occurs at z = 0 and x = I a separation of the transmission channel for the rf signal from any exposure to the beam. Second, &, and even if the x = 0 plane were to locate a point of change of tV, C,, occurring without a circuit reflection (i.e., with no change in circuit characteristic impedance) it would still be true that Eqs. (184b) and
98
W. G . DOW
(1 84d) call fundamentally for total circuit wave continuity us measured by the beam-location potential in the absence of the beam. Thus the value of f v to use would be, even in such a case, that applying between x = 0 and z = 1. In employing the electromagnetic conditions of Eqs. (182), and (183) it will be assumed that, nearly enough, fv is the same for all the first forward space harmonics, whether of forward or backward total waves, and whether perturbed or unperturbed, The mathematical statement of the Eq. (182) output-plane electromagnetic conditions gives, in the established symbolism, after cancelling f,,, and dividing the field equation by
r,:
T h e quantities in the parenthesis on the right have their origin in the Eqs. (184b) and (184d) forms. In the corresponding statements of the Eq. (183) input-plane continuity, there is used the concept that at 2 = I , the factor exp(jwt-Fnz) becomes expCjot-r,Z) ; in the expression providing continuity of course exp jut cancels, as does 5, again. The statements are: pho,
exp ( - r n l l )
+
Pm2
exp ( - r n z O =
‘ff
+ Pms
+ ‘bi;
~ X (-rn30 P
+
‘h04
~ X (P- F n A
beam-location (input-plane voltage continuity
input-planebeam-location axial field continuity
(
).
(186b)
I n the analysis of backward-wave operation, it is convenient to consider the given quantities to be the output voltage Vboand the voltage Vl0 of a forward wave carrying power toward the output, as from a mismatch in the output line. Then the quantities to be found are the voltages
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
99
v b i of the backward wave at the input that will produce v b o at the output, and the voltage V,i of the wave reflected back along the input line from the electrokinetic mismatch at the input plane. This permits treatment by the same general procedure that Johnson used for 3 perturbed waves (2). It is now convenient to replace Eqs. (185a) and (185b) by two other equations obtained by respective elimination of P b o and P between ? the members of the pair, with similar respective elimination of Phi and P , between Eqs. (186a) and (186b). I n the replacing expressions I'JI',, exp( -FmZ),r o / F , , and a0/I', will all be replaced by their equivalents in terms of g, b, a, d, and jS as defined in an earlier section. This change in symbolism includes the following stated for the n = n3 wave as an illustration:
T h e exp( -Fbl) factor is common to all terms, so is moved to the righthand side of the equations, appearing there as exp r b l . There is introduced at this point also a mathematical statement of the Eqs. (175a) and (175b) electrokinetic requirements, in which sums of four Eqs. (180a) and (180b) expressions for oq,,,also for jqo,give the dependence on any Dqo0and f q o o input bunching condition. Thus there appear in combination six equations for the determination, in the linear four-wave backward-wave amplifier study, of six unknowns, these including always PhOl,Pho2, PhO3, P h 0 4 , and Pfi. Either v b o or Phi may be thought of as the sixth unknown, depending on whether one wishes to find the output-voltage for given input, or vice versa. I n all presently interesting cases, P,,, and fqo0are dealt with as knowns rather than unknowns. T h e SIX equations are as follows:
oqoo,
+ ?,,(
b + j d -$3,)
=
-pfo(2g -j 2 d ) ;
(1 88b)
(188~)
100
W. G. DOW
(1 88d)
Pnol(b
+jd
-A) exp (-jalCJbO
+ VaOa(b +j d j Q exp (-$*car&) + +j d -is3)exp ( -j&JarJ) -
VhOa(b
+ PhOl(b+j d -j8,)
exp (-j8&mrb2)
=
-
-j2d) exp (FJ).
(1 88f)
For the types of backward-wave amplifier conditions in which, as discussed in Section XVI, the quartic for jS must be used, the four Eqs. (188a), (188b), (188c), and (188d), are employed to find the four perturbed-wave output-plane voltages PhOl, PhO2,PhO3,Ph04, in terms of a desired output circuit voltage PbO,also of the voltage P,, of the forward wave, if any exists, that is reflected back to the output plane by a circuit mismatch in the output transmission line. The four outputplane perturbed-wave voltages thus found are then used in Eqs. (188e) and (188f), to find the input voltage Pbi required, and to find the voltage Pji at x = I of the wave reflected back toward the input signal source by the electrokinetic discontinuity at the input plane. As discussed in Sections XV and XVI, when with h positive g becomes large, reducing the quartic to the Eq. (157b) backward-wave cubic, jS, approaches the cardinal point T in Fig. 14, becoming in the limit j S 4 = 2g b - jd, which describes coincidence with cold-circuit forward total wave propagation. With this jS, used, and recognizing that g is large, it is easily shown that Eq. (188b) reduces to = PI,, as of course it must, and that the Tho4 terms become negligible or zero in Eqs. (188c), (188d), and (188e). Thus when the quartic reduces to a backward-wave cubic, the three simplified Eqs. (188a), (188c), and (188d), and Phos, in terms of a desired Pbo, and Eq. determine Phol, PAO2, (188e) is used to find the P,, needed. Meanwhile the ph4wave, attenuated according to cold-circuit propagation, becomes P,, at z = 0 and P,,
+
rho,
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
a
d
It Q
10 1
W. G. DOW
102
at z = 1; the electrokinetic action does not cause a reflection back along the input line. Completely parallel reasoning applies when with h negative, g is large, reducing the quartic to the Eq. (158) forward-wave cubic. In this case b = b’ - 2g, and in the limit j S , becomes -2g b’ j d . With g large, this reduces Eq. (188a) to P,,, = pa,, and eliminates the pholterms from Eqs. (188c), (188d), and (188f). Thus the three simplified Eqs. Ph03, PhO4, in terms (188b), (188c), and (188d), permit finding of a given P,,, which is now the rf input signal voltage. The only backward wave in the system is Phi, which becomes Vao at z = 0 and p,, at x = 1. This section’s derivation has up to this point not been limited to the first-order perturbation, and has implied use of the complete Eq. (137a) relations for finding$,. Conversion of Eqs. (188) to the first-order forms applicable when C, is small is accomplished by omitting jSC, from the (1 jSC,) parentheses; this affects only the electrokinetic Eqs. (188c), and (188d). When so converted, the jS values used are those from the Eq. (141b) first-order quartic. The fist-order determinant form for P,,, (used as an illustration) is arranged for convenience by multiplying each column (in both numerator and denominator determinant) by its appropriate Sa u2. The result is shown in Eq. (189a). Similar expressions apply for the other three perturbed-wave voltages. The quantities u2 and b cancel from the denominator determinant; thus
+ +
vhoa,
+
+
Denominator deter(minant in Eq.(189a)) =
-j 2 4 (8,
- 8,) (8,
- 8,) (8,
- 8,) (8,
- 8,) (8,
- 8,) (8,
-
4).(1894
For a simple backward-wave amplifier, having no electrokinetic input at the beam entrance, and with zero mismatch in the output = 0. For this circuit transmission line, ??*, = 0, fqoo= 0, and special condition the jrst-order equations for the perturbed-wave voltages are as shown in Eqs. (190a-d). Although g does not appear in Eqs. (190), and u2, b, and d appear only in the numerator, these parameters are all involved in the determination of the 6’s. With a mismatch in the output line, so that p,, has a value that must be considered, there may be used in addition to Eq. (190) a set of four equations identical with them except that P,, replaces pboJand (2g b - j d ) is used in place of (b + j d ) . The actual behavior considering both and P,, is then described by the algebraic sum of their separate contributions. A knowledge of the location and the complex
v,,
+
rao
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
103
104
W. G . DOW
reflection coefficient of the mismatch permits expressing P,, in terms of P*o. When 6 is significantly negative, so that forward-wave amplification is described, P,, is of course the input signal voltage at x = 0, and as is the reflection from such is under the operator's control. However, the electrokinetically caused mismatch at the input point; as such a requirement that it be zero becomes an item of design synthesis, as it is not directly under the operator's control. It can in general be made zero only by introducing to the right of x = I a mismatch that will is zero. produce the particular r b i that Eq. (188e) calls for when The Eq. (190) first-order expressions, or their more complete counterparts obtained by using Eqs. (188a)) (188~))and (188d) without dropping out the jSC, terms, are appropriate for use in Eqs. (188e) and (188f) to obtain P,, and Bfi; complete forms convenient for this purpose, not limited to the first-order application, are:
rb0
+ p,,, (1 + 6 -j2Rd -j8, ) exp ( -ja4CaI'&)] :
(191a)
(191b)
Addition of the last two equations gives the following simple and rather obvious expression:
P,,
+ Pti
=
[exp (-rJ)][P,,, exp(-ja,CarbZ)
+ P,,
+ P,,,
+
exp (-jSZcZbZ)
exp (-j%GrJ) P m exp ( - j W a r J ) l (191c)
I05
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
In forward-wave amplification a perfect circuit match to the powerreceiving transmission line gives P b , = 0. In that case Eqs. (188b), (188c), (188d), and (188e), are used to find PhOl,Plro2, PhO4,in terms of the input signal voltage P,o; the determinant then contains exponential terms. The voltage output Pt, and the electrokinetic mismatch reflection voltage pb,, are found from Eqs. (188f) and (188a). XXI. PHASOR REPRESENTATION OF BACKWARD-WAVE GAIN-PRODUCING INTERFERENCE Backward-wave amplification in a space-harmonic structure occurs because of the presence of an interference pattern among the several perturbed waves. Although this interference is primarily a total wave phenomenon, it has become common practice, for reasons of convenience, to analyze the behavior in terms of waveIengths hb of the purely fictional propagation described by r b =JPb = j u / U b . Thus there are used: hb
2n1pb,
( 192a)
Nb, = z/& = the number of wavelengths of the fictional r, propagation at radian frequency w corresponding to the distance from z = 0 to z = 2 ; (192b) Nb = llAb = the number of these fictional wavelengths between z = 0 and z = I, (1 92c) rbz = jj$,Z
= j2nZ/hb = j2nNbZ.
(1 92d)
Although the measurement of wavelengths is thus roughly according to the space harmonic behavior (because ub does not differ greatly from U t ) , the voltage equations used deal with total-wave voltages; this is illustrated by Eq. (191c). T o deal with such measurement in a correct conceptual sense, the exp(-Fbl) factor needs restatement in total-wave terms. To accomplish this, note from Eq. ( I lob) and Eq. (106a) that (193a) r, = -Po r, r,,
+ +
where, with reference to the various fils on
w/p
diagrams,
rd= j/3, =j2n/hd, r, = j&
=
j(Pb - &),
measuring frequency offset.
(193b) (193c)
Use of Eq. (193a) permits stating that exp
= exp [(To- F,.) z] exp
(-rdz).
(193d)
106
W. G. DOW
At this point z, and likewise 1, are digitalized into units of length A,, that is, limited to having significance only at each of the sets of like-side vanes. Clearly, with z so digitalized, exp (fFaz) = 1,
(193f)
exp (-Fbl) can be replaced by exp [(Fo- F,) 11.
(193g)
vh
oh = The real part of: ph03
(193e)
exp (--r$) can be replaced by exp [(To- T,)21,
The circuit voltage
+
exp r,l= 1;
at any digitalized z can be expressed as follows:
fi[P,,,
+
exp (6&cUNbz)
Pho,
+ P,, exp (S&CaNb,) exp (S42nca~ta)I~ X [Pj u t + (ro-
exp (S&CaNbz)
r r ) 21.
(I 94a) There can be used the following alternative form of the propagational operator, obtained by noting that PJrn= -b/g: exp [ j u t
+ (Po - I",) z] = {expjd + F0[l + (h/g)]
(194b)
2).
At z = 1, Eq. (194a) becomes Eq. (191c). Definitions as follows will be used in mathematical statement and phasor representation of the variation of v h with the digitalized z: 42 is the complex amplitude, having the attributes of magnitude and spatial phase, of the modulation applied to the circuit voltage propagation having the sine-wave propagational operator exp bwt (rn- r,)z], being modified from the backward-wave cold-circuit propagation by the frequency offset measure = -(b/gYn ; B,,, is the phase angle between Phe and the voltage P,,, Tho, P,, PhM, which is, in general, [from Eq. (185a)l the inputPI,, becoming just for the condition used plane voltage for Eq. (190) of no circuit mismatch in the output line; Ohe is of course described in any statement of Phs. In the light of these definitions, and noting that 8 = x jy, Eq. (194a) separates as follows:
vhe
+
r,
+
+
+
rb0
rb0+
+
ti,4
= The real Phe
part of:
z/z
exp [ j u t
+ (r,- I',)
exp (x12ncaNb~) exp (jy12ncaNbz) Pho2 exp (%2nCuNbx) exp (jy22?rcaNbz) PhO, exp ( x 3 2 ~ C u Nexp d (jYg2~CaNbJ -tp h 0 4 exp (x42&&z) exp (jy42wCuKd.
z];
(195a)
= PhOl
+
+
(195b)
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
I07
For convenience in spatial phasor diagrams there can be symbolized thus the four components of
rhe;
P*e =
p,,,+ P,',
+
V,n
+ p,,,,
( 1 95c)
where each term corresponds to a term in Eq. (195b). These Eqs. (195) relations are not limited to first-order studies. The convenient space phasor representation of PI,, and its several components will be illustrated by use of the g = 2 four-wave condition
8,s- j1.117
8, = - j 1.0
FIG. 19. The complex plane &s, for the two conditions g = 2 and g = Q) when b = a = d = 0. (a) The 8's for the three backward total waves, for g = 2, b = a = d = 0. (b) The 6 for the one forward total wave, for g = 2, 6 = a = d = 0. (c) The d = 0 ; in this case 2g-j6, = 0. three backward-wave a's, for g = Q), b = a
108
W. G. DOW
and the large-g three-wave condition, in both cases for b = a2 = d = 0; the 6's are as in the columns of Eq. (166), and are diagrammed in Figs. 19a and b. The 6, for g = 2 is illustrated by means of its complement (2g - j&,)/j,which is small and will be found to make P,,, correspondingly small in this case. For the four-waveg = 2 condition use of
-a '-
S O L
A
'ho I
*
'ho2
A
--
'h03
A
"h04
FIG.20. Four perturbed-voltage space phasors, at the output z = 0, and at two other values of 2. (a) At the output, 0 = 2rNb,C,, the four perturbed-wave voltage space phasors. (b) The phasors at 2aNb,Ca = 1. (c) The phasors at 2nNb,Ca = 2. These are all forg = 2, a = b = d = 0.
SPACE-HARMONIC TRAVELING- WAVE ELECTRON INTERACTION
I09
Eq. (166) in Eq. (190) gives first-order output-plane voltages as follows:
P,,,,
=
(0.305
= 0.305
P,,, P,,,
=
(0.305
-
j0.0137)
PbO;
(the declining backward wave);
Pb,(cos 2.57" -j sin 2.57");
+j0.0137) Pa,,
= 0.305
Pbo(cos 2.57"
= 0.365
Pbo,
f,,, = 0.0257 Pa,,
(196a)
(the growing backward wave);
+j sin 2.57");
(196b)
(the unchanging backward wave);
( 196c)
(the unchanging forward wave);
( I96d)
Use of the limiting cubic 6's from Eq. (166) employing a large g, and b = a = d = 0, leads to three equal output voltages:
Comparison with the previous equation shows that the important effect
of the presence of forward-wave competition is the increase of P,L,, (the backward unchanging wave) at the expense of the declining and growing backward waves. T h e Fig. 20 and Fig. 21 spatial phasors represent the b = a = d = 0 behavior, respectively for g = 2 (four waves) and g + 00 (three waves). Note in both figures that P h , grows exponentially as the argument 2nC,Nb, increases, and so dominates Thefor large values of the argument. For the moderate values of 27&,Nb, used in Fig. 20b and Fig. 21b the phasor PI,, is less than the output voltage P,, only because the relative angular positions of P,,, and P,,, result in a net subtraction from the increasing P h , V, sum. T h e ultimately dominating P,, describes a declining backward wave, that is, one which declines in the negative-z direction of its total-wave phase and group velocities.
+
XXII. BACKWARD-WAVE GAIN EVALUATION
To aid evaluating gain and studying start-oscillation conditions, let Obz, 8b,, stand respectively for 2nC,Nb, and 2rC,Nb, being also c,$?bz and ca&l, and let 8,, 8,, B,, B,, symbolize respectively the phase angles between p b o and the respective pao's; for example 8, =: -2.57 degrees and 8, = $2.57 degrees, and 8, = 8, = 0, in Fig. 19c; thus
P,,,, is replaced by:
V,,,(cos
-1j sin
el),
(198)
110
W. G . DOW
where arbitrarily 8, = 0, this being the z = 0, time-reference phase angle of v h ; and similarly for the other three terms.
b -
t
'bo
'bo
FIG. 21. The voltage space phasors for the limiting backward-wave amplifier cubic, occurring for zero overdrive, negligible space charge and circuit loss, i.e., b = a = d = 0, g = Q), (a) The three output-plane voltage space phasors adding to pbo,2nNblc, = 0. (b) The three phasors at 2nNa,CL(= 1, to the same scale as in (a). (c) The three phasors at 2rrNblC, = 2, to a modified scale.
1 11
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
Trigonometric manipulation of Eq. (195b) now gives n-4 Phe
=
2
n=4
~ j L o nexp xnebz
*=l
cos (Ynebz
+ +j 2 On)
VhOn ~ X xnebz P
n=l
sin (YnOb:
+
en).
( 199)
T h e local magnitude I Ph,l of the spatial modulation envelope is the square root of the sum of the squares of the Eq. (199) summations, and the ratio of the sine sum to the cosine sum is tan ohe, describing the phase of the modulation. At the z = 1 rf input point, PI,, and Baa become Pfi) and Oil. T o obtain and Pfi separately it is respectively ( GIbi necessary to employ Eqs. (188e) and (188f), which become
+
r,,
(200a)
Obviously the voltage gain is P b o / P b i , and is therefore not in general the same as the v d u e of r b , / p h , at z = 1. This is a consequence of the excitation of the forward total wave as Ph,,. For the Fig. 13 conditions in which d = 0 and a2 is moderately large, all roots of the quartic for j S are wholly real, making all the x,,’s zero. This greatly simplifies Eqs. (200a) and (200b). I n this case all perturbed waves are unchanging (i.e., constant-amplitude) waves; any voltage gain is due exclusively to the waves beating against one another to form a spatial interference pattern. Pb( then has the same amplitude at all of its successive minimumvalue points, occurring at equal spatial intervals of Obi. Primary interest attaches to the magnitudes rather than the phases of The and Poi.
112
W. G . DOW
Straightforward treatment gives the squares; thus from Eq. (199)
+ [x n=4
vtmn ~ X xttebz P
n=l
sin ( Y n L
+
a on)] *
(201)
lo7il
IPi,I and are similarly obtainable from Eqs. (200a) and (200b) after completing the separation into real and imaginary parts. T h e value of O,, for maximum gain is obtained by taking the derivative of lZ.I&l relative to eb,, then using the appropriate minimum-P,, root of the equation obtained by equating this derivative to zero. There may be several such locally minimum points; the one of smallest d,, will usually provide the greatest gain. XXIII.
THESTART-OSCILLATION CONDITIONS
A Carcinotron or backward-wave traveling-wave oscillator is of course a backward-wave amplifier of infinite gain, that is, one in which there is no input signal. This condition can in principle be provided, for the four-wave device, by any one of at least three quite different sets of conditions, as will be itemized. (a) A short circuit at the input termination. If there is a short circuit at the input plane, making v b , V, = 0, there can be a useful value of P b , if the four phasors that make up p,), add to give just zero at eb, = ebl, (b) A matched termination at the input. If the P,; forward total wave that emerges at z = 1 sees a matched termination, the requirement for a useful output signal without an input is that P b , -- 0, rather than that Pb, PI, = 0. (c) ReJlectionfrom a mismatch to the right of z = 1. With a mismatch to the right of z = 1, phi becomes some determinable complex function of 9,. T h e interaction region of the device is of course a backward-wave amplifier, with a non-infinite Pbo/rbi, but the apparatus assembly is an oscillator. This can occur zuithout any reflecting mismatch to the left of z = 0.
+
+
T h e (a) and (b) conditions are limiting extremes of the (c) operation. employing respectively Pb,/p,i = - 1 and pb,/Pfi = 0.
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
1 13
For any operable design plan there appears a “start-oscillation” equation from which all voltages have cancelled. As an illustration, for the Item (b) condition above, the initial step in establishing the startoscillation equation is to require in Eq. (188e) or Eq. (200a) that P b , = 0. Then fihol,PhO2,PhO3,and P,,, are expressed in terms of f i b , from Eqs. (190); the P b o then cancels, to give the “start-oscillation equation.” Its precise meaning requires careful examination. The designer and operator may usually specify directly in terms of design dimensions and dial settings or meter readings the quantities C,, g( = ub/c,uo),a2(= I b / 2 vbubC,c:), the loss measure a,, characteristic impedance z, the reference frequency wb, and the length 1 of the interaction region, but he cannot specify the precise operating frequency-the device itself chooses that, and in so doing specifies also the offset parameter b[=( ub - U,)/C,U,], the loss parameter d[= Ubao/C,W]) and the wavelength h, = 277 ubjo associated with the fictional r b propagation. Therefore r,l,P b l , Nb,and Nb, as defined in Eq. (192) appear as somewhat unprecise measures of distance, because the unit of distance hb cannot be precisely predicted. This compels an inverse employment of the start-oscillation equation. The analyst selects, in accordance with the design and operational plan, values of C,,g, a2, wb, and a,, Z,,these being, nearly enough, frequency invariant. He also selects a frequency offset parameter b, thus precisely specifying by means of Eq. (133) an operating frequency w, which then fully defines P b , rb,and d. He leaves open the value of device length I, and with it Nb and o b i , as being proportional to 1 as well as to p b . A determination is then made, iteratively if necessary, as to what if any discrete values of Oh, satisfy the start-oscillation equation that uses the selected parameters. Each such 8 b l describes, in terms of the fib identified in the selection of b, the length of a device which, with these operating conditions, will start oscillating at the determined frequency. T h e current l b used in selecting C, and a2 is then the “start-oscillation current” for the device as specified originally. The start-oscillation behavior will be illustrated conceptually using the three-wave operation governed by Johnson’s (2) first-order cubic, Eq. ( I 57b). The first-order terminal boundary conditions, assuming an unbunched entering beam, are %,I
+ L+
L 3=
pa,.
(202a)
w. c. now
114
These solve out and combine in a way generally paralleling Eq. (195c), with pi,, becoming Phi at z = in this case, to give, somewhat as in Eq. (191a),
An illustratively useful condition is that of no circuit loss (d = 0), with ua assumed small enough to justify expecting the cubic to have one real root and a complex conjugate pair, and with b as yet not specified. For this case the start-oscillation equation that is obtained from Eq. (203) by requiring that 9, = 0 rearranges into the following:
o = 1 Pie I
= 4Vkl(cosh2a?&,,
- sin2
el) + via,
f
2vholvho3
exp X18be cos [(Yl - Y3) e6z
+
+
2vholvho9
exp ( - X 1 8 b z )
e6z
cos [(Yl - Y3)
- 41.
(2Ma)
As ear becomes substantial, the last term rapidly becomes unimportant relative to the other two, and cosh a?18bs becomes, nearly enough, i$ exp Usually 8, is a moderately small angle, so that in a semiquantitative study sin2 @,can be omitted from the first term, although 8, cannot usually be omitted from the third term. Thus if 8, is not too exp ( - X & , z ) , the start-oscillation condition is: large, and exp Xlf+,,
>
0 = pie
(vhO1
exp X1e6z)2
+
+ v,”O,
2vh03vhOl
exp X18bz cos [(rl
-Y3>
ebz
+
(204b)
This is seen to have the form of the trigonometric cosine law for the resultant of the two Phi, piL3,phasors, of lengths V,,, exp X#b, and Vho9,at an included angle of rr-[(yl - Ya)8bz el], as this angle approaches zero, and the two lengths become equal. When this relation is satisfied, e b , has become d,,,, becoming the measure 2d,Nb of the number N b of Ab wavelenghts the device should have to make phi -+0. From Eq. (204b) it is clear that, for the conditions stated, the startoscillation relation places two requirements ; first that
+
vhOl
exp X18bE
vh03*
(205a)
+ 41 = - 1,
(205b)
and second that cos [(n- y3) 4,
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
that is, (Yl - y3) ObL
+ 4 z 742 + 11,
1 15
(20%)
where k can be zero or any positive integer. With the voltages of the unchanging wave and of the declining backward total wave of essentially equal magnitude at z = I as required by Eq. (205c), it is clear that at z = 0 the latter’s decline must bring Vholto a much less value than V,,,. This requires the frequency offset parameter b to be substantially greater than zero, usually between 1 and 2, and oscillation takes place at a frequency o that is offset from wb as described by Eqs. (133). T h e first and most interesting zero value for occurs at k = 0, and in Eq. (20%) employs a value of b for which
vbi
(Y1 - Y s )
4, = x
(206a)
- 01.
For the next zero (the next “mode”) b has a value such that
(206b)
(YI - us) h, G 37r - 81.
At this k = 1 mode Eq. (205a) must still be satisfied, as a result of a change in b and so in xl, y l , and y3. A study of the Fig. 22 phasors, based on Johnson’s computations partially tabulated below, suggests what occurs. Data for this figure are: u2 = d = 0; the first (k = 0) Pbi = 0 is found to occur at b = 1.522 and at = 1.976; for this b, a,, and 6, are rt 0.725 j0.150, and 8, is -j1.8233; PhOl, Tho, are (0.123 0.105) Pbo, giving 8, the value tan-’ (- 0.105/0.123) = - 40.5”, or -0.78 radian. For the corresponding k = 1 mode, b will have increased somewhat, causing x1 to be smaller with a still further flattening out of the a,, a,, representations in a new Fig. 22a, but without introducing a major change in y 3 - y I . Thus the principal change on the left of Eqs. (205a, b) is in Ob,, not in y1 - y3. Values of y1 - y 3 lie generally between 1.5 and 2.5 for a substantial range of values of b. Therefore, using as an approximate average value (y, - y 3 )e 2, it appears from Eq. (205c) that, roughly,
+
8,,
rrk
+ (n/2)
-
(207)
8.,
Thus the occurrence of modes roughly at intervals of n in the value of would appear to be a result of the somewhat accidental fact that y1 - y 3 is of the order of 2 for the interesting range of values of b. Note that use in Eq. (206a) of the Fig. 22 precise numbers gives
e,
(0.1505
+ 1.8233)0.1976
-
0.78 = 3.99
- 0.78 = 3.21
GX
3.1416. (208)
4"
%,,&=
/
0.7252 + j0.1505
'h03
I
b: THE -
THREE OUTPUT-PHASE VOLTAGE PHASORS AT 2.0.
8,. - j 1.8223
-a. THE
THREE
8% FOR b11.522, a = d = O
3 0
c.THE THREE VOLTAGES AND Vbi AT 2TTCNb=li THERE
.his/
\hi
Vhi3=0.754 exp ( - j 1.822) Vbo
d. THE THREE VOLTAGES AT THE START-OSCILLATION CONDITIONS 2TfCN=1.976, WHERE Vbi =O SO THAT THE GAIN IS INFINITE.
VhI3=0.754
Vb,
exp kj3.6)
Vhl, =Vbo (0.123-j.105) exp(1.433) x exp(j.297) vhi2'vbo
(0.123 +jJ05) exp(-1.433)
x exp(j.299)
FIG.22. Three complex A's, and the three voltage phasors, for the output at z = 0, and two other z-values, one being that for a start-oscillation condition.
U
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
117
The O1 = 0.78 term has a significant effect. The 3.21 number value differs from 7r by the possibly fortuitously small amount of 2 percent. The over-all behavior of the present theoretical model is well illustrated by Fig. 23 based on Rowe's backward-wave amplifier calculations (13) which employ as a parameter the signal frequency as measured by 6. Note that Eq. (133c) rearranges. into (209a) which for first-order perturbation studies is (209b) Here all quantities except w are frequency invariant for small changes in frequency, and in the U, = - U, nondispersive case for large changes in frequency; thus b=
wb
-w
wbCa
(1
for a nondispersive first+$); (order perturbation treatment
Figure 23 shows that the start-oscillation cusps occur at essentially equal intervals of Bbe but at progressively lower values of frequency. Clearly the design model (and presumably therefore also a corresponding device) has as an attribute a parameter other than but closely related to b for which all the start-oscillation cusps lie on a common curve. This fact, together with the indirection of the whole start-oscillation prediction procedure, suggests that there must exist a first-order perturbation reference which would permit a more direct and simpler analysis. As an illustrative suggestion, in a perturbation study relative to the propagational operator r b o = jpb0 (see Fig. 11) the counterparts of d, &, Nb, and f&, would be frequency invariant, with b still a measure of frequency offset. The start-oscillation equation should then have the two variables e b f and 6 , to be determined by using two equations obtained by placing equal to zero the real and imaginary parts respectively of the start-oscillation equation. For this or some comparable choice of perturbation reference one of these two equations would contain only d b l , and would give the spatial start-oscillation periodicity evident in Fig. 23; the other equation would then give the start-oscillation frequency for the several modes.
-451
Wb
b= -30
1
I FOR FIRST-ORDER NON-DISPERSIVE THE FREQUENCY OFFSET PARAMETER b IS GIVEN BY
-w
PERTURBATION
I
c
00
wb
[+:)
1.0 2H
1.25
1.50 38
1.75
2.0 4H
2.25ebl
FIG.23. Chart of Rowe's computational determinations (13) of the decibel-measure voltage gain of a three-wave backward-wave traveling-wave amplifier as a function of the radian-measure description O,, = 2aN,C, of the length 1 of the interaction region, for zero space charge and no circuit loss.
SPACE-HARMONIC TRAVELING- WAVE ELECTRON INTERACTION
1 19
T h e accompanying table giving some of the results of Johnson’s calculations merits scrutiny. PARTIALTABLE OF THEORETICAL START-OSCILLATION CONDITIONS FOR THREE-WAVE BACKWARD-WAVE OSCILLATORS ebL/2.r c,Nb
Ld
b
us = 0
0 4 10 20 30
1.522 1.457 1.375 I .27 1 1.197
0.3141 0.3414 0.3847 0.4627 0.5470
up = 1
0 4 10 20 30
1.501 1.445 1.380 1.314 1.282
0.3434 0.3774 0.4329 0.5348 0.6451
am = 2
0 4 10 20
1.533 1.526 1.543 1.555
0.3990 0.4237 0.5207 0.6344
aa = 4
0 4 10 20
2.072 2.059 2.063 2.065
0.49 I4 0.5413 0.6218 0.7537
=
From H. R. Johnson,&or. I.R.E. 43, 684 (1955).
T h e u2 = 4, L, = 0 values in this table can be obtained very simply and to a very good approximation by assuming that, because jS, -+ - a, the Eq. (202) P,,, wave is not excited, so that in Eq. (203) the exp S,& term disappears. Then at Pbi = 0, using j S , = - a, the two remaining terms must be equal in amplitude, and must for the first “mode” differ by 7r in phase. This permits stating two equations relating jS,, jS3, and Obi. Expression of Eq. (157b) for jS, and for jS, separately, with d = 0 and a2 = 4, gives two more equations that relate jS,, jS,, and b. Solution of these four equations gives jS, = 1.497,jS, = 2.503, b = 2.063, and 4J2.r = 0.496; thus this approximate treatment gives very good agreement with Johnson’s values. Use then of Eq. (157b) expressed for j S , shows that jS, = - 0.969a, which is close enough to jS, = - a to reduce PI,,, to a negligible value, as initially assumed. This solution gives this same value of b for higher modes, cusps in a Fig. 23 type of representation occurring at regular intervals of 2 x 0.496 in 8t,1/27r.
120
W. G . DOW
XXIV. START-OSCILLATION CURRENT AND DEVICELENGTH ; VOLTAGETUNABILITY ; GAIN BANDWIDTH PRODUCT
A dominating gross-aspect attribute of a backward wave oscillator is its start-oscillation dc beam current. If, with given geometry and dc beam voltage, the beam current is less than the start-oscillation value, the device is a narrow-band amplifying filter; when the beam current reaches the start-oscillation value, the device becomes an oscillator. In either case the frequency of operation is voltage dependent; these devices are voltage tunable devices. The magnitude of the excess of the dc current above the start-oscillation value of course affects the amplitude of the oscillation, in a nonlinear way. T h e indirect, iterative process described in the preceding section permits describing sets of theoretical-model start-oscillation conditions as partly identified in the above numerical table and in Fig. 23. Each such set of conditions permits stating a start-oscillation current and coldao, and the circuit loss conditions as contained in tC,.$,, c,, u b , vb, u//3 diagram. Each such set also permits stating similarly a departure W-Wb of start-oscillation frequency from the reference frequency W b , and a statement of the device length 1 as contained in Bb, = c , h / u b . This information provides of course useful inputs to preliminary design studies. T h e voltage tunability of a backward-wave oscillator comes from the fact that the reference frequency is dependent on the beam voltage. Note from Fig. 11 that
z,,
(210a) which rearranges into (210b) for a nondispersive circuit
uob
=
Uo, so that
T I
(nondispersive),
(210c)
where Pd is a property of the circuit structure, and usually ub/uois a minor term in the denominator. This rearranges for frequency f b = 0 b / 2 7 7 into the following for the non dispersive condition: (210d)
SPACE-HARMONIC TRAVELING-WAVE ELECTRON INTERACTION
121
or, numerically, in mks units of course, into (210e)
Note that the reference frequency varies somewhat less rapidly than as the square root of the voltage Vb. I n a genuinely slow-wave structure the dependence comes very close to the square root form, but the likelihood of competitive excitation of the forward total wave is increased. These relations also describe a reference frequency which governs the center frequency of the backward-wave device treated as a voltage tunable amplifier. I n such a device, in principle, the design attributes are such that when Eq. (205c) is satisfied, Eq. (205a) is not satisfied for any value of b, that is, for any frequency-the dc current is too small. It is straight-forward but rather tedious to show that there then exists, in the neighborhood of any of the major dips in the counterpart of Fig. 23, a gain-bandwidth product which is to a first approximation invariant with the dc current.
XXV. CONCLIJDINC COMMENTS This paper’s development has separated out quite completely three distinct conceptual aspects of the small-signal traveling-wave interaction theory, these being: the electrokinetic effect of the space-harmonic electric field in bunching the beam ; the propagational attributes of the circuit as a whole, and the coupling influences via the electromagnetic field between the circuit and the bunched beam. Dealing with the electrokinetic interaction in differential-equation form has made possible quantitative statement of the very-small-signal nature of the smallsignal limitation. By starting the treatment with differential equations rather than with assumed solutions for an equivalent circuit, it has been possible to retain throughout conceptual clarity as to the contrasting contributions to the behavior from the fast total wave propagation and the slow space harmonic; this has made it possible to deal with the true circuit characteristic impedance rather than using the quite unsatisfactory concept of an “impedance plane” property of the circuit. By retaining through most of the analysis the generality of its four-wave attributes, it has been possible to establish quantitative criteria to show under what design and operating conditions there is to be expected electrokinetic interaction affecting both backward and forward total waves, in contrast to the specialization toward one extreme to Johnson’s backward three-
122
W. G . DOW
wave analysis, and toward the other extreme to Pierce’s somewhat different forward three-wave analysis. The four-wave behavior may offer interesting utility, for example as a means of simultaneously generating equal forward and backward signals. By retaining field-theory rather than equivalent-circuit concepts of mutual effects of the bunched beam and the total wave, it has been possible to clarify the reasons why digitalization of the circuit increases the space-charge voltage at the beam, and to evaluate illustratively the reduction in coupling caused by the digitalization. Both of these field-analysis influences are particularly interesting when the beam is maintained in very close proximity to the circuit. Use of an annular beam model is advantageous in simplifying the mathematics while gaining generality, for one may always extend the coverage to a solid beam by treating it as an assembly of annular beams, whereas the inverse is not feasible. By setting up the characteristic equation for the spatial feedback interaction in complete coupled form it was possible to summarize the physical-concept aspects of individual terms, pairs, and groups of terms, to distinguish between undisturbed and interacting space-charge waves and circuit waves, and introduce a resistive-wall gain parameter. T h e description of perturbational relations based contrastingly on the cold-circuit space-harmonic phase velocity at the signal frequency and on the dc beam velocity clarifies the point that Pierce’s choice of the latter is based on mathematical convenience, and leads to the suggestion that for oscillator studies still a different perturbational study, using the dc beam velocity and departures from a fixed reference frequency, might be advantageous. The use of Johnson’s perturbational parameters b and d is extended by adding a space-charge parameter aa = 4Q,C, and a slowness-of-wave parameterg, thus facilitating greatly a qualitative study of the relationship of the roots of the quartic to these parameters. Johnson’s b is identified as a frequency offset parameter measuring the departure of the signal frequency from a design-based reference frequency, thus substantially clarifying its significance. I n establishing the longitudinal boundary-value relations it was pointed out that for the present model the convection current and space charge of any one wave at electron entrance are both measures of that wave’s entrance-moment positional electron offset, and that therefore continuity of convection current carries with it continuity of space charge. By establishing the longitudinal boundary relations for fourwave interaction, the groundwork has been laid for further studies of means either to avoid harm from or usefully employ four-wave interaction.
SPACE-HARMONIC TRAVELING- WAVE ELECTRON INTERACTION
123
T h e illustrations given of spatial phasors for the several waves in combination has emphasized the fact, often lost sight of, that backwardwave gain is completely dependent on the details of a phase interference pattern among the several perturbed waves. T h e cusped pattern of the gain curve is similarly useful. I n a more general sense, this paper has begun the space-harmonic interaction analysis at the beginning, that is, with the physically-based differential equations respectively for fast total-wave circuit and slow space-harmonic interaction phenomena. I n proceeding through the analysis it has preserved separate use of concept as between the differential equations for interaction dynamics and the field-theory analysis of beam-circuit coupling; it has retained the generality of the four-wave perturbational possibility through the longitudinal boundary-value 1 and 1 3 treatment, with appropriate specialization to the 3 backward-wave and forward-wave analyses ; there is emphasized by spatial phasor illustrations and a start-oscillation cusp chart the essentiality of a phase interference pattern to backward-wave amplification and oscillation.
+
+
REFERENCES 1. Pierce, J. R., “Traveling-Wave Tubes.” Van Nostrand, Princeton, New Jersey, 1950. 2. Johnson, H. R., Proc. Z.R.E. 43, 684 (1955). 3. Dow, W. G., Proc. Nut. Electronic Conf. 9, 422 (1954). 4. Rowe, J. E., Proc. Z.R.E. 47, No. 4 (1959). 5. Haeff, A. V., Proc. Z.R.E. 37, No. 4 (1949). 6. Dow, W. G., and Rowe, J. E., Proc. I.R.E. 47, 536 (1959) (letter to the editor). 7. Johnson, W. C., “Transmission Lines and Networks.” McGraw-Hill, New York,
1950.
8. Goldberger, A. K.. and Palluel, P., Proc. Z.R.E. 44, 333 (1956). 9. Slater, J. C., “Microwave Electronics.” Van Nostrand, Princeton, New Jersey, 1950. 10. Stratton, J. A., “Electromagnetic Theory.” McGraw-Hill, New York, 1941. 11. Beck, A. H. W., “Space Charge Waves.” Pergamon Press, New York, 1958. 12. Dow, W.G., Z.R.E. Trans. on Electron Devices ED-7, 123 (1960). 13. Rowe, J. E., Proc. Symposium on Electronic Waveguides, Polytech. Znst. Brooklyn, 1958.
This Page Intentiona lly Left Blank
Thermionic Energy Conversion J . M . HOUSTON AND I4 . F . WEBSTER General Electric Research Laboratory. Schenectady. New York pogc 125 126 129 Idealized Model of a Thermionic Converter ............................ 130 A Current-Voltage Characteristics and Power Output ................... 130 B. Efficiency ...................................................... 136 The Work Function of Various Surfaces .............................. 142 143 A Pure Metals ..................................................... 143 B Thin Films on Pure Metals ....................................... C Oxide and Carbide Cathodes ...................... 8 . . . . . . . . . . . . . . . . 151 D Evaporation of Cathode Materials ................................. 153 Vacuum Thermionic Energy Converters ............................... 154 A History ....................................................... 154 155 B Space Charge Problems ........................................... 163 C Surface Physics Problems ......................................... 164 D Fabrication Problems ............................................. E Experimental Results ............................................. 166 F Three Electrode Vacuum Converters ............................... 167 Cesium Thermionic Energy Converters ............................... 169 A Introduction .................................................... 169 B Low Pressure Converterg ......................................... 171 180 C High Pressure Converters ......................................... Devices Using Auxiliary Discharges .................................. 193 Devices Using Fission Fragments for Ion Production .................... 195 Applications of Thermionic Converters ................................ 195 Summary and Future Trends ....................................... 199 References ......................................................... 201
I. Introduction
.
I1
.
111
....................................................... A . Surface Work Function and the Richardson Equation ................ B. Brief History of Thermionic Conversion ............................
.
. . . . IV. . . . . V
.
. . . .
VI VII VIII IX
. .
. . .
I . INTRODUCTION
A thermionic converter is a device which converts heat energy into electrical energy by utilizing the thermionic emission of electrons. As Fig. 1 shows. it is basically a simple device. consisting of a hot cathode emitting electrons to a cooler anode. These electrons flow back to the cathode through an external load resistance where they do useful work . That almost any thermionic diode can convert heat to electricity with a very low efficiency has been known for many years . However. in the last five years the power output and efficiency of such devices have been greatly improved. now being approximately 5 watt/cm2 at 15yoefficiency. 125
126
J. M. HOUSTON AND H. F . WEBSTER
Thermionic converters appear to be a unique, static, high-temperature source of electricity with a number of potential space and terrestial applications. Some types of thermionic converters are well understood theoretically, while for others the theory is less complete. This paper will survey the situation as of mid-1961. Other review papers on thermionic conversion include those of Dobretsov ( I ) , Morgulis ( 2 ) ,and Cayless (3).
CATHODE
ANWE
TC
Ta
FIG.2. Motive diagram of idealized thermionic converter at maximum power output. Negative potential is plotted upwards.
+
LJV
c
FIG. 1. Sketch giving the basic idea of thermionic energy conversion.
A. Surface Work Function and the Richardson Equation Figure 2 gives an idealized potential energy diagram [strictly speaking, a motive diagram’] for a thermionic converter. Note that negative potential is plotted upwards, a convention widely used in solid-state physics because of the negative charge on the electron. Here +c and +a are the work functions of cathode and anode respectively, the work function of a surface being defined as the work necessary to remove an electron from the sea of electrons within the solid [i.e. from the Fermi level] to a point at rest outside the surface. For a uniform metallic surface, “outside the surface” is defined as the point where mirror-image force [which is the dominant force contributing to the work-function motive barrier] Langmuir and Kingdon (4) suggest that an electrical potential be called a “motive” when the electric field causing the force comes into being as B result of the presence of the test particle. The motive is really a fictitious potential and does not obey many of the laws applicable to true potential, e.g. Laplace’s equation.
THERMIONIC ENERGY CONVERSION
127
becomes essentially negligible. This occurs about 500 A outside the metal surface, i.e., the mirror-image motive changes by less than 0.01 volt between 500 A and infinity. Note that, in general, the work function is not equal to the work required to carry an electron to infinity if either externally-applied or “patch” electric fields are present. The work function of a material depends somewhat on the crystallographic face exposed and also can be greatly changed by even a fraction of a monolayer of adsorbed foreign atoms. Most materials have work functions in the range of 1 to 6 volts ( 5 , 6 , 7). Idealized converter operation can be viewed as follows. By virtue of their random thermal energy the more energetic of the electrons within the cathode can mount the work-function barrier (see Fig. 2) and enter the interelectrode space. This process (i.e., thermionic emission) cools the cathode by an average amount (er,hc 2kT,) per electron. If electron space charge in the gap has been eliminated somehow and all collisions are neglected, the emitted electrons drift to the anode with thermal velocities, and there fall “down” the anode work-function barrier, 2RTc) per electron as heat to the anode. T h e giving up an energy (e+* is thus the output voltage I/ of the device potential difference (+c at maximum power output. T h e fundamental equation governing the thermionic emission of electrons from a solid is the Richardson-Dushman equation (8). If electron reflection at the solid-vacuum interface is neglected,2 this equation is: J = AT2 exp (- e+/kT) (1)
+
+
where J is the emission current density, T is the absolute temperature of the solid, e is the electronic charge, and k is Boltzmann’s constant. A statistical treatment of electron motion within solids predicts that A should be a universal constant (for a homogeneous surface) having a value equal to 120 amp/cmZ0K2,where r,h is the true work function as defined in the previous section. Herring and Nichols (9), Nottingham (I2), and Hensley (13) give extensive discussions of the derivation and applicability of Eq. ( I ) . Experimental measurements of electron emission vs. temperature are often plotted in the form In ( J / T 2 )vs 1/T. Such a Richardson plot For a clean uniform metal surface quantum mechanics predicts a reflection coefficient of approximately 0.05 for therma1 electrons (9). Experiments by Shelton (10) with tan-
talum, and Hobson (11) with tantalum and copper, are in agreement with this estimate in the low energy range, e.g., for electron energies of one volt or less. Thus, electron reflection does not seem to be a large effect, at least for clean metals. However, as Nottingham (12) has discussed, this conclusion is by no means certain since other measurements of electron reflection have yielded large reflection effects.
128
J. M. HOUSTON AND H. F. WEBSTER
usually approximates a straight line, the intercept and slope of which yield the empirical constants A , and 4r which typify the emission from the material in question, As many authors (7, 9 , 1 2 , 13) have pointed out, the constant$, (the Richardson work function) obtained from such a plot is not equal to $, the true work function, if r$ changes with temperature. The temperature variation oft$ can arise in a number of ways which include: (1) inherent temperature dependence of the work function of a physically unchanging surface ; (2) unless vacuum conditions are excellent, gases may adsorb on, or react with, the cathode, especially at cooler temperatures; (3) if the cathode is any type of dispenser cathode (as are most strongly-emitting cathodes), the equilibrium surface coverage tends to vary with temperature. An extreme case of such a dispenser cathode is a refractory metal in Cs vapor. Here, the coverage changes so greatly with temperature that the value of $r can become negative, a rather meaningless concept. However, the same problem exists with all dispenser cathodes, to a greater or lesser degree. Because the Richardson work function often differs considerably from the true work function, it is increasingly common in emission studies to assume the theoretical A-value of 120 amp/cm2"K2and calculate 4 vs T from measurements of J vs T . Hensley (13) suggests calling such a value the "effective" work function. For a uniform surface the effective work function calculated from zero-field J-values should equal the true work function provided that reflection effects are negligible. For a surface composed of patches of differing work function, the effective work function (calculated from true zero-field emission) is always somewhat larger than the area-weighted average work-function, which one might define as the true work function of the patchy surface. However, the difference is usually not large and since detailed information (e.g., patch size and work function) on the patchy surface is usually not available, the effective work function is generally the best value to use in converter analysis and design. A good example of the difficulties that would arise if Richardson work functions were used in converter design can be seen in recent measurements of the electron emission from pure UC and from UC mixtures, summarized in Table 11. T h e values of $r, which were measured, in general, over the range 1400-2000"K, vary by as much as 1.5 volt. However, the effective work functions are more consistent, at 1700°K being about 3.3 volt for pure U C and 3.4 volt for UC,,,ZrC,.,. T h e somewhat lower value of effective 4 given for the "extrapolated Schottky" measurements of Haas and Jensen (24) is not inconsistent with the other data given, but is a consequence of the fact that they alone were able to apply high enough electric field to obtain the theo-
THERMIONIC ENERGY CONVERSION
129
retical Schottky slope in a plot of log J vs Y 1 / 2This , high-field data, when extrapolated to zero field, yields a higher J-value than do measurements extrapolated to zero-field from lower E fields. As Haas and Jensen point out, the work function value determined from the extrapolated Schottky slope is weighted more toward that of the low-+ patches, while that determined from true zero-field measurements is an area-weighted average.
B. Brief History of Thermionic Conversion T h e earliest known analysis .of thermionic energy conversion was given by Schlicter in 1915 (15). He showed (neglecting space charge and assuming +a = c&) that the vacuum converter at maximum power output yields a current density of J12.718 at a voltage of kTc/e, where J is the zero-field cathode emission. T h e theoretical power output was thus a discouraging 0.031 watt/cm2 for J = 1 amp/cm2 and T o = 1000°K. He also built a vacuum converter having a platinum cathode at 1000°C which watt at an efficiency of about Champeix (16) yielded 1.5 x and Ansel’m (27) in 1951 published analyses similar to that of Schlicter, except that Ansel’m extended the analysis to the region where da < dC and considered aIso anode emission. However, there was no widespread activity or interest in the field of thermionic conversion until about 1958 when it became apparent that the power output and efficiency could be greatly improved as a result of two developments. T h e first of these was the realization that by suitably arranging the work-function difference, one could greatly raise power output and efficiency. As Medicus and Wehner (18) pointed out in 1951 in a paper on negative-anode arcs, the desirable arrangement is an anode work function lower than the cathode work function. T h e second improvement was the development of effective methods of overcoming electron space charge. If space charge effects are not eliminated in some way, it is impossible to get current densities of the order of one amp/cm2 or more to flow from cathode to anode. Techniques which have been used to reduce electron space charge effects include close cathode-anode spacing, crossed electric and magnetic fields in a three (or more)electrode device, and the introduction of positive ions into the interelectrode space. T h e details and history of each of these developments is discussed in later sections. One important development was the introduction of cesium vapor into the converter. This gas adsorbs on the cooler anode, thus producing the desired low anode work function. Because of its low ionization potential (3.89 volt) cesium is also readily ionized, either by surface ionization at the hot cathode or by electron impact in the interelectrode space. These
130
J. M. HOUSTON AND H. F. WEBSTER
two advantages were clearly pointed out in papers by Gurtovi and Kovalenko (19) in 1941, and, in particular, by Marchuk (20) in 1956. However, the current upsurge of widespread interest in thermionic conversion really began in 1958 with the publishing of three papers (21, 22, 23a) describing various versions of cesium converters, all demonstrating efficiencies of 5 to 10% and power densities of 3 to 10 watt/cm2. Previous to this, all thermionic converters had yielded efficiencies of less than 1 %. In one of these papers Wilson (22) proposed and demonstrated that the Cs vapor could have a third function, i.e., a partial coating of Cs could be used to control the work function of a refractory metal cathode. I n this way a long-lived, low-emissivity cathode of the optimum +c can be achieved. Another important development was the publication ( 2 3 ) of work done at the Los Alamos Scientific Laboratory indicating that a U C : ZrC mixture can be both a good cathode material for Cs converters and a suitable high-temperature fuel for fission reactors. Cesium converters have been intensively investigated since 1958, and have recently yielded (for a few hours operation) measured efficiencies as high as 17 "/b at Tc 1920°K (24, a figure which may well be obsolete by the time this article is published.
-
11. IDEALIZED MODELOF
A
THERMIONIC CONVERTER
I t is instructive to analyze an idealized model of a thermionic converter, i.e., one in which collisions and space charge have been completely eliminated, for example, by very close anode-cathode spacing. Analysis of such a model yields considerable insight into how a thermionic converter works, indicates the optimum values for electrode work functions and temperatures, and sets upper limits on the heat-to-electricity conversion efficiency one can hope to achieve. Such analyses have been given by many authors, including Hernqvist et al. (21), Houston ( 2 4 , Rasor (26), Dobretsov (I), Ingold (27), Schock (28), and Cayless (3).
A. Current- Voltage Characteristics and Power Output T h e converter electrodes are assumed to have uniform surfaces with saturated electron emissions given by the Richardson equation [Eq. (l)] with A = 120 amp/cmZ0K2.Equal cathode and anode areas are assumed. T h e symbols J c s and Jas will represent the saturated emission of cathode and anode, respectively, while Jc and Ja will represent the cathode and anode emission which reaches the opposite electrode. T h e symbol J will equal the net current flowing from cathode to anode, i.e.,
THERMIONIC ENERGY CONVERSION
131
J
= J c - Ja. Figure 3 gives the motive diagrams for this idealized model for the cases where the magnitude of the load voltage V is greater than, equal to, or less than (r$c Note that while bCand r$a are by
d
-7-
F.L.-
Tr-
i-i 4I I ‘
b
FIG.3. Motive diagram of idealized converter with the magnitude of V greater than, equal to, or less than (& - $a) in drawings a, b, c, respectively.
definition positive quantities, the load voltage V (defined as the anode voltage relative to the cathode) can have either sign, and is negative in all the diagrams of Fig. 3. This convention is necessary in order to make thermionic converter J - V plots have the same polarity convention as those of ordinary vacuum tubes.
FIG.4. T h e three types of 1-V plots which can exist in an ideal thermionic converter. Curve “a” occurs when .$a > +c - kTc/e. Curve “b” occurs when +a < dC - kTc/e and anode emission is negligible. Curve “c” occurs when the anode saturated emission becomes large. T h e circles indicate the point of maximum power output.
132
J. M. HOUSTON AND H. F. WEBSTER
A quantity which will be useful in later discussions is V,, the “internal voltage” of the converter, shown in Fig. 3c. The internal voltage is defined as the potential difference in the inter-electrode gap, and equals the external voltage with a correction for contact potential, i.e., vs =
v+
(+c
-44
(24
Note that in Fig. 3c, V shas a positive value while Y is negative. Figure 4 gives typical J vs V characteristics for various operating conditions. Note that the converter is producing electrical power only in the upper, left-hand quadrant of Fig. 4. For any given J - V characteristic, the point of maximum power production (open circles on Fig. 4) occurs when the J - V product in the upper, left-hand quadrant has a maximum value. Note that there are three types of J - V characteristics in Fig. 4. Let us first consider curve b, the J-V characteristic which occurs when the anode is cool enough so that anode emission is negligible and when +a 5 cjC - kTc/e. The region where V < - (tjjC-+a) (corresponding to Fig. 3a) is the retarding range of the diode, the current increasing exponentially as V becomes less negative according to the equation: IC
= Ics
exp [(+c - +a
+ V)/kTcl.
(2b)
The region where V > - (4, -+a) has a motive diagram similar to Fig. 3c, the cathode saturated emission now reaching the anode. This region is often called the constant-current region. Maximum power output occurs when V = - (#c -+a) as shown in Fig. 3b, the power output being: prn = lc*(+c -+a). (3) The load resistance into which maximum power would be delivered equals: R L = (+c
-4B)IJCSAC
(4)
where A, is the cathode area. For a typical thermionic converter these values might be JcsAc= 50 amp and q5c - +a = 1 volt, yielding R, = 0.02 ohm. Thus thermionic converters tend to be low-impedance power sources. It is obvious from Fig. 4 that as +a is increased, the power output decreases. When +a > - kTc/e, one can still obtain a little power output by operating the diode in the retarding range as shown by curve a on Fig. 4. In this case maximum power output occurs when V = - kT,/e. However, this case is of little interest since power output and efficiency are small.
THERMIONIC ENERGY CONVERSION
133
A more interesting question to ask is, “HOWfar can one decrease+, and still obtain an increase in power output?” As long as Ja remains negligible, a decrease in #a is clearly beneficial. However, when Ja becomes comparable to Jc, power output must go through a maximum since it becomes zero when Ja = Jc. Note that the back emission Ja is not necessarily determined by4a, since the diode can also be operated with an electric field retarding the anode emission, as shown in Fig. 3c. Here, the back emission, Ja, is determined by Eq. (1) with Xa being used as the “work function.” For this reason, the discussion of maximum power output will be given in terms of Xa, where Xa +a. T h e following discussion of back-emission effects is similar to that given by Dobretsov (1). When appreciable back-emission is present, the net current J is the sum of J c and Ja, as shown in Fig. 4c. Here J c and Ja each have a saturated and a retarding range, one being saturated when the other is retarded as indicated in Fig. 3. As shown in Fig. 4c the maximum power point now generally occurs at a voltage smaller in magnitude than (dC- + a ) because a retarding field (as in Fig. 3c) must be present to reduce Ja to less than Jas. Under this condition the power output is: P = [$c - xa]
[]c~
- ATa2 exp (-e~a/kTa)]
(5)
where the first bracketed term is the output voltage and the second is the net current. Let Xm be the value of Xa which yields maximum power output, i.e., Xm is the optimum anode work function for a motive diagram as in Fig. 3b. By straightforward maximization of Eq. (9,Xm is found to be given by the solution of the following transcendental equation:
T h e solution to this equation can be written in the form: Xm = ($cTa/Tc)
+ GTa
(7)
where G is a function primarily of (Ta/Tc).T h e function G is plotted in Fig. 5 for several values of d C / k T ccorresponding [see Eq. (l)] to Jc in the range 1 to 100 amp/cm2. Note that the first term in the right-hand side of Eq. (7) predominates, the term involving G typically amounting to 0.1 volt or less. If maximum power output is desired, Eq. (7) sets a lower limit on the value of +a that it is profitable to use, i.e., power output becomes independent of +a for lower values. Since all strongly emitting cathodes
134
J. M. HOUSTON AND H. F. WEBSTER
-
have (et$c/kTc)
18, the minimum useful
$a
is about 1.67 volt for
Ta = 1000°K. However, it will be pointed out later than converter ejiciency is maximized at a value of Xa slightly lower than that given by Eq. (7). To summarize, the J - V characteristics of ideal converters can be divided into three classes, depending on electrode work functions and temperatures. Table I summarizes these three classes. Note that
I
FIG. 5. The function G [defined by Eq. (7)] vs "&/Tofor various a where a = e&JkTc.
135
THERMIONIC ENERGY CONVERSION
at maximum power output the motive diagram is retarding for cathode electrons in class A, constant in class B, and accelerating in class C. Class C operation corresponds to operating with the optimum value of TAHLE I. T H E THREE CLASSES
Class B
Class A
Limits Typical J - V plot
+n
> +c-kT,/e
OF CONVERTER OPERATION
Xm
< +a < +c-kTc/e
Class C
+n
Xm
Fig. 4a
Fig. 4b
Fig. 4c
Motive diagram at maximum power
Fig. 3a
Pig. 3b
Fig. 3c
Voltage at maximum power
-k Tc/e
- ( 4 c - +a)
-(+c-xm)
back-emission, and yields the maximum possible power output for a given value of bC, Tc, and Ta. I n class C operation the output voltage and power are independent of +a, and decrease as T a is increased. As mentioned earlier, class A operation is of no practical interest because of low power output and -efficiency. Most present-day thermionic converters operate in a manner analagous to class B, although actual converters rarely have the ideal characteristics discussed here. Operation in a manner analagous to class C may become important with Cs-filled converters operating at high anode temperatures. Here, as will be later discussed, the anode saturated emission can become larger than that of the cathode. An example may aid in clarifying the preceding discussion. Assume T a = 1000"K, Tc = 2000"K, and J c s = 10 amp/cm2, corresponding to +c = 3.05 volt. If one were to lower (starting from a high value) one would proceed through all three classes of J - V characteristics as is illustrated in Fig. 6. T h e transition from class A to class B occurs at +a = 2.88 volt. As +a is lowered further, the power output increases, reaching a maximum value at +a = 1.65 volt as given by Eq. (7). This is the transition point from class B to class C operation. Further lowering of +a produces no change in the power output since one would then operate as in Fig. 3c, keeping Xa constant at the optimum value of 1.65 volt. T h e power output is 13.2 watt/cm2, as given by Eq. ( 5 ) . A problem closely related to the example just given is the calculation
136
J. M. HOUSTON AND H. F. WEBSTER
of how output varies as anode temperature is increased, the quantities and Tc remaining constant, At low values of Ta, where anode emission is negligible, the power output is given by Eq. (3), provided
$a, $c,
I
I
I
I
7 I
I
#a = 3.5
I
-3
1
-2
I
I
I
I
-I
0
I
FIG. 6. 1-V curves of the ideal converter for various values of4*, the quantities TB,Tc, and 4o being held constant. The circles indicate the maximum power points.
that $a < $c - kTc/e. As Ta is raised, back-emission grows and one shifts from class B to class C operation when the anode saturated emission exceeds the optimum value of back-emission. This transition corresponds to the point where Xm (which increases as Ta increases) becomes equal to +a. For higher values of Ta, the converter output voltage and power decrease with increasing Ta as given by Eqs. (5) and (7). Figure 7 gives the ideal J - V characteristics for various Ta values, calculated for the case of da = 1.5 volt, +c = 3.05 volt, and Tc = 2000°K (corresponding to Jca = I0 amp/cm2). In this case class C operation begins at 910°K where Jas = 0.5 amp/cm2.
B . Eflciency The efficiency of a thermionic converter is defined as the ratio of electrical power output to cathode heat input. In this section the effi-
THERMIONIC ENERGY CONVERSION
I37
ciency of an ideal converter, i.e., one without electron space charge or resistance in the interelectrode space, is considered. The resulting efficiency will be an upper limit to the efficiency of actual converters. Such an analysis also yields insight as to the necessary properties of the cathode and anode of a high-efficiency converter.
V VOLT
FIG.7. J - V curves of the ideal converter at various values of Ta, the quantities &,dc, and Tc being held constant. The circles indicate the maximum power points.
Let us first consider what can be called the “electronic” efficiency, i.e., the efficiency with all cathode heat losses neglected except the electron emission cooling. This efficiency equals:
where the numerator is the power output and the denominator the net emission cooling of the cathode. As has been discussed by Houston (25) and by Rasor (26), this efficiency is maximized (to a good approximation) when Xa = 4cTaITc (9)
138
J. M. HOUSTON AND H. F. WEBSTER
At this point the back emission is given by:
as can be seen from Eq. (1). By comparing Eq. (9) with Eq. (7) one sees that maximum efficiency occurs at a slightly lower value of ,ya (higher value of J a ) than does maximum power output. It is of interest to calculate the magnitude of the maximized electronic efficiency. If one approximates the denominator of Eq. (8) by the slightly different expression, ( J c s - J a ) (bC 2KTc/e),then Eqs. (8) and (9) give for the maximum electronic efficiency:
+
Since (e+,/kTc)-l 8 for strongly emitting cathodes, the maximum electronic efficiency is thus about 90% of Carnot efficiency. T h e fact that the efficiency is bounded by the Carnot efficiency is, of course, no surprise since the thermionic converter is subject to the same thermodynamic limits as any other heat engine. Next, the efficiency of the idealized converter will be calculated including other unavoidable cathode heat losses, namely radiation and cathode lead losses. For simplicity, the anode will be assumed to be cool enough so that back-emission is negligible. T h e net heat transfer by radiation (neglecting anode radiation) can be written as: R = EnRbb (12) where R b b is the radiation from a black-body at Tc, and en is the net emissivity of the cathode-anode combination. By considering multiple reflections, it is readily shown (29) that: ~n = (EC-'
+
€a-1
- 1)-'
(13)
where ec and ea are the cathode and anode emissivities. For precision, one should apply Eqs. (12) and (13) in a small wavelength range (i.e., use spectral emissivities) and integrate over-all wavelengths to find the net radiation heat transfer. Since spectral emissivities are often not available, Hatsopoulos, Kaye, and Langberg (30) discuss an approximate method for evaluating Eq. (13) using total emissivities. A second small but unavoidable heat loss is due to the metal lead which connects the hot cathode to cooler portions of the converter. As shown in Fig. 1, this lead is often in the form of a thin foil which supports the cathode and completes the vacuum envelope. As been discussed by
THERMIONIC ENERGY CONVERSION
139
numerous authors, including Schlicter (1.9, Houston (25), Rasor (26), Hatsopoulos et al. (30), and Schock (28), an optimum geometry exists for this lead, the optimum being a compromise between heat conduction and I R drop in the lead. If radiation from the lead is neglected and the electrical and heat conductivity of the lead are assumed to be related by the Lorenz number (31),the voltage drop V , and heat conduction per unit cathode area H , of the optimum lead are given by:
v, = [L7)(T,2 - T,")/(2 Hw
= JCVW(1
7))]"2
- d1.1
(14) (15)
where L is the Lorenz number (2.45 x watt ohm/deg2)and T is the converter efficiency. Once V , is known, the optimum lead resistance is given by V w /],A, where A , = cathode emitting area. I n choosing a lead geometry having this resistance, it is desirable to keep the foil thin and short so as to minimize lead radiation. T h e ratio of 7, the efficiency including lead losses, to yo, the efficiency neglecting lead losses, can be shown to be: rllT10 = 1 - W
d 4 C - 441
(16)
Lead losses thus become relatively more important as 7) increases or as decreases. For a typical converter with Tc = 2000"K, Ta = 1000"K, 7) = 15%, and (4c - +a) = 1 volt, the lead voltage drop, Vw,equals 0.077 volt and q / q o = 0.85. T h e efficiency of the idealized converter including radiation and lead losses is given by:
(dc - # a )
This equation is plotted in Fig. 8 for several values of $a and net emissivity, including optimum lead 10sses.~I n addition to the plots of q vs dCat constant Tc (solid lines), the contours of constant J c are indicated (dashed lines). A +a value of 1.7 volt represents a conservative, readilyachievable value, while 1.0 volt is about as low a work function as has ever been produced. Similarly, a net emissivity of 0.3 is readily achieved (e.g., refractory metal such as W facing a black-body anode) whereas a net emissivity of 0.1 would require either a heat-shielded cathode or an anode which remains shiny during converter operation. In calculating lead losses, the anode temperature enters in a minor way. Here Tn 1.7 volt, and 500°K for = 1 volt, were assumed. 900°K for :
=
140 J. M. HOUSTON AND H. F. WEBSTER
8-
UI 0
NOIStIIFIAN03 A3XXNB 3INOIINtIXII.L.
P 0
w
0
$ 4 0
N
-
.b
--
= 1.7 and 1.0 volt and net emissivity 0.3 and 0.1. The anode is assumed to be cool enough so back emission is negligible. The heavy solid lines give the efficiency at constant cathode temperature, while the dashed lines are the efficiency contours at constant cathode current density. =
It71
N-
0
.”
FIG.8. Efficiency of the idealized converter (including optimized lead losses) versus4c assuming+.
142
J. M. HOUSTON AND H. F. WEBSTER
Some of the conclusions which can be drawn from Fig. 8 are:
(1) For a fixed value of Tc the efficiency peaks fairly sharply at an optimum value of dc. This optimum occurs because as +c is lowered (i-e., J c raised), at some point radiation losses become essentially negligible compared to emission cooling. Further lowering of +c then decreases efficiency because Eq. (17) becomes approximately equal to (+c - +a)/&. (2) At high values of Tc, the Jc-values corresponding to the optimum r j c tend to be larger (e.g., 30-100 amp/cm2) than the capabilities of known cathodes. If one compares cathodes at some reasonable current density (e.g., 10 amp/cm2) the efficiency goes through a broad maximum as +c (and T,) are increased. I n general, this maximum occurs in the range of $c = 2.5 to 4 volt for Jc = 3 to 10amp/cm2. (3) At the optimum value of$,, the efficiency is not strongly dependent on the radiation loss. However, at dC values above the optimum (where most converters operate) the efficiency becomes more strongly dependent on the radiation loss. For example, for $a = 1.7 volt, = 3.25 volt, Tc = 2000"K, and Jc = 3 amp/cm2, 7 increases from 11 yo to 21 yoas the net emissivity decreases from 0.3 to 0.1. (4) T h e efficiency increases rapidly as +a is lowered. However, as discussed previously, one is limited in this process either by anode back-emission or by the nonavailability of very low 4 surfaces. ( 5 ) A strongly emitting cathode, e.g., at least 1 amp/cm2 and preferably more, is essential for reasonable efficiency. T h e efficiency increases rapidly as Jc increases, up to the Jc-value corresponding to the optimum value of +c. Conclusions similar to those listed above can be arrived at in a more precise and general form by mathematical analysis of the ideal converter. Such analyses have been given by Rasor (26),Ingold (27), and Schock
(28). 111. THEWORKFUNCTION OF VARIOUS SURFACES As has been described in previous sections, it is necessary to use electrodes of certain specific work functions for the cathodes and anodes of thermionic converters if they are to yield maximum output. This section will discuss the work function of the surface of various materials
THERMIONIC ENERGY CONVERSION
143
and combinations of materials. Since the refractory metals which are mechanically suitable for use as converter electrodes are of relatively high work function, various surface layers must be added to lower these values. A series of electrode surfaces will next be described which have work functions ranging from approximately 1 to 6 volts. T h e discussion begins with simple structures and ends with complex combinations of a number of elements. These electrode surfaces are often referred to as cathodes because of their traditional application as emitters of electrons but it is clear that the same surfaces may also be used as anodes of thermionic converters if their work functions are appropriate for this application. A. Pure Metals T h e simplest electrode is the surface of a pure metal single crystal which is free of facets and physical defects. T h e work function of such a surface of a given metal depends considerably upon the crystal structure of the surface and the lattice spacing. For example, tungsten which is one of the most extensively studied metals, has work functions (32) ranging from approximately 4.3 volts for the 116 plane to 5.3-6.0 volts for the 110 plane. T h e work function of a given surface usually has a small temperature dependence, generally in the range to lo-* volt deg for pure metals and larger for composite surfaces. Most of the tabulated work function values for the various chemical elements (5) result from measurements on polycrystalline samples. Since the value varies so much with crystal face, such measurements are largely determined by the statistical distribution of crystal faces exposed on the various grains. This grain distribution is determined by the processing of the metal and is affected by drawing, rolling, heat treating, the presence of foreign atoms, and both thermal and chemical etching. An extensive literature exists on the crystal statistics of metal grains (e.g., Barrett, 33). Very little has been done in correlating grain statistics and polycrystalline sample work function mainly because of the lack of information on the work function of a sufficient number of single crystal faces both when clean and when contaminated. I n addition, the effects if any, of such physical structures as grain boundaries, facets, and dislocation sites are presently unknown. All of this indicates that further work is needed before we can precisely predict thermionic emission currents even from pure metals.
B . Thin Films on Pure Metals T h e next cathode in the order of increasing complexity is the singlecrystal metal substrate coated with a thin (e.g., approximately one
144
J. M. HOUSTON AND H. F. WEBSTER
atomic layer or less) film of an electropositive material. This cathode will be considered in some detail because of its importance in alkali-metal thermionic converters. At present most converters use polycrystalline base metals, but again it is necessary to know the single crystal results before one can hope to understand the polycrystalline system. T h e electropositive metal used to lower the work function of the substrate crystal can be dispensed to the surface in several ways. If the base-metal crystal is operated in an atmosphere of the film metal, then some of the atoms striking the crystal will adsorb and alter the work function. Alternatively the film can be formed by the film metal diffusing or migrating from within the base metal. I n both cases the film will be lost either by evaporation or by reaction with foreign atoms present in the surrounding atmosphere, and sufficient film metal must be supplied to the surface to maintain the desired coverage. In cathodes which have the electropositive metal dispensed to the surface from within the base metal, both the dispensing rate and the evaporation rate are determined by the cathode temperature. In contrast to this, in cathodes which have the electropositive metal dispensed from the vapor, these two processes are separately controllable, which gives greater flexibility to this system. T h e cathode consisting of a single crystal of refractory metal operating in an atmosphere of alkali metal vapor will now be considered. I t has been known for many years that the different crystal faces of the base metal show different thermionic emission densities when covered by films of cesium metal, e.g., from the measurements of Martin (34) on a single crystal sphere and Bruche (35)on different grains of polycrystalline ribbon as observed by an emission microscope. Recent measurements of Webster (36) using a sphere similar to that used by Martin but modified to allow quantitative measurements to be made have shown that the difference of emission density between different faces may be almost 100: 1 for rhenium in cesium. A graph of these measured values is shown in Fig. 9. In addition, measurements of the positive ion density produced by surface ionization are also shown in the same figure. Measurements have also been made of W, Mo, Nb, Ta, and Ni in cesium, T a in rubidium, and Mo in potassium. All of these metals show the large variation of emission density with crystal face. T h e observed behavior of these cathodes as a function of cathode temperature is such that at low temperature the work function of the surface is close to that of the bulk alkali metal, e.g., 1.8 volts (37) for Cs. As the temperature is raised (keeping the alkali-metal pressure constant), the work function initially falls and goes through a minimum value at some optimum coverage, For Cs on clean metals this minimum is
145
THERMIONIC ENERGY CONVERSION
generally in the range 1.4 to 1.7 volts. As the temperature is further increased, the work function increases until the work function of the clean base metal is reached. At low temperatures the cathode is covered by a complete coat of alkali metal. As the temperature rises, the lifetime 10
10 RHENIUM IN CESIUM TC' = 5 5 o c
10
.
N
t
u
n
a
i0 '
c v)
2 W 0
c
z
W
a LL
0 3
10
10'
lo .5
I .5
1.0
I .9
1000 I T
FIG. 9. Electron emission density vs. reciprocal cathode temperature for four crystal faces of rhenium in Cs vapor: (a) 1010 face; (b) 1011 face; (c) near 1120; (d) near 2131.
146
J. M. HOUSTON AND H. F. WEBSTER
of the alkali atom on the surface decreases and the equilibrium coverage therefore decreases, until finally the base metal is hot enough to remain free of alkali atoms. This behavior can be observed for each of the four measured faces in Fig. 9, but measurements were stopped before emission from pure rhenium was obtained. T h e results of all these tests on single crystals of refractory metals in cesium vapor leads to a simple rule: T h e crystal face which will provide the highest emission density in the region where emission decreases with increasing cathode temperature is the atomically closest packed plane, i.e., the plane which would have the highest work function when not coated with the alkali atoms. Thus the largest emission density for a partial coating of cesium on all of the body-centered-cubic base metals comes from the 110 face, for the face-centered-cubic metal (such as Ni) from the 11 1 face, and for the hexagonal metal (such as Re) from the 1000 face. Some other crystal faces emit strongly but generally they yield peak emission at a lower cathode temperature. This result is in agreement with the theories of Zalm (38), Rasor (39),and Gyftopoulos and Levine (40). T h e next most complex cathode is the alkali-metal film on polycrystalline base metals. This is a cathode of great practical interest and several measurements (42,42,43) of its properties have been made. It will differ from the single crystal cathode riot only in that a statistical average of the work functions of the various grains must be used but also the intergrain electric field may be important and the dependence of emission on applied electric field may be quite different than that of the single crystal. These so-called “patch effects’’ result not only from the difference of work function of various grains but also from the surface geometry of the grains which may deviate greatly from a simple plane. T h e surface may have sharp points and edges as well as concave regions as a result of various surface treatments such as thermal and chemical etching. Thus some field enhanced emission as well as hollow cathode effects may occur. I n addition, when such a polycrystalline cathode is supplying both electrons and positive ions, the electrons will come predominately from the low work function grains and the positive ions from the high work function grains and thus this cathode may behave differently from one made of a single crystal face, where the ions and electrons are generated together. T h e most complete measurements made on this type of cathode were made by Taylor and Langmuir (43)for a tungsten wire in cesium vapor. They measured the emission current as a function of cathode temperature at relatively low cesium pressures and extrapolated their results to higher cesium pressures. Cesium thermionic converters generally
THERMIONIC ENERGY CONVERSION
147
operate at orders-of-magnitude higher Cs pressures than those used by Taylor and Langmuir and there is always some question as to the validity of this extrapolation. Houston (44) has measured the emission of Cs on W at higher Cs pressures using the plasma technique described by Marchuk (45). These measurements are in excellent agreement with the extrapolated Taylor-Langmuir curves (solid lines in Fig. 10). Taylor T *K
FIG. 10. Electron emission density of Cs on W vs reciprocal cathode temperature. Comparison of the measurements of J. M. Houston [Bull. Am. Phys. SOC.[2] 6 , 358 (1961)] (experimental points) with a careful extrapolation of the Taylor-Langmuir results [J. B. Taylor and I. Langrnuir, Phys. Rev. 44, 423 (1933)] (solid lines).
and Langmuir heat treated their wires extensively and they thought that the wire surface was made up almost entirely of 110 facets as a result of this treatment. This, however, is in doubt because Houston's wires were not heat treated for many hours and yet the emission current densities are in good agreement. Also, we now know that the work function of the 1 10 plane of W is considerably greater (32)than the effective work function of 4.6 volt which Taylor and Langmuir found for their W filament. Thus, it is probable that a variety of crystallographic surfaces were present in the Taylor-Langmuir experiment.
148
J. M. HOUSTON AND H. F. WEBSTER
This plasma anode technique was also used by Houston (44) to determine the emission properties of five other materials and the results are shown in Fig. 11. These results are generally in agreement with T
OK
J
AMP
eM2
\
\
\
\
104/T
OK-'
FIG. 11. Thermionic electron emission density from five elements and from Mo-W alloy (50-50 by weight) in Cs vapor.
measurements made by other methods except for rhenium, which appears to have been contaminated, i.e., has too low a work function when free of Cs. I n general the metals having the highest work functions when uncoated again yield the highest emission density at intermediate temperatures when partially Cs-coated. Aamodt (46) has measured the emission of a Mo filament in Cs vapor in the range Tcs = 313 to 374°K using the Taylor-Langmuir (43) technique. His results and Houston's are in fairly good agreement, Fig. 12 giving the Cs-on-Mo electron emission as deduced from both sets of data. It should be
THERMIONIC ENERGY CONVERSION
149
emphasized that the Taylor-Langmuir, Houston, and Aamodt data were all taken on small, round, polycrystalline wires. A cathode made in some other form, e.g., a rolled sheet, might well have a different distribution of surface patches, and therefore yield a somewhat different emission. Because certain crystal faces emit more strongly than others when Cs-covered, it is desirable to find some way of altering the statistics of the surface grains in favor of the optimum crystal face. This can be done either by heat treatment or by etching. For example, D. B. Langmuir (47, 48) has shown that 110 and 112 facets develop on tantalum ribbon lo;
10
I
10-
10-2
lo-? J AMP CM'
lo-'
10-5
lo-(
lo-; 10-8
10001T Pi0
FIG. 12. Thermionic electron emission density from Mo in Cs vapor from measurements of R. L. Aamodt (private communication) and J. M. Houston [Bull. Am. Phys. SOC. [2] 6, 358 (1961)l Figure courtesy of R. L. Aamodt, Los Alarnos Scientific Laboratory.
150
J. M. HOUSTON AND H. F. WEBSTER
when it is heated with either an electric field or a thermal gradient present along the surface. Similarly, Hughes, Levinstein, and Kaplan (49) have shown that etching of tungsten wires with HF-HNO, leaves 110 and 112 facets. Another chemical etch that leaves 110 facets is given in Barrett (33). Webster has shown that low-voltage electrolytic etching of tungsten in NaOH develops 110 facets. Such techniques may be useful for obtaining optimum electrode surfaces. These cathodes of clean, refractory metals in alkali-metal vapor have an advantage in that the workfunction can be adjusted over a wide range by varying the cathode temperature. Thus I # ~can be adjusted to some optimum value or to match some available heat source. At high emission densities the emission is not a strong function of temperature so the same cathode can emit strongly over a considerable temperature range (see Fig. 10) which is an advantage in practical systems where the temperature cannot always be precisely controlled. In contrast, most conventional cathodes only yield high emmision and long life in a narrow temperature range, However, in order to obtain high emission densities high alkali-metal pressures are necessary. For example, for Cs- W, Fig. 10 indicates that for J = 10 amp/cm2 at 2000"K, a Cs reservoir temperature of 615°K is necessary, corresponding to Pcs = 5 mm. Such high Cs pressures are undesirable in thermionic converters in that they increase the resistance of the interelectrode gap, thus necessitating rather small gaps. One way to lower Pc, for a given J (or raise J for a given Pcs) is to add another gas, e.g., oxygen, hydrogen, or the halogens, to the cathode surface so as to increase the binding energy of the Cs atoms to the substrate, and, in some cases, increase the surface dipole per Cs atom. I. Langmuir and Kingdon (50) show, for example, that an oxidizedtungsten surface has a peak emission of 0.35 amp/cm2 (at 1000°K) at T,, = 303°K. This is about lo4 times the peak emission one obtains with clean W in the same Cs pressure. Aamodt (46) has observed a similar effect when flourine or CsF vapor is present as well as Cs vapor. This approach has one problem, however, in that many of the gases which enhance the binding energy of Cs also cause chemical cycles which corrode the cathode, This is especially serious at high cathode temperatures where one must use higher partial pressures of the additive gas (or of some compound which decomposes thermally to yield the desired binding layer) in order to maintain the coating of the additive gas on the cathode, However, at lower cathode temperatures, the required additive-gas pressure may become negligible; e.g., I. Langmuir (51) estimates it takes three years for half of an oxygen layer to evaporate from a W surface at 1500°K.
THERMIONIC ENERGY CONVERSION
151
Complex surfaces such as Cs-0-W or Cs-F-W should be particularly useful for converter anodes because they can have considerably lower work-functions than Cs on clean refractory metals, and also require lower Cs pressures to maintain a low at high anode temperatures. Langrnuir (51) indicates that a Cs-0-W surface at 1OOO"K and T,, = 80°C has an effective4 of -1.4 volt, whereas a Cs-W surface has 4-2.3 volt for the same conditions. Koller (52) describes a cesiated silver-oxide photocathode having an effective 4 of 1.0 volt at 400°K. However, such surfaces are not stable at higher temperatures. T h e next most complex cathode in the series is the thin film on refractory metal cathode in which the film metal is contained in the pores of the base metal matrix and is slowly dispensed to the surface by processes of migration, evaporation, and diffusion. An example of this cathode is the thorium in tungsten matrix (53)cathode which has a work function of about 3.3 volt. A similar cathode is the old and much studied (54,55) thoriated tungsten cathode which contains thoria in the pores. A chemical reduction of the thoria by the base metal is required before thorium can be dispensed to the surface and, in general, this type of cathode will not survive in as poor vacuum conditions as will the thoriumimpregnated cathode because of the smaller dispensing rate of thorium to the surface. Some cathodes which are structurely similar to the above cathode but somewhat more complex chemically are the impregnated cathodes (56, 57). These consist of a matrix of tungsten or other refractory metal the pores of which have been filled with the oxides, aluminates or other compounds of barium, strontium or calcium. Another version of this cathode is the "L" cathode (58,59) in which barium and strontium oxides are supplied from a reservoir behind the porous tungsten emitter. There are many possible combinations and the reader should refer to the original papers for details. I n such complex systems there are many processes going on and it is difficult to describe them with certainty. However, the net result is that some electropositive material is dispensed to the surface crystallites of the matrix and this lowers their work functions. I n addition, some electrons may be emitted by the pore material directly. Reviews of such dispenser cathodes have been given by Stout (@a) and Beck (dob).
C . Oxide and Carbide Cathodes T h e so-called oxide cathodes (61, 62) consist of a layer of small crystallites of barium, strontium, or calcium oxide on a metal base. In these cathodes the electron emission comes predominately from the
J. M. HOUSTON AND H. F. WEBSTER
152
oxide crystallites themselves rather than from the base metal. It appears necessary to provide some free barium metal in the oxide crystallites to control their conductivity and surface work function and many cathode base metals contain reducing agents for this purpose. Since the electric current is carried from the base metal to the emitting surface by semiconducting grains rather than through metal grains, the resistance of this cathode can be relatively high which is a disadvantage in thermionic converters. Another complex cathode which has been used in this application is made of a solid solution of uranium and zirconium carbides (63). A variety of compositions have been tried and the results of some measurements are shown in Table 11. These cathodes have supplied current TABLE 11. THERMLONIC EMISSION FROM Investigator and material
uc
AND
uc
MIXTURES"
(1700" K) (volt)
Ar amp/cm2"K2)
(volt)
7.3 x 106 6.6 x lo4
4.57 4.3
3.29 3.37
33 50
2.94 3.14
3.13 3.21
12
3.1
3.44
90
3.21
3.31
4r
be
Pidd et al. (63)
uc
U G SZ G s Haas and Jensen (14) UC (extrap. Schottky) U C (zero field) Danforth and Williams (64) UCo.*ZrC,,, Abrams and Jamerson (65) UCo.nNbo.2 (by ~01.)
a The columns A, and $r are the zero-field Richardson constants which were evaluated, in general, in the range 1400-2000°K. The effective 4 (assuming A = 120 amp/cm2"K2) for T = 1700°K is also listed. The Haas and Jensen data labeled "extrapolated Schottky" result from extrapolation to zero field of Schottky plots (log 1vs W * )made at high enough fields (50,000 to 230,000 volt/cm) such that the theoretical Schottky slope was observed. The results labeled "zero-field" are their estimate of the true zero field emission. Most of the remaining data result from some form of zero-field extrapolation involving smaller electric fields.
densities as high as 62 amp/cm2 at 2600°K (66) and show promise for use in nuclear reactors. However, their total emissivities are relatively high (63) so radiation losses are larger than with many other cathodes. T h e evaporation rate of pure uranium carbide is high, but by the addition of zirconium carbide the evaporation rate in high vacuum can be reduced to as low as gm/cm2sec (67) at 2120°K ( J = 8 amp/cm2) which corresponds to about a 25p layer evaporated per year.
T H E R M I O N I C ENERGY CONVERSION
153
D . Evaporation of Cathode Materials If thermionic diodes are to have long life it is important that the evaporation rate of the volatile components of the cathode be small. This is necessary for two reasons, (1) if the supply of volatile material is limited, cathode emission will decrease when the supply is exhausted, (2) the evaporated material may deposit on other electrodes and alter their properties in an undesirable way. T h e evaporation rate of atoms from the cathode and the thermionic emission of electrons both increase exponentially with cathode temperature. I t is useful then to compare cathodes on the basis of amount of material evaporated for a given electron emission density. Figure 13 from Stout (60a) presents such a comparison for eight different cathodes I0
I n V W
N I .
z
3 W b4
= 10-
z 0 I4 -
a
2 4
>
W
10-
ELECTRON EMISSION AMP CM-*
FIG. 13. Evaporation of cathode material vs thermionic emission density for several cathodes.
154
J. M. HOUSTON AND H. F. WEBSTER
compiled from the measurements of several authors. It is clear that many of the composite cathodes are better than pure tungsten on this basis. T h e alkali-vapor-refractory-metal cathode has a unique advantage in that the electropositive layer on the cathode is continuously replenished from the alkali metal vapor. Thus the life-determining evaporation rate is that of the refractory substrate. If this evaporation rate is assumed to equal that measured in high vacuum, the life would be extremely long. For example, tungsten at 2000°K has an evaporation rate of 1.8 x 10-13 gm/cm2sec (68)which corresponds to about 0.3p/century. Thus cathode evaporation would probably not be the life-limiting factor in converters using this type of cathode.
