Advances in Electronics and Electron Physics, Volume 13

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME XI11

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Advances in

Electronics and Electron Physics EDITED BY L. MARTON National Bureau of Standards, Washington, D. C.

Assistant Editor

CLAIRE MARTON EDITORIAL BOARD T. E. Allibone W. B. Nottingham H. B. G. Casimir E. R. Piore L. T. DeVore M. Ponte W. G. Dow A. Rose A. 0. C. Nier L. P. Smith

VOLUME XI11

ACADEMIC PRESS A Subsldlery of Heccourt Brace Jovenovlch. Publlshers

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COPYRIQHTO 1960,

BY

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CONTRIBUTORS TO VOLUME XI11 RAYMOND CASTAING, Ddpartement de Physique Gdn6rale) Faculte' des Sciences, Universite' de Paris, Orsay, France

JOHN B. HASTED,Physics Department, University College, London, England

P. A. LINDSAY, Research Laboratories of The General Electric Company Ltd., Wembley, England ERWINW. MULLER,Field Emission Laboratory, T h e Pennsylvania State University, University Park, Pennsylvania PAULK. WEIMER,R C A Laboratories, Princeton, N e w Jersey

V

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PREFACE An editor’s constant worry is the material which goes into his publication. He has to see to i t that the material is useful and good. By useful, I mean that it presents material in which most of the readers are interested; and, in presenting such material, the significant is winnowed away from the insignificant. By good, I mean that the material is presented by a person who can write well and who is able to satisfy the requirements of a useful reading. I n selecting both the materials and the authors, your editor is guided by the advice of the editorial board and of a number of other people, including the book reviewers in scientific magazines. All this gives a very good guideline, but I would appreciate it if the guidance would be on even a broader basis. I would like to invite the individual reader to send me comments on areas which we have neglected, on topics which we have not presented well, making suggestions for improvement and making suggestions about new authors whom we could draw in. T o provide the critical reader with a guideline as to the fields which we intend to cover in the next three or four volumes, I am giving here a brief summary of such subjects. They are as follows: Photoconductive Phenomena The Electron as a Chemical Entity Generation of Microwaves by Cerenkov Radiation Hydrogen Thyratrons Quadrupole Lenses High Power Microwave Tubes The Distribution of Ionization in the Upper Atmosphere hlillimicrosecond Techniques Masers Millimeter Waves Atomic Frequency Standards The Autodyne Detector as Applied to Paramagnetic Resonance Relaxation in Diluted Paramagnetic Salts a t Very Low Temperatures Ultrahigh Vacuum Techniques Scattering in the IJplwr Atmosphere Airglow Thermionic Conversion Elcctroluminescence vii

viii

PREFACE

Capacitance of P-N Junctions Electron Phenomena on the Semiconductor Surface Thermoelectric Phenomena Atomic Collisions Cathode Sputtering

I hope this listing will be of aid to those who are willing to make any new suggestions. I n advance, I am transmitting my best thanks for their comments and would like to use this opportunity to thank all those who have helped in preparing this and all other preceding volumes. Washington, D.C. October, 1960

L.MARTON

CONTRIBUTORS

PREFACE.

TO

VOLUME XI11

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

V

vii

Inelastic Collisions between Atomic 8~8tOmS JOHN B . HASTED I . Introduction . . . . . . . . . . . . . . . . . . . I1. Classification of Collisions . . . . . . . . . . . . . . . I11. Experimental Methods of Study of Inelastic Collisions . . . . . . IV . Collision Cross Sections-The Determining Factors . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . List of Symbols . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .

1 3 4 27 75 75 78

Field Xonization end Field Ion Microscopy ERWINW . M ~ L L E R I . Introduction . . . . . . . . . I1. Field Ionization of Free Atoms . . . I11. Field Ionization near a Metal Surface . IV . Field Ion Emission from a Metal Surface V . Field Ion Microscopy . . . . . . Acknowledgments . . . . . . . References . . . . . . . . . . .

. . . . . . . . . . 83 . . . . . . . . . . 84 . . . . . . . . . . 87 . . . . . . . . . . 100

. . . . . . . . . . 114 . . . . . . . . . . 177 . . . . . . . . .

177

Velocity Dbtribution in Electron Streams P . A . LINDSAY I. Introduction . . . . . . . . . . . . . . . . . . . I1. General Considerations . . . . . . . . . . . . . . . . I11. Probability Considerations . . . . . . . . . . . . . . . IV . Velocity Distribution of the Electrons Emitted by a Thermionic Cathode V . Velocity Distribution in Plane Systems . . . . . . . . . . . VI . Velocity Distribution in Cylindrical Systems . . . . . . . . . VII . Velocity Distribution in t.he Presence of a Magnetic Field . . . . . V I E Experimental Support for the Theoretical Results . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . Referencea . . . . . . . . . . . . . . . . . . . .

ix

182 185 189 197

206 250 294

308 310 311

X

CONTENTS

Electron Probe M i c r o d y e k RAYMOND CASTAING I . Introduction . . . . . . . . . . . . . . . . . I1. General Structure of the Microanalyzer . . . . . . . . . 111. The Fundamentals of Quantitative Analysis by X-Ray Emission IV . The Contribution of Microanalysis to Scientific Research . . . References . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

317 324 360 379 384

. . . . . .

. . . . . .

387 389 390 394 399 406 414 419 423 426 430 435 436

.

Television Camera Tubes: A Research Review PAULK . WEIMER I . Introduction . . . . . . . . . . . . . . . . . I1. Television Pickup with Nonstorage Devices . . . . . . . I11. The Concept of Storage . . . . . . . . . . . . . . IV . The Image Orthicon . . . . . . . . . . . . . . . V . Camera Tubes Based on Photoconductivity . . . . . . . VI . Electron Optical Considerations in Camera Tubes . . . . . VII . Sitpal-to-Noise Considerations in Camera Tubes . . . . . . VIII . Image Intensifier Camera Tubes . . . . . . . . . . . IX . The Search for More Efficient Methods of Video Signal Generation X . Camera Tubes for Special Applications . . . . . . . . . X I . Fundamental Limitations on Camera Tube Performance . . . XI1. Image Pickup Devices of the Future . . . . . . . . . References . . . . . . . . . . . . . . . . . . AUTHOR INDEX .

. .

. .

. .

. .

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SUBJECTINDEX

. . . . . .

439 447

Inelastic Collisions between Atomic Systems JOHN B. HASTED I’h y&cs Department, University College, London, England

I. Introduction.. . . . . . . . . . . . . . .......................... 11. Classification of Collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B. Mass Analysis of the ProjectiIe Ion after Collision.. .................... C. Charge-Collection Methods. . . . . . . . . . . . . . . . . . .......... D. Exchange Collisions at Low Energies.. ...... ............

............

1

8 11 14 21

,4. Individual Quantum-mechanical Calculations B. Adiabatic Theory .................................

I+’. The Behavior of Cross Sections at Ex

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........

78

1. INTRODUCTION I n many branches of the physics and electronics of gases there occur collisions between atomic systems in which the internal energy and electronic structure are affected. It is important to survey the studies that have been made of such inelastic collisions, from thermal energies up to the energy region where the likelihood of collisions becomes comparable with that of nuclear reactions. While not undertaking a complete survey, we wish in this article to point out what we believe to be the significant trends in this field. The cross sections for inelastic collision processes between atomic systems are often in demand in the study of thermonuclear machines; gaseous electronics; particle accelerators; the physics of the ionosphere, night sky, and aurora; thermochemical systems; and mass spectrometry. 1

2

JOHN B. HASTED

In discharges intended to reach thermonuclear temperatures the atoms must carry charge in order that their temperature may be raised electrically. Neutralization of this charge by the processes of charge exchange with unreactive impurities can be a most troublesome factor, since each atom of impurity may react many times, being stripped in the process of all its outer shell of electrons. In the ionized regions of the atmosphere the high rate of recombination must certainly be due to a nonradiative process. Yet the ions initially formed by photoionization are largely atomic. A balance of reactions determines the equilibrium of ions present, for example, the high proportions of NO+ ions found in rocket measurements. Auroral ionization is caused in part by the direct ionizing collisions between protons and the atmospheric gases. Other fields of contemporary interest in which inelastic collisions are important include t,hose of ion-molecule reactions in mass spectrometry and radiation chemistry, the production of chemical compounds in gaseous discharges, the charge equilibrium in flames, the impurity problems in accelerators, the acceleration of charged particles for ion motors in space travel, the kinetic heating of rockets, the development of tandem van de Graaff generators, and the production of polarized proton beams for nuclear collisions. We are very far from being able to calculate by quantum theory the cross sections of more than a limited number of inelastic processes. For hydrogen and even helium, a direct comparison of experiment with quantum theory is possible, and an excellent survey of measurements with theee atoms has recently been made by Allison ( I ) . For more complex atoms, quantum theory becomes far too unwieldy to handle by normal techniques, and we are left with more general, skeleton hypotheses, semiclaesical, statistical, and quantum, to guide the experimentalist and the client who requires to know the cross sections. Nevertheless there is a considerable body of experimental knowledge and theoretical reasoning which can be applied, and it is with this that we shall largely concern ourselves in this article. Since these types of collision have been studied by physicists interested in many different fields, we should not be surprised to find different methods of approach. In studies of the passage of high-energy particles through matter, for example, there is naturally an emphasis upon the absorption and charge equilibrium of the beam of particles. We cannot afford, however, to adopt any but the most general approach, interesting ourselves if possible in the nature, the internal and kinetic energy, and direction of travel of all the products of the reaction. We shall first introduce a more general system of notation of these collisions than has been done in past studies.

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

3

’rhe inter;tc:tiotm of atomic xystems in collision may involve the exch:mge or liberation of internal energy, electromagnetic radiation, or electrons. Inelastic collisions nlay be classified in many ways, the most fundamental being the division into ionizing- and exchange-type collisions. In the exchange collision only internal energy may pass to or from the atom pair, atid no rndiation or electrons. Thus the collision : An+ + Bb+ -+

Am+

+ Bn+

+

+

would be classified as charge exchange provided that a b = m n. If, on the other hand, a b < m n, the collision would be regarded as an ionization, and in the simplest case only one atom is ionized, so that either a = m or b = n. If both atoms are ionized, m > a, n > b. But it is possible that a combination of the two types of collision may occur, that is, m+ n >a b but m < a , b < n. Examples of the purely exchange-type of collision include simple Bf, that is, a = 1, b = 0, m = 0, n = 1. charge transfer A+ B - A We shall describe this collision by the notation 1001, and put its cross section as 1 ~ 0 1 Collisions . of type 2002 have been studied, and collisions such as 2011, 1012,’ which we shall describe as “partial charge transfer.” Collision 001 1 and its reverse, “mutual neutralization” are well known; clearly many more possibilities exist. Collisions of the ionization-type include 101n, “positive ion ionization ,” 10m0, which has been called “stripping,” the “electron detachment” reactions ZOO0 and 1010, and the important “electron loss” process 0010. More general versions of these might be written aOmn (m = a ) , aOmO (m > a), fOmO, OOmO, and even 000n. An important reaction which is a combination of both types is loon, which may play a part in the ionization of neutral atoms by positive ions. More generally, such a dual reaction might be written a h n , where m n >a b, although m < a. We might call such a collision “transfer ionization.” I t will be noticed that b has usually been written as zero; but in collisions between two charged particles, such as mutual neutralization, 100, this is not so; there is thus a justification for the adoption of a symmetrical notation with four subscripts instead of three, or even the more usual two. Reactions may take place between excited atoms or ions and groundstate atoms or ions. The products of ground-state collisions may of course

+

+

+

+

+

+

+

is used to notate the single negative charge.

JOHN B. HASTED

4

be excited, so that we have such processes as; charge transfer between excited ions and ground-state atoms, A+‘ B + A B+, which we shall notate 1’001; charge transfer resulting in excited products, 100’1, 1001’, or even 100’1‘; exchange of excitation between similar atoms 0’000’; excitation of atoms or ions by collision with ground-state ions or atoms lolo’, 101’0, 0000’; and finally, double transitions in which both particles are excited lOl’O’, 000’0’. Reactions between ions and molecules may involve molecular rearrangement, dissociation, or association. We do not consider it worth while at this stage to introduce a more general notation to cover these eventualities, but rather we shall write the chemical equations of the processes. The types of reaction, notation, and nomenclature described above are tabulated for convenience in Table I.

+

+

TABLEI. TYPES OF INELASTICCOLLISION Exchange collision 1001, 2011,

2002, 1012,

etc., etc.,

Charge transfer Partial charge transfer Mutual neutralization Charge transfer to an excited state Excitation exchange

ooii, iioo 1001‘, etc., 0’000’

Ionizing collisions 101n, 10m0, iooo, 0001, lolo’,

aOmn aOmO ioio, OOOn 101’0,

(m = a)

( m > a) iomo

0000’

Positive ion ionization Stripping Electron detachment Electron loss Excitation processes

Composite collisions lOOn

etc.,

Transfer ionization

111. EXPERIMENTAL METHODS OF STUDYOF INELASTIC COLLISIONS For complete knowledge of an inelastic collision it would be necessary to assess the energy state, charge, velocity, and direction of travel of each particle and photon before and after a collision. This is as yet far from being realized, although various parts of this information have been obtained. Let us first consider the methods used for the study of state of charge. The general reaction abmn requires four determinations of charge-to-mass ratio-a quadruple mass spectrometer

INELASTIC COLLIGIONS BETWEEN ATOMIC SYSTEMS

5

in which two ion beams would be crossed, the charge-to-mass ratio of each being determined both before and after crossing. The crossing of ion and atom beams has been carried out in experiments of Fite, Brackmann, and Snow (a), and is also being attempted by Donahue (3). But, in general, experiments have always been done with one beam of mass-analyzed ions in collision with atoms of a gas. Mass analysis of the slow ion formed from the gas atom has been carried out by Fedorenko and Afrosimov (4), Lindholm (5), and Gilbody and Hasted (6). Mass analysis of the impinging ion beam after collision has been carried out by Fogel (7), Kaminker and Fedorenko (B), Flaks and Soloviev (9),Ribe ( l o ) ,Allison ( I I ) , Montague ( l a ) , Whittier (IS),Everhart et al. (14), and Bydin (15).It is also possible to obtain information about cross sections from a collection of the charge due t o the particles formed in collision without actual mass analysis [Keene (16), Hasted (17),Moe (It?), Fedorenko et al. (19)]. At energies of a few electron volts, data concerning i-ielastic ion-molecule collisions can be obtained by measurements in the source of a mass spectrometer [Field, Franklin, and Lampe (do), Stevenson and Schissler (21)]. At near thermal energies, measurements with a mass spectrometer may be made in afterglows of discharges [Sayers and Kerr (2.291. Ion atom excitation cross sections have been studied by examination of the photon emission in gas collisions [Fan and Meinel (El),Sluyters and Kistemaker (2411.