THERMIONIC ENERGYCONVERTERS IV. VACUUM
A. History Work on the vacuum thermionic converter began with the pioneering analysis and experiments of Schlicter (25) in 1915, which have already been described in Section I, B. I n 1951 Champeix (26) presented an analysis of the vacuum thermionic converter essentially identical to Schlicter’s. Space charge was neglected and the results were pessimistic ya. Morgulis (2) describes a detailed theorebecause he assumed pc tical paper by Ansel’m (17) in 1951 in which the efficiency of a vacuum converter (neglecting space charge) was calculated both for pc < rpa and yc > ya. However, (according to Morgulis) no emphasis was placed on the role played by the contact potential difference. There have been many other analyses of the vacuum converter neglecting space charge, which are described in Section I1 of this paper. I n 1956 Hatsopoulos (69) analyzed a vacuum thermionic converter in which the space charge barrier in front of the cathode was reduced by the application of an accelerating electric field created by a third electrode which was maintained at a positive potential with respect to cathode and anode. A magnetic field was applied to minimize collection of electrons by this third electrode. This is the magnetron or crossed-field energy converter. Hatsopoulos discussed both a simple device with a single cathode and anode, and complex devices with multiple cathodes and anodes. His analysis, however, did not consider the anode work function. I n 1957, Ioffe (70) made a brief reference to vacuum thermo-elements in his book on thermoelectric conversion. He, however, did not mention the effects of electrode work functions.
<
T H E R M I O N I C ENERGY CONVERSION
155
Also in 1957, Moss (71) made the first analysis of the vacuum thermionic converter which correctly dealt with the effects of space charge. He calculated the current-voltage characteristic of such a device at a single temperature (1 100°K) from the Langmuir (72) space charge solution with initial velocities. Although his analysis is both extensive and correct, his results are pessimistic because he too assumed that 'pc \< 'pa. In 1958 a number of papers appeared dealing with vacuum converters. Webster and Beggs (73) gave an analysis of the diode converters. based on the Langmuir solution for any electrode work function difference and any cathode temperature. In addition, they presented some results with gas flame heated vacuum converters which had 1.2 cm2 of cathode area and in one test gave 0.13 watt of output power (0.2 volt at 0.65 amp). Hatsopoulos and Kaye (74, 75). described their experiments with two 1 /8-inch diameter L-cathodes spaced lop apart. T h e output power density obtained in one test was 0.76 watt/cm2 at Tc = 1540"K, yielding an efficiency estimated to be 13%. Measurements of L-cathode emissivity by Rittner (76) indicate that this estimated efficiency is somewhat optimistic. Feaster (77) gave an addendum to Moss' paper in which he pointed out the importance of operation with Fa
< vc.
I n 1959, several more papers related to the vacuum thermionic converter appeared. Nottingham (78) presented an approximate analysis of the vacuum converter based on the master curve which appeared in his thermionic emission article (12). Nottingham, Hatsopoulos, and Kaye (79)published a letter which compared their theory and experiments and showed good agreement. Webster (80) published an analysis of the vacuum converter which enlarged upon his earlier presentation (73). Lindsay and Parker (81) presented a solution to the problem of the space charge between two electron-emitting electrodes which they later extended to find effects of an emitting anode on the converter characteristics. In the following years, other papers have appeared which will be considered in more detail in later sections. T h e number of papers now appearing concerning the vacuum converter, however, is small because most work is now being concentrated on gas-filled thermionic converters.
R. Space Charge Problems in the Vacuum Thermionic Converter As has been .previously mentioned, the major problem of the vacuum thermionic converter is that of sufficiently reducing the electron spacecharge barrier so that a large current density can flow from the cathode
156
J. M. HOUSTON AND H. F. WEBSTER
to the anode. To know the height of this barrier it is necessary to be able to calculate the potential distribution in the vacuum gap between cathode and anode in terms of the converter parameters, i.e., the electrode work functions, temperatures, and geometry. Because potential differences in a vacuum converter are relatively small, electron thermal velocities cannot be neglected in any analysis of electron space charge. Langmuir (72) has calculated the potential distribution in a planar diode assuming the electrons issue from the cathode with a Maxwellian distribution of velocities. T h e Langmuir theory is both elegant and simple in that it presents the complete solution with a single curve, in terms of the reduced variables 71 and 4 which are related respectively to potential in the diode, and distance in the diode in terms of the Debye length. This solution does, however, have a shortcoming in that it relates all values to the potential minimum which has an unknown potential and position relative to the diode electrodes.
In
FIG. 14. Effect of space charge on the current-voltage characteristic of planar thermionic vacuum diodes. Natural logarithm of (J/Jcs) vs eVs/kTcfor fourteen values of the space charge parameter R' defined by Eq. (18). The quantity J is the diode current density, JCs is the cathode saturated emission, and V8is the diode internal voltage defined by Eq. (2a).
THERMIONIC ENERGY CONVERSION
157
The converter designer needs to determine the current-voltage characteristic of his device from the potential distribution in the diode and this can be done in several ways. Compton and Langmuir (82) have suggested a point-by-point method which is straightforward but laborious. Rittner (76), has represented the vs 5 curve of Langmuir by several empirical equations which are sufficiently accurate to be useful in programming this problem for a computer. Three other
-I
0
I
2
3
k1
c4
JaD5
6
7
8
9
2
FIG. 15. Current voltage characteristic of thermionic vacuum diodes with large values of the space charge parameter R',compared with the characteristics of diodes with finite values of R'. This limiting curve for large R' is the same as the universal diode characteristic of Ferris or the master curve of Nottingham.
methods have been developed in papers by Ferris (83), Nottingham (78), Webster (80), which yield the current-voltage characteristic directly from the diode parameters. These three solutions are similar but differ in the choice of reference point for the presentation of the currentvoltage characteristic. The first two methods use the potential distribution with the minimum at the anode as a reference and the third uses the potential distribution with the minimum at the cathode. A composite of these three methods is presented on two graphs, the first of which (Fig. 14) gives the current-voltage characteristic valid for all space charge conditions. The second graph (Fig. 15) gives a limiting
158
J. M. HOUSTON AND H. F. WEBSTER
curve which is valid for large emission densities and large diode spacings. I n addition this graph evaluates the error in applying this limiting curve to diodes with smaller space charge effects. Figure 14 presents a plot of the internal voltage of the diode Vs [see Eq. (2a)l in the dimensionless form eVa/kTc as a function of ln(J/Jcs) where J is the diode current density flowing to the anode and J c s is the cathode saturated emission density. T h e current-voltage characteristic has been drawn for several values of the space charge parameter R' which is defined by the equation R' = 84.24 x 1O'O JcBd2/Tca/2
(1 8)
where Jcs is in amp/cm2 and d is in cm. When R' = 0, space charge effects are negligible and the exponential current-voltage characteristic which is obtained when the diode is in the retarding regime meets the saturated-current line at a sharp knee. This current-voltage characteristic is identical to that of Fig. 4b. For larger values of R' space charge effects appear, the knee rounds off, and a larger value of internal voltage is required to draw the saturated emission current. As the load is varied from a high resistance to a low resistance, the curent-voltage characteristic follows a constant R' line from the retarding field region, through the space charge region, and into the region of saturated current. All of the curves of Fig. 14 drawn for large R' have almost the same shape and it can be shown that a simple translation of magnitude ln(Jcs/ Jm) along both the current and voltages axes, causes these curves to cluster. As A' becomes infinite the cluster has a limiting curve which is the universal diode characteristic of Ferris or the master curve of Nottingham. The quantity J m is defined as the current density that would flow if the potential minimum were at the anode and the cathode had an unlimited emission capability (i.e.,qc = 0)
jm = (0.245 x i o - y q (~~/1000)3~~ where J m is in amp/cm2 and d is in cm; following equation Jm
Jm
(19)
is simply related to R' by the
= 6.523J c s / R .
(20)
Figure 15 presents the limiting curve along with curves for R' = 1000, 100, and 20, to indicate the magnitude of the error involved in using the limiting curve. Figure 16 allows rapid evaluation of R' if Tc, Jcs, and d are known. Diodes with d > 0.1 mm (e.g., most conventional vacuum
T H E R M I O N I C ENERGY CONVERSION
159
diodes) have large R-values and thus the limiting curve in Fig. 15 can be accurately used. Most vacuum thermionic converters, however, have spacings of < 0.01 mm, yielding R' < 100 and making the limiting curve inaccurate. Rittner (76) was the first to point out this difficulty with the limiting curve.
d ( IN 001 INCHES)
FIG. 16. Curves useful for evaluating the space-charge parameter R' from known values of the cathode-anode spacing (d), the cathode temperature ( Te),and the saturated emission available from the cathode (Jcs).
All the methods of finding the current voltage characteristics considered here have neglected the effects of series and shunt resistance in the diodes. Since typical vacuum thermionic converters operate into load resistances well under one ohm, a series resistance of a few ohms would be ruinous because most of the output power would then be dissipated in this resistance. T h e series resistance often appears in the coatings applied to the cathode and anode surfaces to lower their work functions. If the coatings are semiconductors, the largest series resistance is likely to appear at the cooler anode surface. Tests made on some very close spaced vacuum thermionic converters built by Beggs (84) indicated that even'with the relatively thin oxide layers (e.g., 2p) which were used on the anode surfaces, frequently more than half of the output converter power was lost in the series resistance, Impregnated cathodes,
160
J. M. HOUSTON AND H. F. WEBSTER
however, are less hampered by series resistance because of the low resistance of the metal matrix. The effect of shunt resistance on the converter performance usually proved to be much less severe than that of series resistance. It affects the current-voltage characteristic mainly near open circuit. This shunt resistance may be produced by a leakage film across the ceramic insulator. CURVE A d z . 0 0 0 6 INCHES re 1160.K bC - + & .5 VOLT k = I em2 J, = 2 amp /em2
CURVE B SAME AS A EXCEPT WITH 10 OHMS SHUNT RESISTANCE
I
CURVE C SAME AS A EXCEPT WITH I OHM SERIES RESISTANCE
CURVE D AS d APPROACmS C T, :1160°K +c -#A : .5 VOLT : I cm2 J, : zamp/cm2
I.
I AMP
VOLT
FIG. 17. Current-voltage characteristic of vacuum thermionic converters showing the effects of shunt and series resistance. A limiting curve which is obtained as the cathodeanode spacing approaches zero is also shown.
THERMIONIC ENERGY CONVERSION
161
Figure 17 shows a theoretical J - V curve for a typical vacuum converter with neither of these resistances, with I-ohm series resistance, and with 10 ohms shunt resistance. Another factor which must be taken into account in determining the complete current-voltage characteristic of a vacuum thermionic diode is the emission of electrons from the anode back to the cathode. This problem has been discussed in section 11 for the case with negligible space-charge effects. T h e effects of anode back emission on the space charge barrier have been calculated in papers by Dugan (85) and by Lindsay and Parker (86). Dugan set up the space-charge equations and solved several specific cases by means of computer calculations. Lindsay and Parker present their solution in the form of a family of Langmuir-type 9 vs 5 curves but the solutions are complex and readers should refer to the original papers for details, Once the complete current-voltage characteristic is in hand it is a simple matter to determine the maximum power point. This maximum
T,
llOOn K
I, 5.0
.'.I
AMP/CM*
10 . SPACING IN ,001 INCHES
FIG. 18. Maximum output power density from a vacuum thermionic converter vs cathode-anode spacing for seven different values of work function difference, 4c - ba, in volts.
J. M. HOUSTON AND H. F. WEBSTER
162
output power density has been evaluated in terms of the diode parameters in three papers, all of which neglect effects of series resistance and anode back emission. Nottingham (78) presents an equation for the maximum output power which is somewhat optimistic because it assumes an emitter of unlimited emission capability. A second paper by Nottingham (87) presents the result for emitters of finite emission capability. I n the third paper, Webster (80) has evaluated the maximum power to the load as a function of R' and the electrode work function difference (& - +a). Since the output power of vacuum thermionic converters depends 10
a 6 4
2
I
.a .6 PMAX 9
WATTS
CM2
.2
.I
.0e
.O€ -04
.Oi .O
x)
1000
1500 CATHODE TEMP. 'K
2000
FIG. 19. Maximum output power density from a vacuum themionic converter vs cathode temperature for four different values of the saturated emission density available from the cathode.
T H E R M I O N I C ENERGY CONVERSION
163
sensitively on so many variables, it is difficult to do more than evaluate a few specific cases. This, however, can be done easily with the curves of Figs. 14, 15, and 16 for each case of interest. T h e maximum power density that a vacuum thermionic converter can deliver to a load has been evaluated (assuming negligible series resistance) from these curves for a variety of device parameters and the results are shown in Figs. 18 and 19.
C . Surface Physics Problems in the Vacuum Thermionic Converter T h e requirements for electrode work functions described in Sections I and I1 apply both to the vacuum and to the gas-filled converters. There are, however, a few requirements which are specific to the vacuum converter and those will be described here. T h e work function of the vacuum converter cathode need not be as high as that of the Cs diode converter because it is not required to supply ions by surface ionization. Thus the vacuum converter can operate at a relatively low cathode temperature which is probably the major advantage of this device since it eases heat source and fabrication problems, Because of the lower pc and Tc values, however, and because space-charge compensation is never complete, the output voltage and power of vacuum converters tend to be smaller than that of Cs diodes. Both the cathode and anode electrodes of the vacuum converter usually have had their work function determined by barium, strontium, and calcium oxides, either as impregnated matrices or simple oside layers. T h e materials used on these electrodes, however, must be compatible because evaporation carries material back and forth between the close-spaced electrodes. Beggs (88) has found that using a tungsten base metal with these materials is advantageous in that the temperature coefficient of the work function is then positive, with the result that cpa is lower than plC because of the temperature difference. A limit of the life of the vacuum device is set by either loss of cathode coating or by alteration of the anode work function by deposited material. T h e evaporation rates of these materials have been the objects of a number of studies (89-92) and Gibbons (92) has suggested the use of strontium-calcium oxides for the cathode and barium-strontium oxides for the anode tp improve the converter life. This combination has been used by Beggs (88). T h e impregnated cathodes used by Hatsopoulos and Kaye (75) have a high evaporation rate but nevertheless have a long life because of the large supply of active material in the pores of the matrix. T h e evaporated material, however, is troublesome in that it may short out the tube or alter the anode work function and emissivity in an unfavorable way.
J. M. HOUSTON AND H. F. WEBSTER
164
D. Fabrication Problems I n order to build successful vacuum thermionic converters it is necessary to space the two electrodes the order of lop apart and have this separation remain reasonably constant for the required life of the tube. In addition, it is necessary to encapsulate the electrodes using suitable materials and seals. I n order to achieve and maintain very small spacings, rigid electrodes and spacers are required which are stable at cathode temperatures. T h e electrodes have usually been planar because planar electrodes having the required flatness and uniformity are relatively easy to prepare by surface grinding. T h e cathodes which have been used are made of tungsten either in the form of impregnated cathodes or solid tungsten blocks. T h e latter appears to resist mechanical deformation with time somewhat better than the sintered mass of tungsten particles which make up the impregnated cathodes. Less refractory materials can be used for the anodes because of the lower operating temperatures, and molybdenum or nickel have been used in a few devices. A reaction between molybdenum and the barium and strontium oxides which yields a relatively high-resistance interface, however, makes this metal a questionable choice, Many of the most successful tubes have used tungsten for both anode and cathode, I n most of the vacuum thermionic converters which have been tested, electrically-insulating spacers mounted between the electrodes have determined the electrode separation. T h e spacers are made of materials which keep the heat leak from cathode to anode small. Figure 20 SPACING METHODS FOR THERMlONlC DIODES CA5HODE
ANODE
CATHODE
w ANODE
FIG.20. Three methods which have been used by Beggs to insulate and separate the cathodes and anodes of vacuum thermionic converters.
T H E R M I O N I C ENERGY CONVERSION
165
illustrates several of the spacer geometries developed by Beggs (84). Figure 20a illustrates a simple approach, namely flakes of mica which have been cleaved to the right thickness and placed around the edge of the electrodes. This method, however, was not completely successful because of the slow decomposition of the mica at cathode temperatures.
\
CERAMIC INSULATOR
CERAMIC INSERT
ELbpgDE
FIG.21. Cross section of a close-spaced vacuum thermionic converter.
Figure 20b illustrates the use of sapphire pins which are mounted in holes in the anode, ground flush with the surface, and then raised the proper distance by a metal shim. Relative expansion of the pin and metal must be considered in this design. Figure 20c illustrates the use of a ceramic insert in the anode which is first ground flat and then a small bit of refractory-metal foil is placed between it and the cathode. High purity metals and ceramics must be used to encapsulate the vacuum converter in order to avoid producing gases which will alter the cathode or anode work functions as the tube ages. Metal-ceramic seals must be used which will operate at relatively high temperatures, and, if the device is to be operated in air, oxidation-resistant metals must be used in the envelope. A converter tube designed by Beggs (88) is illustrated in Fig. 21. T h e ceramic insulator is a special Forsterite with an expansion coefficient that matches titanium. T h e seals are nickeltitanium alloy and the cathode foil is either platinum or a platinumrhodium alloy.
E . Experimental Results Figure 22 presents experimental data taken on vacuum converters with maximum power transfer to the load. They were taken with a variety of converter types which are labeled on the graphs. T h e almost exponential rise of output power with cathode temperature can be seen
J. M. HOUSTON AND H. F. WEBSTER
166
which agrees with the predicted behavior shown in Fig. 19. This desirable increase in power output must be weighed against the exponen-
N
3i
2 v) I-
s *
c v) az
W 0
a W
3
0
a. I-
a n
t
3
0
CATHODE TEMP 'K FIG.22. Measured output power density of twenty vacuum thermionic converters The converters marked with either A or 0had barium-strontium oxide coated anodes Those marked with A had barium-strontium oxide coated cathodes and those marked with 0 had strontium-calcium oxide coated cathodes, The point x was measured on a converter made by Hatsopoulos and Kaye which used impregnated electrodes for both cathode and anode.
THERMIONIC ENERGY CONVERSION
167
tial increase in evaporation rate of cathode coating materials, which has already been discussed. To give some idea of the efficiency, output power density and life of close-spaced vacuum thermionic converters which have been attained prior to the end of 1961, this section will conclude with a summary of results obtained by three groups which have produced these devices in quantity. T h e staff of the Thermo Electron Engineering Corporation has built converters using the Phillips impregnated cathodes for both cathode and anode, and has attained output power densities around 0.5 watts/cm2 at 0.6 volts (93).T h e over-all efficiency was about 5 % at cathode-anode spacings of a few ten-thousandths of an inch. T h e major problems with these devices appear to be warping of the closespaced electrodes and evaporation.of the cathode material onto the anode. Members of the staff of the General Electric Research Laboratory have made converters with oxide-coated tungsten electrodes for both cathode and anode. I n one test a power density in excess of 1 watt/cm2 was measured. T h e best efficiency measured was 4.5 %. These tubes were all built to operate in air and a platinum-rhodium alloy foil was used to protect the cathode. Most of the tubes failed in less than 100 hours because of leaks which developed in the foil. This tube was put in limited production by the Power Tube Department of General Electric Company and described in a paper by Baum and Jensen (94, and their experience has been that they could produce them in quantity with output power densities between 0.3 and 0.4 watt/cm2 at a cathode temperature of 1423°K. T h e optimum anode temperature for this condition was near 870°K and the cathode-anode spacing was approximately 0.0003 inches. T h e efficiency for these devices was less than 4 %. Since the gas-filled thermionic converters have now been demonstrated to have output power density, efficiency and life which is superior to that of the close-spaced vacuum devices, most of the effort is at present being applied to gas-filled devices.
F. Three Electrode Vacuum Thermionic Converters T h e principle of these devices is that an accelerating electric field at the cathode surface produced by the third electrode will allow a larger current to flow than would be possible with the wide-spaced electrodes. Some means must be used to prevent appreciable current collection by the third electrode and a magnetic field is usually used for this purpose. Many such geometries are possible in theory, e.g., a striped cathode aligned opposite a positive grid so that the electrons (confined by a B-field normal to the cathode) flow between the grid wires to the anode. However, only the magnetron or crossed-field geometry discussed by
168
J. M. HOUSTON AND H. F. WEBSTER
Hatsopoulos (69) and by Hernqvist et al. (21) received serious consideration. In this device, illustrated in Fig. 23, electrons are emitted from the CROSSED FIELD THERMlONlC CONVERTER TYPICAL ELECTRON ACCELERATING ELECTRODE
x
x
x / x CATHODE
x
x\x
x
ANODE
FIG.23. Section of a high vacuum crossed field thermionic converter showing a typical electron path from cathode to anode. The line A - - - B indicates the plane where a virtual cathode frequently forms.
cathode, accelerated toward the positive accelerating electrode, and then turned by the magnetic field until they return to the anode and are collected. Unfortunately, not all the electrons are collected there and any that reach the accelerating electrode dissipate an amount of power from an external battery which is frequently larger than the power supplied by the electrons to the anode load. 30 B = I 9 0 GAUSS1 Thus it appears to be something of an accomplishment even to achieve zero efficiency with this device. An analysis of a crossed-field device of this sort has been presented by Welch, Hatsopoulos, and Kaye (954. T h e results, however, are highly optimistic because they neglect the dissipation of energy by the flow of electrons to the accelerating electrode. Figure 24 has been drawn from the data of Peters (95b) and it illustrates the ANODE VOLTS currents to the anode and accelerFIG.24. Typical current-voltage char- ating electrode as a function of the acteristic of a crossed field thermionic anode potential. The measurements converter showing the useful current to were made with a high-workthe anode and the undesirable current to function anode and thus, if this the accelerating electrode, both as a funcelectrode had been coated with a tion of the voltage applied to the anode.
1
THERMIONIC ENERGY CONVERSION
169
low-work-function material, the zero on the anode voltage axis of this graph would be displaced 2 to 3 volts to the right. I t is clear that even at this voltage, most of the cathode current is collected by the accelerating electrode. Peters has found that much of this difficulty results from the occurrence of a virtual-cathode in front of the anode surface. It may be argued that the symmetry of the electron path toward the accelerating electrode and away again indicates that a virtual-cathode cannot occur and this would be true if space charge did not change the potential distribution in the path. It must be remembered, however, that any scattering or interaction of electrons in the beam can lead to an abnormally high space charge in front of the anode in two ways. First, some electrons that are deflected from their motion toward the anode will not reach it and will contribute twice to the space charge near the anode as they arrive and then leave. Second, any low velocity electrons resulting from energy exchange processes will contribute disproportionately to the space charge because such electrons spend a longer-than-average time in the anode region. Since the virtual-cathode occurs so frequently, it appears that processes such as these are common and if a successful converter of this sort is to be produced, such processes must be minimized.
V. CESIUM THERMIONIC ENERGY CONVERTERS
A. Introduction As mentioned previously cesium vapor is often introduced into thermionic converters in order to form the ions needed for neutralization of electron space charge and to lower the anode work function. I n addition, as Wilson (22) has demonstrated, Cs vapor may be used to modify dC. T h e Cs pressure is usually determined by the temperature (hereafter called T,,) of an appendage containing liquid cesium, Taylor and Langmuir (96) have made the most comprehensive study of the vapor pressure of Cs, their result for liquid Cs being: log,,P(mm) = 11.0531 - 1.35 log,,Tc,
-
4041/Tc,.
(21)
Shelton, Wuerker, and Sellen (97) have recently essentially repeated the Taylor-Langmuir experiment in the T,, range 370-415"K, their results agreeing with Eq. (21) to within a few per cent. Figure 25 is a plot of various data pertinent to Cs converters, taken, in part, from a plot prepared by W.B. Nottingham. All of the data
170
J. M. HOUSTON AND H. F. WEBSTER
assume a gas density equal to that in the Cs reservoir. I n using the data one must therefore make a small correction (i.e., increase mean free paths) for the fact that the Cs density N a in the interelect.rode space is less than the density N,, in the Cs reservoir. When the neutral atom mean C s RESERVOIR T E M P .
TCS
[
L
r
l
1
I
k
I
1i-5
1
,
I
I
,
Ib
I 8 1 1 1 1 1
lbll
1
1
I
lb
I
I
500
!b-2
I
1
I
I
I
I
1
Ib 3
I
1
I
I
Ib2
I
lot4
I
I I1
10
I
I
1
1
1
1
1
1
I
I
I
iols
I
,
I
1
600
I
I
IIII
I
I l I I 1 L
1bl7
1016
I
N C s : Cs D E N S I T Y ( A T O M S / CM3 I
550
l ' i ' l ' ' ' ' ~ ' ' ' ' ' t l " I lo-' I 10
1
C s .PRESSURE ( M M H g
L L .
,
.I
I
10-
4
Pc,: t
I
1
O K
450
400
350
1
I
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
l
lo-'
10-2
10-1
NEUTRPL- NEUTRAL M E A N FREE PATH- CM I
1
I
I
I
I
4
:
I
I
I
I
I
l
Ib-
I
l
I
I
1
1
1
1
1
-
ATOM ARRIVAL RATE
1
1
I
I
I
1
I
Ib-2
I
1
1
1
1
1
110'2
lbl8
1bl7
1;-3
1;-4
I
I
lbl6
,
I
1
1
1
1
1
1
1
Ib-3
ELECTRON-NEUTRAL M E A N FREE PATH-CM I I I I I I I I I I I I I I 1 1 1 1 1
I
pa I
1
10
lb15
I
1
I
1
Ib2
1 1 1 1 1 1 1 1 1 1 1
It21
lb20
1bl9
A T O M S / C M e SEC. I
I
I I
1-;
I
I
1
1
1
1
1
1
1
1
I
I
I
10
1
1
1
1
1
1
1
1
1
Ib2
ION C U R R E N T ( I O O O / ~ I O N I Z A T I O N ) - A M P / C M ~
I
I
I
1-;1
I
I
I
I
I
I
I
10
I
I
I
I I I I IbZ
l I I l I l I I l l l 1 1 1 1 1 1 1 1
1b3
10'
492 X ION CURRENT - A M P / C M 2
FIG. 25. Plot of physical quantities pertinent to Cs-filled converters. All the data assume a Cs gas density equal to that in the Cs reservoir. For precision, one must therefore correct (e.g., increase mean free paths) for the fact that the gas density in the interelectrode gap is less than in the Cs reservoir. This correction involves either the Knudsen law or ideal gas law, as discussed in the text.
free path is greater than the cathode-anode spacing, the Knudsen law where T g is the average inter-electrode applies and N a = Ncs(Tc,/Tg)1/2 gas temperature. When the atom mean free path is less than the spacing, the ideal gas law' holds and Na = Nc,Tc,/Tg. I n the Knudsen regime, the random current of atoms pa striking a surface is constant throughout the tube, while in the ideal-gas-law regime pressure is constant. For typical converters with spacings of 0.1 to 1 mm, one sees from Fig. 25 that the transition from the Knudsen to the ideal-gas regime occurs at a Cs pressure of about 1 mm. In calculating the neutral-neutral mean free
THERMIONIC ENERGY CONVERSION
171
path, an atomic radius of 2.6 A was assumed. In calculating the electron mean free path, a collision probability of Pc = 1400 collisions/cm (at 1 mm pressure and OOC) was assumed, this being an “average” of Brode’s data (98) for Cs at electron energies of approximately 1 volt.
B. Low-Pressure Converters By definition “low pressure” operation occurs when the Cs pressure is small enough so that the average electron makes no collisions or energy exchanges in traveling from cathode to anode. The electron energy is not randomized as it passes through the plasma in this mode of operation, and is, in fact, determined solely by the initial electron velocity and the electrostatic potential in the interelectrode space. If one considers only electron-neutral collisions (i.e., neglects electron-ion collisions or other randomizing effects such as electron-plasma interactions), Fig. 25 indicates that low pressure operation occurs for T,, 5 400°K for an electrode spacing equal to 1 mm. In a low pressure converter the principal source of Cs ions is resonance ionization at the hot cathode surface. When a neutral atom (or ion) strikes a surface, it remains there in thermal equilibrium for a short time, and then re-evaporates as an ion or a neutral atom. T h e probability ,9 that the atom will leave the surface as an ion is given by the following
1
SURLACE IONIZATION OFCESIUM I
dl-ll I+
“sc
$IVOLT
FIG.26. Surface ionization efficiency of Cs vs 4 and T of surface.
172
J. M. HOUSTON AND H. F. WEBSTER
well-known expression derivable (99, 100) from the Langmuir-Saha
Here I is the first ionization potential of the gas atom and wa/wp is the ratio of statistical weights equal, in theory, to 2 for alkali metals. Figure 26 is a plot of Eq. (22) for Cs where I = 3.89 volt. Note that @ approaches “one” for #J > 3.89 volt, and falls rapidly as 4 is reduced below 3.89 volt. As Auer (100) has shown, the ion current density Jp emitted from a surface equals: Js = e d ? / ( 1 - Pf) (23) where p a is the neutral atom current striking the surface (given by P(2nMkTg)-1/2from gas kinetics) and f is the fraction of the emitted ions returned to the cathode for any reason (e.g., a retarding potential barrier). The /If term is, in general, only important with high-4 cathodes where /I approaches one. Zandberg and Ionov (101) give a comprehensive review of surface ionization theory and experiment. Auer (100) and Eichenbaum and Hernqvist (102) have calculated the potential distribution in a collision-less diode when both electrons and ions are being emitted. I n general, the potential distributions look qualitatively like those in Fig. 27, shown for the case where
FIG. 27. Theoretical potential distributions in the low-pressure Cs themionic converter, drawn for the case where V = - (4c -4&), Curves 1 and 2 represent electron-rich potential distributions, while 4 and 5 represent ion-rich distributions. At higher plasma densities where the sheath width becomes small compared to the gap, a constant-potential region develops as in curve 6.
V = - (#Jc - #Ja). As the cathode ion-to-electron emission ratio, Jp/Jcs, increases, the potentials change from those similar to 1 and 2, characterized by excess electronic charge, to those similar to 4 and 5,
THERMIONIC ENERGY CONVERSION
173
characterized by excess ionic charge. Note that the cathode saturated emission reaches the anode when the potential is ion-rich, but not when the potential is electron-rich. The boundary between the two types of distributions occurs when Jp/Jcs
=
(4w2
(24)
where m and M are the electron and ion masses respectively. For Cs this ratio is J p / J c s = 1/492. This relation, first given by Langmuir (99), can be readily derived by requiring that electron and ion densities be equal just outside the cathode, and assuming that both particles are emitted with thermal velocities. One might expect that a “flat” potential such as curve 3 in Fig. 27 would result when Jp was just sufficient to neutralize Jcs. However, such a potential is unstable in the absence of collisions. Auer’s analysis (ZOO) indicates that, in the transition region between electron-rich and ionrich potentials, bistable solutions are possible in which the potential oscillates vs distance. Eichenbaum and Hernqvist (202) have analyzed the case in which electrons and ions are emitted symmetrically from both electrodes. They found that the potential had two stable distributions when Jp was within approximately f 20% of JCs/492, one being electron-rich and the other ion-rich. Both potential hills had a magnitude of a few kT. T h e resulting potential depended on whether one approached neutrality from an .electron-rich or ion-rich condition, i.e., hysteresis effects and abrupt potential transition occur as one varies Jp/Jc8. Eichenbaum and Hernqvist describe experiments which are in good qualitative agreement with theory. As the plasma density is increased, the potentials no longer exhibit the continuous curvature of curves 1 to 5 in Fig. 27. Instead, an equipotential (except for I R drop) region of electrically-neutral plasma develops with sheathes at either electrode, as illustrated by curve 6 of Fig. 27. The sheath is the order of the Debye length in thickness, the Debye length (10.3) equalling s = 6.90
(25)
where s is the Debye length (cm), n is the plasma density (cm-a), and T is the plasma temperature. For a typical thermionic converter with n = lo1* ~ m and - ~ T = 2000”K, the Debye length is 0.3p, a distance much shorter than any practical cathode-anode gap. Note that the sheathes can be either retarding or accelerating for cathode electrons, i.e., be either electron-rich or ion-rich depending on the ratio of ion to electron emission as given by. Eq. (24). If, for example, the cathode saturated electron emission is greater than 492 Jp, then an electron-rich
J. M. HOUSTON
174
AND
H. F. WEBSTER
sheath develops with a magnitude such that the electron current J c able to overcome the retarding sheath and reach the anode is limited to approximately 492 Jp. A similar limitation of ion current occurs if Jcs < 492 Jp, the ion current to the negative anode being limited to approximately Jcs/492. The above statements about space-chargecompensated ion or electron currents are only correct, of course, when the electron or ion current to the anode are large compared to the spacecharge-limited currents which would flow in the absence of the opposite particle. This is the case in any practical Cs thermionic converter. Equation (24) has been tested several times in experimental Cs converters, both with high-$ cathodes where M 1, and low-$ cathodes 1. Marchuk (20) heated a tungsten filament to high temwhere peratures in a low pressure of Cs. He found that as Tc was increased, J c began to fall below Jcs at approximately J c / J p = 200, where Jp was calculated from the measured T , value. This result is in approximate agreement with Eq. (24) if one notes that the electron emission from the lowest-# plane of clean tungsten (32) is roughly 2.5 times the average emission from tungsten. The low-$ patches become ion-limited first as Tc is increased, leading to the result observed by Marchuk. When Tc was increased sufficiently so that the electron emission from nearly all the filament area was ion-limited (i.e., such that Jcs 2 5Jc), then the anode current Jc was found to equal 430 J p , which is in satisfactory agreement with Eq. (24) within the precision of the experiment, Houston and Gibbons (204) found that when a W filament was gradually cooled with T,, constant, the anode current at first rose (due to Cs adsorption on the filament) and then fell abruptly very near the temperature where one calculates [from Eqs. (22) and (23)] that
<
Jp =
0.002 J C 6 .
Ranken and Teatum (10.5) measured both the ion emission and the short-circuit electron emission from a planar UC-ZrC cathode opposite a planar guard-ring-equipped anode. Their data (Fig. 28) is in good agreement with Eq. (24). At low Cs pressures (curves 1 and 2) the anode current is limited by insufficient ions at high values of Tc. At Tc = 2200°K and P,, = 6.3 x mm, for example, they measured an ion emission (by operating their anode 5 to 10 volts negative) of 1.5 x amp/cm2 and an ion-limited electron current (Fig. 28) of Jc = 0.65 amp/cma, yielding Jc/ Jp = 430,which is close to the theoretical values of 492. Their ion currents yield [from Eqs. (22) and (23)] a $,-value of 3.6 volt, which is slightly above the effective #-values calculated [Eq. (l)] from the electron emission measured at higher Cs pressures where J c is no longer ion limited. For example, at Tc= 2200°K curve 4 in Fig. 28 yields $, = 3.5 volt and curve 6
THERMIONIC ENERGY CONVERSION
175
yields+, = 3.3 volt. This decrease in the effective4 at higher Cs pressures is probably due to the accelerating electric field which is present at the cathode surface at higher Cs pressures because of the ion-rich sheath. As is well known, the anomalous Schottky effect (9) predicts that a relatively
FIG. 28. Data of W. A. Ranken and E. T. Teatum [Bull. Am. Phys. SOC.[2] 6, 371 (1961)] showing anode short-circuit current density vs. temperature of (LJC),,, (ZrC)”., cathode at various Cs pressures. Cell geometry was parallel plane with 2 cm2 cathode area, 0.19 cmZ anode area, and 3 mm cathode-anode gap. The anode was surrounded by a large, annular guard ring.
small electric field can cause the emission of a patchy cathode to increase considerably above the zero-field emission. T h e decrease in current seen for curve 7 is probably due to some effect (e.g., plasma resistance) which causes the short-circuit current to no longer equal the saturated emission. T h e dashed-line “jump” in curve 3 is probably connected with the abrupt onset of volume ionization and will be discussed later. One concludes that when volume ionization is negligible, Eqs. (22), (23), and (24) seem to correctly predict Cs ion emission and electron space charge compensation, at least to within a factor of two. When ample Cs ions are present, a low-pressure converter is in many respects identical to the ideal .converter discussed in Section 11. Lowpressure Cs converters using clean refractory-metal cathodes have been
TABLE 111. THERMIONIC CONVERTERS USING Cs VAPOR^ V
P e
Reference W W Ta Mo
Ta Ni
cu cu
Mo
Mo W Ta Mo Mo
$0
Ni cu
-
Mo Mo cu Mo Mo Mo Mo Mo
Mo Ta W Ta Mo Mo Mo
Ni cu
10. 1.2 10. 1.o
2. 0.8 0.12 0.1 0.2 0.25 0.25 0.15 0.040 0.046 0.25 0.25 0.15
1.o 1.o
“L”
-
Philips Philips
Mo Ni
2. 2.5 2.5
ThC, Th-W ThC,
Ni kovar
0.75
-
1.o
-
Clean Refractory Metal Cathodes 396 2.4 2500 2910 -420 2.5 570 526 0.7 2900 -470 1.64 2525 Cesiated Refractory Metal Cathodes 1900 564 0.78 1900 900 590 2100 1.33 590 2710 650 623 1.o 1920 1030 573 1600 980 573 0.32 0.6 1800 1110 573 2000 960 640 0.85 1.0 2060 900 643 0.85 2000 900 638 0.3 942 565 1600 0.43 990 590 1770 935 605 1770 0.45 UC Cathodes -770 -550 2200 1.1 -650 -623 2070 1.5 Barium Cathodes 1610 440 0.7 1550 483 0.9 1370 0.6 Tholium Cathodes 2600 550 1.9 500 473 2160 1.26 523 1.1 2100
-
~~
-
1.o 19. 10. 1.8
0.7a 10.4a 5. b 2.7a
3.1 7.5 30. 8.2 2.5
9.2b 12.0a 15. b 15. b 16.3a 11. c 15.5c 13.6a -18. b
5.
6.3 14.7 9.0 2.5 4.7 6.7
14. b
-
aten Wilcnn
rnm
a
Marchuk (20) Hernqvist (21) Grover (23a) Block (106) Wilson . . .._..122) ~
Rasor (i67j Wilson (108) Ranken -(109) Rasor (24) Baum (110) Baum (110) Hatsopoulos (111) Hatsopoulos (93) Hatsopoulos (93) Hatsopoulos (93) Hatsopoulos (93) Hatsopoulos (93)
Y
z
0
3
2 3* 3:
‘II
18. 12. 0.57 5.0 0.6 15. 4.3 16.
10. a
Reichelt (112) Howard (113)
5. a 15. c
Morgulis (114) Silverberg (115) Hernqvist (116)
15.2b 7.5a
Fox (117) Houston (118)b Morgulis (119)
-
10-15
a Usually, the data corresponding to the highest efficiency quoted by an author is listed in the table. In calculating 7. all authors use measured power output but use a variety of methods to estimate power input. These are indicated by the letter affixed to the 9 value. where “a” indicates that the total power input to the converter was used, “b” indicates that measured power input minus stray heat losses (e.g., heat that never reaches the converter cathode) was used, and “c” indicates that the power input was calculated from estimated heat losses. It should be stressed that the data listed in this table should be used with caution. The converter lifetime is often short at the listed power output, and the data is sometimes not repeatable from tube to tube. b
4
5
z!
I77
THERMIONIC ENERGY CONVERSION
tested by several authors (20, 21, 106), the performance being listed in the first part of Table 111. One of the earliest papers in this field was the work of Marchuk (20) who demonstrated the compensation of electron space charge by Cs ions and the shift of the J - V characteristic to negative anode voltages because of contact potential. T h e work of Hernqvist, Kanefsky, and Norman (21) is typical of this type of converter. They used a W ribbon spaced 1.2 mm from a concentric nickel anode. When the Cs pressure was high enough to yield ample ions, they observed a J - V characteristic similar to curve b in Fig. 4. At maximum power output the output voltage was always 2.5 volts (close to that predicted from q5c = 4.5 volt and da = 1.8 volt) while the output current equalled the emission from clean W. At 2910°K a power output of 19 watt/cm2 at an efficiency of 10.4% was observed. However, such converters are not very practical because of excessive cathode evaporation and because the operating temperature is too high for any long-lived heat source. TABLE IV. Cs PRESSURE REQUIRED FOR NEUTRALIZATION Jc =
I
I amplan*
Jc =
10 amp/cmz
CC
(volt) __
4.5 4.0 3.5 3.0 2.5 2.0
-~
~
2550 2295 2030 1765 1495 1220
8.9 x 4.7 x 5.1 x 1.4 x 1.0 x 8.5 x
10-l
10-l lo-*
10-o
2.4 4.2 3.6 1.2 1.5 1.7
x x x x x x
lo-‘ 10-l 10’
lo4
I
2840 2555 2265 1970 1675 1370
8.6 4.5 6.4 2.8 3.5 6.1
x lo-’ x lo-’ x x
x x 10-a
4.8 6.1 3.2 6.5 4.9 2.5
x x x lo-* x lo-’ x lo1 x lo4
To lower the operating temperature one must use a cathode with a lower value of dC.This, in general, requires an increase in P,, because when +c is lowered below 3.89 volts, becomes considerably less than one, and Pcs must be increased in order to maintain sufficient ion generation. One can readily calculate (104, 120) [from Eqs. ( l ) , (22), (23), and (24)] the minimum value of Pcs that is required to yield space-charge neutralization. Table IV gives this result, taken in part from the work of Rittner and Milch (120).T h e ideal gas law was assumed in relating reservoir and gap Cs density. Note that rather high (e.g., > 10 mm) Cs pressures are required for (bc less than -2.7 volt. A similar calculation can be done in which dC, Pc,, and Tc have an additional constraint in that they are related to a particular surface, e.g.,
178
J. M. HOUSTON AND H. F. WEBSTER
cesiated tungsten. Such a calculation for cesiated tungsten has been done by several authors including Langmuir (99) [his Eq. (66)], Moizhes and Pikus (121), and Houston and Gibbons (104). This calculation indicates, for example, that for J = 10 amp/cm2 and Pc, = 10 mm, the minimum temperature for neutralization is 1780°K at c$c = 2.67 volt. In practice, Cs diode converters can be operated at slightly lower temperatures than this (see Table 111) because at high Cs pressures ions can be created by volume ionization as well as surface ionization. I t is sometimes proposed that neutralized converter operation can be achieved at low values of To by using a cathode composed of patches of differing 4. The argument usually given is that abundant ion generation will occur at the high-4 patches and abundant electron emission at the low+ patches. If the patches are large, this scheme will obviously suffer from electron and ion space charge effects which occur before the two types of emission can mingle. However, even with very small patch size the argument is still not valid because, as Dobretsov ( I ) has indicated in his detailed analysis, the ions created at the high-4 patches must overcome the retarding “patch fields” before they can mingle with the copious electron emission from the low-+ patches. These patch fields reduce the “useful” ion current density from the high-4 patches to a value equal to that which would come from a uniform surface having a +value equal to the area-weighted average q5 of the patchy surface. As a result, it can be shown that, for any given T , and Pc,, a patchy surface has a smaller useful ion emission than that of a uniform surface having either the same area-weighted average I$ or the same effective 4 (same electron emission) as the patchy surface. Several qualifications must be placed on the above argument of patchy vs uniform surfaces. First, the argument is not valid if a timevarying electric field is present, especially if the field has a period comparable to the transit time of the ions through the patch fields. Second, the argument is invalid if one electrically biases the high-4 patches positive with respect to the low+ patches, as has been proposed by Hernqvist (222,123).Either of these possibilities can eliminate the effect of retarding patch fields and thus allow neutralized operation with lower-4 cathodes. Low-pressure Cs converters are often observed to oscillate at frequencies generally in the range 100 kc to 5 Mc. This phenomenon, apparently first observed by Gurtovi and Kovalenko (19), has been described by many authors including Marchuk (20), Fox and Gust (217), Eichenbaum and Hernqvist (202), Johnson (224, 125), Garvin et al. (126, 127), Zollweg and Gottlieb (228), Luke and Jamerson (129), Rocard and Paxton (230), and Koskinen (132). The oscillation is
THERMIONIC ENERGY CONVERSION
I79
usually nonsinusoidal and consists of a periodic interruption or modulation of the diode current as a result of space charge fluctuation in the interelectrode space. T h e following are some of the observed features of these oscillations: (1) T h e occurrence and period of the oscillations depends on the electron and ion emission of the cathode, and not directly on the cathode material or temperature (117, 130). (2) As the temperature of a constant-+ cathode (e.g., clean W) is raised, the oscillations commence at some critical emission level, e.g., Tc = 2000°K (128). T h e oscillations generally disappear as P,, is increased, e.g., at Pcs 3 0.008 mm (117). However, oscillations are also observed (128, 129) at high Cs pressures and low Tc, e.g., Pcs = 0.5 mm and Tc = 1200°K. (3) T h e period is comparable to the ion transit time across the diode, and varies linearly with interelectrode gap d at low Cs pressures (127, 128, 130). (4)T h e oscillations are never seen in the retarding range of the diode (20, 117, 124). ( 5 ) T h e period is usually not strongly dependent on the Cs pressure (124, 128) or the cathode saturated emission level (126, 128, 130). Various theories have been proposed to explain these oscillations. Auer (100) proposes that the oscillations may be due to the bistable nature of the class of potentials in which the potential oscillates vs distance. Johnson (125) proposes that periodic reflection of ions at the anode sheath causes the interelectrode potential to flip alternately beween electron-rich and ion-rich potentials of the same general character as those discussed by Eichenbaum and Hernqvist (102). However, this model cannot explain the fact that oscillations are seen under extremely ion-rich as well as electron-rich conditions, and are also seen when the anode voltage is held fixed (131) with respect to cathode. Zollweg and Gottlieb (128) propose that the oscillations are due to Cs ions oscillating in an electron space-charge “well.” They give a theory and experimental data which seems reasonable and self-consistent in many respects. However, calculation of ion current from their values of T,, and Tc indicates that all of their data was taken under ion-rich rather than electron-rich conditions. Thus it is difficult to see how an electron space-charge well could be present. Birdsall and Bridges (132) propose that the space charge instabilities of the type seen in vacuum diodes incorporating electron injection may be pertinent to the oscillations seen in Cs diodes. Their model could apparently have some validity
J. M. HOUSTON AND H. F. WEBSTER
180
when an ion-rich cathode sheath is present since one then can have legitimate electron injection. However, when an electron-rich sheath is present, the electrons have only thermal velocities as an “injection velocity” and it seems unlikely that this type of diode instability can then occur. It is obvious from the preceding discussion that the oscillation phenomenon is not yet completely understood. At present no theoretical model is in very good agreement with the experimental observations.
C. High-pressure Converters As the Cs pressure is increased the electron mean free path becomes much shorter than the cathode-anode spacing, i.e., electron-atom collisions become important. The plasma density is usually high enough so that sheath dimensions are small compared to the interelectrode gap [see Eq. (25)] and the majority of the gap is filled with an essentially neutral plasma. The random electron current in the plasma is usually large compared to the directed current, i.e., the electron random velocity is large compared to the drift velocity. Thus the electrons emitted from the cathode into the plasma rapidly become indistinguishable from the other plasma electrons, and the current to the anode is supplied by a portion of the random electron current in the plasma. This mode of operation is often called the “plasma” mode, and has been theoretically analyzed in papers by Lewis and Reitz (133, / 3 4 ) , Nottingham (135, 136), Moizhes and Pikus ( 1 2 4 , Carabateas et al. (137), Talaat (/38), Gottlieb and Zollweg (139),and Warner and Vernon (140). Note that the A
I
JtT-tJcs
T -
I IC
_I
V
I
- ._
I
FIG.29. Potential
I vs. distance in a high-pressure Cs diode.
THERMIONIC ENERGY CONVERSION
181
“plasma” mode does not require local thermodynamic equilibrium (i-e., electron and ion temperature can differ), and is not to be confused with the “plasma thermocouple” mode proposed by Lewis and Reitz (141). As will be later discussed, the “plasma thermocouple’’ mode only occurs at very high Cs pressures and large gaps. A good starting point in the description of the Cs plasma converter is the simple one-dimensional model first proposed by Lewis and Reitz (233,134). Most of the other above analyses assume a similar model. The potential, shown in Fig. 29, is assumed to consist of two ion-rich electrode sheathes of amplitude Vc and Va, connected by a neutral plasma of density n. A resistive voltage drop Vr may be present in the plasma. The current Jl,flowing from the plasma to the cathode is equal to the random plasma electron current reduced by the appropriate Boltzmann factor, i.e., J1 =
en (KT1/2nm)1/2 exp -(eVc/kTl)
(26)
where TI is the electron temperature in the plasma adjacent to the cathode. Similarly, at the anode:
Jz = en ( / Z T J ~ ~exp ) ’ -(eV,/kT,) /~
(27)
Continuity of current (neglecting ion currents and anode electron emission) requires: (28) Jcs = 1 1 Jz
+
Summing voltages around the circuit yields: Vc - Va
=
V
+
4 c - $8
- Vr
(29)
By making various assumptions about the plasma resistance, electron temperature, and electron energy distribution, various forms of J - V curves (237, 239, 240) can be calculated from Eqs. (26) through (29). Carabateas et al. (237) made the first such analysis. They assumed that the electron gas in the plasma was isothermal [i.e., TI = T, = T,], and that Te was independent of the converter voltage V. They found that if they assumed negligible plasma resistance, an excellent fit occurred beween their theory and experimental data, yielding electron temperatures in the range 3000 to 9000°K. This excellent agreement is somewhat surprising because, as will be later discussed, in a one-dimensional converter the electron temperature should vary considerably with V, and cannot be greater than Tc in the retarding range, i.e., when V 5 - (+c - $8). However, as will be pointed out later, it is possible to have an electron temperature greater than Tc in the retarding range if a
182
J. M. HOUSTON A N D H . F. WEBSTER
certain type of spurious arc is present. Also, the assumption of negligible plasma resistance is certainly not valid for many Cs converters. Thus, when using this approach one must not put too much faith in the electron temperatures which result, since effects such as plasma resistance or patchy electrode surfaces can cause high apparent values of electron temperature. For example, the 9000°K values reported by Carabateas et al. would seem to be energetically impossible near V s = 0 [see Eq.
(3211.
By considering the power flow into the electron gas (not the entire plasma) in the interelectrode region, an upper limit for the electron temperature can be shown (142, 243) to exist which is a function of the internal voltage V8 defined by Eq. (2a). T h e potential model of Fig. 29 is again assumed. Note that Fig. 29 is drawn with V s having a positive value, while V , as usual, is negative. Define a “control volume” with planar boundaries “A” and “B” located at the top of the motive “hill” where the mirror-image motive joins the sheath potential. T h e power flow into and out of the electron gas is given by: 2KTc Jcs
+ JzVs
= 2KTl J1
+ 2KTzJz + Pc
(30)
where K = k/e. T h e power input terms to the left consist of the cathode thermal energy. (2kTc per electron) and the potential energy gained due to the fact that the anode barrier is lower (more positive) than the cathode barrier. When V s 0, this potential energy term is, of course, omitted from Eq. (30). The power output terms on the right consist of the random energy carried out of the electron gas by J1 and Jz, and a term, Pc, representing the net energy lost by the electron gas in collisions (elastic, exciting, or ionizing) with ions or neutral atoms. Note that Pc is always positive because in elastic collisions the atom temperature is always less than or equal to the electron temperature, while in inelastic collisions there is no way that the electron gas can regain much of the energy lost to the atoms, i.e., most of this energy is delivered to the electrodes either by radiation or by ion recombination. For simplicity, assume an isothermal electron gas, i.e., TI = T, = Te. I n the retarding range (i.e., where Vs 0) Eqs. (28) and (30)yield
<
<
2KTcIm
= 2KTe Jcs
+ pc
It is obvious that this equation can only be satisfied with Te
(31)
< Tc.
Even if one does not assume an isothermal electron gas, it can still be shown (143) that, in the retarding range, the electron temperature in the interelectrode gap can nowhere be greater than Tc. This results from the fact that if the electrons develop a temperature larger than Tc at some
THERMIONIC ENERGY CONVERSION
183
point, this random energy must either be delivered to the electrodes or lost from the electron gas by collisions, and the 2kT, (per electron) energy input is too small to maintain such a high electron t e m p e r a t ~ r e . ~ When V s > 0, the electron temperature can rise as a result of converting potential energy into random energy. An upper limit on the electron temperature can be calculated from Eq. (30) by assuming that Pc = 0 and using the lowest conceivable value of TI under this condition, i.e., T I = Tc. Equation (30) then reduces to TZ
=
TC
+ Ir,/2K
==
Tc
+ 5800 Vs.
(32)
Thus, for example, V , = 0.5 volt results in a maximum possible electron temperature of ( T , 2900)”K at the anode. As mentioned earlier, this is also the maximum electron temperature possible anywhere in the gap. Equation (32) thus predicts that moderately high electron temperatures can occur in Cs thermionic converters if a portion of .the output voltage is sacrificed. Johnson (142) was perhaps the first to point out the significance of Eq. (32) in his paper on low-voltage arcs. He also discussed the mechanisms by which the directed velocity of the injected electrons can rapidly become randomized, e.g., plasma oscillations (144). Langmuir (145) discovered that the scattering of an electron beam in a plasma is much greater than would be predicted from a simple collision picture. Gabor, Ash, and Dracott (146) have shown that this is due to plasma oscillations. Nottingham (136) and Lewis (147) have pointed out the pertinence of this mechanism to the Cs converter, the ion-rich cathode sheath furnishing the electron injection. Note, however, that one should not, in general, assume that the gain in thermal energy is equal to the entire cathode sheath potential, a mistake which is often made. That this idea is incorrect can easily be seen by considering a constant-temperature hohlraum containing C s vapor. Here electron emission in the presence of a large ion-rich sheath can readily exist, yet it is obvious from thermodynamic considerations that no electron heating can occur, in agreement with Eq. (32). As a result of the high electron temperature, copious volume ionization can occur in the Cs plasma, presumably caused by electrons in the tail of the Maxwellian energy distribution (142, 136, 147). Gas discharges of this sort are called “anomalous low-voltage arcs” or “hot-cathode arcs” and have been studied for many years (18, 142, 148-152). T h e ionization
+
This statement assumes that thermoelectric voltages in the plasma can be neglected. This is believed to be a good assumption far existing Cs converters, as will be discussed later.
184
J. M. HOUSTON A N D H. F. WEBSTER
mechanism in such discharges in Cs vapor is not completely understood. However, as Steinberg (150) has indicated, there are strong indications that it is a cumulative rather than a one-step ionization process. Probe measurements of Cs discharges (150, 152) indicate electron temperatures in the 2500-3500°K range. At these temperatures there are far too few 3.9-volt electrons in the tail of the electron energy distribution to explain the observed ionization. Steinberg suggests that the first exited state (1.45 volt) of Cs is involved in the cumulative process since trapping of the resonance radiation (8521 A) by the Cs vapor can greatly extend the effective lifetime of the excited state beyond its natural lifetime of about 3 x lo-* seconds. Another indication that a cumulative process is involved is the observation (104) that as Jcs is increased in the range 0.03 to 3 amp/cm2, the value of V , necessary to obtain an arc drops sharply. In a cumulative process the number of ions created varies as a power of the plasma density (e.g., as n2 for a two-step process) and, under ion-rich conditions, the plasma density is determined primarily by the cathode electron emission. Thus, when n is large, lower values of V , suffice to produce abundant ionization. Some of the ions undoubtedly recombine while in the plasma. However, at the spacings and Cs pressures used in typical Cs converters, many of the ions drift to the cathode under the influence of the small electric field in the plasma resulting from IR drop. T h e magnitude of this ion current can be obtained from measurements of the cathode heat balance (152, 104) provided that an estimate of the cathode sheath height can be made. T h e ion current arriving at the cathode increases as V , increases, and can be the order of 0.4 amp/cm2(104) in a Cs converter near short-circuit. As Found (153) has discussed and experimentally demonstrated, the arriving ions accentuate the electric field at the cathode and increase its electron emission, especially if the cathode is patchy (the anomalous Schottky effect). With cesiated cathodes this effect can be particularly pronounced. For instance, Taylor and Langmuir (43) found that an anode voltage of 6 volts (corresponding to roughly 350 volt/cm at the cathode surface) increased the emission from Cs-W as much as a factor of three over the zero-field emission. I n a high-pressure Cs converter the electric field at the cathode can be much higher than this due to ion arrival. For example, Found’s Eq. (13) predicts E = 6100 volt/cm for V , = 1 volt and an ion current (arriving from the plasma) (153) which is only 0. I amp/cm2 in excess of Jcs/492. As predicted by Morgulis and Marchuk (254), anomalous low-voltage arcs are often observed (22, 108, 109) in high-pressure Cs converters. Figures 30 and 31 give typical I - V characteristics showing the
I85
THERMIONIC ENERGY CONVERSION
transition to the low-voltage arc. Figure 30 was measured (104) on a cylindrical diode using an indirectly-heated T a cathode spaced 0.8 m m from a stainless steel anode. Below about 1700°K insufficient surface ionization to neutralize J c s exists (prior to the arc), and the diode current
I AMP
-4
I
1
-3
-2
I
v
-I
1 0
VOLT
FIG. 30. Current-voltage characteristics measured on a cylindrical diode with an indirectly-heated 0.250-inch diameter, tantalum cathode spaced 0.032-inch from a stainless-steel anode. The data was taken with a mechanical x-y recorder with a voltage sweep of about 1 volt/sec.
is therefore limited by the available ions. Carabateas (155) has recently described a theory which correctly predicts the magnitude of the ionlimited current observed prior to the arc. As the anode voltage is swept slightly positive, a discontinuous jump into a low-voltage arc occurs, the diode current abruptly rising toward J c 8 once volume ionization commences. Although the details of this transition into the arc mode are still obscure, it is obvious that an abrupt transition from an electronTc in the plasma) to an ion-rich sheath rich cathode sheath ( T e ( Te Tc ) is involved. Above approximately 1700°K sufficient surface ionization occurs and hysteresis effects or discontinuous jumps in current are no longer observed. This behavior is not surprising because when Vc is sufficiently large (due to abundant surface ionization), a continuous,
>>
<
I
186
J. M. HOUSTON AND H. F. WEBSTER
smooth increase in Te should occur as V , is increased, i.e., the transition to the arc is a gradual, continuous phenomenon. Figure 31 gives I - V data from experiments of Ranken, Grover, and Salmi (109) showing both high and low-pressure operation. Again
FIG.31. Current-voltage characteristics of a planar Cs diode with a 2 cm* Ta cathode spaced 2.5 mm from a serrated copper anode. Data is from W. A. Ranken, G. M. Grover, and E. W. Salmi, J. Appl. Phys. 31, 2140 (1960).
abrupt increases in current are seen at high Pcs and low To.At high Cs pressures this data is complicated by the fact that for V near zero or positive, emission from intermediate-temperature areas of a large Cs-covered cathode support disk could also reach the anode. As Figs. 10 and 12 indicate, the electron emission density from this transition region can be considerably higher than that from the cathode. This stray emission exists in most experimental Cs converters although an attempt was made to minimize it in the tube yielding Fig. 30 by making the intermediate-temperature area small (15 % of &). This stray emission, as well as the anomalous Schottky effect, is apparently what causes the current in Figs. 30 and 31 to continuously rise (at high Cs pressures) as V is made more positive.
THERMIONIC ENERGY CONVERSION
187
In the preceding discussion it was pointed out that in a one-dimensional Cs converter the electron temperature in the retarding range can nowhere be above T,. Yet many experiments indicate otherwise. Ranken and Teatum (105) describe current measurements in a diode with a planar UC-ZrC cathode spaced 3 mm from a planar guard ring anode. T h e ion current was measured by using the anode as a Langmuir probe, i.e., the anode was held more negative than open circuit. As Pc, was increased at Tc = 2200”K, a discontinuous order-of-magnitude jump in the ion current was observed at P,, = 0.002 mm. From the slope of log J vs V plots in the retarding range, electron temperatures of 6700°K were deduced, whereas before the “jump” the slope indicated essentially cathode temperature. This phenomenon is probably due to a low-voltage arc occurring between the cathode and the cool regions of the cathode structure adjacent and electrically short-circuited to the cathode. There is generally ample work-function difference between the Cs-covered cooler areas and +c to produce such an arc. Lewis and Reitz (133) have predicted that such internally-circulating currents would be large. Spurious arcs of this sort have been observed at open circuit by Gibbons (156) and by Wilson and Lawrence (157) in experimental Cs diodes. T h e observations were made both visually and by using a portion of the cathode support as a current shunt to detect the internallycirculating current. Agnew (258) has determined the electron temperature Te and the ion density from spectroscopic measurements of the radiative recombination continuum in a planar Cs diode, His measurements are shown in Fig. 32. He also determined a Boltzmann temperature, TB, from the relative intensities of two optically-thin spectral lines. At high Cs pressures (e.g., I mm) Te % TB,and the ion density was just that predicted by the Langmuir-Saha equation. However, as Pcs was lowered, Te rose while TBfell. Agnew suggests that at low Cs pressures, TBdoes not represent the true electron temperature because line intensities were anomalously high. T h e fact that T e > T , at open circuit suggests that a spurious arc of the type just described was present during Agnew’s measurements. This could also explain the decrease in plasma density seen at open circuit by Agnew as P,, was increased. Because of increasing plasma resistance as Pcs was increased, the spurious arc would diminish in intensity and flow preferentially from the cathode edges, leaving a reduced plasma density at the center of the diode. At short-circuit, however, the arc would flow directly to the nearby anode, so no decrease would be observed. This explanation may also apply to the peak in the ion current at Pcs = 0.1 mm seen by Ranken et al. (109). Agnew’s measurements illustrate the power of spectroscopic techniques, i.e., they yield direct
J. M. HOUSTON AND H. F. WEBSTER
188
measurements of electron temperature and plasma density without perturbing the plasma. Such spurious arcs have several implications. They are undesirable in one respect because they drain heat from the cathode and lower I
I
I
8
I
IIII
!
!
I I 1 Ill
1
t x SHORT CIRCUIT
Y
c
2400
2000 1600
FIG. 32. Spectroscopic measurements of L. Agnew [Bull. Am. Phys. SOC.[2] 6, 343 (1961)l taken at the center of a planar diode consisting of a tantalum cathode at 2470°K
spaced 6.3 mm from a copper anode. The electron temperature Taand the ion density were determined from the radiative recombination continuum. The Boltzmann temperature TB was obtained from the relative amplitudes of two optically-thin spectral lines.
efficiency. However, they create ions which can be useful, especially at low values of T,. When such arcs are present, it obviously becomes impossible to interpret Cs diode measurements with a one-dimensional model. However, spurious arcs should diminish in importance at high Cs pressures and close spacings. T h e resistivity of a Cs plasma is obviously of importance to a highpressure Cs diode. Several authors, including Mohler (159) and Lewis and Reitz (141), discuss this subject and show that at low fractional ionizations electron-neutral collisions determine the resistivity, while for fractional ionizations above roughly 0.02, collisions with ions dominate the resistivity. Mohler (159) found that the resistivity of a Cs
189
THERMIONIC ENERGY CONVERSION
plasma is determined mainly by the electron temperature, varying from roughly 0.3 ohm cm at Te = 3500°K t o 0.1 ohm-cm at Te = 6000°K. 20
~
~
~~
MOLYBDENUM EMITTER COLLECTOR TEMP. 6 CESIUM PRESSURE OPTIMUM
18
16
14 N
5
-
\
u)
c
g
12
I
> k cn
z W a
IC
W
Bn 5
-5
0
X
a
I 6
4
2
0
15
30
45
60
75
I
D
SWCING ( 1 0 - ~ i n
FIG. 33. Data of R. L. Hirsch [J. AppI. Phys. 31, 2064 (1960)l showing the effect of spacing on the maximum power output of a high-pressure Cs thermionic converter. At each value of spacing and T,,both Tcsand T, were optimized.
1 90
J. M. HOUSTON AND H. F. WEBSTER
Roehling (260) has measured the resistivity of an isothermal plasma between two hot, planar electrodes at the same temperature and potential. His preliminary results measured in the range Tc, = 523 to 613°K indicate a resistivity of approximately 5000 ohm cm at T = 1000"K, 50 ohm cm at 1400"K, and 2 ohm cm at 1800°K. These results indicate that voltage drop in the plasma is a serious effect at high Cs pressures unless one either uses close spacings or else somehow raises the electron temperture, e.g., by sacrificing some output voltage so that V , > 0. Hirsch (161) has measured the effect of spacing on the power output of a Cs converter using Mo electrodes. Two sets of his data are shown in Fig. 33. The data for T , 3 1650°C is early data taken under slightly gassy conditions. Recent data taken under cleaner conditions shows qualitatively the same behavior but somewhat lower power output because one no longer obtains the beneficial increase in electron emission because of the foreign gas. The increase in power output as spacing is reduced is presumably because of decreased plasma resistance. As the spacing is reduced, the optimum Pcpgenerally increases (See Table 111), resulting in more cathode emission. Also shown in Fig. 33 is some recent data taken under clean conditions which shows that at low values of Tc an optimum spacing exists. That such an optimum exists was first reported by Jensen (262,110) who found that with planar Mo electrodes at Tc = 1600"K, Ta= 970"K, and Tcs = 573"K,a power output of 2.5 watt/cm2 was obtained at the optimum spacing of 0.23 mm, and less than 1 watt/cm2 at spacings of 0.18 to 0.01 mm. As Jensen indicated, this effect is probably caused by the fact that at low values of Tc (where surface ionization is insufficient) too few ions are produced by the lowvoltage arc if the spacing is made too small. Jensen also suggests that photons from the cathode aid in populating the first excited state in the Cs vapor. I n an early paper, Lewis and Reitz (141) propose that the output voltage of a high-pressure Cs converter results from the thermoelectric power of the plasma. However, as has been discussed by several authors, e.g., Dobretsov (2) and Moizhes and Pikus (124, this approach is only valid when the plasma approaches local thermodynamic equilibrium, i.e., when the mean free path for relaxation of electron-ion temperature differences is much smaller than the cathode-anode gap. This is true only for large gaps or very high Cs pressures. For example, at Tcs = 573°K (a typical operating point) the electron mean free path is cm (Fig. 25) and the relaxation length is approximately about 8 x (M/2rn)'lZtimes larger, i.e., 0.3 cm. The cathode-anode gap would thus have to be large compared to 0.3 cm for the thermocouple approach to be valid, a situation of no practical interest. Thus the name "plasma
THERMIONIC ENERGY CONVERSION
191
thermocouple” which is sometimes applied to existing Cs plasma diodes, would seem to be a misnomer. Cesium thermionic converters can also be classified by cathode type as well as by the Cs pressure range. As described in Section 111, cathodes fall into two types, the essentially constant-+ variety and the cesiated refractory metals. Table 111 gives a fairly complete listing of published Cs converter results, classified by cathode type. In the remainder of this section several typical high-pressure Cs converters will be described. Figure 34 shows a Cs converter built by Wilson (108) [see Table 1111 I4)CATHoDE LEAD
/
(31CERAMC SEAL
(I 1CATHODE
1 ANWE
LEAD
FIG. 34. Cs converter built by Wilson suitable for heating by focused solar heat in space.
with a geometry suitable for heating by focused solar energy in space. The device consists of an I 1.4 cm2 tungsten disk spaced 0.13 mm from a planar nickel anode. A blackened copper radiator was attached to the anode, A Cs pressure of about 3 mm was used so as to keep the cathode partially Cs-covered. When heated by electron bombardment in a vacuum bell-jar, the device yielded a power output of 7.5 watt/cma (5.6 amp/cm2 at 1.33 volt) at an efficiency of 15 ”/o. However, the life of the device was short because the cathode foil developed a leak. Rasor has tested cylindrical Cs converters with Mo electrodes, the cathode being indirectly heated by electron bombardment. An early 10 cma device (see Table 111) yielded 8.2 watt/cma at Tc = 1920°K and measured overall efficiency of 16.3 yo.This high performance is believed to be due to the beneficial action of trace gases (possibly water vapor) which cause increased cathode electron emission and decreased anode
1000
900
L
a oz
k! 700 I W
t-
600
500
I500
1600
I700
1800
1900
CATHODE TEMPERATURE
2000
2100
(OK)
FIG.35. Data of N. S. Rasor (private communication) taken on a Cs diode consisting of a cylindrical, 5 cm*, Mo cathode spaced 0.8 mm from a concentric stainless-steel anode. The cathode was indirectly heated by electron bombardment. At each value of Tc, both Ts and Tcs were optimized. The efficiency was determined by dividing the measured electrical power output by the total filament and bombing power input to the cathode.
THERMIONIC ENERGY CONVERSION
I93
work function as discussed in Section 111, B. However, these gases also caused chemical cycles which corroded the cathode and caused short life. Figure 35 gives some of Rasor’s more recent data (163) taken under cleaner conditions with no corrosion. At the Los Alamos Scientific Laboratory a number of experimental Cs converters have been tested which used UC-ZrC cathodes heated by uranium fission in a nuclear reactor. Reichelt et al. (112) report that a typical diode using a cylindrical slug of UC,., ZrC,,, as the cathode ( A c = 4.7 cm2, d = 1 mm) produced a maximum short-circuit current of 130 amp, a maximum open-circuit voltage of 4.2 volt, and a maximum power output of 85 watt at 1.1 volt. Short circuit currents vs Tc (determined pyrometrically) agreed well with the data of Ranken and Teatum (Fig. 28). In a lifetest the initial power output of 40 watts fell to 20 watts after 250 hours, at which point the cathode fractured. One unknown property of such diodes is how they will behave at high anode temperatures. This will depend on the adsorption properties of Cs on UC, since the anode quickly becomes coated with a U C layer. When Cs thermionic converters were first being developed, converter life was often short because of a variety of non-fundamental causes such as cathode corrosion due to chemical cycles, or the failure of ceramicmetal seals. However, many of these problems have now been overcome and lives of over 500 hours are now not uncommon for converters using cesiated-refractory-metal cathodes. As mentioned in Section 111, D, there is no obvious reason why the life of such a thermionic converter could not be much longer than this. I n this paper no attempt will be made to discuss all the materials used in building cesium thermionic converters. However, high-purity aluminum oxide has proved to be satisfactory as an electrical insulator. Metal-ceramic seals using such ceramics are usable with long life at operating temperatures up to about 1000°K. Single-crystal aluminum oxide (sapphire) is satisfactory for windows in Cs converters. Conventional glass-to-metal seals involving oxide layers are usually unsatisfactory due to reduction of the oxide layer by Cs. A wide variety of metals have been used successfully in Cs converters, including W, Ta, Mo, Nb, Re, Cu, Ni, Kovar, and stainless steels. Gold and graphite are to be avoided, since both are severely attacked by cesium. Further discussion of Cs converters is given in Section IX.
\‘I. DEVICES USINGAUXILIARY DISCHARGES Although cesium-filled diodes are the most efficient thermionic converters thus far investigated, their operation is limited to relatively
194
J. M. HOUSTON AND H. F. WEBSTER
high cathode temperatures, e.g., 1600°K or greater, because of the necessity for a fairly high cathode work function in order to produce Cs ions.5 Many problems, both in heat sources and in materials, could be simplified if the operating temperature could be lowered. Several authors (71,164-267) have proposed that this be accomplished by creating the ions in a small auxiliary gas discharge adjacent to the cathode-anode gap. This approach has the additional advantage that the discharge could be obtained in the rare gases rather than in Cs. I n this way one would avoid the problems caused by the chemical reactivity of cesium and take advantage of the low collision cross-section between low-energy electrons and neutral rare-gas atoms. T h e essence of this idea is that sufficient ions can be produced in a low-current auxiliary discharge to allow a much larger electron current to flow from cathode to anode. That this is feasible has been demonstrated by Found (153) and by Johnson and Webster (268). Gabor (164) has suggested a converter using a cold-cathode rare-gas auxiliary discharge behind the anode which injects ions into the cathodeanode gap of the converter. He indicates that approximately 0.05 to 0.1 watt of auxiliary discharge power is required for each ampere reaching the converter anode. Bernstein (165), Schultz (166), and Bloss (167) have all suggested the use of a small, hot, electron-emitting wire in the cathode-anode gap of the converter. This hot wire is operated negative (e.g., 10-20 volts) with respect to cathode and anode, and serves as the cathode of the ion-producing auxiliary discharge. Bernstein indicates that about 0.14 watt/cm2 of auxiliary power is required for a 1 amp/cm2 converter, while Schultz has found that about 10% of the converter power is required for the discharge. Bloss (167) has demonstrated a thermionic converter of this type which had an output power of 1.3 watts with an auxiliary discharge power of 0.1 watt and a cathode operating temperature of less than 900°C. One problem which plagues all rare-gas thermionic converters is the maintenance of a low anode work function in the presence of cathode evaporation. T h e evaporation rate of most cathodes is such that the anode rapidly becomes coated with cathode material, and then generally has a work function similar to the cathode. Possible solutions are to geometrically shield the anode from the cathode, or to operate the auxiliary discharge in a partial or total atmosphere of Cs. “ven when the ions in a diode are produced predominantly by volume ionization rather than surface ionization, a fairly high 4c is required because if the contact potential difference gets too small the “discharge” mode of operation will not occur when the anode is negative.
THERMIONIC ENERGY CONVERSION
195
VII. DEVICESUSING FISSION FRAGMENTS FOR IONPRODUCTION When one considers operating gas-filled thermionic converters inside nuclear reactors, the first thought which generally occurs is that the necessary ions might perhaps be created by the intense y-radiation. However, calculation shows that even at a pressure of one atmosphere, not enough y-rays are intercepted to create sufficient ions to neutralize a strongly-emitting cathode. However, Jablonski et al. (169) have suggested that sufficient ions could be created in a noble gas by the fission fragments expelled from a UC cathode. They assume a UC cathode with a density of 13.6 gm/cm3, a thermal neutron flux of 5 x 1013 neutrons/cm2 sec, a cathode-anode gap of 1 cm, and a gas filling of neon (with 0.1 yo argon) at 1 atm pressure and 1000°K average temperature. For these conditions, they calculate that about 10'' ions/cm3 sec are created and that the equilibrium plasma density is 1013 ions/cm3, the ions being lost principally by volume recombination. T h e reactor radiation (assumed equal to lo8 r/hr) would produce only lOI3 ions/cm3 sec. They calculate that the plasma resistivity would be about 4 ohm-cm, assuming an electron temperature of 3000°K. An experiment to test this idea has been described by Jamerson (170). He built a tube, filled with Ne : A (1000 : 1) at 20 mm Hg pressure, in which cathode and anode consisted of planar, electrically-heated, Baimpregnated, tungsten electrodes spaced 1 mm apart. A third T a electrode, lightly coated with U235to provide fission fragments, was located circumferentially around the cathode-anode gap. When placed in a reactor with a neutron flux of 5 x 10l2neutrons/cm2 sec, a saturated electron current at the anode was observed which corresponded to a plasma density of 2 x 10" ions/cm3. However, this anode current (of the order of 1 ma) was only 0.1 to 0.02 of the saturated cathode emission, i.e., the performance was limited by insufficient ions. He also observed that fission gases released from hot UC poisoned the barium cathode. It therefore appears that it will be difficult to obtain sufficient ionization with this scheme, even if one succeeds in maintaining an appreciable contact potential difference (a problem in all noble-gas thermionic converters). OF THERMIONIC CONVERTERS VIII. APPLICATIONS
It is too early to predict accurately all the applications of thermionic conversion. However, one promising application is power for space vehicles. T h e thermionic converter has several features which are
196
J. M. HOUSTON AND H. F. WEBSTER
desirable in a device converting nuclear or solar heat into electricity in space. These include light weight, lack of moving parts, and the ability to operate at high anode temperatures (e.g., 1000°K) which reduces the size of the radiator necessary for getting rid of waste heat. A typical solar-thermionic system for space power would consist of a parabolic reflector focusing solar radiation onto an absorbing cavity lined with thermionic converters (171). I n such an application it is desirable to use converters with as low a cathode temperature as possible (e.g., 2000°K or lower) in order to minimize re-radiation from the cavity absorber. As T , is lowered, one can tolerate a larger aperture in the cavity absorber, thus relaxing the accuracy requirements on both the reflector surface and the reflector aiming system. As discussed by Purdy (171) and McClelland (172) a rather precise mirror having a concentration ratio of 2000 is necessary in order to re-radiate only 30% of the incident solar energy at a cavity temperature of 2000°K. This precision can be achieved now in small one-piece mirrors [e.g., 5-foot diameter], but will be difficult to achieve with large, foldable paraboloids. For this reason the space solar-thermionic system of the future may consist of an array of small paraboloids, perhaps unstacked and bolted into place by the inhabitants of a space station. At higher power levels, e.g., 10 kwe (kilowatts electrical) and above, nuclear reactors are a promising heat source for thermionic converters in space. At power levels of the order of 10 to 70 kwe, several designs (122, 173) have been proposed in which no moving parts or fluids are employed. In such a design the converters are arranged on the outer surface of a small, fast-neutron, high temperature (e.g., 2000°K) reactor, the waste heat being radiated to space directly from the converter anodes. As discussed by Perry and Pidd (174, the power output of such a system is limited by the available anode radiating area. At higher power levels, liquid-cooling of the converter anodes becomes essential in order to transport waste heat to the required large radiator. Hirsch and Holland (275), and Perry and Pidd (274) have described conceptual designs for reactors cooled by liquid alkali metals such as Li or NaK alloy. As illustrated by Fig. 36 a fuel rod in such a reactor would consist of many cylindrical cesium thermionic converters electrically connected in series. Perry and Pidd estimate that a 300 kwe system would weigh 4.2 1b/kwe, assuming a converter efficiency of 14 %, a power output of 11 watt/cm2, and an anode temperature of 1100°K. Other assumptions yielded heavier systems, e.g., assuming 6 % converter efficiency at 8 watt/cm2 yielded a system weight of about 8 lb/kwe, a figure not very different from that of Johnson (276). All that can really be concluded at the present stage of development is that nuclear-
THERMIONIC ENERGY CONVERSION
197
thermionic systems for space power are potentially rather light-weight, but that many difficult problems remain to be solved before such systems become a reality.
ANODE COOL ANT
CATHOOE E.G. U C - 2 r C PERHAPS
FIG. 36. Typical design proposed for a nuclear-thermionic fuel rod. The fuel rod is cylindrically symmetrical except for small holes to permit the flow of Cs vapor.
For example, in a nuclear-heated thermionic system, the effect of fission products on the device is not yet certain. These could increase the electron scattering or change the Cs adsorption on either cathode or anode electrodes. Also, if the nuclear fuel such as UC or UO, is clad with a refractory metal (as the cathode), chemical reactions between the cladding and the fuel may be a problem since they can cause mechanical failure. I n addition, diffusion of the nuclear fuel through the cladding material may be troublesome since it can alter the Cs adsorption of the surface. It has been proposed by Yaffee (177) that auxiliary electrical power for rockets be generated by heating thermionic converters with the rocket exhaust. Block et al. (206)have described the construction of a converter
198
J. M. HOUSTON AND H. F. WEBSTER
to explore this application. However, there appears to be little merit in this idea since the converter power per unit weight (0.17 watt/gm) is considerably smaller than that of chemical batteries (about 7 watt min/gm) ( I 78) for typical rocket burning times of a few minutes. Thermionic converters also have potential terrestial application, both in special-purpose power sources and in central-station power. Because a low-temperature heat sink is generally available in terrestial applications, the achieving of long-lived low-work-function anodes (e.g., 1 volt or less) becomes of interest. Alternatively, as proposed by Ioffe (70) one can operate thermionic and thermoelectric generators thermally in series, e.g., the thermionic devices utilizing the range 2000- 1000°K and the thermoelectric 1000-300°K. Small, portable thermionic power sources would probably use chemical energy (e.g., a gas flame) as the heat source. Because heattransfer and materials problems in a combustion-gas environment get rapidly more difficult with increasing temperature, there is a strong incentive to use some form of low temperature converter, e.g., operating at 1500°K or less. Converters using auxiliary electrodes for ion production are therefore attractive. Nuclear-thermionic systems may be of use for special-purpose power sources (e.g., unattended power sources in remote locations) where their compactness, simplicity, or silence is an advantage. Nuclear-thermionic systems are also of interest to nuclear central-station power generation as a “topper” for a conventional steam turbine. In such an application the reactor would be composed of nuclear-thermionic fuel rods similar to Fig. 36. T h e waste heat from the fuel rods would produce steam to operate a conventional turbine. I n this way the over-all system efficiency could be raised, e.g., adding a 20% efficient thermionic converter to a 30 % efficient turbogenerator would raise the system efficiency to
44%.
However, such systems are certainly not foreseeable in the near future because economic as well as technical factors are important. Before it is economically advantageous to add thermionic converters to nuclear power plants, much more reliable, long-lived, and inexpensive thermionic converters must be developed than now exist. Many factors enrichment of the enter the economic decision, such as the increased U235 nuclear fuel required because of the reactor volume occupied by the thermionic converters. Also such a “topper” produces low-voltage direct current which is expensive to convert to alternating current. For this reason such “toppers” will first be economically feasible where low-voltage direct current can be consumed adjacent to the power plant.
THERMIONIC ENERGY CONVERSION
199
IX. SUMMARY AND FUTURE TRENDS T h e close-spaced high-vacuum converter appears to have neared the limit of its development at a power output of 1 watt/cm2 and an efficiency of approximately 5 yo.T h e life is generally short because of the extremely close spacing involved, typically 5p. No other type of highvacuum converter seems promising. Three electrode gas-filled devices should allow relatively low temperature operation, e.g., in the range 1300-1700°K. T h e third electrode (to which a small current flows) is used either as an ion emitter or a source of electrons for impact ionization. Either Cs or rare gases have been proposed as the gas filling. Although the rare gases have a lower electron-neutral cross-section (i.e., lower resistance), a Cs filling has the advantage that a low anode is maintained and that lower auxiliarydischarge voltages are required. T h e addition of a third electrode considerably complicates any system (e.g., a nuclear reactor) in which it is used. Therefore, it only is advantageous for low temperature operation (below roughly 1700"K), or perhaps when periodic interruption of the converter output is desired for generation of A. C. power. Cesium-filled diodes are probably the most promising type of thermionic converter. These use cathodes of two types, the essentially constant-+ cathodes (such as barium, thorium, or U C cathodes), and the cesiated refractory metals, which have a strongly dependent on temperature and Cs pressure. The constant-+ cathodes require smaller Cs pressures and thus electron transport effects (e.g., plasma resistance) are less of a problem. Therefore, larger spacings (e.g., 1 mm) can be used. However, the cathode life is limited, as with most cathodes, by cathode evaporation. Under optimum conditions, nearly all constant-+ cathodes evaporate active material at a minimum rate of approximately to lop9 gm/cm2 sec (60a) when emitting 10 amp/cm2. At this evaporation rate a life of one year is quite feasible. However, to secure long life at high emission density, the cathode must be carefully designed so that the active material is dispensed to the emitting surface at the proper rate. Converters with cesiated refractory-metal cathodes have a number of real advantages which include the following. (1) T h e power output varies less strongly with T , than does that of constant-+, devices, because J c is not strongly dependent on Tc in the operating range. (2) Radiation losses are minimized since the electrodes can be shiny refractory metals. (3) Life can, in theory, be extremely long because the evaporation rate of refractory metals (such as W or Ta) is small at typical operating temperatures (e.g., 2000°K). However, cesiated-cathode devices have one
+
+
200
J. M. HOUSTON AND H. F. WEBSTER
disadvantage in that high Cs pressures are necessary in order to maintain coverage and therefore electron transport effects are serious unless close spacing (e.g., 0.1 mm) is used. Two promising approaches are now being explored for improving cesiated-cathode converters. T h e first of these is the use of electrode surfaces with the optimum crystallographic orientation. T h e second is the addition of gases such as halogens, oxygen, or hydrogen which increase the cathode emission and decrease the anode work function. T h e gas used must be carefully chosen since many of these gases also cause chemical cycles which corrode the cathode. Either of these approaches should allow one to obtain present emission levels at perhaps an order-of-magnitude lower Cs pressure, and thus reduce electron transport effects. Both of these approaches also allow the attainment of low anode work functions, e.g., under 1.5 volt. Cesiated-cathode converters have yielded the highest efficiencies measured to date, i.e., 17% for a short-lived device and approximately 10% for a long-lived [ > 500 hours] device. It is probable that measured efficiencies will exceed 20% within the next few years, judging from the present rapid rate of progress.
ACKNOWLEDGMENTS T h e authors wish to thank the friendly cooperation of many authors who have supplied reprints or manuscripts of their own work and have given permission for reference to such material here. We also acknowledge the generous permission given by several journals to reproduce previously published figures. Special thanks are due V. L. Stout, M. Duval, F, P. Hession, and T. A. Howlett for aid in preparing and correcting the manuscript. Finally, we gratefully acknowledge the cooperation and aid of the General Electric Research Laboratory.
Nomencluture theoretical constant in Eq. ( 1 ) equal to 120 amp/cm2”K2 cathode area “A” constant from intercept of Richardson plot cathode-anode gap electronjc charge net current density to anode anode emission density which reaches cathode saturated electron emission density of anode cathode emission density which reaches anode
THERMIONIC ENERGY CONVERSION Ice
JP
J C O
20 I
saturated electron emission density of cathode ion current emitted from cathode current density that will flow in vacuum diode if the potential minimum is at the anode and the cathode has unlimited emission capability Boltzmann's constant plasma density electron density ion density neutral Cs atom density in gap d neutral atom density in Cs reservoir Cs pressure in reservoir maximum output power density net power density radiated from cathode space charge parameter load resistance into which thermionic converter will deliver maximum power Debye length temperature of anode temperature of cathode temperature of Cs reservoir electron temperature average gas.temperature in gap d anode voltage with respect to cethode internal diode voltage defined by Eq. (2a) surface ionization efficiency efficiency of, thermionic converter flux of neutral Cs atoms striking cathode true work function of a surface work function of anode work function of cathode effective surface work function work function from slope of Richardson plot motive barrier which determines Ja value of xa which yields P m
REFERENCES 1. Dobretsov, L. N., Zhur. Tekh. Fiz. 30, 365 (1960); English transl.;Smiet Phys.Tech. Phys. 5, 343 (1960). 2. Morgulis, N. D., Uspekhi Fiz. Nuuk 70, 679 (1960); English Transl., Soviet Phys.Uspekhi 3, 251 (1960). 3. Cayless, M. A,, Brit.J . Appl. Phys. 12, 433 (1961). 4. Langmuir, I., and Kingdon, K. H., Proc. Roy. SOC.(London) A107, 61 (1925). 5. Michaelson, H. B., J. Appl. Phys. 21, 536 (1950). 6. Riviere, J. C., Proc. Phys. SOC.(London) B70,676 (1957). 7. Wright, D. A,, Proc. I.E.E. 100, 125 (1953). 8. Richardson, 0. W., Phil. Mug. [6] 23, 594 (1912). 9. Herring, C., and Nichols, M. H., Revs. Modern Phys. 21, 185 (1949). 10. Shelton, H., Phys. Rev. 107, 1553 (1957).
202
J. M. HOUSTON AND H. F. WEBSTER
11. Hobson, J. P., Rept. 16th Ann. M.I.T. Conf. on Phys. Electronics p. 31 (1956). 12. Nottingham, W. B., in “Handbuch der Physik” (S. Fliigge, ed.), Vol. 21, p. 1.
Springer, Berlin, 1956. 13. Hensley, E. B., J. Appl. Phys. 32, 301 (1961). 14. Haas, G. A., and Jensen, J. T., Jr., J . Appl. Phys. 31, 1231 (1960). 15. Schlicter, W., Ann. Physik [4] 47, 573 (1915). 16. Champeix, M. R., Vide 6 , 936 (1951). 17. Ansel’m, A. I., “Thermionic Vacuum Thermal Cell.” Acad. Sci. U.S.S.R., Moscow, 1951. 18. Medicus, G., and Wehner, G., J . Appl. Phys. 22, 1389 (1951). 19. Gurtovi, M. E., and Kovalenko, G. I., Fiz. Zapiski 9, 240 (1941). 20. Marchuk, P. M., Candidates Dissertation, Inst. Phys. Acad. Sci. Ukr. S.S.R. (1951); Trudy Inst. Fiz. Akad. Nauk l7kr.S.S.R. 7 , 3 (19561. 21. Hernqvist, K. G., Kanefsky, M., and Norman, F. H., R.C.A. Rev. 19, 244 (1958). 22. Wilson, V. C . , Bull. Am. Phys. SOC.[2] 3, 266 (1958); J . Appl. Phys. 30,475 (1959). 23a. Grover, G. M., Roehling, D. J., Salmi, E. W., and Pidd, R. W., J. Appl. Phys. 29, 1611 (1958). 236. Grover, G. M., Nucleonics 17, 54 (1959). 24. Rasor, N. S., Inst. Aerospace Sci. Meeting, New York Paper 61-72 (1961). 25. Houston, J. M., J. Appl. Phys. 30, 481 (1959). 26. Rasor N. S. J. Appl. Phys. 31 163 (1960). 27. Ingold, J. H., J . Appl. Phys. 32, 769 (1961). 28. Schock, A., J. Appl. Phys. 32, 1564 (1961). 29. McAdams, W. H., “Heat Transmission,” p. 63. McGraw Hill, New York, 1954. 30. Hatsopoulos, G. N., Kaye, J., and Langberg, E., I.R.E. Trans. on Electron Devices ED-7,117 (1960). 31. Kittel, C., “Introduction to Solid State Physics,” p. 245. Wiley, New York, 1953. 32. Dyke, W. P., and D o h , W. W., Advances in Electronics and Electron Phys. 8, 115, Table I11 (1956). 33. Barrett, C. S., “Structure of Metals,” McGraw Hill, New York, 1952. 34. Martin, S. T., Phys. Rev. 56, 947 (1939). 35. Bruche, E., Jahrb. Forschungsinst. A.E.G. 4, 25 (1933-1935). 36. Webster, H. F., Bull. Am. Phys. SOC.[2] 6, 343 (1961); Appl. Phys. 32, 1802 (1961). 37. Kingdon, K. H., Phys. Rev. 25, 892 (1925). 38. Zalm, P., Philips Research Lab., Eindhoven, private communication (1961). 39. Rasor, N. S., Atomics International, Canoga Park, California Report AI-6799 (1961). 40. Gyftopoulos, E. P., and Levine, J. D., J. Appl. Phys. 33, 67 (1962). 41. Killian, T. J., Phys. Rw. 27, 578 (1926). 42. Langmuir, I., and Kingdon, K. H., Phys. Rev. 34, 129 (1929). 43. Taylor, J. B., and Langmuir, I,. Phys. Rev. 44, 423 (1933). 44. Houston, J. M., Bull. Am. Phys. SOC.[2] 6, 358 (1961). 45. Marchuk P., M., Trudy Inst. Fiz. Akad. Nauk Ukr. S.S.R. 7 17 (1956). 46. Aamodt, R. L., Los Alamos Scientific Laboratory, private communication (1961). 47. Langmuir, D. B., R e p . 15th Ann. M.I.T. Conj. on Phys. Electronics, p. 169 (1955). 48. Langmuir, D. B., Acta Met. 5, 13 (1957). 49. Hughes, F. L., Levinstein, H., and Kaplan, R., Phys. Rev. 113, 1023 (1959). 50. Langmuir, I., and Kingdon, K. H., Phys. Rev. 23, 112 (1924). 51. Langmuir I. Ind. Eng. Chem. 22, 390 (1930). 52. Koller, L. R., Phys. Rev. 33, 1082 (1929). ’
r.
THERMIONIC ENERGY CONVERSION
203
53. Goldwater, D. L., and Danforth, W. E., Proc. 4th Nut. Conf. on Tube Techniques, New York, 1958, p. 190 (1959). 54. Langmuir, I., Phys. Rev. 22, 357 (1923). 55. Schneider, P., J . Chem. Phys. 28, 675.(1958). 56. Levi, R., J . Appl. Phys. 26, 637 (1955). 57. Rittner, E. S., Rutledge, W. C., and Ahlert, R. H., J . Appl. Phys. 28, 1468 (1957). 58. Lemrnens, H.J., Jansen, M. J., and Loosjes, R., Philips Tech. Rev. 11, 341 (1950). 59. Rittner, E. S., Ahlert, R. H., and Rutledge, W. C., J. Appl. Phys. 28, 156 (1957). 60a. Stout,V. L., Proc. 4th Nat. Conf. on Tube Techniques,New York, 1958, p. 178 (1959). 60b. Beck, A. H. W., Proc. Z.E.E. 106B,372 (1959). 61. Eisenstein, A. S., Advances in Electronics 1, 1 (1948). 62. Herrmann, G., and Wagener, S:, “The Oxide-Coated Cathode,” Vol. 2. Chapman & Hall, London, 1951. 63. Pidd, R. W., Grover, G. M., Roehling, D. J., Salrni, E. W., Farr, J. D., Krikorian, N. H., and Witteman, W. G., J . Appl. Phys. 30, 1575 (1959). 64. Danforth, W. E., and Williarps, A. J., 111, J . Appl. Phys. 32, 1181 (1961). 65. Abrarns, R. H., Jr., and Jarnerson, F. E., J . Appl. Phys. 32, 1783 (1961). 66. Pidd, R. W., Grover, G. M., Salmi, E. W., Roehling, D. J., and Erickson, G. F., J . Appl. Phys. 30, 1861 (1959). 67. Pidd, R. W., John Jay Hopkins Laboratory, General Atomics, San Diego, private communication (1961). 68. Jones, H. A., and Langmuir, I., Gen. Elec. Rev. 30, 354 (1927). 69. Hatsopoulos, G. N., Sc. D Thesis, Mass. Inst. Technol. (1956). 70. Ioffe, A. F., “Semiconductor Therrnoelernents and Thermoelectric Cooling,” p. 73. Infosearch Ltd., London, 1957. 71. Moss, H., Brit. J . Electronics 2, 305 (1957). 72. Langmuir, I., Phys. Rev. 21, 426 (1923). 73. Webster, H.F., and Beggs, J. E., Bull. Am. Phys. SOC. [2] 3, 266 (1958). 74. Hatsopoulos, G. N., and Kaye, J., J. Appl. Phys. 29, 1124 (1958). 75. Hatsopoulos, G. N., and Kaye, J., Proc. I.R.E. 46, 1574 (1958). 76. Rittner, E. S., J . Appl. Phys. 31, 1065 (1960). 77. Feaster, G. R., Brit. J. Electronics and Control 5 , 142 (1958). 78. Nottingham, W.B., J . Appl. Phys. 30, 413 (1959). 79. Nottingham, W. B., Hatsopoulos, G. N., and Kaye, J., J. Appl. Phys. 30, 440 (1 959). 80. Webster, H. F., J. Appl. Phys. 30, 488 (1959). 81. Lindsay, P. A., and Parker, F. W., Brit. J . Electronics and Control 7 , 289 (1959). 82. Compton, A. H., and Langmuir, I., Revs. Modern Phys. 3, 241 (1931). 83. Ferris, W.R., R.C.A. Rev. 10, 134 (1949);11, 568 (1950). 84. Beggs, J. E., G. E. Research Lab., private communication (1959). 85. Dugan, A. F., J . Appl. Phys. 31, 1397 (1960). 86. Lindsay, P. A., and Parker, F. W., J. Electronics and Control 9, 81 (1960). 87. Nottingham, W.B., R e p . 20th Ann. M.I.T. Conf. on Phys. Electronics, p. 58 (1960). 88. Beggs, J. E., Advanced Energy Conversion 1, (1962), to be published. 89. Leverton, W. F., and Shepherd, W. G., J . Appl. Phys. 23, 787 (1952). 90. Bickel, P. W., and Holroyd, L. V., J . Chem. Phys. 22, 1793 (1954). 91. Wooten, L. A., Ruehle, A. F., and Moore, G. E., J . Appl. Phys. 26, 44 (1955). 92. Gibbons,M. D., G. E. Research Lab. Final Report Contract.AF-l9(604)-5472(1961). 93. Hatsopoulos, G. N., Thermo Electron Engineering Corp., Waltham, Mass., private communication (1958).
204
J. M. HOUSTON AND H. F. WEBSTER
94. Baum, E. A., and Jensen, A. O., Proc. 15th Ann. Power Sources Conf., Atlantic City, 1961. 9%. J. A. Welch, G. N. Hatsopoulos, and J. Kaye, “Direct Conversion of Heat to Electricity”, Chapter 5. Wiley, New York, 1960. 956. Peters, P. H., G. E. Research Lab. Final Report Contract AF-l9(604)-5472 (1961). 96. Taylor, J. B., and Langmuir, I., Phys. Rew. 51, 753 (1937). 97. Shelton, H., Wuerker, R. F., and Sellen, J. M., A.R.S. Meeting, Sun Diego, 1959 Paper No. 882-59. 98. Brode, R. B., Rews. Modern Phys. 5, 257 (1933). 99. Langmuir, I., Phys. Rew. 43, 224 (1933). 100. Auer, P. L., J . Appl. Phys. 31, 2096 (1960). 101. Zandberg, E. Ya., and Ionov, .N. I., Uspekhi Fiz. Nauk 57, 581 (1959); English transl., Swiet Phys.-Uspekhi 67, 255 (1959). 102. Eichenbaum, A. L., and Hernqvist, K. G., J . Appl. Phys. 32, 16 (1961). 103. Spitzer, L., “Physics of Fully Ionized Gases,” p. 17. Interscience, New York, 1956. 104. Houston, J. M., and Gibbons, M. D., Rept. 2lst Ann. M.I.T. Conf. on Phys. Electronics p. 106 (1961). 105. Ranken, W. A., and Teatum E. T., Bull. Am. Phys. SOC.[2] 6 , 371 (1961). 106. Block, F. G., Corregan, F. H., Eastman, G. Y., Fendley, J. R., Hernqvist, K. G., and Hills, E. J., PYOC. I.R.E. 48, 1846 (1960). 107. Rasor, N. S., in “Energy Conversion for Space Power” (N.W. Snyder, ed.), p. 155. Academic Press, New York, 1961. 108. Wilson, V. C., in “Energy Conversion for Space Power” (N.W. Snyder, ed.), p. 137, Academic Press, New York (1961). 109. Ranken, W. A,, Grover, G. M., and Salmi, E. W., J. Appl. Phys. 31, 2140 (1960). 110. Baum, E. A., and Jensen, A. O., PYOC.15th Ann. Power Sources Conf., Atlantic City, 1961. 111. Hatsopoulos, G . N., private communication (1961); see also Electronics 34, 78 (1961). 112. Reichelt, W., Grover, G., Salmi, E., and Schafer, W., Bull. Am. Phys. SOL.[2] 6, 359 (1961). 113. Howard, R. C., Yang, L., Garvin, H. L., and Carpenter, F. D.. in “Energy Conversion for Space Power” (N.W. Snyder, ed.), p. 211. Academic Press, New York (1961). 114. Morgulis, N. D., and Naumovets, A. G., Fiz. Twer. Tela 3, 537 (1960); English transl., Soviet Phys.-Solid State 2, 501 (1960). 115. Silverberg, M., Ford Instrument Co. private communication (1961). 116. Hernqvist, K. G., R.C.A. Rev. 22, 7 (1961). 117. Fox, R., and Gust, W., Bull. Am. Phys. Soc. [2] 4, 322 (1959). 118. Houston, J. M., Rept. 20th Ann. M.Z.T. Conj. on Phys. Electronics, p. 12 (1960). 119. Morgulis, N. D., Uspekhi Fiz. Nauk 70, 679 (1960); English transl., Swiet Phys.Uspekhi 3 , 251 (1960). 120. Rittner, E. S., and Milch, A., J . Appl. Phys. 33, 228 (1962). 121. Moizhes, B. Y.,and Pikus, G. E., Fia, Tver. Tela 2, 756 (1960); English transl., Swiet Phys.-Solid State 2, 697 (1960). 122. Hernqvist, K. G. Nucleonics 17, No. 7 , 49 (1959). 123. Hernqvist, K. G., in “Space Power Systems” (N.W. Snyder, ed.), p. 167. Academic Press, New York, 1961. 124. Johnson, F. M., Rept. 20th Ann. M.I.T. Conf.on Phys. Electronics, p. 88 (1960). 125. Johnson, F.M., R.C.A. Rew.22,21 (1961).
THERMIONIC ENERGY CONVERSION
205
126. Garvin, H. L., Teutsch, W. B., and Pidd, R. W., J. Appl. Phys. 31, 1508 (1960). 127. Garvin, H. L., General Atomics, San Diego, Report GA-2059 (1961); Bull. Am. Phys. SOC.[ 2 ] 6, 358 (1961). 128. Zollweg, R. J., and Gottlieb, M., Bull. Am. Phys. Sac. [2] 5 , 383 (1960); J . AppI. Phys. 32, 890 (1961). 129. Luke, K. P., and Jamerson, F. E., J. Appl. Phys. 32. 321 (1961). 130. Rocard, J. M., and Paxton, G. W., J . Appl. Phys. 32, 1171 (1961). 131. Koskinen, M. F., M.T.T. Research Lab. for Electronics Progress Report 63, p. 55 (1961). 132. Birdsall, C. K., and Bridges, W. B., J . Appl. Phys. 32, 261 1 (1961). 133. Lewis, H. W., and Reitz, J. R., J. Appl. Phys. 30, 1838 (1959). 134. Lewis, H. W., and Reitz, J. R., J . Appl. Phys. 31, 723 (1960). 135. Nottingham, W. B., in “Direct Conversion of Heat to Electricity” (J. Kaye and J. Welsh, eds.), Chapter 8. Wiley, New York, 1960. 136. Nottingham, W. B., Rept. 20th Ann. M.I.T. Conf. on Phys. Electronics, p. 95 (1960); also Thermo Electron Engineering Corp. Report TEE-7002-5 (1960). 137. Carabateas, E. N., Pezaris, S. D., and Hatsopoulos, G . N., J. Appl. Phys. 32, 352 (1961). 138. Talaat, M. E., A.I.E.E. Aerospuce Trunsportution Committee Conf,, Philadelphia Paper CP 61-5050 (1961). 139. Gottlieb, M., and Zollweg, R. J., Westinghouse Research Labs. Interim Sci. Report 1, Contract AF 33(616)-8262, p. 38 (1961). 140. Warner, C., 111, and Vernon, R., Atomics International, Canoga Park, California, Report A1 6799, p. 167 and 177 (1961). 141. Lewis, H. W., and Reitz, J. R., J. Appl. Phys. 30, 1439 (1959). 142. Johnson, E. O., R.C.A. Rev. 16, 498 (1955). 143. Houston, J. M., Proc. Power Information Center Conf.on Thermionic Converters, Los Angeles, December 1961, p. 8-1 (1962). 144. Bohm, D., and Gross, E. P., Phys. Rev. 79, 992 (1950). 145. Langmuir, I., Phys. Rev. 26, 585 (1925). 146. Gabor, D., Ash, E. A., and Dracott, D., Nature 176, 916 (1955). 147. Lewis, H. W., in “Energy Conversion for Space Power,” (N.W. Snyder, ed.), p. 133. Academic Press, New York, 1961. 148. Compton, K. T., and Eckhart, C., Phys. Rev. 25, 139 (1925). 149. Mohler, F. L., J . Research Natl. Bur. Standards 9, 25 (1932); 21, 697 (1938). 150. Steinberg, R. K., J. Appl. Phys. 21, 1028 (1950). 151. Malter, L., Johnson, E. O., and Webster, W. M., R.C.A. Rew. 12, 415 (1951). 152. Morgulis, N. D., and Marchuk, P. M., Ukruin. Fiz. Zhur. 1, 59 (1956). 153. Found, C. B., Phys. Rev. 45, 519 (1934). 154. Morgulis, N. D., and Marchuk, P. M., Ukrain. Fiz. Zhur. 2, 379 (1957). 155. Carabateas, E. N., Proc. Power Information Center Conf. on Thermionic Converters, Los Angeles, December 1961, p. 9-1 (1962). 156, Gibbons, M. D., General Electric Research Laboratory, private communication (1961). 157. Wilson, V. C., and Lawrence, J., General Electric Research Laboratory, private communication (1961). 158. Agnew, L., Bull. Am. Phys. SOC.[2] 6 , 343 (1961). 159. Mohler, F. L., J. Research Nutl. Bur. Stundards 21, 873 (1938). 160. Roehling, D. J,, Los Alamos Scientific Laboratory, private communication ( I 900). 161. Hirsch, R. L., J. Appl. Phys. 31, 2064 (1960).
206
J. M. HOUSTON AND H. F. WEBSTER
162. Jensen, A. O., PYOC.Power Information Center Conf. on Thermionic Conversion, Bedford, Massachusetts, June 1961, Appendix F (1961). 163. Rasor, N. S., Atomics International, Cmoga Park, California, private communication (1960). 164. Gabor, D., Nature 189, 868 (1961). 165. Bernstein, W., Rept. 21st Ann. M.Z.T. Conf.on Phys. Electronics, p. 124 (1961); Bull. Am. Phys. SOC.[2] 6 , 358 (1961). 166. Schultz, R. D., Rept. 21st Ann. M.Z.T. Conf. on Phys. Electronics, p. 146 (1961). 167. Bloss, W., Naturwissenschaften 48, 497 (1961). 168. Johnson, E. O., and Webster, W. M., Proc. I.R.E. 40, 645 (1952). 169. Jablonski, F. E., Leffert, C. B., Silver, R., Hill, R. F., and Loughridge, D. H., J . Appl. Phys. 30, 2017 (1959). 170. Jamerson, F. E., Rept. 2lst Ann. M.Z.T.Conf. on Phys. Electronics, p. 121 (1961). 171. Purdy, D. L., in “Space Power Systems” (N.W. Snyder, ed.), p. 153. Academic Press, New York, 1961. 172. McClelland, D. H., in “Space Power Systems” (N. W. Snyder, ed.), p. 129. Academic Press, New York, 1961. 173. Miller, B., Aviation Week p. 61, July 10 (1961). 174. Perry, L. W., and Pidd, R. W., in “Space Power Systems” (N.W. Snyder, ed.), p. 399. Academic Press, New York, 1961. 175. Hirsch, R. L., and Holland, J. W., in “Space Power Systems” (N.W. Snyder, ed.), p. 389. Academic Press, New York, 1961. 176. Johnson, K. P., in “Space Power Systems” (N.W. Snyder, ed.), p. 409. Academic Press, New York, 1961. 177. Yaffee, M., Aviation Week 71, 92 (November 23, 1959). 178, Morehouse, C. K., Glicksman, R., and Lozier, G. S., Proc. Z.R.E. 46, 1462 (1958).
Thermoelectricity FRANK E. JAUMOT, JR.
Delco Radio Division, General Motors Corporation, Kokomo, Indiana
I. Introduction .........................................................
11. Basic Considerations .................................................. 111. Materials ........................................................... IV. Practical Considerations ............................................... A. Figure of Merit ................................................... B. Materials Fabrication .............................................. C. Device Design .................................................... V. Thermoelectric Applications ........................................... VI. Theory and Problems. ................................................. A. Device Design Criteria ............................................. B. Materials Development ............................................ C. Theoretical and Associated Fundamental Work ....................... References ..........................................................
Page 207 208 215 220 220 224 227 233 235 236 237 238 242
I. INTRODUCTION Thermoelectric effects offer means of refrigerating, heating, and generating electrical power all with the same materials, and without moving parts. Further, the transport of heat required or the generation of electricity is done without intervening mechanisms or media; that is, one can convert heat to electricity or “pump” heat directly. Thus, it is not surprising that there has been much interest, both commercial and scientific, in this field. Also, it is to be expected that since the conversion is achieved in the solid state, the main problem centers on the materials involved. Although, more recently, there has been a shifting of emphasis to the design of devices, the problem has not changed since the Seebeck effect was discovered in 1822. The recent resurgence in this field followed the advent of semiconductor technology which led to materials which made practical applications appear much more promising. Unfortunately, if one reads the material on thermoelectricity available in the various media, one will gain the impression that little progress has been made in the past few years. On close examination the truth is almost the opposite. What has happened is that recent progress is masked by overly optimistic reports of results in the early days of renewed interest in thermoelectricity. The net result is that now the state201
208
FRANK E. JAUMOT, JR.
ments as to what can be done are on a much sounder basis although it is obvious that publicity is still leading technology. However, it is not the purpose of this paper to cite the rather impressive progress that has been made. Rather, it will be concerned more with a review of the principles involved, with particular emphasis on what is lacking in the way of understanding. In ail fairness to the workers in this field, it should be pointed out that, being no different than any other technology, improvement in the understanding of the principles involved as well as in the performance of devices comes at an ever increasing cost in dollars and hours of labor per increment of improvement.
11. BASICCONSIDERATIONS Any phenomenon involving the interchange of heat and electrical energy may be called a thermoelectric effect. However, the term is usually meant to imply the Peltier, the Thomson, and the Seebeck effects. These effects have been defined and the relationships between them have been derived in numerous books and articles (see for example Jaumot, I ) and no attempt will be made to do so here. Rather, it will simply be stated that the Peltier coefficient, T, which relates the rate at which Peltier heat is transferred to the current which flows through a junction of dissimilar conductors, is physically a latent heat per unit charge. The Thomson coefficient, N , which relates the rate at which Thomson heat is transferred into a small region of a conductor carrying a current I and supporting a temperature difference d T , is a specific heat per unit charge. The Seebeck coefficient, S , is the name given to the rate of change with temperature of the electromotive force in a complete circuit and is, physically, an entropy per unit charge. For practical applications, the Seebeck coefficient is the important quantity and is the only one appearing in the equations pertinent to device design. This is because it is directly related to the Peltier coefficient and the absolute temperature ( S = r / T ) and because the Thomson heat is neglected as being small. The latter point will be discussed in more detail below. When one wants to determine the factors involved in the efficiency of a device, he writes down the expression for the efficiency from first princip1es.l Then, through a series of optimizations with respect to internal versus external or load resistances and the geometric factors involved, he finds that his efficiencies are expressed only in terms of A list of useful device equations is given in the appendix. Typical derivations may be found in Jaumot (I) or in Ioffe (2).
THERMOELECTRICITY
209
the temperatures of the hot and cold junctions and a factor containing the materials parameters. It is this latter factor, called the figure of merit, which is of paramount interest. It determines completely the value of a material for practical applications apart, of course, from questions of mechanical properties, melting point, and volatility. The figure of merit is given by z=- s2u (1 1 K
where S is the Seebeck coefficient, u is the electrical conductivity, and K is the thermal conductivity. Although the relations are far from satisfactory for any detailed study, the theory for extrinsic semiconductors gives us relations for the important parameters in terms of the fundamental transport phenomena (2, 3, 4). The Seebeck coefficient is given for a one carrier model by
k s =(6 e
-
6)
where k is the Boltzmann constant, e is the charge on the electron, .$ is the Fermi energy measured from the edge of the band and expressed in units of kT (also called the degeneracy parameter), and 6 is the average energy, relative to the band edge, of the transported electrons. The quantity 6 depends on both the scattering mechanism and the degeneracy parameter. In the nondegenerate case (5 > -2), 6 is two for acoustical scattering and four for scattering by ionized impurities. T h e thermal conductivity is given by K
= Kph $. K e l = Kph = Kph
k 2 + d (T) Tepn,
(3)
+LUl’
where Kph is the lattice thermal conductivity, K e l is the electron thermal conductivity, p is the mobility of the charge carriers, n is the number or density of charge carriers, L is the Wiedemann-Franz ratio, and d is a scattering parameter which depends on the scattering mechanism and has the same numerical values as 6 for similar scattering mechanisms. For the fully degenerate case, d is 7r2/3. The electrical conductivity is u = epn
(4)
and has been used in defining the electron thermal conductivity along with the assumption that the Wiedermann-Franz ratio K e l / # = 2(k/e)2T, holds.
210
FRANK E. JAUMOT, JR.
T h e degeneracy parameter can be related to the number of charge carriers (or vice versa) by
or alternately
n
= Nee(.
Here h is Planck’s constant and ms is the effective (or density of states) mass. If Herring’s definition ( 5 ) is used for this latter quantity the relation holds for multivalley semiconductors so long as f 5 0. By simple algebra, using the above expressions for the various factors, the figure of merit is, for the case of acoustical scattering,
Thus, for maximum z, one wants a large effective mass, a high mobility and a minimum lattice thermal conductivity, assuming that an optimum charge carrier density is present. T h e optimum charge carrier density is proportional to the ratio of the effective mass to the free electron mass and to the absolute temperature T, but is in the range from 1OlDto lozoper cubic centimeter. (This value corresponds to near-degenerate conditions in semiconductors, the most desirable material for thermoelements.) It is the charge carrier density that presents one of the first major problems. Although this quantity can, in principle, be determined from chemical, optical, or thermoelectric measurements, it usually is obtained from the Hall coefficient, R, which is proportional to (ne)-l. But this requires two correction factors. First, one has to employ a correction factor which depends upon the electron scattering and the degree of degeneracy; this factor can usually be estimated fairly well from the behavior of the material and, in any event, generally is in the range from 1 to 2 so that the carrier density will not be wrong by more than a factor of 2. However, there is another correction factor which depends on the shape of the energy surfaces. This is frequently ignored or, perhaps, one should say it is taken as unity. But for multivalley structures, it is less than one and may be of the order of 0.1 for very anisotropic, oblate energy surfaces (6). Thus, the simple Hall coefficient formula may lead to large overestimates of the carrier density. T h e situation is further complicated by the fact that useful semi-
THERMOELECTRICITY
21 1
conductors may be anisotropic in practically all their properties. For example, in bismuth telluride with three possible orientations in which to measure R, one can end up with two different values. (The Onsager reciprocal relations tell us that, although the resistivity may be different, the Hall coefficient must be identical in two of the orientations.) Further, the conductivity u may be markedly different with the current flow along different crystalographic axes so that there must be two different mobilities. With two values of the Hall coefficient and two values of the conductivity, there are four different products of these two quantities (Ru), the idea of a “Hall mobility” loses its usefulness and the determination of the charge carrier density becomes quite complicated. Incidentally, it is of interest that the orientation one would choose intuitively to give the better estimate of the density of charge carriers is the wrong one (7). Thus, it is necessary to know at least something of the detailed band structure of a material in order to determine even the simpler properties. The band structure may be studied through susceptibility measurements, galvanomagnetic effects. optical properties, the anomalous skin effect, or magneto optic effects.2 However, such studies generally require that a model be assumed and the experimental work used to check the model. T o date, very little work has been done along these lines for the more interesting thermoelectric materials. The mobility also presents no mean problem. In the more common semiconductors, such as germanium or silicon, one can measure a “drift” mobility. This is done by injecting a pulse of electrons (or holes) into the semiconductor and measuring its velocity in an applied field. However, in the more promising thermoelectric materials the charge carrier density is so high and the concentration of impurities so large, injection is very difficult and any injected pulse would decay before it could drift a measurable distance. Thus, this technique is not available for these materials. The definition of conductivity [Eq. (4)] also defines a “conductivity mobility.” The conductivity itself is easily measured, so if n can be determined, the mobility is known. Then, since R oc (ne)-l and u is equal to nep, the product Ro is generally referred to as the Hall mobility. Unfortunately, Ru has the same proportionality constant difficulties discussed above for the Hall coefficient alone. It was stated above that a large effective mass was needed for maximum z. This is not always true as are the statements regarding mobility
* See the excellent review of “Experimental Investigations of the Electronic Band Structure in Solids” by Benjamin Lax (84.
212
FRANK E. JAUMOT, JR.
and lattice thermal conductivity. Actually the optimum effective mass is determined by the type of chemical binding and the scattering characteristics, with due consideration for the fact that small masses are usually associated with poor thermal stability and generally less desirable thermal characterisitcs. More specifically, the problem is that the mobility depends on the effective mass; so does the charge carrier density but it is assumed that that parameter is optimized. In particular, in a homopolar lattice, the mobility is inversely proportional to mB-5P; in an ionic lattice p is proportional to mS-8f2. When impurity scattering predominates, the mobility is directly proportional to the effective mass. Finally, the dependence of the mobility on the effective mass of some of the more interesting compounds is not known (see below), but it does appear to be a direct rather than an inverse proportion. It is this variation in dependence of mobility (and thus z ) on effective mass that prevents the various calculated values of an optimum value for z from being more than reasonable approximations, as will be discussed below. For the purposes of this discussion, a large effective mass is generally desirable and this is to be found in association with a complex band structure or, more generally, in materials having low crystal symmetry. This brings us to the thermal conductivity, and it is this parameter which holds the most promise for improvement of presently available materials since, of all the variables involved, it exhibits the least interdependence with the others. Also, it may well be the least understood. It was stated above that the thermal conductivity could be divided into two contributions, one due to the charge carriers, K e l r and the other due to the lattice, Kph. Since o / K ~is~ roughly a constant for any given temperature, there is not much we can do to improve K e l . Further, in interesting materids K e l is usually smaller than Kph. Thus, the discussion here will be concerned primarily with the lattice thermal conductivity. The difficulty of obtaining true values of n and p have been stressed above; unfortunately it is just about as difficult to measure thermal conductivity. (A review of possible methods is given by Krumhansl and Williams, 8b ; specific measurements on thermoelectric materials are described by Goldsmid, 9, and Devyatkova, 10.) For most purposes, phonon transport may be regarded as a scattering problem very similar to electron scattering, with much the same mechanisms applying. The acoustic modes of lattice vibration are the most important influence but impurities and lattice defects can be major factors. Although detailed experimental work is meager, it appears likely that thermal conduction varies less with changes in chemical
THERMOELECTRICITY
213
composition and detailed lattice texture than do the electrical properties. This, combined with the fact that lattice thermal conductivity exhibits the least interdependence of the various important parameters makes it desirable, when choosing a material, to select one that would be expected to have a low thermal conductivity. A low lattice thermal conductivity means a small phonon mean free path. This is obtained as a result of a high degree of anharmonicity in the lattice. High anharmonicity is usually found in conjunction with a large, anisotropic thermal expansion and anomalously low Debye temperatures. From a physical point of view this means that the crystal structure will either be one of low crystal symmetry or a defect structure of a simple lattice; other desirable physical conditions are large atoms of high atomic weight with adjacent atoms in the lattice varying significantly in size and weight. Thermal conductivity is so important it may be well to pursue it further here. At the same time, to consider the subject in detail would require an analysis of a great many subjects (8b) which space will not permit. Thus, the present discussion will be confined to the more important considerations related to semiconductors and the reader is referred to the review of Krumhansl and Williams (8b)(which includes 5 1 additional references) for amore detailed review. For thermoelectricity purposes, the usual order of magnitude calculation is not adequate; a factor of two variation in thermal conductivity can be of major practical value. So far, only charge carrier conduction and phonon conduction by the lattice have been mentioned as mechanisms for thermal conduction. Actually, there are three other significant mechanisms ; hole-electron pair formation, diffusion and recombination (ambipolar heat transfer or diffusion), exciton conduction, and radiative heat transfer. Although the contributions from the first two mechanisms are conceded to be small in most presently useful materials ( I ) , it may well be that as the study of thermoelectricity becomes more refined, these will have to be considered. Ambipolar heat transfer occurs as a result of electron-hole pairs (which do not contribute to the electrical conduction) diffusing from the hot to the cold junction. This mechanism has been discussed by Price ( I I ) and Kittel and Frohlich (12). I t is most important at high temperatures ; the specific temperature depending upon the width of the band gap since the activation energy involved is approximately one-half the band gap energy. Ambipolar conduction is usually negligible and can be ignored in device calculations since it becomes significant with the onset of intrinsicity at which time the material is generally no longer desirable.
214
FRANK E. JAUMOT, JR.
Exciton heat transfer generally can also be neglected, since it almost certainly has a higher excitation energy than ambipolar conduction (13). Thus, for temperatures above which this mechanism is appreciable, semiconductors are no longer useful for thermoelectric devices. Although very little is known of the basic phenomenon itself, particularly in more complicated solids, it is not likely that exciton conduction would be large anyway in useful materials since the number of exciton states could hardly be as large as the number of free carrier states and the optical mean free path for recombination is small in comparison with carrier recombination lengths. Radiative transfer could well be another matter. I n fact, because the index of refraction in semiconductors is high, heat transfer by radiation can be significant at relatively low temperatures. On the other hand, in thermoelectric materials, with their high carrier density, one would expect the free carrier absorption to prevent radiative transport from becoming significant. Thus, at present, the phonon thermal conductivity stands out as the parameter with which to work. However, as the theoretical situation becomes better understood and greater departures are made in fabrication of “tailored” materials, the “tailoring” will have to take the above mechanisms into account. Before concluding the discussion of basic considerations, it may be well to state, briefly, why the discussion has been concerned only with semiconductors. This can perhaps be done most easily by reference to Eq. (1). For insulators, although the Seebeck coefficient S is large, the electrical conductivity is prohibitively small to yield a large value for z. On the other hand, for metals were u is quite large, S is much too small and K is too large. We are thus left with semiconductors. I n choosing a specific semiconductor, the first requirement is a low lattice thermal conductivity. I n addition to the requirements on the lattice discussed above, the band gap should not be too narrow so that the number of intrinsic carriers will be small in the temperature range of interest. Otherwise, ambipolar conduction will be significant. (An exception would be where very large differences exist in the mobilities of the holes and electrons so that the intrinsic contribution is limited to the desirable charge carrier.) Conversely, the band gap must not be too wide or it will prove impossible to maintain reasonable mobilities and effective masses. Given the basic material with these attributes, it must be doped to bring the charge carrier concentration u p to the optimum without incurring degeneracy and destroying mobility. As can be seen from the
215
THERMOELECTRICITY
simplified expressions above, one wants the Fermi level very near the conduction band; this is the basic purpose of the "doping" and the reason for an optimum carrier density. Since the doping will be quite heavy by comparison to the more familiar semiconductor cases, consideration must be given to the solubility of the material for impurities that contribute the desired charge carrier. I t is, of course, desirable that the dopant also contribute to a greater anharmonicity of the lattice. The specifics, the practical aspects and the lack of understanding of the above factors will be the subjects of the rest of this article.
111. MATERIALS Table I summarizes the materials systems of greatest interest at present. Obviously, no attempt has been made to include all the promising systems: in fact, this table is different than it might have been a few years ago. In the table, column 1 gives the system and the most studied combinations. The second column lists the approximate temperature at which the system appears to be most useful. This factor is included because high temperature operation is of great interest in power generation. (Below about 600°K or, roughly, 300"C, the material is useful primarily for refrigeration.) The third column gives the best results (maximum z) obtained in a given system which appear to be TABLE I. MOST PROMISING MATERIALS' SYSTEMS FOR THERMOELECTRIC APPLICATION -
System and most studied combinations
Temperature
I-IV compounds Ag with T e and Se
to 600°C
111-V compounds Ga and In with As and Sb IV-VI compounds Ge and Pb with Se and Te V-VI compounds Bi and Sb with Se and T e I-111-VI (chalcopyrite structure) Ag, .Cu-Ga-Se, T e Rare earth compounds Ce, Sa, Gd, T h with 0, S, Se, T e
to 700°C PbTe to 550°C GeTe to 650°C BiBTe3to 350°C SbzTeoto 350'C
-
Best results Maximum in "C-' AgzTe 2 = 1.3 X Ag,Se 2 = 2.5 X AgSbTe, a = 1.9 x
-
InGaAs Alloy z PbTe z = 3 x G e T e z = 1.2 x Bi2Te3 SbrTe, > 4 x 10-3
+
550°C
CuGaTep z
1oOO"C
Ce3S, z
-
-
3 x
216
FRANK E. JAUMOT,
JR.
reasonably confirmed (there have been a number of startling announcements within the last five years which have not been confirmed or, to the best of this writer's knowledge, reproduced). Before discussing Table I, it may be well to mention the omissions: all of which were omitted because they did not live up to their earlier promise in terms of providing high values of z. T h e 11-V system is omitted although it was one of the earliest to exhibit promise, T h e system includes ZnSb on which Maria Telkes did considerable work and a patent of hers (14) on this material indicates figures of merit as high as 2.2 x. although the data show considerable scatter, This system also includes CdSb, which Justi discussed in a patent application in 1952 (15). T h e transition metals combined with 0, S, Se, and T e have also been omitted in spite of the considerable publicity received by the mixed valency oxides. Although they have not yet lived up to their early promise, they do warrant some discussion since they have several attractive features. First of all, they would be outstanding from the standpoint of high temperature operation, perhaps even to 1500°C. Further, the properties of mixed valency oxides are not appreciably affected by traces of other impurities and they are resistant to radiation damage, in contrast to more common semiconductors, Since efficiencies as high as 5 % have been achieved (16), it is sincerely hoped that the rather formidable difficulties can be overcotne. For additional discussions of these materials, see Zener (16) and Heikes (17). T h e I-VI compounds have produced some encouraging results. As early as 1952, published results described n- and p-type alloys with figures of merit of about per degree (18). Mobilities are high in these compounds and their lattice thermal conductivities are quite reasonable; in fact, the lowest lattice thermal conductivity yet reported for a promising thermoelectric material was exhibited by AgSbTe, (19). (It should be noted that this compound is not truly a I-VI but is really a mixture of a I-VI and a V-VI compound.) T h e 111-V compounds which exhibit the highest mobility of any semiconductor compounds are included in spite of an unpIeasantIy high thermal conductivity which presents a critical problem that may not be solvable. Also, the effective masses are small and the energy gaps are small ( 4 . 4 5 ev). I n general they appear to have potential (exhibit reasonable z-values) only near the upper end of their useful temperature range where the thermal conductivity decreases. Thus, they may be useful in special mechanical configurations where they form a portion of a device which must sustain higher temperatures. Even then only a limited number of 111-V compounds would appear
217
THERMOELECTRICITY
to be useful; that is, those which have large differences in effective mass for electrons and holes and will operate at reasonable efficiency even in their intrinsic temperature range. As in the I-VI compounds, mixtures are used for higher z-values than can be obtained from the "pure" binary compounds (see table). I n addition to the mixture of InAs and GaAs, InAs InP has shown promising results (20). T h e IV-VI system contains one of the two compounds that have been used in attempts to fabricate relatively large size prototype generators and is commercially available. Specifically, reference is to PbTe, which has proved to yield (reproducibly) the second highest z-value of any material. This system also includes GeTe, which exhibits very modest mobility, a large effective mass, low lattice thermal conductivity and quite reasonable total thermal conductivity which is largely electronic (it is desirable that it should be). GeTe is alwaysp-type, apparently having a greater than optimum number of charge carriers, and exhibits its maximum and reasonable values of x (-0.9 x per degree) in the temperature range from 500 to 650°C (20). Thus, GeTe is promising as a high temperature material but, like the 111-V compound, may have to be used in special mechanical arrangements since at room temperature the x-value is quite low. T h e V-VI compounds and mixtures thereof have remained for almost a decade the best materials available in spite of their rather severe temperature limitations, Bi,Te, has been by far the most widely studied single compound, I t has a low lattice thermal conductivity (0.016 watts/ cm°C), a moderate mobility (1 140 cm2/volt-sec for n-type and 485 cm2/ volt-sec for p-type) and a moderate effective mass. (References I , 2, and 20 give extensive bibliographies on this system.) However, it has been combinations of Bi,Te,, Bi,Se,, and Sb,Te, which have given the highest figures of merit. T h e reasons for these mixtures will be discussed below, but perhaps it should be noted here that, in addition to the mixture of two compounds, additional doping is used; in the V-VI compounds CuBr is perhaps the most popular and one of the more effective secondary dopants. Work on the I-111-VI compounds has been relatively recent (21). However, early reports indicate they have moderate to low Seebeck pw/"C) good conductivity ( p 4 x lo-, ohm-cm) coefficients (-100 and reasonable thermal conductivities (-0.02 watts/cm°C). T h e value of z given in the table for the most promising compound is a calculated value based on measured values of S and u and extrapolated values of K . At the moment, the rare earth compounds appear to be the most promising for high temperature applications, principally power conversion. Their particularly desirable features are a low lattice thermal
+
+
-
218
FRANK E. JAUMOT,
JR.
conductivity and good chemical stability above 1000°C. T h e pure sesqui-compounds have Seebeck coefficients in the range from 50 to 600 pv/OC (22) which are good, but the charge carrier mobilities are probably low. I n view of the low mobility, their reasonable conductivities (near lo3 ohm-l cm-1) indicate very high carrier concentrations. There are several disheartening facts illustrated by Table I. First, there is not yet a truly high temperature material sufficiently good to be included. This is not as unfortunate for the power generation outlook as it could be if the applications were to be confined to the more normal combustible fuels. That is, the attractive possibilities of focused solar and nuclear heat sources fall in the range from 300 to 800°C and several of the listed systems could be adequate in special applications. Somewhat more disappointing is the fact that Bi,Te, and PbTe base materials (particularly Bi,Te,) are still the best available. Figures of merit of nearly two were achieved in Bi,Te, in 1955. This is undoubtedly because these materials are the most obvious ones meeting the .empirical guides outlines above and because of the natural (and proper) tendency to attempt to improve and optimize a good material, once it has been fabricated. Significant progress is being made in the other systems to a degree that leads one to believe that with an expenditure of effort equivalent to that applied to Bi,Te, and a better understanding of some of the materials anomalies discussed below, materials other than those with a Bi,Te, and PbTe, base will be the standards. A possible case in point is that of the announcement (23) of gadoliper degree. nium selenide (Gd,Se,) with a figure of merit of 45 x So far, no one appears to have confirmed or reproduced these results. However, there is good reason to believe that the values quoted were not the result of serious measurement errors or gross misinterpretations. I t is quite possible, when the extreme difficulty of preparing and handling these materials are considered, that some metastable form of the material was prepared or perhaps just a fortuitous, highly defect structure. Returning to the possibility of measurement error in this system, the difficulty in determining the parameters must have led to some error because the reported ratio of o to K was significantly greater than the Wiedemann-Franz ratio. However, correcting by this amount still leaves a remarkable material and it is not surprising that there is appreciable effort being applied to it, both here and abroad. Finally, it is disheartening that all of the good alloys and compounds involve Group VI materials and particularly tellurium which is not readily available at low cost. (It is an interesting sidelight that the demand for tellurium for thermoelectric applications appears to be driving up the price of tellurium copper). T o be sure, the rare earth sulfides
THERMOELECTRICITY
219
so far appear to be the best, but more work could easily shift this to tellurium and the requirement for heavy atomic weight elements would indicate it probably will. Obviously, the 111-V compounds do not include Group VI material but it is the opinion here that the thermal conductivity of these materials present an almost insurmountable problem except for applications over a very narrow temperature range. Deferring the discussion of the detailed diffic.ulties and the lack of understanding of certain of the critical materials factors mentioned above until after the practical considerations, the materials discussion can be concluded with a brief statement of the empirically determined trends of these factors. In compound semiconductors, the energy gap decreases with increasing atomic number. This is to be expected since the general rule of thumb is that the energy gap increases with increase in bond strength (e.g., in ionic binding, the energy gap increases with increasing ionic character). Here, the situation is that of the binding energy of the valence electrons becoming smaller as the atomic number increases. T h e temperature dependence of the energy gap for most semiconductors is negative and ev/deg. There is an interesting lies between about -3 and - 5 x and important series of compounds which do not obey either of these empirical rules, which will be discussed below. Mobilities, which depend so strongly on scattering and thus, on the degree of perfection of the crystal lattice are not as easy to categorize. Furthermore, since mobility and effective mass are intimately related, they can by considered together (but only in a general discussion). Generally, in homopolar compounds the electron mobility increases rapidly and the effective mass decreases rapidly with atomic number. T h e hole mobility and effective mass change more slowly so that the ratio of electron to hole mobility increases with increasing atomic number. Possibly, this explains why those compounds with more ionic character exhibit only n-type conductivity and may present serious difficulties in obtaining good $-type homopolar materials. On the other hand, some of the 111-V compounds have very high mobilities, much higher than the more homopolar crystals of the elements, so that one must conclude the influence of ionic type binding is more complicated than the present simple picture. In general, however, a simple band structure introduces certain degeneracies which may be removed in more complex bands; thus, many valleys are usually associated with high mobilities. Also, to a rough approximation, high mobilities are usually found with high Debye temperatures, at least for elements. Since high Debye temperatures mean lattice vibrations of small amplitudes (which is undesirable from the standpoint of lattice thermal
220
FRANK E. JAUMOT, JR.
conductivities) this may be largely a reflection of the dependence of mobility on scattering (crystal perfection). This discussion has been concerned primarily with broad band semiconductors. For compounds with narrow bands (mixed valency oxides), mobility and effective mass have less meaning and conductivity is better treated by another mechanism (27). As a general rule, the useful materials will be highly doped semiconductors or semimetals so that one wants to look for base materials with a certain degree of metallic behavior and reasonable solubility for appropriate dopants. (Roughly, the more metallic compounds are more nearly alloys and tolerate a large range of substitution.) Such compounds can usually be visualized as simple valence compounds obeying rules very similar to the Hume-Rothery rules. Consequently, accurate phase diagrams can be of considerable value in assessing materials potential. A broad temperature maximum and small stability regions imply easy alloying and metallic behavior. Sharp curvature and deep stability regions imply strong compound formation, smaller solubility and less metallic behavior. Along the above lines, it would appear that much could be learned, at the same time a search for practical materials was in progress, by studies of what might be called complex compounds. That is, certain combinations of atoms form very tight bonds (partly ionic) but act in combination as an ion or cation. (An example from the first and second systems of Table I is Ag and In giving (Ag-In)*+ which could be used with, say, selenium or tellurium.) Judicious choice should permit the selection of the proper degree of ionic and covalent bonding to give reasonably tailored energy gaps and mobilities. I t is to be remembered, of course, that generally, the electrical properties of compounds reflect the weakest bond in their structure. Finally, since the trends of the properties of more complex compounds with increasing atomic number, binding, etc., are similar to those of the elements and simpler compounds, these trends could be used to aid the selection of the optimum combinations of compounds.
IV. PRACTICAL CONSIDERATIONS T h e important practical considerations can be grouped into three categories: the figure of merit, materials fabrication, and device design.
A. Figure of Merit Since this quantity is the heart of thermoelectricity from the standpoint of applications, it is not surprising that there has been an enormous
THERMOELECTRICITY
22 1
amount of discussion about the maximum value it can have if, in fact, there is an upper limit. What is surprising is that there are papers still being written, giving very complicated thermodynamic derivations, and stating that the product zT cannot exceed unity. This, in the face of voluminous, well established data showing values well in excess of this over wide temperature ranges. T h e fact is that any attempt to develop a maximum limit short of the Carnot limit by thermodynamic means indicates a lack of understanding of either thermodynamics or thermoelectricity. Actually, it is felt safe to make the general statement that no entirely convincing argument has ever been advanced for applying thermodynamics to transport phenomena. It is possible to construct hypothetical models which would exhibit arbitrarily large values of the figure of merit. Unfortunately, such models have properties that one feels are just not obtainable within the framework of present materials knowledge and experience. Thus, at the moment, the best one can do is to make calculations using such constraints as are indicated by present knowledge, for a particular class of materials. T h e treatment which follows is basically a condensed version of that due to Donahoe (24) and the reader is referred to the original paper for the details. T h e equation for z, Eq. (I), can be written, using Eq. (3), in the form, z=
s 2 0
Kph
+ LUl’
Now, for an extrinsic semiconductor, S is a linear function of the logarithm of the electrical conductivity and the thermal conductivity is a linear function of u. These are the constraints, or interdependencies of the parameters used which permit us to examine three cases in terms of the relative sizes of Kph and K e l . (1) When Kph is much larger than LuT, the maximum of z occurs at the maximum in S2u. This is the situation that led Ioffe (2) to the conclusion that S should be exactly equal to 2(K/e) or 172 mv per degree. With reasonable values of Kph and u, it leads to values for z (max) lower than many measured values. Further, it is not an interesting case since present good materials have Kph sufficiently small that the variation of the thermal conductivity with electrical conductivity cannot be neglected. (2) T h e case when Kph is approximately equal to LaT, occurs for most presently used materials. This differs from (1) in that the maximization of z.with respect to the electrical conductivity yields a value of S given by 2(k/e)(K/Kph); that is, the 172 pvldeg is multiplied by the ratio of total to lattice thermal conductivity. As in the general case, the
222
FRANK E. JAUMOT, JR.
maximum figure of merit will be determined by the mobility of the charge carriers, their effective mass and the lattice thermal conductivity. I n spite of the rather lengthy discussions above citing the relationship between mobility and effective mass, the empirical evidence indicates that these quantities vary independently over rather wide limits. It would appear that the mobility is limited by crystalline imperfections; when these effects are removed, reasonable values of the parameters indicate per that the figure of merit should have values of the order of 6 x degree in the case of Kph comparable to LOT. Bottger (25), treating what he states to be the “absolute” case, also arrives at a value of approximately 6 x low3 per degree for the maximum figure of merit. Actually, what he does is calculate values of the parameters on the basis of simplified semiconductor theory, assume the Wiedemann-Franz ratio and assume a lattice thermal conductivity of 0.005 watt-cm-1 deg-I. He also states categorically that it is hopeless to look for materials for thermoelectric generators outside a conductivity range from lo2to 3 x lo4 ohm-l cm-1. Such statements are unfortunate, even when they encompass a large range, since they are based on theories which can lead to large errors in the estimation of the most common parameters in even the simplest semiconductors. A much better understanding of the complicated nature of transport phenomena in rather peculiar material is needed before such arbitrary statements can be considered well founded. (3) When Kph is much less than LaT, the analysis becomes more complicated and has been treated by Price (26). Using his analysis and the theoretical value for the Wiedemann-Franz-Lorentz number, one can calculate a figure of merit of 17 x per degree at room temperature. To date, semiconductors with normal Wiedemann-Franz ratios are not available. Also, this figure of merit requires a semiconductor with a very small energy gap as well as values of the lattice thermal watt-cm-1 deg-l. On the basis of our conductivity of less than best present materials, either an order of magnitude reduction in lattice thermal conductivity or an order of magnitude increase in carrier mobility, or some combination of these is required to achieve this figure of merit. But this is precisely the difficulty with calculating maximum figures of merit on the basis of models which are inadequate to cover the phenomena. For example, our entire concept could be changed if a new mechanism were found which would reduce lattice thermal conductivity significantly without decreasing carrier mobility. Some of the discussion regarding manipulation of the figure of merit will be left to the practical consideration of materials. However, there is one limiting factor on the figure of merit that applies to any real
THERMOELECTRICITY
223
material. It is the maximum temperature at which a material will have a useful figure of merit. This temperature is limited by: the melting point or chemical stability (volatility) of the material ; the solubility limits of doping material ; the onset of intrinsicity (with the exceptions mentioned above) ; and anomalous thermal diffusion caused by radiative effects, ambipolar diffusion or exciton conduction. All of these factors were discussed previously but what can be done about them in the practical solution ? Very little can be done, fundamentally, about the melting temperature of a useful material unless a dopant which is useful for adjusting the charge carrier concentration increases the melting point. Similarly, volatility problems can only be overcome by encapsulation and this is seldom satisfactory for very long term usage. For example, PbTe tends to loose T e by evaporation and the coatings tried so far have not succeeded in preventing degradation at the proposed 700°C operation temperature; thus, the 550°C listed in Table I. T h e solubility limit of the material for a particular dopant is, again, a fundamental property of the material. However, the use of different dopants generally give one a fairly wide range of possibilities. Finally, doping can frequently be accomplished by adding an excess of one of the constituents of the compound, providing a two phase material is nor formed. T h e temperature at which the material becomes intrinsic can be controlled in several ways. First, and perhaps best, is to choose that member of a homologous series which has the largest energy gap; the larger the energy gap the higher the usable temperature. When a semiconductor becomes intrinsic, the Seebeck coefficient usually falls rapidly since the contribution of the holes and electrons have opposite signs and tend to cancel. At the same time the electronic portion of the thermal conductivity increases, since both hole and electrons transport heat down the temperature gradient. T h e undesirable effects of the minority carriers can be partially suppressed by over-doping the material; this will provide an excess of majority carriers and the material can be heated to a higher temperature before it becomes intrinsic. However, the figure of merit is reduced in the lower temperature region by this over-doping. T h e anomalous thermal conduction items were discussed under Basic Considerations and need no further discussion here. All in all, the best possibility for increasing z is to decrease the lattice thermal conductivity. Perhaps the best way to achieve this is by introducing into the lattice another substance, either element or compound, which crystallizes in a similar lattice and has approximately the same
224
FRANK E. JAUMOT, JR.
lattice constant. (Such a system should exhibit fairly extensive solid solubility). T h e distortion of the basic lattice by the added impurity is then relatively small and is limited to crystal regions in direct contact with impurity atoms. Such distortions are reasonably effective in scattering thermal oscillations, whose wavelengths at normal temperatures are of the order of the lattice constant. As a result, lattice thermal conductivity may be reduced appreciably, but the current carrier mobility is not affected significantly because lattice periodicity is not greatly affected and, thus, the electron waves with their longer wavelengths are not effectively scattered. Thus, this technique can be quite effective and has been experimentally demonstrated; for example, in the InAs-GaAs system (27). I n this system a ten percent addition of either constituent into the respective pure compound lowers the thermal conductivity of the lattice by about 70%. However, it should be pointed out that this technique is most successful when the material has a high figure of merit well below the melting temperature. This is due to the fact that the lattice thermal conductivity of a pure compound generally varies as T-1 while the conductivity due to the alloying is independent of temperature. Consequently, at high temperatures the conductivity may be low enough that alloying will not change it appreciably. There is another, little explored possibility of obtaining a low thermal conductivity. I t is fairly well established that the lattice thermal conductivity of a solid generally exhibits a sharp drop just before the temperature or composition reaches a point where a second phase is ready to form. Thus, it may be possible to adjust the stoichiometry of a semiconducting compound to the point where a second phase is almost ready to form. This will probably lead to an excess of charge carriers over the optimum doping level, but this usually can be corrected by counter doping (if, in fact, it turns out to be intolerable).
B. Materials Fabrication T h e actual details of fabricating materials have been covered many times in the literature and will be little more than touched on here. With respect to growth of the single crystal, and at the present state of the art single crystals are highly desirable when obtainable, there are almost as many variations of methods as there are laboratories. However, the Rridgman technique, or modifications thereof, are generally satisfactory. It is usually quite important to ascertain that the constituents of the material are separately free from unwanted impurities. A common
THERMOELECTRICITY
22 5
example is oxygen. Since most materials contain tellurium it is easy to form a cement or glass of T e O plus a second oxide which will cause severe strain resulting in extreme brittleness. Also, in many cases (e.g., PbTe) oxygen tends to produce acceptor levels which are undesirable. Actually, the problem can be generalized to that of contaminants introduced inadvertently. Such contaminants may act as doping agents. Using the oxide mentioned above as an example, approximately 0.002 mole of TeO, will reduce the Seebeck voltage of Bi,Te, by nearly 20 %(28). Also, the example of the cement formation given above is just one possibility of chemical reaction of an impurity with the material. Finally, an impurity can act as a nucleation site for crystal formation, preventing the growth of a high quality crystal. Obviously, any of the above effects could mask the effects one is looking for either with respect to the original crystal constitution or more likely, with respect to deliberate doping. Once one has a good basic material, stoichiometric or not, it will usually require additional doping. Such doping is needed to produce the optimum carrier concentration mentioned previously. It also can produce other effects which will be discussed specifically below. But first, there are a few general remarks. T h e control of doping concentration is more than a matter of avoiding contaminants. Frequently it is complicated by the fact that the congruent melting temperature does not occur at the melting temperature corresponding to the stoichiometric composition. [e.g., in PbTe the congruent melting temperature lies on the tellurium-rich side (29)]. Thus, with standard Bridgman techniques, it may be impossible to obtain a completely stoichiometric crystal from a stoichiometric liquid. Actually, this is an area that needs further exploration; that is, the question of whether or not one has, or wants, a stoichiometric compound. Usually, higher figures of merit are obtained from nonstoichiometric compositions. Bi,Te, is a good example. But, what does a nonstoichiometric composition mean in a case like this? It is entirely possible, for example, that a bismuth-rich composition is a mixture of Bi,Te, and BiTe. Some workers have started to worry about this problem and it appears that more should be done. It is important to know if the nonstoichiometric composition is a mixture, whether the figure of merit is increased because of doping only, or because of formation of a defect structure, or because the thermal conductivity is reduced as described above. Care must be exercised in making arbitrary assumptions as to what doping will do because of the detailed nature of the transport phenomena involved in thermoelectric effects. For example, in doping a bismuth-
226
FRANK E. JAUMOT, JR.
tellurium-selenium material, Bennett and Wiese (30) found that Pb and I gave the expected results but Ag did not give the obvious result. That is, if silver entered the lattice partially ionized on either the Bi or T e sites, it could be expected that the siver atoms would be acceptors and not donors as they actually are. More specifically, if silver replaced bismuth, its single s electron would be promoted to the p state with two vacant orbitals giving rise to p-type conductivity. However, Bennett and Wiese conclude that the n-type conductivity is due to silver entering the lattice interstitially. Then, because of the high dielectric constant, the Bohr radius of its outer s electron would be of the order of 40 A, its excitation energy would be negligible and it could easily enter the conduction band, causing n-type conductivity. Since CuBr is perhaps the most popular dopant to obtain n-type BiTe and since the assumption has generally been that bromine is the donor, with the role of Cu unknown, an investigation of the type carried out by Bennett and Wiese may change this presumption. T h e rare earth compounds also need similar work. Their mixed d-states in the valence and conduction bands may have very different transport properties than the widely studied semiconductors which are mostly s-p band conductors. I n spite of the extraneous problems mentioned above, doping is usually performed to obtain one or more of three specific effects. These are: changing the width of the band gap, optimizing the number of charge carriers, and introducing scattering centers for charge carriers and phonons. We have discussed the introduction of phonon scattering centers to reduce the thermal conductivity in connection with the improvement of the figure of merit. Similarly, the optimization of charge carriers was a basic consideration. Perhaps the most important effect of doping, widening the band gap, occurs for intrinsic materials, That is, as described above, when a material becomes intrinsic there are too many carriers in the conduction band for optimum figure of merit; this is usually first observed as a too-small Seebeck coefficient. Then, if a doping material which increases the band gap can be found, its addition may increase the gap to the point that the number of carriers in the conduction band is reduced to a number close to the optimum. A good example of this is the addition of phosphorous to indium arsenide (31). At about IOOOOK, the addition of roughly 5?4 phosphorous results in an improvement in the figure of merit of about 15 yo. It has been reported that mobility increases with doping in the HgTeCdTe system (32). This is contrary to what one normally observes but
THERMOELECTRICITY
227
if it be anything more than cadmium filling mercury vacancies it could be an extremely important observation. I n any event, it warrants additional investigation on this, and other, systems.
C. Device Design There is little point in discussing the detailed design of any particular device ; almost any issue of the various trade or engineering journals will provide examples. Suffice it to say that the mechanical and thermal problems are formidable for any operating device of reasonable size. There are some more general considerations, however, that frequently receive too little attention. Each device presents a dual problem; the electrical and the thermal circuits. These circuits interact and the interaction is generally quite complex. Although detailed analyses must consider both, valuable clues as to the design can be obtained from the much simpler process of analyzing them separately. T h e electrical circuit presents little difficulty but the thermal circuit should probably be treated by irreversible thermodynamics [Clingman has given a simplified treatment (33)]. Under the category of heat problems one has to know the thermal operating conditions. For example, the optimum efficiency of a generator operating in a constant temperature situation (34) has one value, but operating under constant heat flux conditions the efficiency is different (35). So is the optimum power output and the optimum ratio of internal-to-load resistance. I n actual operation, the temperatures of the junctions of thermoelectric generators are not constant, but depend upon the electrical loading. More accurately, the operation is between real thermal sources for which the temperature depends on the heat load; since the heat input and output depend upon the electrical load, the junction temperatures depend upon the electrical load. This has been treated by Gray (36). Basically, the analysis shows that the generator is no longer electrically linear and the electrical loading condition for optimum efficiency is different from the constant temperature case. T h e quantitative difference is significant even for materials with modest figures of merit and increases with increasing figure of merit. Aigrain (37) has considered the question as to whether an optimum hot source temperature exists and concludes that it does. He states it should be 4 to 6 times the cold source temperature: for operation with the cold junction close to room temperature, this means a hot source temperature of about 700-800°C. T h e matter of high temperature operation is extremely important
228
FRANK E. JAUMOT, JR.
and, from the materials descriptions above, it is obvious that the best materials available operate only at modest temperatures except for those which operate over a very limited range. Thus, it is necessary to attempt to increase the temperature range of materials which must be used marginally. This can be done only by changing the impurity concentration. The principles involved are given above. Figure 1 gives a possible
I
1FIG. 1. Illustrative plot of the figure of merit versus temperature for different doping levels of the material; generally, n, < n2 < ns.
plot of the figure of merit for a given basic material as a function of the temperature T for several values of the doping level. The usable figure of merit is, of course, the envelope of these curves (the actual device must have either segmented or graded-leg couples as discussed below) unless one wants to use only the narrow temperature range represented by one of the doping levels. Obviously, to calculate the desired doping and/or the temperature range it is necessary to know the efficiency of the device in terms of the parameters of the material for the case of temperature dependent parameters. This becomes quite complicated and, in general, it is easier to determine doping empirically. As implied above, Fig. 1 could just as well represent the z versus T curves for three different materials as for three doping levels. In either case, if a satisfactory means of fabricating the legs of the thermocouple can be found, the couple will work with considerably enhanced efficiency over a greater temperature range. There are two rather obvious ways to fabricate the legs, both presenting some problems. T h e first, and apparently the only one successfully used so far, is to prepare separate crystals of different, or differently doped, materials and solder them
THERMOELECTRICITY
229
end to end to make a single leg of a couple as illustrated schematically in Fig. 2. T h e difficulties here are rather straightforward. T h e contact resistances at the junctions of the materials subtract directly from the efficiency. Also, there is the mechanical problem arising from mis-
FIG.2. Schematic of a thermocouple using segmented legs for maximum efficiency over the temperaturerange involved. MI,M,, and M , are different thermoelectricmaterials.
matched expansion coefficients; this is magnified if M,, M,, and M3 are completely different materials. T h e second and, at first glance, more promising method is to use for each leg a single material in which the doping is graded. Figure 3 shows how such a system could perform. (Actually, the concept of “doping” is being stretched pretty far here.) Alloys are known in which the energy gap increases with concentration of the second material; also, the lattice thermal conductivity behavior shown is known to exist. However, for thermoelectric use, the only system of this nature yet reported is InAs-GaAs (38) and it has the problems mentioned above as associated with 111-V compounds. T h e attractiveness of this method would indicate it will receive considerable attention. For example, none of the problems cited for the case of different materials occur here since M,, M2, and M , are all the same material (or an infinite number of infinitesimal segments having perfect electrical and thermal contact). However, this method has its own problems and they are quite fundamental. First, it is extremely difficult to grow the legs or branches of the thermocouple with the precise composition gradient one would like. Second, the dopant will diffuse in such a way as to level the concentration gradient; in fact, the driving force for the diffusion, the chemical potential, will probably be quite high, making the diffusion appreciable at quite low temperatures. Another, but unrelated, method of using different materials to span a large temperature difference is to cascade multiple junctions as illustrated schematically in Fig. 4. This permits the temperature difference across a given stage to be relatively small, while using each material in the temperature range of maximum efficiency. It also minimizes the
230
FRANK B. JAUMOT, JR.
mechanical distortion. Unfortunately, calculations (39) show that, from the standpoint of efficiency only, there is no substantial advantage i n cascading. Further, for maximum efficiency of a cascade, the individual efficiencies of the various stages should be equal and this is not easy
0
3
W
A
20
60
40
H B-
A
20
60
40
80
XBFIG. 3. Illustrations of how a graded therrnoelement might be prepared, showing change in band-gap energy and lattice thermal conductivity with change in composition.
to achieve in practice, Inevitably, economics must be considered and the cost per watt in generation (or Btu in cooling) is higher when couples are cascaded, because the advantage in efficiency is small and because cascading requires more material and fabrication labor.
THERMOELECTRICITY
23 I
In the area of the actual fabrication, the method of joining the legs of thermoelectric material to the heat sinks is important and provides an area of development in itself. Contact resistance must be kept to a minimum since the resulting joule heating subtracts directly from the
FIG. 4. Schematic of a three-stage thermoelectric cascade.
operating efficiency. T h e contact must be intimate, covering the complete area (free of voids) so that small “reverse” junctions are not formed and so the entire area can be used for the desired purpose. The reduction in efficiency due to contact resistance can be consideused in a refrigerator rable. For example, for a material with z = 2 x with the hot junction at room temperature, a contact resistance equal to one quarter the sum of the branch resistance reduces the maximum temperature difference by 15 %. Present “soldering” techniques are capable of giving reproducible contact resistances as low as 4 x 10-50hm/cm2 with relative ease. Some laboratories are achieving and are aiming for 2 x 1O-’ ohm/cmz. McConnell and Sehr (40) investigated some very fine soldered junctions (10-50hms/cm2) and found that, in these good junctions, there was no rectification at the junction. They also found there was no relationship between the contact resistance and the mechanical strength of the bonds. Further, they found that junctions produced by ultrasonic soldering (one of the most popular methods) ranged from 5 x lop5 to 2 x ohm/cm2, but could not be made as good as soldered junctions. More important to the present discussion, McConneli and Sehr found that the contact resistance may vary as much as IOOYh around the junction when the average is about 10-50hm/cm2. Also, the contact resistance is not quite proportional to the contact area; large samples are more difficult to bond and the nonuniformity increases with increase in contact area.
232
FRANK E. JAUMOT,
JR.
I n spite of the importance of contact resistance, it is only one facet of the junction fabrication. T h e materials used for the heat transfer and any solder must be chosen carefully. Poisoning of the junction material by diffusion can be very serious even at temperatures of the order of 100°C. I n short, very often the materials used for the heat sink and solder are ones which act as donors or acceptors in the branch materials and which have large diffusion coefficients therein. Specific examples are copper in 1ead.sulfide and bismuth telluride (41); in each case a reasonably sized single crystal becomes poisoned in a matter of hours at 100°C. This poisoning problem may be overcome by choosing contact materials which do not significantly affect the pertinent properties of the thermoelectric material or by choosing contact materials whose diffusion coefficients in the thermoelectric materials are so small that a negligible diffusion occurs at the temperature of operation. I n summary, wherever an intermediate material such as solder is used, it must be of high conductivity, extremely thin, not soften at a temperature in the working range, fulfill the diffusivity requirements mentioned, and it should have a thermal coefficient intermediate between that of the junction material and that of the heat transfer material. Indeed, contact problems provide an extensive field for research within the entire field of semiconductor technology. It was mentioned in the section on basic considerations that the formulas there had been based on an optimization of geometric factors and the ratio of load to internal resistance (see ref. 2 for a discussion of these factors). Actually, in device fabrication and use these optimizations can be taken rather loosely. For example, if the geometric factors are not optimized which means, from a practical point of view, the resistances of the branches of the thermocouple are not matched, it is seldom serious. I n fact, a 20% deviation from an ideal match will seldom result in as much as a 1 yo reduction in the figure of merit. Similarly, the optimization of the ratio of load to internal resistance is not at all critical in the neighborhood of the optimum condition. Finally, it was stated, in the same section, that the Thomson heat is neglected as small in the derivation of most useful equations. Actually, the Thomson heat is not always small in semiconductors and much discussion has arisen on the point. However, the assumption that S is independent of temperature over the temperature range in question is equivalent to assuming the Thompson heat vanishes. Even through this is certainly not true in general, the Thomson heat is still partly accounted for by using the mean value of S , which is what is measured in a completed device.
233
THERMOELECTRICITY
V. THERMOELECTRIC APPLICATIONS At the moment, thermoelectric devices are not economically practical, in this country, for the usual consumer devices. This says nothing more than that the American technological development is such that we don't need the combination oil lamp and radio or the small commercial refrigerators that are available in China or Russia. It is estimated that figure of merit of about 7 x lop3 deg-I would make many applications feasible, particularly refrigeration and certain specialized generation applications. A figure of merit several times larger would be required to make commercial power production competitive. T h e best values of z today are between 3 and 4 x 10-3/deg. (In terms of the dimensionless parameter zT, about unity). But what do these values mean in terms of salable devices? This is answered in Fig. 5 and 6 for the two most popular applications, generators and 35
I
25
-
L
0
I
2
3
4
5
FIG. 5. Optimum efficiency of a thermoelectric generator as a function of the figure of merit for various hot junction temperatures.
refrigerators. For example, from Fig. 5, a z of 4 x deg-l, a hot junction temperature of 600°K (about 325"C), and a cold junction at room temperature would give a generator with an optimum efficiency of 15.5 yo.This would make many applications practical. Unfortunately, both p - and n-type materials are needed with this high value of z and many junctions must be put together with very low 12R losses. Also,
234
FRANK E. JAUMOT, JR.
it is very difficult to transfer heat uniformly to all junctions. Consequently, a significant portion of the figure of merit of the original material is lost in the device. Specifically, small generators with over-all efficiencies of 10% have been fabricated relatively reproducibly, but most reasonably useful ones have been between 5 and 10% efficient. 7 6
5 4 In.
0
u
3 2 I
0
FIG. 6. Optimum coefficient of performance of a thermoelectric refrigerator as a function of the figure of merit for various temperature differentials.
For refrigerators, shown in Fig. 6, a x of 4 x deg-' would result in a coefficient of performance of about 1.3 with a hot junction temperature of 300°K and a d T of 30°C. Note that a hot junction temperature of 500°K and a AT of 30°C would yield a coefficient of performance of 4.5, which is good indeed. This illustrates the importance of knowing the hot junction temperature if one is concerned with operational efficiencies. There is another facet to the refrigerator application. T h e usual vapor compression machines do not lower in cost, appreciably, when they are reduced in size. O n the other hand, the cost of thermoelectric refrigerators is almost directly proportional to their size (except for very large devices where complex cooling problems arise). Thus, for small capacities, thermoelectric refrigerators (or thermostated chambers) are actually economical now. I t was relatively obvious from the beginning that the first consumer
THERMOELECTRICITY
235
applications would be specialty devices and that these would involve cooling or heating rather than generation. Such devices are now slowly coming into the market. However, it is in the area of military requirements that thermoelectric devices can be truly practical. A simple listing of the advantages of these devices over more conventional ones should provide the best proof. For convenience, the advantages of thermoelectric generation will be considered. (1) No moving parts. (2) Virtually maintenance free. ( 3 ) Noise free. (4) Will operate at higher temperatures. In general, for high ambient temperature applications, a static thermoelectric generator cannot be touched by rotating machinery. Heat source temperatures of 1100" to 1600°C and sink temperatures of 600°C are conceivable and it is wellknown that the operation of rotating machinery presents serious difficulties at temperatures well below 600°C. ( 5 ) Operates over a wide range of input conditions. Most generators require a fairly limited range of input conditions. In contrast to this, thermoelectric generators will operate over any set of input conditions as long as there is a temperature gradient. (6) Requires no external support except heat. This heat may be derived from any source. I n contrast, a rotating generator requires a motor to drive it, and that motor requires a special fuel. (7) Eficiency is independent of size. All conventional methods of converting heat into electrical power become less and less efficient as the power rating decreases. T h e efficiency of a thermoelectric device is determined solely by the Carnot cycle and by the index of efficiency of the thermoelectric device. As to what can be done today with thermoelectric devices, the answer is almost anything that can be afforded. I n short, there is no particular problem with capacity, weight or size (see Jaumot, I ) ; however, thermoelectric devices are just too expensive (in one way or another) at present.
VI. THEORY AND PROBLEMS T h e parameters of interest in thermoelectricity all involve transport phenomena. Thus, from a theoretical point of view, one would like to set up the Boltzmann equation, in a completely general fashion from first principles and solve it. This has proved, so far, to be impossible
236
FRANK E. JAUMOT, JR.
for all practical purposes. Possible methods of attack and lengthy discussions as to what is wrong with the approaches made to date are given in refs. I and 34 (along with extensive bibliographies) and need not be repeated here. However, there are some general problems involving both the theoretical and experimental aspects of thermoelectricity which merit some discussion and which need further investigation badly; most of these are directly related to items considered in previous sections of this article. There is little doubt that the major advances in this field will come largely through a better understanding of transport phenomena in insulators, semiconductors and semimetals. These are remarkably little understood at present, particularly in the higher temperature ranges. I n fact, it is well-known that the uncritical use, for even the simplest system, of very familiar theories for semiconductor phenomenon can lead to large errors in the estimation of such common parameters as the density of charge carriers. What is needed is more basic knowledge. Theoretical work is needed very badly, coupled with supporting fundamental research on materials. On the other hand, present knowledge is sufficient to continue materials development and device design criteria studies, particularly along the lines of more basic research. Perhaps it would be well to consider these three areas in reverse order.
A. Device Design Criteria T h e literature is full of design criteria for devices and aids to designing devices (see for example Clingman, 33, and StoIl et al., 42). Yet it has never been proved that the present design of the basic thermocouple is really the one to give optimum performance. The problem of low resistance contacts dominates the size and weight of a device (see Section 11, C above) but, so far, much of the work has been done on the roughest kind of “cut and try” basis. T h e diffusion of the material at the contact points as well as the diffusion of the doping agents determines the operating life of a couple. Information is badly needed in this area, particularly good experimental data on the diffusion of solutes at low concentrations. Also, since these devices will be used over long periods of times with a temperature gradient impressed upon them, one needs to know how the thermoelectric material’s behavior will vary as a function of time because of these temperature gradients. Clearly, one would expect more constant performance characteristics if the legs of the couple were doped with a concentration gradient representing the long term steady state distribution (due to diffusion) rather than uniformly doped. (Apparently this
THERMOELECTRICITY
237
has never been tried.) An interesting corollary question is, is this concentration gradient compatible with graded legs or do the diffusion aspects rule out graded couples? T h e answer to the latter half of the question is almost certainly no, but it should be investigated for any specific system. Further, in the area of diffusion, is there a way to arrest or control it in a favorable way as opposed to means which introduce other deleterious effects ? Most attractive thermoelectric materials are very brittle ; in fact, most of their mechanical pwperties are unfortunate. Thus, studies in all areas of mechanical properties are needed to aid intelligent device design. Finally, very little work has been done on the performance of materials except under the most benign environmental conditions. One needs to know how devices work under actual use conditions and the limits to which one must go in packaging them so they are useful. Once more is known about some of these factors, fabrication techniques can be studied more intelligently and production means that will provide quantity production at reasonable costs can be sought.
B. Materials Development I n the materials development area, more detailed studies are needed on systems, other than the IV-VI and V-VI, which appear to have desirable thermoelectric properties. Fortunately, these appear to be going forward. Since nearly all the best materials to date have been ternary or quaternary systems doped with still other materials, a good method is needed for predicting optimum composition, doping level and the probable maximum figure of merit from measurements on one or a few samples. T h e process of finding these conditions by cut and try is much too laborious and expensive. Several computer programs are being worked on and one, in particular, appears to be promising; it is sincerely hoped so. As mentioned in Section IV, A , it has been shown that the thermal conductivity of a material can be reduced by alloying; what is needed now is confirmation and explanation of the evidence of Blair (32) and Lawson et al. (43) that mobility may also be increased by alloying. Particularly, one needs to know if this is, in any way, a general possibility. Positive evidence of the importance of extreme chemical homogeneity and perfect crystal texture is needed. Certain evidence supports the general feeling that they are important, but just how important is a significant factor to production ease and costs.
238
FRANK E. JAUMOT, JR.
C . Theoretical and Associated Fundamental Work I t is in this area that the field is in the worst shape and, perhaps, has the greatest need. As mentioned above, what is really needed is to set up the general transport problems with all the electrical, magnetic and thermal effects and solve it. Considerable work has been done on transport phenomena due, in part, to the economic importance of semiconductors, but a fair portion of it leaves much to be desired ( I , 34). Still, much remains to be done even within the present framework and within the bounds of present understanding. T h e discussion here will be concerned with items in these areas, the understanding of which appear to be vital to the future of the thermoelectricity. First of all, the thermal properties are i n bad shape. I n spite of the work of Krumhansl and others, we have no exact theory of thermal conductivity (but ref. 8b will provide an introduction to the subject and additional references). As to a specific item, there is one thermal “fact” which has almost universal concurrence. That is, the higher the melting point of a material, the larger is its Debye characteristic temperature and hence, its lattice thermal conductivity at a given temperature. Yet, a thermal conductivity of less than 0.01 watt/cm deg C has been reported for samarium sulfide with a melting point near 2000°C (44). If this and similar data can be confirmed, it badly needs explaining since if the field has to face high thermal conductivity in high melting point materials, it would have to face a severe restriction on efficiency of high temperature devices. As to the explanation needed, it does not appear enought to say simply that the concept of T , means high Kph arises primarily from the assumption that phonon-phonon interactions are the main source of thermal resistance at elevated temperature. T h e Seebeck coefficient itself is far from understood as a transport phenomenon. To cite one little discussed example, condider the case of the silicides of iron, cobalt, manganese, and chromium. T h e Seebeck coefficient of some of these compounds are higher than would be expected from a classical model, either for a semiconductor or metal. Generally, the behavior is such as to suggest contributions arising from sources other than the simple presence of charge carriers. Two possible effects, among those available, stand out because of the relatively low thermal conductivity of these silicides. T h e first is the phonon drag effect (45). T h e second is related to the magnetic properties of the materials. Friedel (46) has shown that contributions to the Seebeck effect can arise from the fact that the two directions of spin are no longer equivalent when considering the interaction between electrons and quasi-
THERMOELECTRICITY
239
bound states, as they exist near the Fermi level of transition metals. Perhaps a more detailed investigation of Friedel’s model could provide valuable information. Another area that needs work is the energy gap. The minimum energy gap of a semiconductor is usually obtained from either the infrared absorption edge or from measurement of the resistivity or Hall coefficient as a function of temperature. In elemental semiconductors, the optical and thermal energy gaps agree quite well but they do not in the heavy compounds. Excess charge carriers won’t explain the difference and although defect structures might, more detailed knowledge is needed, particularly as to what the meaning of this departure is in terms of other properties. T h e familiar parameters including energy gap, effective mass, dielectric constants and mobility, although exhaustively studied and discussed, still are without any simple theory that can describe their change with temperature and other external forces. There are a number of empirical rules but why are the interesting lead compounds PbS, PbSe, and PbTe, exceptions to nearly all of these rules? T h e positive energy gap dependence on temperature, the T5I2dependence of mobility, and the virtually unchanged energy gaps in this homologous series of lead compounds are very desirable for thermoelectricity, making it even more important that they be understood. One final item: a high dielectric constant is desirable in order to have less scattering of the charge carriers. But the well-known Moss relation correlates a high dielectric constant with a small energy gap. Fortunately, there are deviations from the Moss relation and understanding these deviations would be of considerable value in searching for new materials. Many more examples of the understanding needed could be cited. But those given should be sufficient to indicate that, whether large scale applications of thermoelectric devices ever appear or not, this field will certainly yield materials which will find applications in many fields of technology. But the most important effect the interest in thermoelectricity will have is the increased understanding of solid state physics. FOR PRACTICAL APPLICATIONS OF APPENDIX. USEFULFORMULAS THERMOELECTRIC EFFECTS
T h e following formulas are based on approximate treatments but yield numerical answers sufficiently in agreement with experiment to be of va1ue.l Their derivations are, in many cases, obvious but may be The assumptions used in the derivations of these formulas are, loosely: a single stage
240
FRANK E. JAUMOT, JR.
found most easily in the Russian literature. A more extensive list of formulas is included in ref. 1.
1. The figure of merit for a single material is
where p is the resistivity and the other quantities have been defined in the text. For a couple composed of two materials,
2. For generators ( 4 If r
pi4 Pa4 = r1+ r2 = --
A1 A,
is the internal resistance of the thermocouple, R is the load resistance, Rlr = m, and if the physical dimensions of the, elements obey the relation (3)
KlP2
where I is the length of the element and A the cross sectional area, the efficiency of a generator is given by m
I=
Tl - To 1
Tl
m+l lm+l Tl-To 1 2T1 m f l Tl
+;--
(4)
where TI and To are the temperatures of the hot and cold junctions, respectively. (b) For maximum power delivery to the load, R = r or m = 1, and
~
~~
system ; the resistivity, thermal conductivity, and Seebeck coefficient are constant within a branch; the Thompson coefficient is zero; any current flowing through the thermocouple is constant; and, in some cases, the contact resistances at hot and cold junctions are negligible in comparison with the sum of the branch resistances. Note that the assumptions are more reasonable for refrigerators than for generators.
24 1
THERMOELECTRICITY
(c) For maximum possible efficiency,
m=M= and Vopt
=
1;- To M - 1 TO T1 &I + TI
3. For refrigerators (a) T h e optimum coefficient of performance is m
Copt
TO = TI-TO
M--
1 0
Tl M+1
where all parameters are defined as above. (b) Copt is reached for values of current and voltage given by
Eopt = ZOPtY
+ ( S , - S,) (To - TI)
where Y is the resistance of the thermocouple, equal to resistance is neglected. (c) T h e temperature difference AT mum value at a current given by
=
(10) yl
+
y2
if contact
( T , - To) reaches its maxi-
(d) T h e number of individual thermoelements (couples) required for a given supply voltage V is V N = ( S , - S,) AT IY
+
TI
Note: Equation (12) gives N for the refrigerator requiring the minimum input power if Eq, (9) is used for I . Similarly, Eq. (13) gives the number of couples required for the refrigerator using the minimum material.
242
FRANK E. JAUMOT, JR.
4. For heating (a) T h e optimum coefficient of performance is reached when the applied voltage is
(b) T h e optimum coefficient of performance is
REFERENCES 1. Jaumot, F. E., Jr., in “Foundations of Future Electronics” (I. Langmuir, ed.), Chapter 11. McGraw-Hill, New York, 1961. 2. Ioffe, A. F., “Semiconductor Thermoelements and Thermoelectric Cooling.” Infosearch, Ltd., London, 1957. 3. Goldsmid, H. J., and Douglas, R. W., Brit. J. Appl. Phys. 5, 386 (1954). 4. Stil’bans, L. S., Iordanishvili, E. K., and Stavitskaya, T. S., Izvest. Akud. Nuuk S.S.S.R., Ser. Fiz. 20, 81 (1956). 5. Herring, C., Phys. Reo. 96, 1165 (1954). 6. Herring, C., Bell System Tech. J . 34,237 (1959). 7. Wolfe, R., in “Thermoelectricity” (P. H. Egli, ed.), Chapter 7. Wiley, New York, 1960. 8a. Lax, B., Revs.Modern Phys. 30, 122 (1958). 86. Krumhansl, J. A., and Williams, W. S., “Thermoelectricity” (P. H. Egli, ed.), Chapter 5. Wiley, New York, 1960. 9. Goldsmid, H. J., Proc. Phys. SOC.(London) B69, 203 (1956). 10. Devyatkova, E. D., Zhur. Tekh. Fiz. 27, 461 (1957). 11. Price, P. J., Phil. Mag. [7] 46, 1252 (1955). 12. Kittel, C., and Frohlich, H., Physica 20, 1086 (1954). 13. Davydov, B. I., and Shmushkevitch, I. M., Uspekhi Fiz. Nauk 24, 21 (1940). 14. Telkes, M.,U. S. Patent 2, 229, 482, (1941). Z.5. Justi, E., U. S. Patent 2, 685, 608 (1954). 16. Zener, C., in “Thermoelectricity” (P. H. Egli, ed.), Chapter 1. Wiley, New York, 1960. 17. Heikes, R. R.,in “Thermoelectricity (P. H. Egli, ed.), Chapter 6. Wiley, New York, 1960. 18. Schwarz, E., Research 5, 407 (1952). 19. Wernick, J. H., in “Properties of Elemental and Compound Semiconductors” (H. C. Gatos, ed.), p. 69. Interscience, New York, 1960. 20. Ure, R. W., Jr., Bowers, R., and Miller, R. C., in “Properties of Elemental and Compound Semiconductors” (H. C. Gatos, ed.), p. 245. Interscience, New York, 1960. 21. Bobone, R., Kendall, L. S., and Vought, R. H., Joint Tech. Soc.-Dep. of Defense Symposium on Thermoelectric Energy Conversion, Dallas, Texas, 1961.