A . Mass Analysis of the Ion Formed jrom the Target In one type of experiment the collision is carried out in an electric field perpendicular to the ion beam, small enough not to distort its path unduly, but large enough to extract the slow charged particles formed in the collision region. These are accelerated and passed into a conventional mass spectrometer, which may be sector (Fig. 1) (4) or 180' (Fig. 2) (6). From the measurement of the ions as a function of collision gas pressure, provided this is low enough for only a small fraction of the ion beam to make collisions, there can be deduced a cross section for the formation of ions of a given charge-to-mass ratio, in the present notation

for values of n from one to four (4). The cross section is made up almost entirely of a sum of lOln (positive ion ionization) and lOOn (transfer ionization), which for n = 1 is simple charge transfer; less important collisions lOmn, combinations of stripping

6

JOHN B. HASTED

and ionization, must also be taken into account. Fedorenko's equipment permits the movement of the mass anaIyBer in such a way that the angle of scattering of the target atom ions (nearly 90") may be measured, and he also measures their energies.

cliiced in rollisions bet.ween ions and atoms.

Lindholm's 180" mass-analysis equipment is used primu,rily for the study of ions formed in the dissociation of molecular gases by ion beams, This might take place either by an ionization-type or an exchange-type collision, but there is a large internal energy difference between the two types of reaction for the same systems. On the grounds of adiabatic theory (see Section 1V.B) it is possible to be reasonably certain which cross section is being measured at the low energies a t which the experiments are being conducted.

7

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

There is clearly a lower energy limit set in Lindholm’s apparatus (Fig. 2) by the extraction field, which will not only distort the primary beam but also.introduce an energy spread in the secondary beam which may interfere with the mass analysis. Lindholm succeeds in measuring

X

(b) FIG.lb. Apparatus used by Fedorenko el al.for the measurement of angular distribution and elm of slow ions produced in collisions between ions and atoms. The primary ion beam passes along the dotted line.

large cross sections down to an energy of 25 ev, but the small cross sections encountered in multiple ionization collisions have only been measured down to a few kilovolts. There is scope for the design of experiments with good ion optics for the separation of primary beam and product; this might bc achieved with a radio-frequency mass spectrometer of small path length [Boyd (S6),Kerr (SS)].An alternative has been proposed by Donahue (3); the collision takes place between two superposed fast beams in motion, one charged and the other neutral; when these have nearly the same velocity the collision is taking place at n low energy; the neutral beam after ionization is capable of being mass analyzed. Some of the inliwriit, diffii:iilties iri t h t h m:~ss;LnxIysix of t:trgc.t :Ltoni iotis :m as follows: 1. There niust at the time of oollisiori IK? at least two orders of ningrii-

8

JOHN B. HASTED

tude difference between the kinetic energies of incident ion and target atom ion, in order that the electric field chosen for the collision region may satisfy the requirements outlined above. This places a lowenergy limit on the method of about 25 ev, which may be inconvenient in the investigation of ion-molecule reactions, and also in the study of the “onset” of multiple ionization by positive ions. Ion beam ‘A’ in vertical plane

Ion beam

‘B’in

horizontal plane

FIQ.2. 180” double mass spectrometer used by Lindholm for studying collisions in which the target atom ion is mass analyzed.

2. It is not always possible t o collect ions formed with appreciable kinetic energies, such as may occur in the dissociation of molecules in antibonding states. 3. The usual difficulties of mass discrimination and angle of acceptance are encountered in the mass spectrometer. There is also the possibility of change in calibration of the electron multipliers with type of ion and kinetic energy. The measurements are far from easy, and it is encouraging, therefore, t o find comparatively good agreement between the data of Fedorenko and Afrosimov (4), and those of Gilbody and Hasted (d’?’), using the simpler charge collection method. (Section 1II.C).

B. Mass Analysis of the Projectile I o n after Collision 1. Equilibrium Experiments. Many of the experiments in which the primary heam is niitss analyzed iLfter collision have been performed using

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

9

the ohargequilibrium metohod,which has heeti fully discussed in Allison's review mentioned above (1). I n a two-c:omponent system (for example, H", H+ beams a t high energies) the ratio of charged to neutral components in a primary beam passing through a gas, is determined by the balance of charge exchange 1001and loss 0010 cross sections; their ratio may be determined from the charge equilibrium a t a sufficiently high gas pressure for it to be pressure independent, provided there is no multiple ionization or energy degradation. At pressures or path lengths too small to equilibriate the beam, a single cross section may be measured by determining the ratio of charged t o neutral components. As far back as 1912, Wien (28) devised the equations:

where F1, is the equilibrium fraction of singly charged ions in the beam and [MI is used to represent the number of atoms of target gas per square centimeter traversed by the beam. If the beam passes through high vacuum each side of a collision chamber of length 1 containhg a gas having ( atoms per molecule, at a pressure P dyne/cm2, then [ M ]= N l ( P / R T where N is Avogadro's number, T absolute temperature, and R the gas constant. At this pressure Fl is the fraction of singly charged ions. The separation of charged from neutral beam components may be achieved by magnetic deflection, as in the work of Montague (12),Ribe ( l o ) ,or by electrostatic analysis, as in the experiments of Stier and Barnett (29), Bydin and Bukhteev (15). In earlier experiments, Bartels (SO) and Meyer (31) employed the separation of charged and neutral components purely by detection methods, but mass analysis, eve11 of a crude type, is greatly superior. Since the only change in the experimental conditions during measurement is a change of pressure, calibration of the detectors is unnecessary, which represents a great advantage. The errors that might arise from elastic scattering and from pressuic variations outside the collision chamber have been fully investigated and it'is now clear that what a t firet sight appears an unsatisfactory method is in t i 0 way inferior to the single-collision experiments otherwise in general use. There are, however, still unresolved differences between certain cross sections determined in this way and those determined by single-collision methods. The main disadvantage of the method is that it is limited to the determination of 1 0 ~ 0 1 , WCIO, iouoo, and muil in systems where multiple ionization is impossible; moreover the charge state of the target :itom is uiicertain. Undoubtedly,

10

JOHN

n.

HASTED

tlio future I d o i IBSto ~iiigl~~-c:olli.uio~~ c!xporiirlent,x, with target gas pressure^ low o11011gh to study all the prodrirtn of collision. 2. Sin,gle-C'ol/isiot~Rxpe~~imenls. In these experiments the m a analysis of the primary bcmi, after collision at low enough pressures, permits the measurement of stripping cross sections n

[Everhart et al. (14) Kaminker and Fedorenko (S)].In the experiments of Fogel et al. (7) iwiz and i ~ i are o measured; and in the experiments of Flaks and Soloviev (Q), w11and w02are measured. The movement of the mass

(a)

FIQ.3a. Apparatus used by Fedorenko el al. for the mass-analysis of a primary ion beam after collision. The equipment for the acceleration and mass-analysis of the beam before collision are also shown.

spectrometer and detector permits the study of the angular distribution of the ions formed. Integration of the differential cross sections will yield the total cross sections for multipIe stripping. The experiment gives no information on the charge state of the target atom. Typical instruments of this type arc shown in Figs. 3n and 3b.

INELASTIC COLLIBIONS BETWEEN ATOMIC SYSTEMS

11

Target pus inlet pressure gouges

Ion

(b)

FIQ.3b. Apparatus used by Everhart et d.for the measurement of angular scattering and elm of a primary ion beam afhr collision.Hole sizes: a-0.046 in. diameter: ~-0.012 in. X 0.070 in.; d 4 . 0 2 4 in. diameter.

C . Charge-Collection Methods In these methods the total charge due to the target atom is collected i n a low-pressure single-collision region. There is no information obtained about the charge state of the incident particle. It is possible to separate the effects due to the exchange collision 1001 and the ionization collision 1011, so that these cross sections can be measured, provided that multiple ionization can be ignored, which is the case a t low enough energies. Otherwise the measurements would give us

and

The method was devised many years ago by Goldmann (32) and was used by such workers Wolf (33) and Rostagni ($4) before its modern use by Hasted and co-workers (35), Dillon et al. (%), de Heel- (YY), Keeiw ( I G ) , and others. The charge separatioii is usually carried out by meatis of an electric field transverse t o the primary beam; a transverse magnetic

JOHN B. HASTED

12

field, which makes no essential difference under correct conditions, is sometimes added. Under the influence of these fields the slow ions formed by inelastic collisions travel down the lines of force. The charge exchange cross section 1001 is obtained by measuring the sum of charges collected at po2itive and negative electrodes, and the ionization cross section 1011 is obtained by measuring the electron current at the positive electrode, suppressing secondary electrons formed at the negative electrode by means of a negative grid. A typical electrode system of this type, as used by Gilbody and Hasted (6),is shown in Fig. 4. The difficulties in this method arise from gold-plated braae

I

X,

resistive SPeCeCE

/

beam path I

I

0

I

X

rm

section X X

?

FIQ.4. Electrode system used by Gilbody and Hasted for the collection of total charge produced in ion-atom collisions.

the effects at electrode surfaces, and the necessity of ensuring that all the ions or electrons formed in certain path length I of gas are collected. When this is so, the absorption of the primary beam I0 is given by the equation 1, = loe-"[M1'where I , is the final primary beam. At low enough pressures, in the so-called "single-collision region," this becomes 11/12 1, = u[M]l where I , is the charge collected. This equation may be used to calculate cross sections in experiments on the mass analysis of the target ion. Single-collision processes will produce currents with a linear pressure dependence, but double reactions such as

+

+ H2 -+ H + Hz+ Hgf + H2+ Ha+ + H H+

followed by

mould produce a current of Ha+ dependent upon the square of the pressure. The limitations of the method a t higher energies, arising from its inability to distinguish multiple charges, have been fully realized by Fedorenko, Afrosimov, and Kaminker (38).Its principal use at high energies is now in combination with mass-spectrometric analysis, as a check and calibratioii of the mass spectrometer and multiplier. A t,ype of experiment by which positive ion ionization 1011 hss been

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

13

studied a t energies low enough for the (wss sections to be extremely small, is that due to Varney (39) and most recently carried out by Moe ( I @ , under ultrahigh vacuum conditions. In the electrode system shown in Fig. 5, a magnetic field of a few hundred gauss parallel to the axis of the tube confines the paths of electrons to tight helixes along this axis, but has little

a

n

I7

0000

0000

0000

-

0

1

2

3

crn

FIG.5. Apparatus used by Moe for the measurement of ionization by positive ions 1001. Ions emitted by the Kunsman source S are accelerated through grid GI,traverse the equipotential ionization chamber, and are collected on P. The slit system S1, Sz,and & prevents reflected ions and secondary electrons originating a t P from reaching the ionization collector C. The Helmholtz coils shown in cross section produce an axial magnetic field which constrains electrons to move in tight spirals along the tube axis. The tube is usually operated with a 45-volt ion stopping potential between C and Ss, and a &volt electron stopping potential between Sa and Sz.

effect upon the paths of ions of energies of tens or hundreds of electron volts. Thus the ions from Kunsman source S are accelerated by grid G , and collected by plate P. Electrons formed by ionization in the gas are collected at C,those produced by SSbeing suppressed before reaching S2. With this type of experiment using powerful sources, cross sections as low as 10-23 cm2 can be detected. The primary ion beam is, however, limited in species and

14

JOHN B. HASTED

possibly impure. A feature of this experiment is the nieasiireiiicnt of lh! derivative of the cross Nection-energy curve hy .superimposing a sninll audiofrequency signal on the ion accelerating potential, so that the a+! component of the ionization current, measured on a phase-eensitive detector, is proportional to this derivative. In charge-collection techniques certain difficulties arise from the fact that the ions or electrons formed in the collision may have appreciable kinetic energy. The normal practice is to increase the transverse electric field until its variation produces no change in charge collected (saturation field). Fields as large as 100 volt/cm are not uncommon and a t energies of tens of kilovolts a fraction of the electrons produced by ionization may have energies of several hundred electron volts. In the detachment collision, 1000,studied by Hasted (40) and Bydin and Dukelskii (41, &), it is necessary to distinguish between electrons produced and negative ions which might be produced in charge transfer TOOL Normally such a process is impossible or unimportant at low energies, but for cases in which it is not, an electron filter has been designed by Bailey (43).This is a version of Loeb’s alternating current grid (4) in cylindrical symmetry. The primary beam is surrounded by two cylindrical grids, the inner one to screen it electrically, the outer one to accelerate the negative particles; outside this are plates arranged as in the vanes of a paddle steamer; to alternate vanes, potentials varying at a radio frequency are applied to filter out the electrons.

D . Exchange Collisions at Low Energies 1 . Mass-Spectrometer Sources. There are two methods of overcoming the special difficulties that surround the measurement of exchange cross sections at low energies. The first is to content oneself with measuring the cross sections integrated over an energy spectrum, which may be done by conventional mass-spectrometer technique. The second is to measure the time dependence of ion densities in afterglows of discharges, a technique comparable to chemical kinetics. The information obtained by the first method consists of rates of ionmolecule reactions of the type A* BC + A+B C. It is possible to deduce the reaction rates and the cross sections by measurement of the ratio of ion currents IA+B/IA+ issuing from a mass-spectrometer source a t different pressures. In these sources the gas is ionized by an electron beam in a transverse electric field (Fig. 6). The ions formed may react with the gas to form new ions whose abundance is proportional to the square of gas pressure, and whose appearance potential indicates that they are not formed by simple electron impact. Variation of the repeller potential changes the maxi-

+

+

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

15

mum ion energy in the source, but the collision may occur a t any energy below this value, or even above it, owing t o the issue of gas from the source and the fringing electric field. Assuming an ideal source, it is poscible to calculate mean cross section by the method of Field, Franklin, and Lampe (20)which follows.

u

CILIYPNT

FIG.6. Mass-spectrometer electron bombardment source. (a) Vinw from ion esit slit 8.(b) View from electron accelerating slit 5.

The ion A+ produced by electron impact in an ion source can travel a distance do to the extraction slit, under the influence of a potential gradient V(volt/cm). In collision with a molecule BC we have: kr

A+

+ BC -* ABC+ )S,+

ABC+ --+

Sj+

+ Fi

16

J O H N B. HASTED

where S,+ can be AB+, BC+, B+, C+; kl and ks; are the rate constants; the currents of secondary ions of the j t h type that are formed will be:

where [MI is the number of molecules per unit volume, ? A t is the time that the primary ion remains in the gas chamber, and IAt is the current of primary ions. Where only one secondary ion is formed, j = 1, Neglecting the magnetic field in the source ? A t can be shown t o be ?At = - 1 / D 2

2c

+D + 2Cdo + do In d DD22 +-+ 2Cdo 2Cdo - D

The ratio of currents of different ions will give us the rate constant kl. The number of complexes formed kI?At is equal to the product of the experimental cross section ueand the number Q’ of collisions made: k l T A t = u,QI. For unit concentration Q’ can be shown to be

where C = eV/Ml, D = d 8 k T / n M 1 , M1 being the mass of the primary ion. ?A+ must be calculated as before. It will be agreed that the Field, Franklin, and Lampe treatment is far in advance of the calculations of Stevenson and Schissler (21); the latter calculate cross section from the simple equation 1st =

~.f~+do[M].