THERMOELECTRICITY
243
22. Vickery, R. C., and Muir, H., Joint Tech. SOL-Dep. of Defense Symposium on Thermoelectric Energy Conversion, Dallas, Texas, 1961. 23. Vickery, R. C., and Muir, H., Am. Rocket Soc. Space Power Systems Conf., Santa Monica, California, 1960. 24. Donahoe, F. J., Elect. Eng. June (1960). 25. Bottger, O., Z. Physik 151, 296 (1958). 26. Price, P. J.. Phys. Rev. 104, 1223 (1956). 27. Abrahams, M . S., Braunstein, R., and Rosi, F. D., in “Properties of Elemental and Compound Semiconductors” (H. C. Gatos, ed.), p. 275, Interscience, New York, 1960. 28. Horne, R. A., J . Appl. Phys. 30, 393 (1959). 29. Miller, E., Komarek, ,K., and Cadoff, I., Trans. Met. Sac. A.I.M.E. 218, 382 (1950). 30. Bennett, L. C., and Wiese, J. R., J. Appl. Phys. 32, 562 (1961). 31. Folberth, 0. G., 2. Naturforsch. 10a, 502 (1955); N. A. Goryunova and N. N. Fedorova, Zhur. Tehh. Fiz. 25, 1339 (1955); Weiss, H. 2. Naturforsch. Ila, 430 (1956). 32. Blair, J., Sc. D. Thesis, Mass. Inst. Techno]., 1960. 33. Clingman, W. H., Proc. Z.R.E. 49, 1155 (1961). 34. Jaumot, F. E., Jr., Proc. Z.R.E. 46, 538 (1958). 35. Castro, P. S., and Happ, W. W., J. Appl. Phys. 31, 1314 (1966). 36. Gray, P. E., Trans. A.I.E.E., Commun. and Electronics 47, 15 (1960); also, Proc. Nut. Electronics Conf. 16, 123 (1960). 37. Aigrain, P. R., in “Thermoelectricity”, (P. H. Egli, ed.). Chapter 10. Wiley, New York, 1960. 38. Rosi, F. D., and Ramberg, E. G., in “Thermoelectricity” (P. H. Egli, ed.), Chapter 8. Wiley, New York, 1960. 39. See for example Rittner, E. S., J. Appl. Phys. 30, 702 (1959). 40. McConnell, G., and Sehr, R., Solid State Electronics 2, 157 (1961). 41. Bloem, J., and Kroger, F. A., Philips Research Repts. 12, 281 (1957); also, White, D. C., and Woodson, H. H., Tech. Progr. Rept. No. 1, Contract A F 33 (616) 3984. 42. Stoll, G. C., Eichhorn, R. L., and Sicked, R. G., in “Thermoelectricity” (P. H. Egli, ed.), Chapter 3. Wiley, New York, 1960. 43. Lawson, W. D., Nielson, S., Putley, E. H., and Young, J., J. Phys. Chem. Solids 9 , 325 (1959); also, Lawson, W. D., Nielson, S., and Young, A. S., Solid State Phys. in Electronics and Telecommuns. 2, 8 10 (1 960). 44. Houston, M. D., Joint Tech, Soc.-Dept. of Defence Symposium on Thermoelectric Energy Conversion, Dallas, Texas, 196I . 45. Gurevitch, L., J . Phys. U.S.S.R. 9,477 (1945). 46. Friedel, J., J. phys. radium 17,27 ( 1956).
This Page Intentiona lly Left Blank
Impact Evaporation and Thin Film Growth in a Glow Discharge ERIC KAY International Business Machines Corporation, Research Laboratory, San Jose, California Page I. Introduction ....................................................... 245 11. Emission of Charged Particles from Metal Surfaces by Energetic Particles . , . 247 A. Secondary Electron Emission . . . . . . . . . . . . . . . B. Secondary Positive and Negative Ion Emi .................... 251 .................... 255 C. Reflection of Metastable Atoms and Ions ) .................. 257 111. High Vacuum Impact Evaporation (Cathode A. Theoretical Models .............................................. 257 B. Angular Distribution of Ejected Particles ............................ 263 C. Energy of Ejected Particles . . . . . . . . . . . . . D. Experimental Criteria . . . . . . . . . . . . . . . . . . . IV. Glow Discharge Characteristics . . . . . . . . . . . . . . . . . . V. Nucleation and Film Growth in High Vacuum Envir A. Film Growth Evaluation Tools .............. B. CondensationPhenornena .......................................... 290 VI. Film Growth in Glow Discharge Environment ......................... 297 A. Chemically Reactive Sources of Contamination ....................... 299 B. Inert Gases as Possible Sources of Con ' C. Contamination by Chemisorption ......................... 308 VII. Reactive Impact Evaporation .......... A. Advantages ..................................................... 311 B. Effect of Electronegative Gases on Glow Discharge Characteristics . . . . . . 313 ...................................... 314 ...... 317 References ..........................................
...
I. INTRODUCTION It is the purpose of this paper to describe in some detail the impact evaporation, or cathode sputtering, process and to discuss the important parameters of this process with respect to thin film nucleation and growth in a glow discharge environment. Impact evaporation occurs when a surface (the target) is bombarded with energetic particles which cause the ejection of surface atoms. T h e ejected atoms can then be condensed on a substrate to form a thin film. Only those investigations of impact evaporation which have appeared in the literature since 1955, the date of the last extensive survey articles on this subject (I, 2u), will be described in any detail. T h e references cited are chosen as typical and are not intended to be complete. 245
246
ERIC KAY
Interest in thin film studies has increased sharply over the past decade, largely as a result of the applications foreseen for thin film devices in the electronic industry, but also because fundamental research on thin films, with their special geometry, has given better insight into a number of solid state phenomena. Common to most thin film studies has been an evaluation of the method of preparation of the thin film. Of the numerous methods available, vacuum deposition has been most thoroughly examined. T h e high vacuum technique, in principle at least, lends itself best to an investigation of thin film nucleation and growth, which must be understood in order to control the properties of a film. With the increasing demand for more kinds of materials in thin film form, however, the limitations of the normal evaporation process have become manifest. Complementary methods of film preparation, in spite of their relative complexity, have had to be critically re-examined. For example, although understanding of the processes taking place in electrodeposition is still primitive, this method has gained greatly in popularity because it produces useful surfaces. Study of such other techniques as chemical vapor deposition, electrophoretic deposition, electroless plating and impact evaporation, the method to be treated here, have gained impetus in the last few years, largely For the same reasons. Ideally, in thin film research, one would like to have an exact description of the substrate and its immediate environment ; this description requires definition of parameters such as the type, energy, and direction of the particles arriving, as well as the microscopic surface condition of the substrate. Since simultaneous determination of all these parameters during deposition is difficult, one is usually forced to examine the substrate, its environment, and the material source influencing its environment separately, and then attempt to correlate these observations. Largely because of the difficulty of accurate description of the conditions prevailing at the substrate during deposition, the voluminous film deposition literature of the past is quite controversial and must be evaluated with caution. However, recently developed high resolution electron microscopy and diffraction techniques are making some headway in the investigation of film nucleation and growth during deposition under high vacuum conditions. Similar experiments on films prepared by impact evaporation would be considerably more complex ; evaluation of such films by means of simultaneous use of an ion beam and an electron diffraction camera during deposition is nevertheless quite conceivable. Trillat (3-5)has published reports of the behavior of thin films which were examined by an electron diffraction camera while being bombarded with an ion beam.
IMPACT EVAPORATION AND T H I N FILM GROWTH
247
Most of the work published on films prepared by impact evaporation, however, has been done in a glow discharge environment, even though an accurate description of conditions prevailing in the glow discharge system is fraught with additional complexities. The use of a relatively high mm Hg) glow discharge, rather than an ion beam pressure (lo-’ to in a high vacuum environment, has been partially justified by the relative simplicity, versatility, and economy of equipment required, These considerations will continue, in all but a few specialized areas of application, to recommend the glow discharge environment for large scale thin film production by impact evaporation. It therefore becomes imperative to study film formation in such a system, while attempting to reduce system complexity. Naturally, the type of particles and the manner in which they leave the source (target) will influence the films deposited at the substrate. Therefore the processes taking place at the target will be discussed first. The transport of ejected particles to the substrate will be discussed in later sections.
11. EMISSION OF CHARGED PARTICLES FROM METALSURFACES BY ENERGETIC PARTICLES Generally, collisions between particles are divided into two main types-elastic and inelastic. In an inelastic collision kinetic energy is lost in exciting internal motion of the participating particles. The type of internal excitation-vibrational, rotational, or electronic-is then used to further classify such collisions. In an elastic collision, no kinetic energy is lost to or gained from internal motion of the atoms. Superelastic collisions, electrons colliding with metastable atoms in an excited state, are also possible. Here energy is actually gained from the internal motion of the atoms. When an energetic particle collides with a material, all these kinds of collisions can take place. What actually occurs depends on the “collision cross section”-the probability of different events-which in turn, is determined by such parameters as size, charge, energy, and type of the particles participating. In collisions of energetic positive ions with materials in the condensed phase-the condition most prevalent in the work to be reported herethe following phenomena are known to occur: secondary electron emission; secondary positive and negative ion emission; reflection of metastable atoms and ions; and ejection of atoms or clusters of atoms, i.e., impact evaporation or sputtering. In order to set impact evaporation in its right perspective, the other phenomena will be discussed briefly.
248
ERIC KAY
A. Secondary Electron Emission
I . Kinetic Electron Ejection. In the kinetic ejection of electrons from a surface bombarded by ions it is thought that the electrons in the valence band, rather than in the conduction band of the metal, play a decisive role. Morgulis(6), Ploch(7), and Roos (8) pointed out that an ion can impart only a negligible part of its kinetic energy to a “free” electron of the conduction band because of the unfavorable ratios of the masses. It may be assumed that, since there are a large number of vacant energy levels in the conduction band, very little additional momentum is necessary to permit an electron of this band to leave the sphere of interaction with the ion and to take no further part in the processes that take place. Such experimental evidence (9) as the much larger yield of electrons by kinetic ejection, from dielectrics than from metals, where there is no energy loss due to interaction with conduction electrons, would lend credence to this idea. T h e excitation of bound electrons from a filled valence band of a metal, for example, is possible only when the energy imparted to these electrons is greater than a certain threshold energy Et, i.e., 15 to 20 ev. Since only a small fraction of the kinetic ion energy will be transmitted to any given bound electron, the energy for kinetic ejection has to be much greater. Petrov (10) suggests that the kinetic ejection of electrons by ions of rare gases, Cd+, and Znf with energies to -1 kev does not occur. He further reports (11) a very small electron emission coefficient y in the bombardment of metallic surfaces by ions of the alkali elements with energies up to several hundred electron volts. Analytically, the dependence of the electron emission coefficient y on ion energy E can be written as y = ypot C ( E - Et), where ypot is the coefficient y associated with the potential emission of electrons (discussed below), independent to a first approximation of the ion energy, and C = dy/dE. C = 0 for E < E t ; for E > Et, C is almost independent of E in a certain energy interval, but then decreases with increasing ion energy. T h e constant C depends on the kind of ion and ordinarily amounts to several per cent per 1 kev.
+
2. Potential Electron Ejection. T h e potential ejection of electrons has been the subject of recent intensive study by Hagstrum (Z2).It is postulated that, when the approaching ion has come close enough to the surface for the electronic wave functions of atom and solid to overlap, one electron from the valence band tunnels into the ground state of the approaching ion, neutralizing it. A second electron picks up the energy released by the first and becomes an excited (Auger) electron which will
249
IMPACT EVAPORATION AND THIN FILM GROWTH
leave the solid if it has energy above the surface barrier and is properly directed. Hagstrum further demonstrated that, since excitation of the internal secondary electrons by incident ions takes place just outside the metal surface, an interposition of an adsorbed gas layer can change electron emission drastically. Recent measurements of angular distribution of emerging electrons by Abbott and Berry (13) corroborate these These authors report a cosine distribution for ion-induced 10.32 - 1 c.
z
----- Hogctm Data
o’12B 0
0.6
0.4
ii
-
0.08
h
h 0.04
0 1 2 3 4 5 8 7 INCIDENT ION ENERGY (kw)
Xe i-
8
b
OO 200 800 1000 INCIDENT ION ENERGY(ev1
a c
’:’
I
1
- 14-
I
2 0
I3
1
a. l o -
t ; -
2
-
h
T
-Hg’
Y ;-
Y
1
HI+-
1
\H+ I
1
1
1
4
Frc. 1. Variation of secondary electron emission coefficient y with incident ion energy. (a) y versus incident ion energy for singly charged ions of the noble gases on atomically clean tungsten [H. D. Hagstrum, Phys. Reo. 96, 325, 336 (1954); 104, 1516 (1956); 122, 83 (1961); J. Phys. Chem. Solids 13, 33 (1960); J. Appl. Phys. 42, 1015, 1020(1961). (b) y versus incident ion for energy for Ar and He ions on tungsten [N. N. Petrov, Sowiet Phys.-Solid State 2, 857, 1182 (1960)l. (c) y versus incident ion energy for He+, Hz+, and H + on molybdenum [H. S . W. Massey and E. H. S. Burhop, “Electrons and Ion Impact Phenomena.” Oxford Univ. Press (Clarendon), London and New York, 1956.1.
250
ERIC KAY
secondary electron emission, T h e dependence of the secondary electron emission coefficient on angle of incidence of the primary ions is also well established. Figure 1 shows the variation with energy up to 400 kev of the number of ejected electrons per incident ion, y , observed in a number of experiments covering a variety of incident ions and target surfaces.
.
isL
I
I
1
I
I
I
I
I
I
I
I
1
I
I
I
-
-
0.08
2 9 0.06 -
E
0.04
J
3 0.02 k
-
0
I
400
200
600 800 INCIDENT ION ENERQY ( O V ]
1000
a
0.20
z
CI
00.l8
2
v)
!i
Y
0*16
w' 0.14
Y
b
0.12 0
200 400 600 800 INCIDENT ION ENEROY (av)
lo00
b FIG. Variation of secondary electron emission coefficient y with sui -_ce characteristics. ( p ) y versus incident ion energy for Cd ions on tungsten: ( I ) fresh, not previously heated target; ( 2 ) after 3 hours in vacuum at 2000" K ; (3) after many hours at 2000" K [N.N.Petrov, Swiet Phys. -Solid State2,857,1182(1960)]. (b) y versus incident ion energy for He+ ions on a clean Ge (1 I 1) face, and this face covered with CO and 0,under conditions described by H. D. Hagstrum [Phys. Rev. 96, 325, 336 (1954); 104, 1516 (1956); 122, 83 (1961); J . Phys. Chem. Solids 14, 33 (1960); J. Appl. Phys. 32, 1015, 1020 (1961)l.
IMPACT EVAPORATION AND THIN FILM GROWTH
251
I n Fig. 1b, Hagstrum and Petrov’s data coincide at 1 kev, indicating the predominance of electron extraction at the expense of potential energy of the ion metal system, The kinetic extraction of electrons by Art ions began at ion energies of the order of 1.5 kev. In Fig. 2a, y becomes practically independent of the ion energy as the surface becomes cleaner, thus again indicating the potential nature of electron extraction. Figure 2 shows very clearly that meaningful secondary electron emission measurements require careful description of the surface characteristics of the target.
B. Secondary Positive and Negative Ion Emission
A good deal has been done since 1958 in the investigation of the nature of secondary ion emission by means of mass spectrometry. Honig ( I d ) , analyzing the particles ejected from an Ag and Ge target as a result of Xe ion bombardment (30 to 400 ev), demonstrated that a small fraction, < 1 yoof the ejected particles, had either a positive or negative charge. He also studied the positive ions driven out of the surface when a target of Ge-Si alloy was bombarded by positive ions of rare gases in order to compare the yields of Gef and Si+ from the alloy (see Tables I and 11). TABLE I. PARTICLES SPUTTERED FROM Ag SURFACE BY Xe+ IONS” (I+/A = 4 ~A/cM*,E+ = 30@400 EV)
Neutrals
HZO
Positive ions
Aglt Na Fe+ AgH-
Negative ions
0-
co
COP
&a+
&a+
Mg+ 160 Ag,OH-
Al+ 185 Ag,O-
AgaO’ S+ Hg+ .4g,o,-
F-
26
26*
50
51
K+
Cat
S-
SH-
~ _ _
52
SH,-
c1a
R. E. Honig, -7. Appl. Phys. 29, 549 (1958). Mass number.
Bradley (15) extended the work of Honig to refractory metals Mo, Ta, Pt, and Cu using inert gas ions up to 1000 ev and found, as Honig did, secondary ion species characteristic of the metals themselves and of certain impurities, Neither of these authors found any multiply charged ions other than “reflected” inert gas ions. Stanton (16) bombarded Be targets and other metals and alloys with inert gases and N,, 0,, and CO ions. In conformity with earlier investiga-
252
ERIC KAY
TABLE 11. PARTICLES SPUTTERED FROM Ge SURFACE BY RAREGAS IONS (Xe+, Kr+, A+, Ne+: ENERGIES: 30-400 ev). Neutrals Positiveions Negative ions a
Gel H,O Gel+ Na+
Ge,
26b
27
co
NZ
Ge,+ Al+
GeH+
GeOH+ Rb+
Ge,O+ Hg+
F-
24-
26-
c1-
K+ 25-
0 2
co,
Hg
R. E. Honig, J. Appl. Phys. 29, 549 (1958).
* Mass
number.
tions, it was found that an appreicable fraction of the ions was liberated with energies less than 5 to 10 ev although some secondary ions of more than 200 ev were observed. The distribution appeared to be at least Maxwellian in character. Furthermore, his data suggest an independence of ion emission of the mass of the incident ion (see Figs. 3 and 4). 8000
1
0
-
$6000 K
E4000
-
v)
=0 2000 0
I
0
Xe He
Ar
Kr
0
0
0
0
a
0
f .
I
I
10
I00
-
ION MASS
FIG.3. Variation of secondary ion emission rate with mass number of incident ions for the noble gases on lead-208 at two target currents. Points shown as open circIes were obtained with a target current of approximately 5 x 10-0amp, and the points shown as closed circles represent a similar set of observations with a target current of 3 x 10-D amp, with a fixed bombarding ion energy of loo0 ev [H. E. Stanton, J. Appl. Phys. 31,
678 (1960)l.
Leland and Olson (16a) studied secondary ion emission at much higher incident ion energies-] 00 kev-in vacuum systems in which the total pressures were to lo-' mm Hg. Bombarding A1 with N2+, Fe with Kr+ and Au with N,+ at these high energies, they found the majority of negative ions to be typical of system impurities, i.e., 0-, OH-, C-, C2H,-, C1-, C4-, and relatively few species belonging to the metal target. Positive ion emission, however, consisted largely of the
253
IMPACT EVAPORATION AND THIN FILM GROWTH
+s
I
4
I
/ 0
0
LL
0 0
J
w
; 1.0
I
I
FIG.4. Variation of secondary ion yield with mass number of incident ions for various gases on beryllium. The cluster of 3 points close together at the middle of the curve were values found from the chemically active gases N,, O,, and CO. The curve is corrected for secondary emission of electrons from the target, with a fixed bombarding ion energy of 1000 ev [H. E. Stanton, J. Appl. Phys. 31, 678 (1960)l.
metallic species in the target. Other measurements included the effects of such other parameters as primary current density, primary ion energy, and type and temperature of target surface. It is interesting to note that at high energies these authors find a very definite dependence on ion type in contrast to Stanton’s low energy bombardment work (see Fig. 5). 17 16 15 14 13 u)
= o_
12-
II 10
-
0 LL J
*
I
I
@K;e
-
-
-
.C02 .Ar+
-
-
.CO+
3 2 1 -
-
-
5 4 -
0
I
-
9 -
n
f
I
-
8 7 6 -
w
f
-
+J
a
I
-
"2 @N+
I
...---
I
I
I
I
..-...--. .
FIG. 5. Variation of secondary ion yield with mass number of incident ions for various gases having 150 kev energy on aluminum [W. T. Leland and R. Olson, Proc. Atomic and Molecular Beams Conf., Univ. of Denver, 19601.
254
ERIC KAY
T h e most recent Russian contribution comes from Veksler (17), who studied the energy spectra of “reflected” and emitted positive ions obtained by bombarding Mo and T a targets with Cs+ ions with energies up to 2200 ev. Energies of 30 to 35 ev for Mo+ and 35 to 50 ev for T a + are reported. T h e values are considerably higher than those quoted previously in the literature. Fogel’, Slabospitskii, and Rastrepin ( I S ) made measurements of the dependence of the number of ejected positive and negative ions from various metals on the energy of the incident ions in the 5 to 40 kev range. They conclude that the secondary negative (K-) and positive ion (K+) emission and the ion reflection ( R ) are of the same order of magnitude (see Fig. 6), and vary as a function of mass and velocity of the ion from several tenths of a per cent to several per cent. (Stanton, on the other hand, in his experiments at lower energies with I
0.03
L
a
0.02
0.W
oio1
0.01
0
e
0.04
0.03
II11111
-
0.1
z
I
0.2
0.4 0.6 0% I
2.108
r--l
tx 0.02 0.01 - 0
0.1
0.2
04 OBQBl
2108
FIG. 6. Variation of K+, K-, y, and R with incident ion velocity for (1) H+, (2) He+, (3) Ne+, (4) Ar+, (5) Krf, and (6) O+ p a . M. Fogel’, R. P. Slabospitskii, and A. B. Rastrepin, Sooiet Phys. - Tech. Phys. 5, 58 (1960)l.
IMPACT EVAPORATION AND T H I N FILM GROWTH
255
lead and beryllium, observed a dependence only on energy and not on mass of the incident ions, as shown in Figs. 3 and 4.) T h e secondary electron emission coefficient y was found to be up to two orders of magnitude larger than the above coefficients.
I
a
0.4
0.2
200 400 600
800
1000 1200
1400
INCIDENT POSITIVE ION ENERGY (ev)
FIG.7. Variation of reflection coefficient R with incident ion energy for argon, nzon and helium ions incident normally on nickel [H. S. W. Massey and E. H. S. Burhop “Electrons and Ion Impact Phenomena.” Oxford Univ. Press (Clarendon), London and New York, 19561.
Hagstrum (19) recently published a paper on the detection of metastable ions (i-e., metastable with respect to radiative transitions to the ground state) by means of Auger de-excitation and neutralization at a solid surface. T h e metastable ions are detected by their greater ability, with respect to unexcited ions, to eject electrons from the metal. T h e dependence of all these emission effects on temperature and surface characteristics has been amply demonstrated by all these authors. All authors so far agree that use of secondary ion emission for quantitative analysis of the target presents very serious difficulties.
C . Reflection of Metastable Atoms and Ions A number of investigators have attempted to study this problem. Oliphant (20), who studied reflection coefficients for metastable helium atoms from a molybdenum surface, showed that this reflection falls off sharply as the energy of the incident metastable atom is increased as shown in Table 111. Oliphant also demonstrated the effect of angle of incidence on reflection coefficients of metastable helium atoms from molybdenum surfaces, as shown in Table IV.
256
ERIC KAY
TABLE 111. REFLECTION COEFFICIENTS FOR METASTABLE HELIUMATOMSFROM Mo SURFACE^ Energy of ions which produce the metastable atoms (ev)
A
Reflection coefficient
2100 800 400 120
0.05
0.10-0.20 0.40-0.50 0.50
" M . L. E. Oliphant, Proc. Roy. SOC.A124, 228 (1929).
TABLE IV. EFFECTOF ANGLEOF INCIDENCEON REFLECTION COEFFICIENT FOR METASTABLE He ATOMSFROM A Mo SURFACE'" Energy of positive ions producing metastable atoms (ev) 600
600 600 200 200 200
Angle of incidence 0" 45" 75" 0"
45" 75"
Reflective coefficient 0.10-0.30 0.15-0.40 0.40-0.60 0.40 0.40-0.70 0.60-0.90
M. L. E. Oliphant, Proc. Roy. SOC.A124, 228 (1929)
Gurney (21) also demonstrated the same general dependence of the reflection coefficient R on the angle of incidence; R is small for normal incidence but increases rapidly with increasing angle of incidence. Most of the observed scattering occurred in a forward direction. The reflection of unneutralized positive ions has been found by Paetow and Walcher (22) to be small in the case of alkali metals, the reflection coefficient R being of the order of 0.02 to 0.03 for incident ion energies up to 1000 ev. For rare gas ions, Healea and Houternans (23) find much greater values of R as illustrated in Fig. 7. Recently, Bradley (24) published some data on inert gas ions "reflected" from copper targets. The energy spread of the reflected ions observed was invariably less than 1 ev even for bombqrdment energies as high as 1000 ev; in certain systems directional effects were evident despite this low ejection energy. The yield (defined as the ratio of reflected ion current measured at the final collector to incident ion
IMPACT EVAPORATION AND T H I N FILM GROWTH
257
current) increased with target temperature and was reversible. T h e yield was proportional to the background pressure of inert gas and increased with increasing bombarding energy up to 250 ev and then slowly declined. He states that a true reflection cannot be understood on the basis of Auger neutralization processes and suggests that the reflected ions originate from the inert gas absorbed on the target surface. Detailed discussion of the mechanism of absorption, ionization and ejection of this kind has not appeared in the literature. T h e most recent publication of reflection of ions with large ionization potentials, e.g., He+, Ne+ and Arf on W, Mo and Si comes from Hagstrum (25). He distinguishes between ions reflected as ions and as metastable atoms as in (19). He found the reflection coefficients of ions to ions to be smaller than previously reported (0.0004 to 0.002) and essentially independent of incident ion energy. T h e coefficient of ions to metastable atoms was found to be dependent on incident energy, however, ranging from 0.0004 at 10 ev to values as high as 0.04 at 1000 ev.
111. HIGHVACUUMIMPACTEVAPORATION (CATHODE SPUTTERING) A. Theoretical Models Impact evaporation has been studied since 1850, but only within the last few years have sufficiently consistent data been collected to permit some insight into the mechanism of the process. T h e erosion of solid surfaces by ionic or molecular bombardment has recently attracted a great deal of interest, because a detailed knowledge of the behavior of solids in very hot gas streams and discharge plasmas is necessary for the design of suitable components for nuclear fusion and jet propulsion. A comprehensive review of the subject up to 1955 was given by Wehner ( I ) and by Massey and Burhop (2); a brief review by Moore (26) appeared in June 1960 and most recently a report by Thompson appeared (2a). Two theoretical models have been proposed for impact evaporation: (a) According to the vaporization theory, the incident ion causes a local hot-spot in the solid from which vaporization occurs ; the ejection ratio, i.e., the number of ejected particles per incident ion, should therefore depend only on the energy of the incident particles, and not on the ratio of their mass to that of the target atoms. (b) According to the momentum theory, elastic impacts occur between incident ions and atoms in the solid, with transfer of momentum to the target; particles
258
ERIC KAY
may be ejected by the initial impact or by subsequent collisions between displaced atoms within the crystal lattice. In this model the sputtering rate does depend on the ratio of masses of the incident ion and target atom, as well as on the energy of the incident ion. Only a few quantitative treatments of the vaporization theory have appeared in the literature (e.g. Townes, 27). His equation predicts that sputtering yield S is dependent on incident energy and independent of mass ratio. Considerable evidence has now been gathered indicating that such a simple energy transfer does not suffice to explain the experimental data of impact evaporation. Not only does S depend on the mass ratio of incident ion to target atom, as shown in Fig. 8, but ample
I
2
3
10
20
MASS NUMBER FIG. 8. Variation of sputtering yield S with mass number for 5 kev ions incident normally on silver [log - log plot; W. J. Moore, Am. Scientist 48, 109 (1960)l.
evidence has also been gathered that S depends on the angle of ion incidence (see I), which would indicate that the process is not governed merely by a transfer of energy, but by transfer of momentum from the ions to the solid as in hard sphere collisions. On further consideration it becomes evident, however, that a classical momentum transfer mechanism can only be invoked in a certain energy range. Three types of interaction are generally recognized, classified according to energy range. (1) Particles with high energies interact through the Coulombic repulsion of their positive nuclear charges. This is the type of collision treated by Rutherford in his original work on alpha particle At intermediate energies the electron clouds that surround scattering. the nuclei partially screen the positive charges, since the particles are not moving fast enough to penetrate the electron cloud completely.
(a)
IMPACT EVAPORATION AND THIN FILM GROWTH
259
(3) At low energies there is little penetration of the electron screen and classical hard sphere collisions between atoms occur. While electronic excitation is the principal mode of energy dissipation for the rapidly moving charged particles in a solid, the conversion of translational energy into vibrational energy of the solid structure must be considered for ions moving with velocities less than the Bohr velocity of the electrons. I n the last five years a number of extensive theoretical papers on the modes of energy dissipation in irradiated solids in various energy ranges and the dynamics of radiation damage have appeared in the literature. Those by Kinchin and Pease (28) and Seitz and Koehler (29) and, more recently, by Gibson et al. (30) are perhaps the most noteworthy. Moore (26), in his recent review, begins by referring to Bohr's (32) original monograph, published in 1948, which deals with the general theory of these various collisional processes. Bohr showed the regions in which the various treatments were applicable in both classical and quantum mechanical forms. T h e interaction potential, V(?),in the shielded Coulombic model for the collision is V ( r )= (ZlZ2e2e-r/a)/r, where 2, and 2, are the nuclear charges of the ion and of the target atom, r is the separation of charges, and a is the characteristic radius of the screening electron cloud, called the shielding parameter. T h e Thomas-Fermi model for the atom yields =
u,/(zy+ 231'2,
where a, is the Bohr radius. Moore refers to a simplified theory proposed by Pease (32) which is in fair agreement with experimental results. Pease assumed that most of the impact evaporation was due to atoms displaced in the first collision of the ion occurring in a thin layer beneath the surface with a depth of 1 mean free path. H e obtained, for the sputtering yield,
This equation contains no adjustable parameters. E is the average energy transferred from ion to target. Es, the latent heat of sublimation, is the energy necessary to remove an atom from the surface. T h e displacement cross section in collisions is u,,, and n is the number of atoms per unit volume. T h e values of E and up are calculated from the law of repulsion
260
ERIC KAY
that governs the collisional interaction. ED is the energy required to displace an atom from its lattice site. For silver, as an example, ED is about 21 ev. This value may seem surprisingly high, but not if one considers the following qualitative argument. The energy of sublimation, E,, of a typical atom in a tightly bound solid is in the neighborhood of 5 ev. Since atoms sublime from the surface, where only about half the binding forces active in the interior are operative, it can be concluded that the energy required to remove the typical atom or ion from an interior site in an adiabatic or reversible manner should be nearer 2E,. If, however, the atom is removed and forced into the lattice in an irreversible way, as when it is struck by a fast incident particle, a considerable barrier of activation must have been surmounted. This energy is usually estimated at 4Ec. On the basis of such an argument the above value of 21 ev for ED seems quite reasonable. Pease goes on to estimate the ranges of validity of the different collisional models as follows: The Rutherford collisions predominate when the incident energy E is greater than a critical value LB. For E between L, and LA, the effect of electrons in screening the nuclei is important. These collisions have a roughly constant collision cross section given by cr = ,a*. For energies below LA, a better estimate of the cross section is u = 7x2 where X , the distance of closest approach, is the solution of E = (Z,Z,e2/X) e-xla These are hard sphere collisions of the classical type. The cross sections in this energy range are not accurately known. The final critical energy Lc is that below which electronic excitation is negligible. For metals it is roughly given by m L - L E F , - 16m, where EF is the Fermi energy of the conduction electrons and mo is the electron mass. Using the approach by Pease, Moore estimated the various critical energies for ions incident on silver (see Table V). Keywell (33), doing both experimental and theoretical work on impact evaporation in the 4000 to 6100 ev energy range, with various gas-metal combinations, interpreted his data by treating the incident ion as a hard sphere which “cools” somewhat in the manner of a neutron losing energy by collisions in a lattice, where each collision produces recoil atoms and atomic displacements near the surface. According to this model, the sputtering yield S is expected to rise quite rapidly from a certain threshold value’as energy increases. The initial rise is
26 1
IMPACT EVAPORATION AND THIN FILM GROWTH
steep because of the rapid increase in the probability of collisions sufficiently energetic to cause atomic displacements. This probability levels off at higher energies to an approximately logarithmic dependence TABLE V. APPROXIMATE CRITICAL ENERGIES FOR IONIC COLLISONS I N SILVER"
a
Ion
LA(k4
LB(k4
LC(k4
H+ D+ He+ N+ O+ Ne+
4.8 4.9 10.1 41.0 48.0 64.0
10 20 170 820 1250 2620
0.1 1.3 2.6 9.1 10.4 13.1
W. J. Moore, Am. Scientist 48, 109 (1960).
on incident energy. On the other hand, with increasing energy the range of the ions in the solid also increases; i.e., the cross section for collisional interaction decreases with energy. As a result of these two opposite effects of increasing energy the yield vs. energy curve will display a maximum. This has been demonstrated in a number of experiments (see Fig. 12). By use of the neutron cooling theory and the Seitz formula for displacements produced by a recoil atom within a solid, Keywell was able to derive a formula for the ejection coefficient 8, i.e., number of atoms ejected per incident ion, which agrees fairly well with his experimental data and that of Timoshenko (34, who worked in the energy range of 200 to 7000 ev. Thommen (35)refined the method of Keywell to obtain 8 = (J%/EB)$(Ml/M2)!
where E, is the energy of the incident ion, E, is the energy of sublimation of the solid, and M I and M , are the respective masses of ion and target atom. T h e function t,b is defined by a rapidly converging series with one term for each collision of the decelerating ion. T h e resulting function represented quite well the form of the experimental curves of 0 vs. MI, M , for rare gas ions, with E,, less than 8 kev incident on single crystals of PbS, Ge, and FeS,. It was again assumed, as in Pease's model, that only the first collision was effective; i.e., that sputtering was a true surface phenomenon. T h e models of Keywell and Thommen (35) assumed hard sphere collisions and no energy transfer to the electrons in the solid. T h e
262
ERIC KAY
velocity of a helium ion at 5000 ev, say, is small compared to a 5-volt electron, and therefore the energy transfer to that bound electron is thought to be negligible. Although energy transfer to conduction electrons within a solid is possible at electron energies ranging from 0 to the Fermi energy EF, Keywell and Thommen considered this process unlikely for two reasons. First, transfer of energy to electrons at the lower levels of the Fermi sea is forbidden by a combination of the exclusion principle and the small energy transfer which is allowed, in classical theory, for an ion electron collision. Second, as a result of the above, the only electrons which can receive energy from ions are those near the top of the Fermi sea and these will have large velocities compared with those of the ions. Data by Seitz (29) and by Pease (32, see Table V) would suggest that in the case of 5000 ev helium ions bombarding silver these assumptions are marginal. T h e mathematical methods of the neutron diffusion theory used by Keywell were also applied to impact evaporation by high energy ions (up to 10 kev) in a more detailed treatment by Harrison (36). This theory is also in fair agreement with experimental data. Like most of the others, it compounds a number of approximations of as yet unknown atomic parameters. Keywell’s model also formed the basis of a Monte Carlo calculation of the sputtering yield by Goldman et al. (37). T h e many difficulties of the surface processes were avoided by Goldman and Simon (38) who worked out a theory applicable above about 50 kev. I n this energy range they assumed the emergent particles to originate at depths in the material which are small compared with the range of the incident particles. T h e displacement rate is nearly constant over this region, so that relatively simple solutions of the diffusion problem for emission can be obtained. They found an expression for the ejection ratio, where @ is the angle between the incident beam and the normal to the target surface. So far few experimental results in the range covered by this theory have been published; Skiff (39) recently reported data up to 160 kev which seems in fair agreement with the above. Langberg (40) postulated that for impact evaporation in the low energy (“threshold”) region the interatomic energy and distance relation can be represented by a Morse potential model and that the energy transfer occurs at the target surface and essentially by two particle collisions. Collisions of the ion, or of a knocked-on upper surface atom, with a
IMPACT EVAPORATION AND THIN FILM GROWTH
263
lower surface atom which must be assumed to explain certain anisotropic particle ejection features (see focusing effects), are not considered at all by Langberg. Henschke (41) also assumed two-body collisions between free particles for collisions which lead directly to the ejection of upper surface atoms. However, collisions of ions with target atoms in the direction to the inside of the target, he did not treat in this way. H e suggests that the bulk of the target is behind the struck atom and produces a very large “effective” mass compared with the mass of the ion or the mass of the target atom, so that a rebound of the striking particle occurs in such a collision no matter whether its mass is lighter, heavier, or equal to the mass of the target atom. I n this theory a derivation of the threshold and ejection yield formulas, as well as an explanation of the details of the deposit spot patterns, is given. I n Henschke’s latest publication (42), his theory is extended to explain the periodicity of threshold energies as seen in a plot of energy vs. atomic numbers. Most recently Harrison and Magnuson (43) have attempted to get a coherent definition of the controversial term “sputtering threshold.” They propose two distinct models, one generally applicable when the mass ratio is less than one, and another when it is greater than one. They obtained single crystal threshold laws, and, by averaging the single crystal forms, derived polycrystalline laws for fcc crystals. In all cases their theoretical thresholds are less than or comparable to experimental thresholds, including those observed in the most sophisticated recent work by Wehner and Stuart (44) and McKeown (45). Harrison and Magnuson (43) point out that there is still no direct evidence that a true experimental threshold actually exists.
B. Angular Distribution of Ejected Particles 1. Single Crystals-Focusing Effects. Wehner (46) first observed in 1956 that, when a single crystal of a metal is bombarded with low energy ions (300 ev), atoms are preferentially ejected in the directions of close packing, i.e., the (1 10) in the fcc and the (1 11) in the bcc system. Since then numerous articles have appeared reporting the anisotropic spread of energy from a collision center and supporting the concept of focusing collisions in nearest and next neighbor directions. At low energies, only those directions appear which require the least directional change in momentum. I n 1956 Henschke (47) examined the spot patterns produced from low index planes in some detail and proposed a theory explaining these preferential ejections. He refers to an attenuating influence of the
ERIC KAY
264
100
80-
6
Q, 0
60
-
2
!!! 0
40-
W
20
I
I
0
I '
10
20 ANGLE
I
30
B
Q
2
0 \ v)
r 0
s
0 CU SINGLE CRYSTAL
0 CU POLYCRYSTALLINE 60'
50'
40°
36
ZOO
loo
Oo
loo
20°
30°
4 6
500
60°
KRYPTON IONS (45 kev)
b
FIG.9. (a) Focusing efficiency in solid state collisions [W. J. Moore Am. Scientist 48, 109 (1960)l.(b) Sputtering ratio- for a Cu single crystal turned around on [110] axis in the [ 1 1 11 plane. The [ll I] direction is 3' from zero. [Almen and Bruce (53a)l.
IMPACT EVAPORATION AND THIN FILM GROWTH
265
different electron densities in the paths of the ion within the lattice and attempts to show that these densities are highest in the closepacked rows of the lattice. I n 1957 Silsbee (48) published a paper dealing with this problem. He points out that most of the previously proposed collision theories neglect the fact that crystalline materials have well-ordered structure and that the distribution of energy and momentum among the atoms in a sequence of collisions may well be influenced by this structure. A focusing parameter was defined, u = d/ro, where d is the spacing between centers of atoms and y o is the diameter of the atom in a hard sphere model. Consider a line of spheres along the x-axis in which the first sphere is given an impulse in the xy-plane, the velocity making an angle /3 with the x-axis. T h e efficiency of focusing is defined as the ratio of the energy focused along the line to the energy imparted to the first sphere. Some calculated efficiencies are plotted in Fig. 9. From Silsbee’s criterion for focusing, namely, that cr = d/ro < 2 cos /3, it can be shown that the close-packed chains are of greatest interest. He shows that a significant fraction of the original energy can be focused into closepacked chains and can be propagated along them until the momentum pulse reaches some sort of defect. I n 1960 Leibfried (49) extended this theory. He calculated a range of distribution of focusing collisions and the number of Frenkel pairs produced, assuming that Frenkel pairs are, in fact, produced by focusing collisions of sufficiently high energy which encounter a stacking fault area of an extended dislocation, He also calculated the extent to which the Frenkel pairs so produced pin a dislocation. A rough estimate was made of the energy loss due to effects of thermal lattice vibration, of interaction with neighboring chains, and of alloying. Interaction with neighboring chains is apparently the predominant mechanism whereby energy is lost along a given chain. Momentum pulses focused up to about IOQ) atomic distances are thought feasible. Leibfried developed the theory for the (110) case of focusing while Gibson et al. (30) found theoretical evidence of focusing along the (100) and (1 11) directions by means of a machine calculation assuming a dynamical model. Several experimental papers have appeared in the literature in the last three years which lend support to the focusing collision concepts mentioned above, even in energy ranges much higher than the 300 ev used in Wehner’s original observation. Thompson (50) irradiated a thin gold foil, whose grains had been oriented by rolling, with a proton beam of such energy that the beam penetrated the foil emerging with an average energy of 300 kev. T h e geometry of the gold deposit collected from the back of the foil definitely
2 66
ERIC KAY
indicated preferential ejection in the close-packed crystal directions of the fcc gold target similar to the geometry obtained at much lower bombarding energies from the front of a (100) Au surface. More recently Koedam and Hoogendoorn (51), using an Argon ion beam of up to 2 kev, on single crystals of copper; Yurasova et al. (52) at up to 50 kev; and Skiff and Reynolds (39) at up to 170 kev found further confirmation of such anisotropic ejection of atoms. T h e most extensive description of atom ejection patterns from single crystals was published by Anderson and Wehner (53). Most recently AlmCn and Bruce (53a) published data on sputtering ratios for copper single crystal targets which were turned around an [110] axis in the [l 113 plane. Similar data obtained on polycrystalline Cu again demonstrated the focusing in the single crystaI case very effectively (see Fig. 9a). Work by Molchanov et al. (53b) further confirmed this data. More direct evidence is now being gathered by field ion microscopy. Mueller (54) observed focusing in a bcc lattice using a 50% neutral 20 kev He beam in his field emission microscope. Radiation damage was greatest along close-packed directions. Dislocations were observed only in the surface layers (-10 layers) of the target. This is hard to understand since the vacancies from within the lattice should not have enough mobility to form such dislocations. Rued1 et al. (55), examining Pt foils by electron microscopy after neutron irradiation, found that defects (measles) appear preferentially along coherent twin boundaries. They conclude that momentum transfer and the theory of focusing collisions can readily account for their experimental observations.
2. Polycrystalline Targets--“Cosine” Distribution. A number of other papers have appeared recently dealing with the angular distribution of particles leaving a polycrystalline target surface as a result of impact evaporation in various energy ranges. According to Knudsen’s cosine law, the angular distribution of particles ejected by a thermal evaporation process is independent of the angle of incidence. I n impact evaporation, however, “under cosine” distribution has been reported by several authors ; Wehner (56), for example, found such a distribution of deposited material when a polycrystalline material was bombarded perpendicularly with ions of energy ranging up to 1000 ev. Under oblique ion incidence, atoms were sputtered preferentially in the forward direction, that is, along the direction parallel to that of specular reflection of the incoming beam (see Fig. 10a). Recently Stein and Hurlbut (57) reported qualitatively similar findings in the low energy range when potassium surfaces were bombarded
IMPACT EVAPORATION AND T H I N FILM GROWTH
267
with inert gases. Gronlund and Moore (58) Fig. 10b bombarded silver with a well-defined beam of D+ ions in a higher energy range (2 to 12 kev) at an angle of incidence of 60".I n this energy range they found fairly good agreement with cosine distribution except, again, for a
b FIG. 10. (a) Angular distribution of molybdenum ejected by obliquely incident Hg+ ions at 250 ev IG. K. Wehner, J . Appl. Phys. 31, 177 (196O)l.(b) Angular distribution of silver ejected by Dtions incident at 0 = 60" and 9 kev; dotted curve is cosine distribution [F. Gronlund and W. J. Moore, J. Chem. Phys. 32, 1540 (1960)l.
ERIC KAY
considerable amount of additional sputtering in the forward direction (see Fig. lob). They expected this latter effect to be much more pronounced with higher mass species Ne+, because of its lesser penetration into the target and its more effective momentum transfer. However, the net results seem to follow the cosine distribution about the normal to the surface almost perfectly. This observation can as yet be only tentatively explained, but it is probably related to changes in target morphology as pointed out by Holmstrom and Knight (59) and by Molchanov (60). As Moore suggests, a particularly sensitive test of the theoretical models will be their ability to predict the variation of sputtering yield with the angle of incidence of the energetic ions. Rol et al. (61) bombarded polycrystalline copper with various ionic species into an even higher energy range ( 5 to 25 kev) and found an (4 over cosine” distribution for the intensity profile of the deposit. Rol was working with an ion beam whose converging angle was 20”, so that his angular distribution data must be considered with caution. Rol very recently reported (61) that he does observe, after an extended period of bombardment, a dependence of angular distribution of ejected material on the angle of incidence of the beam. He suggests that this is the result of a gross change in the surface characteristics of the originally smooth target caused by preferential etching, so that preferred ejection directions result entirely from the new physical geometry of the surface. The results obtained by Holmstrom and Knight (59), using high current density He or Ar beams at energies up to 30 kev on polycrystalline Ag1l0 targets, have shown an “over cosine” distribution which is typical in this high energy range when the angle of incidence of the beam is normal to the target. At oblique angles greater than 45” from the target normal, the results were quite contrary to expectation and in sharp contrast to results already cited. Instead of an increase of material sputtered in the forward direction, an increase in the backward direction was found with the maximum deviation from the cosine distribution occurring at the angle of incidence. Knight is now examining the possibility of an energy dependence of this deviation (62). No significant differences are noted between 10 and 30 kev. Experiments with energies up to 60 kev are contemplated. Preliminary results would indicate that resputtering of Ag from the collecting substrates by energetic particles reflected from the target is not involved in contrast to a mechanism suggested in a paper by Molchanov and Tel’kouskii (60). It has been established that the distribution asymmetry of ejected particles can be correlated to the target sur€ace morphology during bombardment using electron micrographs of target surface replicas. Most recently Balarin
IMPACT EVAPORATION AND THIN FILM GROWTH
269
and Hilbert (64a) treated this topic in detail for incident ion energies up to 40 kev. Their comparison of thermal etching versus ion impact surface etching clearly corroborates the postulates by previous authors that most of these directional effects are being interpreted as an indication that impact evaporation is a vectorial process involving momentum transfer rather than simple energy transfer. This asymmetry changes with degree of surface roughness. Under somewhat similar high energy sputtering conditions, Molchanov and Tel'kovskii (60) recently established a dependence of sputtering ratio on surface roughness and angle of incidence using an argon beam on polycrystalline Cu targets. Directional etch effects from oblique incidence ion bombardment of polycrystalline targets at low energies (up to 500 volts) were discussed in detail by Magnuson et al. (63) and at energies up to 8 kev by Cunningham et al. (64) (see Fig. 11). T h e only recent evidence contradicting the momentum transfer idea comes from Wolsky and Zdanuck (65) who reported sputtering yields for double ionized argon on Si to be much higher than one would expect; instead of SE(Ar+2)= SZE(Ar+)as expected, they claim SE(Ar+2)= 4SZE(Ar+).
FIG. 1 1. Electron Micrograph (Rhodoid replica) of polycrystalline gold bombarded with Art of 8 kev for 120 minutes at 70" incidence [R. L. Cunningham et al., J . Appl. Phys. 31, 839 (1960)l.
270
ERIC KAY
It is difficult to reconcile this interpretation of these experimental findings with any of the existing theoretical models of energy exchange. Until more direct evidence of the concentration of multiply charged argon ions in Wolsky's discharge-like mm Hg) apparatus is available, these results must be considered with caution. The applicability of Bleakney's findings (M), upon which all of Wolsky's interpretation depends, is questionable. Bleakney used a much better defined monoenergetic electron beam to obtain the concentration of Ar2+ and Ar+, and his experiments were performed at a much lower pressure, so that only a small fraction of the electrons collided with the argon atoms along their path. The electron beam provided a linear source of positive ions of practically uriiform density which were subsequently mass analyzed.
C. Energy of Ejected Particles Wehner (67),using a torsion balance immersed in a low pressure Hg plasma, measured the forces occurring on ion-bombarded electrodes. The force curves obtained for 22 metals in the incident energy range of 20 to 900 ev indicate that the Hg+ ions are completely accommodated ( a = 1) on clean metal surfaces and the forces originate essentially from the ejection of sputtered atoms. He further demonstrated (68) that atoms ejected from an inclined target at angles less than 30" to the surface normal have higher velocities (28 ev) than atoms ejected normal to the target surface (12 ev) presumably because the former require less directional change of momentum. AlmCn and Bruce reported some preliminary data on mean energies (see Table VI) of sputtered particles with the incidence angle of a 45 kev Kr ion beam being 45". They report very small variations with ion energies in the 25 to 50 kev range. Mean TABLE VI. ENERGIES (EV) OF SPUTTERED PARTICLES LEAVING VARIOUS TARGET MATERIALS BOMBARDED WITH 45 KEV Krf BEAM AT 45" ANGLE OF INCIDECE." Target material cu Ag Ta Au
a
Energy of sputtered atoms (ev)
45
36 250
60
Almln, 0. and Bruce, G., Nuclear Instr. & Methods 2, 257 (1961).
IMPACT EVAPORATION AND THIN FILM GROWTH
271
velocities of 7.1, 6.3, and 4.1 x 103 meters/sec were calculated for ejected Ag atoms when Ar, Kr, and Xe ions of 45 kev were used. This and the fact that the average kinetic energy of these ejected particles is much higher in magnitude (in some cases as much as 100 times) than thermal evaporation energies can certainly be considered as damaging evidence against the thermal evaporation theory. As indicated in the last few pages, numerous theoretical treatments have been given to the impact evaporation process. However, since quantitative agreement with experiment has only been fair, it is obvious that both further refinements in the collision model and in the treatment of the complex problem of diffusion of the displaced atoms and better defined experimental parameters are necessary.
D. Experimental Criteria I t is the conviction of the author that none of the more refined recent theoretical models have as yet been put to a critical test, largely because few if any of the experiments performed have met the standards of definition considered necessary for subsequent meaningful interpretation. T h e experimental criteria necessary for significant quantitative study of the impact evaporation mechanism can be summarized as follows: (1) T h e energetic paricles striking the target should have a uniform known velocity. (2) All particles should strike the target from the same direction. (3) The composition of the beam should be known; preferably it should contain only one species. (4) T h e target surface should be pure and crystallographically and morphologically well defined. ( 5 ) An accurate measurement of the incident particle flux and vacuum conditions at the target is essential. T h e beam diameter should be a minimum. (6) T h e emitted particles should be unable to return to the target. (7) T h e direction of the particles leaving the target should be measured. (8) T h e identity and energy distribution of the particles leaving the target should be determined. (9) T h e temperature of the target should be known. Naturally, for order of magnitude determinations, not all of these requirements must be satisfied. However, for more quantitative work,
17
-5
-
I
I
I
I
I
I
1
1
1
I
I
I
1
1
1
I
1
++
A L M ~ NR B R U C E ( ~ ~ ~ )
- -6- ROL ET AL ( 5 7 )
-
12w I1
-
2
1
+ YONTS, NORMAND, 8 HARRISON (69) + COLOMBIE (70) -
-
16 15 1413-
I-
1
8 7
ArtonPd (53a)
32-
-
-
-
-l
Arton Mo (53a) l , l l , , y , , , , , 0 5 10 I5 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
I -
1
1
1
l
1
INCIDENT ION ENERGY (kev)
b
0.35
I
1
1
I
1
I
2 W 0
0.25
1
I
-
-
-
-
0.30 I-
I
E a a
W
v)
5
INCIDENT ION ENERGY (kev)
d FIG. 12. Variation of sputtering yield S with incident ion energy for several ion-target combinations, (a) sputtering yields of Ar+ on Cu by various authors; (b) sputtering. yields of Art on.Ag, Pd, Sn, and Mo; (c) sputtering yields of He+and Df on Cu. Comparison of (a) and (c) demonstrates maximum in yield-energy curves; see p. 18. [O. C. Yonts, C. E. Normand, and D. E. Harrison;J. Appl. Phys. 31,447 (1960).]. (d) sputtering yield of Xe+ on vitreous silica [R. L. Hines and R. Wallor, J. Appl. Phys. 32, 202 (1961).
274
ERIC KAY
good definition of every one of these parameters is important, as will be seen in the following sections. To strive for this goal is not unreasonable, since such definition of experimental parameters is feasible.
&P/
NORMAL INCIDENCE
I
4
2
I .8 .6
A
.2
NORMAL INCIDENCE .I
0
I
2
3
4
5
6
7
9
ENERGY ( k r v )
a FIG. 13. Variation of sputtering yield S with incident ion energy for incidence (each Nf with 1/2 the
275
IMPACT EVAPORATION AND THIN FILM GROWTH
I . Energy, Direction, and Composition of Incident Particles. The dependence of the number of atoms being ejected from the target on (a) the ratio of mass of the incident particles, M I , to that of the target
It
COPPERa-
iU/d /
/MOLYBDENUM
0 0 E .-
\
E
0 c U
N:
2Nt 4 5 O INCIDENCE 0
I
I
I
I
I
1
I
t
3
4
5
6
I
2
7
8
9
t IRON
TUNGSTEN
't P
Ni
o 0
2N'
45O INCIDENCE I
'0
I
2
3
4
5
6
7
ENERGY (kev)
monatomic and diatomic ions: (a) at normal incidence; (b) at 45" N,+ energy). [Bader et al. (dla).]
8
-
276
ERIC KAY
.-
( N O i lN3013NI M3d SW01V)S
(NO1 lN3013NI Y3d S W O W ) S
FIG.14. Variation of sputtering yield S with incident ion energy for metals of different periods [G. K. Wehner and D. Rosenberg. -7. Appl. Php. 32, 887 (1961)l. (a) 4th-period metals. (b) 5th-period metals. (c) 6th-period metals.
atoms, M,;(b) the energy of the incident particles; and (c) the angle of incidence has been well established. Inspection of Figs. 12-15 and of Table VIa serves to illustrate the experimental justification for requirements (l), (2), and (3) above (see 69-73).
TABLE VIa. SPUTTERING RATIOAS A FUNCTION OF ANGLE OF INCIDENCE"
2
*w * z*cd
c)
Sputtering ratio of copper by bombardment with:
Ion energy (k-9
Nf
0"
50"
Si+
N,+ ratio
0"
50"
ratio
0"
50"
TI+
Ar+ ratio
0"
50"
ratio
0"
50"
ratio
0
50 Z
5
1.7
3.0
0.57
3.4
4.6
0.74
3.0
5.5
0.55
5.6
8.7
0.64
10
1.9
3.4
0.56
3.5
5.9
0.59
3.6
7.4
0.49
6.3
11.0
0.57
15
2.1
3.4
0.62
3.6
6.5
0.54
4.0
7.8
0.51
6.4
11.6
0.55
20
1.7
3.4
0.50
3.6
6.8
0.53
4.3
8.2
0.52
6.4
12.1
0.53
3.7
7.0
0.53
6.5
12.4
0.52
* 3 0.46
2
P. K. Rol, J. M. Fluit, and J. Kistemaker, Proc. Symposium on Isotope Separation, Amsterdam, 1951, p. 657 (1958); Physica 26,
3z
25
lo00 (1960).
13.8
30.0
278
ERIC KAY
3
2
4
incident ions at 400 ev [N. L. Laegreid and G . K. Wehner, J. Appl. Phys. 32, 365 (1961).]: (a) neon, (b) argon; (c) mercury.
w m 5
L
a
FIG.15. Variation of sputtering yield S with atomic number of target material for neon, argon, and mercury
2. T a r g e t Surface Effects. The necessity for both chemical and crystallographic characterization of the target surface during bombardment is unquestionable, and certainly there is extensive experimental evidence for requirement (4).What happens to the properties of the target surface during bombardment will depend mainly on how the
IMPACT EVAPORATION AND THIN FILM GROWTH
2 79
crystallographical, morphological and chemical nature of the surface changes during the bombardment, which in turn is determined by the energy of the incident particles [requirement (8)] and the chemical properties of the target and incident particles. A number of authors have demonstrated that an energetic incident gaseous species can increase the gas content in the target several orders of magnitude over that to be expected from the normal “solubility” of the gas, with profound changes in the physical properties of the target material. These gaseous impurities can in part be considered as large solid defects which undoubtedly affect the diffusion of energy pulses and subsequent impact collisions, as well as the etching mechanism on the target surface. Morrison and Lander (74) report that, when clean nickel was bombarded with hydrogen ions, no excess hydrogen was introduced, but when nickel was coated with NiO or BaO, the concentration of dissolved hydrogen was increased by several orders of magnitude. T h e hydrogen ions appear to penetrate the oxide layer and become trapped beneath it. Lander (75) found that hydrogen concentrations lo5higher than normal were produced in zinc oxide. This indicated a surface barrier to the escape of hydrogen atoms from the crystals of about 15 kcal, with consequent important effects on the semiconducting properties. Numerous other recent articles (76) on changes of electrical properties of semiconductor surfaces produced by ion bombardment in the 1 to 100 kev energy range have appeared. Rourke et al. (77) used 10 kev ion bombardment doping to make shallow p - n and n-p junctions by impregnating silicon with group I11 or V elements. Doping concentrations of 10’8 atoms/cm3 have been achieved with very small ion currents (< lo-’’ amps). Further interesting results have been reported by Barnes et al. (78), who actually observed vacancy sources in a metal on a microscopic scale as follows. Energetic helium ions were injected into a metal by bombardment. O n subsequent heating, the helium precipitated out as small gas bubbles in those parts of the metal where the helium atoms can capture vacancies to make room for bubble formation. Numerous instances of inert gas bubble formation in irradiated metals have been cited since then (79-84). Other examples of change in the target surface properties by incident energetic ions have been reported (85,86). Gillam (87) bombarded various alloys, chiefly Cu3Au, with 4 kev argon ions and, by careful electron diffraction work, clearly showed that after the very first surface layer has been eroded, the next layer, about 40 A deep, has an altered, but constant, composition during prolonged erosion. T h e rate of removal
280
ERIC KAY
of particles is apparently never as great as the number of target particles affected by the incident ions. The thickness of the altered layer is taken to be a measure of the range of the incident ions. It may be more correct to describe the depth of such a layer as the net depth of detectable change in crystal structure, since some of this damage may well have occurred beyond the depth of penetration of the incident particles, as a result of the focusing effects mentioned earlier: A report by Davies et a/. (88) on range of radioactive labelled alkali metal ions up to 100 kev in aluminum appeared recently. According to Davies, the median range increases linearly with energy as predicted from the Bohr-Nielsen equation, but has a value twice that predicted by the equation. Ogilvie and Thomson (89) more recently described the influence of temperature and bombardment rate on disorientation of Ag single crystals. It is suggested that, during bombardment, point defects are introduced in large numbers into a thin layer at the surface of the crystal. I t has been established that such defects can aggregate to form stacking faults and the electron diffraction patterns show evidence of considerable fault density. There is a possibility that the annealing can be accelerated by the presence of point defects, since the observed grain growth due to simple annealing is not fast enough to account for the decrease in disorientation with increasing temperature of bombardment. Evidence has accumulated to show that a high concentration of point defects can accelerate local or microdiffusion. With this sort of evidence of surface damage, one may wonder how generally the exact description of the original crystallographic properties of the target can be used in correlations with the angle of incidence of ions drawn after the experiment. T o the number of instances already described under “focusing effects” may be added a rather convincing example from Rol ( 9 4 , who demonstrated that the ejection ratio 0 for single crystal surfaces goes through minima and maxima corresponding to a more or less open or transport crystal orientation presenting itself to the incident beam as the crystal rotates. T h e peaks of these minima and maxima coincide very well with the path of the beam falling on specific crystallographic planes. This effect is directly related to the longer mean free path of the energetic particles along the more transparent directions. However, confusing exceptions to these general findings have been reported. For example, Anderson and Wehner (53) found ejection patterns from Ge single crystals strikingly similar to those from bcc crystals. This could be explained by assuming that, under ion bombardment, so many interstitials are formed near the surface that the atom arrangement of the Ge crystal resembles that of a bcc lattice as in Si for example, They also reported that, besides the expected focusing effects
IMPACT EVAPORATION AND THIN FILM GROWTH
28 1
along the close, packed hexagonal direction of Ti, focusing occurred along other directions. T h e possibility of a phase change was considered. Further information about target surface changes on ion bombardment comes from recent metallurgy studies. Ion bombardment as a method of etching single crystals, as well as of obtaining atomically clean surfaces, has become an extensive field of study. Only a few typical recent references (since 1959) will be cited here. A lengthy report by Farnsworth et al. (91) has appeared which deals with conditions resulting from ion bombardment of semiconducting surfaces, particularly silicon. Another extensive report on the same subject came from Arnold (92). Gianola (93) measured damage and depth of ion penetration of silicon targets subjected to 30 kev He ions. Both Ogilvie (94) and Sosnovsky (95) studied surface structure of Ag crystals of various orientations after bombardment with positive ions having energies up to 4 kev. Bierlein (96) discusses etching effects similarly produced. Wolsky (97) also examined similarly bombarded surfaces, paying particular attention to oxidation kinetics. Trillat et al. (3-5) recently demonstrated that, in some ion bombardment systems, both single crystal and polycrystalline thin targets can completely change their crystallographic orientation.
3. Incident Particle Flux and Vacuum Conditions at Target. T h e species which arrive at the target depend largely on the ion beam composition and the background pressure at the target. Weiss et al. (98) showed that, in certain cases, by the time an ion beam reaches the target, it contains, in addition to ionic species, a large proportion of energetic neutrals. T h e extent to which electrons picked up by the ion beam at the source or in subsequent flight account for these beam compositions is not accurately known. However, in making quantitative measurements of such often-mentioned quantitties as “sputtering yield” per incident ion it becomes important to know how much of this yield is caused by impact of neutrals with the target, which an ion-current measuring device would not see. I n this connection, it is also important to note that secondary electron emission currents must be adequately accounted for in any current measurement of ion flux at the target. T h e best solution to requirement (5) so far is the application of a beam which is magnetically analyzed (e.g., mass spectrometrically) so that any particular ion species can be directed at the target. Unfortunately, current densities in these devices are often low. A neutral high energy kev beam can be achieved by magnetically bending the ions away from the target. T h e energy spectrum of such a neutral beam might be broad and difficult to measure. Since momentum transfer is known to be involved in the energy
ERIC KAY
range of interest here, it would often be more meaningful to measure the particle flux in some other way, possibly by a momentum balance, rather than by measuring ion current at the target. Since impact evaporation is a surface phenomenon, it is to be expected that vacuum conditions at the target will pose a problem. In most ion beam impact evaporation work reported to date, the actual pressure at the target was held to about at best. At this pressure the rate of arrival of residual gas species is such that approximately a monolayer per second could condense on the target surface (if one assumes an accommodation coefficient of 1). What actually happens to the residual gas species arriving at a target simultaneously being struck by the much more energetic ions is open to conjecture. Some specific evidence comes from Bradley (25). He bombarded a very carefully precleaned platinum sample with 10 ma/cm2 Xe+ ions at 1000 ev in a mass spectrometer arrangement. He was able to pick up a PtO, peak after 10 minutes of bombardment; under the same vacuum conditions (10-8 to lo-' mm mercury) without bombardment, it took roughly a whole day before this peak appeared. That is, ion bombardment increased the room temperature oxidation rate of platinum by more than a hundred times. Naturally, the fewer unknown species arriving at the target, the better defined will be the experiment. Thus, the best possible vacuum should be maintained at the target. Much improvement in this direction can come from preferential pumping near the target surface. Closely related to this problem, of course, is requirement (I), the rate of arrival of energetic ions. In principle, one would like to work with very high current density beams so that the ratio of ions to unknown residual gas species arriving is as large as possible. All but a few authors so far have used very low current densities, i.e., micro-amperes per square centimeter, although densities of milliamperes per square centimeter are quite feasible. A current density of 20 ma/cma is equivalent to about 10'' ions striking a square centimeter per second. An ambient impurity gas pressure of mm Hg is equivalent to about 4 x lOI4 atoms arriving per second per square centimeter. A monoenergetic high energy (up to 60 kev) ion beam having a current density of 20 ma/cm2, extracted from a von Ardenne (99) Duoplasmatron ion source into a mm target area (ratio of ions to ambient gas arriving at target -100 at best), is now being used in this laboratory (ZOO). Since to focus a monoenergetic parallel beam of much higher current density becomes very difficult, better definition in the sense discussed above can only come from better vacuum conditions. Furthermore, the vacuum should be such that the mean free path of
IMPACT EVAPORATION AND THIN FILM GROWTH
283
any particle leaving the target is large enough to condense on some surface before the particle collides with another gaseous species, so that it cannot be scattered back to the target and thereby change target characteristics. This latter condition can, however, readily be met at mm Hg. 10
I
I
I
I
I
1
h
z
0
I-
ez
9-
0
I a W a. In
8-
I 0
F
5 In
7 I I 1 I I I OD2 0.03 0.04 OD5 0.06 0.07 OD8 0.09 MANIFOLD PRESSURE (microns of Hg)
FIG. 16. Variation of sputtering yield S with manifold pressure for 30-kev argon ions on copper [O. C. Yonts and D. E. Harrison, Jr., J. Appl. Phys. 31, 1583 (1960)].
Bradley’s oxidation work on platinum (15) points up the general futility of trying to clean the surface by bombardment unless the current density is large enough to remove all adsorbed impurities as fast as
! i
-
x- 7 - 0 x 6 -p -
0 0
“
0
0 0
K
f 5 -
g 4
I
I
I
-
I
FIG. 17. Variation of sputtering yield S with current density for 20-kev argon ions on copper [P. K. Rol, J. M. Fluit, and J. Kisternaker, Proc. Symposium on Isotope Separation, Amsterdam, 1957 p. 657 (1958)l.
284
ERIC KAY
they form. Yonts and Harrison (ZOZ) recently presented evidence that surface contamination may be a significant factor in quantitative ejection measurements as can be seen from the sputtering ratio dependence on background pressure obtained in their experimental setup (see Fig. 16). On the other hand, Rol et al. (61), under approximately similar conditions, found this ratio to be independent of current density (see Fig. 17). Obviously, only more specific definition of the target environment can resolve these problems.
IV. GLOWDISCHARGE CHARACTERISTICS Having some appreciation now of the processes taking place at the target being bombarded by energetic particles, one can proceed to study the influence of a glow discharge environment on these processes and on the subsequent behavior of cathode ejected particles. Since no detailed insight has yet been gained into the impact evaporation process in a glow discharge, it is only possible to infer, from experimental findings, the influence of such an environment on particles leaving the cathode. T h e type o f discharge which occurs between two electrodes depends upon: (1) the value of the gas pressure, (2) the value of the applied voltage, and (3) electrode geometry, the discharge path length, and the resultant current density. T h e current-voltage characteristics of a
FIG. 18. Schematic of a self-sustaining gas discharge. Voltage: linear scale; current: logarithmic scale; Vb: breakdown voltage; Vn: normal cathode fall of potential; v d : arc voltage [F. M. Penning, “Electrical Discharge in Gases.” Macmillan, New York, 19571.
IMPACT EVAPORATION AND T H I N FILM GROWTH
285
discharge between flat plates are shown in Fig. 18 (see Penning, 102). T h e discharge of interest here is that in the milliampere range, i.e., the glow discharge. I n the pressure range between 1 and 200 microns of Hg the discharge will take the form shown in Fig. 19 (103). 1st 2nd Cathode Layers Positive Column Negative Glw Anode Glow
Space
Cathode Dark Space
,
2 ‘ >
Light Intensity Potent ia1
+m I
I
i 7:
Gas Temperature
+ -
FIG. 19. Variation of discharge parameters along the length of the discharge [S. Flugge, ed., “Handbuch der Physik,” Vol. 22. Springer, Berlin, 19561.
T h e discharge is maintained by electrons produced at the cathode as a result of positive ion bombardment. In the Aston dark space there is an accumulation of these electrons which gain energy through the Crookes dark space, also called cathode fall, or cathode dark space. When the positive ions are neutralized, the decay of their excitation energy gives rise to the cathode glow. T h e electrons which have passed through the Crookes dark space enter a practically field-free space, some of them with high velocities. There the electrons lose energy in further inelastic collisions in which some of them ionize, and some excite, atoms. I t is the decay of these atoms to the ground state by means of a radiative transition which gives rise to the negative glow. T h e sharp visible boundary between the negative glow and the Crookes dark space is largely due to low energy electrons from the field free negative glow region which diffuse toward the cathode and run up against the potential wall at the edge of the Crookes dark space. The anode end of the negative glow corresponds to the range of electrons with sufficient energy to produce excitation. I n the weak field of the Faraday dark space, the electrons once more gain energy. T h e positive column is the ionized
286
ERIC KAY
region that extends from the Faraday dark space almost to the anode and is followed by the short anode dark space and anode glow. Reduction of pressure causes the cathode dark space to expand at the expense of the positive column because electrons must now travel farther (mean free path is greater) to produce efficient ionization. This phenomenon shows that the ionization processes in the cathode dark space are essential for maintenance of the discharge, and the positive column merely fulfills the function of a conducting path between the anode and negative glow region. T o maintain the discharge, an electron must, in its passage through the gas, produce enough positive ions to release a new electron at the cathode. When this requirement is not met, the cathode dark space will expand until it contacts the anode electrode and the discharge will be extinguished. T o increase the number of inelastic collisions leading to ionization, the electron path can be elongated by imposing a magnetic field such that electrons travel toward the electrodes in spirals rather than in straight lines. By using a magnetic field of 850 oersted, Penning and Moubis (104) were able to maintain a glow discharge at 500 volts and 1.65 amp in a mm Hg. Using larger fields small apparatus at a pressure of 3 x and electrode geometries not too practical for thin film work, discharges can be sustained at much lower pressures. An additional advantage of using a magnetic field is that now the three variables, current density, pressure, and potential across the cathode fall, can be varied independently within certain limits. As might be expected, the “temperature” across the discharge changes drastically with position in the plasma. T h e concept of temperature in a plasma is somewhat difficult to delineate. I n simple kinetic theory temperature is .related to the mean translational kinetic energy of the gas molecules. I n nonequilibrium situations one refers to other modes of energy storage and defines rotational, vibrational, electronic, spin and translational temperatures. In addition, different constituents of plasmas can have different temperatures: Temperature differences of several orders of magnitude often exist between electrons, ions, and neutrals, particularly under the influence of electric fields. Furthermore, similar constituents may fall into several distinct energy groups within a plasma, So, for example, the energy source in the negative glow region (of most interest in this report) is the kinetic energy of the primary electrons entering from the cathode fall region. Energy losses are due to collisions, heat conduction and radiation. I n contrast to a plasma in equilibrium, the velocity distribution of electrons in the negative glow is not Maxwellian, but distributed in three energy groups: the primaries, the secondaries, and ultimate electrons. The primaries arise at the
IMPACT EVAPORATION AND THIN FILM GROWTH
287
cathode and enter the negative glow with beam properties and velocities gained in crossing the electric field of the Crookes dark space. T h e secondaries are formed by various collisional processes and have energies of 6 to 10 ev. T h e density of ultimate electrons, which are the predominant energy group, increases towards the Faraday dark space and also increases with pressure and current density. Their velocity distribution is approximately Maxwellian. T h e net result is a decrease of electron energy towards the anodic boundary of the negative glow region. Further complications come from the interaction of charged particles with the confining walls of the discharge. These interactions cannot be treated as simple specular reflections with no transformation of energy. Various possible processes already discussed in earlier sections must be considered. The recombination processes at the walls can greatly influence the chemical reactions in the discharge, which will be of significance in “reactive sputtering,” to be discussed in a later section. Such restriction of freedom of motion facilitates rapid recombination of ions and electrons. T h e rate of such recombination is, however, much higher than the ambipolar diffusion rate of ions and electrons out of the bulk plasma, which consequently is the limiting process for loss of charged particles from the plasma at the walls. For a more detailed treatment of this and other basic characteristics and of the necessary equipment to produce a glow discharge, the reader is referred to such books as those by Fliigge (103), Loeb (105), Holland (106), Penning (102), Brown (107),Mayer (108), and Ecker and Muller (109). A number of papers, e.g., by Guentherschulze (110) and Wehner ( I1I ) , should also be consulted. T h e relatively high pressure necessary to maintain a glow discharge is probably the most important factor affecting the ejection of particles from the cathode target and their subsequent diffusion to a collecting substrate. T h e work by Hagstrum (12), Honig ( 1 4 , and Bradley (24, mentioned earlier, clearly demonstrated the physical and chemical effects of adsorbed gas layers on impact evaporation, even when the target is held in a high vacuum environment of 10-o mm Hg. A target mm Hg would be more subject to these effects. T h e held at lo-’ to type of “unwanted” chemical reactions at the cathode depends largely on the purity of the cathode material itself as well as on the purity of the gas species. T h e rate of such reactions depends upon the kinetics of the reactions of charged high energy ions as well as on interactions of excited and thermally diffusing neutrals with the target material. This rate can be greatly affected by the current density and energy of incident ions which will determine the rate of removal of the “reaction product” from the cathode. T h e electrodes can also act as catalysts for reactions
288
ERIC KAY
which occur with low yield in the bulk plasma. For example, the yield of ozone is enhanced in a glow discharge when the electrodes are of aluminum. Problems of this kind must be considered when surface oxidation is of importance. Another target-surface variable in a glow discharge, not present in a high vacuum environment, is the back diffusion of ejected particles, which continually changes the physical characteristics of the cathode surface. T h e cause of this back diffusion is that only some of the neutral ejected particles get through the cathode fall region without collision, because of their much shorter mean free paths at higher pressures. At 10-1 to mm Hg, a common glow discharge working pressure, the mean free path of neutral gaseous species with thermal velocities ranges from 0.1 to 5 mm. Von Hippel (112) calculated that in a plane electrode arrangement without a magnetic field and with the collector just beyond the cathode fall region, at a pressure of 0.1 mm Hg, approximately 90% of the ejected material diffuses back to the cathode. T h e return material may be deposited in peculiar forms [dust, as shown by Blechschmidt. (113), or cones, as shown by Guentherschulze (114)]. Some of the ejected particles, however, may become charged in the Crookes dark space and their transport will be very different from normal diffusion; i.e., their mean free paths will be different from the mean free paths of neutral gas atoms. A theoretical treatment of this problem by Ecker and Emeleus (115) appeared in 1954. Unpublished experimental data being collected by the present author indicate that the back diffusion problem is not as severe as formerly thought. Numerous serious efforts have been made in the past to use glow discharge data as the basis for a mechanistic interpretation of the impact evaporation process itself. These must now be almost entirely discounted. T h e description of the target surface is, as noted, very complicated and constantly changing. Further, the incident energetic particles may have a large spectrum of energy, depending on where in the cathode dark space they first became ionized before acceleration to the cathode. So far we have only qualitative knowledge that variously ionized, as well as excited and, of course, neutral thermal species are present among the particles arriving at the target. The direction of many incident species can only be approximated. Charged particles arrive at and leave the cathode orthogonally to the plane of the cathode surface. This ray-like property is quite significant in the preparation of uniform thin films in a discharge environment. Cathode edge effects greatly influence the direction of the particles. T h e quantitative collection of all ejected particles is almost impossible at the higher operating pressures. I n spite of these uncertainties, a large number of surprisingly consistent results have been
IMPACT EVAPORATION AND T H I N FILM GROWTH
289
published. Most of the significant papers in this category have attempted to correlate such parameters as the geometry of the electrodes, and its relation to that of the entire apparatus, gas pressure and density, positive and negative current carriers, and the axial and radial particle energy profile. T h e cross sections for production of current carriers in a volume of a neutral gas produced by electron or ion collisions, photoionization, or collisions of the second kind are known only for a limited number of systems. T h e influence of these phenomena on cathode ejected atoms or clusters of atoms is not well known for any system; however, electron-neutral collisions are undoubtedly the dominant process in a glow discharge.
V. NUCLEATION AND FILMGROWTH IN HIGH VACUUM ENVIRONMENT Since the remaining remarks will ‘deal with thin films produced by impact evaporation in a glow discharge system, one must consider the conditions at the collecting substrate in such a system. With the advent of high vacuum techniques and electromagnetically analyzed ion beams, it is no longer advisable to study the mechanism of impact evaporation itself in a complicated glow discharge system. No such encouraging alternative to the glow discharge system exists, however, for the study of thin film growth in a relatively high pressure discharge environment. Since films so grown are often significantly different from films prepared by other methods and are potentially useful in a number of areas, this study becomes an unavoidable task despite its complexity. Furthermore, although a complete understanding of the problem lies in the distant future, the consistency of results so far obtained is sufficiently encouraging to warrant such a study. Within the last few years some headway has been made in studies of nucleation and growth of thin films prepared under more favorable (i.e,, high vacuum) conditions. Perhaps the most significant recent source of information dealing with this subject is the “Proceedings of the International Conference on Structure and Properties of Thin Films” (116). I n view of this extensive, excellent source material, this paper will deal only with aspects most relevant to film growth in a glow discharge.
A. Film Growth Evaluation Tools Such sensitive tools as the high resolution electron microscope, the field emission microscope, and high resolution electron diffraction techniques have obviously made an important contribution to progress in film growth studies. Resolution of better than 10 A, though difficult
290
ERIC KAY
to obtain, is very valuable in obtaining meaningful results in such studies. Most of the high resolution studies with an electron microscope have been made on films prepared by thermal evaporation. Although these methods are potentially applicable to impact evaporated films as well, no such studies have been reported. A few preliminary high resolution electron microscope observations were, however, made by Pashley (117) on electrodeposited films, which are qualitatively similar in growth to thermally evaporated films. More detailed information about individual crystallites can now be obtained by the lattice resolution method developed by Menter (i18),which allows direct resolution of the periodic structure of some crystal lattices, but is limited by the resolving power of the present-day electron microscope (theoretically and in practice about 5 A). This technique can therefore not be used for the study of simple elements and inorganic compounds, whose crystal spacings are usually less than 3 A. This limitation has now been overcome to some extent (119) by the application of the well known optical principle of forming MoirC patterns for imaging of crystal gratings with electrons. This technique, by which spacings less than 5 A have become indirectly discernible, has made it possible to observe various kinds of defects in the solid state which are known to influence the growth and subsequent physical properties of thin films. Elastic bending and variations in crystal orientation are now readily revealed. With the application of the kinematic and subsequently the more sophisticated dynamical theory of electron diffraction, much progress has been made in the interpretation of contrast effects such as dislocations and stacking fault contrast, extinction contours, and absorption, in electron micrographs of thin crystals. However, a great deal needs to be done in the interpretation of certain imperfections in images of periodic objects. I n the study of adsorption phenomena, which are obiously involved in film growth, such authors as Ehrlich (120), Gomer (122), and Muller (122) have made valuable contributions using field emission microscopy.
B. Condensation Phenomena There is no direct experimental evidence concerning the initial nucleation. It has been shown by several observers that, if the vapor density arriving at the substrate is below a certain critical value, films cannot be formed. This critical density for deposition has been interpreted as a critical supersaturation for heterogeneous nucleation of a condensed phase. I n ordinary sublimation processes only small supersaturations are required to condense the sublimate ; whereas, on cold substrates, for example, the beam pressure must be many orders of
I M P A C T EVAPORATION A N D T H I N FILM G R O W T H
29 1
magnitude greater than the vapor pressure of the condensate at substrate temperature. Frenkel (123), in his classic theory, has explained this by assuming that atoms arriving at the substrate move over the surface and eventually re-evaporate from the substrate. When a collision occurs between two of the atoms moving on the surface, an atom pair is formed which has a much longer lifetime on the surface than a single atom. Associated with this pair formation is a liberation of the energy of association which decreases the probability of re-evaporation of these complexes compared to the re-evaporation of single atoms. A pair, once formed, can act as a nucleus for further collisions, which can lead to the formation of larger aggregates. Once the size of the aggregates is above a certain value not yet defined, there is high probability that interaction with the substrate will cause a stationary nucleus to form from the mobile aggregate. Experiments on the reflection of molecular beams can be used to distinguish between direct reflection and adsorption followed by reevaporation. Specular reflection cannot involve temporary adsorption ; diffuse reflection, according to the cosine law, almost certainly involves temporary adsorption. While the latter situation has been abundantly demonstrated in many systems [e.g., Wood (124, Knudsen (125), Taylor (126)],only very few examples of specular reflection have been reported [e.g., Knauer and Stern (127)]. It has been pointed out recently by Sears and Cahn (128) that, though qualitatively correct, the Frenkel model fails to account for the tremendous ratios of critical beam intensity to evaporation rate per unit area of condensed phases, because it does not distinguish between substrate and adsorbate temperatures. T h e true critical supersaturation for deposition is the ratio of the impingement rate to evaporation rate of condensate phase at the adsorbate temperature rather than at the substrate temperature, the latter temperature possibly being much lower than the former when a hot gas hits a cold substrate. Gafner (129), in a recent theoretical discussion of the thermal aspects of thin film growth, concluded that the temperature of an overgrowing film will be essentially that of the substrate when growth is by a twodimensional monolayer or continuous film mechanism. High film temperatures necessitate low substrate thermal capacities in order that the heat of sublimation released on formation of the film may elevate the temperature of the system appreciably. In the initial stages of growth of a film, where a continuous film mechanism is hardly applicable, Gafner was able to show that at high rates of deposition onto a substrate of low thermal diffusivity and with rapid migration of incident particles on the substrate, the possibility of high transient temperatures definitely
292
ERIC KAY
exists. High temperatures may be caused by an entirely different thermal situation when growth occurs though a three-dimensional isolated, aggregate mechanism, which is known to be the case initially for most thin film condensates. It is certainly to be expected that growth temperature will influence such important properties as crystallite size and perfection, as well as the tensile stresses which might result from differential thermal contraction of the substrate and overgrowth. Such stresses would have a marked effect on adhesion and epitaxial growth. From some thorough experimental work published by Levinstein (130), it would appear that the rate of arrival of incident particles at the substrate affects the crystal structure only indirectly by affecting the grain size of the deposit. When antimony is evaporated rapidly, small patches are formed which at first exhibit no crystalline properties. They are, however, highly unstable in that form and crystallize rapidly when sufficient atom layers are present to overcome the disordering influence of the substrate. Patches which are formed by slower evaporation, and therefore cover a larger substrate area, are much more stable and crystallize slowly, if at all. Levinstein suggests that this dependence of particle size on rate of arrival of particles may be explained by expanding Frenkel’s concept of surface mobility, making two additional assumptions: (1) the number of atoms in motion on the surface is proportional to the number of atoms arriving at the surface per unit time; and (2) atoms and molecules will move over the surface until they suffer collision with other atoms or molecules and thereby lose their mobility. T h e large extent to which mobility of atoms on a substrate affects the structure has been very clearly observed by a number of authors. T h e effect of mobility becomes particularly evident when the collecting substrate is changed. While the mobility can be quite high on a neutral substrate such as glass or collodium, it can be very low when, for example, antimony is deposited on gold. Here it has been shown (120) that rate of evaporation does not affect particle size, the microcrystals remaining of uniform size for all rates of evaporation. This is probably because antimony atoms react with gold and remain near the place they originally hit, Needless to say, great care must be taken to prevent contamination of surfaces, since this has been shown to affect the surface mobility of atoms considerably. Numerous other theoretical studies on the kinetics of condensation have been reported. A few of these, showing different approaches, will be mentioned here. According to Lennard-Jones (134, an atom on a surface is there by a field of force which has a certain potential at every point. T h e surface has pockets of low potential separated by potential barriers; at very low temperatures any deposited atoms will vibrate about
IMPACT EVAPORATION AND THIN FILM GROWTH
293
the minima of these potential wells, as the energy of the atoms depends directly on the temperature. The latent heat of vaporization of the atom from the surface, Ls, is the average energy which is required to remove an atom completely from the influence of the surface, assuming the atom is initially at its lowest vibrational level, I n order to move freely over the surface an atom needs much less kinetic energy than Ls. If the atom is given sufficient kinetic energy A for it to surmount the highest potential barrier along the surface, it may move freely over the entire surface as long as it retains this energy. Since A is less than L,, free motion of the atoms on the surface is possible at temperatures well below those needed to evaporate the atom. Both A and L, depend on the nature of the surface and the atom. On the basis of the above theory, Appleyard (132) classifies metallic films into three broad groups, depending on the stability of their first layers. I n Type I surfaces, Ls > Lm, where Lm is the latent heat of evaporation of the metal from itself, and the first layer is inherently stable. I n Type I1 surfaces Ls < L m , but L, is 1 ev or more and A is a few tenths of an electron volt. Fairly coherent films of this type may be formed at temperatures below 180"C, but as the temperature rises agglomeration takes place as a result of the increased energy and hence increased mobility of the atoms. In Type I11 surfaces, Ls < L m , but L, is about 0.1 ev and A is only a few hundredths of an electron volt. Such films show rapid agglomeration, even at extremely low temperatures of deposition, because A (which determines the maximum height of potential barriers) is so small. A somewhat more empirical classification was made according to melting points by Levinstein (130). He showed that: (1) Metals of high melting points, i.e. greater than 1900"C, such as tungsten, tantalum, iridium, columbium, and rhodium, give very diffuse electron diffraction patterns which are, however, not completely amorphous, judging from the fact that the lattice constants agreed well with those computed from X-ray data. T h e crystal sizes were estimated to be 15 A. (2) Metals having melting points of 600 to 1 9 W C , such as gold, silver, copper, nickel, iron, cobalt, chromium, manganese, titanium, beryllium, lead, tin, paladium, and platinum, gave very sharp line diffraction patterns usually associated with crystallites larger than 100 A. (3) Metals with melting points below 650°C produce films whose microcrystals are oriented with respect to the substrate. Those examined were: antimony, bismuth, tellurium, cadmium, zinc, manganese, indium, and thallium. Although this classification is obviously oversimplified, it nevertheless serves as a good rule of thumb. Levinstein also investigated the effect of atomic velocity of incident
294
ERIC KAY
particles on crystal structure. That velocity should be important seems intuitively clear. Unless the incoming atom transfers its kinetic energy to the surface during the collision, adsorption cannot occur and the atom will again return to the gas phase, even though the bound state on the surface may be more stable thermodynamically. During the time of collision, the atom must lose to the lattice an amount of energy equal to or greater than its kinetic energy, in order to be bound in one of the higher vibrational states. This energy may be transferred to the lattice by excitation of phonons or, in a conductor, it may be accommodated by populating higher levels in. the Fermi distribution of electrons. If this first step is successful, the atom will still be in a highly excited state and must make transitions to one of the lower vibrational states in order to achieve the Boltzmann distribution over the vibrational levels chaTacteristic of the temperature of the surface. Some recent detailed calculations have led to an estimate of the residence time of an ad-atom on the surface T and the relaxation time for thermal equilibration. Rapp, Hirth, and Pound (233), in contrast to Langmuir (234) and Lennard10-l2;i.e., very much Jones’ (235)earlier work, estimate values of T greater than the period of one vibration. They therefore conclude that, once adsorbed, an ad-atom will probably equilibrate thermally with the surface rather than re-evaporate. Aziz and Scott (236),on the other hand, examined silver films grown in a high vacuum environment and at a mm H g and were able to demonstrate nitrogen pressure of 3 x consistent growth patterns in these two regions. In the high pressure region, where incident particles were thermally “cooled,” less particle aggregation on the substrate was observed. T h e influence of adsorbed nitrogen and changes in angle of incidence on the mode of film growth were not adequately accounted for. From a comparison of the actual velocity distribution and that anticipated from a Maxwell-Boltzmann distribution, Levinstein was able to show that the arriving species were not always monatomic as one might have naively assumed. This fact has, of course, become very clear since the pioneering work by Inghram et al. (237) on high temperature thermodynamics, which indicates that a large number of unusual gaseous molecules are quite stable in polymeric form, as well as in oxidation states not known at all at room temperature. I n view of Frenkel’s ideas on the effect of atom pairs on nxleation, it is important to recognize that the stable vapor species arriving at a substrate can be polymeric. It has been clearly demonstrated by Kay and Gregory (238) and others that whenever the structure of the gaseous species is unlike that of the condensed phases, the condensation efficiency is low. While the
>
IMPACT EVAPORATION AND THIN FILM GROWTH
295
condensation coefficient ac for metals vaporizing as monatomic species has recently been shown unequivocally to be equal to unity, this is not the case for such materials as carbon, phosphorous, arsenic, and arsenic oxide all of which have stable polymeric vapor species. I n cases like potassium chloride and iodine the interatomic distance in the stable molecular species in the vapor phase is considerably different from that in the condensed phase and again ac < 1. So far no solid-vapor system in which the vapor species is chemically different from that in the condensed phase, such as MX,s, + M(s, X(g),have been found with ac = 1. Other theoretical treatments of condensation processes and their measurements can be found in reviews by Hirth and Pound (139), Knacke and Stranski (140) and Beeck et al. (141). T h e effect of the substrate structure on crystallographic orientation properties of a deposited crystal growing upon it, usually referred to as epitaxial growth, has been the subject of literally dozens of publications [for a recent review, see Pashley (142)]. However, there is at present no thorough understanding of the phenomenon. T h e assumption of a good match in symmetry and spacing between the substrate and deposit crystals at the interface has been the basis of most of the theories so far advanced-the most detailed, that of Frank and Van der Merwe (143), is an outgrowth of Finch and Quarrell’s (144) earlier concept of basal plane pseudomorphism. But recent systematic work has provided no support for the theory of pseudomorphic growth, and epitaxial growth with very high lattice mismatch (-39% to +90% for alkali halides) has been demonstrated. In fact, although little is known about the state of strain at the substrate deposit interface, the direct evidence always indicates that, when stationary nuclei are first formed on the substrate surface, the lattice spacings are the same as for the corresponding bulk material. T h e role of the microscopic irregularities of the surface of the substratum, such as steps and exposed edges of screw dislocation, on epitaxial growth has only been superficially examined (145). Thus, none of the theories so far advocated have general application. It would seem that present theories are based upon unrealistic models for the mode of nucleation of one substance on the surface of another, and that more insight into the initial stages of growth is necessary. T h e situation with respect to understanding the growth of a film after nucleation is not quite so vague. Pashley and Menter (146) and Bassett (147) for example, have followed the sequence of events rather carefully in several cases. T h e results show that the initial observable stages of growth often involve the formation of three-dimensional discrete nuclei even when the average deposit thickness is less than an
+
296
ERIC KAY
atomic diameter. Nuclei as small as about 10 A across, consisting of only about 50 atoms, have been observed by electron microscopy. Under these conditions the technique is sufficiently sensitive to allow deposits of no more than 0.5 A in average thickness to be detected. It has also been clearly demonstrated (Bassett, 147) that preferential nucleation occurs at steps on a substrate surface (e.g., freshly cleaved rock salt), giving rise to pronounced decoration effects which are invaluable for revealing the step structure of the substrate crystal. T h e steps acts as preferential nucleation sites even when they are only monatomic in height. In Bassett’s experiments, with Au on NaCl, the prefernetial growth at the steps occurred for both oriented and nonoriented deposits, and there was no evidence that orientation of the nuclei on the steps was any different from that of the nuclei which formed on the parts of the surface which were found by refraction to be atomically smooth. Once a sufficiently high density of the initial nuclei is formed on the substrate surface, further deposition gives rise to growth in the following ways (according to Pashley and Menter, 146): (1) direct deposition onto the nuclei ; (2) interchange of atoms between neighboring nuclei, causing large nuclei to grow at the expense of small nuclei; (3) physical growing together of neighboring nuclei. All these processes have been observed by high resolution microscopy. I n order to deposit a continuous, very thin film, as would be desirable in many practical applications, there must be a strong preference of the initial nuclei of a deposit to grow parallel rather than perpendicular to the surface. This preferential direction of growth is determined by a balance between the surface energies of the substrate and the deposit, the energy of the interface between the deposit and the substrate, and any volume strain energy associated with misfit at the interface. T h e interfacial energies also include a term associated with a misfit, so that a low misfit, other factors being equal, favors the formation of uniform continuous films. This is presumably the explanation of the Schulz (148) finding that alkali halide deposits on alkali halides form thin plate-like nuclei when the misfit is less than 20 %, and three-dimensional nuclei for larger misfits. I n a few studies, thin films grown in a glow discharge environment have been observed to consist of smaller crystallites packing more continuously; i.e., less island formation was observed than is normally seen for thin films prepared by high vacuum evaporation [Koedam (149) and Ells and Scott (ZSO)]. Although continuous and uniform films have been prepared so that they closely resemble a thin slice of the corresponding bulk material, they always contain a large number of lattice imperfections which are appa-
IMPACT EVAPORATION AND THIN FILM GROWTH
297
rently an intrinsic part of the growth process. Dislocation lines, stacking faults, twins, and smaller irregularities, possibly associated with gaseous impurity or vacancies, are commonly observed. No direct evidence of the mechanism producing these imperfections in thin films has been obtained so far, although it has been established that they are not extensions of similar imperfections in the substrate. Continued electron microscopy studies of growth and structure of films deposited on a single crystal substrate will undoubtedly lead to a better understanding of the mechanisms of nucleation and growth. Studies of the more complex phenomena associated with growth on amorphous or polycrystalline substrates can then be attempted. ENVIRONMENT VI. FILMGROWTHIN GLOWDISCHARGE Keeping these qualitative remarks about nucleation and film growth in mind, one can now consider the type of problems that may arise when the film grows, not in a high vacuum environment, but in a glow discharge. Two of the glow discharge conditions are, by definition, different from high vacuum conditions. T h e substrate is in a relatively high pressure atmosphere and is being bombarded by an assortment of charged as well as neutral particles which, depending on where they were formed, may have traversed an electric field. T h e description of the energy and direction of incidence of these particles is very complex. In attempting to predict these parameters the elementary elastic sphere concepts used in classical kinetic theory of gases are no longer applicable and must be modified. If the internal structures of the molecules are sensitive to the presence of electric and magnetic fields, collisions will occur with different frequencies in the presence of such fields. Furthermore, the complete randomness in the distribution of separate particles with respect to velocity, implicit in the classical kinetic theory, no longer holds here. T h e motion of the negative charges is, to some extent, ordered with respect to that of the positive charges so that the essential premise of the classical theory of molecular chaos does not apply. T h e effects of long-range Coulomb forces tend to couple the ionized gas into a quasi continuum. Qualitatively, then, particles can come from various directions with a large spectrum of energy, depending on the zone of the glow discharge in which the substrate is situated, and whether or not a magnetic field is being used. If one considers means of sorting out these effects, one is forced back to the monoenergetic parallel ion beam in a controlled high vacuum environment. This is not only a commendable, but, to answer some
298
ERIC KAY
questions, the only approach. However, all things considered, it is probably not the most efficient future route to an understanding of the finished product-the film grown under glow discharge conditions. After all, the individual parameters affecting film growth in the welldefined system are being studied intensively, one by one, by a number of people working in the area of evaporated thin films. For example, the effect of energy of incident particles is being studied by such people as Levinstein (130) and Trillat ( 4 , 5 ) ; the effect of substrate characteristics by many workers, including Pashley (142) and Collins (151); the effect of energetic incident electrons by such people as Heavens (252); the effect of direction of incident particles by Evans and Wilman (153); and the effect of gas pressure by Aziz and Scott (136) and Wilson el al. (154). Monitoring these efforts is certainly most important. But, at the same time, more quantitative studies of the diffusion process of cathode ejected material through the plasma of a glow discharge and more quantitative description of the types of species, their energy spectrum and their direction on arrival at the substrate must be attempted so that the influence of superposition of these various effects on film growth in a plasma can be better understood. T h e discharge system need not be used as in the past, where very little if any attempt was made to optimize conditions in line with contemporary insight into thin film technology. It is believed that a glow discharge environment can be made more compatible with the requirements of film technology without losing the versatility and simplicity of operation so essential for any practical applications. In vacuum evaporation studies of nucleation and film growth, the philosophy of working at the best possible ultimate vacuum has rightly been advocated, if for no other reasons than to eliminate another variable, that of surface adsorption of undesirable contaminants. I n glow discharge studies this is an unreasonable criterion. Even with the help of a magnetic field to increase ionization efficiency, as pointed out earlier, one can only hope in practice to achieve about mm Hg with a planar, parallel electrode system. At this pressure about lo1* residual neutral gas atoms arrive per square centimeter of substrate per second. At first sight this seems like an intolerable situation; in fact, it is not nearly so serious. I n “physical impact evaporation,” where only an inert gas is used as the high energy bombarding species, the problem of film growth with respect to the residual gaseous species arriving at the substrate may be reduced to that of physical adsorption, as will become evident from the discussion to follow. Physical adsorption is here somewhat arbitrarily defined as those interactions arising from van der Vaal’s forces between the incident
IMPACT EVAPORATION AND THIN FILM GROWTH
299
atom and the substrate lattice, which are similar in magnitude to the heat of sublimation of the rare gases. Chemisorption, on the other hand, will denote the stronger interactions, with energies comparable to those of ordinary chemical bonds (greater than 10 kcal per mole), which arise from significant rearrangement of the electronic structure of the interacting entities.
A. Chemically Reactive Sources of Contamination Where inert gas is the bombarding material, it is assumed that the gas is the only species besides the impact evaporated material and electrons arriving at the substrate, so that no chemisorption need be considered. Perhaps a more realistic way of stating this situation would be to say that the degree of chemisorption need not be any more serious here than in a high vacuum environment. That chemisorption of residual gases must be minimized, or at least controlled, in thin film studies has been amply demonstrated. For example, Schlier and Farnsworth (155) have used the low energy electron diffraction technique to study the deposition of copper on titanium at mm Hg. They conclude that a monolayer of adsorbed oxygen on the surface of titanium inhibits epitaxial growth of the copper. Heidenreich et al. (156) relate magnetic anisotropy in films to small amounts of oxygen located at specific crystallographic planes of the film. T h e effect traces of reactive gases have on transition properties of such superconductors as tantalum has been recognized for some time by Ittner and Seraphim (157), Budnick (Z58), Caswell (159) and many others. T h e question is, how can the partial pressures of offending reactive gases be kept as low as in high vacuum work. One obvious way is to go through the same elaborate and costly procedure of complete outgassing at high temperature with subsequent prolonged pumping to reach the mm Hg pressure required before leaking the pure inert gas into the apparatus. But such a procedure would lose much of the relative simplicity and economy of the glow discharge technique, and it is not at all certain that is would be the only way of achieving the desired end. There is a school of thought which maintains that as good a way as any available for precleaning a fairly large practical system is by ion bombardment with an inert gas. Whether this method is in fact equivalent or superior to the high temperature bake-out, with ultra-high vacuum pumping equipment to reduce the partial pressure of reactive gases, has never been adequately determined. Available data would indicate that this may not be unreasonable. A recent publication by Hagstrum and
ERIC KAY
D’Amico (160) demonstrated that ion bombardment can produce an atomically clean surface on such a reactive metal as tungsten. The surface-sensitive phenomenon used in this work was the ejection of electrons by the Auger type of neutralization of slowly moving positive ions at the surface. This method was demonstrated to be capable of detecting changes in surface concentration of foreign atoms which amount to a few per cent of a monolayer. Hagstrum (161) more recently showed that the damage of the surface produced by such ion bombardment has very little effect on the Auger results. Similar findings have come from a number of workers (86,87, 162, 163) in semiconductor research. The atomically clean surface desirable for the study of film growth is not likely to be attained in any existing practical fair-size glow discharge system. Nevertheless, much has already been learned, though few, if any, studies on thin films reported so far have met this condition. Using somewhat less sensitive criteria, such as reflection coefficients of visible light, coefficients of friction and wetting characteristics, or adhesion of anodically oxidized films on metal surfaces, such authors as Holland (164), Hines (165), and Florescu (166) have shown that a glow discharge system can be used to clean surfaces more effectively than the far more common mythical methods of using acids, detergents and cotton, etc. However, it has become quite clear that a glow discharge cannot be used indiscriminately for precleaning an apparatus. An understanding of the discharge characteristics with relation to the geometry of the apparatus to be cleaned is absolutely necessary. Holland (267) was able to demonstrate that high energy electrons being emitted from the cathode have an entirely different effect on the surface from that of high energy ions, depending on the location of the surface to be cleaned with respect to the discharge. Electrons in the relatively low energy range, i.e., less than 5 kev, will not eject contaminants from a surface by impact evaporation, since they cannot transfer enough momentum. However, the resulting heating of the surface can have deleterious effects. At high gas pressures a glass substrate held inside or on the fringe of the cathode dark space was coated with an organic deposit, presumably from the pump oil vapor, whereas glass held in the positive column remained clean. Glasses exposed within the positive column of the high pressure glow discharge may remain free from contamination because: (1) the electrons accelerated within the positive column possess energy sufficient only to excite molecules but not decompose them, which is thought a necessary mechanism for organic contaminant deposition, based on work by Ennos (168); or (2) decomposed material is removed by the bombardment of ions and neutral
IMPACT EVAPORATION AND THIN FILM GROWTH
30 1
molecules at high velocity. Later experiments indicate that insufficient energy of electrons to dissociate hydrocarbon molecules is the more likely reason. Substituting silicone vapor for hydrocarbon gave the rather misleading result that the decomposed layer, unlike the hydrocarbon deposits, had a high coefficient of friction and could easily be wetted-two properties associated with clean glass or silica. It is, of course, well known to electron microcopists that electron irradiation of silicone molecules in high vacuum produces silica-like deposits. All this suggests that the high energy electrons should be kept away from the substrate and oil vapor content in the system minimized. Using appropriate electrode geometry and placing substrates only in those regions of the glow discharge where the electron temperature is low, as well as using high-surface-area Biondi (169) traps may achieve this end. Proper design of the electrode geometry together with high pumping speeds should permit adequate precleaning of the entire system while protecting the substrates from contamination by placing them at the appropriate location. Holland (167) recently published an evaluation of various electrode designs which have not yet been examined with sufficient care to warrant further comment here. Cleanliness during actual film deposition can also be improved as follows. After removing most of the contaminants, mainly water, from the various surfaces with an uncontained glow discharge, great care should be taken to allow only very pure inert gas to enter the system during the actual film deposition. As large a throughput of inert gas as the pumping speed of the system allows, in keeping with discharge pressure requirements, should be used at all times to help flush out the desorbed material. Naturally, some contaminant gases continue to diffuse from within various surfaces, but, at the high pressures necessary to maintain a discharge, desorption from such surfaces will not be any more serious than in an ultrahigh vacuum system. Evidence exists in the case of metals that gas held in the interior can only be completely out-gassed in a reasonable length of time if the metal is taken near its melting point. Even though glass, which is most frequently used in high vacuum systems, has a more open structure than metals, only part of the occluded gases from the interior structures is pumped off at the normal bakeout temperature of 450°C. (The softening point of Pyrex glass is -525°C.) The attainment of subsequent good vacuum is only possible because the rate of desorption of surface contaminants greatly decreases during a period of 12 hours or so at 450°C, and the pumping speed of the
302
ERIC KAY
system can then keep ahead of the diffusion of gases from the interior of the walls. Hydrogen and helium are often the major remaining species. In thin film growth studies it is well to remember that the conditions prevailing directly at the substrates are of prime importance and cannot be accurately described from pressure measuring devices situated a considerable distance away, particularly if intermediate conductance problems could give rise to pressure gradients. Many commercial high vacuum units have their ionization gauges very near the throat of the diffusion pump, which is rarely where a substrate is held. I n a glow discharge, where mean free paths are much shorter, it becomes reasonable to assume that a smaller fraction of desorbing species will diffuse to the substrate area, particularly if the throughput of pure inert gas is kept at a maximum, than is the case in a high vacuum where such species travel by molecular flow. T h e growing thin film is unfortunately often a good trap for reactive vapor species because of its gettering properties. Another way to minimize desorption from the container walls, which usually represent the largest surface area, is by confining the glow discharge in such a way that neither ions nor electrons bombard these walls during film preparation. This situation can be approached by proper design and/or by the use of a magnetic field. (If a magnetic field is used, it must be remembered that not only are the electrons affected, but, in consequence, the whole structure of the discharge changes). Thus, during the film deposition little material will be ejected from the container walls by impact evaporation, and, furthermore, the walls will not get heated as a result of electron bombardment. T h e other potentially serious source of film contaminants is the cathode material which is to be transported to the substrate by impact evaporation. T h e surface of any reactive metal usually has an oxide coating which is difficult to remove by evaporation since many metal oxides have comparatively low vapor pressures. Here again, ion bombardment may have the edge over high vacuum bakeouts; however, it must be remembered, as mentioned earlier, that surface oxidation kinetics may be enhanced in a discharge environment, so that the ion energy must be such that the rate of ejection of particles exceeds the rate of oxidation at the target. This should easily be possible, since after a thorough precleaning of the apparatus the rate of arrival of contaminants such as oxygen will be orders of magnitude less than the rate of arrival of inert gas ions if current densities of inert gas of several milliamperes per square centimeter are used. T h e number of metal atoms ejected from a metal cathode may exceed the number of incident heavy
IMPACT EVAPORATION AND THIN FILM GROWTH
303
inert gas ions in the 1 to 5 kev range; however, this situation will not be quite so favorable for certain surface oxide coatings. Ion bombardment does not, of course, remove the occluded and dissolved gases and other impurities from deep within the cathode surface. T h e depth of ion penetration in the energy range of 1 to 5 kev is in the order of 100 A (84, f70,171, 172), depending, of course, on the mass ratio of incident ion to target. [Batholomew and LaPaluda (174, using a radioactive K r tracer, found the depth of penetration two orders of magnitude greater than reported elsewhere.] I n this 100 A or so layer, atoms are displaced and surface bonds broken while impact evaporation is taking place, while such is not the case deep within the lattice. T h e diffusion of gas from within the target to the surface in a glow discharge system should be negligible compared with that in a high vacuum system because the bulk temperature of any metal cathode material can readily be kept at 100°C or lower by external cooling. Consequently, the diffusion rate of offending species from within the target will be low, To obtain a practical rate of vaporization of a metal in thermal evaporation in a high vacuum environment, the bulk temperature of the metal must be raised to somewhere near the melting point or even above. I n the case of more refractory metals, the melting point can be up to 3000°C, where the diffusion of gases to the surface is a much more serious problem. Here, therefore, the out-gassing of the bulk sample is obviously necessary before a pure film can be deposited. T h e main source of trouble in an impact evaporation experiment, as far as desorption from the cathode is concerned, is its surface layer, which is affected by the incident energetic particles. Needless to say, a fraction of the impurities in the layers, which are actually being eroded off the cathode surface during film deposition, will arrive at the substrate together with the other ejected materials. How much of the cathode impurities so ejected will actually condense on the substrate depends on the chemical nature of the system involved and on the rate of arrival. Since thousands of cathode surface layers have to be ejected to form a film deposit on a substrate of 1000 A or so, this source of contamination may be a problem. Even the purest materials commercially available usually contain incorporated gas species in excess of the content tolerable for some applications. T h e concentration of occluded gases in the lattice layers at the surface of the cathode may actually be a little greater during ion bombardment than it was before bombardment, because a small fraction of gas in this damaged layer of 1 0 0 A or so, which will always be beyond the layer being eroded, may be forced into the very top layer as a result of internal collisions between primary and knock-on particles.
304
ERIC KAY
A common cause of contamination encountered in high vacuum evaporation is alloying of the source material to be evaporated with the container, which is usually made of W, Ta, Mo, or C. This problem is entirely nonexistent in an impact evaporation experiment. No container is necessary for the source material, i.e., the cathode. I n most situations, alloying would not take place even if the cathode were in contact with other metals because its bulk temperature can be held below 100°C. B. Inert Gases as Possible Sourses of Contamination So far only the chemically reactive sources of contamination have been discussed. It is believed that these may be minimized to the same degree as in a high vacuum system. However, the inert gas is ever present in large quantities. One must now examine the effects of this gas on film growth in a glow discharge. As mentioned earlier, Wehner’s (68) work has shown that the cathode-ejected material leaves the cathode with energies greater than usually associated with thermal evaporation. This has recently been corroborated by Veksler (17#),who observed energies up to 50 ev for sputtered particles with incident energies up to 2000 ev. Further, since charge transfer can take place in the gaseous phase with little loss of kinetic energy, one can have both fast and slow moving inert and cathode-ejected neutrals, as well as ions. T h e extent to which the energy of these various particles has been dissipated by the time they reach the substrate depends largely on the position of the substrate in the glow discharge and the pressure in the system, which determines their mean free path. At the high pressures used in a discharge, neutral inert atoms moving with thermal velocities, as well as some much more energetic inert neutrals and ions, may arrive at the substrate along with the material being ejected from the cathode, which is presumably also largely uncharged. If one considers only thermal neutrals, then the substrate can be brought within the mean free path of these particles if the discharge is operated near its low pressure limit, i.e., lop4 mm Hg. More important than the energy of the material arriving at the substrate is the rate of arrival of cathode-ejected material compared with the rate of arrival of inert species. This will largely depend again on pressure, but also on the energy of the bombarding ions at the cathode, which determines the ejection ratio at the cathode. Furthermore, insight is required into relationships of discharge characteristics with the direction of cathode-ejected material travelling through the plasma, which is not regulated by a simple diffusion process. T o determine the effect of the various glow discharge characteristics, the “similarity laws” which relate potential, current, gas density, cathode ejection ratios and
IMPACT EVAPORATION AND THIN FILM GROWTH
305
geometry of vessel and electrodes are very useful. (For an excellent review see Francis, 175.) Only a few papers [Von Hippel (176) and Seeliger (177)] have appeared in which the motion of the cathode-ejected species through the various zones of a discharge is discussed in any detail. There are large discrepancies between these calculations and experimental findings. Any difference in mass, velocities (including directional components), and charge, between the cathode-ejected material and the inert gas species must be taken into account. Ecker and Emeleus (178) have attempted this and derive an expression for the total cathode sputtering and its radial variation, and the density distribution in the gas as a function of distance from the cathode. This density distribution was found to be a maximum at a short distance in front of the cathode, the position depending on parameters mentioned above. T h e curves so obtained agree qualitatively with values measured by Guentherschulze ( I 79).
1. Contamination by “Compound” Formation. One must now consider how inert atoms may become incorporated into the film. I n the past, a number of authors have claimed to have produced compounds of the inert gases. T h e existence, for example, of HeH+, He:, HeH:, and A; has been well established. T h e existence of various other chemical compound ions of inert gases involving simple organic systems has also been demonstrated by Franklin (179a). Damianovich et al. (180) published a large number of papers relating to compounds of inert gases and metals produced in a glow discharge at the cathode. Boomer (181) also claims to have formed “tungsten helide” in the same way. However, recently Waller (182) re-examined processes reported by these authors and found no evidence of compound formation with metals. Although the author does not consider Waller’s analysis as final, it wiil be assumed for the remainder of this report that any such compounds, if formed, would be unstable. 2. Contamination by Clathrate Formation. I t must be mentioned here that the existence of clathrate compounds, in which an inert gas is retained in closed cavities or cages provided by the crystalline host structure, has been well established. These compounds, however, consist of two distinct components, which have not reacted chemically with one another, i.e., electrons of the inert gas atom are not involved in the binding forces which are responsible for their existence. T h e only inert gas clathrates known are the so-called gas hydrates. Under conditions of a glow discharge these unique water complexes would be completely unstable and therefore need not be considered further. (For a
306
ERIC KAY
recent review article on these and other clathrates see Mandelcorn,
183.) Although presumably these inert gases do not enter chemical bonding, they may nevertheless be trapped in some manner within a film under certain conditions. The phenomenon usually described as “gas cleanup” in a glow discharge has been known for many years. The many contradictory descriptions of this phenomenon by various authors indicate the complexity of the mechanism involved. One of the latest accounts was published by Blodgett and Vanderslice (284) who showed that the “cleanup” of rare gases in a glow discharge is governed largely by two factors: first, the rate at which the cathode material is being ejected, and second, the potential of the surface on which it lands. This seems an oversimplification, but can serve as a good rule of thumb. These authors found conditions at which the collecting surface has a net cleanup rate of 0. Less than a monolayer of inert gas was detected on the anode by a mass spectrometer, even though the anode was covered with metal ejected from other parts of the apparatus. Precise lattice parameter measurements of a nickel electrode “containing” argon showed no difference from pure nickel within an accuracy of 0.01%. However, metal deposits on surfaces held at different potentials did contain considerable quantities of argon. Using a radioactive tracer krypton-85 in the gas of the glow discharge operated at 150 volts, Bartholomew (173) also found no evidence of inert gas incorporation at the anode. Not enough data were given to permit extrapolation of their findings to other systems. These efforts clearly indicate, however, that the gas incorporation at the anode can vary over a large range depending on position and potential of substrate as well as the pressure in the system. The object of future studies must be to learn to control this gas content and, with numerous applications in mind, to learn to optimize or minimize this incorporation. If the gas is trapped in such a way as to strain the lattice, one might indeed hope to determine changes in unit lattice parameters by extremely careful high resolution diffraction experiments. Naturally, a very accurate density measurement would also give information about gas content. Another method of determining the gas content of a thin film is by “boiling” the gas out of the film by a vacuum fusion and then analyzing the gas with a mass spectrometer. Great care must be taken here to insure that all the gas has been evolved. The validity of this procedure has been questioned by several workers (e.g., 284). Perhaps the more reliable and more sensitive method is the use of radioactively tagged inert gas in the glow discharge with subsequent counting of the deposited film. Argon-37, for example, has a half-life of 110 min, which
IMPACT EVAPORATION AND THIN FILM GROWTH
307
is barely long enough to be practical. Krypton-85, however, has a half-life of ten years and emits beta radiation (680 kev) and some gamma radiation (500 kev) and therefore lends itself well to such studies. Since it is assumed that any trapping of inert gases involves a physical rather than chemical phenomenon, a distinction will undoubtedly have to be made between epitaxially grown single crystal films and amorphous or polycrystalline films of known particle size. I n the latter case, grain boundary diffusion becomes effective. How closely the theoretical density of the film material is approached will be of significance. Many authors have claimed the theoretical limit of density was achieved in thin films produced by various techniques. However, in a number of cases, the accuracy of their evaluation techniques is open to question.
3. Contamination by Physical Adsorption. I t has been well established that rare gases can be physically adsorbed on metals at very low temperatures with heats of adsorption of the order of 1 to 8 kcal/mole. Some very interesting results by Mignolet (285) in 1957 demonstrated some changes in contact potential attendant on covering metal films with a xenon film; i.e., a lowering of the work function for a xenon-covered surface below that of a clean metal film was found. This indicated that some modification of the electron distribution of rare gas atoms does occur upon adsorption, suggesting that such earlier theories as Bardeen’s (286) need partial modification. Mignolet’s results were corroborated by field emission microscope studies by a number of workers [e.g., Ehrlich (120)and Gomer (221)].A marked dependence of the adsorption on the crystallography of the surface was demonstrated by emission photographs. Observation by field emission microscopy of the distribution of the adsorbed gas on various crystallographic sites is possible because of the large changes in work function that accompany adsorption, giving rise to intensity changes in the pattern of the electron emission map of the metal tip. Although physical adsorption does take place in inert gases at low temperatures, it is negligible for neutral inert species at the temperature encountered in a glow discharge system. A natural consequence is that actual solubility of inert gases in metals is negligible. Since inert atoms are not adsorbed on the entrance surface or taken into solution by a metal, it follows that the metal must be impermeable to the inert gas. This has been amply confirmed for He, A, and Ne (187).However, this does not mean that an inert gas will not diffuse through the lattice of a metal if it can be forced into solution in the metal lattice. I n fact, an excellent report by LeClaire and Rowe (187) indicates that an inert
308
ERIC KAY
gas like argon can be forced into the silver lattice of a film deposited in a static glow discharge system by a mechanism similar to the “cleanup” process referred to earlier, and that these atoms will then diffuse. In fact, indirect evidence showed that the argon is incorporated in the lattice substitutionally, and that it diffuses by a vacancy mechanism. Diffusion coefficients and activation energies are given: D
= 0.12
exp (- 33,600/RT)cm2/sec.
The diffusion coefficient D is of the same order of magnitude as that of other elements in silver, but slightly larger because the argon atom is larger than the other atoms (see Table VII). 4. Contamination by Occlusion. Assuming for the moment that such “forced solubility” can exist at the collecting substrate, there are still ways in which such gas atoms could fail to reach the film surface and escape. Any closed pores or other defects produced as the film grows will trap the gas atoms, because, again, atoms entering these cavities by diffusion from the lattice would remain in them permanently. The process of adsorption onto the surface of the cavity, necessary before re-entry and subsequent diffusion through the lattice, does not occur with inert gases. Inert gases will continue to accumulate in such imperfections until the pressure rises to a value sufficient to cause a fracture; subsequently they will flow out. Near the surface a sudden eruption of gas from such pockets could occur, leaving craters. In some cases this has been observed. The precipitation of inert gases at lattice defects was already referred to in work done by Cottrell. This inability of inert gases to re-enter the lattice and diffuse without disrupting the film could be a real problem. To expel inert gases rapidly from a metal requires temperatures of 300 to 400°C according to work done by Thulin (288). C . Contamination By Chemisorption
So far the feasibility of controlling the concentration of chemically reactive species in a glow discharge has only been demonstrated on paper. Until expected experimental verification is available, chemisorption must be considered even in “physical” sputtering. Furthermore, since chemisorption is particularly significant in thin films prepared by reactive sputtering, to be discussed briefly in the next section, it may be well to consider this subject a little further at this point. Inspecting a model of a metal crystal shows that different surface metal atoms M’s have a different number of nearest neighbors. An
IMPACT EVAPORATION AND THIN FILM GROWTH
309
310
ERIC KAY
atom in the interior of a crystal of tungsten, for example, has eight nearest neighbors. Densely packed planes on the surface are made up of atoms having five or six nearest neighbors. Most other planes have four. Edge atoms have four; corner atoms may have three. An atom which sits on top of a plane may have three or four nearest neighbors. It is obvious that any M which has a small number of nearest neighbors will be bonded less firmly to the surface. Further, it will be much more eager to make bonds with absorbing atoms or molecules, and these bonds will be stronger. One would also expect that when an exposed M makes a bond with a gaseous atom, the bond between M and other metal atoms will be weakened. As the metal crystals become heated, the surface atoms tend to rearrange themselves so as to decrease the number of M’s having few nearest neighbors. That is, maximum binding occurs on those sites for which the number of lattice atoms surrounding the ad-atom are maximized. Consider adsorption mechanisms for molecules such as H,, 0,, CO. When such molecules first strike the surface, there are no electron transfers but only polarization forces. Hence, they exist as physisorbed molecules with weak bonding to the surface, and, as a result, can move very freely over the surface at room temperature. T h e probability is appreciable that they will leave the surface as molecules before they become “activated.” If they do become activated, the H-H bond is broken and replaced by two H-M bonds. I n other words chemisorption has taken place; the atoms are now much more strongly held to the surface. T h e probability of chemisorption is some times called the sticking probability (s). For clean surfaces values of s range from 0.5 to 0.05. T h e value of s remains constant until there is one adsorbed atom for each active M. Thereafter, s takes on a lower value which again may be constant until each M has made second valence bonds. After several monolayers have been adsorbed the value of s may be as low as A steady state is reached when the arrival rate times s is equal to the evaporation rate. T h e heat of chemisorption has its highest value while the M’s with the lowest number of nearest neighbors react with the adsorbed H. It is constant until each M has reacted, then it takes on a lower value and reamins constant until the next most exposed M’s, including those M’s which now make bonds with two adsorbed H atoms, are satiated. T h e heat of adsorption is still lower when M’s combine with three adsorbate atoms. T h e desorption mechanism is also a two-step process; it is the inverse of the adsorption process. Two ad-atoms combine to form an adsorbed molecule which is only physisorbed and hence evaporates at a high rate.
IMPACT EVAPORATION AND T H I N FILM GROWTH
311
I t may, however, decompose before it evaporates. Hence the concept of s enters in the desorption mechanism too. When complex molecules, or groups of molecules, are adsorbed, there is a greater number of possible interactions. I n this case, too, if we start with a clean surface, probably all the molecules, with the possible exception of CO, are decomposed into their constituent atoms. After the most exposed M’s are chemically satisfied, however, a complex molecule may be decomposed only into simpler molecules. Naturally, in this decomposition the weakest bonds are the first to be broken. As the amount of total adsorbate increases, the surface will consist of a larger number of all kinds of intermediates or all kinds of “surface free radicals”: At this stage many of these intermediates will be bound so loosely that at room temperature they will migrate freely over the surface. They will also continuously form and decompose. Which intermediate or which compound will evaporate will depend on its concentration, on its value of s, and on the desorption energy, Note especially that s enters at this stage. Of course, s and the desorption energy depend on the concentration of all adsorbates and the degree to which they have satiated the M’s. When one considers medium sized and large molecules, the arrangement of the metal surface atoms may become important. Field emission microscopy has provided quite definite experimental evidence for all these adsorption concepts mentioned above, except in the case of the adsorption of complex molecules.
VII. REACTIVEIMPACTEVAPORATION I n this process a partial pressure of reactive gas such as 0,, H,, N,, H,S, etc. is deliberately introduced into the glow discharge system to produce thin films of oxides, hydrides, nitrides, and sulfides. Although this is probably the most attractive application of impact evaporation, it is also the most difficult to describe accurately. However, again, enough consistency has been achieved in the past to make this a very worthwhile area of study.
A. Advantages T h e advantages of this process over thermal evaporation are quite pronounced. There are many compounds that cannot be transported under high vacuum conditions at a reasonable rate without various degrees of dissociation. I n impact evaporation the bulk temperature of both cathode and collecting substrate can be kept comparatively low (100°C or so), and the dissociation pressure will thus be very much lower,
3 12
ERIC KAY
in some cases negligible. The partial pressure of oxygen in the glow discharge can, for example, be held well above the dissociation pressure, so that the oxides once formed are quite stable. There are few, if any, compounds in their highest oxidation state which, though perhaps thermodynamically stable under high vacuum conditions, would be immune to chemical reduction by the usual source containers used in thermal evaporation, such as W, Mo, Ta, C. Many of these high oxidation state compounds are very refractory and must be heated to very high temperatures in order to achieve practical evaporation rates. Alloying with a container is another serious problem. Since no container is necessary for the cathode in a glow discharge, and in any event, temperatures at the electrodes are low, these particular problems are completely absent. There are many alloys whose constituents have greatly varying vapor pressures, which cannot be evaporated without preferential vaporization of the more volatile components. Although the problems in impact evaporation have not been completely resolved, there is considerable evidence in the literature and in this laboratory that alloys containing species of highly varying vapor pressures have been transported without changing chemical composition (189). Another rather significant potential advantage of reactive impact evaporation in thin film technology can be seen best from the following example. Suppose metal oxide films such as Fe30a, Ta20,, Also3, or SiO,, are to be produced. One of the most important requirements is that they be, if at all possible, one-component films; i.e., films in which there is only the one desired oxidation state, rather than a heterogeneous mixture, or a solution of the metal in various oxidation states, including zero. As already noted, oxides are quite difficult to transport by evaporation. Normally the metal is deposited and then oxidized by subjection to a controllable oxygen-containing system, such as C0,gl-C02(g),or HZO(gl-H2(g). T h e oxygen must now diffuse to every crystallite at every level of the film, then be chemisorbed and subsequently participate in the diffusion mechanism associated with surface oxidation. T h e kinetics of this process are so complex that-even though, because it is of prime importance in corrosion studies, many papers have been published in the past on the subject-the controlling mechanism is still under lively discussion (for a recent review see Gatos, 190). If to the chemical restrictions one adds stringent restrictions on the physical properties of the resultant oxide film, then the problems multiply to such an extent that this approach has, except in a few systems, not been too encouraging. Reactive impact evaporation might overcome this problem of multi-
IMPACT EVAPORATION AND THIN FILM GROWTH
313
component mixtures, since, with this method, each iron atom is arriving at the substrate at a controllable rate and can immediately participate in its oxidation with oxygen arriving at the substrate. In a more macroscopic sense the oxide film will be formed layer by layer as the film grows in thickness so that the problems of diffusion into the film are alleviated. To a first approximation this statement has validity; however, it is obviously a gross oversimplification. Complete understanding of this complex process is difficult to attain, as can be seen in the following sections.
B. Effect of Electronegative Gases on Glow Discharge Characteristics Discharge characteristics are quite different when electronegative gases are used in addition to inert gases. Among the most important electronegative gases, here defined as those which have a pronounced tendency to attach electrons and thus form heavy negative ions, are oxygen, the halogens, and some organic molecules. According to Massey and Smith (191), the destruction of negative ions in the gas phase is a very inefficient process. A publication by Emeleus and Sayers (292) and a monograph by Massey (293) outlined the main effects of the negative ions in a discharge. Most of the new characteristics of the discharge due to negative gases can be traced to the decrease in the average mobility of negative carriers, which now consist of both electrons and negative ions. For example, there is an apparent increase in pressure: the drift velocity of electrons is reduced by their temporary attachment to gas molecules. This effect is equivalent to a more continuous retardation such as would be caused by more frequent collisions with nonattaching gas molecules, i.e., an increase in gas density. T h e resulting contraction of the negative zones and the properties of the positive column are characteristic of a discharge containing only electrons and ions at considerably higher gas pressures. Since the negative ions are more massive than the electrons, they are less likely to diffuse away. Consequently, they tend to produce regions of negative space charge. There is also a change in the distribution of negative carriers in the plasma regions (see Fig. 20). Because of their low mobility, negative ions will tend to move along the lines of electric force; their direction in the positive column will be towards the axis, where, if the pressure is not too low, a concentration of these ions will thus be built up. At very low pressure such concentration will be less marked, because the ions will be carried by their own momentum across the center to the opposite wall. An analysis of an oxygen discharge has been made by Lunt and
3 14
ERIC KAY
Gregg (294), at pressures of approximately 30 microns. I n the positive column they found almost equal concentrations of oxygen ions and electrons, oxygen being present only as 0-. I n the Faraday dark space and
F 0 0:
I
I
I
I
1
1
so
w n
v v)
w
3
25
v)
w
z0
P 0
I .o
2.0 3.0 DISTANCE FROM CATHODE (Cm)
FIG.20. Negative ion distribution in an iodine discharge [S. Fliigge, ed., “Handbuch der Physik,” Vol. 22. Springer, Berlin, 19561.
negative glow hardly any negative ions were detected. It appears that in the positive column molecular ions are readily destroyed, whereas elsewhere both molecular and atomic negative ions are destroyed. T h e distribution of negative ions in an oxygen discharge is depicted in Fig. 21 as a function of current and pressure. Keeping some of these discharge characteristics in mind, one must now find answers to such questions as: What energy should the bombarding ion have, and, in what pressure range, to give rise to a controlled oxidation at the various sites ? Perhaps the simplest method of study would be to arrange conditions so that oxidation is suppressed everywhere except at the substrate, if this is possible.
C . Complexities
As yet, it is not quantitatively known either in what form-ions or neutrals, monatomic or polymeric-or with what energy distribution various species arrive at the substrate in a glow discharge environment. Such knowledge is essential in order to establish meaningful rate laws. Further, it is not at all clear to what extent oxidation at the cathode (which takes place under entirely different conditions) or oxidation in is short) may influence the nature of flight (mean free path at particles arriving at the substrate. Another rather unusual complexity
IMPACT EVAPORATION AND THIN FILM GROWTH
315
arises in a glow discharge environment with regard to chemical activity. It is not enough to refer to a nonequilibrium situation; in addition, it must be pointed out that the expected concentration ratios between reactants and products can differ greatly from those obtained in purely
I
I 16
11
\
I 18
I
I
I
I
I 1 I 2.2 24 2.6 PRESSURE(x d3mm Hql I
20
1
I
2.8
a
c.
4
-
m
B
0.5
I
1.5 2 RADIUS, r (em)
2.5
3
b
FIG. 21. Negative ion distribution in an oxygen discharge [S. Fliigge, ed., “Handbuch der Physik, ” Vol. 22. Springer, Berlin, 19561. (a) At the center, with different currents i and pressures p. (b) Variation across the radius Y .
316
ERIC KAY
thermal reactors. Also, in a discharge environment electrons colliding with other species may be able to overcome the energy of activation necessary to initiate formation of chemical bonds, particularly at low pressures where electron temperatures are high. Thus, for example, an electron colliding with a neutral species can transfer an electron into a normally unpopulated shell, thereby increasing its chemical activity. This has been realized in the case of inert gas, for example helium, where the 1s electrons are unpaired, promoting one electron to the 2s state. This excitation requires 460 kcal of energy per gram atom and has only been observed spectroscopically under conditions of electric discharge or electron bombardment where such species as HeH+, and He: exist. Although these compounds are not stable in the gaseous form and will decompose in the absence of the discharge, they can act as catalysts and affect the kinetics of other discharge reactions. O n the other hand, it is conceivable that the chemical reactivity of atoms with nearly filled valence orbitals may be inhibited. It is possible for electron collisions to move electrons from deep-lying orbitals and increase the valence orbital population, thereby giving the appearance of temporary chemical inertness. T h e superficial analogy to photochemical reactions certainly suggests itself in these problems; however, some of the simplifying feature of photochemistry are unfortunately absent in high energy radiation chemistry. In studying photochemically initiated reactions, a specific wavelength of incident light can be selected to produce a desired excited state of one of the reactants. No such choice exists in high energy radiation chemistry. T h e energy of a single particle can be so great relative to excitation and ionization energies that the energy transfer is not a single event but the result of a great number and variety of events. All possible excited states and ions may be produced and little or no selectivity is possible. T h e nature and relative numbers of these excited states are, in general, not known, I n addition to primary electronic excitation, the possibility of such reactions as dissociation of molecules or free radicals-which may also be electronically excited ; fluorescence ; internal conversion ; predissociation ; and inter- and intramolecular energy transfer must be considered. Ions may undergo unimolecular dissociation or rearrangement, intra- or intermolecular charge transfer, reaction with neutral molecules, or neutralization by reactions with other charged particles. While it is known that fast charged particles excite optically allowed states almost exclusively, slow electrons (electrons with velocity comparable to that of the Bohr velocity of orbital electrons) can induce optically forbidden transitions by electron exchange. Because of the intense chemical activity associated with free radicals
IMPACT EVAPORATION AND THIN FILM GROWTH
317
and high energy yields resulting from their combination, their possible existence in a discharge must be emphasized. T h e breakdown of everpresent water vapor into H and OH radicals, for example, is well known. An intense band emission in the far ultraviolet and far infrared, associated with rotation-vibration transitions of the diatomic OH system, has been studied. T h e dependence of OH emission on current density is known. Work on nitrogen and oxygen radicals has also been reported. During the thermalization of electrons by various inelastic collisional processes, there is a very real possibility of the formation of triplet states. This is in contrast to the probability of singlet-triplet transitions resulting from photoexcitation, which is generally very low because such a change in multiplicity (i-e., a change in total spin momentum of the electrons) is forbidden by spin-momentum conservation. Excitation of triplet states may also result directly from neutralization of positive ions by electrons or indirectly from higher excited states. I n fact, the number of triplet states can be comparable to the number of ions formed. Molecules in the triplet state are particularly interesting chemically because they are di-radicals and have long radiative lifetimes. Even if all the variables affecting the growth of thin films in a glow discharge were well understood, the difficult problem of adjusting the glow discharge characteristics to create the desired conditions would still exist. I n view of the many unknowns, further discussion here of other complexities would not be profitable. To end this paper on a more encouraging note, it should be stated that fair success in growing films of many compounds, such as oxides, sulfides, nitrides, and hydrides, has been achieved by a semiempirical approach, without the benefit of a more profound insight into the reaction kinetics in a glow discharge. Thus hope exists that the rate-controlling processes at the substrate are amenable to examination. ACKNOWLEDGMENT T h e author wishes to express his gratitude to Drs. A. G. Anderson and J. P. Goldsborough for interest in and constructive criticism of this report. T h e author is also obliged to several publishing agencies for the use of some of their material. REFERENCES 1 . Wehner, G. K., -4dwances in Electronics and Electron Phys. 7 , 239 (1955). 2. Massey, H. S. W., and Burhop, E. H. S., “Electrons and Ion Impact Phenomena.” Oxford Univ. Press (Clarendon), London and New York, 1956.
318
ERIC KAY
2a. Thompson, M. W., Brit.J. Appl. Phys. 13, 194 (1962). 3. Trillat, J. J., Terao, N., Tertian, L., and Gervais, H., J . Phys. SOC.(Japan) 11, 406 (1956). 4. Trillat, J. J., Mihama, K., and Terao, N., Congr. intern. microscopie electronique, Berlin 1958. 5. Trillat, J . J., and Mihama, K., Compt. rend. acad. sci. 248, 2827 (1959). 6. Morgulis, N. D., Zhur. EksptZ. Teoret. Fiz. 9, 1484 (1939). 7. Ploch, W., 2.Physik 130, 174 (1951). 8. Roos, O., 2.Physik 147, 210 (1957). 9. Batanov, G. M., and Petrov, N. N., Zhur. Fia. tverdogo tela 2, No. 6 (1960); Transl., Soviet Phys.-Solid State. IO. Petrov, N. N., Soviet Phys.-Solid State 2, 857, 1182 (1960). 11. Petrov, N. N., Izvest. Akad. Nauk S.S.S.R., Ser. Fiz. 24, 682 (1960). 12. Hagstrum, H. D., Phys. Reo. 96, 325, 336 (1954); 104, 1515 (1956); 122, 83 (1961) J . Phys. Chem. Solids 14, 33 (1960); J . Appl. Phys. 32, 1015, 1020 (1961). 13. Abbott, R. C., and Berry, H. W., J . AppZ. Phys. 30, 871 (1959). 14. Honig, R. E., J. Appl. Phys. 29, 549 (1958). 15. Bradley, R. C., J. Appl. Phys. 30, 1 (1959). 16. Stanton, H. E., J . AppZ. Phys. 31, 678 (1960). I6a. Leland, W. T., and Olson, R., Proc. Atomic and Molecular Beams Conf. Univ. of Denver, 1960, p. 293. 17. Veksler, V. I., Soviet Phys.-JETP 11, 235 (1960). 18. Fogel’, Ya. M., Slabospitskii, R. P., and Karnaukhov, I. M., Sooiet Phys.-Tech. Phys. 5, 777 (1961); Fogel’, Ya. M., Slabospitskii, R. P., and Rastrepin, A. B., ibid. 5, 58 (1960). 19. Hagstrum, H.’D.,J . Appl. Phys. 31, 897 (1960). 20. Oliphant, M. L. E., Proc. Roy. SOC. A124. 228 (1929). 21. Gurney, R. W., Phys. Rev. 31, 629 (1928). 22. Paetow, H., and Walcher, W., 2. Physik 110, 69 (1938). 23. Healea, M., and Houternans, C., Phys. Rev. 58, 608 (1940); Massey, H. S. W., and Burhop, E. H. S.,“Electrons and Ion Import Phenomena.” Oxford Univ. Press (Clarendon), London and New York, 1956. 24. Bradley, R. C . , Phys. Rev. 93, 719 (1954); Bradley, R. C., and Ruedl, E., Bull. Am. Phys. SOC.[2] 5, 16 (1960). 25. Hagstrum, H. D., Phys. Reo. 123, 758 (1961). 26. Moore, W. J., Am. Scientist 48, 109 (1960). 27. Townes, C. H., Phys. Rev. 65, 319 (1944). 28. Kinchin, G. H., and Pease, R. S., Repts. Progrs. in Phys. 18, 1 (1955). 29. Seitz, F., and Koehler, J. S., Solid State Phys. 2, 305 (1956). 30. Gibson, J. B., Goland, A. N., Milgram, M., and Vineyard, G. H., Pkys. Rev. 120, 1229 (1960). 31. Bohr, N., KgZ. Danske Videnskab. Selskab, Mat.-Fys. Medd. 18, No. 8 (1948). 32. Pease, R. S.,Rend. Scuola Intern. Fisica Corso XIII S E T T 1959, pp, 151-165. 33. Keywell, F., Phys. Rev. 97, 67 (1955). 34. Timoshenko, G., J . Appl. Phys. 12, 69 (1940). 35. Thommen, K., 2.Physik 151, 144 (1958). 36. Harrison, D. E., Phys. Rev. 102, 1473 (1956); J . Chem. Phys. 32, 1336 (1960). 37. Goldman, D. T., Harrison, D. E., and Coveyou, R. R., Oak Ridge National Laboratory Report ORNL-2729 (1959). 38. Goldman, D. T., and Simon, A., Phys. Rev. 111, 383 (1958).
IMPACT EVAPORATION AND T H I N FILM GROWTH
3 19
39. Skiff, P. D., and Reynolds, H. K., 13th Ann. Gaseous Electronics Con!. Octo6er, 1960, Abstract A- 1. Monterey, California, 1961. 40. Langberg, E., Phys. Rev. 111, 91 (1958). 41. Henschke, E. B., Phys. Rev. 105, 737 (1957). 42. Henschke, E. B., Phys. Rev. 121, 1286 (1961). 43. Harrison, D. E., Jr., and Magnuson, G. D., Phys. Rev. 122, 1421 (1961). 44. Wehner, G. K. ef al., Phys. Rev. 108, 35 (1957); J. Appl. Phys. 29, 217 (1958); 32, 365 (1961); Phys. Rev. Letters 4, 409 (1960). 45. McKeown, D., Rev. Sci. Instr. 32, 133 (1961). 46. Wehner, G. K., Phys. Rev. 102, 690 (1956). 47. Henschke, E. B., J. Appl. Phys. 28, 411 (1957). 48. Silsbee, R. H., J. Appl. Phys. 28, 1246 (1957). 49. Leibfried, G., J . Appl. Phys. 31, 117, 1046 (1960). 50. Thompson, M. W., Phil. Mag. [8] 4, 139 (1959). 51. Koedam, M., and Hoogendoorn, A., Physica 26, 351 (1960). 52. Yurasova, V. E., Pleshivtsev, N. V., and Orfanov, I. V., Zhur. Eksptl. Teoret. Fiz. 37, 966 (1959).
53. Anderson, G. S., and Wehner, G. K., J. Appl. Phys. 31, 2305 (1960). 53a. A l m h , O., and Bruce, G., Nuclear Instr. @ Methods 2, 257 (1961). 536. Molchanov, V. A., Tel'kovskii, V. G., and Chicherov, V. M., Soviet Phys.JETP 6, 222 (1961). 54. Miiller, E. W., Bull. Am. Phys. SOC.[2] 6, 373 (1961). 55. Ruedl, E., Delavignette, P., and Amelinckx, S., Phys. Rev. Letters 6, 530 (1961). 56. Wehner, G. K., J. Appl. Phys. 31, 177 (1960). 57. Stein, R. P., and Hurlbut, F. C., Phys. Rev. 123, 790 (1961). 58. Gronlund, F., and Moore, W. J., J . Chena. Phys. 32, 1540 (1960). 59. Holmstrom, F., and Knight, R. D., Bull. Am. Phys. Soc. [2] 5, 503, No. 7 (1960). 60. Molchanov, V. A., and Tel'kovskii, V. G., Soviet Phys.-JETP 6, 137 (1961). 61. Rol, P. K., Fluit, J. M., and Kistemaker, J., Proc. Symposium on Isotope Separation, Amsterdam, 1957. p. 657 (1958); Physica 26, 1000 (1960). 61a. Bader, M., Witteborn, F., and Snouse, T. W., Natl. Aeronaut. Space Admin. Report N A S A TR-RIOS (1961). 62. Knight, R. D., Private communication, 1961. 63. Magnuson, G. D., Meckel, B. B., and Harkins, P. A., J. Appl. Phys. 32, 369 (1961). 64. Cunningham, R. L., Hayman, P., Lecomte, C., Moore, W. J., and Trillat, J. J., J. Appl. Phys. 31,839 (1960). 64a. Balarin, M., and Hilbert, F., J. Phys. Chem. Solids 20, 138 (1961). 65. Wolsky, S. P., and Zdanuk, E. J., Phys. Rev. 121, 374 (1961); J. Appl. Phys. 32, 782 (1961).
66. Bleakney, W., Phys. Rev. 36, 1303 (1930); see also ibid. 35, 139, 1180 (1930); 34, 157 (1929).
67. Wehner, G. K., J. Appl. Phys. 31, 1392 (1961). 68. Wehner, G. K., Phys. Rev. 114, 1270 (1959). 69. Yonts, 0. C . , Normand, C. E., and Harrison, D. E., Jr., J. Appl. Phys. 31, 447 (1960).
70. Colombie, N., Compt. rend. mad. sci. 252, 2108 (1961). 71. Hines, R. L.,and Wallor, R., J. Appl. Phys. 32, 202 (1961). 72. Laegreid, N. L., and Wehner, G. K., J . Appl. Phys. 32, 365 (1961). 73. Wehner, G. K., and Rosenberg, D., J. Appl. Phys. 32, 887 (1961). 74. Morrison, J., and Lander, J. J., J. Electrochem. SOC.105, 145 (1958).
320
ERIC KAY
Lander, J. J., J. Phys. Chem. Solids 3, 87 (1957). Bredov, M. M., Soviet Phys.-Solid State 3, 195 (1961). Rourke, F. M., Sheffield, J. C., and Waite, F. A,, Rev. Sci. Znstr. 32, 455 (1961). Barnes, R. S . , Redding, G. B., and Cottrell, A. H., Phil. Mag. [8] 3, 97 (1958). Rich, J. B., Redding, G. B., Barnes, R. S . , J . Nuclear Materials 1, 96 (1959). Ells, C. E., and Perryman, E. C. W., J. Nuclear Materials 1, 13 (1959). Lillie, D. W., Trans. A.Z.M.E. Canada 218, 270 (1960). 82. Churchman, A. T., Barnes, R. S., and Cottrell, A. H., J. Nuclear Energy 7 , 88 (1958). 83. Murray, G. T., J. Appl. Phys. 32, 1045 (1961). 84. Goland, A. N.; Phil. Mag. [8] 6, 189 (1961). 85. Hirsch, P. B., Silcox, J., Smallman, R. E., and Westmacott, K. H., Phil Mug. [8] 3, 897 (1958); Silcox, J., and Hirsch, P. B., ibid. 4, 72 (1959). 86. Leibfried, G., J. Appl. Phys. 30, 1388 (1959). 87. Gillam, E., J. Phys. Chem. Solids 11, 55 (1959). 88. Davies, J. A., McIntyre, J. D., Cushings, R. L., and Lounsbury, M., Can. J. Chem. 38, 1535 (1960). 89. Ogilvie, G. J., and Thomson, A. A,, J. Phys. Chem. Solids 17, 203 (1960). 90. Rol, P. K., et al., A . R. S. Electrostatic Propulsion Conf., 1960 Paper No. 1395-60. 91. Farnsworth, H. E., Dillon, J. A., Jr., Schlier, €2. E., and Hanernan, D. ASTIA Document No. AD-229060 (1959). 92. Arnold, S. R.,ASTIA Document No. AD-210839 (1959). 93. Gianola, V. F., J. Appl. Phys. 28, 868 (1957). 94. Ogilvie, G. J., J. Phys. Chem. Solids 10, 217 (1959); 10, 222 (1959). 95. Sosnovsky, H. M. C., J. Phys. Chem. Solids 10, 304 (1959). 96. Bierlein, T. K., and Mostel, B., Rw. Sci. Znsh. 30, 832 (1959). 97. Wolsky, S. P., Phys. Rev. 108, 1131 (1957). 98. Weiss, A., Heldt, L., and Moore, W. J., J. Chem. Phys. 29, 7 (1958). 99. von Ardenne, M., “Tabellen der Electronenphysik,” Vol. 11: Ionenphysik, Deut. Verlag Wissenschaften, Berlin, 1956. 100. Knight, R. D., Trans. 2nd Intern. Congr. Intern. Org. for Vacuum Sci. and Technol., Washington D. C., October, 1961. Abstract p. 253. 101. Yonts, 0. C., and Harrison, D. E., Jr., J. Appl. Phys. 31, 1583 (1960). 102. Penning, F. M. “Electrical Discharges in Gases,” Macmillan, New York, 1957. 103. Flugge, S., ed., “Handbuch der Physik,” Vol. 22, p. 56. Springer, Berlin, 1956. 104. Penning, F. M., and Moubis, J. H., Koninkl. Ned. Akad. Wetenschap. Proc. 43, 41 (1940). 105. Loeb, L. B. “Basic Processes of Gaseous Electronics.” Univ. of California Press, Berkeley, California, 1955. 106. Holland, L., “Vacuum Deposition of Thin Films.” Wiley, New York, 1956. 107. Brown, S. C., “Basic Data of Plasma Physics.” Wiley, New York, 1959. 108. Mayer, H., “Physik dunner Schichten.” Wissenschaftl. Verlagsges., Stuttgart. 1955. 109. Ecker, G., and Muller, K. G., ASTIA Document No. AD-251976 (1961). 110. Guentherschulze, A., Vacuum 3, 360 (1953). 111. Wehner, G. K., Phys. Rev. 108, 35 (1957). 112. von Hippel, A., Ann. Physik [4] 81, 1043 (1926). 113. Blechschmidt, E., Ann. Physik [4] 81, 999 (1926). 114. Guentherschulze, A., and Tollimen, W., Z. Physik 119, 685 (1942). 115. Ecker, G., and Emeleus, K. G., Proc. Phys. SOC.(London) B67, 546 (1954). 75. 76. 77. 78. 79. 80. 81.
IMPACT EVAPORATION AND THIN FILM GROWTH
32 1
116. “Proceedings of the International Conference on Structure and Properties of Thin Films” (C. A. Neugebauer, J. B. Newkirk, and D. A. Verrnilyea, eds.). Wiley, New York, 1959. 117. Pashley, D. W., p. 40, Ref. 116. 118. Menter, J. W., Proc. Roy. SOC.A236, 119 (1956). 119. Hashimoto, H., and Uyeda, R., Acta Cryst. 10, 143 (1957). 120. Ehrlich, G., pp. 423-475, Ref. 116. 121. Gomer, R., Field Emission Sympwium, Washington, D. C,‘., 1959. 122. Miiller, E. W., Advances I n Electronics and Electron Phys. 13,83- 177 ( 1960). 123. Frenkel, J., 2. Physik 26, 117 (1923). 124. Wood, R. W., Phil. Mag. [6] 30, 300 (1915); 32, 324 (1916). 125. Knudsen, M., Ann. Physik. [4] 48, 113 (1915). 126. Taylor, J. B., Phys. Rev. 35, 375 (1930). 127. Knauer, F., and Stern, O., 2. Physik 53, 779 (1929). 128. Sears, G. W., and Cahn, J. W., J. Chem. Phys. 33, 494 (1960). 129. Gafner, G., Phil. Mug. [8] 5, 1041 (1960). 130. Levinstein, H., J. Appl. Phys. 20, (4) 306 (1949). 131. Lennard-Jones, J. E., and Dent, B. M., ?’roc. Roy. SOC.A121, 247 (1928). 132. Appleyard, E. T. S., Proc. Phys. Sac. (London) 49, El18 (1937). 133. Rapp, R. A., Hirth, J. P., and Pound, G. M., J . Chem. Phys. 34, 184 (1961). 134. Langmuir, I., Phys. Rev. 8, 149 (1916). 135. Lennard-Jones, F. R. S., and Strachan, C., Proc. Roy. Sac. A150, 442 (1935). 136. Aziz, R. A., and Scott, G. D., Can. J . Phys. 34, 731 (1956). 137. Inghram, M. G., Chupka, W. A., and Berkowitz, J., Mim. soc. roy. sci. Likge 18,
513 (1957). 138. Kay, E., and Gregory, N. W., J . Phys. Chem. 62, 1079 (1958); J. Am. Chem. SOC. 80, 5648 (1958). 139. Hirth, J. P., and Pound, G. M., ASTIA Document No. AD-229060 (1959). 140. Knacke, O., and Stranski, I. N., Prop. in Metal Phys. 6, 181 (1956). 141. Beeck, O., Givens, J. W., and Smith, A. E., Proc. Roy. Sac. A177, 62 (1940). 142. Pashley, D. W., Advunces in Phys. 5, 173 (1956). 143. Frank, F. C., and Van der Merwe, J. H., Proc. Roy. SOC.A198, 205 (1949); 200,
125 (1949). Finch, G. I., and Quarrel], A. G., Proc. Phys. Sac. (London) 46, 148 (1934). Vermount, P., and Dekeyser, W., Physicu 25, 53 (1959). Pashley, D. W., and Menter, J. W., pp. 12-46, Ref. 116. Bassett, G. A., Phil. Mag. [8] 3, 1042 (1958). Schulz, L. G., Acta Cryst. 5, 130 (1952). Koedam, M., Philips Research Repts. 16, 101 (1961). Ells, C. E., and Scott, G. D., J . Appl. Phys. 23, 31 (1952). Collins, L. E., and Heavens, 0. S., Proc. Phys. Soc. (London) B70, 265 (1957). Heavens, 0. S., “Optical Properties of Thin Solid Films.” Butterworths, London, 1955. 153. Evans, D. M., and Wilman, H., Actn Cryst. 5, 731 (1952). 154. Wilson, J. N., Voge, H. H., Stevenson, D. P., Smith, A. E., and Atkins, L. T., J . Phys. Chem. 63,463 (1959). 155. Schlier, R. E., and Farnsworth, H. E., J. Phys. Chem. Solids 6, 271 (1958). 156. Heidenreich, R. D., Nesbitt, E. E., and Burbank, R. D., J . Appl. Phys. 30, 995 (1959). 157. Ittner, W. B., and Seraphim, D. P., Bull. Am. Phys. SOC.[2] 4, 224 (1959).
144. 145. 146. 147. 148. 149. 150. 151. 152.
322
ERIC
KAY
158. Budnick, J. I., Phys. Rev. 119, 1578 (1960). 159. Caswell, H. L., J . AppZ. Phys. 32, 105 (1961). 160. Hagstrum, H. D., and D’Amico, C., J. Appl. Phys. 31, 715 (1960). 161, Hagstrum, H. D., J . Appl. Phys. 32, 1015 (1961). 162. Law, J. T., J . Phys. Chem. Solids. 14, 9 (1960). 163. George, T. H., and Burger, R. M., J . AppZ. Phys. 29, 1150 (1958). 164. Holland, L., Adwances in Vacuum Sci. und Technol. 2, 755 (1960). 165. Hines, R. L., J . Appl. Phys. 28, 587 (1957). 166. Florescu, N. A., Vacuum 8, 46 (1958). 167. Holland, L., Advances in Cucuum Sci. and Technol. 2, 753 (1960). 168. Ennos, A. E., Brit. J . Appl. Phys. 5 , 27 (1954). 169. Biondi, M. A., Reu. Sci. Imp. 30, 831 (1959). 170. Warshaw, S. D., Phys. Rew. 76, 1759 (1949). 171. Young, J. R., J . Appl. Phys. 27, 1 (1956). 172. Hines, R. L., Phys. Rev. 120, 1626 (1960). 173. Bartholomew, C. Y., and La Padula, A. R., J. Appl. Phys. 31, 445 (1960). 174. Veksler, V. I., Soviet Phys.-JETP 11, 235 (1960). 175. Francis, G., in “Handbuch der Physik” (S. Fliigge, ed.), Vol. 22, pp. 53-203.
Springer, Berlin, 1956. 176. von Hippel, A., Am. Physik [4] 81, 1043 (1926). 177. Seeliger, R., Z. Physik 119, 482-492 (1942). 178. Ecker, G., and Emeleus, K. G., Proc. Phys. SOC.(London) B67, 546 (1954); Natur-
wissenschaften 41, 185 (1954). 179. Guentherschulze, A., Z. Physik 119, 79 (1 942). 179a. Field F. H., Head H. N., and Franklin J. L., J. Am. Chem. SOC. 84, 1118 (1962). 180. Damianovich, H., et ul., Proc. 8th Am. Sci. Congr. (Phys. and Chem. Sci.) 7 , 137 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194.
(1940); and more than 30 other papers. Boomer, E. H., Nuture 115, 16 (1925); Proc. Roy. SOC.A109, 198 (1935). Waller, J. G., Nuture 186, 429 (1960). Mandelcorn, L., Chem. Revs. 59, 827 (1959). Blodgett, K. B., and Vanderslice, T. A,, J. Appl. Phys. 31, 1017 (1960). Mignolet, J. C. P., “Chemisorption,” p. 118. Buttenvorths, London, 1957. Bardeen, J., Phys. Rev. 58, 727 (1940). LeClaire, A. D., and Rowe, A. H., A.E.R.E. Report M/R 1417 (1957). Thulin, S., Arkiv Fysik 9, 107 (1955). Hanav, R., Phys. Rev. 76, 153 (1949). Gatos, C., “The Surface Chemistry of Metals and Semiconductors.” Wiley, New York, 1960. Massey, H. S. W., and Smith, R. A., P y a . Roy. Sot. AIM, 472 (1936). Emeleus, K. G., and Sayers, J., Proc. Roy. Irish Acad. A44, 87 (1938). Massey, H. S. W., “Negative Ions.” Cambridge Univ. Press, London and New York, 1950. Lunt, R. W., and Gregg, A. H., Trans. Furuduy Soc. 36, 1062 (1940).
Ultrahigh Vacuum P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
Radio and Electrical Engineering Division, National Research Council, Ottawa, Canada
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Physical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. T h e Significance of Surface Effects .................................. B. Physical Adsorption ............................................... C. Chemisorption .................................................... D. Sources of Gas . . . . . ....................... E. Positive Ion Impact on ..................... F. Electron Interactions ..................... G. Photo and Chemical R ..................... 111. Technology of Ultrahigh Vacuum . . . ........................... A. Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Measurement of Total Pressure ..................................... D. Measurement of Partial Pressure ................................... E. Measurement of Pumping Speed, Leak Rate, and Gauge Sensitivity . . . . . . F. Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Applications .................................. A. Surface Physics and Chemistry . . . . . . . . . . . . . . B. Thin Films . . ............... C. Thermonuclear a ............... D. Space Simulation ........................ References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Page 323 325 325 328 337 343 358 368 369 371 371 386 391 405 412 414
418 419 420 422
I. INTRODUCTION Ultrahigh vacuum may be considered as the region of pressure below
lop8 Torr.* Prior to 1950, no adequate method of measuring pressures below Torr existed because the ionization gauges then available were limited by a residual current to the ion collector. In 1950, Bayard and Alpert (1) developed an ionization gauge which reduced the residual current by two to three orders of magnitude. T h e development of this gauge permitted a systematic study of the processes limiting the ultimate pressure in vacuum systems, and led to the rapid improvements in ultrahigh vacuum technology of the last ten years. I t is presently possible to obtain pressures less than 10-l2Torr in small systems, and pressures of the order of 10-lo Torr have been achieved in large metal systems.
*
1 Torr = I m m H g .
323
324
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
T h e initial requirement for ultrahigh vacuum (u-h-v hereafter) arose in the fields of surface physics and chemistry. I n experiments where adsorbed gases have an appreciable effect on surface properties, it is necessary to maintain the surface in a clean condition for a reasonably long time. A monolayer of adsorbable gas will form on a surface in about one second at 10-6 Torr. Since cleaning a surface usually involves heating, the pressure of adsorbable gases must be less than about 10-O Torr to permit measurement of the properties of clean surfaces with observation times reasonably long compared to the cooling time of the sample. U-h-v is also necessary in any system where gases of very high purity have to be introduced (e.g., thermo-nuclear machines) or where very pure evaporated films are to be prepared. More recently u-h-v techniques have been applied to the construction of vacuum chambers for the simulation of extra-terrestrial conditions. Several reviews of the techniques and general problems of u-h-v have appeared (2-8);some discussion of u-h-v is included in more general review articles (9, 10, 11).The use of u-h-v techniques in thermonuclear machines is discussed by Munday (12) and Grove (13). T h e ultimate pressure achieved in a vacuum system is established by an equilibrium between (a) the rate of arrival of gas from various sources, and (b) the rate of removal of gas from the evacuated volume (either by removal of gas molecules from the system, or by transfer of gas to the adsorbed phase within the system). T h e partial pressure ( p ) in Torr of any one gas in the steady state ( p and temperature constant) is given by
where
L is the leak rate through holes in the envelope (molecules/sec),
FK is the permeation rate through the walls of the system (molecules/sec),
F,, is the rate of evolution of gas absorbed in the walls and component parts of the system (molecules/sec), FA is the rate of evolution of previously adsorbed gas (molecules/sec), FR is the re-emission rate of gas molecules previously pumped by being ionized and driven into a solid surface (molecules/sec),
no = 3.27 x lO1O molecules/liter at p = 1 Torr, and T = 295"K, S is the speed of momentum-transfer pumps in the system (i.e., diffusion or molecular-drag pumps) (liters/second), S, is the speed of the ionization pumps (liters/second),
ULTRAHIGH VACUUM
325
is the specific arrival rate of gas molecules at a surface ( u = 1.98 x 1021M-1/2cm--2sec-1at 295°K and 1 Torr for a gas of molecular weight M ) , A, is the area of the j t h adsorbing surface, v
ai the adsorption probability on the j t h surface. (The adsorption probability is the fraction of the incident molecules which are adsorbed on a surface.) T h e various gases present in the system cannot, in general, be treated independently. Interaction between different gases can affect the two emission rates FA and FR,the electronic pumping speed Si and the adsorption probabilities a j . Further interaction between gases can occur through chemical reactions occurring principally at heated surfaces. T h e existing data concerning the various competing processes listed in Eq. (1) are not sufficiently extensive to permit detailed calculations of partial pressures, nor are such calculations usually necessary. T h e considerations of these various processes, which is the subject of Section 11, is of value in determining the best processing methods, materials and techniques to be used in obtaining u-h-v in a specific system. Section I1 is concerned with the various physical processes which (a) control the behavior of sources and sinks of gas and (b) affect the measurement of pressure. Section 111 describes the methods of production and measurement of u-h-v. T h e applications of u-h-v techniques are described very briefly in Section IV.
11. PHYSICAL PROCESSES A . The Significance of Surface Effects T h e pressure in u-h-v systems is very significantly affected by phenomena occurring on the solid surfaces of the system (chemical and physical adsorption, surface migration, and evolution of previously sorbed gases). T h e importance of surface effects may be demonstrated by some simple considerations. Table I shows some of the important parameters governing surface effects, as a function of pressure at T = 295°K. T h e third column gives the rate of impingement of gas molecules of molecular weight 28 on one square centimeter of surface. T h e fourth column shows the time required to form a layer of adsorbed gas containing 5 x IOl4 molecules/cm2 assuming that every molecule that stirkes the surface is adsorbed (i.e., a sticking probability of unity).
326
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
This time is an approximate measure of the time it takes for the pressure to come to equilibrium (i,e., for adsorption to cease) after a surface has been cleaned of adsorbed gas. T h e fifth column shows the ratio of the TABLE I
Pressure (Torr)
1 10-8 10-11
Molecular density in gas phase n, (molecules/cms)
Impingement rate v (molecules/cma/sec)
3.3 x 1016 3.3 x 1010 3.3 x 106
3.8 x lopo 3.8 x 1014 3.8 x 109
Monolayer time tm
NdNe
1.3 X
[email protected] X 7.5 x 108 1.3 sec 7.5 x 108 36 hr
number of molecules in the adsorbed phase (Na)to the number in the gas phase ( N g ) in a one-liter sphere, assuming an adsorbed density of 5 x 1014molecules/cm2. If the monolayer of gas adsorbed on the surface of the one-liter sphere was completely desorbed, the pressure would Torr. These simple considerations indicate the increase to 7.5 x enormous ratios of the numbers of molecules in the adsorbed and gaseous phases at u-h-v and the profound effect on the system pressure produced by the desorption of only a minute fraction of the adsorbed gas. T h e adsorbed gas which desorbs spontaneously at room temperature can be easily removed by baking. T h e average time that an adsorbed molecule remains on a surface is given approximately by (14), ta, = to exp (Ed/RT)
(2)
where Ed is the activation energy of desorption and to is the period of the thermal oscillation of the adsorbed molecule normal to the surface. Table I1 shows how ta varies as a function of Ed and T,taking to = sec. Measured values of Ed lie between 20 cal/mole (the heat of vaporization of liquid helium) and 236,000 cal/mole (the activation energy of desorption for 0, on Ti). This range is often divided into two parts, called “physical adsorption” (below 8 kcal/mole) and “chemical adsorption” (above 8 kcal/mole). T h e dividing line between the two ranges is somewhat arbitrarily chosen. The nature of the bonding forces for these types of adsorption is different. T h e bonding forces of physical adsorption are always present, They occur, even in the absence of permanent electric dipole or quadrupole moments in adsorbent or adsorbate, because the charge distributions in molecules are time-dependent, and
327
ULTRAHIGH VACUUM
cause a net attraction. I n chemisorption, electron exchange occurs between the adsorbent and adsorbate and the bonding forces are similar to normal chemical bonds. Chemisorption is specific; that is, chemisorption does not occur to a measurable extent for certain combinations of adsorbent and adsorbate.
ta
Ed (kcal/mole) T = -196°C 0.1 1.0 10
1.9 x 10-13 6.9 x 8 x lo6 centuries
T
=
25°C
1.2 x 10-1s 5.4 x 2 x 10-6
50
m
100
m
03
150
m
03
9
T
(sec)
=
500°C
1.9 x 6.7 x lo-”
x 1013 centuries
14 6
x lo6 centuries m
T = 1000°C T
=
2000°C
1.5 x 5.2 x lo-’% 9.2 x
4
x
4.2 hr 2
x 103 centuries
6.4 x 4.2 x lo-‘ 21
If a molecule approaching a surface from the gas phase has to surmount a potential barrier before falling into the potential well of depth E d , then this barrier height is called “the activation energy of adsorption” (Ea). T h e relation between E d and Ea is written, Ed
E.8
3-q
(3)
where q is called “the heat of adsorption” and is a quantity that arises in thermodynamic studies; E d is a quantity that arises in studies of rates of desorption; in physical adsorption Ea = 0 and Ed = q. Atomically clean surfaces within an u-h-v system have a maximum pumping speed of 3.638 (T/M)”*
(4)
liters/sec/cm* if all molecules striking the surface are adsorbed. Here M is the molecular weight of the gas. Thus, for a system with a few hundred square centimeters of clean surface, the pumping speed caused by adsorption on the surface may greatly exceed the speed of the pumps attached to the system. The pumping action by adsorption is selective, at room temperature only the chemically active gases are pumped. As
328
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
the surfaces are covered by gas, and multiple adsorbed layers are formed, Ed decreases and the rate of desorption increases. An equilibrium is finally reached, and the pressure in the system becomes constant when Ed decreases to such an extent that adsorption effectively ceases or reversible adsorption occurs. These surface phenomena are discussed in more detail in the following sections. B . Physical Adsorption General discussions of this subject have been given by Brunauer (/.5), de Boer ( I d ) , and many other authors. We now separate the various processes which take place when a molecule from the gas phase is incident upon a surface [see also Tompkins (15a)l. This separation is useful because in practical systems one process or another may be made to dominate.
1. Condensation Coeflcient, Sojourn Time, and Accommodation Coeficient. A molecule colliding with a bare surface may do one of two things on the initial impact: (1) it may rebound; (2) it may become adsorbed on the surface. The probability that a molecule will execute (2) is known as the "condensation coefficient" (c). For physical adsorption a = c [see Eq. (I)]. If the number of molecules striking a surface per cm2 per sec is v [see Eq. ( l ) ] then the number adsorbed per cm2 per sec is du = cv
dt
where cr is the number adsorbed per cm2. Foner et al. (16) have measured c for argon atoms impinging on an argon surface at 4.2"K and found a value of c = 0.6. Schafer and Gerstacker ( 1 7 ) quote measured values of c for gases on glass surfaces. These results are given in Table 111, where we have used our notation. TABLE 111. CONDENSATION COEFFICIENT OF GASESON A GLASSSURFACE IN TEMPERATURE RANGE0- 100"C.a
THE
Gas
C(O0C)
c(50"C)
c(1Oo"C)
Xenon Argon Oxygen Nitrogen Neon Hydrogen Helium
0.987 0.888 0.835 0.823 0.483 0.623 0.030
0.980 0.853 0.789 0.768 0.402 0.555 0.022
0.975 0.8 13 0.729 0.71 7 0.340 0.468 0.015
After Schafer (1 7).
329
ULTRAHIGH VACUUM
Other evidence concerning the magnitude of c comes from a rather unexpected source. It is found in chemisorption that the sticking probability of many gases on metals-in particular, tungsten-lies between 0.1 and 0.9, and moreover is independent of coverage up to nearly a monolayer. This latter property has been interpreted (18) to indicate that adsorbed molecules exist in a physically adsorbed state prior to becoming chemisorbed. If this is so, then c is at least as large as the sticking coefficient and hence lies in the same range as the results of Table I11 for glass. It may be noted that the experiments on tungsten have all been done in u-h-v systems. Mickelsen and Childs (19) have given a theoretical discussion applicable to calculation of pumping speeds in practical systems since they consider multiple collisions with wall surfaces which is the usual practical situation. Hurlbut (20) finds that for most cases of scattering of nitrogen at surfaces of steel, aluminum, and glass, the cosine distribution is obtained as has been generally assumed. Cabrera (22) has examined analytically a simple one-dimensional model of the impact 1 for between a gas molecule and a surface, and has concluded that c practical values of the parameters. Zwanzig (22) draws a similar conclusion from a similar model. Littlewood and Rideal (23) discuss the general problem of measuring c and conclude that heat transfer effects can cause serious discrepancies in measured values. They quote their own vaIues of c for long-chain fatty acids and alcohols between 0.36 and 0. I. Our conclusions from this evidence is that c lies between 0.1 and 1 for the great majority of collisions of gas molecules with surfaces, with a preference for values nearer 1. It may be noted in the above discussion that no dependence of c on the coverage of the surface with previously adsorbed gas was explicitly mentioned. I n the derivation of his famous isotherm equation, Langmuir (24)made the assumption that an incident molecule striking a site already occupied was elastically reflected. Thus if a fraction B of the available sites of a surface are already occupied, then the number of molecules adsorbed per cm2 per second is just
+
da
--vc(i -
dt
e)
In a generalization of Langmuir’s theory to multilayer adsorption, Brunauer et al. (25) permitted the condensation of molecules striking other molecules already adsorbed, and with this assumption, which is in qualitative agreement with the results above, they built up a theory of multilayer adsorption which has been widely used.
330
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
Following adsorption there is a statistical chance that the molecule will desorb from the surface. T h e mean time of sojourn on the surface is given by Eq. (2). There is an analogous relation for the mean time of sojourn on a single site before sideways or surface diffusion ts
=
to exp E,/RT
(7)
where Es is the energy barrier between adjacent sites. De Boer (14) emphasizes that since Es is normally less than E d ; an adsorbed molecule makes many sideways excursions before returning to the gas phase. Kruyer (26) has given a detailed discussion of ts. T h e magnitude of t s is determined in an analogous way to that of ta in Table 11. When the value of ta becomes comparable with the flight time between wall collisions then quantitative effects of adsorption will become noticeable in vacuum systems. I n different systems this onset will appear in different ways. For example Hayashi (27) has applied the basic work of Clausing (28) to the problems of production and measurement of high vacuum. I t may be noted that Clausing (28a) has very recently corrected an error in his earlier work. A step function of pressure is applied at the end of a pipe and Hayashi gives the solutions for the resulting pressure as a function of time and distance down the pipe, taking into account adsorption at the walls. Formally the equations are similar to diffusion equations (Section 11, D , 4). I n his calculations Hayashi uses the latent heat of vaporization to compute ta. As will be seen in Section 11, B, 2, heats of adsorption are usually several times the heats of vaporization. This will greatly emphasize adsorption effects. Hayashi recognizes this point in an application of these ideas to leak detection. De Boer (14, p. 138) discusses modifications of the considerations as a result of surface diffusion, Using methods similar to Hayashi’s, Eschbach et al. (28b) have measured the time of sojourn of a helium 8 x atom on glass at 20°K to be I sec. Provided ta is long enough at the temperature of adsorption, it is possible to trap molecules on a surface and later to cause them to desorb by raising the temperature of the surface. I t follows directly from Eq. (2) that the rate of desorption is given by
A measure of desorption rate, even an approximate one, is sufficient to establish E d from this equation. Measurements of this type have had wider application in chemisorption (Section 11, C) where in many cases Eq. (8) is modified by second-order effects, but some notable measure-
33 I
ULTRAHIGH VACUUM
ments of low adsorption energies have been made in this way. In all cases mentioned below these measurements have employed u-h-v techniques. Ehrlich (29) reports a low temperature ( T -100°K) state of binding of nitrogen on tungsten with E d = 9 kcalimole, which has a complex interaction which Ehrlich does not interpret as true van der Waal adsorption. In a companion paper, Ehrlich (30) reports no analogous binding states for carbon monoxide even at T -1 15°K. From field emission studies, Ehrlich and Hudda (31) deduce the heat of adsorption of xenon on tungsten to be 8 kcal/mole, and also report a number of activation energies of surface diffusion for inert gases on tungsten. These were all below 5 kcal/mole. Ehrlich et al. (32) g’ive a desorption energy for argon on tungsten, measured by flash filament methods, of 5 kcal/mole. Gomer (33) has studied the adsorption of neon, argon and xenonon tungsten in a field-emission microscope. Gomer gives the activation energy of desorption for argon from tungsten, obtained from an Arrhenius plot, as 1900 & 200 cal/mole. Gomer comments that physisorption on clean metals results in energies not very different from other substrates. Table IV below is taken from Gomer’s paper. We have omitted from the table the results on work function changes which do not concern us directly here, but which are normally obtained from field emission experiments. TABLE Iv. DIFFUSION AND D E S O R P T I O N TEMPERATURES AFTER GOMER (Temperatures refer to completion of diffusion or desorption in 10-100 sec; “mono” and “mult” refer to monolayer and multilayer, respectively) ~~
Diffusion temperature
Desorption temperature
(OK)
(“K)
Gas Neon Argon Xenon
5 18-21
40 mono
10
28 mult 60 mult
50 mult
35 mono 100 mono
T h e temperatures given in Table IV may be converted to energies with Eq. (1 3). In a subsequent paper, Gomer (34) discusses his work on the adsorption and diffusion of argon on tungsten in detail. T h e accommodation coefficient is a measure of the energy exchange between an impinging gas molecule and a surface; it thus combines the effects of the condensation coefficient and the sojourn time. If the temperature of the surface is T s O K , while that of the desorbing gas is T,, then the accommodation coefficient is defined as:
x = ( T , - T,)/(Ts
-
To)
(9)
332
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
where To is the temperature of the impinging gas molecules. Reviews of accomodation coefficients have been given by Devienne (3.9, and Hartnett (3%) who quote results for a variety of gases and solids. Thomas and Golike (36) have compared two methods of measuring accommodation coefficients (the temperature jump and low pressure methods) and quote results for helium, neon, and carbon dioxide on platinum. A summary of these results has been taken from their paper and is shown in Table V, where we have disregarded the small temperature dependence observed by Thomas and Golike. TABLE V. ACCOMMODATION COEFFICIENTS OF HELIUM, NEON,AND CARBON DIOXIDE ON PLATINUM BY Two METHODS
Accommodation coefficient Gas Helium Neon Carbon dioxide
Method LP
method TJ
0.1747 & 0.009 0.433 0.779 j~0.006
0.1764 f 0.012 0.326 0.787 & 0.017
l'O1
z
-
0
0.4-
0
0
5
I
-
0 0.20 0
a
-
-
-
250
350
300
Th
400
l
FIG. I . Accommodation coefficients of various gases on glass as a function of temperature [after K. Schiifer and H. Gerstacker, 2. Elektrochem. 60, 874 (1956)l.
ULTRAHIGH VACUUM
333
Schafer and Gerstacker (37) have measured the accommodation coefficient of various gases on glass, and these are shown in Fig. 1. T h e order of the accommodation coefficient follows approximately that of the boiling points of the gases, and a similar pattern emerges in the adsorption equilibrium results of Section 11, B, 2. Schafer and Cerstacker relate these accommodation coefficients to the results on condensation coefficients given in Table 111. T o our knowledge, no accommodation coefficient measurements have been done in u-h-v systems, although the results of Schafer and Gerstacker explicitly show that the accommodation coefficient depends upon the degree of outgassing of the surface. This point is also stressed by Hartnett (352).