This gives a correct mean cross section averaged over distance, in contrast t o which the cross section of Field, Franklin, and Lampe is averaged over time. But Stevenson and Schissler jgnore the initial thermal energies possessed by the projectile ions. This may account for certain disagreements, such as the energy dependence of the CH,+ forming reaction, which Field et al. find not to be proportional to V-%, although Stevenson and Schissler find that it is. Although Stevenson and Schissler’s cross sections are roughly correct, it seems to the author that the rate constants they calculate from them are wrong by an arithmetical factor. If the rate constant k = ueij and 8 the mean velocity is given by 5 = vfinal/2 where

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

17

then

where B is the experimentally determined factor of energy dependence of cross sections; it is found that for many reactions uC = BV-%. Stevenson and Schissler state that k = B . 600[M1/edo].H Moreover by this method it is not possible to determine the rate constants when the energy dependence of cross section differs from u. = BV-%, whereas using the Field, Franklin, and Lampe method of averaging, the rate constant may always be determined. As will be seen later, it is coincidental that ud = BV”; there is no reason why rate constants should be temperature independent. An entirely different method of analysis is used by Saporoschenko (45), whose experimental arrangement is essentially similar t o those outlined above, and is used in the same range of pressures.* Using nitrogen in his source he obtained ions N3+,N4+,both quadratically pressure dependent, and from the appearance potentials he deduces that N3+ is formed from an excited molecular ion N*+ NP--f N3+ N, while N4+ is simply formed, in a vibrationally excited state, from N2+and N2.SHe finds that the factor determining the abundance of these ions, for a fixed electron energy, that is, fixed abundance, is the familiar ratio of field strength t o pressure V/doP. Variation of both V and P confirms this result. This quantity is the X / P , well known in the physics of glow discharges and electron swarms. It can be related experimentally and theoretically to the mean electron temperature, on which it depends monotonically, the power being not very different from unity. Thus, as in Fig. 7, Saporoschenko is able to plot ion abundances as a function of a parameter which may be taken as mean ion

+

+

* Although many workers have the impression that ion-molecule reactions are studied a t lower premures, it will be seen from the following that this is not always the c m . Authors

Pressure ranges (mm+f-Hg)

-4 x lo-‘ 10-3-10-~ unknown, presumably the same aa those of Field el al. Saporoschenko (45) -10” 3 Franklin points out that a reaction of this type is unknown; therefore i t is more likely that the nitrogeh molecular ion is excited. Alternatively, Magee suggests that an excited nitrogen molecule may react with a ground-state nitrogen molecule to give N,+, the excess energy contributing to the kinetic energy of the electron liberated. Despite the evidence of the appearance potentials, it seems that “sticky” reactions are unlikely; Hamill and Pottie have observed several for alkyl halide ions and molecules, but never for diatomic molecules, whose degrees of freedom are far fewer.

Talroze and Lyubimova (46) Field, Franklin, and Lampe (20) Stevenson and Schissler (21)

18

JOHN

n.

HASTED

energy. I t will he sliowii l:bt,er that the cross srctioiis w l i i d i w i t 1)e drducod from these nbundances behave differently from t8hose of other workers, which may he of great, t,heoretical interest. It is questioiiable whcther Inass-spectrometer experimelits are aMe to produce reliable cross section-energy variation curves, because of uncertainty in the methods of energy analysis; this is therefore a key problem for experimentalists t o solve.

I00 90

-

80 .c Y)

a

.-e e

70

L

60

0

c .-

E

.-m

so

2 40

;30 20 10

0

FIO.7. Ion abundances obtained by Saporoschenko as a function of potential between ion repeller and exit slit.

It will be agreed that a more exact definition nf ion energy is required before the cross sections can be considered as accurate as those obtained by other methods; a refinement due to Baldock (47) is of importance in this connection. In his mass spectrometer (Fig. S), the ions issuing from the source r k c t with a crossed molecular beam in the region between the source chamber and first slit. The energy is thus defined more exactly than in the mass-spectrometer source. The use of a particles is also proposed (48). 2. Measurement i n Afterglows. It is of the greatest importance to study ion-molecule reactions not in mass spectrometers built for other purposes,

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

19

but in specially designed experiments. Moreover, if rate constants are considered to be as important as cross sections, a sound technique a t thermal energies is required. Such a technique is that due to Sayers and Kerr (22). It depends upon the development of the radio-frequency mass spectrometer, in which the ion has to travel only a short path length (-1 cm) [Boyd (26),Kerr (SS)]. By means of this instrument, ion currents proceeding from comparatively high-pressure regions of discharges may be measured without superposing electric fields for extraction. A pulsed d-c discharge is struck in a gas or

ION REPELLER

FIQ.8. Ion-molecule reaction system devised by Baldock.

mixture of gases and the time dependence of the ions issuing from an orifice is measured in the afterglow. Where there is already information available about the types or coefficients of recombination of the ions, the decay constants of certain ions will yield information about the “charge exchange” cross sections for the reactions by which they are destroyed. For example, in the oxygen afterglow the reaction O+ 0 2 --+ 0 Oz+ removes the O+ ions quicker than recombination would. Thus the decay constant for O+ ions, divided by the partial pressure of oxygen, gives us l@ol for this reaction. A nonreactive diluent gas, such as helium, is added in order to lengthen the inconveniently short decay times.

+

+

Ki 0

CHOPPER WHEEL LIGHT AND PHOTOCELL

LLEL PLATES

OE F:LECTOR PLATE

4

ION COLLECTOR

z =!!

S SPECTROMETER

FIG.9. Apparatus used by Fite, Brackmann, and Snow for the study of collisions of ions with atomic hydrogen.

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

21

&. Experinients with Crossed Beanis

Only in the experiments of Fite, Brackniann, and Snow (2) with atomic hydrogen have crossed-beam experiments been used to study inelastic ion-atom collisions. The method becomes feasible because of the “chopping)’ technique, by means of which, among other things, the high vacua ueually associated with atomic beams have been avoided. The atomic beam is chopped a t an audio frequency by means of a rotating shutter which controls the phase of a narrow bandwidth amplifier used to detect the product of the collision. The noise, especially that due to ion collisions with residual gas, is thus kept to a very low figure. The apparatus used in charge-exchange studies is shown diagrammatically in Fig. 9. I t includes both a mass spectrometer for the analysis of the primary beam (after collision) and a “condenser” for the collection of unaiialyzed charged products. The mass spectrometer is used mainly for ensuring that the primary beam is reasonably pure atomic hydrogen, which is obtained from a furnace a t 2700°K. When mass analysis is being carried out, a crossed electran beam (not shown in the diagram) serves to ionize the hydrogen. By changing the furnace temperature from 2700°K to a temperature low enough for the beam to be purely molecular, the collision cross sections of atomic and molecular hydrogen can be compared. The relative measurements must be calibrated against previously determined cross sections a t an arbitrary ion energy (1800 ev). The significance of these experiments lies in two facts: 1. Collisions with atomic hydrogen are of prime importance from the point of view of exact quaiiturn theory calculations; they can be used as ;t test of the relative reliability of the various methods of calculation. 2. Certain other collisions with atomic gases (for example, 0) are of importance in geophysics, and the possibility of studying these is opened up. It is important to perfect the chopped-beam technique because of its value in measurements with crossed charged beams, hitherto unknown except in the fields of electron, photon, and nuclear physics. By chopping each beam a t a different frequency and detecting the products with an amplifier sensitive only to the beat frequency, an especially low signal-tonoise ratio might be obtained.

F . Excitation o j Atoms by Ions Interest in this field was first shown in connection with the interpretation of auroral spectra. An examination of the spectra of oxygen and nitrogen through which fast (50 kev) protons were passed was made by Fan and Meiiiel (23). In order to obtain sufficient light intensity the gas pressure was such that multipIe collisions could occur. As a result the data were not easy t o interpret in terms of single-collision processes, and it be-

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

23

came imperative to desigu experinieiits at lower pressures. This was achieved hy Carleton and Lawrence (49), by Clarke (50),and by Sluytcrs and Kistemaker (24). Clarke has used Kunsman ion sources and long exposure photospectrograms to examine the excitation of mercury :It hundreds of electron volts energy. Carleton and Lawrence used interference filters to examine the optical emission. With this technique it is only poesible to examine certain regions of wave length, which proved adequate for the collision process relevant to the auroral studies, namely protons in nitrogen. Their energy region was 1.5-4.5 kev. But Sluyters and Kistemaker have not only worked over a wider energy range, 5-25 kev, they have used a normal incidence grating vacuum monochromator (51), by means of which the Balmer series can be examined from 1600 A and a wider range of rare gas excitation can be studied. Their apparatus is shown in diagram in Fig. 10. Both in this experiment and in the work of Carleton and Lawrence charge-collection electrodes are incorporated, so that a check can be made on the collision-chamber gas pressure, and the physics of the system can be verified. But it is noticeable that the charge-transfer cross sections obtained by Carleton and Lawrence are smaller than those of Stier and Barnett, while the ionization cross sections obtained by Sluyters and Kistemaker are smaller than those of Gilbody and Hasted, and Fedorenko and co-workers. Measurements are expressed as cross sections for the emission of one photon of a particular wave length, but only rarely does this correspond to a real excitation cross section. The excitation cross section for argon ions in collision with rare gases (101’0, to the 2 P s 0 level) has been measured by Sluyters and Kistemaker. Usually the emission will indicate the excitation either of ion or atom, 101’0 or 1010’, plus a correction for cascading from other states, which cannot at present be calculated but may often be small. These reactions are characterized by a linear pressure dependence, but where the pressure dependence is quadratic the excitation presumably is due to fast atoms, 0000’; these are formed by charge transfer. In one case Carleton and Lawrence have found a complex pressure variation, made up of a linear pox tion due to charge transfer 100’1, and a quadratic portion due to excitation of the fast neutrals 0000’. As yet there are many fewer data available for these reactions than for the corresponding charge-changing collisions, and there is considerable scope for the experimentalist.

G. General Remarks It is clear from the foregoing that our experimental technique for the study of these collision processes is a t an intermediate stage. At high energies where multiple ionization is possible we need to develop a new

24

JOHN B. HASTED

niethod of studying thr: individual process 10mn. We iieed to extwd and develop the extxeme lowenergy end of the spectrum. It will become clear from later discussion that we need to develop ion sources capable of producing beams whose population is solely in the ground state. We need to develop methods of studying the kinetic energies of the particles after collision. We need to extend the studies of collisions between two charged particles and between charged particles and dissociated (atomic) gases. We need a more detailed knowledge of the interaction of atoms with surfaces, and we need to develop vacuum and other physical techniques to eliminate the effects of impurities, so that very small cross sections can be more reliably determined. The prospects for the solution of these problems will now be discussed. 1. Coincidence Counting Techniques. The individual processes lOmn clearly can be studied only by single-collision techniques similar to those used in nuclear physics. These should not present undue difficulties now that electron multipliers have been found suitable for the detection of ions at energies of 1 kv or greater. Workers in the field of mass spectrometry have found it possible to count single ions, and several experiments in ionatom collisions have been made using electron multipliers to detect beams of the order of thousands of ions per second. With the gain of 106-107 available from 11-15 stages of beryllium-copper or silver-magnesium, such beams will yield direct currents suitable for measurement by electrometers, particularly of the vibrating reed variety. Development of coincidence counting techniques will make it possible to distinguish between collisions of type lOOn and 101n, and to solve many similar problems. It is necessary to use a triple mass spectrometer, both primary beam and secondary product being analyzed and detected, respectively, by multipliers Am+ and Bn+.Coincidences Am+Bn+ then give the relative importance of cross section 10mn. It is possible to take m = o but not n = 0, since neutral particles may be detected by electron multipliers, but not extracted or accelerated by fields. The measuiement must, therefore, be taken in conjunction with the older techniques described above, for certain cross sections can be found only by difference methods. Such a coincidence counting program is being developed at University College, London. It is significant that Fedorenko and Afrosimov (4) have already found indirect evidence that the transfer ionization processes 1002 and even 1003 may be important. The energydependence curves of the double and triple ionization cross sections for He+ in A and Kr show maxima which can be interpreted only on the adiabatic maximum rule (Section 1V.B) in terms of an internal energy difference much lower than that of a straightforward double ionization. 2. Energies of a Few Electron VoEts. The problem of determining cross

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

25

sections at the lowenergy end of the spectrum (1 ev) is, like so many problems in experimental physics, a matter of sensitivity. The problems of this energy region consist, first, of obtaining large enough ion beams, and second, of extracting the products of the reaction. Mass-analyzed ion beams of -1 ev energy were first obtained by Simons et al. (52) for experiments on elastic scattering. The technique of retardation, now further developed by Lindholm (53), could be applied more widely to inelastic scattering. The difficult problem appears to be the type of ion lens most suitable for retardation of the beam without overfocusing. Extraction of reaction products without disturbing the collision region may best be achieved in the long run by pulse methods similar to those used in the electron impact measurements of Fox et al. (6Sa). A pulse of primary beam may be passed into the collision chamber and collected without any field being applied to extract until collection is complete; the pulse of transverse field then applied will not disturb the path of the primary beam through the gas.

FIQ.11. Ion Bource for production of ions solely in the ground state.

9. Ground-State Ion Beams. The development of ion sources producing solely ground-state ions is, again, a problem of sensitivity, since such a source would not be difficult to make provided that a large reduction of intensity could be tolerated. Probably the best approach would be to bombard the gas with electrons whose energy was sharply limited by magnetic or electrostatic analysis. In such an arrangement as Fig. 11, a magnetic field of a few gauss would be sufficient to keep electrons circling at energies of a few electron volts; the upper limit of energy can be sharply defined, and the electron path would pass as near as possible to a slit from which ions could be extracted by fringing field alone.