2. Systems in Thermodynamic Equilibrium. When a quantity of gas is adsorbed on a surface, and an equilibrium has been reached in which there is no net transfer of matter from one portion of the system to another, there will be in the system a pressure analogous to a vapor pressure, but differing in one essential respect. T h e pressure above an adsorbed layer depends in general upon the density of molecules in the layer. Thus there is a relation between p , T, and the relative coverage 8, and for a given system we may write:
f(& P , T ) = 0
(10)
There are two important cases where the dependence of pressure on 6' at fixed T becomes simple. The first occurs when many layers have been adsorbed and the adsorbent is indistinguishable from the liquid or solid phase of the molecular species concerned. In this case, p is identified with the vapor pressure or sublimation pressure respectively, and there is no dependence on the number of molecules involved. T h e second occurs when the adsorption of one molecule does not alter the adsorption of another, as has been assumed in most of our preceding discussion. I n this case p is proportional to 8, a condition known as Henry's Law. With T fixed Eq. (10) is termed the adsorption isotherm and many different relations have been proposed for it. Dushman (38) reviews a number of these. As may be seen from this simple discussion, Eq. (10) combines the kinetic processes which we have discussed separately in Section 11, B, 1 and it is often difficult to say which of the kinetic parameters is the essential variable. From a practical viewpoint, a knowledge of the relation f (0,p , T ) permits an estimate of the limiting pressure achievable in a given system. Halsey and his co-workers (39-42), in a series of investigations have
334
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
performed experiments in which a small reduction in pressure was measured when the adsorbent was cooled. Halsey carried out experiments at temperatures well above the boiling point of the adsorbate and at pressures of the order of hundreds of Torr. While these conditions appear far removed from those of an u-h-v system, the experiments yield a parameter E* which we may identify with the heat of adsorption q and which we believe applicable to the same gas-solid combinations in u-h-v systems. Heats of adsorption obtained by Halsey and co-workers are listed in Table VI. TABLE VI. VALUES OF
c * (KCAL/MOLE) FOR
VARIOUS GAS-SOLID COMBINATIONS~ ~
lGas
Solid
He
H,
Ne
Porous glass Saran charcoal Carbon black Alumina Graphitized Carbon black P. 33 (2700")
0.68 0.63 0.60
1.97 1.87
1.54 1.28 1.36 0.87
NO 4.26 3.70
~~~~
A
3.78 3.66 4.34 2.80 2.46
0% CH,
Kr
Xe
4.09 4.64 3.46 3.30
4.23
After Halsey and Co-Workers (39-42).
T h e gases have been arranged in the order of increasing boiling points and we might expect the heats of adsorption to vary in the same order. Generally this is true, but hydrogen and nitrogen appear to be exceptional, possibly because of an electric quadrupole contribution to their heats of adsorption (43).While the heats of adsorption of a given gas on porous glass, saran charcoal and carbon black are approximately the same there appears to be a marked reduction for alumina and the particular graphitized carbon black, P. 33, which is specially prepared to yield a uniform surface. T h e great bulk of physical adsorption measurements have been carried out in the pressure range from several hundred Torr to about Torr with temperatures generally less than those of Halsey and coworkers. Relative surface coverages in these experiments were usually greater than B = 0.01. Dispersed adsorbents were generally used (an exception is the work of Rhodin, 44) and the data were usually analyzed with the B. E. T. theory (25) to yield a value of the true adsorbing area. There appears little doubt that the value of the area obtained in this way is essentially correct, and is probably the most widely used method for the measurement of the surface area of disperse solids (25). However,
335
ULTRAHIGH VACUUM
heats of adsorption obtained from such conventional data have to be examined carefully if they are to be used to predict the behavior of u-h-v systems. An example of this type has been examined by Hobson (45) for the physical adsorption of helium on pyrex at 4.2"K in an u-h-v system. T h e highest value of the heat of adsorption of helium on any adsorbent quoted from conventional adsorption measurements at or below the boiling point (2' 4.2"K) was 148 cal/mole (46)for helium on NiSO, . 7H,O. T h e immersion in liquid helium of 100 cm2 of adsorbent in a one-liter system containing He gas at a pressure in the u-h-v range should have given a barely perceptible drop in pressure if the heat were of the order of 150 cal/mole ; whereas, if the heat were of the order of that given in Table VI, the pressure should have dropped to an immeasurably low value. There was indeed a spectacular drop in pressure (see also Hobson and Redhead, 47), but the final pressure (approximately 10-l2 Torr) still appeared measurable. Hobson's conclusion was that this final pressure was the result of a dynamic flow of helium in the system and he placed a lower bound on the heat of adsorption of helium on Pyrex at 250 cal/mole, in qualitative agreement with Table VI and other measurements at temperatures well above 4.2"K (48). This result suggested that u-h-v techniques might make significant contributions to the existing data on physical adsorption and this idea has been pursued (49,50). T h e latter reference was a measurement of the adsorption
<
--
I
I
I
I
I
I
I
I
I
I
I
I
HOBSON (PYREX)
w DRAIN 8 MORRISON IRUTILE)
0
0
P
-I
-
-2
-
LOPEZ-GONZALEZ E T AL.(CHARCOAL) STEELE 8 HALSEY (POROUS GLASS)
0
-
CONVENTIONAL DATA
-3 -
I
'
-5 -10
ULTRA-HIGH VACUUM 1
I
-8
I
I
-6
I
I
-4
I
,
-2 log,, p [Torrl
DATA I
I
0
I
I
2
'
-5
FIG.2. Adsorption isotherms for N, on various adsorbents [after J. P. Hobson and R. A. Armstrong, Rept. 2lst Ann. M.Z.T. Conf. on Phys. Electronics p. 236 (1961)l.See J. P. Hobson, J . Chem. Phys. 34, 1850 (1961); L. E. Drain and J. A. Morrison, Trans. Faruduy Soc. 29, 654 (1953); J. de Dios Lopez-Gonzales, Carpenter, F. G., and Deitz, V. R., J . Research Nutl. Bur. Standards 55, 11 (1955); W.A. Steele and G. D . Halsey. Jr., J . Phys. Chem. 59, 57 (1955).
336
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
isotherms of nitrogen on Pyrex at seven different temperatures between
63.3"K and 902°K. A selection from these results is compared with other adsorption isotherms obtained by other techniques in Fig. 2. T h e results of the u-h-v experiments have caused Hobson and Armstrong (51) to propose a general hypothesis in which to describe physical adsorption isotherms. This hypothesis rests at present on rather sketchy data, although it has been reinforced recently by measurements by Hansen (5lu) of the adsorption of A, Kr, and Xe on zirconium. In principle the hypothesis may be used for calculating the result of a wide variety of adsorption isotherms including those in the u-h-v range. An interesting result arises when the hypothesis is applied (51b) to an experiment which has been performed by Gomer et al. (52). Their experiment is idealized to the immersion in liquid helium of a sphere of Pyrex, volume 500 cm3, which had been exhausted to 5 x 10-l0 Torr of helium gas ; the predicted pressure is Torr. Gomer, et al. had no means of checking this prediction. At this pressure, in any interval of time a single atom of helium is present in the gas phase for 10-13 of the interval, and the average gaseous density is about grams/cm3. Allen (53)quotes the mean density of interstellar gas as 1 x grams/cm3. Gomer (54) has developed a universal gas source based on physical adsorption of gases at 4.2"K. While this method of achieving very low pressures appears very powerful, it has serious limitations. T h e central difficulty is that the pressures are achieved only at very low surface coverages, and in practical systems the amounts of gas involved may be sufficient to drive LIP the pressure, or to cause limitations due to dynamic flows of gas in the system. Of course reduction of T below 42°K can always produce extremely low pressures, the vapor pressure of helium at T = 0.1"K Torr. being 3 x Hengevoss and Huber (542) describe experiments in which gases are absorbed on a cold trap in an u-h-v system and note that residual gases desorb from the trap at different temperatures. They suggest that this result may have practical application as a rough method of residual gas analysis. Garbe et al. (55) have published isotherms for water on glass in an u-h-v system, but they mention serious hydrogen evolution and it appears uncertain whether these data can be used as a check on the hypothesis mentioned above. They find the heat of adsorption of water on glass to be about 12 kcal/mole at low coverage. This value decreases with coverage with an average value 6-7 kcal/mole. Tuzi and Okamoto (552) have measured the adsorption of water vapor on glass in highvacuum apparatus and find it t o consist of two processes, the first being
ULTRAHIGH VACUUM
337
physical adsorption with a heat of adsorption of 11 kcal/mole, and the second being activated adsorption with an activation energy of 9 kcal/ mole and a heat of adsorption of 1 3 4 0 kcal/mole. In another paper Tuzi (5%) discusses the diffusion of water molecules into a hygroscopic surface layer. At first sight it might appear possible to lay down a new adsorbing surface on top of an adsorbed layer in a manner analogous to that used in getter-ion pumps (see Section 111, A, 3). T h e results of Becker (56)on additional pumping by mercury vapor in a cold trap, do not appear encouraging. However the results of Balwanz (57) indicate that some additional pumping can be achieved in this way. Penchko et al. (58)have immersed a tetrode ionization gauge in liquid helium at 13°K and obtained pressures of about 10-lo Torr. T h e gauge was sealed off at relatively high pressures (4 x Torr) and the authors give evidence that several gases were probably present at the time of immersion. The results then appear to be a case of physical adsorption of a complex mixture of gases. Adsorption of gases on preadsorbed layers of other gases has been studied [e.g., Keesom and Schweers (59); Singleton and Halsey (60)] and one systematic work in the u-h-v range has appeared ( 5 1 ~ ) .
C . Chemisorption Chemisorption is the dominant process controlling the partial pressures of chemically active gases (i.e., gases other than the inert gases) in most u-h-v systems. Chemisorption of neutral molecules of the chemically active gases occurs predominatly on the metal surfaces in the system. T h e adsorption of ions or other active species will be considered in a later section, Chemisorption of gases on metals has been reviewed recently by Gundry and Tompkins (61). The rate of adsorption into a chemisorbed layer at constant temperature may be written as
where u is the number of adsorbed molecules per cm2, and s(u) is the sticking probability which is a function of surface coverage (s = a in this case). For the common active gases a clean metal surface has a sticking probability of 0.1 to 0.5, thus a few square centimeters of a clean metal surface has a very appreciable pumping speed [see Eq. (3)]. The sticking probability of a gas on a clean metal surface remains constant until a certain fraction of the available sites are filled; this
338
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
fraction depends on the gas-metal combination, on the crystal face and on temperature. The initial constancy of the sticking probability implies that chemisorption takes place from a reservoir of physically adsorbed molecules, As the coverage is increased the sticking probability drops very rapidly. The best measurements of sticking probability have been made on tungsten; Fig. 3 shows the variation of sticking probability for 1 .o
lo-’
~
t
-I -
m 4
m
0
CL
a 0 10-2
z
Y
I!
I-
w
II In
lo5
0
1
2
TOTAL NUMBER
3
4
5
6
7 X 10j4
OF ADSORBED MOLECULES PER CM2
FIG. 3. Sticking probability as a function of surface coverage of various gases on the 411 plane of tungsten at room temperature [after J. A. Becker, Solid State Phys. 7, 379 (l958), courtesy of the Bell Telephone Laboratories].
various gases as a function of surface coverage on a tungsten ribbon exposing the 41 1 plane predominantly. Becker (62) interprets these data as indicating that the sticking probability changes abruptly when the adsorbed species has made a first set of valence bonds with every surface metal atom and again when they have made second valence bonds with all the surface atoms. At higher coverages, adsorption takes place in the outer layers and bonds are being made to the adsorbed atoms in the first and second layers. The sticking probability and bond strength is much smaller in these outer layers. Sticking probability curves measured
ULTRAHIGH VACUUM
339
on polycrystalline adsorbents do not show the second region of constant s. Adsorption into the outer chemisorbed layers is quite slow because the sticking probability has dropped to very low values. Considerable quantities of gas can, however, be adsorbed into these outer layers and although the effective pumping speed may be very low, a very long time is required to reach an equilibrium condition in the u-h-v pressure range, Of the four gases for which most data are available, 0,, N,, and H, are dissociated upon adsorption on tungsten and are chemisorbed in the atomic form. T h e atoms recombine and desorb as molecules, thus the desorption reaction is second order. T h e fourth gas, carbon monoxide does not dissociate on tungsten. T h e chemisorption of the above four gases on tungsten, and most other metals, is unactivated, i.e., the activation energy of adsorption approaches zero. Thus for most cases of interest in vacuum problems the heat of adsorption approximates the activation energy of desorption. T h e sticking probability is observed to decrease with increasing temperature for most cases. For nitrogen on polycrystalline tungsten the temperature dependence of the sticking probability is shown in Fig. 4(a). T h e sticking probability of carbon monoxide on tungsten is not strongly affected by temperature [see Fig. 4(b)]. T h e time taken to reach a monolayer coverage (tm) is sometimes a useful method of estimating the pressure of a chemically active gas. T h e monolayer time is here defined as the time until the sticking probability starts to drop from its initial constant value. This time can be measured by initially flashing a tungsten filament at a temperature high enough to desorb all gases (-2400°K) and then cooling the filament. T h e pressure drop on cooling the filament, remains constant and then starts to rise slowly. T h e time at which the pressure starts to rise, from its initial constant value, is the monolayer time. This “flash-filament” method has been widely used for the measurement of sticking probabilities (see for example Becker, 62). T h e rate at which the number of molecules per unit area (u)is depleted by desorption is given by - da - u7vr ~ exp ~
dt
[-
1-RT -Ed
at constant temperature. Here x is the order of the desorption reaction and Ed is the activation energy of desorption; v, is the rate constant, independent of temperature. For a first-order reaction (x = I ) , v1 is approximately 10’“ sec-I.
340
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
A useful rule of thumb is that desorption becomes significant at a temperature T where T("W ,20 Ed (kcal/mole) Recent measurements of the heat of adsorption of the simple gases on metal single crystals show that it is constant until one "layer" of gas is adsorbed. The heat then decreases for the second layer and again for the outer layers. For example, Becker (62) has found that the heat of adsorption of hydrogen on the 320 plane of tungsten is 53, 37,
0.12
: 9 8 OK
I I
I
0.08
SAMPLE # 16
I
\ 0.04
0
0
0.5
I .o
c [MOLECULES/
I.5
2.0
C M ' ~10-1~1
FIG.4(a). Sticking probability of nitrogen on polycristalline tungsten as a function of surface coverage for various temperatures of absorption ( T o ) .
ULTRAHIGH VACUUM
34 1
37, and 25 kcal/mole for the first, second, third, and fourth stages, respectively. T h e first stage corresponds to one hydrogen atom per surface tungsten atom and the following stages are multiples thereof. The measured values of the important parameters of chemisorption on metals (sticking probability and heat of adsorption as a function of coverage) are not complete and there is considerable divergence between the results of different experimenters. Part of this experimental uncertainty is caused by differences in the crystal structure of the surfaces of the various experimental samples and by contamination of the surfaces, the latter is particularly true of the older measurements made with inadequate vacuum techniques [see Becker (63) for comments on this problem]. Measurements made on filaments or ribbons of refractory metals which can be rigorously outgassed are likely to be least affected by 0.5
0.4
>
t
-1 0.3
m
2
0 [L
a
a
z $ 0.2 $
0.1
0
FIG. 4(b). Sticking probability of carbon monoxide on polycrystalline tungsten as a function of surface coverage for various temperatures of adsorption (To).
342
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
contaminations. Measurements on evaporated metal films may be in doubt unless very great care was taken in the preparation of the films. Measurements made with oxygen and hydrogen using hot-filament ionization gauges may be in error because of decomposition of the gas by the action of the hot filament (see Section 11, G). T h e predominant impurity is usually carbon monoxide with the above gases. Table VII lists the measured values of the initial heats of adsorption of various gases on several metals. Most of these data were obtained on evaporated films. I n the case of oxygen, the initial heats of chemisorption are very similar to the heats of formation of the corresponding stable oxides (64). Measurements on evaporated films show that the total heat of adsorption decreases as the coverage increases. Recent measurements on polycrystalline wires of metal show that adsorption occurs into several TABLE VII. EXPERIMENTAL VALUESOF INITIALHEATSOF ADSORPTION qo kcal/mole) ON POLYCRYSTALLINE SAMPLESO 0 2
No
< 3 < 3 < 3
Ag
A1 Au Cd
21 1
co
100
H2
co
C*H,
8.7b
< 3
174
Cr cu Fe
136
40"
Hg In
45 9 32 < 3
102
9.3b 32"
68
31 < 3 27 30 27 45
32"
58
45
80"
< 3
150
Mn
Mo Nb Ni Pb
172 208
Pd
67 67 115 212 236 194
41
107
Pt Rh Ta
Ti W
140'
140' 95c
Zn
138
< 3
data from Brennan (64); Hz data from Ehrlich (65); CpH4data from Beeck ( 6 6 ) . Trapnell (67). Beeck (68).
OI
Bagg and Tompkins (69). Baker and Rideal (70). f Beeck et al. (71). 0 Redhead (72).
102
343
ULTRAHIGH VACUUM
different phases with different heats of adsorption. The phase with the highest heat is filled first, followed in order by the phases of lower heats of adsorption. These various phases correspond to adsorption on the different crystal planes exposed on the surface. Thus the total heat of adsorption, for all phases, appears to decrease with coverage. Trapnell (67) has examined the adsorption of several gases on evaporated metal films and shows that the activity of. the various gas-metal combinations can be tabulated as shown in Table VIII. TABLE VIII."
W, Ta, M o , Ti, Zr, Fe, Ca, Ba Ni, Pd, Rh, Pt Cu, A1 Zn, Cd, In, Sn, Pb, Ag Au
+ -
-
+ + -
-
+ + ++
+ + + +
+ + ++
+ + + +-
~
a
+ gas chemisorbed; - gas not chemisorbed.
It can be seen from this table that the order of activity of these gases for all metals is 0 2 > CzH2 > C2H4 > CO > H, > N, , if a metal chemisorbs a particular gas it will also chemisorb all gases higher on the scale, whereas if it does not chemisorb the gas it will not chemisorb gases lower in the scale. The only exception to this rule is gold. The adsorption of CH4 and C,H, on evaporated metal films has been examined by Trapnell(73). He finds that W, Mo, Ta, Cr, Rh, Ti, and Pd adsorb CH, and C,H, strongly; Fe, Co and Ni adsorb no CH, and little C2H,. Roberts (73a) finds that C,H, decomposes on Rh to yield CH, and an adsorbed hydrocarbon residue. Two processes of great importance in u-h-v systems have received little attention, they are: (a) the simultaneous adsorption of two or more gases, and (b) the replacement of one adsorbed gas by the introduction of a second gas. The second process is frequently observed in u-h-v systems, but little quantitative data is available.
D. Sources of Gas This section is concerned in the main with the sources of background pressure in an u-h-v system.- A number of authors (74-78) provide
344
P. A. REDHEAD, J. P. HOBSON, E. V. ICORNELSEN
extensive data and methods of analysis for the degassing of materials under vacuum. These discussions, while not normally directed toward u-h-v, nevertheless are applicable to u-h-v problems. Rather than examine these works in detail, however, we seek below to isolate the various physical processes which give rise to the degassing of materials and to illustrate their orders of magnitude with simple examples. As a specific example we consider a system of volume 1 liter, with an applied pumping speed S = 1 liter/sec, and we shall discuss the mechanisms involved and the procedures .required in reducing the partial pressure Torr. The total input leak-rate for each of any gas to 10-lo, IO-l5, of these pressures is given in Table IX [derived from Eq. (l)]. TABLE IX. PERMISSIBLE LEAKRATESFOR V = 1
P (Torr)
FI(
ema (STP) s e c -
molecules/sec
I
!1
lo-'"
LITER,
10-16
1.32 x 10-l"
1.32 x
3.3 x lo*
3.3
x 10'
S =1
LITER~SEC
10-*0
1.32 x 0.33
Next, we examine individually the various sources of background pressure in an u-h-v system which has been made leak-tight. These sources are usually critically dependent on particular choices of components. Often the main parameters are very sensitive functions of temperature, and in these circumstances detailed numerical agreement between various authors is not found, and it is difficult to make precise design calculations for a particular case. We have omitted from the list of processes treated below the re-emission of gas already pumped by the various pumps described in Section 111, A, and also gases which arise directly from pump action, such as hydrocarbons in systems using oil-diffusion pumps (79). An interesting practical discussion of some of the processes considered is provided by Farkass and Barry (80).
1. Desorption of Adsorbed Gases. Most materials, when exposed to the atmosphere, acquire one or more surface layers of gas held to the surface by chemical or physical forces (see preceding sections). When these materials are placed under vacuum, these layers tend to desorb, but the process may not go to completion (81) and if precautions are not taken, this desorption may continue for long periods, making the achievement of u-h-v impossible. As an example we consider below an idealized model in which a solid surface of area A = 100 cm2 is initially covered with a monolayer of gas bound to the surface with an activation energy of desorption of E d cal/mole. We imagine the desorbing gas from this
ULTRAHIGH VACUUM
345
surface to be pumped away by a pump speed I liter/sec. We disregard repumping by the surface; the latter assumtion greatly simplifies the calculation but is unrealistic in practice. We also neglect surface diffusion effects which may be of importance in the nonequilibrium situation we are examining. However, the conclusions we draw from the model appear so decisive from a practical viewpoint that even these poor assumptions will not alter them. By using Eqs. (1) and (8) we compute the time necessary to achieve the design pressures of Table IX as a function of E d . T h e results of this calculation are given in Fig. 5(a) for a temperature ( T I )of 295°K. Figure 5(a) demonstrates that for E 5 20 kcal/mole, desorption takes place quickly and the surface becomes bare, while for E 2 40 kcal/mole the surface binding is sufficiently strong to permit u-h-v to be reached without the surface becoming bare. However there is a range of heats between these limits which can make the achievement of u-h-v a very lengthy process under the conditions described. Also, it seems quite probable that the troublesome range of heats will be found in practice in the second chemisorbed layer. T o investigate the effect of raising the temperature of the desorbing surface we have calculated from the model the heating time necessary at T, = 300°C so that design pressures can be reached after cooling the surface instantaneously to room temperature at the end of the heating time, T h e result is given in Fig. 5(b). It is readily seen that the time scale has been drastically reduced. Thus even modest outgassing appears adequate to prepare surfaces approximated by our model for u-h-v purposes. Variations on these considerations have been given by Venema (81a) and Hobson (81b). However, the work of Todd (82) shows that not all surface gas can be treated as simply as we have done above. Todd finds that to remove surface water on Corning 0800 glass requires outgassing for several hours at 480°C. He postulates a mechanism involving formation of hydrates. Garbe et al. (55) give qualitatively similar results for the desorption of water from lime glass, although their times for the desorption of surface water are shorter. Their results on the adsorption isotherms (see Section 11, B , 2) of H,O on glass also show that glass once cleaned will act as quite a rapid adsorption pump for small amounts of further water vapor even at room temperature. This result is in essential agreement with Todd (83) who demonstrates that the adsorption of water by glass is a reversible process. Garbe and Christians (83a) have made a detailed study of the gases evolved by glass upon heating, and find CO,CO,, N,, H,, A, and CH, given off, in addition to H,O. Dayton (836)has discussed the desorption of gas from porous surface layers. Hunt et nl. (83c) argue that surface desorption can be significantly reduced
/
TI =295'K
P 5 10-20 Torr
10-15 irr
10
30
20
Ed
7
kcal /mole1
FIG. 5. Time required to reach given pressures as a function of E d ( A = 100 cma, S = 1 literlsec, V = 1 liter): (a) desorption occurs at room temperature; (b) desorption occurs at 573°K.
347
ULTRAHIGH VACUUM
by evaporating a molybdenum film on the interior surface of the apparatus. I t should be emphasized that the problem we are discussing in this section is distinct from that of obtaining atomically clean surfaces. T h e latter is a field in which u-h-v has found wide application. T h e specific problem of producing and demonstrating clean metal surfaces has been discussed by Hagstrum and D’Amico (84,and Dillon (842) has given a general discussion of the clean surface approach to chemisorption studies.
2. Gus Permeation. I n general, gases permeate or pass through solid who materials. Reviews of this subject are given by Norton (85) (8.5~) draws two general conclusions: (1) no rare gas permeates any metal; (2) diatomic molecules dissociate into atoms when permeating metals. It is consistent with these conclusions to write for the permeation rate: FK =
KAP 7 gases through nonmetals,
FK =
KAP 7 gases through metals,
(1 4 4
where K is the permeation constant and has the dimensions of cm2/sec and where n is usually about 0.5 (see Barrer, 86). I n these units K is the quantity of gas in the cm3 (STP) passing per second through a wall of area 1 cm2, thickness 1 cm, when a pressure difference of 1 atm exists across the wall. Equations 14 (a) and (b) break down at high pressures but appear adequate for this discussion. For a wall of A = 100 cm2 and d = 0.1 cm, the values of K to give the design pressures of Table IX must not exceed the values of Table X. TABLE X. MAXIMUM VALUESOF K FOR A WALLOF AREA 100 C M ~ ,THICKNESS 0.1 CM GIVEDESIGN PRESSURES IN A SYSTEM OF VOLUME1 LITER, WITH PUMPING SPEED 1 LITER;SEC AT 1 ATM EXTERNAL PRESSURE
TO
10-10
K
1.3 x
1043
10-15
1.3 x 10-18
10-20
1.3 x
10-25
Torr crn*/sec
If the external gas pressure is less than 1 atm, the permissible values of K are increased accordingly. Figure 6(a) gives values for the permeation constant for various gasinon-metal combinations, and Fig. 6(b) for various gaslmetal combinations. Altemose (87a) has recently published permeation constants for He in 20 different glasses as a function of temperature. Less extensive results are given for hydrogen, neon,
348
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
10-10n 0
,c E
&I10-12-
Y
10-l~
-
FIG.6(a). Permeation constants for various gas-nonmetal combinations as a function of temperature. Units of K are cms/sec [quantity of gas in cm3 (STP) passing per second through a wall of area 1 cmz, thickness 1 cm, when a pressure difference of 1 atm exists across the wall]. Curve number
Gas-nonmetal combination
-
1. Oa or N, Pyrex 2. Air - Pyroceram 3. Air - 97% alumina ceramic 4. Air-Pyrex 5. He - Lead borate glass G 6. He - 97% alumina ceramic 7. Ne - Vycor 8. N8 -50, 9. He 1720 glass 10. He - Pyroceram 9606 11. H, - S O p 12. He - Pyrex 7740 13. He - Vycor 14. H, -Pyrex 15. Air - 1720 glass
-
Reference
85 876 876 87b 85 876 107
86 85 87b 86 92 107
85 87b
349
ULTRAHIGH VACUUM
methane, argon, and nitrogen in glasses. The results of Altemose have not been included in Fig. 6(a). Similarly the results of Altemose on the solubility and diffusion of He in various glasses has not been included in Figs. 7(a) and 9(a). Permeation data are available for the combinations given below: H, - Pd (87c);0, - Ag (87d); H, - Ni, D, - Ni (87e); H, - Ni, H, - OFHC Copper, H, - Kovar, H, - stainless steel, H, - cold drawn steel, H, - gas-free iron, H, - Inconel (87f).From Fig. 6 it may be seen that at room temperature choice of wall material will generally meet the requirements of Table X for P = 10-lo Torr; for P = 10-l6Torr only particular combinations will meet the requireTorr scarcely any will meet the requirement. ment; for P = However, cooling the walls or evacuation outside the walls (88,89, 89a) may solve the problem. A further consideration in permeation problems is the time required to reach the steady-state condition of Eq. (14).
0
1
2
3
FIG. 6(b). Permeation constants for various gas-metal combinations as a function of temperature. (Units as in Fig. 6(a).) 1 . Ha-Pd 5. N, - M o 2. HZ- Ni 6. CO - Fe 3 . Ha-Mo 7. H, - F e 4. NP- Fe 8. H, -CU Ref.: 96.
350
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
Y
0
0
FIG. 7(a). Solubilities for various gas-nonmetal combinations as a function of temperature. Units of C, are dimensionless [quantity of gas in cm3 (STP) in 1 cm3 of material at 1 atm pressure]. Curve number
Gas-nonmetal combination
1. H,O - 0800 glass, Co 2. Ha -SIOa 3. He -Pyrex 7740
Reference
0.6 at 300°K
82 not shown on graph 86 92 107 107 107 not shown on graph
4. He -Vycor
5. H, -Vycor 6. Na - Vycor, lo-&< C,
<
at 673°K
Glass composition in
yo (108)
~~
SiOg BpOs Ala0, Na,O
--
7.
,%I
11.
12.
\ He-glass
76.1 75.9 64.1 75.3 56.2 69.1
16.0 16.0 23.2 7.6 -
-
1.75 0.4 4.0 6.2 1.2 3.3
5.4 4.9 4.0 5.7 7.6 13.2
KaO PbO
4.1 0.8 0.6
0.8
4.5 1.7
30.0
-
i
\
I
Io3
-
3
2
T (‘K)
FIG.7(b). Solubilities for various gas-metal combinations as a function of temperature [units as in Fig. 7(a); values From E. Waldschmidt, Metal1 8, 749 (1954)l. 1. Ha-Ti 2. Ha- Ta 3. Ha-Pd
4. o a - c u 5 . Ne - Fe 6. HZ-CU
10
-4
10
\ I
Io3 T ( K)
2
3
a
FIG.7(c): Solubilities for various gas-metal combinations as a function of temperature [units as in Fig. 7(a); values 1 - 5 from E. Waldschmidt, Metoll 8, 749 (1954), and 6 from S. Dushman, “The Scientific Foundations of Vacuum Technique.” Wiley, New York, 19491. 1. H , - N i 4. H,- Fe 2. Np-Mo 5. HZ-MO 6. N z - W 3. Op-Ag
352
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
This time may be very long. All these considerations have been quite thoroughly investigated for the helium-Pyrex system by the Westinghouse group of Alpert and associates (90, 91, 92). This system received wide attention because the permeation of helium, present in the atmosphere with a partial pressure of 5 x Torr, through the walls of a Pyrex system determines the final pressure of this system when other processes have been eliminated. McAfee (93, 94) has investigated the dependence of helium diffusion in Pyrex under various conditions of stress in the glass.
3. Diffusion of Gas from Inside Solids. General texts on this subject are Jost (95) and Barrer (86). In general, gases are soluble in solids, the amount of gas in solution depending upon the pressure of gas outside the solid. This phenomenon is closely related to that of permeation, and indeed the solubility equations are similar in form to the permeation equations (14a, b), with the same qualifications: C
= COP gases
in nonmetals,
C = C,Pn gases in metals,
(1 5 4 ( 15b)
where C,,is the solubility and is dimensionless. I n these units it is the quantity of gas (cm3 at STP) in 1 cm3 of solid at 1 atm external pressure; C in Eqs. (15a, b) is in similar units. For example, metal parts exposed to an atmosphere of nitrogen over a long period, will contain a concentration of nitrogen in solution given by Eq. (15b). Values for C, for typical gassolid combinations are given in Figs, 7(a) and 7(b) as a function of temperature. When parts containing dissolved gases are placed under vacuum this dissolved gas diffuses to the surface and desorbs until a new equilibrium governed by Eq. (15a or b) is established. During the transition between the two equilibria the diffusion is generally governed by Fick’s law of diffusion [for an exception see McAfee, (93, 94)]. T h e transition times, as will be shown, may be very long, particularly when the second equilibrium pressure is in the u-h-v range. Often the second equilibrium state is never achieved in practice, and it is necessary to consider the desorption from the solid as a time dependent intermediate between the two equilibrium states. T h e analysis of Todd (82)in his work on the desorption of water vapor from glass provides some simple and useful relations for the desorption problem. At t = 0, the pressure outside a body is reduced to a low value. Figure 8 is reproduced from Todd’s paper and shows the ratio of the amount of material VI, which has desorbed from the body after time t to
3 53
ULTRAHIGH VACUUM
the original amount V,,. T h e curves have an initial linear portion and it is of interest to examine the time required before deviations from the
0.5
a/ a
I .o
FIG. 8. Ratio of the amount lost by diffusion to the initial amount of diffusing material plotted against (Dt)'l2/iafor a slab of thickness 2a, an infinite circular cylinder of diameter 2a, and a sphere of diameter 2a [after B. J. Todd, J . Appl. Phys. 26, 1238 (1955)l.
linear portion take place. At this time approximately half of the dissolved gas has desorbed. For a slab of thickness d this time is given by: '
0.28 d2
tc = __
D
where D is the diffusion constant and has units of cm2/sec. Values for the diffusion constant are shown in Figs. 9(a) and 9(b). Diffusion data are available for the following combinations: He - fused quartz (9.52); Ne - fused quartz (9%); H, - Pd ( 9 5 , 9.54; H, - Ge (95e); H, hardened steel (9Sf);H, - mild steel (95g); He - T i (9%); 0, - Si, 0, - Ge (95i);A - Ag, A - Au, A - Al, A - Pb (95k). I n Table XI we have calculated values of tc from this formula for the three combinations: H,O - 0080 glass, N, - Fe, and H, - Ni for a slab 0.1 cm thick, surface area 100 cm2 at room temperature. For times less than those given by tc in Table XI desorption will be
3 54
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
-
,610
I
2
I o3
3
4
T (OK
FIG. 9(a). Diffusion constants for various gas-nonmetal combinations. Units are cm*/sec. Curve number
Gas-nonmetal combination
Reference
1.
He - Pyrex 7740 He-Vycor Hp-Vycor NI-Vycor HI-SiOl 6. He - Duran glass
92 107
2. 3. 4. 5.
107 107 86 108 Glass composition in yo (108) .-
SiO,
B,O,
Al,O,
Na,O
K,O PbO
76.1 75.9 64.7 75.3 56.2 69.1
16.0 16.0 23.2 7.6
-
1.75 0.4 4.0 6.2 1.2 3.3
5.4 4.9 4.0 5.7 7.6 13.2
0.6 0.8 4.1 0.8 4.5 1.7
76.1 75.9
16.0 16.0
1.75 0.4
5.4 4.9
0.6 0.8
30.0 -
ULTRAHIGH VACUUM
355
Id4
-Y
10-6
N '
E
Y
a
10-8
-10
7'
10
I
2
3
FIG. 9(b). Diffusion constants for various gas-metal combinations. Units are cm*/sec. Reference
1. 2. 3. 4.
H, -Pd NP - F e CO - N i H2 - N i
86 86 38 86
Reference
5. Ha- Fe 6. 0 2 - N i 7. O p - F e
86 38 86
governed by the linear region of Fig. 8. In this region the rate of desorption is given by, F - dVL A (D/t)"2 dt - 2d TABLE XI. OUTGASSING A SLAB 1
vo
AREA100 C C = 0.1 INITIALLY
MM THICK, OF
FOR
(17) M ~AT ROOM
TEMPERATURE;
F,,[cma(STP)/sec] Gas-solid
H,O - 0080 glass N, - Fe H2 - Ni
tc
(hr)
After 1 hr
x 10-10
6
x
2
1 o=e
2 6
400
4.5 x 10-8
1089
After 10 hr x x 10-18 1.5 x 10-6
356
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
where X is a geometric factor dependent on the shape of the specimen and is 1.13 for a slab. Values of dV,/dt have been calculated from Eq. (17) for the slab, and are given in Table XI for t = 1 hr and t = 10 hr. T h e value of C used in calculations of this type is a subject of some interest and is discussed below. T h e initial value of C used for the table was 0.1. A comparison of Table XI with Table IX shows that the desorption rates of the H,O - 0080 and N, - F e systems are sufficiently low to permit the achievement of u-h-v, whereas the H, - Ni system provides a very serious source of gas from the point of view of u-h-v. T h e dependence of the desorption rate on the time is so weak that it does not represent a practical solution to the outgassing problem. Thorough outgassing of metal parts thus appears mandatory for most u-h-v components. Waldschmidt (96) has given the time/temperature combinations for typical metal samples for outgassing to 5 % of their original gas content. I n Table XI we used a value of C = 0.1. As may be seen from Figs. 7(a) and 7(b), this corresponds more closely to the solubility of gases in metals at temperatures nearer their melting points than room temperature. T h e figure is a rough upper limit on the solubility found in practice, based on data of the type given by von Ardenne (97). Hashimoto et al. (98)and also Bliss (99)give data similar to those of von Ardenne and discuss the methods of measuring outgassing rates. A method for automatic determination of vacuum outgassing rates which might be applicable to the u-h-v ranges has recently been described by Fish (100) Varadi (101) has constructed an apparatus for following partial pressures as a function of time during thermal outgassing. T h e simple discussion of diffusion above has assumed that a certain gas had a clearly defined diffusion constant, While there is no difficulty with this assumption when the gas consists of one element, the diffusion of a gas containing two elements, such as CO, is more complex, and it has been shown (102) that the carbon and oxygen atoms which are released as carbon monoxide diffuse at different rates. Also, reactions may take place after a gas is released into the gas phase. These are discussed in Section 11, G and serve to emphasize the complex nature of gas-metal reactions found in practice. Della Porta (103, 104) has shown that adsorption and diffusion are closely related in the gettering process.
4. Vapor and Dissociation Pressure. For many materials in a vacuum system the residual pressure will be the vapor pressure of the material. Von Ardenne (97) and Honig and Hook (104a) give useful data on the vapor pressure of solids, liquids, and gases at various temperatures, and for pumping fluids and sealing materials. Balwanz et al. (57) summarize the vapor pressure of gases commonly found in u-h-v systems. For most
ULTRAHIGH VACUUM
357
solids likely to be used the vapor pressure is exceedingly low at room temperature. The vapor from an incandescent filament, however, may be a factor in u-h-v systems. For example, Alpert and Buritz (90) have found that the equivalent nitrogen pressure of tungsten vapor in a Torr. Bayard-Alpert gauge with a tungsten filament at 2300°K was (A particular vapor pressure of interest is that of Hg in a liquid N, trap. This is about Torr.) Recently Borovik et al. (105) have applied u-h-v techniques to the measurement of the vapor pressure of nitrogen at the normal boiling point of liquid hydrogen (2.2 x 10-l1 Torr) and the vapor pressure of hydrogen at the normal boiling point of liquid helium (3.5 x lo-’ Torr). Clearly, u-h-v techniques provide wide opportunities for the extension of vapor pressure data. One of the causes of limiting pressure in high-vacuum systems has been described as “dissociation pressure” (106, 38). At a given temperature every compound MO can be in equilibrium with its components
If one or both of the components M and 0 are volatile they will create a pressure known as a “dissociation pressure”. An example of a compound (in which both components are volatile) is HgO, whereas the more common compounds encountered in vacuum work are metal oxides, nitrides, and hydrides. The basic parameter governing the magnitude of the dissociation pressure is the heat of dissociation which, for metallic oxides, has been shown to be approximately equal to the heat of chemisorption of oxygen (64). Thus the dissociation pressure of oxygen will be closely related to the pressure above a chemisorbed layer of oxygen on the metal and we believe the analogy is sufficiently close for practical purposes to eliminate the need for further discussion of dissociation pressure.
5. Measurement of Diffusion Coeficient, Solubility, and Permeability of Gases in Solids. T h e central conclusion drawn from the foregoing discussion of the sources of background pressure in u-h-v systems is that at room temperature the diffusion of gases from the interior of solids represents the limiting process in the achievement of u-h-v, provided the permeation problem has been solved. Thus the main parameters required for. design are the diffusion coefficient, the solubility, and the permeability. Rogers et al. (92) have described a method for measuring these three quantities with a single experimental arrangement. T h e method has been used by Leiby and Chen (107) and by Eschbach (108).
358
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
The arrrangement used u-h-v techniques and we review it briefly here because the method appears to have great value in u-h-v studies. We estimate that leak rates of 10-l2 Torr liters/sec are measurable with mass spectrometers operating continuously, and that leak rates of Torr liters/sec are measurable by integrating the entering gas on a filament by chemisorption, or on a cold surface by physical adsorption, and then releasing it in a burst whose height can be measured. If sufficient time is available to establish steady-state permeation across a slab 100 cm2 in area and 1 mm thick, with a pressure differential of I atm, these leak rates suggest that values of K = cm2/sec and K = lO-l* cm2/sec are measurable with continuous and integrating techniques, respectively. T h e measurement of D and C however require a measurement of the time dependent approach to the steady state. For the “late approximation’’ method of Rogers et al. (92), if we restrict the time duration of the measurement to several weeks and the value of tc to 1 week = 6 x lo5 sec (where t c is the time of onset of the late approximation), then for a 1 mm slab the measurable value of D is restricted to D 2 d2/6t, = 3 x cm2/sec. (19) by the Since C = K/Dthe lower limit on C is given by C = 3 x integrating method of measurement. T h e method thus appears potentially able to give values of K , D , and C several orders below those given in Figs. 6, 7, and 9. Such data would be useful for design calculations for u-h-v systems.
E. Positive Ion Impact on Surfaces T h e range of ions in solids for energies usually encountered in u-h-v systems ( < 30 kev) is less than -lo3 lattice distances (I09),and approximately proportional to ion energy. Thus surface effects can be expected to play a significant role in almost any ion impact phenomenon, their influence becoming greater as the ion energy decreases. Aside from the application of u-h-v to their detailed study, ionic impact phenomena are of interest in u-h-v systems for two reasons: (1) they affect the vacuum conditions in the system ; (2) they affect ionic current measurements (notably that of pressure) which depend on the neutralization of the ion’s charge at a solid surface. I n the former category entrapment and sputtering are the most significant processes; in the latter one needs to consider electron ejection, ion reflection, and the production of secondary ions.
1. Entrapment. Ionic entrapment results in the transfer of gas
ULTRAHIGH VACUUM
359
molecules to the solid phase where they can remain for relatively long periods. A pumping mechanism is thus provided which requires no direct connection to the external atmosphere. There are, however, two ways in which trapped particles can be returned to the gas phase: ( 1) by spontaneous re-emission, a process probably involving both diffusion and desorption; (2) by the sputtering action of incoming ions, For the energy range of interest here (0-30 kev), an ion striking a solid transfers most of its momentum to the lattice atoms by elastic scatter in their composite electrostatic (i.e., time average) potential field (110, 111). T h e atoms which have absorbed the momentum recoil, causing lattice vibrations, defects, or sputtering, depending on the momentum absorbed and the direction of recoil. For initial ion energies greater than -10 ev, there exists a significant probability that the original particles will come to rest in the solid. The trapped particles are then bound to the solid by forces that can be considered electrostatic. I n most cases the trapped particle possesses considerable potential energy (up to -10 ev) indicating that severe distortion of the lattice must occur (112). T h e time spent by the particle in the trapped state depends strongly on the height of the potential barrier which separates it from the gas phase, and on the temperature of the solid. Penetration depths of ions in solids have been measured by Young(113) for H+ and He+ in aluminum, and by Beliakova and Mittsev (114) for Li+ ions aluminum and gold. In both cases the transmission of ions of energy 1 kev and higher through thin evaporated films to cm) was observed. T h e results indicated that ions of 1 kev energy were a b k t o penetrate about ten lattice distances into aluminum. Depths obtained indirectly in this laboratory from ionic pumping onto thin evaporated titanium layers (115) are in general agreement with this result. For somewhat higher energies (45 kev), Nielsen (116)has derived range expressions from the general relations of Bohr (110) which he concludes are in faire agreement with the available data (109). Bredov and co-workers (117) have done a Monte Carlo type of calculation, based on Bohr’s potential, to find the depth distribution of ionically pumped potassium ions (4 kev) in germanium. They obtained a curve of approximately the form predicted by Nielsen, but with a characteristic cm (180 atomic layers). This is larger depth of approximately 5 x by about a factor of five than would have been predicted by Nielsen, but is still smaller than their experimental value obtained from etching the germanium target and measuring the (radio-active) potassium remaining in the target. Davies and Sims (117~2)have used similar techniques to obtain the depth distribution of ionically pumped alkali
3 60
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
metals in aluminum for ion energies in the range 0.7 kev to 60 kev. They have slightly modified Nielsen’s theory to account for the large energy decrement per collision, and obtain good agreement with their experiments. It is generally conceded ( 2 2 1 ) that the equation for the atomic potential field suggested by Bohr gives values which are too small at separation distances larger than about 0.5 A (i.e., energies less than about 1 kev). Further calculations of the type done by Nielsen, but using more accurate potentials would yield useful expressions for the penetration of low energy ions. Ion entrapment probabilities at low concentration have been reported for only a few gas-metal combinations. Varnerin and Carmichael (118) measured the ionic pumping speed and the target ion current to determine the “trapping efficiency’’ of positive helium ions on molybdenum at energies of 150 ev to 2600 ev. Similar measurements were made by Buritz and Varnerin (129) for the combination A+ -+ Mo. Carter and Leck (120) studied the trapping of inert gas ions in glass by measuring the quantity of gas re-evolved during subsequent heating of the target to 350°C.They used ion energies of 80 ev to approximately 1000 ev, and measured the desorbing gas with a mass spectrometer. I n another paper (121) the same authors describe similar pumping and desorption experiments on metal ribbons. They found that the total amount of gas recovered, while dependent on the number and energy of the bombarding ions was “essentially the same” for the metals nickel, tungsten, molybdenum, platinum, and aluminum. I n both of these papers, however, the number of ions used in bombardment (> 1014/cm2)was too large to allow the trapping probabilities for low coverage to be determined. A similar technique has been used in this laboratory to measure the entrapment probability of positive argon ions on tungsten between 80 ev and 4.0 kev (122). Except for the 80 ev case, the number of bombarding ions was < 1013/cm2which was found to ensure proportionality between number desorbed and number incident. T h e results of these measurements and some of those mentioned above are summarized in Fig. 10. T h e most notable features are: the fairly distinct energy threshold at -100 ev for A+ + W, the almost energy-independent region above -1.5 kev, and the close similarity of the A+ + Mo data of Buritz (119) with that for A+ -+ W. Of considerable interest in the application of entrapment to pumping is the maximum number of particles which can be trapped per unit target area, As the concentration of trapped atoms in the target increases, the pumping speed is observed to decrease, falling to very nearly zero at some maximum trapped concentration. It is probable that this saturation effect, rather than involving the reflection of incoming ions, is
361
ULTRAHIGH VACUUM
the result of an equilibrium in which the ion releases on the average one previously pumped atom by sputtering or direct ejection. Convincing measurements of such replacement phenomena have been made by
10
*
10
103
10
ION ENERGY ( e v )
FIG. 10. The probability of entrapment of inert gas ions at metal surfaces as a function of ion energy. Reference
-0-0-0
A+ -+W
-0-0-0
He+ + Mo
-x-x-x
A+ + M o
122 I 18 I19
Carmichael and Trendelenburg (123) using mass spectrometric techniques. They induced the re-emission of one ionically pumped inert gas from a nickel target by bombardment with ions of another, Figure 1 I , taken from their paper, shows the number of atoms of a trapped gas which are re-emitted (An,) when no trapped atoms of this gas are bombarded with 5 x I O I s ions of a second gas. In all cases the ion energies were spread over the range 40 to 180 ev. This same replacement or exchange process was observed qualitatively by Schwarz (124) as early as 1944. Bills and Carleton (125) measured pumping saturation in a normally operating ionization gauge for the gases nitrogen and oxygen, and
362
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
found that about 2 x Torr liter of gas was required for saturation (about 3 x IOI5 molecules/cm2 of bulb surface). They developed a theory based on a fixed number of adsorption sites and assigned sticking
FIG. 11. The ion induced re-emission of ionically pumped inert gases [after J. H. Carrnichael and E. A. Trendelenburg, J . Appl. Phys. 29, 1570 (1958)l.
probabilities to particles striking occupied and unoccupied sites. Reasonable agreement with their experimental data resulted. T h e data of Carter (120) and of Leck (121) give saturation numbers of about 1014/cm2to lOI5/cm2on metals for argon ions of energies 400 ev to 5000 ev. Saturation values for a number of materials and ions in the energy range 5 kv to 65 kv have been reported by AlmCn and Bruce (126). A theory of ionic pump saturation based on simultaneous penetration, diffusion and sputtering has been given by Kuchai and Rodin (127). These authors make the following simple assumptions: all ions penetrate to the same depth; the diffusion coefficient of the pumped particles is independent of coordinates and concentration; and because of diffusion, the maximum allowable concentrations are never approached. (This assumption is made implicitly.)
363
ULTRAHIGH VACUUM
They then solve a “Fick’s law’’ diffusion equation to obtain the steadystate depth distribution and the total amount pumped, in terms of the diffusion coefficient, the penetration depth, the bombarding flux, and the sputtering rate. While their particular assumptions may require modification, the general form of their resulting expressions should provide valuable information on the saturation characteristics of ionic pumps. Predictions of the spontaneous re-emission rate could be obtained from the same formulation by deriving an expression for the concentration gradient at the metal surface, although this is not done by these authors. Carter et al. (227a) have developed a theory which allows them to derive a penetration probability function from the saturation characteristic of an ionic pump. This model takes no account of diffusion of the pumped gas, but rather considers the saturation to be due to sputtering alone. The spontaneous re-emission of previously pumped gas begins to affect the performance of an ionic pump before saturation of the pumping action. Varnerin and Carmichael (118) first reported quantitative data on re-emission probabilities of ionically pumped atoms. They observed the time variation of pressure in a closed volume during the ionic pumping of helium into a molybdenum target. Such pump-down curves have been observed by many workers and are of the form shown in
W
i g P
W
t la
-1 W
a
I0
o;
30 40 TIME WIN)
50
FIG.12. The variation of pressure with time in a sealed I-liter system when a coldcathode discharge is turned on ( t = 0) in the presence of a pressure P,, (approximately 10-4 Torr) of argon.
364
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
Fig. 12. By interrupting the pumping, Varnerin and Carmichael were able to show that for large t, where dp/dt had become small, the pressure was determined by an equilibrium between (a) a slowly decreasing gas source proportional to the total amount of gas pumped, and (b) the original pumping speed. The source was attributed to spontaneous re-emission of the gas trapped earlier in the pump down. A calculation ) the relation, of re-emission probabilities ~ ( tyielded 1 < t < lo3 min (20) where t was measured from the moment of pumping. This re-emission work was extended by Carmichael and Knoll (128) to the four lightest inert gases on targets of nickel and molybdenum. In every case they found Eq. (20) to hold for t > 40 sec. They give values of “k” ranging from 2 x lop3 to 0.15. More recently, Fox and Knoll (129) found, by a “pulse pumping” technique, that for the combination A+ -+ Mo, Eq. (20) was valid for t > 1.5 sec. It is clear, however, from integration, that Eq. (20) cannot hold rigorously to zero time nor to infinite time: in each case an infinite amount of gas would be re-emitted. The limits of applicability of the equation in time are as yet unknown. The physical basis of the observed decreasing n(t) is also uncertain. It is not consistent with desorption from states of a single energy. Diffusion of ionically pumped inert gases does take place within metals (130), even though these gases do not spontaneously permeate metals (85). However, if diffusion alone is to account for the change of re-emission with time, a specific form of depth distribution, valid for all gases, targets and energies, must be postulated. Experiments in this laboratory using a “desorption spectrometer’’ technique (1.31) have shown ionically pumped argon to be bound in rigorously clean tungsten with several preferred energies. Desorption spectra for various bombarding energies are shown in Fig. 13 (122). Assuming that Eq. (8) is applicable, the calculated desorption energies of the peaks in the figure range from 40 kcal/mole to 110 kcal/mole. T h e mean lifetimes of atoms bound with these energies should be extremely long (> IO*’sec) and the re-emission rates completely negligible. A small fraction of the gas must exist in sites of much lower desorption energy to account for the usually observed re-emission. It is not known to what extent the presence of adsorbed gases on the target affects the phenomena of trapping and re-emission. For low energies such as were used in the reemission and particularly the induced desorption work described (118, 123) the ion penetration depths are expected to be only a very few lattice spacings, and variation with the amount and type of gas adsorbed would not be surprising. ~ ( t= ) kt-1
365
ULTRAHIGH VACUUM
T h e application of ionic entrapment to u-h-v pumping is discussed in Section 111, A, 3. Ionic pumps have the cardinal advantage of containing no fluids which might be sources of gas. Their main shortcoming is the limited amount of gas which they can remove. When pumping action is
i S ii'
Y)
-
c
c rJ
) .
=e
-e 0
i" i il'
0
x X
0
W
X
250 e v
z 0
I-
150 e v
z
a 0 cn
100 e v
9 -
W
+ a a
z
P n
G a
ev
k a
ev
W 0
ev
W
n
0 I TEMPERATURE
2 (OK
3
x 10-3)
is i i d
2 Ea X
z
0
! i a 0
v)
W
n
1
0
1
I
2
3
FIG. 13. Desorption rate spectra of ionically pumped argon from clean tungsten for various ion energies. The rate of temperature rise was SO"K/sec [after E. V. Kornelsen, Trans. Natl. Symposium on Vncrrzrm Tech. 8 , 281 (1961).]
366
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
undesirable, as for example in ionization gauges, Fig. 10 shows that with proper choice of ion energy and target material, trapping probabilities can be drastically reduced.
2. Sputtering. Sputtering is the ejection of atoms of a target from its surface upon the impact of positive ions. This transfer of solid material from one place to another within the system affects the vacuum in the following ways: (1) It removes chemisorbed layers from the target. (2) It releases gas atoms previously dissolved or trapped in the target. ( 3 ) It forms, in other parts of the system, metal films capable of chemisorbing active gases and burying trapped gases. The extent to which the vacuum conditions are altered will depend on the sputtering rate, the gas composition and the previous history of the target. For recent developments in this field, the reader is referred to a comprehensive series of papers by Wehner and co-workers dealing with low energy sputtering (132-135). Some work has also been published recently on sputtering by ions of higher energy (136-138, 126). At low ion energies (< 1000 ev) it seems well established that the sputtering mechanism involves momentum transfer between individual atoms (132). The most important parameters governing threshold and yield (atomslion) are the mass ratio of ion to target atom, the elastic constants and sublimation energy of the target, and the type of crystal lattice (133). During the sputtering of metal single crystals, atoms are ejected with strong preference along directions of closest atomic spacing in the lattice (239, 136) in which direction the most efficient momentum transfer would be expected to occur. For higher ion energies, it is thought that evaporation from small regions of intense local heating may also contribute to the sputtering (140). Using the most sensitive detection method, Stuart and Wehner (135) have found that sputtering threshold energies lie in the range 15 ev to 45 ev. They find that, with few exceptions, the product of threshold energy and momentum transfer coefficient is a constant:
where Et is the threshold energy in ev, mi is the mass of the ion, and mt is the mass of the target atom. As the energy is increased above about 100 ev sputtering yields (atomslion) increase with gradually decreasing slope to a maximum at energies in the range 10-100 kev. For
ULTRAHIGH VACUUM
367
300 ev normally incident A+ ions the yields for 28 pure metals fall between the limits 0.3 (for Si and Re) and 2.2 (for Ag) (234). Yields for glasses do not appear to be available, but Hines and Wallor (241) report, for vitreous silica under Xef ion bombardment, a yield of 0.5 at 20 kev and a peak yield of 2.0 at -45 kev. These yields are considerably lower (about a factor ten) than would be expected from metal targets. T h e peaks in the yield curves for a given metal target seem to occur at approximately constant ion velocity but are higher (in atoms/ion) for the heavier ions. Thus, for example, on a copper target (237) the sputtering yields for positive argon ions reaches a broad maximum of about 9 at 30 to 50 kev, while for positive helium ions a maximum yield of 0.20 occurs at approximately 16 kev. It is generally agreed that the presence of adsorbed gases on the target can strongly influence sputtering yields even at energies of several kev (132, 137). For accurate sputtering measurements the sputtering rate must therefore greatly exceed the rate of adsorption of atoms on the target surface. Either high current densities (as used by Wehner) or very low active gas pressure (142) must be employed. T h e removal of chemisorbed gases or “cleaning” of a surface by sputtering has been used with some success. Farnsworth and co-workers (143, 144) have produced clean silicon and germanium surfaces by sputtering with argon ions. T h e equivalence of cleaning by sputtering and high-temperature outgassing was recently demonstrated by Hagstrum and d’Amico (84) using the secondary electron ejection yield of tungsten under helium ion bombardment. By sputtering it is possible to clean surfaces which are covered by a chemisorbed layer that cannot be thermally desorbed below the melting point of the metal. T h e release of previously trapped or dissolved atoms by sputtering can cause serious gas sources and the saturation of ionic pumps. These effects were briefly described earlier (see Section 11, D, 1).
3. Secondary Effects. A number of other impact phenomena can affect the detection of ions in vacuum systems even though they exert no strong influence on the vacuum conditions. One of the most important such effects, the ejection of electrons from solids by positive ions, has received considerable attention. A careful examination of Auger or “potential” ejection by noble gas ions has been carried out by Hagstrum (145, 246, 247) and the reader is referred to his excellent series of papers for quantitative data. T h e ejection yields were found to vary only slightly with energy in the range 0-1 kev (for clean metals) and decreased with decreasing ionization potential. Thus, on clean tungsten, the yields dropped from approximately 0.25 for positive
368
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
helium ions to approximately 0.02 for positive xenon ions. For ions whose ionization potential is less than twice the target work function, Auger ejection cannot occur. Kinetic electron ejection is still possible, but for energies < 2 kev the yields are generally much lower than for potential ejection, and are strongly energy-dependent (248, 249). A great many papers have been published in which the state of cleanliness of the target surface is either definitely bad or uncertain [see, for example, the review paper of Little (ISO)].Ejection yields are extremely sensitive to the presence of adsorbed gases (146) and even for energies in the Mev range the sensitivity is still evident (152). Auger yields are in general lower for gas covered surfaces than for clean ones and show a different energy dependence. T h e ion reflection which occurs at metal surfaces also shows a strong dependence on the relative values of ionization potential V , and work functionrj. For V , > 24, as is the case for noble gas ions, Auger neutralization of the ion occurs with high probability unless the energy is very large, and reflection coefficients are correspondingly small (152). For alkali metal ions, on the other hand, V, < 24, and reflection coefficients rise sharply with decreasing energy below 1 kev (253). Brief mention should be made of the production of secondary ions by positive ion impact. Such effects have been reported by Bradley et al. (154) and by Stanton (255) who used mass spectrometric methods to study secondary positive ions. In general these effects do not occur with large probability and do not often cause serious measurement errors. All the secondary effects discussed above show radical dependence on the presence of adsorbed gases, making a large fraction of early experiments of questionable quantitative value. Only within the past decade have proper vacuum techniques, in particular, u-h-v techniques, been applied to the study of ion effects at truly clean surfaces. Bills (156, 257) has reported results in which alkali ions were produced during the flashing of a filament in an u-h-v system after the system glassware had been subjected to heating or charged particle bombardment. The source of these ions was assigned to decomposition of the glass and subsequent transfer of the decomposition products to the filament. Bills points out the possible errors that may arise in physical electronics measurements as a result of these effects. Further details on glass decomposition are given by Donaldson (257~).
F . Electron Interactions Ionization of gas molecules by electrons, emission of soft X-rays by electron impact on surfaces, secondary electron emission, the elastic
ULTRAHIGH VACUUM
,
3 69
reflection of primary electrons, and the modification of work function by absorbed layers of gas, are all important in u-h-v systems, but these phenomena are not special to u-h-v systems and have been treated extensively in other publications. A bibliography of these subjects has been compiled by Nottingham (158). In this article we examine these phenomena only where they play an important role in a particular u-h-v instrument. There is, however, one general field of electronic interactions in which little work appears to have been done, but which is of direct interest in u-h-v. This is the evolution of gas by electron bombardment of a surface. I n a mass spectrometer, Moore (159) has studied the fragments emitted when electrons bombard carbon monoxide adsorbed on molybdenum and tungsten. T h e dominant fragment emitted is the positive oxygen ion, and the threshold incident energy for this process is about 17.5 ev. Moore also found other fragments (particularly with new filaments) wich were independent of the carbon monoxide. T h e time-dependence of the positive oxygen peak under various bombarding conditions was studied, and it was found that this peak reached a maximum near monolayer coverage of CO and decreased thereafter. The reactions investigated must presumably take place at the grid of every BayardAlpert gauge, and data on other gases would appear to be of importance. Reynolds (160) has reported multiply charged ions of tungsten and mercury in a mass spectrometer, which he attributed to bombardment of the walls of the ion source by secondary electrons. Jacob (161) describes the bombardment with electrons of barium on an anode and the resulting physical transfer of the barium to the cathode. T h e threshold for the reaction was 300 ev primary energy. Todd et al. (162) have studied the outgassing caused by electron bombardment of glass in a television tube with 20 kev electrons. Ninety-five per cent of the evolved gas was oxygen for five types of glass examined.
G. Photo and Chemical Reactions When an adsorbed layer of gas is irradiated, three processes may occur, viz, (a) photodesorption of the gas molecules, (b) photosorption of gas under the influence of light adsorbed by the solid, and (c) photodecomposition of the adsorbed gas. A review of Russian work on these photoeffects has been published by Terenin (163). T h e effect of principal interest in u-h-v systems is the photodesorption and decomposition of gases from metals and glass, these processes are only significant in u-h-v systems where a reasonably high flux of ultraviolet radiation exists. Valnev (164) has shown that carbon monoxide is photodesorbed from
3 70
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
nickel by the action of light of wavelength less than 240 mp. Valnev also showed that water vapor is photodesorbed from cadmium and zinc at wavelengths less than 250 mp, only 50% of the desorbed gas was condensable at - 180°C, indicating that photodecomposition of the water was occurring. Data on the photodesorption of gases from glass are very scanty (1642). Kenty (165) has shown that irradiation of Pyrex glass by light of wavelength less than 300 m p caused photodecomposition of water and carbon dioxide in the body of the glass, the hydrogen and carbon monoxide so formed then diffused out of the glass. This process is particularly noticeable in mercury arc lamps. Experiments in these laboratories have shown that hydrogen and carbon monoxide are desorbed from Pyrex glass surfaces when illuminated by a xenon flash lamp, in this case the gas was desorbed from the surface only, since the flash was too short (approximately 1 msec) for diffusion to occur (see Fig. 9a). T h e threshold wavelength for this effect was found to be about 270 mp. Interactions in which one gas is converted to another will be referred to as “chemical” reactions. The predominant chemical effects occur when the gas is exposed to an incandescent filament. Hydrogen, oxygen, water, and some hydrocarbons are dissociated at a hot tungsten filament, the dissociated fragments are extremely reactive, and may react with impurities in the filament or with other surfaces of the system to produce gaseous products. Hickmott (166) has studied the interaction with a glass surface of atomic hydrogen, formed at an incandescent filament. It was found that carbon monoxide, water, and methane were formed in a system containing hydrogen and a tungsten filament operated above 1000°K. As an example, in a system initially filled to 1.5 x lo-’ Torr of hydrogen the partial pressure of carbon monoxide was about 1.4 x Torr with the filament cold, with the tungsten filament heated to 2000°K the CO partial pressure had risen to about 3 x lo-* Torr. Becker (167) has shown that if the carbon impurities in the tungsten are reduced by prolonged heating of the tungsten in oxygen, the amount of carbon monoxide produced in a hydrogen atmosphere is greatly reduced. T h u s the dominant source of carbon must be the impurities in the filament while the oxygen in the carbon monoxide and water come from the glass. A modified water cycle may be involved in which the atomic hydrogen reacts with the glass to produce water, the water then decomposes on the hot tungsten to form carbon monoxide with carbon impurities on the tungsten surface. It is also possible that the atomic hydrogen releases some carbon monoxide directly from the glass. It has been observed in these laboratories that whenever two glass surfaces in an
ULTRAHIGH VACUUM
371
u-h-v system are rubbed together, carbon monoxide is evolved. This indicates the presence of an adsorbed layer of carbon monoxide on the glass surface which can be released by “tribodesorption” and which may also be evolved by the action of atomic hydrogen. Young (168) has examined the reaction of oxygen with hot filaments of tungsten, rhenium and molybdenum, and found that carbon monoxide and carbon dioxide are formed by interaction with the carbon impurity in the filaments. All three metals produced carbon monoxide and carbon dioxide in almost identical amounts. When the system was filled to lop6 Torr with oxygen and a tungsten filament heated to 2000°K the partial pressure of CO increased to 1.2 x lo-’ Torr and the CO, pressure to 6 x Torr. Before heating the tungsten filament the pressures of CO and CO, were extremely small. T h e glass walls were shielded from the hot filaments by a tantalum cylinder. When the shield was removed no change in the amount of carbon monoxide and carbon dioxide was observed, thus the source of carbon was established as the impurity in the filaments. Rhodin and Rovner (169) have studied the interaction of oxygen with a hot tungsten filament. Becker (167) has examined the reactions of oxygen with tungsten in detail. Water vapor reacts with a hot cathode to form hydrogen, carbon dioxide, carbon monoxide, and methane, it also reacts with a barium getter to give hydrogen, methane, and higher hydrocarbons (55). These chemical reactions which occur at hot metal filaments can cause drastic changes in the active gas composition in an u-h-v system. T h e problems raised by these effects in the measurement of pressure with hot-filament gauges will be discussed in Section 111, C.
111. TECHNOLOGY OF ULTRAHIGH VACUUM The apparatus and techniques for the production and measurement of u-h-v will be described in this section.
A . Pumps
I. Diffusion and Molecular-Drag Pumps. Oil and mercury diffusion pumps are widely used for the production of u-h-v. No special design of pump is necessary to achieve u-h-v; an adequate trapping system is however essential to prevent back-streaming of the pump fluid into the system. T h e choice of a diffusion pump for any particular u-h-v application is controlled by two requirements: (a) adequate pumping speed for the application, and (b) minimum rate of back-streaming of the pump fluid. Large systems, requiring very high pumping speeds, are usually
372
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
pumped with diffusion pumps, sometimes in combination with cryopumps (see Section 111, A, 4). Systems of a few liters volume can equally well be pumped by diffusion pumps, but for many applications getter-ion pumps may be more convenient (see Section 111, A, 3) and avoid the problem of contamination from pump fluids. In general it is possible to make the compression ratio of a diffusion pump sufficiently high that the ultimate pressure is not determined by diffusion from the fore-vacuum back through the vapor jet. Typically, the compression ratio is lo4 for hydrogen and 10l2for nitrogen in an oil diffusion pump. Thus the ultimate pressure that can be obtained in practice is limited by the outgassing rate of the system and the vapor pressure of the pump fluid or any of its breakdown products. Mercury diffusion pumps have three principal advantages for u-h-v use; (a) the pump fluid is very easily trapped at liquid nitrogen temperatures and if the trap is inadvertently allowed to warm up, the mercury can be readily removed from the system by baking, (b) the mercury pump will operate against a high backing pressure, permitting the backing pump to be turned off for long periods, and (c) the fluid is stable, i.e., there are no products produced by thermal breakdown. Pump designs will not be discussed here since pumps used at high vacuum are quite suitable for u-h-v, and have been described frequently elsewhere (see for example Pollard, 9). The use of mercury diffusion pumps to achieve pressures as low as 10-l2 Torr has been described by Venema (4). Mercury diffusion pumps are to be preferred over oil diffusion pumps when it is essential to avoid contamination from hydrocarbon vapors. Oil diffusion pumps are more widely used than mercury pumps because (a) the pump fluid is less dangerous, and (b) recently developed traps for oil vapor (copper foil or molecular sieve traps) do not require refrigeration. As in the case of mercury pumps, no special design of oil pump is necessary for u-h-v, it is however advisable to use a pump with the lowest possible backstreaming of the pump oil to reduce the trapping problem. Smith (Z70)has considered the thermodynamics of the processes occurring in the pump fluid of an oil diffusion pump and has obtained considerable improvement in diffusion pump performance by superheating the vapor in the stacks, I t is suggested that backstreaming would also be reduced by this technique but no experimental data has been reported. Dreyer (171) has reported measurements of the backstreaming rate of several oil and mercury pumps of 25 cm diameter. T h e design of high speed oil pumps, where a low backstreaming rate has been obtained by adding a cooled shield around the high vacuum jet, has been described by Hablanian (172).
ULTRAHIGH VACUUM
373
Hickman (173) has recently investigated polyphenyl ethers as pump fluids. These compounds have very low vapor pressures at room temperature and it is expected that the surface migration rates of these materials should be much smaller than for conventional pump oils. Hickman claims pressures of about 5 x 10-lo Torr with an untrapped, three-stage glass diffusion pump without baking the system. Molecular-drag pumps have not been widely used in u-h-v systems; however, some new designs show considerable potential for u-h-v applications. Becker (274) has described a molecular-drag pump which is capable of achieving an ultimate pressure of 5 x 10-10 Torr without trapping. T h e residual gas was mostly hydrogen because of the low compression ratio of a molecular-drag pump for gases of low molecular weight. This pump resembles an axial-flow turbine using nineteen stages of flat plate blades and a rotor-tip speed of 145 meters/sec; the spacing between rotors and stators has the very large value of 1 mm. T h e speed of the pump is 140 liters/sec for air and 170 liters/sec for hydrogen. T h e apparent pumping speed for hydrogen decreases at low pressures (below lo-' Torr) because the compression ratio for hydrogen is only 250. ) T h e compression ratio for air is as high as 5 x 10'. Becker ( 1 7 5 ~ has developed the theory of these pumps and Pupp (275b)has evaluated the commercial pump. A design of bakable molecular pump using a magnetically suspended rotor has been described by Williams and Beams (176). A steel rotor is freely suspended by the axial magnetic field of a solenoid. A sensing coil is used to regulate the solenoid current so as to maintain the desired vertical position of the rotor, T h e rotor is spun by a rotating magnetic field produced by the drive coils, at low pressures the rotor will coast for many days without the drive. Pumping takes place between the upper surface of the rotor and a spiral groove in the stator. A pump with a rotor diameter of 23 cm was operated at 300 revolutions/sec (peripheral speed, 220 meter/sec) and gave a compression ratio of lo2 at input pressures of 6 x lo-* Torr with clearances as large as 0.5 mm. T h e limitation on the clearance of rotor and stator was caused by vibration of the building and supports. T h e pumping speed was 6-10 liter /second. T h e lowest pressure obtained was 4 x 10-lo Torr. T h e biggest advantage of diffusion and molecular-drag pumps over getter-ion pumps lies in their ability to pump inert gases with speeds commensurate with the speed of pumping active gases. The pumping speed of getter-ion pumps for inert gases is always considerably lower than their speed for chemically active gases. T h e speed of diffusion pumps increases with decreasing molecular weight of the gas, which is
374
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
particularly advantageous for u-h-v applications where the principal residual gases are frequently of low molecular weight (hydrogen and helium). T h e compression ratio of molecular-drag pumps decreases with decreasing molecular weight, thus these pumps are at a disadvantage in most u-h-v applications.
2. Getters. I n many u-h-v systems it often becomes practical to pump by transferring the gases to the solid surfaces of the system rather than to the external atmosphere. Pumps can then take extremely simple forms (a getter, an ionic pump, or a cold surface), and problems of the evolution of pump fluids into the system are completely avoided. Since the pumped gas remains within the system, however, the problems of saturation and re-evolution become of prime importance in determining the range of applicability of these pumping methods. I n this section the use of evaporated films as getters in the production of u-h-v will be considered. Only a limited number of getter materials has been used to any extent in the production of u-h-v and these are all pure metals (Ti, Zr, Ta, Mo, W). There does not appear to have been any thorough investigation to determine the getter materials most suitable for u-h-v. T h e materials listed above have been found adequate, by various experimenters, to produce u-h-v in their particular systems. A review of the properties of various getters has been given by Wagener (177). Titanium, in the form of evaporated or sputtered films, has been found to be a very convenient getter for pumping the chemically active gases in u-h-v systems. Klopfer (178) has measured the maximum capacity of evaporated titanium films for various gases at 20°C (see Table XII). TABLE XII. GETTER CAPACITY IN
ToRR-LITERs/mg X
T h e adsorption of air and oxygen by evaporated films of titanium and zirconium has been studied by Zdanuk (179). Luckert (180) has studied the physical and chemical adsorption of oxygen and nitrogen on ) studied the formation of evaporated titanium films. Holland ( 1 8 0 ~has hydrocarbons from carbon and hydrogen impurities in titanium getters. Stout has measured the characteristics of bulk titanium metal as a getter (181) and showed that, above 70O0C,0,, N,, and CO, were rapidly adsorbed by titanium metal. Hydrogen was adsorbed in the range 25 to 400°C, and released at higher temperatures. Hydrogen was
ULTRAHIGH VACUUM
375
the only gas found to be released on heating the titanium. Gibbons (182) has reported on the gettering properties of alloys of titanium and zirconium in bulk form. These alloys are capable of dissolving any oxide film and sorbing hydrogen at temperatures below 400°C. Evaporated films of molybdenum and tantalum have also been used successfully in u-h-v systems. Milleron (183) has evaporated molybdenum films by electron bombardment of a molybdenum wire and obtained pumping speeds to hydrogen greater than lo4 liters/sec with a film area of about 1.5 x lo5 cm2. Measurement of the pumping speed of evaporated molybdenum films (produced by heating a molybdenum filament) have been made by Hunt ( 8 3 ) . Maximum speeds to hydrogen of about 4 x lo4literslsec were observed with a film area of 3 x lo4cm,. Pressures of about Torr could be obtained in an unbaked stainless steel system of 85 liters volume by means of the molybdenum getter. There is insufficient experimental data to permit a decision on the most suitable getter material for any specific u-h-v application. Titanium has been proved adequate for most purposes ; the principal disadvantage of titanium is its behavior with hydrogen, i.e., at temperatures about 500°C hydrogen is re-evolved from titanium.
3. Ionic Pumps. This section will be concerned with pumps which employ electrical excitation of the gas to enhance its transfer to the solid. In addition to the ionic entrapment discussed in Section 11, E, I, another phenomenon can play an important role in practical ionic pumps; the electrons which produce the ions can at the same time produce considerable numbers of excited neutral particles which may (particularly in the case of the active gases N,, 0,, H,, etc.) be able to adsorb when neutral molecules would not (184, 185). T h e performance of ionic pumps is further modified by sputtering (Section 11, E, 2) and in some cases by deliberate evaporation of metal onto the pumping surface. T h e terms “electrical clean-up” or “getter-ion pumping” have been applied to pumps which utilize various combinations of the phenomena mentioned above. We first discuss pumps in which deliberate sputtering and evaporation are absent, and then proceed to those in which these processes play dominant roles. T h e production of u-h-v by the use of an ionization gauge as the major pump was reported by Alpert (186, see also 90) in his early definitive work in this field. By ionically pumping the atmospheric helium which permeated the glass envelope, he was able to maintain a Torr even though the gauge pumping speed for pressure of about helium was only approximately lo-, liter/sec.
376
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
For inert gases the total pumping speed is a measure of the ionic pumping speed because chemical effects are then absent. In a BayardAlpert gauge most of the ion-pumping occurs at the envelope; ion current to it being 5 to 10 times the collector current. An evaporated film of metal on the glass is necessary for appreciable pumping of helium to occur (187, 188). Such a film is usually produced during outgassing of the grid structure. Young (188) has measured a maximum liter/second for helium in a Bayard-Alpert pumping speed of 4 x gauge with a grid voltage, V , = 145 volts and electron current, 1- = 10 ma. Hobson (189) has measured the ionic-pumping speed of nitrogen in a Bayard-Alpert gauge and finds an initial speed of 0.25 liter/sec (at V g = 250, 1- = 8 ma) which remains constant until lo1' molecules have been pumped, and then decreases rapidly. Cobic et al. (189a) have reported extensive data on the ionic pumping of both inert and active gases in a Bayard-Alpert gauge. Re-emission and saturation characteristics (see Section 11, E, I ) directly determine how much gas can be pumped by an ionization gauge before the vacuum conditions begin to deteriorate. T h e maximum number of molecules N s which can be pumped by a gauge will depend on the type of gas, the material and area of the pumping surface, and the ion energy. Some measurements of pumping saturation are summarized in Table XIII. Normally pumping speeds are not seriously reduced from their maximum value until the number of molecules pumped is approximately 0.1 Ns. Re-emission effects are difficult to assess because of the strong dependence of the re-emission probability on the time since pumping took place. For the simple case of pumping away a known quantity of gas ( N molecules) in one relatively short interval, the system pressure (in the absence of other gas sources) at a later time t can be predicted from the equation, S N dP -- - vp _ + -kKt-' dt no v
in which the last term represents a rate of re-emission of the pumped gas and is obtained from Eq. (20). When gas is pumped continuously or over some extended period of time, account must be taken of the contribution to the re-emission at time t of pumping at all earlier times. In general, numerical methods would be required to carry out such calculations, although certain special cases can be solved analytically. For the maximum gas input rate likely to occur due to re-emission, let us consider the rapid pumping of a quantity of gas by an ionization gauge. If the t-l law [see Eq. (20)]holds, and K (a typical value),
TABLE XIII.
Authors
Type of gauge
Bills and Carleton (125)
BAG (WL5966)
Brown and Leck (190)
cylindrical anode Penning
Carter and Leck (120)
BAG
Kornelsen
IMG
(115)
(IMP)
Pumping area (cm')
-
200
Material of pumping surface glass W on glass
Maximum ion energy (ev)
Ns, Saturation number (molecules x lo-'') ~
He
Ne
A
Kr
Xe
150
Nz -60
-60
2.2
1.o
-6
-
4Ooo
10
4.0
2.3
0.17
glass
250
> 10
2.0
2.0
2.0
titanium
6Ooo
A1
200
26
75
2.0
4.0
w
4 4
378
P . A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
Torr-liter (Le., a number of molecules then 100 sec after pumping N = 3.3 x 1016) the re-emission rate would be r ( t ) = khijt =
3.3 x 1012 molecules/sec, or 10-7 'I'orr-liter/sec
(23)
I n contrast, the atmospheric helium permeation rate in a one-liter Pyrex system is typically 10-l2Torr-liter/sec. If the available pumping speed is a constant S = V/r,the course of the system pressure with time in the above example will be given by
P
=*--
kN
t n,V
=-
St
fort>r
Torr
where T is the pumping time-constant of the system (sec). When the total quantity of gas to be pumped ionically exceeds the saturation amount (see Table XIII) modifications of the pumping method are necessary. One of the simplest methods of increasing the capacity of an ionic pump is to deliberately evaporate, when required, a fresh layer of metal onto the pumping surface. This provides new sites in which incoming ions can be trapped and at the same time greatly reduces the rate of re-emission of previously pumped gas. The total capacity of the pump should then depend only on the amount of metal available for evaporation. Alpert (292) has briefly described titanium evaporation in a Bayard-Alpert gauge type of structure for the pumping of nitrogen. T h e high pumping speeds achieved in his experiment (up to 20 liters/sec) indicate that electrical excitation of the gas, other than ionization, contributes significantly to the pumping action. A more decisive demonstration of such "activated" pumping has been given by Holland (185) who pumped oxygen onto an evaporated titanium getter which had been saturated with unactivated 0,. Similar observations with nitrogen have been made in this laboratory using a cold-cathode discharge and a titanium getter. Increased capacity for helium and argon has been obtained in this laboratory (ZZ5) using titanium evaporation within an inverted-magnetron gauge (47). The pump is shown schematically in Fig. 14. It was found that 7.5 x 10l8helium atoms or 7.5 x 10'' argon atoms could be pumped for each milligram of titanium evaporated. This implies that the number of evaporated titanium atoms per gas atom pumped was only 2.0 for helium and 20 for argon. Pumps of this design used recently in this laboratory have had approximately 30 mg of
AMOUNT PUMPED (TORR LITER)
FIG. 14. Schematic diagram of inverted-magnetron pump.
lo-'
I 1
I
-I W
2
ZE 10-9
a
W
n
cn cn
3
w
OZ
0
-n n -I-
10-8
10-i
-
10-6
IO-~
10- 4
HELIUM PRESSURE ATTAINABLE WITH 16 HR OF PUMPING
I O - ~
10-2
IMP
D
Voc-Ion TYPE (4 CELLS1
titanium available for evaporation and have pumped approximately 0.5 Torr-liter of argon, an increase of more than lo2 over the saturation number for the same gauge operated without metal evaporation. Perhaps of greater significance, the re-emission rate of the pumped atoms
380
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
was sufficiently low that a base pressure below 2 x 10-lo Torr could be maintained regardless of the amount pumped (up to 0.5 Torr-liter), even though the pumping speeds for helium and argon were only 0.03 and 0.25 liter/sec, respectively. A pump for the noble gases with very much higher speed and capacity has been described by Alexeff and Peterson (292). It is based on the “Evapor-Ion” principle first exploited by Herb (293). Using a magnetically confined electron flow to produce the ions, and titanium evaporation, speeds of 90 literslsec for helium and 260 liters/sec for argon at Torr have been achieved. While these speeds are much lower than those for the active gases in similar pumps (approximately lo4 literslsec), they are nevertheless among the highest entrapment pumping speeds yet reported for the noble gases. With appropriate modification and processing of the pump components, and thorough bakeout, there seems no reason to expect the performance of this type of pump to be inferior in the u-h-v range. During the past five years, extensive development work has been done on cold-cathode discharge pumps which rely on sputtering to avoid saturation of the pumping action (194, 195,196). These pumps are presently sold in a variety of sizes from fractions of a liter/sec to thousands of liters/sec. They all consist of parallel arrays of Penning discharge cells with titanium cathode plates. Ions created in the discharge strike the cathodes with several kev kinetic energy, causing sputtering of the titanium onto the electrodes. Generally speaking, inert gases are pumped by ionic entrapment at locations on the cathodes where there is a net accumulation of sputtered material, while chemically active gases are, in addition, pumped by chemisorption predominantly at the anode. A discussion of the main pumping mechanisms for various gases in such pumps has been given by Rutherford et al. (297). Rutherford concludes that penetration depths, sputtering efficiencies and chemical properties of the gas are involved in rather complex combination in the pumping action. Two modified “sputter-ion” pumps have been described recently (195, 196). These pumps do not show the pressure instabilities during argon pumping which were sometimes troublesome in the original type (294). T h e ultimate capacity of these “sputter-ion” pumps is limited only by the amount of titanium which is available for sputtering from the cathodes. In the nominally 10-literslsec pumps described by Hall (29#), the cathode plates weighed approximately 160 grams. Assuming that 30 grams of this is available for sputtering, the expected capacity might be of the order of lo3 times that of the “IMP” (115) in which 30 mg of titanium could be evaporated. Thus Hall’s pump should be able to
38 1
ULTRAHIGH VACUUM
remove as much as 500 Torr-liters of argon or 5000 Torr-litres of helium, assuming comparable ratios of gas to titanium atoms. Since the pumping speed per discharge cell does not vary greatly, this capacity should be proportional to the number of cells for other pumps of the same type. In the u-h-v range the performance of the pumps described above is fundamentally limited by the re-emission of a portion of the previously CATHODE SAPPHIRE ROD TITANIUM EVA PORATOR
GLASS BASE
I
\
2
3
CM SCALE
FIG. 15. T h e helium pressure obtained with two types of ionic pumps after 16 hr of pumping following the introduction of various amounts of gas.
pumped gas (see Section 11, E, I). It is useful to imagine two types of area on the cathode surface: one in which a net build-up of sputtered metal occurs (area A), and one in which a net excavation takes place (area B). I n area A, gas can be removed permanently through burial by subsequently sputtered metal. T h e ions trapped in area B, however, give rise to spontaneous re-emission, as discussed earlier (Section 111, A , 3)
382
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
and are also liable to be re-evolved by the sputtering action of later arriving ions. T h e dynamic equilibrium between the pumping action in area A, and the re-evolution in area B, results in an apparent “limiting pressure” which decreases only very slowly with time after a large amount of gas has been pumped. Similar arguments apply to more recent modified pumps (195, 196) although quantitatively they may show improvements. In addition, measurements in this laboratory have indicated that, for argon and helium, the ratio of discharge current to pressure in a typical diode “sputter-ion” pump decreases by about a factor of 10 as the pressure decreases from Torr. This Torr to has the effect of accentuating the apparent “limiting pressure” mentioned ) recently reported that the pumping speed above. Klopfer ( 1 9 6 ~ has does not decrease at low pressures if the cathode surfaces are kept free of adsorbed gas. It should be noted that this internal equilibrium limitation is independent of the pumping speed (i.e., of the number of cells in the pump), being determined only by the pumping and reemission characteristics of the individual pump cells. Figure 15 demonstrates the limiting effect in a four-cell diode pump for helium. The pressure obtained after 16 hr of pumping in a sealed system is plotted against the amount of helium removed by the pump near the start of the 16-hr interval. Similar results for an IMP (ZZ5) are included for comparison. The results for argon were qualitatively similar except that the upper plateau occurred at a pressure of approximately 3 x Torr, as compared with 2 x Torr for helium in Fig. 14. The significance of the pumping re-emission equilibrium in a “sputterion” pump becomes clearly evident if the pump voltage is turned off after pumping inert gases. The pump then immediately becomes the Torr-liter/second per pump cell of the source of approximately gas previously pumped. As mentioned earlier, helium premeation in a Pyrex glass u-h-v system (the dominant gas source) amounts to approximately 10-l2 Torr-literlsecond in a one-liter system. I n summary, pumps of the “Hall” type are extremely convenient for pumping systems of low or moderate gas through-put in the pressure range Torr to 10-s-10-9 Torr. For the most demanding u-h-v conditions (approximately 10-lo Torr), however, their performance can be considered adequate only when the amount of gas they are required to pump is extremely small (less than Torr liter/cell). Ionic pumps have also been described (198, 199), which operate by the more conventional method of transferring the gas from the system to the external atmosphere. Compared with diffusion pumps, the power consumed per unit pumping speed is extremely high. No attempt has yet been reported to use ionic pumps of this type in the u-h-v range,
ULTRAHIGH VACUUM
383
but such extension may prove feasible with a number of pumping stages in series, as is the common practice in diffusion pumps.
4. Traps and Cryogenic Pumps. Physical adsorption and/or condensation has two main applications in u-h-v systems: traps and pumps. Both rely upon the physical processes described in Section 11, B and the classification is made on the basis of application and design rather than upon the physical principles involved. A clear distinction between the two is not always possible. T h e principles of trap design are not special to u-h-v systems but care may be required for special problems such as surface migration of the adsorbate (200). An example of refrigeration trapping in the u-h-v range is provided by the work of Venema and Bandringa (4). These authors use three liquid nitrogen traps in series with a mercury diffusion pump to reach pressures in the range lo-” to 10-l2 Torr in a small glass system. They give reasons why more than one trap is necessary. Ullman (200) has described a liquid nitrogen trap used with an oil diffusion pump in a large u-h-v system (570 liters). This trap is designed to prevent surface migration along the warmer parts of the trap. Smith and Kennedy (201) describe mechanical refrigeration systems for high vacuum traps and baffles, Caswell (202,203) has used a liquid helium trap in an apparatus for thin film evaporation and finds that it is about 100 times faster than an evapor-ion pump. It is difficult to classify his helium trap as a trap or a pump. Caswell estimates the condensation coefficient of nitrogen and carbon monoxide on this trap as 0.2 to 0.3, which is lower than the results of Section 11, B, I suggest. However, Caswell’s incident molecules come from a source which is hotter (1400°C) than is normal. Henderson et al. (204) have discussed refrigerated trap design in the u-h-v system of the Model C Stellarator. T h e volume of this system was 1500 liters and pressures in the 1O-lo Torr range were achieved with oil diffusion pumps and liquid nitrogen traps. Milleron (205) has achieved a base pressure below 1 x 10-lo Torr in a metal system of volume 70 liters, using liquid nitrogen traps and oil diffusion pump. Simons (206) reports a base pressure of approximately Torr in a volume of 1100 liters, with oil diffusion pumps and liquid nitrogen traps. Metal foil traps (207) are effective in trapping oil at room temperature and maintain their effectiveness for considerable periods (208, 209). Small copper-foil traps have a “stay-down” time of about 15 days. T h e stay-down time is the useful life of the trap before the surface of the trap becomes saturated and the pressure in the system rises. Traps employing molecular sieves (zeolite or activated alumina) have a con-
384
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
siderably longer stay-down time than metal foil traps. Biondi (210, 211) has shown that small glass traps using molecular sieves have a stay-down time of more than 100 days and that large metal traps are effective for periods ranging from 20 to 100 days. Goerz (212) has described a molecular sieve trap with an internal heater. The ion trap described by Haefer (212a) has not been used in u-h-v but might find application in this range because of its similarity to an ion pump (Section 111, A , 3). Thus there is no doubt that a variety of trap systems exist with which Torr may be achieved with diffusion pumps, u-h-v pressures of The room temperature traps exhibit saturation with time which is a qualitative result to be expected from the discussion of adsorption isotherms given in Section 11, B, 2. Refrigerated traps, in principle, also exhibit saturation properties, but the vapor pressure (po) of the component being trapped may be so low at the temperature of the trap that the effects are not observed. A feature of traps that becomes increasingly important in the u-h-v range is that they may hold relatively large quantities of the gas being pumped, as well as undesirable vapors. The general considerations governing the quantity adsorbed as a function of the pressure are similar to those of the adsorption isotherm (Section 11, B, 2) and account must be taken of these effects when any sudden changes of pressure occur in u-h-v systems containing traps, particularly when measurements of gas quantity are important. Cryogenic pumps or “cryopumps,” have found application in several large scale systems such as thermonuclear machines, space simulation apparatus, and wind tunnels. As will be shown below, these pumps can reach and maintain u-h-v pressures. Surveys which contrast the general properties of cryopumps with the other types of u-h-v pumps are given by various authors (10, 57, 213, 214). This last reference gives data on a simple liquid helium pump in a 1000-liter system. The results show that good agreement with experimental pressures can be obtained by assuming an accommodation coefficient of unity on the pumping surface. Lazarev et al. (215), in work completed in 1951, have described two designs of cryogenic pumps using liquid hydrogen. In the first the pumping surface is the exterior of a 20-cm diameter sphere which contains liquid hydrogen. A schematic diagram of this pump is shown in Fig. 16. With this pump Torr at a pumping speed of they obtained a base pressure of 6 x 13,200 liters/sec. T h e maximum possible pumping speed for nitrogen was 14,500 liters/sec, indicating a value near unity for the condensation coefficient, in agreement with the results of Table 111. The loss of hydrogen was 0.25 liters/hr. The second pump was smaller (4000 liters/sec) and differed in geometry but not in design principle. I n a later publication Borovik et al. (216) point out that the limiting pressure of a
ULTRAHIGH VACUUM
385
hydrogen condensation pump should be lower than 6 x Torr, since the vapor pressure of the main components of air is about 10-l1 Torr at 20°K. They describe a careful experiment in which elaborate precautions were taken to cut off impurities arising in the diffusion LIQUID N,
EVACUATED SPACE
FIG.16. Schematic diagram of hydrogen cryogenic pump [after B. G . Lazarev et al., Ukrain. Fiz. Zhur. 2, 175 (1957)l.
pumps and to reduce the base pressure to its limiting value, and they then achieved a base pressure of 10-lo Torr with a hydrogen pump, and Torr with a helium pump. They present an interesting but inconclusive discussion of why the pressures were not lower still. While molecular sieve pumps such as those described by Jepsen et al. (227), and Varadi and Ettre (218) would appear to have application in the u-h-v range, the central difficulty at present is probably one of thermal conduction. It is difficult to cool a disperse adsorbent when gaseous thermal conduction tends to zero. Degras (219), following the work of Prugne and Garin (220), has combined evapor-ion and liquid helium pumping in a small system and has achieved maximum speeds of 2000 liters/sec for hydrogen and 1000 liters/sec for air. An interesting contrast illustrating the range of cryogenic pumps is afforded by the work of Ames et al. (221) versus that of Hobson and Redhead (47). Both use typical small u-h-v systems,
386
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
but the former employ liquid helium pumping to reduce the pressure from atmosphere to Torr, and then proceed to 10-lo Torr by ionic pumping. The latter reach 10-lo Torr with a mercury diffusion pump and ionic pumping, and employ liquid helium pumping to reduce the pressure to Torr. Behrndt (222) describes an u-h-v evaporation apparatus in which pressures of 7 x 10-lo Torr were achieved with an oil diffusion pump and liquid nitrogen traps, and a final pressure of approximately 1 0-lo Torr with liquid helium pumping.
B. Processing The objective of processing is to reduce the emission of gas into the gas phase from the system parts below a tolerable maximum. The value of the tolerable maximum will vary with the particular system being used, and will depend on such variables as the available pumping speed, the type of gas being emitted, the type of experiment being done, etc. We give here a general description of system processing, which is common to all u-h-v systems, and in part common to other high-vacuum devices such as electronic tubes. Specific procedures for particular materials may be found in recent books by Kohl (223) and by Knoll (224). Espe (225) is publishing a comprehensive series on materials used in high vacuum technology. For large-scale operations it is advisable to adopt the general rules of cleanliness used in the electron tube industry: minimum handling of parts, reduction of dust level to a minimum, use of lint-free clothing by personnel, etc. A description of these precautions is given in the transactions of a recent symposium on cleaning electron device components and materials (226).Papers of this symposium also discuss other related problems in the preparation of device components. After parts have been machined or otherwise prepared, they are generally chemically cleaned. This involves careful rinsing in a grease solvent such as trichlorethylene, and possibly other chemical procedures. Following chemical cleaning, parts should no longer be handled. For laboratory operations we have found the foregoing procedures less critical than the thermal procedures described below. The next stage in processing is usually heating in hydrogen or vacuum. The main purpose of hydrogen firing is cleaning, annealing, or brazing, but this firing may also contribute to outgassing. Firing in vacuum is essentially an outgassing procedure. Conditions for vacuum and hydrogen firing are given by Knoll (224, who also shows several designs of vacuum and hydrogen furnaces. Vacuum outgassing can be done by suspending the work in metal container in a nonmetallic vacuum
387
ULTRAHIGH VACUUM
chamber and heating the work by ardio-frequency induction from outside the vacuum. No radiation shielding of the work is usually possible in this case, and the energy efficiency of the operation is low. Resistance-heated vacuum furnaces offer a more efficient use of the available power. Resistance furnaces have been described by Libin and Rocklin (227), Kramers and Dennard (228), and Kornelsen and Weeks (229). These furnaces all used voltages 5 20 volts and currents 2 100 amp. Figure 17 taken from Varadi (101) shows a typical partial pressure versus time plot during the outgassing of a small sample of 220 nickel.
DEGASSING OF 220 NI TEMPERATURE 850'C
TOTAL PRESSURE
I 1
2
3
4
5
6
7
TIME minutes]
FIG. 17. Degassing of 220 Nickel [after P. F. Varadi, Trans. Nutl. Symposium on Vucuum Tech. 7, 149 (196O)l.
After vacuum outgassing, it is advisable to keep the parts in a vacuum dessicator until final assembly. Leak-testing after assembly and prior to the final u-h-v processing often saves time in the long run and is essential in industrial applications. T h e physical appearance of an u-h-v system after assembly will of course depend upon the application. Two forms which at present appear to be limiting cases are shown in Figs. 18(a) and 18(b). Figure 18(a) shows a small glass system of the type used in our laboratory. This system is similar in general to other small glass systems, but differs in
388
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
that it is pumped during bake by a Vac-Ion pump, which may subsequently be removed, making the system thereafter movable from one bench to another. Redhead and Kornelsen ( 5 ) have given the operating procedures used with these systems. Figure 18(b) shows a large thermonuclear machine.
FIG. 18(a). Photograph of a small, glass u-h-v system including a mass spectrometer, an inverted-magnetron pump and a Bayard-Alpert gauge.
T h e next step in the achievement of u-h-v is the baking under vacuum of the entire high-vacuum portion of the apparatus to as high a temperature as the components will tolerate. (An exception to this procedure is provided by the work of Hunt et al. (83c) who achieved u-h-v conditions in an 85 liter metal system by evaporation of Mo onto the water cooled walls of the apparatus.) Usually the baking temperature does not exceed 500°C and a bake lasts from 2 to 24 hr. For small glass systems baking is done underneath an oven. Several methods are used. Alpert (2) describes a method in which w e n s consist of panels which are assembled around the apparatus. In our laboratory we use an oven weighing about
ULTRAHIGH VACUUM
389
3 90
P. A. REDHEAD, J. P. HOBDON, E. V. KORNELSEN
30 l b which can be lifted over the apparatus by two men. T h e latter ovens consume an average power of 1 kw with the system at 500°C. For large systems a pre-formed muffle may be assembled around the apparatus (183) or individual heater elements may be used for different parts of the apparatus. The complexity of outgassing arrangements is well illustrated by Mark and Dreyer (230) who describe the bakeout arrangements for the Model C Stellarator as follows: “Because each flange, observation window and section of tubing is to be maintained within 25°C of the desired bake temperature, 126 separate heater units are required to bake the vessel.and pumping system. Each heater circuit has its own temperature controller and each controller is monitored by the temperature recorder. Each bakeout through 450” Crequired approximately four days. T h e vessel components and traps are raised to 450°C in 25°C increments, and are held at 450°Cfor 18 hr. T h e vessel’s pressure at 450°C with one trap cold is Torr.” T h e baking procedure may be carried out in steps as in the work of Venema ( 4 ) who used a glass system with two diffusion pumps and three liquid nitrogen traps in series arranged vertically, with an oven which could be lowered continuously over the column. During the first part of the overnight bake at 450°C,the upper diffusion pump was not operating and was partly included in the baking region. During the second part of the bake the oven was raised and this pump was operating while the rest of the systemstayedin the
I
,‘u~ A S E PRKSSURE
10-4
J
BEFORE BAKE OUT MAX SYSTEM TEMP: 400°C ON FIRST BAKE
I
I
CUT HEAT 1st. B A Y l S E T E M
3 ANSI
-
WEEKS /
BAKE OUT STARTED
TIME [Hours]
FIG. 19. Pressure versus time during bake of an u-h-v system [after N. Milleron, Trans. Natl. Symposium on Vacuum Tech. 5, 140 (1958), courtesy of Lawrence Radiation Laboratory, Livermore, California, under auspices of U.S.A.E.C.].
ULTRAHIGH VACUUM
39 I
oven at 450°C. After a few hours the oven was again lifted and the first liquid nitrogen trap put into action. Then, with intervals of some hours, the other traps were cooled. T h e final pressure was less than Torr. T h e results of a bake upon the system’s pressure are illustrated by a curve due to Milleron (205) which is shown in Fig. 19 and which appears to be typical of most u-h-v systems, large or small, on the initial bake. Holland (231) has provided an interesting comparison between bakeable and unbakeable metal systems. Normally the final step to u-h-v is the individual outgassing of parts which are likely to be gas sources, such as the grid structures of hot cathode gauges. This is usually accomplished by electron bombardment, ohmic heating, or radiofrequency induction. This localized outgassing is sometimes executed during baking, and following the outgassing further baking may be required.
C.Measurement of Total Pressure Only the ionization gauge, in its various forms, has adequate sensitivity for the measurement of total pressure in the u-h-v region. Various types of viscosity gauges have been proposed for u-h-v, but none has so far been proven useful. Measurements of total pressure at these low pressures must be treated with great caution since the residual gas composition is strongly dependent on the processing methods and the type of system being used. In most cases the total pressure, as indicated by an ionization gauge, can only be used as a rough indication of the general performance of a system and of the effectiveness of any changes in processing methods or system design. Great care must be taken if the ionization gauge readings are to be used as an accurate measure of the gas pressure. I n the latter case it is preferable to use a mass spectrometer as a partial pressure measuring device [see Section 111, D). It should be noted that the output from an ionization gauge is proportional to the gas density within the ionizing region. Conversion of the ionization gauge readings to pressure requires a knowledge of the gas temperature. Leck (232) has reviewed the methods of total pressure measurements, and Brombacher ( 2 3 2 ~ )has published a bibliography of pressure measurements. Review of u-h-v pressure measurements have been published by PQtjr(233) and Grigor’ev (234). 1. Hot-Cathode Ionization Gauges. Nottingham (235) first suggested in 1947 that a lower limit existed to the pressure measureable with a hot-cathode ionization gauge ; he suggested that there was a photocurrent, independent of pressure, produced at the ion-collecting electrode by the
392
P. A. REDHEAD, J . P. HOBSON, E. V. KORNELSEN
action of soft X-rays caused by electron bombardment of the grid. Thus , when the pressure was sufficiently low that the positive ion current and the photocurrent were of the same order, pressure measurement was no longer possible. T h e hot-cathode ionization gauges available at that time had a lower limit of about Torr. Within a few years, Nottingham's suggestion was confirmed and gauges were designed to minimize the X-ray effect by Bayard and Alpert (I), Lander (236) and Metson (237). T h e designs of Bayard and Lander reduced the X-ray effect by decreasing the solid angle subtended by the ion collector at the X-ray source. Metson's design used a suppressor grid in front of the collector to prevent the photoelectrons from leaving the collector. T h e design of Bayard and Alpert was the simplest and most effective, and is now most widely used. T h e existence of the X-ray effect, and its reduction by the new gauge design, were clearly demonstrated by Bayard and Alpert ( I ) by a comparison of the ion-collector current ( I c ) vs. grid voltage ( V g ) characteristics at different pressures for a conventional ionization gauge and a Bayard-Alpert gauge. For the conventional ionization gauge with a large collector, the I c vs. V , curves at high pressures have a shape similar to that of ionization cross section curves. At lower pressure (about Torr) the I , vs. V , curve is a straight line on a log-log plot with a slope between 1.5 and 2, as would be expected for an X-ray induced photocurrent. T h e curves for the Bayard-Alpert gauge show that the X-ray effect has been reduced to such an extent that even at pressures of less than 5 x 10-l' Torr the log I , vs. log V , curve is not a straight line; i.e., the true ion current has not been completely obscured by the residual current even at these low pressures. T h e design of a typical Bayard-Alpert gauge is shown in Fig. 20. T h e ion-collector consists of a fine tungsten wire (typically 150 micron diameter) on the axis of the cylindrical grid structure. T h e filament is outside the grid cage. This inverted geometry has two predominant advantages: (a) the surface area of the ion-collector is minimized, thus reducing the X-ray effect, and (b) the potential distribution between the grid and collector is such that almost all the volume within the grid is available for ionization and the collector acts as an efficient ion-trap. T h e efficiency of ion-trapping is increased, and hence the sensitivity improved, by closing the ends of the grid as shown in Fig. 20. Gauges of the design shown in Fig. 20 have a sensitivity (k)to nitrogen of
k
=
i+ x -1 N 20 Torr-I 2-
P
(25)
for a grid voltage, V , = 105 volts and an electron current, i- = 8 ma.
ULTRAHIGH VACUUM
393
T h e residual current is about 6 x 1 0 ~ amp, ' ~ corresponding to a pressure of 3.6 x lo-" T o r r (equivalent, nitrogen). Electrode potentials throughout are given with respect to the filament. Very rigorous outgassing of the electrodes of the gauge is essential in the u-h-v region. Outgassing is usually accomplished by electron
FIG. 20. Schematic cut-away diagram of modulated Bayard-Alpert gauge.
bombardment of the grid and ion-collector (typically 300 ma at 1 kv for a molybdenum grid). Bayard-Alpert gauges have been designed where the grid is a spiral, without supporting side-rods, which can be directly heated by passing a current through the grid wire. Adequate degassing of the grid is difficult with this design without causing the grid to sag. Various modifications to the original three-electrode design of Bayard and Alpert have been described. Addition of another grid, outside the filament, prevents surface charging of the glass bulb from affecting the gauge calibration and increases the gauge sensitivity slightly (238). These improvements must be balanced against the increased difficulty of thoroughly outgassing the additional grid. T h e electron current can be controlled by an additional control grid between the filament and cylindrical grid; this arrangement simplifies the design of the electronic regulator for the electron current (239). T h e X-ray effect can be reduced below the level of the normal BayardAlpert gauge design by decreasing the diameter of the ion-collector wire to the smallest practical size ( 4 ) ( 2 3 9 ~ )Van . Oostrom ( 2 3 9 ~has ) achieved an estimated X-ray limit of below 10-l2 Torr with a collector diameter of 4 microns. For measurements of pressure below about 5 x Torr it is
394
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
essential to know the residual current accurately. The residual current is the current to the ion-collector which is independent of pressure; it consists of two components: (a) the X-ray induced photocurrent, and (b) the photocurrent caused by radiation from the hot filament and external light sources. T h e residual current can be estimated from a plot of the collector current (Ic) versus V , taken at low pressures where the residual current is a large fraction of the total ion-collector current. T h e I c vs. V gcharacteristic is plotted on a log-log scale, and the straight line portion at high voltage is extrapolated downwards, the extrapolated value of I , at the normal operating grid voltage is a measure of the residual current (ir).T h e accuracy of this method is poor because the large grid voltages, which must be applied to establish the upper portion of the curve, cause changes in the pumping speed of the gauge and thus may change the pressure in the system. A second method for the measurement of the residual current (244, which does not cause any change in the pumping action of the gauge during the measurement, requires the addition of one electrode to the standard Bayard-Alpert gauge. A modulator wire (see Fig. 20) is inserted parallel to the ion-collector inside the grid of a standard Bayard-Alpert gauge. When the modulator is at grid potential, the sensitivity of the gauge is the same as that of the unmodified gauge. When the modulator potential (V,) is equal to the ion-collector potential (V,), the modulator collects a fraction of the positive ion current. By switching the modulator from V g to V,, the positive ion current can be modulated by 30 to 40% without any change in the residual current. T h e ion-collector current in the two cases is given by
Zl = i,
+ ir ( V , = V,)
and I ,
= mi,
+ ir (Vm = Vc)
(26)
where i, is the positive ion current and a is the modulation factor, which is independent of pressure, Thus the true ion current is given by, i, = (4 - Z,)/U
-4
(27)
and can be measured by a difference method once (Y has been found from measurements at higher pressures where i, ir. T h e residual current is given by, ir = (I, - aZ,)/(l - a). (28)
>
T h e validity of this method depends on the escape probability of a photoelectron from the ion-collector being independent of the modulator potential in the range from Y , to Vc. This escape probability is determined by the shape of the potential well around the ion-collector.
ULTRAHIGH VACUUM
395
Because of the large ratio of grid to ion-collector diameters, this potential well is very steep and is relatively unaffected by changes in the modulator potential. Using this method it has been found that the residual current in a Bayard-Alpert gauge may change by large factors in a few hours, particularly in the period following outgassing of the gauge. T h e cause of these changes has not been definitely established. As an example, in one case the residual current was 1.5 x 10-l2 amp after outgassing, and increased to 6 x 10-l2amp in 14 h. Riemersma (241) has designed an ionization gauge in which the source of electrons is the output of a photomultiplier whose first dynode is illuminated with ultraviolet light. A mercury vapor lamp illuminates the first dynode through a quartz window. T h e multiplier is operated at 200 to 300 volts per stage to produce an electron current of the order of amp in the ionizing region. By careful adjustment of the voltages, the output of the gauge can be made linear with pressure from lop5 to Torr. T h e sensitivity of this type of gauge is about 2 x amp/Torr for air. T h e pumping speed of the gauge is about lop3 liters/sec. This type of gauge would be most useful in experiments where the presence of a hot filament is undesirable because of its interaction with the gas in the system. T h e sensitivity of the hot-cathode ionization gauge has been greatly increased by applying a magnetic field to increase the electron path length. Figure 21 shows Lafferty's design (88) which consists of a cylindrical magnetron operated in a magnetic field of 250 gauss (2.5 times the cutoff field). T h e ion current is measured at one of the two negative end plates. In normal operation the anode potential is +300 volts, the shield potential is - 10 volts, and the ion-collector potential is to lop9amp, -45 volts. Very small electron emission, in the range is used to ensure stable operation and to prevent the production of high-energy electrons which would reach the ion-collector. Below lo-* Torr the ion-collector current is a linear function of pressure. For an electron emission of lo-' amp the ion current is 9 x loe2 amp/Torr. T h e X-ray induced photocurrent is given by
ix
= 1.4 x lo-* i,
(29)
where io is the electron emission. Thus the gauge should be linear to pressures of about 4 x 10-14 Torr and could detect a pressure of 10-15 Torr. T h e sensitivity of this type of gauge has been increased by . of about 10-15 the addition of an electron multiplier ( 2 4 1 ~ ) Pressures Torr can be detected with an output current of 10-l' amp.
396
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
The electron emission from the filament in Lafferty’s gauge can only be directly measured by removing the magnet, thus the electron emission cannot be directly regulated in normal gauge operation. At pressures below 10-8 Torr the anode current becomes independent of pressure and
FIG.21. Hot-cathode magnetron gauge [after J. M. Lafferty, 424 (1961)l:
I.
Appl. Phys. 32,
dependent only on the electron emission, thus at pressures below Torr the electron emission can be maintained constant by regulating the anode current. At pressures above Torr continuous regulation is not possible.
2. Cold-Cathode Ionization Gauges. The Penning type of ionization gauge, a cold-cathode discharge in a magnetic field, has been modified to permit operation in the u-h-v region (47,242,234). T h e advantages of a Penning type of gauge are: (a) there is no X-ray limit because the
ULTRAHIGH VACUUM
397
electron current which produces the X-rays is proportional to pressure ; (b) the gauge contains no hot filament, thus photoeffects and chemical changes produced by a hot filament are absent; (c) the sensitivity is higher than that of any hot filament gauge. T h e attendant disadvantages are: (a) stable operation can only be achieved over a limited range of voltage and magnetic field; hence the pumping speed of the gauge can only be changed over a small range; (b) the ion current versus pressure characteristics are, in general, nonlinear ; thus the Penning gauge must be calibrated over a fairly wide range of pressure; (c) at very low pressures (below 5 x lo-" Torr) the Penning gauge takes some minutes to strike; (d) oscillations occur in the discharge at all pressures, and care must be taken to prevent these oscillations from causing errors in the measuring circuit. For a Penning discharge to be useful for u-h-v pressure measurements, two requirements must be met: (a) the electrodes must be designed to trap electrons in the discharge so that the discharge will be maintained at very low pressures, and (b) the ion-collector must be shielded from the high electric fields so that no field emission can occur from the ion collector. Two types of modified Penning gauge have been developed which are capable of measuring to at least 10-l2 Torr. T h e first type of gauge, the cold-cathode magnetron gauge (169, 242), is shown in Fig. 22. T h e anode consists of a cylinder (20 mm long and 30 mm diameter) which is B ?
ION CURRENT AMPLIFIER
FIG. 22. Schematic diagram of cold-cathode magnetron gauge.
398
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
perforated to improve gas flow through the gauge. T h e ion collector is shaped like a spool, consisting of an axial cylinder (3 mm diameter by 20 mm long) joined to two circular end-discs. The end-discs are shielded from the high electric field by the auxiliary cathodes, which consist of two annular electrodes shaped and polished so as to reduce field emission to a minimum. T h e auxiliary cathodes are operated at ground potential. This gauge is normally operated with an axial magnetic field of 1000 gauss and an anode voltage of 5 to 6 kv. T h e ion current in the magnetron gauge is linearly proportional to pressure in the to 10-lo Torr range, and the sensitivity for nitrogen is about 10 amp/Torr. In some cases a change of slope of the ion-current versus pressure curve occurs at about 10-lo Torr, and a relationship, I = up**', is observed for p < 10-lo Torr. T h e cause of the change of slope has not been established and in some gauges of different dimensions it does not occur. T h e second type of modified Penning gauge, the inverted-magnetron gauge (47,243) is shown in Fig. 23. T h e ion collector is a cylinder (30 mm diameter by 20 mm long), partially closed at both ends, with its axis parallel to the magnetic field. T h e anode is a tungsten rod (1 mm diameter) passing axially through the holes in the end-plates of the ion-collector. T h e auxiliary cathodes are circular discs with short spouts placed between the anode and the end-plates. T h e auxiliary cathodes prevent field-emission from the edges of the holes in the endplates. This gauge is normally operated with the auxiliary cathodes at ground, the anode at 5 to 6 kv, and a magnetic field of 2000 gauss. T h e inverted-magnetron gauge has been calibrated from to 10-l2 Torr and the ion current obeys the relationship,
I = bp" (30) where b is a constant and the exponent n is about 1.10. T h e value of b varies slightly from gauge to gauge and has a value of about 10 amp/(Torr)n for nitrogen. For the measurement of pressures below 10-lo Torr, these gauges must be completely shielded from ambient light and operated from a well stabilized high-voltage supply. A cold-cathode gauge without magnetic field has been described by Barness (244) which uses a set of fine tungsten points in a very high electric field as field-ion emitters. T h e positive ions are detected by scintillation counting when they strike a fluorescent screen. Barnes shows that in his unbaked system the counting rate from his gauges reaches a constant value while the pressure, as indicated by a BayardAlpert gauge, decreases by several orders of magnitude (245). Barnes
399
ULTRAHIGH VACUUM
interprets these results as indicating that the pressure readings of the Bayard-Alpert gauge are in error by several orders of magnitude, thus casting doubt on the use of Bayard-Alpert gauges at u-h-v. Measurements by many other investigators are in disagreement with this interpretation (see for example Crawford, 246). A more probable explanation -ANODE
COLLECTOR
FIG. 23. Schematic diagram of cold-cathode inverted-magnetron gauge.
of Barnes’ results is that water molecules, adsorbed on the shank of the field-emitter tips, migrate to the point and are there ionized and counted by the scintillation counter. This background of ions formed from adsorbed water could cause a constant counting rate from the coldcathode gauge independent of pressure. This effect renders this gauge useless for the measurement of low pressures.
3. Ionization Gauge Sensitivity to Various Gases. Table XIV lists the measured sensitivity ratio of various ionization gauges to several gases, the sensitivity to nitrogen being taken as unity. T h e values given for the chemically active gases (hydrogen, oxygen, and water) should be used with caution since the hot filament of a hot-cathode ionization gauge causes decomposition of these gases (see Section 111, C , 3).
400
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
TABLE XIV. RELATIVE SENSITIVITY OF IONIZATION GAUGES TO VARIOUS GASES Gas
Nz
HZ 0%
co
COZ
H2O He Ne A Kr Xe Hg
1
2
3
4
1 1 1 1 0.46 0.47 0.38 0.53 1.14 0.85 1.06 1.36 0.9 0.16 0.16 0.25 0.23 0.25 1.16 1.23 1.06 1.86 1.80 2.69 2.78 3.18 3.93
5 1 0.42
6 1
7 1
8
9
10
0.2808
1 0.52 0.99
1
1.25 1.25 1.09 0.823 1.163 0.21 0.20 0.17 0.33 1.5 1.25 1.33
0.1283 0.2407 1.000 1.333 2.72 2.190
11
I
1.29 0.24
0.15
1.76
2.5
SFB
CH4 CsHa
1.015 1.992 -
Column Reference
10 11
(254) (47)
Type of gauge
Operating Conditions
Vg = 125 v, 1- = 0.5-5 ma Hot-filament triode (FP-62) Vg = 150 v, 1- = 0.5-5ma Hot-filament triode (VG-1) Hot-filament triode (BAR type 507)Vg = 145 v, 1- = 5 ma Hot-filament triode Bayard-Alpert (WL-5966) Vs = 140v,I- = 0.1 ma V, = 125 v, 1- = 100 pa Bayard-Alpert (RG-75) Va = 125 v, 1- = 100 pa Hot-filament triode (826 A) Bayard-Alpert (WL-5966) V, = 145 v, I- = 0.5 ma Vg = 18Ov,I- = 1 ma Hot-filament triode (Leybold IM-1) Cold-cathode magnetron gauge Va = 6 kv, B = los gauss Cold-cathode inverted-magnetron Va = 6 kv, B = 2 x lo3 gauss magnetron gauge
4. Pumping Effects in Ionization Gauges. All ionization gauges behave as pumps to some extent, the removal of gas from the volume is caused by four processes: (1) Ionic pumping: the entrapment of ions that impinge on any solid surface. This is the only pumping mechanism for inert gases. (2) Chemicalpumping:the removal of neutrals by chemisorption on the electrodes or the bulb. ( 3 ) Activated chemical pumping: the removal of excited or dissociated molecules by chemisorption. Excitation or dissociation is caused by the electrons in the discharge.
ULTRAHIGH VACUUM
40 1
(4) Pumping at an incandescent jilament: in the case of hot-filament ionization gauges, an additional pumping process is produced by dissociation of gas molecules (in particular, hydrogen, oxygen, and water) at the hot filament, and reaction with impurities in the filament. The dissociated fragments may then be readily chemisorbed at any solid surface. Ionic and activated chemical pumping in ionization gauges was discussed in Section 111, A, 3. The remaining two types of pumping, insofar as they affect the measurement of pressure, will be discussed briefly in this section. Chemisorption of active gases (Nz, H,, 0,, etc.) on the metal surfaces of an ionization gauge may produce a chemical pumping speed which greatly exceeds the ionic pumping speed. Hobson (189)has measured an initial chemical pumping speed for nitrogen in a Bayard-Alpert gauge ( V , = 250 volts, I - = 8 ma) of 2 liters/sec when the gauge was first operated after the initial outgassing. After molecules of nitrogen had been pumped the chemical pumping speed dropped to zero. It was concluded that most of the chemical pumping took place at the grid by adsorption into the second layer, where the sticking probability is about Bills and Carleton (125) have reported a maximum pumping speed for nitrogen in a Bayard-Alpert gauge of 0.5 liter/sec, and Young (188) has reported a speed of 0.1 liter/sec. These measured values are total speeds but are predominantly caused by activated and ionic pumping. A tungsten filament operated at a temperature exceeding 1100°K will dissociate hydrogen, the atomic hydrogen so formed is readily adsorbed at metal or glass surfaces. Hickmott (255) has shown that for pressure below Torr and filament temperature exceeding 1475"K, the fraction of hydrogen molecules striking the hot surface which are dissociated is constant at 5 yo.Thus, if the sticking probability of the atomic hydrogen is unity and the area of the hot filament is 0.2 cm2, a maximum pumping speed of 0.1 liter/sec for hydrogen will be produced at pressures less than Torr and temperatures greater than 1475°K. This process is complicated by reactions between the atomic hydrogen and carbon impurities in the tungsten filament and the glass which produce CO, H,O, and CH, (166). Eisinger (256) has measured the pumping efficiency of a hot tungsten surface in an oxygen atmosphere. T h e pumping efficiency is defined as the probability that an oxygen molecule striking the hot surface is Torr this efficiency removed from the gas phase. At pressures below has a maximum value of 2 % at a filament temperature of 1700°K. Thus
402
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
for a filament area of 0.2 cm, the maximum pumping speed for oxygen will be 0.04 liter/sec. The pumping efficiency decreases quite rapidly at higher temperatures; at 2100°K the efficiency drops to 0.2%. Oxygen also reacts with the carbon impurities in the hot tungsten filament to produce carbon monoxide and a small amount of carbon dioxide (268). Bistable behavior of the pumping speed of Bayard-Alpert gauges has been observed (121,257) and attributed to changes in potential of the inner surface of the glass bulb. T h e glass surface can stabilize in potential at one of two values, controlled by the secondary emission characteristics of the glass surface. This effect can be prevented either by adding an additional grid between the gauge structure and the glass (238) or by placing a conductive coating on the glass and controlling its potential externally. Only a limited number of measurements has been made of the pumping speed of cold-cathode gauges (258, 259, 269) and in most cases only total pumping speeds have been measured, i.e., chemical and ionic pumping speeds have not been separated. Barnes (254) has measured the pumping speed of the cold-cathode magnetron gauge (242) for various gases and obtains the following results: He-0.17, N,-2.5, H,-2.0, 0,-3.4, A- 1.7, C0,-2.04 liters/sec. For the inverted-magnetron gauge the speed for helium is 0.02 to 0.04 liter/sec, for argon 0.25 liter/sec, and for nitrogen (when the chemical pump is saturated) 0.5 to 1.O liter/sec. Rhodin (269) has measured the pumping speed of the cold-cathode magnetron gauge for various gases and finds that the speed depends on the previous treatment of the gauge. For a gauge of the dimensions given in ref. (242), immediately after bakeout the speeds were 0.14 liter/sec for N,, 0.15 liter/sec for 0,, and 0.2 liter/sec for argon. After operation of the gauge for 16 h in 10-6 Torr of O,, the speeds had dropped to 0.1 liter/sec for N,, 0.12 liter/sec for 0,, and 0.05 liter/sec for argon. I t can be seen from the above data that the pumping speed of an ionization gauge may be comparable with the speed of other pumps in a system, For a hot-filament ionization gauge the ionic pumping speed can be reduced by decreasing the ionizing electron current and/or the electrode potentials. Electron currents of 100 p or less are desirable in a Bayard- Alpert gauge when accuracy of pressure measurement is required. Chemical pumping speeds cannot be controlled and, to obtain accurate measurements of pressure, these speeds must be measured and taken into account. 5. Factors Causing Errors in Pressure measurement. Some of the sources of' error in pressure measurement with ionization gauges have been discus-
ULTRAHIGH VACUUM
403
sed by Redhead (260). T h e commonest source of error is the pressure drop across the gauge tabulation caused by the pumping action of the gauge. T h e high pumping speed of cold-cathode gauges is particularly troublesome in this respect. Significant changes in gauge calibration can result from changes in the secondary emission coefficient (for ions and/or electrons) of the gauge electrodes. I n particular, changes in secondary emission coefficient (electrons per ion) of the ion-collector of a triode ionization gauge may cause significant changes in gauge sensitivity (see Section 11, E, 3). Another source of error is the possible production at hot filaments of alkali metal ions arising from glass decomposition products (156, 157) (Section 11, E, 3). T h e glass bulb of an ionization gauge assumes a potential dictated by secondary emission effects at the glass surface. Uncontrollable changes in potential of the bulb cause changes in the gauge sensitivity (257).This effect can be prevented by depositing a conducting film on the bulb which is held at a known potential. High-frequency BarkhausenKurtz oscillations (40 to 80 Mclsec) occur in Bayard-Alpert gauges under almost all conditions (260). In gauges with an uncoated glass bulb these oscillations can attain sufficient amplitude so that some electrons gain enough energy from the rf field to reach electrodes considerably negative with respect to the filament. If these electrons strike the glass bulb they cause it to go negative, producing a change in gauge sensitivity. A conductive coating on the bulb reduces the amplitude of the Barkhausen-Kurtz oscillations and prevents the above effects. Oscillations also occur in Penning-type gauges (47). T h e efficient trapping of electrons in a cross-field discharge results in strong plasma oscillations, typically in the frequency range 1 to lo3 kclsec. These plasma oscillations are the cause of the sudden breaks and instabilities that are frequently observed in the ion-current versus pressure characteristics of Penning gauges. No way has been found to prevent these oscillations and they represent the biggest difficulty in using a Penning type of gauge for accurate pressure measurements. Pressure measurements with a hot-filament ionization gauge in a system containing appreciable amounts of chemically active gases (H,, O,, H,O or hydrocarbons) are complicated by the chemical changes in gas composition caused by the hot filament; these effects were discussed in Section 11, G. These unwanted reactions at the hot-filament can be reduced by three methods: (1) T h e hot filament can be surrounded with a metal surface at which atomic species recombine readily. T h e production of carbon
404
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
monoxide by a hot filament in an atmosphere of hydrogen or oxygen can be greatly reduced this way (166). (2) The carbon impurity content of a tungsten filament can be reduced by heat treatment in an oxygen atmosphere. The filament Torr of oxygen should be heated at 2200°K for 10-60 h in
(167).
(3) The operating temperature of the filament can be lowered by the use of a low work-function coating. Rhenium filaments coated with lanthanum boride (261) are suitable for u-h-v applications. A typical coated filament for a Bayard-Alpert gauge gives 10 ma electron emission at 1300°K. Another advantage of the lanthanum boride coating is that its vapor pressure is only that of tungsten for the electron emission density necessary to give 10 ma in a Bayard-Alpert gauge. Thoria coatings on tungsten produced by cataphoresis are also suitable (262). Care must be taken when using low work-function filaments to ensure that the ion-collector is kept clean of evaporated material, A reduction in work-function of the ion-collector by evaporated material causes large increases in the residual current because of photoeffects. Mizushima and Oda (263) have reported observing a non-linear relation between the positive ion current and the electron current of a Bayard-Alpert gauge. It has been shown by Baker (264) and others that this effect is caused by a change in gauge temperature when the filament temperature is used to control the electron current. Baker shows that when the electron current is controlled by an additional grid and the filament temperature maintained constant, then the positive-ion current is always linearly related to the electron current. In a system pumped by oil diffusion pumps and containing some oil contamination, there may be large differences in the pressure indicated by: (a) an ionization gauge enclosed in a bulb and connected to the system by a small diameter tube, and (b) an ionization gauge whose electrodes are directly inserted into the chamber in which the pressure is to be measured (“nude gauge”). This effect has recently been studied in some detail by Haefer and Hengevoss (265). In Haefer’s experiments it was found that immediately after bakeout the reading of the two gauges was identical, but after about 24 h the reading of the nude gauge (NG) was an order of magnitude higher than that of the gauge connected externally to the system through a small tube (EG). Haefer interprets his results an indicating that the NG measures the pressure of permanent gases and the oil vapor pressure, whereas the EG measures essentially the pressure of permanent gases
ULTRAHIGH VACUUM
405
alone. This interpretation leads to the conclusion that the tube joining the EG to the system has a conductance to oil molecules which is lower by a factor of lo4 than its conductance for permanent gases. Although the exact mechanism of this effect is not certain, it is clear that in any system where oil pumps are used and trapping of the oil vapor is not complete the pressure should be measured with a nude ionization gauge.
D. Measurement of Partial Pressure I n u-h-v systems, residual gas sources and pumping speeds vary widely for different pumping methods, structural materials, temperatures, previous history, and different gases (see Section 11, D). In addition, serious interdependent effects can occur ; the introduction of a known gas causing large variations in the partial pressures of other gases through chemical conversion, ionic replacement, or pump saturation. It is evident that in such systems measurement of the partial pressures of the individual gases has decisive advantages over total pressure measurement which indicates only the net equivalent pressure resulting from these numerous and complex processes. The most important and most widely used instrument for the measurement of partial pressures in high vacuum is the mass spectrometer in one of its many forms. Two additional advantages of mass spectrometers over ionization gauges are: (1) they usually have lower ionic pumping speeds and thus cause less disturbance of the vacuum conditions, and (2) they have very small residual currents to the ion collector-usually < amp. In the u-h-v region it becomes extremely difficult to provide a mass spectrometer of adequate sensitivity and sufficiently low outgassing rate to fully exploit the advantages of partial pressure measurement. Consider first the problem of sensitivity. An electron current i- (amp) passing a distance I through a gas at pressure p will produce ions at a rate corresponding to a current i+ = LpQl (amp)
(31)
where values of the differential ionization cross section Q lie between 1.0 and 10 cm-' Tor+ for the most common gases (97). Typical operating parameters for a high sensitivity mass spectrometer might be i- = amp, I = 1 to 2 cm. One would then expect the total to ion current produced to lie in the range p . I n practice, due p to to imperfect focusing and ion transmission through the analyzer, or to low i-, collected ion currents seldom exceed approximately lod4 p . A
406
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
partial pressure sensitivity of 10-13 Torr is adequate for most u-h-v applications, thus requiring that ion currents of approximately 1O-l' amp must be registered with reasonable accuracy and speed. Such performance is beyond the capabilities of simple current collection techniques, and it becomes necessary to make use of the kinetic energy of the individual ions in counting or current amplification systems. Even if the ions are detected individually, 1O-I' amp corresponds to an arrival rate of only 62 ions/sec. Errors of approximately 10% due to statistical fluctuations are therefore to be expected for an observation time of 1 sec; or approximately 3 % for 10 sec. This fundamental limit to partial pressure measurement can be avoided onIy by increasing the ion source yield (i.e., increasing i+/p). A general discussion of the outgassing of solids is given in Section 11, Z I of this paper. I n broad outline, one can draw the conclusion that if pressures due to outgassing are to be reduced to below approximately Torr, the components must first be raised to the highest allowable temperature to remove dissolved gases. Such treatment is particularly important for the mass spectrometer ion source which is normally heated to a temperature of a few hundred degrees by the filament input power. Metal components which receive no heat treatment other than the vacuum processing (450-500°C) are frequently major sources of residual gas. Prior heating in a vacuum furnace to about 1000°C has been found in this laboratory to be extremely effective in reducing the outgassing of spectrometer components (266). An additional gas evolution can occur in mass spectrometers due to the sputtering effect of the ions striking solid surfaces. Adsorbed and dissolved gas atoms are released into the gas phase at rates which depend on the sputtering ion flux and energy. Sputtering has been considered in more detail in Section 11, El 2. Because of the relatively low rate of ion production in spectrometers, the effect of sputtering on gas composition is not usually large unless a relatively high pressure of some gas is introduced. Reynolds (160) describes yet another source of outgassing: the impact of high energy electrons on gas-covered surfaces. He found multiple-charged mercury ions to be present due to electrons produced in the analyser and accelerated by the ion beam potential into the source. U-h-v systems do not usually contain residual gases with mass numbers higher than 44 (carbon dioxide). Thus, unless heavier gases are deliberately introduced, the resolution of adjacent masses up to mass 50 will usually suffice. Indications stable to about 1-2 yo are normally acceptable, so that very precise electrical stabilization is not required. It is, of course, important that the collected current vary linearly with
ULTRAHIGH VACUUM
407
partial pressure. This does not appear to present any serious difficulty for permanent gases below about to Torr. Gunther (267) has reported linear operation of an electric quadrupole spectrometer up to 3 x Torr. T h e calibration of a mass spectrometer in terms of gas pressure presents a number of problems. Because operation is limited usually to pressures below Torr, absolute calibration is especially difficult. Most frequently comparisons are made with a previously calibrated ionization gauge, thus limiting the accuracy to that of the gauge calibration (see Sections 111, E and 111, C, 5 ) . T h e main characteristics of a number of reported u-h-v mass spectrometers is summarized in Table XV. A short discussion of the advantages and shortcomings of the various types seems in order. T h e omegatron has been popular for partial pressure measurements because of its small size, relatively simple construction and ease of degassing. Its optimum operating conditions, however, are rather critical, and reproducibility is difficult to achieve without frequent adjustment. Variations of electrode surface potential are thought to be responsible for these effects. Also, though its collection efficiency is high, the necessity for using low electron currents (approximately loF5amp) limits the sensitivity to about lo-" Torr. T h e geometry of the omegatron makes the addition of current multipliers, which would increase sensitivity, almost impossible. T h e magnetic-sector deflection spectrometer (276) which has come into such wide use as an analytical instrument has been adapted by several workers to u-h-v operation. This type of spectrometer has two important advantages: (1) a great deal of experience in design, construction and operation is readily available; (2) both the source and collector regions are easily accessible, making the addition of sensitive current detectors or special ion source assemblies relatively easy. T h e 60" sector spectrometer, which is the most commonly used, tends to be a rather large assembly of a shape which makes attachment to small u-h-v apparatus difficult. When a glass envelope is used, careful dimensioning and jigging are required to obtain proper alignment of the source and collector assemblies. Subsequent performance is dependent to some extent on the stfuctural stability of the glass during processing. All the 60" sector spectrometers mentioned in the table have resolving powers of 150 or higher, analyser radii of 10 to 15 cm, and overall lengths of approximately 60 to 100 cm. Recently a much smaller bakable 90" sector spectrometer has been reported by Davis and Vanderslice (269).T h e tube, shown in Fig. 24, has a maximum dimension of 35 cm, an analyzer radius of 5 cm, a resolving power of 100, and a minimum
TABLE XV. CHARACTERISTICS OF SOME ULTRAHIGH VACUUMMASSSPECTROMETERS
Reference
(90)
Spectrometer type Omegatron Diatron 20 (180" magnetic deflection) 90" sector deflection 60" sector
Approximate partial pressure sensitivity (Ton) 10-0
2x
< 10-15 10-1'
10-13
60"sector
3 x 10-10
Omegatron
lo-" 5x
-
Resolving power 20 at mass 20
-
30
10-10
60" sector
Trochoidal double focusing 60" sector
special current detection
Electron multiplier (Ag-Mg)
60" sector
10-14
Omegatron
lo-"
Omegatron
5 x 10-10
Time-of-flight
10-12
5 x 10-9
x
-2
envelope Glass Stainless steel (gold wire gaskets)
Metal
?
diffusion pump Oil diffusion P-P Oil diffusion Pump Ba getter
.?r
Ti. getter and ion gauge Getter and ion gauge
5
5 150
6 x lo-*
Glass
7 150
3 x 10-9
Glass
3 x
Glass
-
Electron multiplier or scintillation counter Electron multiplier (Ag-Mg) -
Electron multiplier (Ag-Mg)
150
5
150
-
30 at m a s s 30 30 at mass 30
-
-40
1.5
Glass
X
2 x 10-8
5 x
Glass Glass (Pyrex)
Hg diffusion pump
5 x 10-9
Soft glass
Ba getter
2 x 10-9
Glass
Hg diffusion pump
9 x lo-'"
Stainless steel (Cu gaskets)
Ti getter and ion pump
10-l0
* EU
Hg and oil
Electron multiplier (Ag-Mg)
10-9
'd
P
3 x 10-9
30 at mass 30 100
pumps Ionization gauge Vac Ion Ion pump
s: 150
-
52
Glass and metal
-
10-12
10-13
100
Lowest total pressure (Tom) 10-9
P
U
3: 0
Ez
m
c
P
3E 9
ULTRAHIGH VACUUM
409
Trans. Natl. Symposium on Vacuum Tech. I , 417 (1960)l.
FIG.24. 90” sector bakable high-sensitivity mass spectrometer [after W. D. Davis and T. A. Vanderslice,
detectable partial pressure below Torr when using an electron multiplier ion collector. The total pressure during operation is in the low Torr range when connected to the diffsion pump; approximately Torr when valved-off and pumped with an ionization gauge. The trochoidal mass spectrometer reported by Kornelsen (266) is shown, without its glass envelope, in Fig. 25. I n normal operation it Torr to the total pressure in a getter-ion contributes less than 2 x gauge pumped system (see Table XV).
410
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
FIG. 25. Trochoidal ultrahigh vacuum mass spectrometer [after E. V. Komelsen. Rept. 19th Ann. M.I.T. Conf. on Phys. Electronics p. 156 (1959)l.
The ion current multipliers used in the sector mass spectrometers in Table XV have been the secondary electron type with electrostatic focusing (277). These have the disadvantage of introducing into the vacuum system a rather large quantity of material which is difficult to outgas. An attractive alternative method of counting collected ions has
41 1
ULTRAHIGH VACUUM
been used by Bernhard et al. (278). In their method the ions fall at approximately 20 kev energy onto a metal plate, and the secondary electrons produced are accelerated by the same potential onto a fluorescent screen. T h e scintillations are counted by a sensitive external photomultiplier. Pikus (274) has reported the use of such a detector in an u-h-v mass spectrometer with results comparable to those with the secondary electron multiplier. T h e composition of the residual gas can vary considerably with the type of construction and processing. Residual compositions for the twelve representative systems described i n Table XV are presented in Table XVI. T h e numbers refer to relative ion currents except in the case of Wagener (75) in which case true pressures were used. T h e small number of gases which form the bulk of the residual atmosphere for all TABLE XVI. RESIDUAL GASCOMPOSITION I N THE SYSTEMS OF TABLE XV Approxiniate percentage composition
Reference
Total pressure (Torr)
co HZ ~~
z
x
10-9
5 x 10-0 4 x 10-10 3 x JO-D 6 x lo-@
70 17 30 70
CH,
He
-
5 x 10-10 5 x 10-9 2 x 10-9 9 x 10-10 ~~~
a
~
6 I5 4
-
90 17
65 90
7 2 40
26 50 42 15 2
52
5 4
3
~~
In addition, mass numbers 19 and (35
-
31
30
2
coz
10 30 25
5
10
69
Nz
15
3 x 10-9
3 n 10-9 1.5 > 2 k. 10-8
H,O
+ 37) contributed
-
50 2 10
80
15
21 100
2
19
7
-
30% and 3% respectively.
the various types of systems is most striking. With one exception (269), only seven gases are significant. Components contributing less than 2 yo of the total pressure have been omitted. In most cases, the mass spectrometers themselves have been the major sources of residual gas in the systems in which they were used. I t has, however, been shown that more vigorous processing of components can afford drastic reduction in the
412
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
gas evolution rates, The sensitivities obtained with sector field instruments equipped with ion-current multipliers (approximately 10-13 Torr) seem quite adequate for measurements presently of interest. I n the immediate future the study of residual gases with spectrometers adequate in both sensitivity and “cleanliness” seems likely to greatly improve our understanding of u-h-v systems and processes. A less general, but highly sensitive method for partially analyzing the gases in u-h-v has been described by Redhead (231). I t consists of a modified flash filament technique in which a tungsten wire is heated at a uniform, relatively slow rate. Chemically active gases previously adsorbed onto the wire are thus made to desorb at times (i.e., temperatures) characteristic of their chemical binding energy to the tungsten. T h e pressure transients are recorded by a total pressure gauge. By introducing known gases, or by employing a mass spectrometer on initial tests, individual peaks can be identified with particular gases. Unfortunately, a single gas gives rise, in general, to more than one desorption peak (see Section IT, C ) . Despite this complication and very limited resolution, this technique provides a useful monitor of the active gas pressures in a system. Its sensitivity is such that with a 2-min adsorption time, partial pressures of approximately Torr can be detected. Because of its extreme simplicity, such a desorption spectrometer is routinely included in almost every u-h-v system built in this laboratory.
E. Measurement of Pumping Speed, Leak Rate, and Gauge Sensitivity T h e most convenient method of speed measurement in u-h-v systems is to observe the change in pressure following a step-function change in gas pressure or pumping speed. I n general, the partial pressure ( p ) of a specific gas in a system is given by, dt
where L is the rate of influx of the gas into the system from all sources (molecules/sec), S is the total speed of all pumps (literslsec), V is the volume of the system (liters), and no = 3.27 x 1019 molecules/liter at 1 Torr and 295°K.T h e solution of the above equation yields,
P -P m
= (Pi- PaJ exP (-
tl4
(33) where p m is the steady pressure reached when dpldt = 0, pi is the pressure at t = 0, and r = V / S is the characteristic time of the exponential pump-down curve. At time t = 0 a rapid change is made in either the leak-rate ( L )or the speed (S) and the exponential pressure-time curve is measured. T h e
ULTRAHIGH VACUUM
413
speed is obtained by estimating the characteristic time of the exponential pump-down curve. T h e sudden change in conditions at t = 0 can be made in several ways. If gas is being fed into the system through a valve, at t = 0 the valve can be rapidly closed. A second method, which is only applicable to gases which can be chemisorbed, is to allow the gas to adsorb on a metal wire; the wire is suddenly heated to desorb a burst of gas, and the subsequent pump-down curve measured. T h e wire must be maintained at a high temperature ( > 2000°K) during the pump-down, to prevent readsorption of gas on the wire. A third method is to physically adsorb gas onto a cold surface and then rapidly heat the surface at t = 0. A fourth method is applicable when one of the pumps in the system is an ionization pump (or an ionization gauge behaving as a pump). T h e ionization pump is switched off, and at t = 0 is switched on again. T h e pump-down curve so obtained has a characteristic time determined by the sum of the speeds of all pumps inthesystem, notthespeedof the ionization pump alone. T h e rate of influx of gas (leak rate) can be found from a measurement of speed ( S ) and ultimate pressure (pa). Referring to Eq. (32) it can be seen that when dpldt = 0, and p = pm, then L
=
nos&
=
V no7p
(34)
T h e most sensitive method of measuring small leak rates is to accumulate the gas that has leaked in and then measure the pressure of the accumulated gas. Gas may be accumulated in the gaseous or adsorbed phases. I n a system using ionization pumps the leak rate for inert gases can be easily obtained by switching off the ionization pumps (or gauges), thus the pumping speed for inert gases becomes zero. T h e inert gas is allowed to accumulate for a known time and then an ionization gauge is turned on momentarily to measure pressure. This method is particularly suitable for measuring the rate of permeation of helium through the glass walls of a system. In some cases, the gas can be accumulated by adsorption onto a cold surface (physisorption) or by chemisorption onto a metal surface at room temperature. T h e gas is later released by suddenly heating the adsorbing surface. As an indication of the sensitivity of this method, leaks of about lo3molecules/sec of hydrogen have been detected by adsorption and storage on a tungsten filament. T h e accurate calibration of ionization gauges in the u-h-v range is a problem which awaits a completely satisfactory solution. Calibration with the inert gases can be achieved with reasonable accuracy but calibration with the chemically active gases is beset with many of the difficulties
414
P. A. REDHEAD, J . P. HOBSON, E. V. KORNELSEN
described in Section 111, C, 5 . The only satisfactory standard is the McLeod gauge, thus direct calibration against a standard is only possible at pressures above about Torr. Having calibrated the gauge against Torr range, the pressure vs. ion-current a McLeod gauge in the relationship for the lower pressure range must be established by an indirect method. T h e method described by Alpert (90) is satisfactory for Torr. Three volumes are used in calibration in the range 10-lo to this method, separated by adjustable valves. T h e pressure in the third volume will increase proportionally to the square of time. T h e pumping speed of the gauges must be essentially zero; this implies that the gauges must be operated intermittently and all chemical pumps must be saturated. By plotting the ion-current of the gauge in the third volume against t2, the ion-current vs. pressure relationship can be obtained with the use of calibration points obtained with a McLeod gauge in the Torr region. Hobson (45) has described a method for calibrating one ionization gauge against a second pressure gauge, which can be a McLeod gauge if desired. This method is not restricted by the pumping speed of the gauges. Two volumes are used: the first containing the gauge to be tested and a high-speed pump (Hobson used a liquid-helium-cooled finger); the second containing the gauge to be used as a standard. T h e second volume is filled to a pressure near the lower limit of the standard gauge. A valve between the two volumes is opened slightly and the pressure allowed to equilibrate, T h e pressure in the first volume (pl), is then proportional to that in the second volume (p,), independent of the gauge pumping speed, provided that the speed of the high-speed pump greatly exceeds that of the gauge, T h e valve opening is kept fixed and the pressure in the second volume is increased. p , is then plotted against p,. Provided the current-pressure relation is known for the gauge in the second volume in the higher pressure range, then the current-pressure relation for the gauge in the first volume can be found in the lower pressure range. This method has been used to calibrate gauges from 10-11 to 10-7 Torr. Bayard-Alpert gauges have been found to have a linear ion-current vs. pressure characteristic for inert gases from the lowest measurable pressures to about Torr. T h e cause of non-linearity at higher pressures has been discussed by Schulz (250).
F. Components T h e main components of u-h-v systems (pumps, gauges, traps) have been discussed in some detail in other sections of this paper. Descrip-
ULTRAHIGH VACUUM
415
tions of other components specifically designed for u-h-v applications are quite numerous and have been reviewed by others (2,12) ( 2 7 8 ~ )This . section will be devoted to a brief survey of some auxiliary components not discussed elsewhere in this paper.
1. Valves. Considerable attention has been given to the design of valves suitable for gas handling and isolation functions in u-h-v. T h e primary requirements for such valves are bakahility and very small closed leakage conductance. Because of the bakability requirement, all valves employ metal or glass rather than organic compounds for the seat material. Closure is achieved either by transmitting force to a metal valve seat through a flexible portion of the vacuum envelope or by moving a pair of precisely mating glass surfaces into contact by an external magnetic field. Small all-metal u-h-v valves (with open conductances -1 liter/sec) employing flexible metal diaphragms have been described by Alpert (279), Bills and Allen (280), Thorness and Nier (281), and several others. Somewhat larger all metal valves (-100 liters/sec open conductance) which use metal bellows rather than diaphragms have also been developed (282). More recently a much larger valve with a throat diameter 8-3/4 in. and an open conductance -2000 literslsec has been used successfully (283). I n metal seat valves, closing forces must be rather large since the closure depends on plastic flow of one of the seat metals. T h e valves must consequently be structurally very strong and solidly mounted. Glass seat valves (284) are frequently used to isolate volumes within a vacuum system. T h e closure, which depends on the precise mating of two polished glass surfaces, does not give extremely low conductances (usually liter/sec) so that such valves are unsuitable for isolation against the external atmosphere. I n contrast to metal valves, seating forces are very small, and open conductances up to -20 literlsec are easy to realize in a small assembly. Another class of magnetically actuated valves use the sealing action of a molten metal of low vapor pressure such as indium or tin (285,286). A magnetically moved body is made to displace the molten metal in such a way that isolation is achieved and the metal is then usually allowed to cool and solidify, Closed conductances are reported to be immeasurably small (287) and no large forces are involved. T h e metal to be melted must, however, be of very high purity to ensure proper “wetting” and avoid excessive gas evolution during opening or closing. T h e small all-metal valves first described in this section have conductances which can be fairly easily controlled in the range lO--’O to literlsec. They can thus serve as excellent controlled leaks for gas
416
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
handling in u-h-v. A valve of low closing torque (-lo4 cmgm) specifically for such applications is available commercially. It is .claimed that to lo-, liter/sec can be reproduced with with it conductances of good accuracy. Other controlled leaks have been described which rely on the permeation properties of specific gas-solid combinations, as for example 0,-Ag (288), H,-Ni (289), H,-Pd (290), and He-SiO, (291). Such leaks have the advantage of preferentially transmitting the desired gas, thus effectively enhancing its purity. Leak rates are limited to Torr liter/sec), controlled by the temperrelatively small values (< ature of the solid. Electrical control of the flow rate (through the heater power) lends itself to pressure regulition in some cases.
2. Gaskets, Seals, and Transitions. The problem of demountably connecting sections of metal envelope together and of providing transitions from metal to ceramic or glass requires special attention in u-h-v systems. Again, the most important requirements are very low leakage and bakability. For high temperature baking (> 4OO0C), gaskets of gold wire, copper, and aluminium of a variety of shapes and sizes have been successfully used (205,230,292-294). The large forces necessary to produce low leakage seals at metal gaskets require rather heavy flanges and a ring of evenly spaced bolts. Surface finish and machining tolerances on the flange structures vary somewhat but are in general only moderately stringent. The use of neoprene O-rings cooled to - 25°C in systems with maximum bakeout temperatures -250°C has been reported to give pressures as low as 1.5 x 10-10 Torr in a moderate-sized metal system evacuated by a trapped diffusion pump (80). This technique should be adaptable to very much larger systems of similar design. For electrical or optical reasons it is frequently necessary to provide, in a metal u-h-v envelope, a transition to a ceramic or glass section. The technology of effecting seals between metals and glasses or ceramics has been extensively developed in the electronic tube industry [see Kohl (223), Chapters 13 and 141. No special techniques are normally required for u-h-v apparatus. An interesting compression seal technique has been used (295) to produce large diameter joints between cylindrical metal sections and either high alumina ceramic cylinders or sapphire discs. These were components of the u-h-v system of the Princeton C Stellarator.
ULTRAHIGH VACUUM
417
IV. APPLICATIONS
A . Surface Physics and Chemistry T h e applications of u-h-v technology to the problems of surface physics and chemistry are much too numerous to allow a detailed enumeration here. I n general, as we have seen in earlier sections, whenever measurements are to be made on an atomically clean surface, u-h-v techniques are essential. Some of the recent work on the adsorption of gases on solids using u-h-v technology has already been mentioned in Sections 11, B and 11, C. Becker (63) has evaluated the effects of poor vacuum techniques on some of the earlier work on the chemisorption of gases on metals and clearly establishes the need for u-h-v techniques in this field. I n particular, he points out the difficulties of obtaining clean metal surfaces by evaporation, a method very widely used for the measurement of chemisorption on metals. Hickmott (296) has also demonstrated that evaporated films, in particular tungsten, do not have surfaces of the same degree of cleanliness as can be achieved with filaments or ribbons of the metal. T h e use of u-h-v methods for obtaining stable field emission has been demonstrated by the work of the group at Linfield Research Institute (297,298, 299). T h e general problems of field emission have been reviewed by Dyke (300)and the u-h-v methods employed have been discussed by Martin (301). By the use of alumino-silicate glass (Corning 1720) the rate of permeation of helium into the field emission tubes has been reduced and very low pressures can be maintained for extended periods (about Torr of adsorbable gases). Field emission work is one of the few cases where the helium present in the residual gas is troublesome; in this case the helium causes sputtering of the field emitting tip and a slow increase in its radius. T h e effect of surface contamination on the measurement of secondary emission has been pointed out by Jonker (302) and he shows that if care is taken to ensure adequate surface cleanliness the secondary emission coefficient data for many metals will fit a universal curve. T h e extreme care necessary to prepare clean films for work-function measurements was demonstrated some time ago by Anderson (303) in what was probably the first realization of u-h-v conditions. T h e exoemission of ,electrons is another surface phenomenon affected by the presence of surface gas ; very few experiments on exo-emission have becn carried out under vacuum conditions, which would ensure reproducible surface conditions. For a review of exo-emission phenomena, see
418
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
Grunberg (304) and Huguenin (305). U-h-v techniques have been used in recent work on the interaction of slow electrons with atomically clean surfaces (306-312). U-h-v techniques have also been used in studies of the condensation of atomic beams of metals on surfaces. Rapp et al. (313, 324) find that the condensation coefficient of silver, cadmium, and'zinc from the vapor is essentially unity, and that the ambient pressure does not affect this conclusion at high incident beam flux from pressures of 10-lo to Torr.
B. Thin Films T h e important role played by vacuum conditions in determining the physical properties of evaporated metallic films has recently become widely recognized. I n brief summary, the presence of residual active gases prior to and during the deposition of a film (a) affect the nucleation site density, the mobility, and the adhesion of the depositing atoms by forming adsorbed layers on the substrate, (b) tend to create a high density of imperfections within the film by adsorption during the course of deposition. T h e electric and magnetic properties of t h e deposited film can, in consequence, show serious deviations from those expected from the nature and temperature of the substrate and the annealing procedure subsequent to deposition. Since film deposition rates are usually from one to a few hundred atomic layers per second, the residual gas conditions required to avoid these complicating effects are similar to those for measurements on metal surfaces free of adsorbed gases, namely u-h-v conditions in most cases. T h e maintenance of u-h-v during film deposition is difficult due to the outgassing of the evaporating metal, Rigorous prior outgassing of the evaporator is in general necessary to achieve the desired conditions. An excellent cross section of recent contributions to thin film research can be found in the proceedings of a recent conference (315). U-h-v techniques have been used in several of the reported investigations and it can be said that an understanding of the fundamental factors which determine thin film properties is beginning to emerge. Among the most important factors seem to be:
(1) The nature and state of roughness of the substrate. (2) T h e temperature of the substrate during deposition. (3) T h e annealing temperature of the film material relative to the substrate temperature. (4) T h e mobility of the deposited atoms on the substrate.
ULTRAHIGH VACUUM
419
Once ambiguities due to residual gas influence have been removed, it appears possible to obtain films of specific desired properties by proper choice of film, substrate, deposition temperature and annealing procedure. Thus Mayer (326) has studied the fundamental problem of electronic conductivity of metals using u-h-v eyaporated alkali metal films at 90°K. Evans and Mitchell (317) have studied the resistivity of thin copper films on glass during oxygen adsorption, and the saturation magnetization of very thin pure nickel films on glass has been examined (318).A detailed examination of the effects of various residual gases on the superconducting properties of thin tin films has been made by Caswell (329). T h e effects appear to be strongly dependent on the particular gas, as might be expected from their chemisorption characteristics. I n a recent review of the application of thin films to electronic component production, Greenland (320) expresses the opinion that u-h-v techniques may be of great practical importance in this field if they allow the production of thin films possessing nearly bulk properties. In no such application is the anomalous structure resulting from residual gas adsorption considered to be anything but detrimental. Such techniques are expected to undergo rapid development in the near future, and thin film electronic component production may well become the first large scale commercial application of u-h-v.
C . Thermonuclear and Plasma Devices T h e effect of impurities in the operation of controlled thermonuclear devices has been discussed by several authors (13, 321). T h e impurities in the plasma increase the rate at which energy is radiated from the plasma. This effect increases with the atomic number (2)of the impurities. T h e presence of very small amounts of impurities of large z can prevent the attainment of the required plasma temperature. T h e vacuum problem in a controlled thermonuclear device is twofold; (a) to achieve u-h-v in the system before the introduction of the plasma gas (usually deuterium or tritium), and (b) to minimize contamination of the plasma caused by its interaction with the walls of the chamber. Very great advances have been made in the past few years in achieving u-h-v in the large and complex systems required for controlled thermonuclear machines. T h e general technical problems involved have been discussed by Munday (12). Work on the Stellarator project at Princeton has produced significant improvements in u-h-v technology, and has been reported in considerable detail (23,230,204, 271). T h e Model C Stellarator has a vacuum chamber with a 400-liter volume, which is
420
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
bakable at high temperatures, and which consists of a stainless steel tube 20 cm inside diameter, in the shape of an oval with a peripheral length of 120 meters. Pressures below 3 x 10-lo Torr have been obtained with a 25 cm diameter mercury diffusion pump. Fig. 18(b) shows an over-all view of the Model C Stellarator. Sledziewski (322) describes a French thermonuclear device which has a volume of 400 liters and is pumped by six oil diffusion pumps with a total speed of 1400 liters/sec. The system can be baked only at 100°C because of the use of indium for some of the seals. The minimum pressure Torr. obtained is about 3 x The Russian experimental thermonuclear device “Ogra” has been described by Kurchatov (323). This device can be baked at 450°C and is pumped to Torr by mercury diffusion pumps,. Titanium getterion-pumps are then turned on and the pressure reduced to lo-* Torr. Titanium is evaporated directly onto the walls of the chamber so that the walls of the confining chamber become an adsorbing getter surface. Vacuum systems for the Mirror Machine and associated experiments at Berkeley have been described by Milleron (205, 183).
D. Space Simulation I n the last few years the need has risen to simulate extraterrestrial conditions in the laboratory. We are concerned here only with the simulation in the laboratory of the pressure conditions of outer space. The chief requirement for such systems is in the testing of complete space-vehicles and their components under the pressure conditions they will experience in flight. This requirement has led to the design of very large u-h-v systems, in the largest of which it is possible to test complete artificial satellites. There is still considerable uncertainty in the value of pressures to be expected at a given altitude and in the gaseous composition of the upper atmosphere, but the generally accepted pressureversus-altitude curve is expressed by the “model atmosphere” curves Torr is [ARDC Model Atmosphere, 1959 (324)l. A pressure of reached at an altitude of about 320 km. Construction of the space chambers is a particularly difficult job, not only because of their large size but also because of the large number of observation ports, doors, and other openings that must be provided in the walls of the chamber. Moreover, in some systems the problems are further compounded by the introduction of mechanical motion, vibration, high radiant flux, etc. All these facilities must be provided in a chamber capable of being pumped to pressures of about 10-lo Torr. In discussing pressure simulation of outer space Santeler (325) points
42 I
ULTRAHIGH VACUUM
out that molecules emitted from a vehicle in space rarely return to the vehicle. If this condition is to be simulated in the laboratory, then it is necessary for the walls surrounding a test vehicle to have an effective condensation coefficient approaching unity. To achieve this with diffusion pumps for a test chamber 25 meters in diameter, would be prohibitively expensive. T h e cost for direct liquid hydrogen or liquid helium cooling of the walls would also be too high. Santeler (325) proposed as a solution walls consisting of panels cooled to liquid hydrogen temperatures but shielded from radiation from the test vehicle by panels cooled with liquid nitrogen. With this arrangement the wall condensation eficiency is reduced by a factor of two, but the cost is reduced by about a factor of thirty. Bennett (326) has proposed a system in which the test object is placed in a volume surrounded by the vapor stream from the annular jet of a large diffusion pump. Molecules from the test object would be entrapped in the vapor stream and removed from the system. w 5 0 FEETSOLAR LIGHT SOURCE
DIFFUSION P U M P S VACUUM WALL
100' K COLD WALL
20" K CRYOGENIC PUMPS
looo K
CRYOGENIC P U M P SHIELDS
FIG. 26. Schematic diagram of space simulation chamber. (Courtesy of R.C.A.).
A chamber of about 1300 liters volume, capable of reaching a pressure of 4 x 10-1O Torr, has been described by Simons (206). This system consists of a stainless steel cylinder 1 meter in diameter with a removable door at one end. T h e system is baked at 200°C. A 25 cm diameter oil diffusion pump is used to evacuate the system. Some of the technical developments leading to the design of this system have been described by Farkass (2Z4).
422
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
Considerably larger chambers for space simulation are now being built, technical details of their design and performance have not yet been published. Figure 26 shows a schematic diagram of a large space simulation chamber.
REFERENCES I. Bayard, R. T., and Alpert, D., Rew. Sci. Znstr. 21, 571 (1950). 2. Alpert, D., in “Handbuch der Physik” (S. Flugge, ed.), Vol. 12, p. 609. Springer, Berlin, 1958. 3. Schweitzer, J., Le Vide 14, 165 (1959). 4. Venema, A., and Bandringa, M., Philips Tech. Rev. 20, 145 (1958). 5. Redhead, P. A., and Kornelsen, E. V., Vukuum-Technik. 10, 31 (1961). 6. Yanvood, J., J . Sci. Instr. 34, 297 (1957). 7. Kleint, C., Expte. Tech. Physik 8, 193 (1960). 8. Lafferty, J. M., and Vanderslice, T. A,, Proc. Z.R.E. 49, 1136 (1961). 9. Pollard, J., Repts. Progr. in Phys. 22, 33 (1959). 10. Men’shikov, M. I., Pribory i Tekh. Eksptl. No. 4, 3 , July-Aug. (1959); English transl., Instr. and Exptl. Tech. No. 4, 511, July-Aug. (1959). 11. Jancke, H., Exptl. Tech. Physik 7, 241 (1959). 12. Munday, G. L., Nuclear Instr. & Methods 4, 367 (1959). 13. Grove, D. J., Trum. Nutl. Symposium on Vucuum Tech. 5, 9 (1958). 14. de Boer, J. H., “The Dynamical Character of Adsorption.” Oxford Univ. Press, London and New York, 1953. 15. Brunauer, S . , “The Adsorption of Gases and Vapors.” Princeton Univ. Press, Princeton, New Jersey, 1945. 15u. Tompkins, F. C., Le Vide 17, 72 (1962). 16. Foner, S. N., et ul., J. Chem. Phys. 31, 546 (1959). 17. Schafer, K., and Gerstacker, H., Z. Elektrochem. 60, 874 (1956). 18. Becker, J. A., in “Structure and Properties of Solid Surfaces” (R. Gomer and C.S . Smith, eds.), p. 459. Univ. of Chicago Press, Chicago, Illinois, 1952. 19. Mickelsen, W. R.,and Childs, J. H., Rew. Sci. Instr. 29, 871 (1958). 20. Hurlbut, F. C., J. Appl. Phys. 28, 844 (1957). 21. Cabrera, N., Discussions Furaduy SOC.28, 16 (1959). 22. Zwanzig, R. W., J. Chem. Phys. 32, 1173 (1960). 23. Littlewood, R., and Rideal, E., Trans. Furaduy SOC.52, 1598 (1956). 24. Langrnuir, I.,-J. Am. Chem. SOC.54, 2798 (1932). 25. Brunauer, S., Emmett, P. H., and Teller, E., J. Am. Chem. SOC.60, 309 (1938). 26. Kruyer, S., Proc. Koninkl. Ned. Akud Wetenschap. B58, 73 (1955). 27. Hayashi, C., Trans. Nutl. Symposium on Vacuum Tech. 4, 13 (1957). 28. Clausing, P., Ann. Physik [5] 7. 489 (1930). 28u. Clausing, P., Physicu 28, 298 (1962). 28b. Eschbach, H. L., Jaeckel, R., and Mliller, D., T~uns.Nutl. Symposium 011 Vucuum Tech. 8, 1110 (1961). 29. Ehrlich, G., J . Chem. Phys. 34, 39 (1961). 30. Ehrlich, G., J . Chem. Phys. 34, 29 (1961). 31. Ehrlich, G., and Hudda, F. G., J . Chem. Phys. 30, 493 (1959). 32. Ehrlich, G., Hickmott, T. W., and Hudda, F. G., J . Chem. Phys. 28, 977 (1958).
ULTRAHIGH VACUUM
423
33. Gomer, R., J . Chem. Phys. 29, 441 (1958). 34. Gomer, R., J . Phys. Chem. 63, 468 (1959). 35. Devienne, F. M., Mimorial Sci. phys. (Acad. sci. Paris) No. 56 (1953). 35a. Hartnett, J. P., “Rarefied Gas Dynamics,” p. 1. Academicpress, New York, 1961. 36. Thomas, L. B., and Golike, R. C., J. Chem. Phys. 22, 300 (1954). 37. Schafer, K., and Gerstacker, H., 2. Elektrochem. 59, 1023 (1955). 38. Dushman, S., “The Scientific Foundations of Vacuum Technique.” Wiley, New York, 2”“ edition 1961. 39. Steele, W. A., and Halsey, G. D., Jr., J. Chem. Phys. 22, 979 (1954). 40. Steele, W. A., and Halsey, G. D., Jr., 3. Phys. Chem. 59, 57 (1955). 41. Freeman, M. P., and Halsey, G. D., Jr., J. Phys. Chem. 59, 181 (1955). 42. Sams, J. R., Jr., Constabaris, G., and Halsey, G. D., Jr., J. Phys. Chem. 64, 1689 (1960). 43. Drain, L. E., Trans. Faraday SOC.49, 650 (1953). 44. Rhodin, T. N., Jr., J. Am. Chem. SOC.72, 5691 (1950). 45. Hobson, J. P., Can. J. Phys. 37, 300 (1959). 46. Stout, J. W., and Giauque, W. F., 3. Am. Chem. SOC.60, 393 (1938). 47. Hobson, J. P., and Redhead, P. A., Can. J. Phys. 36, 271 (1958). 48. Aston, J. G . ,Mastrangelo,S.V.R., andTykodi, R. J.,J. Chem. Phys. 23, 1633 (1955). 49. Hobson, J. P., Can. J . Phys. 37, 1105 (1959). 50. Hobson, J. P., J. Chem. Phys. 34, 1850 (1961). 51. Hobson, J. P.,and Armstrong, R. A., Rept. 21st Ann.M.Z.T. Conf. on Phys. Electronics p. 236 (1961). 5 l a . Hansen, N., Vakuum Technik (in press). 516. Hobson, J. P., Trans. Natl. Symposium on Vacuum Tech. 8, 146 (1961). 52. Gomer, R., Wortman, R., and Lundy, R., J . Chem. Phys. 26, 1147 (1957). 511. Allen, C. W., “Astrophysical Quantities,” p. 227. Univ. of London Press (Athlone), London, 1955. 54. Gomer, R., J . Chem. Phys. 28, 168 (1958). 54a. Hengevoss, J. and Huber, W. K., Trans. Natl. Symposium on Vacuum Tech. 8, 1332 (1961). 55. Garbe, S., Klopfer, A., and Schmidt, W., Vacuum 10, 81 (1960). 55a. Tuzi, Y. and Okamoto, H., J. Phys. SOC.Japan 13, 960 (1958). 556. Tuzi, Y., J. Phys. SOC.Japan 17, 218 (1962). 56 Becker, G. E., Trans. Natl. Symposium on Vacuum Tech. 6, 197 (1959). 57. Balwanz, W. W., Singer, J. R., and Frandsen, N. P., Trans. Natl. Symposium on Vacuum Tech. 6, 159 (1959). 58. Penchko, E. A., Khavkin, L. P., and Borodkin, A. S., Pribory i Tekh. Eksptl. No. 4, p. 146, July-Aug. (1959); English transl., Znstr. and Exptl. Tech. No. 4, 667, July-Aug. (1959). 59. Keesom, W. H., and Schweers, J., Physica 8, 1032 (1941). 60. Singleton, J. H., and Halsey, G. D., Jr., J. Phys. Chem. 58, 330 (1954). 61. Gundry, P. M., and Tompkins, F. C., Quart. Revs. (London) 14, 257 (1960). 62. Becker, J. A., Solid State Phys. 7 , 379 (1958). 63. Becker, J. A., Adwances in Catalysis 7 , 135 (1955). 64. Brennan, D., Hayward, D. O., and Trapnell, B. M. W., Proc. Roy. SOC.(London) A256, 81 (1960). 65. Ehrlich, G., 3. Chem. Phys. 31, 1111 (1959). 66. Beeck, O., Discussions Faraday SOC.8, 118 (1950). 67. Trapnell, B. M. W., Proc. Roy. SOC.(London) A218, 566 (1953).
424
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
68. Beeck, O.,Advances in Catalysis 2, 151 (1950). 69. Bagg, J., and Tompkins, F. C., Trans. Faraduy Soc.. 51, 1071 (1955). 70. Baker, M. McD., and Rideal, E. K., Trans. Faraday SOC.51, 1597 (1955). 71. Beeck, O., Cole, W. A., and Wheeler, A., lX.~cussionsFaraday Soc. 8, 314 (1950). 72. Redhead, P. A., Trans. Faraday SOC.57, 641 (1961). 73. Trapnell, B. M. W., Trans. Faruduy Soc. 52, 1618 (1956). 73a. Roberts, R. W., Nature 191, 120 (1961). 74. Collins, R. H. and Turnbull, J. C., Vacuum 11, 114 and 119 (1961). 75. Wagener, J. S. and Marth, P. T . , J . Appl. Phys. 28, 1027 (1957). 76. Dayton, B. B., Trans. Natl. Symposium on Vacuum Tech. 6, 101 (1959). 77. Jaeckel, R., and Schittko, F. J., “Forschungsberichte des Wirtschafts und Verkehrministeriums Nordrhein-Westfalen” No. 369. Westdeut. Verlag, Koln and Opladen, 1957. 78. Santeler, D. J., Trans. Natl. Symposium on Vacuum Tech. 5, 1 (1958). 79. Wolsky, S. P., and Zdanuk, E. J., Vacuum 10, 13 (1960). 80. Farkass, I., and Barry, E. J., Trans. Natl. Symposium on Vacuum Tech. 8, 66 (1961). 81. Koenig, H.,Vacuum 3, 3 (1953). 81a. Venema, A., Trans. Natl. Symposium on Vacuum Tech. 8, 1 (1961). 81b. Hobson, J. P., Trans. Natl. Symposium on Vacuum Tech. 8. 26 (1961). 82. Todd, B. J., J. Appl. Phys. 26, 1238 (1955). 83. Todd, B. J., J. Appl. Phys. 27, 1209 (1956). 83a. Garbe, S. and Christians, K., Vakuum Technik 11, 9 (1962). 83b. Dayton, B. B., Trans. Natl. Symposium on Vacuum Tech. 8, 42 (1961). 83c. Hunt, A. L., Damm, C. C., and Popp, E. C., J ’. Appl. Phys. 32, 1937 (1961). 84. Hagstrum, H. D.. and d’Amico, C., J. Appl. Phys. 31, 715 (1960). 84a. Dillon, J. A. Jr., Trans. Natl. Symposium on Vumum Tech. 8, 113 (1961). 85. Norton, F. J., J. Appl. Phys. 28, 34 (1957). 85a. Norton, F. J., Trans. Natl. Symposium on Vacuum Tech. 8. 8 (1961). 86. Barrer, R. M., “Diffusion In and Through Solids.” Cambridge Univ. Press, London and New York, 1951. 87a. Altemose, V. O., J. Appl. Phys. 32, 1309 (1961). 87b. Muller, C. F. and Shepard, R. W., Vacuum 11, 58 (1961). 87c. Young, J. R. and Whetten, N. R., Rew. Sci. Instr. 31, 1112 (1960). 87d. Whetten, N. R. and Young, J. R.. Rev. Sci. Instr. 30, 472 (1959). 87e. Belyakov, Yu. I. and Ionov, N. I., Zh. Tekh. Fi2 ( U S S R ) 31, 204 (1961. Soviet Phys.-Tech. Phys. 6, 146 (1961) [English Trans.] 87j. Gorman, J. K. and Nardella, W. R., Vacuum 12, 19 (1962). 88. Lafferty, J. M., J. Appl. Phys. 32, 424 (1961). 89. Rivera, M.,and Le Riche, R., Trans. Natl. Symposium on Vacuum Tech. 6, 55 (1959). 89a. Metcalfe, R. A. and Trabert, F. W., Trans. Natl. Symposium on Vacuum Tech. 8, 1211 (1961). 90. Alpert, D., and Buritz, R. S., J. Appl. Phys. 25, 202 (1954). 91. Varnerin, L. J., and White, D. H., J . Appl. Phys. 25, 1207 (1954). 92. Rogers, W. A., Buritz, R. S. and Alpert, D., J. Appl. Phys. 25, 868 (1954). 93. McAfee, K. B., Jr., J. Chem. Phys. 28, 218 (1958). 94. McAfee, K. B., Jr., J. Chem. Phys. 28, 226 (1958). 95. Jost, W., “Diffusion in Solids, Liquids, Gases.” Academic Press, New York, 1952. 9Sa. Swets, D. E., Lee, R. W., and Frank, R. C., J . Chem. Phys. 34, 17 (1961).
ULTRAHIGH VACUUM
425
95b. Frank, R. C., Swets, D. E., and Lee, R. W.,J. Chem. Phys. 35, 1451 (1961). 9 5 . Hurlhert, R. C. and Konecny, J. O., J. Chem. Phys. 34, 655 (1961). 95d. Tiedema, T. J., De Jong, B. C., and Burgers, W. G., Proc. I(. Ned. Akad. Wetensch. B 63, 422 (1960). 95e. Frank, R. C . and Thomas, J. E. Jr., J . Phys. Chem. Solids 16, 144 (1960). 95f. Nechai, E. P., Popov, K. V., and Panenkova, L. S., Fiz. Metallov i Metallovedenie ( U S S R ) 10,838 (1 960). 95g. Foster, P. K., Naiure 188, 399 (1960). 95h. Rodin, A. M . and Surenyants, V.V., Fiz. Metallow i Metallovedenie ( U S S R ) 10, 216 (1960). 9.52’. Haas, C., J . Phys. Chem. Solids IS, 108 (1960). 95j. O’Keefe, M. and Moore, W. J., J. Phys. Chem. 65, 1438 (1961). 9Sk. Tucker, C. W. and Norton, F. J., J. Nuclear Materials 2, 329 (1960). 96. Waldschmidt, E., Metalf. 8, 749 (1954). 97. von Ardenne, M., “Tabellen der Elektronenphysik, Ionenphysik und uberrnikroskopie.” Deut. Verlag. Wiss., Berlin, 1956. 98. Hashinioto, K., Iwayanagi, H., and Fukushima, H., Vacuum 10, 92 (1960). 99. Bliss, B. J., Trans. Natl. Symposium on Vacuum Tech. 7, 145 (1960). 100. Fish, I. P. S., Rev. Sci. Znstr. 30, 889 (1959). 101. Varadi, P. F., Trans. Natl. Symposium on Vacuum Tech. 7, 149 (1960). 102. Smithells, C. J., and Ransley, C. E., PYOC.Roy. Soc. (London) A155, 195 (1936). 103. della Porta, P., and Ricca, F., Trans. Natl. Symposium on Vacuum Tech. 5, 25 (1958). 104. della Porta, P., Trans. Natl. Symposium on Vacuum Tech. 6 , 317 (1959). 104a. Honig, R. E. and Hook, H. O., R C A Rev. 21, 360 (1960). 105. Borovik, E. S., Grishin. S. F., and Grishna, E. Ya., Zhur. Tekh. Fiz. 30, 539 (1960); English transl., Soviet Phys.-Tech. Phys. 5 , 506 (1960). 106. Wagener, S., Vucuum 3, I 1 (1953). 107. Leiby, C. C., Jr., and Chen, C. L., J . Appl. Phys. 31, 268 (1960). 108. Eschhach, H. L., in “Advances in Vacuum Science and Technology” (E. Thomas, ed.), Vol. 1, p. 373, Pergamon Press, New York, 1960. 109. Thulin, S., Arkiv. Fysik 9 , 107 (1955). 110. Bohr, N., K g l . Danske Videnskab. Selskab, Mat-fys. Medd. 18, 8 (1948). 111. Gibson, J. B., Goland, A. N., Milgram, M., and Vinegard, G. H., Phys. Rev. 120, 1229 (1960). 112. Rirnrner, D. E., and Cottrell, A. H., Phil. Mag. [8] 2, 1345 (1957). 113. Young, J. R.,J. Appl. Phys. 27, 1 (1956). 114. Beliakova, V. V., and Mittsev, M. A., Zhur. Tekh. Fiz. 27, 803 (1957); English transl., Soviet Phys.-Tech. Phys. 2, 729 (1957). 115. Kornelsen, E. V., Trans. Natl. Symposium on Vacuum Tech. 7, 29 (1960). 116. Nielsen, K. O., in “Electromagnetically Enriched Isotopes and Mass Spectrometry” (M. L. Smith, ed.), p. 68. Butterworths, London, 1956. 117. Bredov, M. M., Lang, Z. O., and Okuneva, N. W., Zhur. Tekh. Fiz. 28,252 (1958); English transl., Souiet Phys.-Tech. Phys. 3, 228 (1958). 117a. Davies, J. A. and Sims, G. A,, Can. J. Chew. 39, 601 (1961). 118. Varnerin, L. J., and Carmichael, J. H., J. Appl. Phys. 28, 913 (1957). 119. Buritz, R. S., and Varnerin, L. J., Dept. of the Air Force (U:S.) Report 71F191-R4 (1955). 120. Carter, G., and Leck, J. H., Trans. Natl. Symposium on Vacuum Tech. 7, 339 (1960).