26

JOHN B. HASTED

The alternative of allowing sufficient time for the excited states to decay in transit becomes unduly cumbersome a t high energies, but it is not to be ruled out in the low-energy region. The neutral beams used in electron loss measurements may also contain metastable atoms, the proportions of which have been found, under some circumstances, t o vary with the pressure of the gas used for production of the neutral beam from positive ions by electron capture. There is (55%) a wide region of pressure of the converter gas in which the electron loss probability of a He” beam is independent of the pressure a t which it was formed; but whether there is a pressure-invariant fraction of metastable atoms remains unknown. Normally the electron capture conversion is carried out a t high energies, but it would surely be desirable to avoid metastable production by conversion a t energies of a few hundred electron volts, where the symmetrical process will dominate. 4. Kinetic-Energy Measurements. The kinetic-energy measurement of collision products has already been attempted by Fedorenko el al. (Q), and needs to be developed in all double mass spectrometers. The experience obtained by such workers as Hagstrum (54) in the field of electron impact should be of great value here. The ion energies are even lower than those studied in the electron impact experiments, and the measurements need to be undertaken in conjunction with studies of angular distribution, in order that such theoretical data as those obtained in the classical mechanics of Fan (56) be checked. 5 . Instrumental Di5culties. The techniques of high vacuum, the mcasurement of beams of charged, neutral, and excited atoms, and the suppression of surface effects are all of importance in the development of this brtinch of physics. Key points seem t o he the particle multiplier and the insulating film or other impurity. The “electron” multiplier is fast beconiing a robust and stable instrument; from the point of view of surfaces of high and stable gain, impervious to atmospheric gases, there appears to be little t o choose between silver-magnesium and beryllium-copper dynodes. But there are as yet very few published data upon the sensitivity of multipliers to ions of different mass, charge, species, and energy. It would seem important, therefore, to calibrate each instrument as each variable is changed in the experiment. A safer alternative would be to count single particles with comparatively little discrimination. When used in this way, only the minimum ion energies necessary for registering a rount need be determined. The vacuuni technique employed in ionic collisions is limited by the necessity of using differentially pumped enclosures. The complex and accurately positioned vacuum chambers have driven nearly all experimeiitalist!: to the use of metal systems with pumps whose speeds are of the order

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

27

of hundreds of liters pcr s:econd. Suvh systems are not yot pr;ic.tic*:rhle with ultrahigh-vacuum technique, therefore, most workers decide to “live with” their impurities. This only hecomes a serious drawback when oollisions with background gases are of comparable likelihood to the particular collision being studied, even though the gas under investigation is a t a much higher pressure than the background impurities. The measurement of such small collision cross sections may require a fast pumped ultrahigh-vacuum system, and it is apparent that such a system could only be achieved with the use of liquid helium traps, or possibly getter-ion pumps [see review by L. Holland (56)l. Another drawback of the conventional metal system is the electrostatic deflections of slow particles produced by insulating films. Such films may form even in the absence of oil pumps by the vaporization and deposit of vacuum or other grease; they may also form by metal oxidation. They are, of course, largely avoided by the use of bakeable vacuum systems, metal gaskets, and nonoxidieing, heat-resistant metals such as nonmagiietic FeCoNi alloys, gold, or rhodium electroplating on brass.

IV. COLLISION CROSSSECTIONS-THE DETERMINING FACTORS We shall now attempt to review the experimental data in relation to the various theoretical approaches that have been made to inela3tic ion-atom collisions. These may be summarized as in the following sections.

A . Individual Quantum-mechanicalCalculations 1. The Born Approximation and Refinements. It is seldom possible to describe fully a n inelastic collision using the methods of classical mechanics; we must therefore commence with a general picture of the quantum theory of a collision between two atomic systems. There are two distinct ways of applying quantum theory to such a collision, namely, the impact parameter treatment and the wave treatment. Although these start from what appear to be different points, they can be shown mathematically to be equivalent, and it is becoming customary to formulate the two methods of calculation side by side. I n the impact parameter treatment first formulated by Mott (56a),the nuclei are assumed to behave like classical particles. Taking the target nucleus as the origin of coordinates, the projectile nucleus is supposed to move with a constant velocity v along a line a t a distance p (the “impact parameter”) from the z-axis. Quanta1 perturbation theory is applied to determine the chance of a transition from one electronic state to another. The effects of mutual interaction on the motion are normally ignored, but have been treated by Bates, Massey, and Stewart (56b).

JOHN B. HASTED

28

The wave treatment has been discussed in “Theory of Atmnic Collisions,” by Mott and Massey (67), under the heading of “rearrangement collisions,” a term which involves the transfer or removal of actual particles as distinct from energy. We attempt to calculate from the wave equation representing the collision the probability of a change of state in either system. To do this the wave equation is written in terms of the relative coordinates p of the centers of mass of the final systems, and the internal coordinates r, and rd of these systems:

+

where p is the reduced mass M,M2/(M1 M z ) of the final systems, H,, Hd the Hamiltonian operators of the internal motion of the final systems, and V(rc,rd, p) the interaction energy between them. Now the wave function $ of a single incoming particle must have the asymptotic form $ eik’ rle”f(e), and the outgoing wave, when this particle is scattered by the center of force, is given by

-

+

where

n is a unit vector in direction r, and dr is an element of volume. To solve this equation and obtain an angular distribution of the scattered particles we make the assumption that the particle wave is not much diffracted by the scattering center, and repIace $(r) by the unperturbed wave function exp (ikz). This approximation is valid only for fast particles and is known as the Born approximation. Without the use of the Born approximation it is extremely difficult to solve problems with more than the simplest atomic systems. Great importance, therefore, attaches to the question: “What are the energy limits over which the Born approximation may be safely used?” The question can really be answered only by making calculat,ions and comparing them with experiment. This explains the great importance of making measurements with the simplest systems, that is, atomic hydrogen. For nonexchange coIlisions the Born approximation has been used for the study of many reactions, such as: H+ (or H 1s)

+ H (152)

--t

H+ (or H Is)

+ H (28, 2 p , 38, 3 p , 3 4 or H+ continuum),

calculated by Bates and Grif€ing (68).Another typical calculation is that made by Moiseiwitsch and Stewart (69) for the collision between atomic

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

29

hydrogen and helium. They fidd,that double transitions are likely to occur, both atoms emerging in an excited state (000'0'). Calculations were carried out for the processes: H (Is)

+ He (Is")

---t

H (2.9, 2 p , 39, 3 p , 3 4 or Hf continuum)

+ He (lsZp,lP).

We would expect the Born approximation to be valid only in the highenergy region where e2/hu2 5 1, and in general this is so; but Bates and Griffing found that it can sometimes account for the general features of the cross section (that is, that it is emall in the "adiabatic" region) a t far lower energies. For exchange-type collisions Bates (59a) has shown that in the Born approximation formulation an unacceptable indeterminacy arises because no account is taken of the fact that the initial and final eigenfunctions are not orthogonal. The complete electronic wave function

x$(r,t)

=

a

i - 9(r,t) at

may be satisfied by adding any function of R to the dcfiiiition of thc Hamiltonian X, =

1 -zvz

+ VA(rl)+ VB(r2).

In this impact parameter formulation R is the position vector of nucleus B relative to nucleus A. The quantities rl, rz and r are the position vectors of the transferred electron relative to A, B, and the midpoint of AB. With the above definition and suitable approximations, a formula for the cross section is derived which may be shown to be equivalent to the original formula derived by Brinkmann and Kramers in 1930 (60) using the wave treatment. But by the addition to the Hamiltonian of a term describing the nuclear-nuclear interaction, an equation is obtained which is equivalent to the wave formula used originally by Bates and Dalgarno (61), Jackson and Schiff (629,and subsequently by Bransden, Dalgarno, and King (6$),and by Gerasimenko and Rosentweig (64) for the reaction 1012. Now it is clear from the physical point of view that the mutual repulsion of the nuclei cannot directly influence the probability of an electronic transition. Nevertheless, the modified Born approximations show reasonable agreement with experimeiit. For example, calculations for the charge transfer of He+ in He are compared with experimeiital data in Fig. 12. The calculations of Bransden et al. for H+ in He show reasonable agreement above 30 kev. This agreement is understandable because Bates has reformulated the problem in such a way that an addition of a function ill R to the Hnniiltonian does not affect the cross section; when this is done the new

30

JOHN B. HASTED

10-1s

10 20

Particle energy (kev) 40 80 100120 160 200

5

2 c

E 10-16 a

c

0

fw 5 0 c

.-c

M

MOlSElWlTSC

T

WSTED AND STEDEFORD HIGH ENERGY MEASURED LOW ENERGY MEASURED HIGH ENERGY COMPUTED LOW ENERGY COMPUTED

2 2 u)

10-17

b’

0

5

0

0

2

-

Particle velocity (crn/sec)

(a) FIG.12a. Comparison of measurements of charge exchange cross section ~ O U Ofor I He+ in He, due to Stier and Barnett, and other workers, with the Born approximation calculations of Moiseiwitsch, Jackson, and Schiff.

transition matrix element is found to be closely equal to that previously used with the nuclear correction. The cross section for capture from state i of A to state j of B is given by:

with

where

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

4 8

PARTICLE ENERGY (rev) 60 80 400 140

$6 25 40 I

S

-

e

I

1

I

I I I

I

I

I

!

,

31

200 I

:---t----c---l

s

v)

0

a

-

2

BATES AND DALGARNO, THE0

a STEDEFORD

V

O

HIGH ENERGY COMPUTED

!

3

2

4

PARTICLE VELOCITY (cm/sec a

5

10-9

6

7

(b)

FIG.12b. Comparison of measurements of charge exchange cross section IOUOIfor H+ in H?with Born approximation calculations.

in which

and

+

Eigenfunctioiis are denoted by and eigenenergies by a;,p+ It is to be hoped that this new formulation will eliminate the ambiguity in choice of interaction between the wave functions, prior to transition or

32

JOHN B. HASTED

post), :tiid :tlso t h ! us(! of ‘‘CtTwtitw’’ pertiirI1:i tioils t80rtqil:ic:c t,eriiis i i i R, th:tt, haw Iweii features of a iiumher of previous calciilatioiis. Ail attempt to (!;hul;tte a “sewlid Born approximation” by the niethod

of back substitution has proved to be cuiiibersonieand liniited to high eiiergies (65). A good indication of the limits of the Born approximation is seen in the calculation of Bates and Dalgarno (66) for the charge exchange of protons in atomic hydrogen. These may be compared with the measurements of Fite, Brackmann, and Snow, and with the perturbed stationary-state calculations of Dalgarno and Yadav (67) for the same process (Fig. 13).

FIG.13. Charge transfer cross sections for protons in atomic hydrogen, due to Fite, Brackmann, and Snow. Curve C-experimental points, calibrated at 1800 ev. Curve %Born approximation. Curve %perturbed stationary states method. Curves A and B refer to other processes.

It is seen that the lowenergy limit of agreement of the Born approximation is 10 kev, that is, projectile velocity 9 X loF8cm/sec. This is the identical limit to that obtained for the electron ionization and excitation of atomic hydrogen, so that the authors conclude that the lower limit occurs when the projectile velocity is -4 X orbital electron velocity. However Bates points out that the limit is likely to be lower for smaller cross sections, since the Born approximation neglects the effect of the inverse reaction. While it may be said that in the Born region the comparison of most hydrogen and helium data is merely a matter of time, the extension into a

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

33

lower energy region is a different matter. There is still uncertainty about the best methods of making calculations; in Belfast and elsewhere a continual process of questioning assumptions and improving methods of computation is in process. 2. Calculations at Lower Energies. The first formulation of a method designed for energies below the Born approximation was made by Mott and Massey (57‘);it was formulated using wave equations and is known as the “perturbed stationary states method” (pSS). The kinetic energy of relative motion is regarded as a perturbation and the wave equations are calculated a t a certain internuclear separation, the nuclei being regarded as at rest. The probability of transition from state i to state f due to a perturbation energy U is proportional to (U,,)2, where Uif is the average of over the wave functions of the initial and final states. Now U = -h2v2/p, and the chance of a transition per collision will be proportional to

IIJFi(R)tli(r,R)vR2{ U,*(r,fW’,*(R) 1 d7 - &nI2 where the wave functions are represented by $, R is the internuclear separation, and r refers to the electronic coordinates relative to the center of mass. The rate of change of the wave functions with nuclear separation results in a pronounced “near-adiabatic region’’ in which the probability of transition rises exponentially with increasing energy (see Section 1V.B). Early computations using this method include the excitation of the 21P state of helium by protons, the capture of electrons from helium atoms by protons, and the excitation of lithium by a-particles (68). Now Jackson (69) and Moiseiwitsch (YO) treated the problem along impact parameter lines, but it is shown in a full discussion by Bates, Massey, and Stewart (56b), that this is essentially similar to the pSS method. Recently Bates (7‘1) has shown that the pSS tends, especially at higher energies, to another approximation, known as the “perturbed rotating atom,” (pRA). In the Born approximation the perturbed eigenfunctions of the target system are presumed to remain in fixed orientation throughout the encounter; but in the pRA they are presumed to follow the rotation of the internuclear line. Thus the pRA is a poor approximation for head-on collisions and very close encounters. Moreover the pRA method ignores the coupling between states differing only in magnetic quantum number, and therefore, except in S-E transitions, does not tend to the Born approximation a t Pigh energies. The pRA is valid only at fairly low energies, and here it differs from the pSS. Thk is rather an unsatisfactory state of affairs, especially for the pSS method. I n the last year or two, therefore, few calculations have been made using the method. A certain measure of agreement with experiment has recently been ohtairied using this method by Haywood (7‘2) for the charge transfer of protons in helium (with certain

34

JOHN B. HASTED

remarks on other rare gases). The cross section is remarkably sensitive to z, the effective nuclear charge of the helium atom, and z may be chosen to produce a maximum cross section a t the correct energy; the cross section found by the modified Born approximation shows no maximum. Dalgamo and McDowell (73) have made pSS calculations of the charge transfer of negative hydrogen ions with atomic hydrogen which show good agreement with recent measurements by Fite and co-workers (74). To remedy this state of affairs Bates has recently introduced a new “distortion approximation” (75), extending the Born approximation to lower energies but taking into account effects ignored in it; he uses impact parameter formulation, and the target atom is taken to he the origin of coordinates. The electronic wave function is expanded in terms of coefficients a,(t), which represent the probabilities that the target atom, after a time t, is excited into the nth state. Thus with ao(- w ) = 1, a,(- m ) = 0, s # 0 the cross section louoo~ is

Expanding the electronic wave function in terms of unperturbed eigenfunctions &(r) and eigenenergies en, and proceeding in the customary manner (56a), it may be shown that

a

i -a&) az

‘c

=2,

a8(z>Vm exp ( -iaanz)

with

v n 8 = J&*@)v(r,R)M)dr, aan

=

(€a

- en)/v;

and the zero of time chosen so that z = vt; R is the relative position vector of the nuclei; and r the position vector of the active electron. Assuming that a&) is unity and that a&), s # 0 can be neglected, the impact parameter version of the Born approximation is obtained (76). But in the distortion approximation we take instead a&) = exp

[ - f 1’V ,

*

dz]

and retain the a,(z) terms; this allows for distortion both in state 0 and state n. It is found that

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

35

with

Unfortunately the main integrations must a t the present time be carried out by numerical methods, so that much labor is involved in repeating the work done using the pSS method. Nevertheless, the 1s - 2s excitation of H by H+ and He2+ has been calculated, and work on the proton excitation of ~ ~ He+ (2s), and Na ( 3 p ) is in progress. Delay is introduced He ( 1 . ~ 2ls3p), by the fact that the collisions most suitable for quantum theory calculations are seldom those most suitable for experiment. Firsov (77) has applied quantum theory to the study of resonance charge transfer, obtaining cross sections which can be compared with experimental data with considerable success, When the relative velocity v is much smaller than the electron orbital velocity v., the probability of capture from s-state to s-state can reach the value unity at internuclear distance T~ which exceeds the size of the atom. For r < T~ the probability P(r2)oscillates between zero and unity, and is taken as equal to 56.For T > T I , P(P)is taken as zero, and the distance rl for which the charge-transfer cross section is 5hn-rlZis calculated from the formula

for an ionization potential Ei; A is a normalizing factor, taken as equal to unity for the approximation calculation. For rare gas ions this formula gives reasonably good agreement with the data of Flaks and Soloviev (9),Gilbody and Hasted (6),and Dillon et al. (36); but the differences between experimentalists make it necessary that further measurements be made. In particular, it has long been unexplained why the cross sections for helium and neon should lie so close together, while those for neon, argon, et.c., are more evenly spaced out. This arises because of the importance of the ionization potential in the above formula and may be seen most clearly in Fig. 11 of Flaks and Soloviev's publication, where the cross sections at v = 2 X lo7 cm/sec are plotted against ionization potential; experimental and computed data for helium may easily be added to the published curve. However, it becomes of prohibitive difficulty to use quantum theory calculations for atomic systems of normal complexity, or for molecules; theref ore it is important to develop semiempirical analysis of the general

36

JOHN B. HASTED

of cross sec t h i s , usiiig such gciientlizations as can be made by clria.iit8un~ theory and classical mechanics. Typical of these are the adiabatic l ) t h i \.ior

thcory, the st:hstical theory of multiple ionization, and a number of other argulnents which will bc discussed in later sectioiis.

B . Adiabatic Theory The adiabatic theory was first formulated by Massey and is extensively discussed in Chapter VIII of “Electronic and Ionic Impact Phenomena” by Massey and Burhop (68). There is usually a region of impact energy where the relative motion of the atoms is so slow that the electron motion of transition can adjust itself to small changes of internuclear distance; this adiabatic adjustment makes the transition an unlikely event, as it would be if there were no energy of relative motion. But as the energy rises out of this “adiabatic region” and the time of transition becomes comparable to the time of collision, the likelihood of transition increases rapidly, The time of collision is taken as a/v, where a is a parameter of atomic dimensions and v the relative velocity; the time of transition as h/AE, where AE is the internal energy defect of the reaction. Thus v << aAE/h is the adiabatic region of energy, and v = aAE/h marks its termination. In the adiabatic region we would expect the cross section to be proportional to exp (-KaAE/v) where K is a constant. The exponential variation of cross section with energy was at first thought to be contradicted by experiment. This was due to the difficulty of choosing reactions unobscured by the existence of excited states in the impacting ion and product, and to the difficulties in measuring small cross sections. But it is now known (78, 79) that simple charge-transfer cross sections vary exponentially with impact energy; when this is taken in electron volts, E, we have a

exp ( - K’IAElmts/EJ5),

a being arbitrarily taken as constant to fit the experimental data (see Fig. cm2) they assume 14a,b). As the cross sections become very small larger values than this relationship would suggest, this may be a real effect, or it may result from other cross sections being falsely included in the measurements. The perturbed stationary states method of calculation shows no departure from the exponential relationship in this region. The “adiabatic maximum rule” (80) is a simple but crude method of formulating the general behavior of cross section-energy curves, which nevertheless can be easily compared with experimental measurements, even if inaccurate. At high energies the cross section invariably falls off with increasing energy, as the chance of interaction becomes smaller. There-

INELASTIC COLLISIONS BETWEEE; ATOMIC SYSTEMS

37

fore most cross sections pass through a maximum a t a certain energy. Let us take this maximum as marking the end of the adiabatic region, that is, where 11 = aAE/h, and let us further assume that a does not vary much in 30-

2.0

O x 10''

--c 0.5 0.4 0.31.0 0.9 0.8 0.7 0.6

0.2

-

i

30

zc

\"

IC

FIG.14a. Charge transfer crom section for H+ in Ne e x p d in the form u = A exp (-K/*). FIQ.14b. Charge transfer rate of rise constant K as a function of adiabaticparameter. Data of Hasted and Stedeford, and molecular data of Lindholm added in the form of dotted lines.

different reactions, taking it as constant. The energy of maximum cross section Ern=% should then be proportional to m%AE. For a large number of charge-transfer and other reactions this is found to be the case within wide limits of accuracy (6, 79). The rule is thus mainly important for predicting the rough general behavior of a cross section. It was found using logarithmic plots that a power law dependence, in which the parameter a had a certain influence, was to be preferred; but as there is no theoretical basis for this, it, is not pursued. Figure 15 shows a fairly comprehensive logarithmic plot of Emxx against AEm", for various different types of reaction. Included for the first time ixi this plot me nmiy receiit calculations and ineasurenieiit.s,inoludirig those of Fogel discussed below. Since them is a zcro ciiergy clcfeot for CCrttLiii

m

Y

I

m <

n

E*

3T

0

w-

0-

N

0

-

a-

u-

N-

- u1 I

0

N

U

0

0

0

a

G

8

M

0

In

0

0

0

8

0

yr)

8

In

0 0

-

0 0 0

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

mm

e

0 M (u

-

39

40

JOHN B. HASTED

+

+

symmetrical reactions such as He+ He 6 He He+ this cross section is found to rise continuously as the energy falls. Thus the plot of Fig. 15 will approach the origin. The maximum rule provides us with a starting point in the study of collisions of the above types. Many complicating factors are found. There is, for example, the possibility of capture into excited states, although this may not contribute a dominating amount to the cross section.

KEYTO FIQ. 15 Charge exchange 1001

0 Positive ion ionization 1011 Positive ion excitation 1010' Excitation by neutral atom 0000' o Reverse neutralization 0011 A 1012 and 1013 reactions 1. HI+ A Total electron production; that is, positive ion ionization plus multiple ionization (4) Total electron production; that is, positive ion ionization plus multiple 2. H*+ H2 ionization (4) 3. H+ A Total electron production; that is, positive ion ionization pIus multiple ionization (4) 4. H+ HZ Total electroil production; that is, positive ion ionization plus multiple ionization (4) 5. H+ H 1s - 2s excitation, by Born approximation calculation (68). Comparison of this point with point 25, taken from a recent, distortion approximation calculation (76),shows that there is a considerable difference in position of maximum. Points 6-24 are likely, therefore, to show a similnr error. 6. Hf H 1s - 2p excitation (68) 7. Hf H l a - 3s excitation (68) 8. H+ H 1s 3 p excitation (68) 9. Hf H 1s - 3d excitation (68) 10. H+ H Ionization (68) 11. H H l a - 2s excitation (58) 12. H H 1s - 2p excitation (58) 13. H H 1s - 3s excitation (68) 14. H H 1s - 3p excitation (58) 15. H H 1s - 3d excitation (68) Ionization (68) 16. H H l s 2 p 1P excitation; Born approximation calculation (69) 17. H+ He 18. He+ He ls2p 'P excitation; Born approximation crtlculation (59) 19. H He Double transition; H e excited to ls2p 'P, H ionized (69) 20. H He Double transition; He excited to l ~ 2 pIP, but I3 excited to 2p (69) 21. H He Double transition; He excited to ls2p ' P , but H excited to 3p (59) Double transition; He excited to ls2p ' P , but H excited to 2s (69) 22. H He Double transition; He excited to ls2p 1P, but H excited to 3s (69) 23. H He llouble tranuition; He exrited to lu2p I P , but 1-i excited to 3d (li!)) 24. H He Excitation 1s - 2s; distortion approximatioil calculation (76) 25. €I+ H

A

-

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

26. TIC++ I1 27. HP+ 112 28. He’ Nz 29. He+ Oy 30. H+ A 31. A+ A 32. Ne+ Ne 33. H,+ HZ 34. Hz+ A 35. Nz+ A 36. H+ Kr 37. H+ CO 38. Nn+ A 39. A+ Nz 40. H+ Xe 41. H+ NH3 42. O+ Kr 43. Hz+ Xe 44. A+ HI 45. C+ Hz 46. N+ H, 47. N+ HZ 48. H+ NHx 49. H+ Ns 50. C+ Xe 51. C+ A 52. Hz+ Xe 53. Br+ Xe 54. HS+ Kr 55. N+ Kr 56. H+ A 57. H+ Nz 58. H+ Ho 59. H+ CO 60. Hz+ Xe 61. He+ Ne 62. Nz+ A 63. He+ Kr 64. Hz+ A 05. H+ CO 66. Hz+ Ne 07. H+ H

41

Excitation 1s - 2s; distortion approximatioil crtlculation (76) Charge transfer ($9); transition aa nearly vertiral aa pomihle Charge transfer; Na+ Zzp+presumably (29) Charge transfer; OZ+‘11. presumably (29) Charge transfer; subsidiary “maximum” A+ W (29) Charge transfer; A+ *Po,J = 135 >5 Charge transfer; Net *Po,J = 155 F! 56 Charge transfer; forming possibly HJ+and H (6) Charge transfer (81, 33) Charge transfer (81) Charge transfer (8f) Charge transfer (6) Charge transfer (6‘) Charge transfer (6) Charge transfer (81) Charge transfer NH3 in (ire)4 state (6) Charge transfer (6) Charge transfer (81) Charge transfer (6) Charge transfer; c+4Pstate (6) Charge transfer, N+ 3P (6) Charge transfer, N+ *D (6) Charge transfer (6) Charge transfer Nz’, AZIIu(6) Charge transfer (6) Charge transfer, C+ ‘P (6) Charge transfer (81) Charge transfer (6) Charge transfer (81) Charge transfer (6) Charge transfer (81) Charge transfer (6) Charge transfer (16, 8f) Charge transfer, Cot A z L (6) Charge transfer (81) Charge transfer (81) Charge transfer, N*+AzJTu (6) Charge transfer (81) Charge transfer (81) Charge transfer, CO+ B22-(6) Charge transfer (81) Charge transfer, H raised to 4f level; Born approximation calculation

*

(66)

Charge transfer, H raised to 2s level; Born approximation calculation (66) Charge transfer (16, 81) H+ He Charge transfer, &+AzII. (6) A+ Ns C+ NH, Charge transfer, C+‘P (6) )~ H+ NHI Charge transfer NHI ( s a ~(6) Ionization; Born approximation calculation (82) He+ He

68. H+

69. 70. 71. 72. 73.

H

JOHN B. HASTED

42 74. H+ 1It! 75. He+ A 76. He+ A 77. C+ 78. H2+ 79. c+ 80. N+ 81. Nz+ 82. A+ 83. H+ 84. H 85. H 86. H 87. H 88. H 89. H 90. c 91. C 92. C 93. 0 94. 0 95. 0 96. 0 97. H+ 98. H+ 99. H+ 100. H+ 101. H+ 102. Hf 103. H+ 104. H+ 105. Of 106. C+ 107. Of 108. O+ 109. Cl+ 110. F+ 111. F+ 112. Ff 113. Cl+ 114. O+ 115. F+

Hz He A

Hz A Nz

N2

Hz He 0 2

Ne A N2 Xe Kr A Xe Kr A 0 2

He Nc A A

Kr Kr Hz N2

Xc Xc Kr A Xe A Kr Xe Nz H2 H2

Ionization; Horn approximation calculation (89) Charge transfer (81) Charge transfer (81) (maximum energy lies between these two limits) Charge transfer (83) Charge transfer (16, 81) Charge transfer (81) Charge transfer, N+ I S (6) Charge transfer, Nz+B22. (6) Charge transfer, Nz+Bzzu(6) Charge transfer, Nz+BZzu(6) 0011 reaction (79,16) 0011 reaction (79,16) O O i l reaction (79, 16) 0011 reaction (79, 16) 0011 reaction (79, 16) 0011 reaction (79, 16) ooii reaction (79) mil reaction (79) ooii reaction (79) ooii reaction (79) ooii reaction (79) ooii reaction (79) ooii reaction (79) m i 2 reaction (79) i o i 2 reaction (79) ioi3 reaction (79) 10iZ reaction (79) i o i 3 reaction (79) i o i 2 reaction (79) 1012 reaction, with dissociation (79) 1012 reaction, without dissociation (79) 10i2 reaction (79) 10i2 reaction (79) 10i2 reaction (79) 10i2 reaction (79) 10i2 reaction (79) 1012 reaction (ground states only) (79) lor2 reaction (ground states only) (79) 1012 reaction (ground states only) (79) i o i 2 reaction (79) 1 O r Z reaction (79) 1012 reaction (ground states only) (79)

There is also the fact that the ion beams as used in all experiments hitherto carried out contain proportions of ions in excited states of long enough lifetime to pass into the collision region; t,he proportions are significant when these states lie within a few electron volts of the ground state, Until pure ground-state beams are studied, a composite cross section

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

43

will be measured, which destroys the value of measurements; this is particularly troublesome with atomic ions of groups 5-8 of the periodic table. On the other hand, quite a number of subsidiary maxima have been found corresponding t o energy defects arising from reactions with excited-state ions (80, 81). The production of pure ground-state beams would be of great value. No doubt selection rules will be found determining the relative probabilities of formation of different excited statep. An investigation of the energy levels of molecules using this method of estimating energy defects has been made by Lindholni ( 5 ) . Owiiig to the form of potential-energy diagrams and the Franck-Condon principle, it is possible for molecular reactions to take place with a nonsingular energy defect. Thus for the reaction H+

+ H*-+ €I + H2+

we should expect an energy defect 16.4-17.4 ev. As far as is possible with the limits of accuracy of interpretation, this is found to be so. Lindholni uses a large number of bombarding atomic ions in the study of each molecule, and working a t low energies it is often possible to deduce the molecular energy by resonance with the atomic ionization energy. There are, however, some cases whose interpretation is not clear. It should be pointed out that in the study of vertical ionization energies for molecular orbital theory, the order of effectiveness as bombarding tools is decidedly: 1. Photon 2. Electron 3. Ion This is because the relations

are generally valid, that is, photon impact curves tend to behave like the differentials of electron impact curves. A situation which remains unclear arises out of the report by some workers that the symmetrical charge-exchange cross sections for A+ in A, Nef in Ne, have maxima a t energies of 150 ev and 120 ev respectively. Normally the cross section for a symmetrical exchange reaction falls off monotonically with increasing energy. I t is claimed (6) that these maxima arise from the splitting of the ground states of these ions 2P,J = 145 and =P,J = 55. The laboratory ion beam will contain ions of spin 46 and 145, and the reactions may be either symmetrical or with energy defects of 0.018 ev (A) and 0.097 ev (Ne). The experimental cross section will be made

JOHN B. HASTED

44

up of the superposition of different reactions, and the maxima conform to the rule. However, this makes the cross sections at near-thermal energies too small to account for experimentally determined ionic mobilities on the theory of Dalgarno (84). The observation of low-energy maxima in cross sectionenergy curves is not usually to be regarded with great confidence. One such maximum to be expected on theoretical grounds, however, is the excitation transfer cross section for helium 3S atoms, O*OOO* (86). Calculation by the method of perturbed stationary states arrives a t a cross section which is very small a t thermal energies, but rises to a maximum "hm2 at a temperature of ~ 1 8 0 K. 0 ~Experimental verification of this is to be found in the work of Stebbings (86), whose determination of the total collision cross section is in fair agreement with theory. This maximum arises from the peculiar nature of the potential-energy curves, which are repulsive at large separations but attractive between 2 and 4 A (Fig. 16) ;

FIG.16. Energy of interaction of a normal helium atom and metastable helium atom in *z1 state. Curves EA, EE are based on firsborder perturbation calculations; curve M is Morse interactions, derived from experimental results (Milliken 1932); curves 1, 2 represent E A - lO/R'j, EA 20/R6 respectively; curve 3 gives possible energy variation linking M and 1.

-

it may be a feature common to collisions between normal and metastable inert gas atoms. It is not impossible that the potential-energy curves in symmetrical - X,) contain such energy molecular ion with molscule collisions (XZ+ barriers, resulting ill anomalous charge-transfer cross sections a t nearthermal energies.

45

INELASTIC COLIJSIONS BETWEEN ATOMIC S>-STEMS

Sonic interesting measrlreinents which relate t,o the adiabatic riinxiiiiritn rule have been made by Fogel and co-workers (79).These are for the process 0011, which is the reverse process to mutual neutralization, discussed in the next section, and for the processes 1012 and 1013 which he calls “double electron capture.” It is true that in Fogel’s experiment only the final charge state of the projectile is studied, and not the charge state of the target atom; nevertheless it is difficult to see what other processes could be involved; the cross sections almost certainly represent those for the single processes listed above. Fogel finds that the maximum cross sections conform to the rule, but the values of the atomic parameter a are apparently much smaller than for other reactions: a = 3 for 0011 and a = 1.5 for 1012 and possibly 1013. This is shown in Fig. 15,where Fogel’s data are plotted on the same graph as the other measurements, and the best fit is found to be for lower values of a. Now there is no reason why the interaction distance should be smaller, but there is a reason why the energy defect should be unexpectedly different. In reactions such as 1001 the potentialenergy curves are separated by approximately A E over a large part of the 8 A interaction range; but in reactions where the final products have opposite charges and are of greater energy than the initial atoms, the curves will be as in Fig. 17a because of the long-range Coulomb interaction. For a large part of the collision range AE will be smaller than AE,. For a strict application of the adiabatic theory we would have t o integrate over the whole range, to obtain a mean EE. This may be done graphically for certain hydrogen potentialenergy curves in the literature, as follows; 1. Fig. 15,reaction 16,H H -P H+ H e, potentialenergy curve by Massey and Burhop (87). AE, taken from 0.5 to 8 A, is found to be 13.93 ev, comparatively little displacement from AE ,= 13.53 ev 2. Fig. 15, reaction 10, H+ H + H+ H+ e, potentialenergy curve by Massey and Burhop (88) z\E, over the same range, is 18.66 ev decidedly larger than AE = 13.53 ev 3. For reaction H H --f H+ H-, potentialenergy curve by Herzberg (89). over the same range, is 16.35 ev while AE- is 13 ev. This reduction is sufficient to bring Fogel’s 0011 reactions into line with 1001 reactions; the effect for 1012 reactions will be much larger still. It is of interest to find that for the positive ion ionization reactions AE is so much larger than A E , ; the maximum rule plot will require modification accordingly. This interpretation lends support to the curiously large value of a found

+

+ +

+ +

+

ID

+

a,

+

46

JOHN B. HASTED

FIG.17. (a) Potential energy curve8 for 0011 reaction. (b) Pseudocrossing potential energy curves. (c)Potential energy curves for 2011 reaction. (d) PoBntiaI energy curves for Auger type of ion ionization. (e) Energy of activation.

originally, as well as its apparent weak dependence on “atomic radius.” Most of the interaction appears to take place at fairly large separations; the close approach of curves at small distances has comparatively little influence. Where AE- is very small, All may differ from it over a large range, so that departures from the maximum rule are observed; it is precisely these reactions together with those studied by Fogel that have the largest scatter about the appropriate parameter line. It may even be that the poor correspondence of molecular charge-transfer reactions to adiabatic theory is due to a similar effect; as we shall see later, it is not always possible to distinguish between a charge-exchange process and an ion-molecule reaction. An outstanding anomaly in an atomic collision that may be clarified by a similar approach is the 2011 partial charge transfer for Xe++ in Ne measured by Flaks and Soloviev (9). It is often the case that potential-

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

47

FIQ. 18. Allison's summary of the data for the chargeehanging crosa sections for H-, Ho, and H+ in collision with H1.

energy curve crossing dominates such reactions, but here the reaction is slightly endothermic (AE = -0.27 ev), and should behave normally (mMAE= 2.2, V,, = 900 ev). Instead the cross section is found to be small and rises monotonically to a maximum obviously much higher than 60 kev. The Coulomb repulsion of the final products elevates the mean energy defect, by an amount presumably of the same order as that for tho proton-proton repulsion considered above (-5 ev). Thus mMAE can be taken as 45, raising V,,, to 300 kev. It is, of course, true that the adiabatic theory gives no idea of the relative values of cross sections at their maximum point. Because of the falloff in the high-energy range, cross sections are likely to be largest when the adiabatic region is smallest; the most likely processes are those closest to resonance. But when a number of different processes are possible, all of them having adiabatic parameters of the same order, large differences of maximum cross section occur, as can be seen from Fig. 18, taken from Allison's

48

JOHN B. HASTED

review article (1).We know that processes 1001,0010,~000,lOmn are likely ones, while 1010, 0011, 1012 are much less likely. It is important to see how far the data for excitation reactions that are at present available are in accordance with what would be expected from adiabatic theory. In no case has an exponential rise of cross section with energy been observed, although Carleton and Lawrence's proton excitation of the first negative bands and Meinel bands of N*+ behave roughly in the expected manner, as does their cross section for the H emission due to the reaction 100'1 for protons in nitrogen. Their proton excitation cross sections for the N I lines around 8216 A are, however, anomalous in that they remain invariant in the lo-'' om2range down to an energy as low as 1.5 kev. This type of anomaly is not unknown in ionization and even charge-transfer reactions, and may be due to curve crossing, possibly with a two-electron excitation (see Section 1V.C). The cross sections for the excitation of argon ions by rare gases 101'0 found by Sluyters and Kistemaker are well in the adiabatic range, and are of the order of 1O-IB cm2;but they certainly show an anomalous energy behavior, which cannot be explained on the basis of cascade emission. The emission data of Clarke, while not absolute, exhibit similar features.

C . Pseudocrossing Potential-Energy Curves It is well known that when the potential-energy curves of the initial and final states of an inelastic enounter are such that they might be expected to cross over one another, the conditions are very different from the collisions for which no crossing point exists. Although the curves cannot actually cross, a region exists where the energy separation between them is small, and there is finite probability of transition from one t o the other (Fig. 17b). This probability is given by a well-known quantum mechanical formula due to Landau (90) and Zener (91). If inelastic collisions are found with a small number of crossing points at calculable internuclear separations, such collisions can be treated fairly exactly if the wave functions are known, and some calculations can be made even if they are not. 1. Coulmb Repulsion. It was pointed out by Bates and collaborators in a series of papers that such conditions will often exist in transfer-type collisions, such as mutual neutralization 1100, etc. (92).For theee systems one state has a strong Coulomb interaction and the other a weak polarizat8ion interaction. It is difficult to find simple processes which may l x investigated experimentally and theoretically, but interesting possibilities arise in the case of partial charge transfer 2011 and 3021. Here the multiple transfer collisions 2002 may be of negligible importance because they are so far in the adiabatic region, and the partial process cross sections may be measured by charge-collection techniques. When the process 201 1 is exo-

INELASTIC COLLISIONS BETWEEN ATOYIC SYSTEMS

49

thermic the coiiditioiis of Fig. 17c exist, and if the energy separation of the pseudocrossing curves is AU(R,) a t an internuclear separation R,, then the adiabatic region will he defined hy the condition

This may be a much smaller energy region than the usual one (h/AE << a/v), in fact it may be of negligible importance. Certain exothermic reactions (AE 6 ev) may thus be distinguished from similar endothermic reactions since they exhibit abnormally large cross sections in the usual adiabatic range. This was found to be the case by Hasted and Smith (9S),working in the energy range 504000 ev. Only a small number of reactions uncomplicated by excited states can be found in this region. Boyd and Moiseiwitsch (94) have made a test of the Landau-Zener formula on the basis of these experiments, and they have also made calculations for certain triply charged ions in helium, for which exact wave functions have been determined. The cross section is given by the formula

-

hR,2PI(v), where I(q) =

A"

exp (-qz)[l - exp (-qz)]z4

- dx

and

where u is 1 for process 2011 and 2 for process 3021, p is the probability that the particles approach along the specific potential-energy curve, all energies being in electron volts. Now the Coulomb interaction is such that an approximation may be made for R,: R, = 27.2u/AE in units of a. provided A E is not too large. In addition, the integral has been tabulated (96) and is found to have a maximum value 0.113 when q = 0.424. We can thus get a very good idea of what the maximum value of a cross section due to the Landau-Zener crossing should be. Figure 19 gives Moiseiwitsch's tabulation of the integral I ( q ) as a function of q-2 = BE where

Using experimentally determined maxima it is possible to calculate cross sections at other energies; the agreement is good, but the energy range so far used is small and the number of cross sections available is limited to

JOHN B. HASTED

50

three. However, when more accurate wave fuiictioiw are available for the ions, as is the case for Be3+, Li3+, Aat, M@+, it is possible to calculate All(&) and the overlap integral h’,, between initial and final states. It will he of great interest to compare these calculations with experimental data. Moreover, log AU(R,) is a smooth function of 1/R, as was found for mutual and R g U ( R , ) / S U is approximately neutralization by Bates and Boyd (M), constant [Magee (8771, though perhaps better expressed as a linear function of R,. Thus within certain limits it is possible to estimate the crossover

Lop 7)-*

FIG.19. Moiseiwitsch’s calculations of the integral I(v).

contribution to other cross sections where the wave functions are not known. There is plenty of scope for work in this field, for instance stripping reactions such as 02+ H+ 09+ H which are of importance in thermonualear reactors. Calculations of a similar nature have been made by Bates and Moiseiwitach (98),Da€garno (99), and Bates and Lewis (100). Mea@rernen€sof the cross sections 2002 and 2011 have been made from 6 to 60 kev for ’mfe gases by Flab and Bdoviev (9). By mass analysis of the projectile beam after-collision the two crom sections may be separated, Most of the 2011 collisions are exotbemic and crossovers are to be expected, even thongh the target ion may be formed in an excited state. In general,

+

+

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

51

these crossovers, and not simple adiabatic theory, are the dominating influence in this energy range. In one case, A++ in Ne, the effect is extremely marked. In other cases, Ne++ in He, Xe++ in Ne, the crossovers have a negligible influence. A more detailed analysis of these data, with measurements over a much wider energy range, is needed. Hasted, Scott, and Chong have made some unpublished measurements for krypton multiply charged ions in neon, from 50 t o 8000 ev, which conform to crossover theory. The reactions include 2011, 3021, and reactions with four and five charged ions, which are probably 4031 and 5041. It seems that crossovers contribute markedly to the charge degradation of these highly ionized atoms. 2. Ionization and Detachment. I t is also possible that potential cnergy curve crossing plays a part in the ionization of complex atomic systems by neutrals and positive ions, 0001, 1011. I t will be seen from Fig. 15 that in several cases of positive ion ionization the maximum energy falls in the region we would expect on the adiabatic maximum rule. Yet the rate of rise of rross sections with energy is far from being exponential in the adiabatic region (78). Moreover there are several features of the data which are not a t all understood. It has long been considered (101) that positive ions are peculiarly effective in ionizing atoms of approximately the same inass. The extreme low-energy end of the region of positive ion ionization has been studied by several workers. At one time it was thought that an onset potential for ionization would exist, just as in the electron ionization of atoms. While there are strong theoretical grounds for believing that this is not so, a t least for simple atomic systems, there might be energy regions in which the rate of rise was much faster than in other regions. These would appear in earlier experiments to be onset energies, since the important improvement in each new experiment is in sensitivity. In the most recent measurements, due to Moe (18), regions of comparatively sudden rate of rise are found, and the first differential of cross section-energy curve, obtained electronically, has a complex structure, as in Fig. 20. It appears that there are different mechanisms of ionization, each one becoming effective in a different energy range. The simplest explanation of this is one which was first put forward by Weizel (102). It is not unreasonable to suppose that ionization can occur by an Auger-type process in which two electrons are excited in the. atom, the energy being subsequently distributed in such a way th&, one electron is no longer bound, while the other falls into the ground state. Now the potentialenergy curve of the doubly excited state might cross both the ion-atom and ion-ion curves, a s in Fig. 17D. Fui ther, it is possible that several doubly excited states might contribute, each one over a different energy range, giving cross-section curves of the type observed by Moe. The comparatively linear rates of rise observed a t rather

52

JOHN B. HASTED

higher energies (34,103) may be due to the superposition of several different types of reaction curve. In the ionization of atoms by neutral atoms 0001, the potential-energy curves may show a crossover without any double process being involved. There has for several years been discussion about the interpretation of meteor ionization measurements [Opik ( I O d ) , Evans and Hall (105)l. Greenhow and Hawkins (106)have calculated from radar echoes obtained from ionization trails that a single meteor atom has a 0.2 chance of producing an electron during its passage through the atmosphere, after an initial

1 0 ton energy in ev

FIG.20. First derivative of the positive ion ioniration cross section IOUU observed by Moe.

velocity of 60 km/sec. Massey and Sida (107') have calculated the momentum loss cross sections for calcium in neon, and with the aid of these they show that the radar data indicate an ionization cross section of the order of 0.3 to 1 2 ~ in ~ the ~ 2 range 20-1000 ev. This is unexpectedly though not impossibly large. Great interest, therefore, is aroused by the measurements of Bydin and Bukhteev (15) who find large ionization cross sections foi beams of neutral potassium in oxygen and nitrogen. In oxygen the ionization of potassium reaches a flat maximum of mW1"om2at 300 ev; in nitrogen a maximum of 2.5 X lo-" cm2 is approached at 2000 ev, while in hydrogen and argon the cross sections rise slowly but monotonically. The curve crossing appears to be occurring only for certain molecular collisions. Sluyters and Kistemaker, in the course of their current excitation measurements, have studied the ionization of rare gases and hydrogen by

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

63

argon atoms froin 5 to 24 kev; their findings caiiiiot be regarded as anomalous except in the case of the argon-argon collision, for which the cross section persists as high as 1.7 X 10-l6 cm2a t 5 kev. They find that ionization by neutrals is always a less likely process than ionization by pwitive ions. The detachment process 1000 may also take place by curve crossing. This arises because the energy defects are not large and the interaction is different between the initial states from what it is in the final states. There may even be associative detachment at thermal energies [Bates and Massey (92)],the molecular oxygen negative ion, for example, being formed from the impact of 0- on 0. Experimental investigations have been carried out by Hasted (108), Bydin and Dukelskii ( l o g ) , Stier and Barnett (29). The detachment cross section exhibits a fairly flat maximum, always at 2 kev or rather less. It has been observed (110) that for all heavy incident ion detachment cross sections a t present known, the maximum value is proportiona1 to reduced mass p ; that is, to the maximum kinetic energy of motion available for excitation > i p v 2 ; v is the velocity corresponding to 2 kev incident ion energy. A constant proportion of this “available kinetic energy” is thus consumed in the inelastic process. For collisions with the light incident ions H-, H2-, this observation does not hold. Fogel et al. (111, 112) have observed that the double detachment process 1010, while unlikely compared to 1000, is still unexpectedly large; it does not conform to adiabatic theory and shows low-energy maxima, which may possibly be due to some two-electron process. Bates and Massey (92) have described a mechanism of detachment in which the two nuclei on being brought together have no bound state for an excess electron. If the affinity becomes zero a t an internuclear separation R p o , the excess electron is free to leave the system at closer separations, and the cross section is of the order (gR,2)fao2where g is the appropriate ratio of statistical weights. It should be noted, however, that the united limits which may be reached adiabatically may all be doubly excited and therefore bounded even though the normal state is unbounded. Such detachment through lack of a f h i t y wiIl not occur in many cases where it might at first be expected. The onIy experimental measurements known which might with certainty be attributed to this cause are those with atomic hydrogen as a target. 3. Multiple Crossovers. The importance of multiple crossovers has been considered by Magee (11.9)for the particular case of mutual neutralization (negative ion recombination) 110’0’. The Coulomb interaction between the initial states results in the potential energy curves having the form of Fig. 17f. At the distant crossing point a; the interaction with the final states is negligible, and at c the interaction is too strong for the likelihood of transfer

JOHN B. HASTED

54

to be great; but at crossing point b a transitioii is probable. The approximate values of R, are given by:

&(A+, B-) - E’(A’, B’) = ez/R,. The reasonably small number of crossing points for the reactions O+ with 0-, O+ with 0,- were considered separately, although the electron affinities were not so accurately known at that time; but in addition a number of generalizations about high state densities were made. For a classical hyperbolic orbit intersecting the 2 N crossing points of neutral states, the probability of transition at each crossing being p t , the probability of emergence as a neutral P N ,is given by:

For large N and small p we have the limit pN

%

1

- e-ZNn;

and as 2 N p , becomes large, PN approaches unity. For N moderately large and p t > >$, PN approaches 2p41

- pt) + 2pdl

- pd3,

which a8 p , 3 1 becomes proportional to p , . Practically all the neutralization occurs at the first transition point so that many crossing points have a negligible neutralization probability. To determine the cross section for neutralization, it is necessary to average over all the poesible classical hyperbolic orbits; expressed in terms of the distance of closest approach ro, the cross section “(V)

= 2*

[+

-,P(r0) 1

e2 v’rO] ro * dro.

While in the publication quoted above Magee and Hourt only considered rate constants a t thermal energies, the method is clearly applicable over a wider energy range. Recently they have studied the energy dependence of charge transfer between protons and hydrocarbon molecules, whose ionization potentials are such that a nearly exact resonance can occur with a region of high density of rotational and vibrational states. For collisions without crossover, such as can occur only at glancing incidence they consider the cross section to be inversely proportional to impact velocity; for collisions with many crossovers, occurring “head-on,” the cross section

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

55

appears to be velocity independelit. Experimentd data obtained by :L rather rudimentary method in the course of elastic scattering experiments of Simons et al. (114) show a fair conformity to this relation: a(v) =

A V

+ B.

The constant term seems to be of greater importance than the inverse velocity term. It is to be hoped that there will be a more detailed experimental study of these reactions. Where there is no Coulomb attraction it is difficult to see exactly how the positions of the crossovers can be calculated. However, the single crossover cross sections calculated by Boyd and Moiseiwitsch show an energy dependence not very different from l / v . Possibly the crossover contribution is dominant. For glancing collisions the cross section for transition to any single state will presumably rise exponentially with energy, as outlined in the previous section, and will fall less sharply, l / v or l/v2. For a state density linear with energy, we have been unable by algebraic summation to obtain a total cross section that does not rise with increasing energy. The strong resonance of the C ~ H molecule B may be responsible for its unusual behavior in charge transfer. Melton (48) has found by mass-spectrometer source studies that at near-thermal energies the charge transfer with the C2H2+ion (AE= f2.2 ev) goes strongly, while between C2H6+ and CzH4(AE = +1.1 ev) and between CzHzf and CzH4 (AE = -0.9 ev) the reaction is very much weaker. It is apparent that collisions with multistate systems may be one of the most difficult as well as the most important fields for study.

D. Ion-Molecule Reactions We shall now consider the factors determining the probability of reactions such as A+ BC -+ AB+ C, an exchange of an atom rather than an electron. Where BC is the diatomic gas from which the ion A+ is derived, the reaction A+ A, -+A Az+ becomes indistinguishable from charge exchange 1001, and could only be separated from it by the use of isotopes. Bates (115) first drew the attention of physicists to the behavior of the reaction, in terms which are familiar to theoretical chemists. The rate constant k, equal to the product of cross section and intcgrsted relative vclocity, will be appreciable only at energies greater than an activation energy AEact,which may be a fraction of 1 ev or even vanishingly small. It can be expressed in the standard Arrhenius form k = A exp ( - A E a o J , but a t energies much above AEaot the cross section must fall off because of the growing difficulty of redistributing the momentum in the required manner.

+ +

+

+

56

JOHN B. HASTED

In the light of subsequent experimental investigations let us ask ourselves the following questions: 1. What activation energies can be expected and how can they be determined from the experimental data? 2. At energies below the activation energy, will there be an appreciable rate constant, and will it rise with increasing energy in the manner typical of the Arrhenius equation, and of charge transfer? 3. Will there be an appreciable energy range over which the rate constant is invariable? 4. What is the form of the falloff of rate constant at higher energies? Stevenson and Schissler (21) found that a number of reactions showed cross sections proportional to E-%, that is to say, rate constants invariable with energy, but showing “zero” activation energies, i.e., much smaller than the experimental energy range. In other reactions the rate constant is observed to fall off with increasing energy. But in view of the difficulties, discussed earlier, of assessing the relative energy of impact, it would be unwise to make too far-reaching deductions. The E-% dependence of cross section requires a strictly r-4 potential function, which is only expected at rather large distances. One would therefore expect the E-” dependence at low energies, and deviations at high energies. When the cross section is comparable with gas-kinetic the considerations leading to the E-% dependence are inapplicable. Field, Franklin, and Lampe (20) find rate constants which fall off slowly at ion field chamber strengths greater than 10 volts, at which energy there is a maximum, with a sharp falloff at lower energies. They do not consider this maximum to be reliable, and it is quite possibly instrumental. But in view of the fact that other workers, particularly Saporoschenko (45) observe maxima which are at different energies for different reactions, it should perhaps be taken more seriously than they suggest. The energies of the maxima cannot at the present stage be related to assumed activation energies, nor are there sufficient data to analyze the rate of rise at still lower energies. Since there is uncertainty in the energy scales, me shall discuss the different sets of data separately. Field, Franklin, and Lampe have discussed the cross section in terms of the intermediate complex ABC+, the formation of which they suggest is the rate governing process. Eyring, Hirschfelder, and Taylor (116) have treated the cross section for the formation of this complex along classical lines, in a manner similar to Langevin (117). They assume that the cross section is determined by the distance at which the centrifugal force separating ion and molecule is counterbalanced by the attractive force due to polarization; also it is assumed that all of the initial energy is converted into rotational energy of the complex, not

57

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

into translational energy. Field, Franklin, and Lampe obtain the relation u =

em&

d6kT

+ (2eVdo/3)

for a molecule of polarizability a.The agreement with experiment is shown in Fig. 21a, and in order to obtain a much better agreement, it is assumed that the experimental cross section U, is modified by an efficiency factor j , which varies inversely with the velocity of the complex a t formation; the agreement is then as shown in Fig. 21b. Extrapolation permits the evalua-

0

(a)

FIG.21a. Comparison of cross sections for the process CzHP measured by Field, Franklin, and Lampe, with theory.

+ CtH,

3

[C,H,' i

(b) FIG. 21b. Comparison of some cross sections with theory after modification. K E Y : 0 0 0 experimental; -authors' equation (31); 0 = 1.1 X 106; T = 423' K.

58

JOHN B. HASTED

tion of rate constants at thermal energies, which may be compared with those calculated according to the formula

due to Eyring, Hirschfelder, and Taylor; K is the transmission coefficient. This formula gives a rate constant of 1.3 X lo+’ cm3/mole sec for the hydrocarbon reactions studied ( K = l), in comparison with the values between 1.2 and 3.5 obtained by extrapolation. It is clear that the general treatment gives a good over-all picture of what is happening, but although the polarization theory gives the same cross section for different ions reacting with the same molecule, very considerable differences are observed in practice. Thus the ethylene molecule a t V = 6 volts presents a cross section of cm2 to C2+ ions, 32 X 10-I’’ cm2 to C2H+ ions. In the measure98 X ments of Stevenson and Schissler, a t V = 0.1 volts, this moIecuIe presents a cross section of 359 X lo-’* cm2 to C2Ha+ions, 24 X 10-l8 cm2 to C2H2+ or C2H,+ ions, forming C3H3+.Moreover, certain very small reaction rates are observed by Stevenson and Schissler for molecules of large polarizibility, e.g.

A+

+ HCI

AH+

+ C1,

kl

< .04 X

10W cni3/niole sec.

These facts point to the existence of other perhaps equally important factors that determine the ion-molecule reaction rate. The transmission coefficient K , assumed to be unity, may actually be much emaller, so that the complex dissociates partly into its original components. In many cases there is no actual potential-energy hole, so that the “complex” has such a short lifetime as to be unreal. Following Stearn and Eyring (118), the potential energies are represented as surfaces, drawn with contours in Fig. 22. As the ion A+ approaches BC, the twodimensional potential-energy curve of BC is raised; likewise the potential-energy curve of A+B is raised when C approaches (Fig. 17e). The activation energy represents the energy location of a region of crossover, which would be represented in Fig. 22 by the termination of lines drawn a t (T - 0)/2 to the appropriate contours, where 0 is the angle between the axes. The calculation of cross section at any giveLienergy thus reduces to the problem of summing the effects of many transitions, the likelihood of each one being deterniined by the Landau-Zener formula. When one of the potential-energy troughs is particularly shallow there may be very few crossovers or even none, so that the rate constant and cross section will be correspondingly low. This explains the small rate constants for the rcmtions forming rare gas hydride ions AH+. The dissoch

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

59

r, Angstrijms -+ FIG.

N#e)

22. Potential energy surface calculated in kcal/mole for the reaction N ~ 0 ( ~ 2 )

+ O(1D) by Stearn and Eyring.

tion energies of these are likely to be small; that of HHe+ has been shown to be 1-1.5 ev by Coulson and Duncanson (129). The cross section-energy curve will normally have the shape of the integral 1 ( q ) for a single crossover. But there is a marked dependence of the crossover energy defect AU(R,) on its separation R,; thus R, exerts a determining influence on qr1(q)and hence u. Only the crossovers in a small range of separation, therefore, will contribute much t o the cross section a t any given energy. In the extreme case there will be an energy range where there is always a crossover such that I(?) is at a maximum, and in this range the cross section will be governed mainly by the variation of RZ2.But outside this range fewer and fewer crossovers contribute, until we reach a region where the cross section is made up of contributions mainly from one crossover. In this region the cross section-energy curves will look like those obtained by Boyd and Moiseiwitsch, reproduced here as Fig. 23; there may be no maximum as in curves 8 and 9, or alternatively the maximum may be as real as in curve 7. It is worth noticing that the experimental cross sections of Field, Franklin, and Lampe fall off precieely in the manner of curves 8 and 9 when plotted on a logarithmic scale. An alternative way of looking at the problem is by considering the motion of the particle along the potential-energy surfaces, statistical allowance being made for vibrational energy and the relative orientations of ion and molecule as they approach. In the case where a potential basin

60

JOHN B. HASTED

exists, the particle may spend an appreciable time losing energy inside it, before it eventually leaves by one pass or the other; the relative chances will be determined by the cross section or depth of the two passes, the ratio of which will influence K. For example, by symmetry, K = $5 for the HZ - H+ collision, but might be appreciably less for H+ Dz + HD+ H. Even when there is no basin, a classical model might give a good indication of the dependence of K on the relevant parameters. The density of vibrational states in the troughs must be included in this type of reasoning; t,his amounts to considering the likelihood of resonance between vibrational levels of initial and final states, just as is done for charge transfer.

+

+

[

log Impact energy of incident ion (ev)]

FIQ.23. Boyd and Moiseiwitsch’s calculations of some partial charge transfer processes as a function of energy. The ion abundance curves of Saporoschenko (45) Iead to rather sharper energy dependences than those outlined above. As has been mentioned in Section III.D.1, the passage of low-energy electrons through a gas may be described in terms of the “reduced electric field” X / p . The electron energy distribution, on the assumption that all collisions are elastic and there is no interchange of energy between the charged particles, can be shown to be proportional to X / p (Druyvestein distribution). Experimental data on electron swarms are available from which a comparison of mean energies with X / p values may be made. Positive ion energy distributions

61

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

0

I

1

3

4

Mean electron energy (ev)

FIG.24. Ion-molecule reaction cross sections as a function of “energy.”

have been investigated by Boyd ( I W , and in a uniformly positive column of argon at 4 microns pressure the mean ion temperature is found to be 0.3 of the mean electron temperature.‘ The ratios of Saporoschenko’s ion abundances, N4+/N2+and N3+/N2+, at each value of X / p , will be proportional t o cross sections for the processes of formation of theso ions. Neglecting the effect of the magnetic field and 4

Probe studies of mass-spectrometer sources have been made by Bohm et ul. ( 1 . 2 h ) .

JOHN B. HASTED

62

converting X / p to mean electron energy after the data of Dear and Emeleus (121)and Townsend (122), the relative values of the rate constants may easily be calculated as a function of mean electron energy. The relative rate constants we have so calculated from the data of Fig. 7 are shown in Fig. 24. The maximum is clearly not instrumental in the case of Na+. This ion appears to be formed in a reaction whose activation energy is of the order of 1 ev (0.3 ev if Boyd's ion energy measurements apply to the present situation) ; it is not impossible that the Arrhenius equation may be used to describe the situation. The region of falloff of rate constants has

1

60 80 100

Je Y FIQ 25. Comparison of cross sections for the inverse processes H+ due to Fite, Brackmann, and Snow.

200

+ HZ+ H + H2+,

not been reached. On the other hand the Ns+ ion is formed in a reaction whose cross section shows a sharp falloff with increasing velocity of impact ( u v - 4 ) , since there is no additional product (N) to remove the excess momentum, which must theref ore be distributed in the vibrational and rotational degrees of freedom. As has been discussed earlier it is difficult t o accept this mechanism of formation of N4+. These result8 are qualitatively different from the data described earlier, and are more in hie with the original idea of Bates. But they depend entirely upon the interpretation of energy in terms of X / p . An interesting situation arises in the charge-exchange reaction €L+ I1 4 H, H+, which has been examined in the experiments of Fitc, Brack-

+

+

63

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

nia1111,and Snow (9).The croa section is shown in Fig. 25, togc~thrwith tho cross sections for the reverse reaction, and while the maxima fall at, &out, the same energy, it is seen that a t much lower energies the forward cross section is far larger than the reverse. I t is just possible that this may be due to the reaction H+H H +B+ 4-HH, which is indistinguishable experimentally, except for kinetic energies. In the reverse process H+ HP4 H H2+, the ion-molecule reaction is much less likely because the pass depth is likely to be smaller for H2+ than H2 (trough depths a t infinite separation are 2.648 and 4.48 ev, respectively). The two types of reaction could be distinguished in a mass spectrometer by the use of D2+:

+

+

+

DI+ D2+

+H +H

+

DI H+, + HD D+. 4

+

But if the interpretation is correct it would seem that this low atomic weight ion-molecule reaction can persist up to much higher energies than previously had been thought. In the light of the experimental data discussed above, we conclude that it would be a little premature to claim that the questions set out at the beginning of this section can yet be answered satisfactorily.

E. The Classical Treatment of Collisions Certain important features emerge from the application of classical momentum and energy equations t o inelastic collisions. Fan (123) has presented classical arguments t o show that the mass of the electron plays a not insignificant part in charge-exchange collisions. There will always be a finite amount of energy transfer in these collisions because of the electron mass m,.It is shown from the energy and momentum equations that this energy is related to a quantitjr q, defined as

AE E

me + Mi -1

where E is the energy of the incident particle. For the important case q < 0 (endothermic reactions in a certain energy range), the minimum energy transfer is found to be

The final energy of the target particle, E2, must be a t least as large this energy. But it can also be shown that

$8

JOHN B. HASTED

64

where 6, is the angle of scattering of the incident particle in center of mass coordinates. Thus

em 2 B l i r n i where

The primary particle may not be forward scattered within a shadow cone determined by this equation.

THEORETICAL

LINE

053

046

s:8"

02

04 I0

100

1000

I

Emax

FIG.26. Comprehensive plot of ( m A E ) s against Em,,$.

Fan argues that if the variation of cross section with B1 is a smooth curve, it will be cut off at eli, when q < 0; when q 6 0 there is no limitation. Hence the cross section will fall as q passes through zero, that is, there will be a maximum at E = AE . Ml/m,. In practice it is found that this gives values of Em,, of the right order, but this may be because the criterion is

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

65

remarkably similar to the adiabatic rule, from which it differs only by a factor of AEN. In Fig. 26 a comprehensive plot of (rnAE)% against Em? is shown. The points represent proton collisions, for which the condition reduces to the adiabatic maximum rule; it may be significant that the heavier ion collisions 0 fall farther off the theoretical curve.

FIO.27. Vector representation of inelastic collision used by Fedorenko. For exothermic reactions no sudden reduction of cross section is predicted, but the position of the maximum does not seem to be much changed in practice. This may be evidence that the classical limitation does not occur a t an angle at which the contribution to total cross section is very large; hence this method of reasoning may not be ae important as appears a t first. It is a classical description of an effect already explained by Brjnkmann and KIamers. Fedorenko and co-workers (124) in studies of the angular distribution of scattered idns in the multiple stripping and ionization process, have reached some interesting conclusions by means of classical arguments. Figure 27 represents the velocity diagram for a collision of particles of equal mass M I = M2 at energies large enough for the inelastic energy loss AE to be a negligible proportion of the incident ion energy E, the target particle

JOHN B. HASTED

being considered as initially a t rest. The scattering angles and final velocities of the projectile ion and target atom are represented by 01, 02, V I , v 2 ; the initial projectile velocity by v. The difference between the diameter of the two circles represents the inelastic loss; it is greatly exaggerated in the figure, For a single value of e2 there are two values of v2 (v'z and v"2), each corresponding to a definite primary velocity v f l and v"1 and scattering angle 0'1 and B"l. The case vf2, vfl, Of1, in which target particles are scattered a t small energies, while projectile ions are scattered through small angles and do not lose much energy, is referred to as "soft'' scattering. The case v " ~ , vffll in which target particles are scattered with larger energies, while projectile ions are scattered through larger angles, without much loss of energy, is referred to as "hard" scattering. In the limiting case where e2 = 90" the tangential line ensures that there is only one value of v2 hence of O1 and vl. In the simpler types of inelastic collision only the soft scattering is observed, the hard scattering is negligible. But Fedorenko finds evidence which may be interpreted as being due to hard scattering; it occurs in the multiple ionization of both projectile and target. First, the energies of the scattered target ions, as determined mass spectrometrically, split into two groups when O2 is less than 86". Second, the abundance of multiply charged primary ions as a function of scattering angle determined by mechanical movement of the mass spectrometer, shows not only a maximum a t 0" but a hard scattering maximum, actually of smaller energy, of the order of 6". As the charge on the scattered primary ion decreases, the hard scattering maximum disappears. Third, the target ions are scattered into one group at -90" when the charge is small, but as it increases the scattering angle decreases, falling to 81" for As+. From this data Fedorenko is able to estimate the effective inelastic energy loss AE,a; this is found to exceed AE, and the excess energy is, believed to contribute to the kinetic energy of the electrons produced. This is confirmed by the work of Blauth (125) who finds a surprisingly wide energy distribution of electrons: Blauth was able to show that some had energies of hundreds of electron volts. This supports the view that the ionization process involves more electrons than are necessary, the excitation energies being pooled, so that a smaller number are able to escape, but with excess energy; the evidence for this view (102) has been discussed in Section 1V.C. How far does this classical picture approximate reality? Everhart (14) finds essentjally no important contribution to the total cross section froin ions scattered in the range 0-1". But there is no other measurement of the angular distribution of target ions besides that of Fedorenko; it certainly seems that the deviations from 90" scattering are large enough to result in

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

67

splittiiig of sv:il t wcc1 c0111poiieiits,if the classical iuoc1c.I holtls. 1’ctlort:itko’s c!riorg.y splitltiig uf target ions would also seem at. first sight to lie decisive, hiit it must he rcwiemhered that the measuremcii1,s ;LIC rin;hle to distill:I

guish bctwccn t:Lrget ions formed by ionization and those formed in exchange-type collisions; these two groups might well have different energies. The intensive work a t present being carried out will no doubt throw a light on the physical significance of the “hard” and “soft” components of the recoil. A serious at tempt to apply classical mechanics to inelastic collisions has been made by Gryzinski (126); since it has so far been applied only to stopping power and electron-atom collision calculations, we shall not discuss it in detail, but only outline the ideas which may be of interest here. Collisions are regarded entirely as binary encounters between target orbital electrons and projectile nuclei, whose Coulomb fields interact strongly only when their velocities are comparable. The interaction is treated in the manner worked out for gravitational fields by Chandrasekhar, using an impact parameter formulation. A complex expression is obtained for the stopping power of two helium electrons of orbital energy 39.5 ev interacting with protons. In a later publication the author succeeds in eliminating the adjustable parameters used in the earlier formulation, and obtains a stopping cross section in good agreement with the experimental data of Reynolds et al. (127).It is assumed that all the exchange of energy is used in raising the helium atom to the first excited state *S, and if other interactions were taken into account the agreement would probably not be so good. Nevertheless, it is likely that the density and momentum distribution of electrons in the medium (particularly in the lower energy region, the valence electrons), plays a large and possibly dominant role in stopping. In the calculation of inelastic cross sections, however, the theory is open to more serious criticisms, on the physical grounds that the energy spectrum in the target system is not always a continuum (see Section 1V.G). In the case of electron-electron collisions, inelastic cross sections are found to depend on a function which varies with impact energy in a manner similar to electron-atom ionization cross sections. It is a temptation to fit this undoubtedly valuable function to experimental data without due care. Thus the type of assumptions which the author has used to fit the electron ionization cross sections of Hz and He may easily be seen to give but poor agreement with the recent data of Fite and Brackmann on the ionization of atomic hydrogen (128). A semiclassical treatment of charge transfer has been given by Takayanagi (Id9),which succeeds in giving a reasonable agreement with experiment for the case H+ in He. However, it requiies the use of wave functions and falls more into the class of an individual calculation (Section IV.A.2).

68

JOHN B. HASTED

It is more than likely, however, that there are collisions and energy regions in which classical considerations will prove to be of value. F. The Behavior of Cross Sections at Extremely High Energies As the energy of collision increases into the million electron volt region, the inelastic atomic cross sections become so small that they are comparable with nuclear cross sections. They are nevertheless of importance in the study of the passage of high-energy particles through matter, and it would be convenient to be able to write a simple expression for the falloff of cross sections with energy. Such an expression might be obtained from the Born approximation. It is a t once apparent (150) that there is a wide difference between collisions of the ionization or excitation type and collisions of the exchange type. The former fall off comparatively slowly, being proportional to: v - ~In av for optically allowed excitation 1010'; V-2 for optically forbidden excitation not involving a change of mu1tiplicity ; v - ~In a'v for ionization 1011; but exchange collisions fall off very fast, proportional to v-12 for electron capture 1001. However, the energy range a t which this is valid is extremely high because the K-shell electron capture reaches a maximum a t a much higher energy than the L-shell. This has led to the fitting of empirical laws to experimental data, as follows: Rutherford (131) found that the capture of a-particles was represented by a v-6.6 law. Stier and Barnett (89) find that the charge exchange of H+ in Hz is best represented by v-lol but that of H+ in A by only v-~.'. The loss cross sections H in HP follow a much less powerful law v+, while H in Nz, A, even fit v-1. Until the electron capture into different shells is separated, and experimental data exist beyond the 200-kev range of Barnett and Stier, and the 1-Mev range of Barnett and Reynolds (132),the empirical laws have a certain value, mainly for use in calculations with ionized gases. Thus for the capture Stier and Barnett use u = A_'", and over a smaller enelgy range Hasted has used umax- u = Cumax In (8 - Dmax). The qualitative difference between the falloff of ionization and that of exchange-type collision is, however, borne out by the experimental data a t present available. A considerable amount of data has been collected on the charge-changing collisions of such ions as N+, N3+, N6+, N6+, O+, B+, Ne+ and fission fragment ions in the Mev range; this is summarized in an article by Stier (133). The usual experimental arrangement consists of acceleration of the ion (which includes mass analysis), collision with a solid film or gas a t sufficiently high pressure for the projectile charge to be equilibriated, and

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

69

subsequent analysis of the charge equilibrium. Thus while the measurements are of greater value to nuclear physicists than much of the data described above, it is difficult to separate single cross sections from them, because so many components are present. There is an additional complication of great interest: Lassen (154) has found that electron loss cross sections are abnormally high at large gas pressures and projectile velocities, where the time interval between successive collisions is much shorter than the time required to rebind the disturbed electron by radiating electromagnetic energy. Theoretical treatment of this has been given by Bohr and Lindhard (136) and by Bell (1%). In view of the fact that Melton (48) has proposed the use of 5-Mev a-particles to dissociate hydrocarbons in mass-spectrometer source studies of ion-molecule reactions, it is important to consider the falloff of more complex reactions, such as molecular dissocjation. Melton has found that the fragment ion proportions are reduced in comparison with parent ion abundance5 when polonium a-particles are substituted for electrons. Thus = loo), the abundance of C*H+is reduced from 20.7 with acetylene (C2H1+ to 7.3, and with methane (CHI+ = loo), the abundance of CHI+ is reduced from 83.3 to 42. Since the more dissociated fragment ions are reduced to a level below the sensitivity of the instrument, the complexities of source studies of ion-molecule reactions are much reduced. But it is difficult to say why the particle should produce more parent ions than the electron of equivalent velocity, in a reaction which is almost certainly of ionization and not of exchange character.

G. Statistical Theory of Multiple Ionization A method of understanding the multiple ionization scattering of heavy ions by atoms lOmn at high energies has recently been developed by Russek and Thomas (187). It gives a reasonable fit to the data of Everhart and co-workers ( l d ) , for the probabilities of various states of ionization as functions of the energy and scattering angle. The authors calculate statistical distribution among the orbital electrons of the energy transferred to the internal degrees of freedom of the target atom. This energy may then be obtained as a function of the collision parameters. It is assumed: 1. That the energy transferred is statistically distributed only among the eight outer electrons; the energy scale is divided into cells of width e; all cells are taken to have the same statistical weight. 2. That the ionization energy for each electron is identical and independent of how many other electrons have escaped. These aseumptions are justified only because the energy region is not adiabatic; hence the time of collision is short enough for the energy lsvels to be broadened by the uncertainty principle to a half-width Ah' = Av/a; they are thus smeared

70

JOHN B. HASTED

into a continuum above the first excited state. The broadening of the K and L energy levels is negligible by comparison, therefore these electrons can adjust adiabatically to the changing potential during the collision. Although these assumptions, particularly the second, are no more than moderately plausible, they are justified by the fact that they are necessary in order to achieve agreement with the data; the authors consider the serious reduction of agreement when they are altered in various ways. The ionization energy is taken as a convenient, multiple of el namely 4e; the energy transferred, En is considered to be mc if me 6 ET < (m l)e, where m is an integer. It follows from the four assumptions that the probability P,(m)that a neutral atom will become n times ionized if an energy me is transferred to its internal motion, is given by the number of ways in which the energy me can be divided among the eight outer electrons so that n and only n electrons have 4e or more, divided by the total number of ways that the energy me can be divided among the eight electrons. We have:

+

where

(:)

js

the binomial coefficient, n -1

2 3

Q(n-i)(m) =

Qn(m - r ) ,

r=O

+

Qz(m)= m 1 for 0 6 m 6 3 =7-mfor 4 6 m 6 6 =.O for 7 6 m

In collisions between a singly charged ion and a neutral atom there is assumed to be an even chance that, upon separation, this single electron deficiency will be associated with either atom; hence the probability that the projectile will be singly charged even before electron evaporation is t:tkcii into consideration, is one half. The modified ionization probability knis given by

P,,On) = ) iP,'(?/l)3. ,!;P( ,,-,) ( r n ) . l'lots of' Il',,(nr)for 0 6 n

6

8 as functioiis of 1 $ ' ~