426
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
121. Leck, J. H., and Carter, G., in “Advances in Vacuum Science and Technology” (E. Thomas, ed.), Vol. 1, p. 463, Pergamon Press, New York, 1960. 122. Kornelsen, E. V., Trans. Natl. Symposium on Vacuum Tech. 8, 281 (1961). 123. Carmichael, J. H., and Trendelenburg, E. A., J. AppZ. Phys. 29, 1570 (1958). 124. Schwarz, H., Z. Physik 122, 437 (1944). 125. Bills, D. G., and Carleton, N. P., J. AppZ. Phys. 29, 692 (1958). 126. Almkn, O., and Bruce, G., Nuclear Instr. @ Methods 11, 257 (1961). 127. Kuchai, S. A., and Rodin, A.M., Atomnaya Energ. 4, 202 (1958). English transl., Soviet J . Atomic Energy 4, 277 (1958). 127a. Carter, G., Colligon, J. S., and Leck, J. H., Proc. Phys. SOC.79, 299 (1962). 128. Carmichael, J. H., and Knoll, J. S., Trans. Natl. Symposium on Vacuum Tech. 5, 18 (1958). 129. Fox, R. E., and Knoll, J. S., Trans. Natl. Symposium on Vacuum Tech. 7, 364 (1960). 130. Le Claire, A. D., and Rowe, A. H., Rev. mPt. 52, 94 (1955). 131. Redhead, P. A,, Trans. Natl. Symposium on Vacuum Tech. 6,. 12 (1959). 131a. Reich, G., Vakuum Technik 10, 109 (1961). 132. Wehner, G. K., Advances in Electronics and Electron Phys. 7, 239 (1955). 133. Wehner, G . K., Phys. Rev. 102, 690 (1956). 134. Laegreid, N . , and Wehner, G. K., J. Appl. Phys. 32, 365 (1961). 135. Stuart, R. V., and Wehner, G. K., Phys. Rev. Letters 4, 409 (1960). 136. Rol, P. K., Fluit, J. M., Wiehbock, F. P., and de Jong, M., in “Proceedings of the Fourth International Conference on Ionization Phenomena in Gases, 1959” (N. R. Nilsson, ed.). Vol. 1. p. 257. North Holland, Amsterdam, 1960. 137. Yonts, 0. C., Normand, C. E., and Harrison, D. E., Jr., J. AppLPhys. 31, 447 (1960). 138. Wehner, G. K., and Rosenberg, D., J. Appl. Phys. 32, 887 (1961). 139. Anderson, G. S . , and Wehner, G. K., J. Appl. Phys. 31, 2305 (1960). 140. Keywell, F., Phys. Rev. 97, 1611 (1955). 141. Hines, R. L., and Wallor, R., J. Appl. Phys. 32, 202 (1961). 142. Wolsky, S. P., and Zdanuk, E. J., Trans. Natl. Symposium on Vacuum Tech. 6, 6 (1959). 143. Farnsworth, H. E., Schlier, R. E., George, T. H., and Burger, R. M., J . Appl. Phys. 29, 1150 (1958). 144. Dillon, J. A., Jr., and Farnsworth, H. E., J. Appl. Phys. 29, 1195 (1958). 145. Hagstrum, H. D., Phys. R w . 104, 672 (1956). 146. Hagstrum, H. D., Phys. Rev. 104, 1516 (1956). 147. Hagstrum, H. D., Phys. Rev. 119, 940 (1960). 148. Waters, P. M., Phys. Rev. 109, 1466 (1958). 149. Waters, P. M., Phys. Rev. 111, 1053 (1958). 150. Little, P. F., in “Handbuch der Physik” (S. Fliigge, ed.), Vol. 21, p. 574. Springer, Berlin, 1956. 15I. Aarset, B., Cloud, R. W., and Trump, J. C., J. Appl. Phys. 25, 1365 (1954). 152. Hagstrum, H. D., Phys. Rev. 123,758 (1961). 153. BrunnCe, C., Z . Physik 147, 161 (1957). 154. Bradley, R. C., Arking, A., and Beers, D. S., J. Chem. Phys. 33, 764 (1960). 155. Stanton, H. E., J. Appl. Phys. 31, 678 (1960). 156. Bills, D. G., Pkys. Rev. 107, 994 (1957). 157. Bills, D. G., and Evett, A. A., J. Appl. Phys. 30, 564 (1959). 157a. Donaldson, E. E., Vacuum 12, 11 (1962).
ULTRAHIGH VACUUM
427
158. Nottingham, W. B., “Bibliography on Physical Electronics.” Addison-Wesley, Reading, Massachusetts, 1954. 159. Moore, G. E., J. Appl. Phys. 32, 1241 (1961). 160. Reynolds, J. H., Rev. Sci. Znstr. 27, 928 (1956). 161. Jacob, L., Proc. Phys. SOC.(London) B70,235 (1957). 162. Todd, B. J., Lineweaver, J. L., and Kerr, J. T., J. Appl. Phys. 31, 51 (1960). 163. Terenin, A., and Solonitzin, Yu., Discussions Faraday SOC.28, 28 (1959). 164. Valnev, J., Zhur. Fiz. Khim. 30, 1308 (1956). M a . Lange, W. J. and Riemersma, H., Trans. Natl. Symposium on Vacuum Tech. 8, 167 (1961). 165. Kenty, C., Rept. 13th Ann. M.I.T. Conf. on Phys. Electronics p. 138 (1953). 166. Hickmott, T. W., J. Appl. Phys. 31, 128 (1960). 167. Becker, J. A,, Becker, E. J., and Brandes, R. G., J . Appl. Phys. 32,411 (1961). 168. Young, J. R., J. Appl. Phys. 30, 1671 (1959). 169. Rhodin, T. N., and Rovner, L. H., Trans. Natl. Symposium on Vacuum Tech. I . 228 (1960). 170. Smith, H. R., Trans. Natl. Symposium on Vacuum Tech. 6, 140 (1959). 171. Dreyer, K., and Mark, J. T., R.C.A. Rev. 21, 508 (1960). 172. Hablanian, M. H., and Landfors, A. A., Trans. Natl. Symposium on Vacuum Tech. 7, 55 (1960). 173. Hickman, K. C.D., Trans. Natf. Symposium on Vacuum Tech. 8, 307 (1961). 174. Becker, W., in “Advances in Vacuum Science and Technology” (E. Thomas, ed.), Vol. I , p. 173, Pergamon Press, New York, 1960. 17.5~.Becker, W., Vakuum Technik 10, 199 (1961). 175b. Pupp, W., Vakuum Technik 10, 195 (1961). 176. Williams, C . E., and Beams, J. W., Trans. Natl. Symposium on Vacuum Tech. 8, 295 (1961). 177. Wagener, J. S., Proc. 4th Natl. Conf. on Tube Techniques p. 1 (1958). 178. Klopfer, A., and Ermrich, W., Va’akuum-Technik8, 162 (1959). 179. Zdanuk, E. J., and Wolsky, S. P., Trans. Natl. Symposium on Vacuum Tech. 8, 188 (1961). 180. Luckert, J., Vakuum Technik 10, 1 and 40 (1961). 180a. Holland, L., Laurenson, L., and Allen, P.G.W., Trans. Natl. Symposium on Vacuum Tech. 8, 208 (1961). 181. Stout, V . L., and Gibbons, M. D., J. Appl. Phys. 26, 1488 (1955). 182. Gibbons, M. D., and Stout, V. L., Rept. 19th Ann. M.I.T. Conf. on Phys. Electronics p. 117 (1959). 183. Milleron, N . , and Popp, E. C., Trans. Natl. Symposium on Vacuum Tech. 5, 153 (1958). 184. Jaeckel, R., and Teloy, E., Trans. Natl. Symposium on Vacuum Tech. 8,406 (1961). 185. Holland, L., Nature 185, 911 (1960). 186. Alpert, D., J. Appl. Phys. 24, 860 (1953). 187. Varnerin, L. J., and Carmichael, J. H., J. Appl. Phys. 26, 782 (1955). 188. Young, J. R., J . Appl. Phys. 27, 926 (1956). 189. Hobson, J. P., Vacuum 11, 16 (1961). 1890. Cobic, B., Carter, G., and Leck, J. H., Brit. J. Appl. Phys. 12, 288 and 384 (1961). 190. Brown, E., and Leck, J. H., Brit. J. Appl. Phys. 6, 161 (1955). 191. Alpert, D., in “Advances in Vacuum Science and Technology” (E. Thomas, ed.), Vol. 1, p. 31. Pergamon Press, New York, 1960.
428
P. A. REDHEAD, J. P. HOBSON, E. V. KORNELSEN
192. Alexeff, I., and Peterson, E. C., Trans. Natl. Symposium on Vacuum Tech. 2, 87 (1955). 193. Herb, R. G., Davis, R.H., Divatia, A.S., and Saxon, D., Phys. Rev. 89, 897 (1953). 194. Hall, L. D., Rev. Sci. Instr. 29, 367 (1958). 195. Brubaker, W. M., Trans. Natl. Symposium on Vacuum Tech. 6, 302 (1959). 196. Jepsen, R. L., Francis, A. B., Rutherford, S. L., and Kietzmann, B. E., Trans. Natl. Symposium on Vacuum Tech. 7, 45 (1960). 196a. Klopfer, A,, Vucuum Technik 10, 113 (1961). 197. Rutherford, S. L., Merser, S. L., and Jepsen, R. L., Trans. Natl. Symposium on Vacuum Tech. 7, 380 (1960). 198. Foster, J. S., Jr., Lawrence, E. O., and Lofgren, E. L., Rev. Sci. Instr. 24, 388 (1953). 199. KovalSkii, G . A., and Kuchai, S. A., Pribory i Tekh. Eksptl. No. 2, 110, MarchApril (1960); English transl., Instr. and Exptl. Tech. No. 2, 293, March-April (1960). 200. Ullman, J. R., Trans. Natl. Symposium on Vacuum Tech. 4, 95 (1957). 201. Smith, H. R., and Kennedy, P. B., Trans. Natl. Symposium on Vacuum Tech. 6, 271 (1959). 202. Caswell, H. L., Trans. Natl. Symposium on Vacuum Tech. 6, 66 (1959). 203. Caswell, H. L., Rew. Sci. Instr. 30, 1054 (1959). 204. Henderson, W. G., Geiger, G. S., and Mark, J. T., Trans. Natl. Symposium on Vacuum Tech. 6, 170 (1959). 205. Milleron, N., Trans. Natl. Symposium on Vacuum Tech. 5, 140 (1958). 206. Simons, J. C., Jr., Trans. Natl. Symposium on Vacuum Tech. 6, 48 (1959). 207. Alpert, D., Rev. Sci. Instr. 24, 1004 (1953). 208. Trendelenburg, E. A., Le Vide 14, 74 (1959). 209. Carmichael, J. H., and Lange, W. J., Trans. Nail. Symposium on Vacuum Tech. 5, 137 (1958). 210. Biondi, M. A., Rev. Sci. Instr. 30, 831 (1959). 211. Biondi, M. A., Trans. Natl. Symposium on Vacuum Tech. 7, 24 (1960). 212. Goerz, D. J., Jr., Trans. Natl. Symposium on Vacuum Tech. 7 , 65 (1960). 212a. Haefer, R. A., Trans. Natl. Symposium on Vacuum Tech. 8, 1346 (1961). 213. Beecher, N., and Hnilicka, M. P., Trans. Natl. Symposium on Vacuum Tech. 5, 94 (1958). 214. Farkass, I., and Vanderschmidt, G. F., Trans. Natl. Symposium on Vacuum Tech. 6, 42 (1959). 215. Lazarev, B. G., Borovik, E. S., Grishin, S. F., and Tsin, N. M., Ukrain. Fix. Zhur. 2, 175 (1957). 216. Borovik, E. S., Grishin, S. F., and Lazarev, B. G., Pribory i Tekh. Eksptl. No. 1, 115 , Jan.-Feb. (1960); English transl., Znstr. and EJcptl. Tech. No. 1, 128, Jan.-Feb. ( 1960). 217. Jepsen, R. L., Mercer, S. L., and Callaghan, M. F., Rev. Sci. Instr. 30, 377 (1959). 218. Varadi, P. F., and Ettre, K., Trans. Natl. Symposium on Vacuum Tech. I , 248 (1960). 219. Degras, D. A., Le Vide 14, 128 (1959). 220. Prugne, P., and Garin, P., in “Advances in Vacuum Science and Technology” (E. Thomas, ed.), Vol. 1, p. 439. Pergamon Press, New York, 1960. 221. Ames, I., Christiansen, R. L., and Trule, J., Rev. Sci. Instr. 29, 736 (1958). 222. Behrndt, K. H., Trans. Natl. Symposium on Vacuum Tech. 6, 255 (1959). 223. Kohl, W. H., “Materials and Techniques for Electron Tubes.” Reinhold, New York. 1960.
ULTRAHIGH VACUUM
429
224. Knoll, M., “Materials and Processes of Electron Devices.” Springer, Berlin, 1959. 225. Espe, W., “Werkstoffkunde der Hochvakuumtechnik,” Vols. I , 2, and 3. Deut. Verlag Wiss., Berlin, 1960. 226. A.S.T.M. Spec. Tech. Pu61. No. 246 (1959) (21 papers). 227. Libin, I. Sh., and Rocklin, G. N., Pribory i Tekh. Eksptl. No. 3, 150, May-June (1959); English transl., Instr. and Exptl. Tech. N o . 3, 497, May-June (1959). 228. Kramers, W. J., and Dennard, F., Vacuum 3, 151 (1953). 229. Kornelsen, E. V., and Weeks, J. O., Rev. Sci. Znstr. 30, 290 (1959). 230. Mark, J. T., and Dreyer, K., Trans. Natl. Symposium on Vacuum Tech. 6, 176 (1959). 231. Holland, L., Trans. Natl. Syyposium on Vacuum Tech. 7, 168 (1960). 232. Leck, J. H., “Pressure Measurements in Vacuum Systems.” Chapman & Hall, London, 1957. 232a. Brombacher, W. G . , Natl. Bur. Standards ( U S . ) Monograph 35 (1961). 233. Paty, L., Pribory i Tekh. Eksptl. No. 6, 3 , Nov.-Dec. (1959); English transl., Instr. and Exptl. Tech. No. 6, 863, Nov.-Dec. (1959). 234. Grigor’ev, A. M., Pribury i Tekh. Eksptl. No. 6, 10, Nov.-Dec. (1959); English transl., Instr. and Exptl. Tech. No. 6, 870, Nov.-Dec. (1959). 235. Nottingham, W. B., 7th Ann. M.I.T. Con$ on Phys. Electronics (1947). 236. Lander, J. J., Rev. Sci. Instr. 21, 672 (1950). 237. Metson, G. H., Brit. J. Appl. Phys. 2, 46 (1951). 238. Nottingharn, W. B., Trans. Nail. Symposium on Vacuum Tech. 1, 76 (1954). 239. Penchko, E. A., and Khavkin, L. P., Pribory i Tekh. Eksptl. No. 1, 128 (1959); English transl., Instr. and Exptl. Tech. No. 1, 132 (1959). 239a. Van Oostrom, A., Trans. Null. Symposium on Vacuum Tech. 8, 443 (1961). 240. Redhead, P. A,, Rev. Sci. Instr. 31, 343 (1960). 241. Riemersma, H., Fox, R. E., and Lange, W. J., Trans. Natl. Symposium on Vacuum Tech. 7, 92 (1960). 241a. Lafferty, J. M., Trans. Natl. Symposium on Vucuum Tech. 8, 460 (1961). 242. Redhead, P. A,, Can. J. Phys, 37, 1260 (1959). 243. Redhead, P. A., Can. J. P h y ~ .36, 255 (1958). 244. Barnes, G., Rev. Sci. Instr. 31, 608 (1960). 245. Barnes, G., Rev. Sci. Insty. 31, I121 (1960). 246. Crawford, C. K., Rev. Sci. Instr. 32, 463 (1961). 247. Dushman, S., and Young, A. H., Phys. Rev. 68, 278 (1945). 248. Riddiford, L., J. Sci. Znstr. 28, 375 (1951). 249. Wagener, S., and Johnson, C. B., J. Sci. Instr. 28, 278 (1951). 250. Schulz, G. J., J. Appl. Phys. 28, 1149 (1957). 251. McGowan, W., and Kerwin, L., Can. J . Phys. 38, 567 (1960). 252. Ehrlich, G., J. Appl. Phys. 32, 4 (1961). 253. Moesta, H., and Renn, R., Vakuum-Technik 6, 35 (1957). 254. Barnes, G., Gaines, J., and Kees, J., Vacuum, 12, 141 (1962). 255. Hickrnott, T. W., J. Chem. Phys. 32, 810 (1960). 256. Eisinger, J., J. Chem. Phys. 30, 412 (1959). 257. Carter, G., and Leck, J. H., Brit.J . Appl. Phys. 10, 364 (1959). 258. Leck, J. H., J. Sci. Instr. 30, 271 (1953). 259. Pitj., L., Czechodov. J. Phys. 7, 113 (1957). 260. Redhead, P. A., Trans. Natl. Symposium on Vacuum Tech. 7, 108 (1960). 261. Lafferty, J. M., J. Appl. Phys. 22, 299 (1951). 262. Weinreich, 0. A., and Bleecher, H., Rev. Sci. Instr. 23, 56 (1952).
430
P.
A. REDHEAD, J. P. HOBSON, E.
V. KORNELSEN
263. Mizushima, Y.,and Oda, Z., Rev. Sci. Instr. 30, 1037 (1959). 264. Baker, F. A., Rev. Sci. Instr. 31, 911 (1960). 265. Haefer, R. A., and Hengevoss, J., Trans. Natl. Symposium on Vacuum Tech. 7, 67 (1960). 266. Kornelsen, E. V., Rept. 19th Ann. M.Z.T. Conf. on Phys. Electronics p. 156 (1959). 267. Giinther, K. G., Vacuum 10, 293 (1960). 268. Caswell, H. L., I.B.M. Journal 4, 130 (1960). 269. Davis, W. D., and Vanderslice, T. A., Trans. Natl. Symposium on Vacuum Tech. 7, 417 (1960). 270. Drawin, D. W., and Brunnbe, C., Vakuum-Technik, 9 , 65 (1960). 271. Huber, W. K., and Trendelenburg, E. A,, Trans. Natl. Symposium on Vacuum Tech. 6, 146 (1959). 272. Kanomata, I., Oguri, T., Kaneko, Y.,Hayakama, T., @S Butsuri ( J . Appl. Phys., Japan) 28,584 (1959). 273. Klopfer, A., Garbe, S.,and Schmidt, W., Trans. Natl. Symposium on Vacuum Tech. 6, 27 (1959). 274. Pikus, G . Ya., Pribory i Tekh. Eksptl. No. 2, 104, Mar.-Apr. (1960); English transl., Instr. and Exptl. Tech. No. 2, 286, Mar.-Apr. (1960). 275. Kendall, B. R. F., PYOC. 9th Ann. Meeting on Mass Spectrometry, A S T M Committee E-14, yune 1961. 276. Nier, A. 0. C., Rev. Sci. Instr. 11, 212 (1940). 277. Allen, J. S., Phys. Rev. 55, 966 (1939). 278. Bernhard, F., Krebs, K. H., and Rotter, I., Z. Physik 161, 103 (1961). 278a. Papirov, I. I., Pribory i Tekhnika Eksper. 7, 5 (1962). 279. Alpert, D., Rev. Sci. Instr. 22, 536 (1951). 280. Bills, D. G., .and Allen, F. G., Rev. Sn'. Instr. 26, 654 (1955). 281. Thorness, R. B., and Nier, A. O., Rev. Sci. Instr. 32, 807 (1961). 282. Lange, W. J., Rev. Sci. Znstr. 30, 602 (1959). 283. Parker, W. B., and Mark, J. T., Trans. Natl. Symposium on Vacuum Tech. 7, 21 (1960). 284. Vogl, T. P., and Evans, H. D., Rev. Sci. Instr. 27, 657 (1956). 285. Pdtjr, L. and Schurer, P., Rev. Sci. Instr. 28, 654 (1957). 286. Axeirod, N. N., Rev. Sci. Instr. 30, 944 (1959). 287. Pritjr, L.,Nature 185, 674 (1960). 288. Whetten, N. R., and Young, J. R., Rev. Sci. Instr. 30, 472 (1959). 289. Hobbis, L. C. W., and Harrison, E. R., Rev. Sci. Instr. 27, 238 (1956). 290. Landecker, K., and Gray, A. J., Rev. Sci. In&. 25, 1151 (1954). 291. Young, J. R., and Whetten, N. R., Rew. Sci. Instr. 32, 453 (1961). 292. Lange, W. J., and Alpert, D., Rev. Sci. Instr. 28, 726 (1957). 293. Holden, J., Holland, L., and Laurenson, L., J . Sci. Instr. 36, 281 (1959). 294. Brymner, R., and Steckelrnacher, W., J. Sci. Znstr. 36, 278 (1959). 295. Zollrnan, J, A., Martin, I. E., and Powell, J. A , , Trans. Natl. Symposium on Vacuum Tech. 6, 278 (1959). 296. Hickmott, T . W.. and Ehrlich, G., Phys. and Chem. Solids 5, 47 (1958). 297. Dyke, W. P., I.R.E. Trans. on Military Electronics MIL-4, 38 (1960). 298. Martin, E. E., Trolan, J. K., and Dyke, W. P., J. Appl. Phys. 31,782 (1960). 299. Dyke, W. P., Charbonnier, F. M., Strayer, R. W., Floyd, R. L., Barbour, J. P., and Trolan, J. K., J. Appl. Phys. 31, 790 (1960). 300. Dyke, W. P., and Dolan, W. W., Advances in Electronics and Electron Phys. 8, 89 (1956).
ULTRAHIGH VACUUM
43 1
301. Martin, E. E., and Collins, F. M., Proc. 4th Nut. ConJ on Tube Techniques p. 21 302. 303. 304. 305. 306. 307. 308. 309. 310. 311. 312. 313. 314. 315. 316. 317. 318. 319. 320. 321. 322. 323.
324. 325. 326.
(1958). Jonker, J. L. H., Philips Research Repts. 9, 391 (1954). Anderson, P. A., Phys. Rev. 47, 958 (1935). Grunberg, L., Brit. J. Appl. Phys. 9, 85 (1958). Huguenin, E. L., and Valat, J. G., J . phys. radium 17, 965 (1956). Bronshtein, I. M., and Roschin, V. V., Zhur. Tekh. Fiz. 28, 2200 (1958); English transl., Sooiet Php-Tech. Phys. 3, 2023 (1958). Bronshtein, I. M., and Roschin, V. V., Zhur. Tekh. Fia. 28,2476 (1958), English transl., Soviet Phys.-Tech. Phys. 3, 2271 (1958). Dobretsov, L. N., and Matskevich, T. L., Zhur. Tekh. Fiz. 27, 734 (1957); English transl., Soviet Phys.-Tech. Phys. 2, 663 (1957). Fowler, H. A., and Farnsworth, H. E., Phys. Rew. 111, 103 (1958). Gorodetskii, D. A., Zhur. Eksptf. Teoret. Fiz. 33, 7 (1958); English transl., Soviet Phys. J E T P 34, No. 7, 4 (1958). Germer, L. H., Scheibner, E. J., and Hartman, C. D., Phil. Mag. [8] 5, 222 (1960). Farnsworth, H. E., Schlier, R. E., and Tuul, J., Phys. and Chem. Solids 9, 57 (1959) (and many other papers). Rapp, R. A., Hirth, J. P., and Pound, G. M., Can. J. Phys. 38, 709 (1960). Rapp, R.A., Hirth, J. P., and Pound, G. M., J. Chem. Phys. 34, 184 (1961). Neugebauer, C. A., Newkirk, J. B., and Vermilyea, D. A., eds., “Structure and Properties of Thin Films.” Wiley, New York, 1959. Mayer, H., in “Structure and Properties of Thin Films” (C. A. Neugebauer et al., eds.), p. 225. Wiley, New York, 1959. Evans, C. C., and Mitchell, 1. W., in “Structure and Properties of Thin Films” (C. A. Neugebauer et a/., eds.), p. 263. Wiley, New York, 1959. Neugebauer, C. A., in “Structure and Properties of Thin Films” (C. A. Neugebauer et al., eds.), p. 358. Wiley, New York, 1959. Caswell, H. L., J. Appl. Phys. 32, 105 (1961). Greenland, K. M., J. Sci. Inttr. 38, 1 (1961). Spitzer, L., Astrophys. J. 95, 329 (1942). Sledziewski, Z., and Torossian, A., Le Vide 14, 107 (1959). Kurchatov, I. V., Atomnayu Energ. 5, 105 (1958); English transl., Sowiet J. Atomic Energy 5 , 933 (1958). Minzer, R. A., and Ripley, W. S.,U.S. Air Force Surveys in Geophys. No. 115 ( 1959). Santeler, D. J., Trans. Natl. Symposium on Vacuum Tech. 6 , 129 (1959). Bennett, W. H., Trans. Natl. Symposium on Vacuum Tech. I , 378 (1960).
This Page Intentiona lly Left Blank
Author Index Numbers in parentheses are reference numbers, and are included to assist in locating reference when the author's name is not cited at the point of reference in the text. Numbers in italics indicate the page on which the full reference is listed.
a
A
B
Aarset, B., 368 (151), 426 Abbott, R. C . , 249, 318 Abrahams, M. S., 224 (27), 243 Abrams, R. H., Jr., 152,203 Agnew, L., 187, 205 Ahlert, R. H., 151 (57, 59), 203 Aigrain, P. R. 227, 243 Alexeff, I., 380, 428 Allen, C. W., 336, 423 Allen, F. G., 415, 430 Allen, J. S.. 410 (277), 430 Allen, P. G . W., 374 (180a), 427 AlmCn, O., 264, 266, 319, 362, 366 (126), 426 Alpert, D., 323, 324 (2), 348 (92), 350 (92), 352 (90, 92), 354 (92), 357 (92), 358 (92), 375, 378, 383 (207), 388, 392, 408 (90), 411 (90), 414, 415 (2), 416 (292), 422, 424, 427, 428, 430 Altemose, V. O., 347, 424 Amelinch, S., 266 ( 5 9 , 319 Ames, I., 385, 428 Anderson, G. S., 266, 280, 319, 366 (139), 426 Anderson, P. A., 417, 431 Ansel'm, A. I., 129, 154, 202 Appleyard, E. T. S., 293, 321 Arking, A., 368 (154), 426 Armstrong, R. A., 336, 423 Arnold, S. R., 281, 320 Ash, E. A,, 183, 205 Aston, J. G., 335 (48), 423 Atkins, L. T., 298 (154), 321 Auer, P. L., 172, 173, 179, 204 Axelrod, N. N., 415 (286), 430 Aziz, R. A,, 294, 298, 321
Bader, M., 275, 319 Bagg, J., 342, 424 Baker, F. A., 404, 430 Baker, M . McD., 342, 424 Balarin, M., 269, 319 Balwanz, W. W., 337, 356, 384 (57), 423 Bandringa, M., 324 (4), 372 (4), 383, 390 (4), 393 (4), 422 Barbour, J. P., 417 (299), 430 Bardeen, J., 307, 322 Barnes, G., 398 (245), 429 Barnes, R. S., 279 (79, 82), 320 Barrer, R. M., 347, 348 (86), 350 (86), 352, 354 (86), 355 (86), 424 Barrett, C. S., 143, 150, 202 Barry, E. J., 344, 416 (80), 424 Bartholomew, C . Y . , 303, 306, 322 Bassett, G. A., 295, 296,321 Batanov, G. M., 248 (9), 318 Baum, E. A., 167, 176, 190 (IIO), 204 Bayard, R. T., 323, 392, 422 Beams, J. W., 373, 427 Beck, A. H. W., 30 ( I I ) , 36 ( I I ) , 57 ( I I ) , 123, 151, 203 Becker, E. J., 370 (167), 371 (167), 404 (167), 427 Becker, G. E., 337,423 Becker, J. A., 329 (18), 338, 339, 340, 341, 370, 371, 404 (167), 417, 422, 423, 427 Becker, W., 373, 427 Beecher, N., 384 (213), 428 Beeck, O., 295, 321, 342, 423, 424 Beers, D. S., 368 (154), 426 Beggs, J. E., 155 (73), 163, 165, 203 Behrndt, K. H., 386, 428 Beliakova, V. V., 359, 425
433
434
AUTHOR INDEX
Belyakov, Yu. I., 349 (87e), 424 Bennett, L. C., 226, 243 Bennett, W. H., 421, 431 Berkowitz, J., 294 (137), 321 Bernhard, F., 411, 430 Bernstein, W., 194 (165), 206 Berry, H. W., 249, 318 Bickel, P. W., 163 (90), 203 Bierlein, T. K., 281, 320 Bills, D. G., 361, 368, 377, 401, 403 (156, 157), 415, 426, 430 Biondi, M. A., 301, 322, 384, 428 Birdsall, C. K., 179, 205 Blair, J., 226 (32), 237, 243 Bleakney, W., 270, 319 Blechschmidt, E., 288, 320 Bleecher, H., 404 (262), 429 Bliss, B. J., 356, 425 Block, F. G., 176 (106), 177 (l06), 197,204 Blodgett, K. B., 306 (184), 322 Bloem, J., 232 (41), 243 Bloss, W., 194 (167), 206 Bobone, R., 217 (21), 242 Bottger, O., 222, 243 Bohm, D., 183 (144), 205 Bohr, N., 259, 318, 359 (1 lo), 425 Boomer, E. H., 305, 322 Borodkin, A. S., 337 ( 5 8 ) , 423 Borovik, E. S., 357, 384 (215), 425, 428 Bowers, R., 217 (20), 242 Bradley, R. C., 251, 256, 282, 283, 287, 318, 368, 426 Brands, R. G., 370 (167), 371 (167), 404 (167), 427 Braunstein, R., 224 (27), 243 Bredov, M. M., 279 (76), 320, 359, 425 Brennan, D., 342 (64), 357 (64), 423 Bridges, W. B., 179, 205 Brode, R. B., 171, 204 Brombacker, W. G., 39 I , 429 Bronshtein, I. M., 418 (306, 307), 431 Brown, E., 377,427 Brown, S. C., 287, 320 Brubaker, W. M., 380 (195), 382 (195), 428 Bruce, G.,264,266,319,362,366( l26), 426 Bruche, E., 144, 202 Brunauer, S., 328, 329, 334 ( 15, 2 9 , 4 2 2 Brunnke, C., 368 (153). 408 (270), 41 1 (270), 426, 430 Brymmer, R., 416 (294). 430
Budnick, J. I., 299, 322 Burbank, R. D., 299 (156), 321 Burger, R. M., 300 (163), 322, 367 (143), 426 Burgers, W. G., 353 (95d), 425 Burhop, E. H. S., 257, 317 Buritz, R. S., 348 (92), 350 (92), 352 (90, 92), 354 (92), 357 (92), 358 (92), 360, 361 (119), 375 (90), 408 (90), 411 (90), 414 (go), 424, 425
C Cabrera, N., 329, 422 Cadoff, I., 225 (29). 243 Cahn, J. W., 291,321 Callaghan, M. F.,385 (217), 428 Carabateas, E. N., 180, 181 (137), 185, 205 Carleton, N. P., 361, 377, 401, 426 Carmichael, J. H., 360, 361 (118), 363, 364 (118, 123), 376 (187), 383 (209), 425 Carpenter, F. D., 176 (113), 204 Carter, G., 360 (121), 362 (121), 363, 376 (189a), 377, 402 (121, 257), 403 (257), 425, 426, 427, 429 Castro, P. S., 227 (39, 243 Caswell, H. L., 299, 322, 383, 408 (268), 411 (268), 419, 428, 430, 431 Cayless, M. A., 126, 130, 201 Champeix, M. R., 129, 154, 202 Charbonnier, F. M., 417 (299), 430 Chen, C. L., 348 (107), 350 (107), 354 (107), 357, 425 Chicherov, V. M., 266 (53b), 319 Childs, J. H., 329, 422 Christians, K., 345, 424 Christiansen, R. L., 385 (221), 428 Chupka, W. A., 294 (137), 321 Churchman, A. T., 279 (82), 320 Clausing, P., 330, 422 Clingman, W. H., 227, 236, 243 Cloud, R. W., 368 (151), 426 Cobic, B., 376, 427 Cole, W. A., 342 (71), 424 Colligon, J. S., 363 (1278). 426 Collins, F. M., 417 (301). 431 Collins, L. E., 298, 321 Collins, R. H.,343 (74), 424 Colombie, N., 276 (70), 319 Compton, A. H.,157, 203
435
AUTHOR INDEX
Compton, K. T., 183 (148), 205 Constabaris, G . , 333 (42), 334 (42), 423 Corregan, F. H., 176 (106), 177 (106), 197 (106), 204 Cottrell, A. H., 279 (78, 82), 320, 359 (112), 425 Coveyou, R. R., 262 (37), 318 Crawford, C. K., 399, 429 Cunningharn, R. L., 269, 319 Cushings, R. L., 280 (88), 320
Drawin, D. W., 408 (270), 411 (270), 430 Dreyer, K., 372, 390, 416 (230), 419 (171, 230), 427, 429 Dugan, A. F., 161, 203 Dushrnan, S., 333, 355 (38), 357 (38), 400 (247), 423, 429 Dyke, W. P., 143 (32), 147 (32), 174 (32), 202, 417 (297, 298, 299), 430
D
Eastman, G. Y., 176 (106), 177 (106), 197 (106), 204 Ecker, G., 287, 288, 305, 320, 322 Eckhart, C . , 183 (148), 205 Ehrlich, G., 290, 292 (120), 307, 321, 331. 342, 400 (252), 417 (296), 422, 429 Eichenbaurn, A. L., 172, 173, 178, 179,204 Eichhorn, R. L., 236 (42), 243 Eisenstein, A. S., 151 (61), 203 Eisinger, J., 401, 429 Ells, C. E., 279 (80), 296, 320, 321 Erneleus, K. G . , 288, 305, 313, 320, 322 Emmett, P. H., 329 (25), 334 (25), 422 Ennoa, A. E., 300, 322 Erickson, G. F., 152 (66), 203 Ermrich, W., 374 (178), 427 Eschbach, H. L., 330, 350 (108), 354 (108), 357, 422, 425 Espe, W., 386, 429 Ettre, K., 385, 428 Evans, C. C., 419, 431 Evans, D. M., 298, 321 Evans, H. D., 415 (284), 430 Evett, A. A., 368 (157), 403 (1571, 426
Darnianovich, H., 305,322 D’hrnico, C., 300, 322, 347, 367, 424 Damm, C. C., 345 (83c), 375 (83c), 388 (83c), 424 Danforth, W. E., 151 (53), 152, 203 Davies, J. A., 280, 320, 359, 425 Davis, R. H., 380 (193), 428 Davis, W. D., 407, 408 (269), 411 (269), 430 Davydov, B. I., 214 (13), 242 Dayton, B. B., 343 (76), 345, 424 de Boer, J. H., 326 (14), 328, 330, 422 Degras, D. A., 385, 428 De Jong, B. C., 353 (95d), 425 de Jong, M., 366 (136), 426 Dekeyser, W., 295 (145), 321 Delavignette, P., 266 ( 5 9 , 319 della Porta, P., 356, 425 Dennard, F.. 387, 429 Dent, B. M., 292 (131), 321 Devienne, F. M., 332, 423 Devyatkova, E. D., 212, 242 Dillon, J. A., Jr., 281 (91), 320, 347, 367 (144), 424, 426 Divatia, A. S., 380 (193), 428 Dobretsov, L. N., 126, 130, 133, 178, 190, 201, 418 (308), 431 Dolan, W. W., 143 (32), 147 (32), 174 (32), 202, 417 (300), 430 Donahoe, F. J., 221, 243 Donaldson, E. E., 368, 426 Douglas, R. W., 209 (3), 242 Dow, D. G . , 83 (12), 123 Dow, W. G . , 2 (3), 4 (3), 5 (6), J 7 (3), 18 (3), 52 (31, 55 (3). 123 Dracott, D., 183, 205 Drain, L. E., 334 (43), 423
E
F Farkass, I., 344, 384 (214), 416 (80), 421, 424, 428 Farnsworth, H. E., 281, 299, 320, 321, 367 (143, l44), 418 (309, 312), 426, 431 Farr, J. D., 152 (63), 203 Feaster, G. R., 155, 203 Fedorova, N. N., 226 (31), 243 Fendley, J. R., 176 (106), 177 (106), 197 (106), 204 Ferris, W. R.,157, 203 Field, F. H., 305 (179a), 322 Finch, G. I., 295, 321
436
AUTHOR INDEX
Fish, I. P. S., 356, 425 Florescu, N. A , , 300, 322 Floyd, R. L., 417 (299), 430 Fluit, J. M., 268 (61), 284 (61), 319, 366 (136), 426 Fogel', Ya. M., 254, 318 Folberth, 0. G., 226 (31), 243 Foner, S. N., 328,422 Foster, J. S., Jr., 382 (198), 428 Foster, P. K., 353 (95g), 425 Found, C. B., 184 (153), 194, 205 Fowler, H. A., 418 (309), 431 Fox, R. E., 176, 178, 179 (117), 204, 364, 395 (241), 426, a29 Francis, A. B., 380 (196), 382 (196), 428 Francis, G., 305, 322 Frandsen, N. P., 337 (57), 356 (57), 384 (57), 423 Frank, F. C., 295, 321 Frank, R. C., 353 (95a, 95b, 95e), 424,425 Franklin, J. L., 305, 322 Freeman, M. P., 333 (41), 334 (41), 423 Frenkel, J., 291, 321 Friedel, J., 238, 243 Frohlich, H., 213, 242 Fukushima, H., 356 (98), 425
G Gabor, D., 183, 194 (164), 205, 206 Gafner, G., 291, 321 Garbe, S.,336,345, 371 (53,408 (273), 41 1 (273), 423, 424, 430 Garin, P., 385, 428 Garvin, H. L., 176 (113), 178, 179 (126, 127), 204, 205 Gatos, C., 312, 322 Geiger, G. S., 383 (204), 419 (204), 428 George, T. H., 300 (163), 322, 367 (143), 426 Germer, L. H., 418 (311), 431 Gerstacker, H., 328, 333, 422 Gervais, H., 246 (3), 281 (3), 318 Gianola, V. F., 281, 320 Giauque, W. F., 335 (46), 423 Gibbons, M. D., 163 (92), 174, 177 (104), 178, 184 (104), 185 (104), 203, 204, 205, 374 (181), 375, 427 Gibson, J. B., 259, 265, 318, 359 (1 1 I), 360 (1 I l ) , 425
Gillam, E., 279, 300 (87), 320 Givens, J. W . , 295 (141), 321 Glicksman, R., 198 (178), 206 Goerz, D. J. Jr., 384, 428 Goland, A. N . , 259 (30), 265 (30), 279 (84), 303 (84), 318, 320, 359 ( I l l ) , 360 ( I l l ) , 425 Goldberger, A. K., 17 (8), 123 Goldman. D. T., 262, 318 Goldsmid, H. J., 209 (3), 212, 242 Goldwater, D. L., 151 (53), 203 Golike, R. C., 332, 423 Gomer, R., 290, 307, 321, 331, 336, 423 Gorman, J. K., 349 (87f), 424 Gorodetskii, D. A., 418 (310), 431 Goryunova, N . A., 226 (31), 243 Gottlieb, M., 178, 179 (128), 180, 181 (139), 205 Gray, A. J., 416 (290), 430 Gray, P. E., 227, 243 Greenland, K. M., 419, 431 Gregg, A. H.,314, 322 Gregory, N. W., 294, 321 Grigor'ev, A. M., 391, 396 (234), 429 Grishin,S.F., 357(105), 384 (215), 425,428 Grishna, E. Ya., 357 (105), 425 Gronlund, F., 267, 319 Gross, E. P., 183 (144), 205 Grove, D. J., 324, 419 (13), 422 Grover, G. M., 130 (23a, 23b), 152 (63, 66), 176 (109, 112), 184 (log), 186, 187 (109), 193 (112), 202, 203, 204 Grunb.erg, L., 418, 431 Gunther, K. G., 407, 430 Guentherschulze, A. Z., 287,288, 305,320, 322 Gundry, P. M., 337, 423 Gurevitch, L., 238 ( 4 9 , 243 Gurney, R. W., 256, 318 Gurtovi, M. E., 130, 178, 202 Gust, W., 176 (117). 178, 179 (117), 204 Gyftopoulos, E. P., 146, 202
H Haas, C., 353 (95i), 425 Haas, G. A., 128, 152, 202 Hablanian, M. H., 372, 427 Haefer, R. A., 384, 404, 428, 430 Haeff, A. V., 5 (51, 123
437
AUTHOR INDEX
Hagstrum, H. D., 248, 255,257 (19), 287, 300, 318, 322, 347, 367, 368 (146, 152), 424, 426 Hall, L. D., 380 (194), 428 Halsey, G. D., Jr., 333, 334, 337, 423 Hanav, R., 312 (189), 322 Haneman, D., 281 (91), 320 Hansen, N., 336, 337 (51a), 423 Happ, W. W., 227 ( 3 9 , 243 Harkins, P. A., 269 (63), 319 Harnett, J. P., 332, 333, 423 Harrison, D. E., Jr., 262 (37), 263, 276 (69), 284, 318, 319, 366 (137), 367 (137), 426 Harrison, E. R., 416 (289), 430 Hartman, C. D., 418 (31 I), 431 Hashirnoto, H., 290 (119), 321, 356, 425 Hatsopoulos, G. N., 138, 139, 154, 155, 163, 168, 176, 180 (137), 181 (137), 202, 204 Hayakama, T., 408 (272), 41 1 (272), 430 Hayashi, C., 330, 422 Hayman, P., 269 (64), 319 Hayward, D. O., 342 (64),357 (64), 423 Head, H. N., 305 (179a), 322 Healed, M., 256, 318 Heavens, 0. S . , 298 (151), 321 Heidenreich, R. D., 299, 321 Heikes, R. R., 216, 220 (17), 242 Heldt, L., 281 (98), 320 Henderson, W. G., 383, 419 (204), 428 Hengevoss, J., 336, 404, 423, 430 Henschke, E. B., 263, 319 Hensley, E. B., 128 (13), 202 Herb, R. G . , 380,428 Hernqvist, K . G., 130 (21), 168, 172, 173, 176, 177 (21, 106), 178, 179, 196 (122), 197 (106), 202, 204 Herring, C . , 127 (9), I28 (9), I75 (94 201, 210 (6), 242 Herrmann, G., 151 (62), 203 Hickman, K. C . D., 373, 427 Hickmott, T. W., 331 (32), 370, 401 (166), 404 (166), 417, 422, 427, 429, 430 Hilbert, F., 269, 319 Hill, R. F., 195 (169), 206 Hills, E. J., 176 (106), 177 (106), 197 (106), 204 Hines, R. L., 276 (71), 300, 303 (172), 319, 322, 367, 426
Hirsch, P. B., 279 (85), 320 Hirsch, R. L., 190, 196, 205, 206 Hirth, J. P., 294, 295, 321, 418 (313, 314), 431 Hnilicka, M. P., 384 (213), 428 Hobbis, L. C . W., 416 (289), 430 Hobson, J. P., 127, 202, 335 (49, 50), 336 (51b), 345, 376, 378 (47), 385, 396 (47), 398 (47), 400 (47), 401, 403 (47). 414, 423, 424, 427 Holden, J., 416 (293), 430 Holland, J. W., 196, 206 Holland, L., 287, 300, 301, 320, 322, 374, 375 (185), 378, 391, 416 (293), 427, 429 Holmstrom, F., 268, 319 Holroyd, L. V., 163 (90), 203 Honig. R. E., 251, 287, 318, 356, 425 Hoogendoorn, A , , 266, 319 Hook, H. O . , 356 425 Horne, R. A., 225 (28), 243 Houston, J. M., 130, 137, 139, 147, 148, 174, 176, 177 (104), 178, 182 (143), 184 (104), 185 (104), 202, 204, 205 Houston, M. D., 238 (44),243 Houternans, C . , 256, 318 Howard, R. C . , 176, 204 Huber, W. K., 336, 408 (271), 411 (271), 423, 430 Hudda, F. G., 331 (32), 422 Hughes, F. L., 150, 202 Huguenin, E. L., 418,431 Hunt, A. L., 345, 375, 388, 424 Hurlbert, R. C . , 353 (95c), 425 Hurlbut, F. C . , 267, 319, 329, 422 1
Inghram, M. G., 294, 321 Ingold, J. H., 130, 142, 202 Ioffe, A. F., 154, 198,203,208,209 (2), 217 (2), 221, 242 . Ionov, N. I., 172, 204, 349 (87e), 424 Iordanishvili, E. K., 209 (4), 242 Ittner, W. B., 299, 321 Iwayanagi, H., 356 (98), 425
J Jablonski, F. E., 195, 206 Jacob, L., 369, 427 Jaeckel, R., 330 (28b), 343 (77), 375 (184), 422, 424, 427
438
AUTHOR INDEX
Jamerson, F. E., 152, 178, 179 (129), 195, 203, 205 Jancke, H., 324 ( I l ) , 422 Jansen, M. J., 151 (58), 203 Jaumot, F. E., Jr., 208, 213 (l), 217 (l), 227 (34), 236 ( I , 34), 238 ( I , 34), 242,243 Jensen, A. O., 167, 176 ( I l O ) , 190, 204 Jensen, J. T., Jr., 128, 152, 202 Jepsen, R. L., 380 (196, 197), 382 (196), 385, 428 Johnson, C. B., 400 (249), 429 Johnson, E. O., 182 (142), 183 (142, 151), 194, 205 Johnson, F. M., 178, 179 (124), 204 Johnson, H. R., 2, 8, 17 (2), 52, 5 5 , 58 (2), 61, 69, 81, 99 (2), 113, 123 Johnson, K. P., 196,206 Johnson, W. C., 9 (7), 123 Jonker, J. L. H., 417, 431 Jones, H. A., 154 (68), 203 Jost, W., 352, 424 Justi, E., 216, 242
K Kanefsky, M., 130 (21), 168 (21), 176 (21), 177 (21), 202 Kaneko, Y.,408 (272), 411 (272), 430 Kanomata, I., 408 (272), 4.1 I (272), 430 Kaplan, R., 150, 202 Karnaukhov, I. M., 254 (18), 318 Kay, E., 294, 321 Kaye, J., 138 (139 (30), 155, 163, 168, 202, 203 Keesom, W. H., 337, 423 Kendall, B. R. F., 408 (275), 411 (275), 430 Kendall, L. S . , 217 (21), 242 Kennedy, P. B., 383, 428 Kenty, C., 370, 427 Kerr, J. T., 369 (162), 427 Kerwin, L., 400 (251), 429 Keywell, F., 260, 318, 366 (140), 426 Khavkin, L. R., 337 (58), 393 (239), 423, 429 Kietzmann, B. E., 380 (196), 382 (196), 428 Killian, T. J., 146 (41), 202 Kinchin, G. H., 259, 318 Kingdon, K. H., 126, 144 (37), 146 (42), 150, 201, 202 Kistemaker, J., 268 (61), 284 (61), 319 Kittel, C . , 139 (31), 202, 213, 242
Kleint, C., 324 (7), 422 Klopfer, A., 336 ( 5 3 , 345 ( 5 5 ) , 371 ( 5 5 ) , 374, 382, 408 (273), 41 1 (273), 423, 427, 428, 430 Knacke, O., 295, 321 Knaver, F., 291, 321 Knight, R. D., 268, 282 (IOO), 319, 320 Knoll, J. S., 364, 426 Knoll, M., 386, 429 Knudsen, M., 291, 321 Koedam, M., 266, 296, 319, 321 Koehler, J. S . , 259, 262 (29), 318 Koenig, H., 344 (81), 424 Kohl, W. H., 386,416,428 Koller, L. R., 151, 202 Komarek, K., 225 (29), 243 Konecny, J. O., 353 (95c), 425 Kornelsen, E. V., 324 (9,359 (115), 360 (122), 361 (122), 364 (122), 377, 378 (115), 380(115), 382(115), 387, 388, 406 (266), 408 (266), 409, 411 (266), 422, 425, 429, 430 Koskinen, M. F., 178, 179 (131), 205 Kovalenko, G. I., 130, 178, 202 Kovalskii, G. A., 382 (199), 428 Kramers, W. J., 387, 429 Krebs, K. H., 411 (278), 430 Krikorian, N. H., 152 (63), 203 Kroger, F. A , , 232 (41), 243 Krumhansl, J. A., 212,*213 (8b), 238 (8b), 242 Kruyer, S., 330, 422 Kuchai, S. A., 362, 382 (199), 426, 428 Kurchatov, I. V., 420, 431
L Laegreid, N. L., 276 (72), 319, 366 (134), 367 (134), 426 Lafferty, J. M., 324 (8), 349 (88), 395 (241a), 404 (261), 422, 424, 429 Landecker, K., 416 (290), 430 Lander, J. J., 279, 319, 320, 392, 429 Landfors, A. A., 372 (172), 427 Lang, Z. O., 359 (1 17), 425 Langberg, E.,138, 139 (30), 202, 262, 319 Lange, W. J., 370 (164a), 383 (209), 395 (241), 415 (282), 416 (292), 427, 428, 429, 430 Langmuir, D. B., 149, 202 Langmuir, I., 126, 146 (42, 43), 148, 150,
439
AUTHOR INDEX 151 (54), 154 (68), 155, 156, 157, 169,
172, 173, 178, 183, 184, 201, 202, 205, 294, 321, 329, 422 LaPadula, A. R., 303, 306 (173), 322 Laurenson, L., 374 (180a), 416 (293), 430 Law, J. T., 300 (162), 322 Lawrence, E. O., 382 (198), 428 Lawson, W. D., 237, 243 Lax, B., 21 I , 242 Lazarev, B. G., 384 (216), 428 Leck, J. H., 360 (121), 362 (120), 363 (127a), 376 (189a), 377, 391, 402 (121, 257, 258), 403 (257), 425, 429 Le Claire, A. D., 307 (1 87), 322, 364 (1 30), 426 Lecomte, C., 269 (64), 319 Lee, R. W., 353 (95a. 95b), 424, 425 Leffert, C. B., 195 (169). 206 Leibfried, G., 265, 279 (86), 300 (86), 319, 320 Leiby, C. C., Jr., 348 (107), 350 (107), 354 (107), 357, 425 Leland, W. T., 252, 318 Lemmens, H. J., 151 (58), 203 Lennard-Jones, F. R. S., 294, 321 Lennard- Jones, J. E., 292, 321 Le Riche, R., 349 (89), 424 Leverton, W. F., 163 (89), 203 Levi, R., 151 (56), 203 Levine, J. D., 146, 202 Levinstein, H., 150, 202, 292,293, 298,321 Lewis, H. W., 180, 181, 183 (147), 187, 188, 190, 205 Libin, I. Sh., 387, 429 Lillie, D. W., 279 (81), 320 Lindsay, P. A., 155, 161, 203 Lindy, R., 423 Lineweaver, J. L., 369 (162), 427 Little, P. F., 368, 426 Littlewood, R., 329, 422 Loeb, L. B., 287, 320 Lofgren, E. L., 382 (198), 428 Loosjes, R., 151 (58), 203 Loughridge, D. H., 195 (169), 206 Lounsbury, M., 280 (88), 320 Lozier, G . S., 198 (178). 206 Liickert, J., 374, 427 Luke, K. P., 178, 179 (129), 205 Lundy, R., 336 (52), 423 Lunt, R. W., 314,322
M McAdams, W. H., 138 (29), 202 McAfee, K. B., Jr., 352, 424 McCleIland, D. H., 196, 206 McConnell, G., 232, 243 McGowan, W., 400 (251), 429 McIntyre, J. D., 280 (88), 320 McKeown, D., 263, 319 Magnuson, G. D., 263, 269, 319 Malter, L., 183 (151), 205 Mandelcorn, L., 306, 322 Marchuk, P. M., 130, 147, 174, 176, 177 (20),178, 179 (ZO), 183 (152), 184 (152), 202, 205 Mark, J. T., 372 (171), 383 (204),390, 415 (283), 416 (230), 419 (171, 204, 230), 427,428,429,430 Marth, P. T.,343(75),408(75),411(75),424 Martin, E. E., 417 (298), 431 Martin, I. E., 416 (295), 430 Martin, S. T., 144, 202 Massey, H. S. W., 257, 313, 317, 322 Mastrangelo, S. V. R., 335 (48), 423 Matskevich, T. L., 418 (308), 431 Mayer, H., 287,320,419,431 Meckel, €3. B., 269 (63), 319 Medicus, G., 129, 183 (18), 212 Men’shikov, M. I., 324 (lo), 384 (lo), 422 Menter, J. W., 290, 295, 296, 321 Merser, S. L., 380 (197), 385 (217), 428 Metcalfe, R. A., 349 (89a), 424 Metson, G. H., 392, 429 Michaelson, H. B., 127 (3,143 (3,201 Mickelsen, W. R., 329, 422 Mignolet, J. C. P., 307, 322 Mihama, K., 246 (4, 5), 281 (4, 5 ) , 298 (4, 51, 318 Milch, A., 177 (120), 204 Milgram, M., 259 (30), 265 (30), 318, 359 ( I l l ) , 360 ( I l l ) , 425 Miller, B., 196 (173), 206 Miller, E., 225 (29), 243 Miller, R. C., 217 (ZO), 242 Milleron, N., 375, 383, 390 (183), 391, 416 (205), 420, 427, 428 Minzer, R. A., 420 (324), 431 Mitchell, J. W., 419, 431 Mittsev, M. A., 359, 425 Mizushima, Y., 404,430
440
AUTHOR INDEX
Moesta, H., 400 (253), 429 Mohler, F. L., 183 (149), 188, 205 Moizhes, B. Y., 178, 180, 190, 204 Molchanov, V. A., 266, 268, 269, 319 Moore, G. E., 163 (91), 203, 369, 427 Moore, W. J., 257, 259, 267, 269 (64), 281, 318,319,320, 353 (95j), 425 Morehouse, C. K., 198 (1781, 206 Morgulis, N. D., 126, 154, 176, 183 (152), 184 (152), 201, 204, 205, 248, 318 Morrison, J., 279, 319 Moss, H., 155, 194 (71), 203 Mostel, B., 281 (96), 320 Moubis, J. H., 286, 320 Miiller, C. F., 348 (87b), 434 Miiller, D., 330 (28b), 422 Mueller, E. W., 266, 319 Muir, H., 218 (22, 23)’ 243 Muller, E. W., 290, 321 Muller, K. G., 287, 320 Munday, G. L., 324, 415 (12), 419, 422 Murray, G. T., 279 (83), 320
N Nardella, W. R., 349 (87f), 424 Naumovets, A. G., 176 (114), 204 Nechai, E. P., 353 (95f), 425 Nesbitt, E. E., 299 (156), 321 Neugebauer, C . A., 419 (318), 431 Nichols, M. H., 127 (9), 128 (9), 175 (9), 201 Nielsen, K. O., 359, 425 Nielson, S., 237 (43), 243 Nier, A. O., 407 (276), 415, 430 Norman, F. H., 130 (21), 168 (21), 176 (211, 177 (21), 202 Normand, C. E., 276 (69), 319, 366 (137), 367 (137), 426 Norton, F. J., 347, 348 (85), 353 (95k), 364 (85), 424, 425 Nottingham, W. B., 127, 128 (12), 155 (12), 157, 162, 180, 183 (136), 202, 203, 205, 369, 391, 393 (238), 402 (238), 427, 429 0 Oda, Z., 404, 430 Ogilvie, G. J., 280, 281, 320 Oguri. T., 408 (272), 411 (272), 430
Okamoto, H., 336, 423 O’Keefe, M., 253 (95j), 425 Okuneva, N. W., 359 (117), 425 Oliphant, M. L. E., 255, 318 Olson, R., 252,318 Orfanov, I. V.,266 (52), 319
P Paetow, H., 256, 318 Palluel, P., 17 (8), 123 Panenkova, L. S., 353 (95f), 425 Papirov, I. I., 415 (278a), 430 Parker, F. W., 155, 161, 203 Parker, W. B., 415 (283), 430 Pashley, D. W., 290, 295, 296, 298, 321 Pit)i, L., 391,402 (259), 415 (285,287), 429 Paxton, G. W., 178, 179 (130), 205 Pease, R. S., 259, 262, 318 Penchko, E. A., 337, 393 (239), 423, 429 Penning, F. M., 285, 286, 287, 320 Perry, L. W., 196, 206 Perryman, E. C . W., 279 (80), 320 Peters, P. H., 168, 204 Peterson, E. C . , 380, 428 Petrov, N. N., 248 (9, 1 l), 318 Pezaris, S. D., 180 (137), 181 (137), 205 Pidd, R. W., 152 (63, 66), 176 (23a), 178 (126), 179 (126), 196, 202, 203, 205 Pierce, J. R., 2, 8, 17 ( I ) , 55, 56, 58 (I), 61, 69, 81 (I), 123 Pikus, G. E., 178, 180, 190, 204 Pikus, G., Ya., 408 (274), 411 (274), 430 Pleshivtsev, N. V., 266 (52), 319 Ploch, W., 248, 318 Pollard, J., 324 (9), 422 Popov, K. V., 353 (95f), 425 Popp, E. C., 345 (83c), 375 (83c, 183), 388 (83c), 390 (183), 420 (183), 424, 427 Pound, G. M., 294, 295, 321, 418 (313, 314), 431 Powell, J. A., 416 (295), 430 Price, P. J., 213, 222, 242, 243 Prugne, P., 385, 428 Pupp, W., 373, 427 Purdy, D. L., 196 (171), 206 Putley, E. H.. 237 (43), 243
Q Quarrel], A. G., 295, 321
441
AUTHOR INDEX
R Ramberg, E. G., 229 (38), 243 Ranken, W. A., 174, 176, 184 (109), 186, 187, 204 Ransley, C. E., 356 (102), 425 Rapp, R. A., 294,321,418,431 Rasor, N. S., 130 (24), 137, 139, 142, 146, 176, 202 Rastrepin, A. B., 254, 318 Redding, G. B., 279 (78, 79), 320 Redhead, P. A., 324 (9, 335, 342, 364 (131), 378 (47), 385, 388, 394 (240), 396 (47, 242), 397 (242), 398 (47, 243), 400 (47), 402 (242), 403 (47, 260), 422, 426, 429 Reich, G., 412, 426 Reichelt, W., 176, 193, 204 Reitz, J. R., 180, 181, 187, 188, 190,205 Renn, R., 400 (253), 429 Reynolds, H. K., 262 (39), 266, 319 Reynolds, J. H., 369, 406, 408 (la), 411 (la), 427 Rhodin, T. N., 371, 397 (169), 402 (169), 427 Rhodin, T. N., Jr., 334, 423 Ricca, F., 356 (103), 425 Rich, J. B., 279 (79), 320 Richardson, 0. W., 127, 201 Riddiford, L., 400 (248), 429 Rideal, E. K., 329, 342, 422, 424 Riemersma, H., 370 (164a), 395, 427, 429 R i m e r , D. E., 359 (112), 425 Ripley., W. S., 420 (324), 431 Rittner, E. S., 151 (57, 59), 155, 157, 159, 177 (IZO), 203, 204, 230 (39), 243 Rivera, M., 349 (89), 424 Riviere, J. C., 127 (6), 201 Roberts, R. W., 343, 424 Rocard, J. M., 178, 179 (130), 205 Rocklin, G. N., 387, 429 Rodin, A. M., 353 (95h), 362, 425, 426 Roehling, D. J.. 152 (63, 66), 176 (23a), 202, 203, 205 Rogers, W. A., 348 (92), 350 (92), 352 (92), 354 (92), 357, 358, 424 Rol, P. K., 268, 280, 284, 319, 320, 366 (136), 426 Roos, O., 248, 318 Roschin, V. V., 418 (306, 307), 431
Rosenberg, D., 276 (73), 319, 366 (138), 426 Rosi, F. D., 224 (27), 229 (38), 243 Rotter, I. Z . , 411 (278), 430 Rourke, F. M., 279, 320 Rovner, L. H., 371, 397 (169), 402 (169), 427 Rowe, A. H., 307 (187), 322, 364 (130), 426 Rowe, J. E., 5 (4, 6), 117, 118, 123 Ruedl, E., 256 (24), 266, 287 (24), 318 Ruehle, A. F., 163 (91), 203 Rutherford, S. L., 380 (196), 382 (196), 428 Rutledge, W. C . , 151 (57, 59), 203
S Salmi, E. W., 152 (63, 66), 176 (23a, 109, 112), 184 (109)’ 186, 187 (IOY), 193 (I 12), 202, 203 Sams, J. R.,Jr., 333 (42), 334 (42), 423 Santeler, D. J., 343 (78), 420,421, 424, 431 Saxon, D., 380 (193), 428 Sayers, J., 313, 322 Schafer, K., 328, 333, 422 Schafer, W., 176 (112), 193 (112), 204 Scheibner, E. J., 418 (311), 431 Schittko, F. J., 343 (77), 424 Schlicter, W., 129, 139, 154, 202 Schlier, R. E., 281 (91), 299, 320, 321, 367 (143), 418 (312), 426, 431 Schmidt, W., 336 ( 5 5 ) , 345 ( 5 3 , 371 ( 5 5 ) , 408 (273), 411 (273), 423, 430 Schneider, P., 151 ( 5 3 , 203 Sohock, A., 130, 139, 142, 202 Schiirer, P., 415 (285), 430 Schultz, R. D., 194 (166), 206 Schulz, G. J., 400 (250), 414, 42! Schulz, L. G . , 296, 321 Schwarz, E., 216 (18), 242 Schwarz, H., 361, 426 Schweers, J., 337, 423 Schweitzer, J., 324 (3), 422 Scott, G . D., 294, 296, 298, 321 Sears, G. W., 291, 321 Seeliger, R., 305, 322 Sehr, R., 232, 243 Seitz, F., 259, 262, 318 Sellen, J. M., 169, 204 Seraphim, D. P., 299, 321 Sheffield, J. C., 279 (77), 320
442
AUTHOR INDEX
Shelton, H., 127, 169, 201, 204 Shepard, R. W., 348 (87b), 424 Shepherd, W. G., 163 (89), 203 Shmushkevitch, I. M., 214 (13), 242 Sickert, R. G., 236 (42), 243 Silcox, J., 279 (85), 320 Silsbee, R. H., 265, 319 Silver, R., 195 (169), 206 Simon, A., 26?, 318 Simons, J. C., Jr., 383, 421, 428 Sims, G. A , , 359, 425 Singer, J. R., 337 (57), 356 (57), 384 (57), 423 Singleton, J. H., 337, 423 Skiff, P. D., 262, 266, 319 Slabospitskii, R. P., 254, 318 Slater, J. C., 21 (9), I23 Sledziewski, Z., 420,437 Smallman, R. E., 279 (85), 320 Smith, A. E., 295 (141), 298 (154), 321 Smith, H. R., 372, 383, 427, 428 Smith, R. A., 313, 322 Smithells, C. J., 356 (102), 425 Snouse, T. W., 275 (61a), 319 Solonitzin, Yu.,369 (163), 427 Sosnoveky, H. M. C., 281, 320 Spitzer, L., 173 (103), 204, 419 (321), 431 Stanton, H. E., 251,318, 368,426 Stavitskaya, T. S., 209 (4), 242 Steckelmacher, W., 416 (294), 430 Steele, W. A., 333 (39,40), 334 (39, a),423 Stein, R. P., 267, 319 Steinberg, R. K., 183 (150), 184 (150), 205 Stern, O., 291, 321 Stevenson, D. P.,298 (154), 321 Stil’bans, L. S., 209 (4), 242 Stoll, G. C., 236, 243 Stout, J. W., 335 (46), 423 Stout, V. L., 151, 153, 199 (60a), 203, 374 ( I N ) , 375 (182L 427 Strachan, C., 294 (135), 321 Stranski, I. N., 295, 321 Stratton, J. A., 21 (lo), 123 Strayer, R. W;, 417 (299), 430 Stuart, R. V., 366 (139, 426 Surenyants, V. V., 353 (95 h), 425 Swets, D. E., 353 (95a, 95b), 424, 425
T Talaat, M. E., 180, 205
Taylor, J. B., 146 (43), 148, 169, 184, 291, 202, 204, 321 Teatum, E. T., 174, 187, 204 Telkes, M., 216, 242 Tel’kovskii, V. G., 266 (53b), 268 (60), 269, 319 Teller, E., 329 (25), 334 (25), 422 Teloy, E., 375 (184), 437 Terao, N., 246 (3,4), 281 (3,4), 298 (4), 318 Terenin, A., 369, 427 Tertian, L., 246 (3), 281 (3), 318 Thomas, J. E.. Jr., 353 (954, 425 Thomas, L. B., 332, 423 Thommen, K., 261, 318 Thompson, M.W., 245 (2a), 257,265,318, 319 Thomson, A. A., 280, 320 Thorness, R. B., 415, 430 Thulin, S., 308, 322, 358 (109), 359 (109), 425 Tiedema, T. J., 353 (95d), 425 Timoshenko, G., 261, 318 Todd, B. J., 345, 350 (82), 352, 369, 424, 427 Tollimen, W., 288 (114), 320 Tompkins, F. C., 328, 337, 342, 422, 423, 424 Torossian, A,, 431 Townes, C. H., 258,318 Trabert, F. W., 349 (89a), 424 Trapnell, B. M. W., 342 (64). 343, 357 (64), 423 Trendelenburg, E. A., 361, 364 (123), 383 (208), 408 (271), 41 1 (271), 426, 428,430 Trillat, J. J., 246,269 (64), 281 (5), 298,318 Trolan, J. K,, 417 (298, 299), 430 Trule, J., 385 (221), 428 Trump, J. C., 368 (ISl), 426 Tsin, N. M., 384 (215), 428 Tucker, C. W., 353 (95k), 425 Tuetsch, W. B., 178 (126), 179 (126), 205 Turnbull, J. C., 343 (74), 424 Tuul, J., 418 (312), 431 Tuzi, Y.,336, 337, 423 Tykodi, R. J., 335 (48), 423
U Ullman, J. R., 383 (200), 428 Ure, R. W., Jr., 217 (20), 242 Uyeda, R., 290 (119), 321
AUTHOR INDEX
443
V Weinreich, 0. A., 404 (262), 429 Weiss, A,, 281, 320 Valat, J. G., 418 (305), 431 Weiss, H., 226 (31), 243 Valnev, J., 369, 427 Welch, J. A., 168, 204 Van der Merwe, J. H., 295, 321 Vanderschrnidt, G. F., 384 (214), 421 Wernick, J. H., 216 (19), 242 Westrnacott, K. H., 279 ( 8 5 ) , 320 (214), 428 Vanderslice, T. A,, 306 (184), 322, 324 (8), Wheeler, A., 342 (71), 424 Whetten, N. R., 349 (87c, 87d), 416 (288, 407, 408 (269), 41 1 (269), 422, 430 29 I ), 424, 430 Van Oostrorn, A., 393 (239a), 429 White, D. H., 352 (91), 424 Varadi, P. F., 356, 385, 387, 425, 428 Varnerin, L. J., 352 (91), 360, 361 (118, Wiehbiick, F. P., 366 (136). 426 119), 363, 364(118), 376(187), 424,425, Wiese, J. R., 226, 243 Williams, A. J., 152, 203 427 Williams, C. E., 373,427 Veksler, V. I., 254, 304, 318, 322 Williams, W. S., 212, 213, (8b), 238 (8b), Venema, A., 324 (4), 345, 372, 383, 390, 242 393, 422, 424 Wilrnan, H., 298, 321 Verrnount, P., 295 (145), 321 Wilson, J. N., 298, 321 Vernon, R., 180, 181 (140), 205 Wilson, V. C . , 130 (22), 169, 176, 184 Vickery, R. C . , 218 (22, 23), 243 (22, 108), 191, 202, 204, 205 Vinegard, G. H., 259 (30), 265 (30), 318, Witteborn, F., 275 (61a), 319 359 ( I l l ) , 360 ( I l l ) , 425 Witternan, W.G . , 152 (63), 203 Voge, H. H., 298 (154), 321 Wolfe, R., 211 (7), 242 Vogl, T. P., 415 (284), 430 von Ardenne, M., 282, 320, 356, 405 (97), Wolsky, S. P., 269, 281, 319, 320, 344 (79), 367 (142), 374 ( 1 79), 408 (142), 41 1 (142), 425 424, 426, 427 von Hippel, A., 288, 305,320, 322 Wood, R. W., 291, 321 Vought, R. H., 217 (21), 242 Wooten, L. A., 163 (91). 203 Wortman, R., 336 (52), 423 W Wright, D. A., 127 (71, 128 (7), 20Z Wagener, J. S., 343 (79, 374,408 (75), 41 1 Wuerker, R. F., 169, 204 (75), 424, 427 Wagener, S., 151 (62), 203, 357 (106), 400 Y (249), 425, 429 Waite, F. A., 279 (77), 320 Yaffee, M., 197, 206 Walcher, W., 256, 318 Yang, L., 176 (113), 204 Waldschrnidt, E., 349 (96); 356, 425 Yarwood, J., 324 (6), 422 Waller, J. G., 305, 322 Yonts, 0. C . , 276 (as), 284, 319, 320, 366 Wallor, R., 276 (71), 319, 367, 426 (1371, 367 (137), 426 Warner, C., 111, 180, 181 (140), 205 Young,’A. H., 400 (247), 429 Warshaw, S. D., 303 (1701, 322 Young, J. R., 237 (43), 243, 303 (171). 322, Waters, P. M., 368 (148, 149). 426 349 (87c, 87d), 359, 371, 376 (188), 401, Webster, H. F., 144, 155 (73), 157, 162, 402 (168), 416 (288, 291), 424, 425, 427, 202, 203 430 Webster, W. M.,183 (ISl), 194,205 Yurasova, V. E., 266,319 Weeks, J. O., 387, 429 Wehner, G. K., 129, 183 (18), 202, 245 ( l ) , Z 257, 258 ( I ) , 263, 266, 270 (68), 276 (72, 731, 280, 287, 304, 317, 219, 320, 366 Zandbern. E. Ya.. 172. 204 (132, 133, 138, 139), 367 (132, 134), 426 Zdanuk,-E. J., 269, 319, 344 (79), 367
444
AUTHOR INDEX
(142), 374, 408 (142), 411 (142), 424,
426, 427 Zener, C., 216 (16), 242 Zollman, J. A., 416 (295), 430
Zollweg, R. J., 178, 179 (128), 180, 181 (139), 205 Zwanzig, R. W., 329, 422
Subject Index A Accomodation coefficient, 328-333 Adsorption on evaporated metal films, 343 physical, 298, 326, 328-337 activation energy, 327 contamination by, 307-308 rate of, into a chemisorbed layer, 337 Amplifiers, backward-wave, voltage gain, 95-105 Anode back emission, 161 work function, optimum, for maximum power output, 133 Arcs anomalous low-voltage, 183-186 hot-cathode, 183 Arrival rate, particle size dependence on, 292 Atom ejection from polycrystalline targets, 266-270 Attenuation coefficient, resistive, 59-60 Auger electrons, 367-368
B Bakeout of vacuum system, 388-391 Barkhausen-Kurtz osciHation in BayardAlpert gauge, 403 Bayard-Alpert gauge, 376 ff., 392 ff. “activated” pumping in, 378 Beam variables, complex exponential notation, 14-17 Bismuth telluride, 218 Bunching equations, electrokinetic, 5-9 C Capacitance mutual, annular bunched beam in drift tube, 35-36 self, beam-circuit, 29-42 Capacitances, beam-circuit, 3-5 Carcinotron, 17, 112 Cathode
carbide, 152-153 dispenser, 151 material evaporation, 153-154 oxide, 151-152 oxidized metal, in Cs vapor, 150-151 polycrystalline, with alkali metal film, 146- 150 single crystal, in alkali vapor atmosphere, 144-146 Cesium plasma converter, 180 ff. one-dimensional model, 181 thermionic energy converter, 169-193 high pressure, 180-183 low pressure, 171-180 Charge continuity equation, small signal form, 9 -ratio beam-circuit coupling coefficients, 42-45 Chemisorption, 299, 327, 337-343 contamination by in sputtering, 308-3 11 pumping in ionization gauges, 401 Circuit charge, self-induced, cold-circuit, 12, 13 loss parameter, space-harmonic structure, 53-60 variables, complex exponential notation, 14-17 Clothrate compound formation, contamination by, 305-307 Collisions hard sphere, 260 ionic critical energy for, 260-261 Keywell model, 260-262 between particles, 247 Condensation coefficient, 328-333 Conductivity, thermal, 209, 212-214 lattice, decrease of, 223-224 Contamination in impact evaporation, 299308 Convection current continuity, 91 Converter, nuclear heated thermionic, 196197, 198 Copper sputtering rate, 277
445
446
SUBJECT INDM
Coupling coefficient, beam circuit, 3-5 for spatially periodic structure, 17-29, 42-45 Cryopumps, 384-386 Crystal structure, effect of particle atomic velocity, 294 Current constant, region of thermionic coverter, 132 density dependence of sputtering yield, 283 diode, 158-159 ion, emitted from a surface, 172 residual, in- Bayard-Alpert gauge, 393395 start-oscillation, 120-121 -voltage characteristics planar diode, 156 ff. thermionic converters, 130-136
D Debye length, 173 Degeneracy parameter, 210 Density, charge carrier, 210-21 1 Desorption activation energy, 326-327 gas, from cathode in impact evaporation, 303 rate, 330, 339 Diffusion back, of ion ejected particles, 288 coefficients for elements in silver, 308, 309 of gases in solids, 357-358 constants for gas-metal combinations, 355 for gas-nonmetal combinations, 354 of gas from inside a solid, 352-356 Diode, planar, current-voltage characteristics, 156 ff. Dislocations in thin films, 296-297 Doping concentration in thermoelectric materials, 225-227 reasons for, 226 Drive quartic, forward total wave, 76-79
E Efficiency thermionic converter, 136-142
thermoelectric generator, 240-241 optimum, 233 Ejection coefficients for ion bombardment, 261-262 Electron -beam rf motions ballistic equation, 5-9 density radian frequency, 56-57 ejection, kinetic by ions, 248 by positive ions, 367-368 potential, 248-251 emission, secondary, 248-251 entrance into rf field electrokinetic boundary conditions, 87-93 electromagnetic conditions, 93-95 interaction, space-harmonic traveling wave, 1-123 Electrons, ion ejected, angular distribution, 249-250 Emission charged particles from surfaces by energetic particles, 247-257 coefficient, electron, 248 density, electron of cesium on molybdenum, 148-149 of cesium on tungsten, 148-149 of rhenium in cesium vapor, 144-145 thermionic, 127 Emissivity, net, of cathode-anode combination, 138 Energy cpnversion, thermionic, 125-206 brief history, 129-130 idealized model, 130-142 vacuum, 154-169 converters, thermionic applications, 195-198 cesium, 169-193 fabrication, 164-165 space charge problems, 155-163 surface physics problems, 163 three electrode, 167-169 vacuum, 154- 169 diagram, potential, thermionic converter, 126 gap in compound semiconductors, 219 of sputtered particles, 270-271 Entrapment, ionic, 358-366 Epitaxy, 295
447
SUBJECT INDEX
Equilibrium, thermodynamic, systems in, 333-337 Evaporation, impact in glow discharge, 245-322 reactive, 31 1-317 sources of contamination, 299-308
F Field continuity, axial, 98 Figure of merit for case of acoustic scattering, 210 thermoelectric device, 209, 220-224, 240 Film growth, thin in glow discharge, 245-322 in high vacuum environment, 289-297 thermal aspects, 291-292 Films, thin, 418-419 work function on pure metals, 143-151 Flash-filament technique, 339 Flight-line diagram, 87-90 Flux density, radial electric, 34-35 incident particle, at sputtering target, 28 1-284 Focusing effects in atom ejection from single crystals, 263-266 parameter, Silsbee, 265 Frenkel model of vapor condensation, 291 Frequency offset effects, 79-84 nearly cubic quartic case for zero values, 84-86 parameters, 60, 63-66 forward-wave, 77 G
Gadolinium selenide, 218 Gain in backward-wave device, 17 -bandwidth product, 121 evaluation, backward-wave, 109-112 growing-wave hot-circuit, 66-68 parameter resistive-wall, 58-60 transmission line, 57-58, 60-66 -producing interferenee, backward-wave, 105-109 Gas cleanup in a glow discharge, 306
content increase of sputtered target, 279 diffusion from inside solids, 352-356 discharge, self-sustainink, 284 evolution by surface electron bombardment, 369 sources in ultrahigh vacuum system, 343-358 Gases absorbed, desorption of in ultrahigh vacuum, 344-347 electronegative, effect on glow discharge. 313-314 ionically pumped inert, ion-induced re-emission, 362-364 Gaskets, vacuum, 416 Gauges Bayard-Alpert, 376 ff., 392 ff. ionization calibration in ultrahigh vacuum region, 413-414 cold-cathode, 396-399 hot-cathode, 391-396 “nude”, 404-405 sensitivity to various gases, 399-400 magnetron cold-cathode, 397-398 inverted, 398 Penning, 396-398 sensitivity measurement, 412-414 Getter-ion pumping, 375-383 Getters, 374-375 Glow discharge characteristics, 284-289
H Hall mobility, 211 Heat of adsorption of helium, 335 initial, of gases on metals, 342 for various gas-solid combinations, 334 ff. of chemisorption, 310 transfer ambipolar, 213 exciton, 214 radiative, 214 of vaporization, latent, of atom on a surface, 293 Henry’s Law, 333
I
M
Impedance cold-circuit characteristic, 10-1 1 -ratio parameter, 56 Impurities in thermoelectric materials, 224-
Magnetron energy converter, 154, 167-169 Metal film classification, 293 work function, 143 Miscroscope, electron, observation of impact evaporated films, 290 Mobility, 21 1-212 in compound semiconductors, 219 Molybdenum ejected by Hg+ ions, 267 Motive, 126 diagram of idealized converter, 13 1 mirror image, 127
225 Interaction, electron, in ultrahigh vacuum,
368-369 Intrinsicity, temperature onset of, 223 Ion bombardment surface damage, 278-281 current produced by electron current in gas, 405-406 counting in mass spectrometer, 410-41 1 emission, secondary, 251-255 incident ion mass dependence, 252-253 incident ion velocity dependence, 254-
255 impact, positive, on surfaces, 358-368 penetration, depth of, 303 pumping, 375-383 gas re-emission rate, 376-378 Ionization efficiency, surface, of cesium, 171-172 gauge Bayard-Alpert, 376 ff., 392 ff. pumping, 375-378 probability, atom, 172 volume, in cesium plasma, 183-186 Ions penetration depths in solids, 359-360 rare gas, sputtering from silver by, 252 reflection coefficient, 254-255 secondary, production by positive ion impact, 368 Xet, sputtering from silver by, 251
N Nucleation film growth after, 295-297 in high vacuum environment, 289-297
0 Occlusion, contamination by, 308 Omegatron, 407 Onsager reciprocal relations, 21 1 Oscillation of low-pressure cesium converter, 178-
180 in Penning gauge, 403 Oscillations, plasma, 183 Oscillators, backward-wave, voltage gain,
95-105 Outgassing of ionization gauge electrodes,
393 Oxygen discharge, ion content of, 314 impurity in thermoelectric materials, 225
J
P
Junctions, thermoelectric, cascading of,
Patch effects, 146 Patch fields, 178 Pease sputtering yield equation, 259-260 Peltier coefficient, 208 Penning gauge, 396-398 Permeability of gases in solids, 357-358 Permeation gas, 347-352 rate, 347 Perturbation quartic equation, graphical study, 70-76
229-230
L Langmuir isotherm equation, 329 Lead resistance, optimum, of thermionic converter, 139 Lead telluride, 218 Leak rate measurement, 412-414 Leaks, controlled, for gas handling in ultrahigh vacuum, 415-416
449
SUBJECT INDEX
Phase relationship, spatial, bunched beamI and vanes, 30-31 Phasor representation of backward-wave gain-producing interference, 105-109 Phonon transport, 212 Photodecomposition of gases, 369-371 Photodesorption of gases, 369-371 Plasma devices, 419-420 mode of high pressure thermionic converter, 180 ff. oscillations, 183 Potential distribution . in a cesium thermionic converter, 172173 Laplacian I,,, open-channel continuous, 27 interaction, in shielded Coulombic model, 259 open-channel, Laplacian expression for, 34 Power loss, cold-circuit, 66-68 output of cesium converter, spacing effect, 190 thermionic converters, 130-136 vacuum converters, 161-163, 165-167 Pressure dissociation, 357 limiting, in Bayard-Alpert gauge, 392393 measurement errors, 402-405 partial of a gas in vacuum, 324 measurement of, 405-412 total, in ultrahigh vacuum region, 391405 ultimate achieved in vacuum system, 324 vapor of cesium, 169 of materials, 356-357 Propagation constant complex, 63 hot-circuit total-wave, 46 perturbation due to space harmonic interaction, 68-70 Pumping cryogenic, 384-386
effects in ionization gauge, 400-402 getter-ion, 375-383 hydrogen condensation, 384-385 at an incandescent filament, 401-402 ionic, 375-383 Pumps, vacuum, 371-386 cold-cathode discharge, 380-382 diffusion, 371-373 “evapor-ion”, 380-382 “Hall” type, 380-382 inverted magnetron, 378-380 molecular-drag, 373-374 “sputter-ion”, 380-382 Pushing measure, 61 perturbation, complex, 46-47
R Rasor cesium converter, 191-192 Recombination continuum, radiative in planar Cs diode, 187-188 Reflection coefficient, thermal electron, 127 of metastable atoms and ions, 255-257 Refrigerator, thermoelectric, performance, 234, 241 Resistance, contact, in thermoelectric devices, 231 Resistivity of cesium plasma, 188-190 Richardson equation, 126-129 plot, 127-128
S Saturation, ionic pump, 361-363 Schottky effect, anomalous, 184 Seals, vacuum, 416 Seebeck coefficient, 208, 209 Semiconductors, compound, 21 5-220 Slip perturbation complex, 46-41 measure, 61-62 Solubilities for gas-metal combinations, 351 for gas-nonmetal combinations, 350 Solubility of gases in solids, 357-358 Space charge nearly cubic quartic case for zero values, 84-86
450
SUBJECT INDEX
neutralization, cesium pressure for, 177-178 problems in vacuum thermionic conversion, 155- 163 reduction factor, 57 space-harmonic structure, 53-60 voltage, beam circuit self-capacitance governing, 29-42 voltage terms, 55-57 wave, 55-57 harmonic interaction characteristic equations, 45-53 space charge voltage affecting, 33 harmonic quantities, first forward, 16. harmonic structure having low cutoff, 51 power, solar-thermionic system for, 196 simulation, 420-422 Spectrometer mass calibration in terms of gas pressure, 407 gas evolution in, 406 measurement of partial pressure, 450 residual gas composition in, 41 I sector, for ultrahigh vacuum operation, 407-409 trochoidal, 409-4 10 Speed, pumping atomically clean surface in ultrahigh vacuum, 327 of cold-cathode gauge, 402 of ionization gauge, 401-402 measurement of, 412-414 Sputtering, cathode, 257-284, 366-367 angular distribution of ejected particles, 263-270 energy of ejected particles, 270-271 surface cleaning, 367 theoretical models, 257-263 threshold energy, 263, 266 Start-oscillation conditions, 83-84, 112-120 Sticking probability, 310 of gases on tungsten, 338 ff. Surface cleaning, glow discharge, 299-301 effects in sputtering, 278-281 in ultrahigh vacuum, 325-328 physcis problems in vacuum thermionic conversion, 163
T Temperature across a glow discharge, 286 desorption, 331 diffusion, 331 electron, in Cs plasma converter, 181-183 junction, of thermoelectric generator, 227 outgassing, 345 Thermocouple cesium using auxiliary discharges, 193-194 using fission fragments, 195 leg fabrication, 228-230 plasma, 190-191 Thermoelectricity, 207-243 applications, 233-235 device design, 227-232, 236-237 materials for, 215-220 fabrication, 224-227 practical considerations, 220-232 theory, 235-239 Thomas-Fermi model for atom shielding parameter, 259 Thomson coefficient, 208 heat, 232 Threshold, sputtering, 263, 366 Time adsorbed molecule remains on a surface, 326, 327 dependence of pressure in ion pumping, 378 desorption, of gas from a solid,. 352-356 sojourn, 328-333 Titanium getters, 374-375 Transmission cicuit equations, 9-14 Trapping efficiency, 360-361 refrigeration, 383 Traps, vacuum, 383-386 metal foil, 383 molecular sieve, 383-384 saturation, 384 Tunability, voltage, backward-wave oscillator, 120-121
V Vacancy production by ion bombardment, 279 Vacuum
45 1
SUBJECT INDEX
conditions at sputtering target, 281-284 system bakeout, 301-302 sputtering, precleaning of, 299-300 ultrahigh, 323-431 applications, 417-422 physical processes controlling, 325-371 processing, 386-391 technology, 371-417 Valves, ultrahigh vacuum, 415-416 Vapor condensation phenomena, 290-297 deposition density, critical, 290-291 polymeric stable species, 294-295 Voltage gain, backward-wave amplifiers, 95-105 internal, of thermionic converter, 132 -ratio beam-circuit coupling coefficients, 17-29, 42-45
W Wave backward cubic, 80-81 gain evaluation, 109-1 12 lossless cold-circuit propagation, 23
forward cubic, 81 frequency offset parameters, 77 total, drive quartic, 76-79 parameters, total backward, 16 propagation in spatially periodic structure, 21-22 Wiedemann-Franz ratio, 209 Wilson cesium converter, 191 Work function metal, 143 Richardson, 128 surface, 126-129 temperature variation, 128 of various surfaces, 142-154
X X-ray current in ionization gauge, 391-392
Y Yield, sputtering current density dependence, 283 incident particle dependence, 275 manifold pressure dependence, 283 Pease equation, 259-260 silver, mass dependence, 258
This Page Intentiona lly Left Blank
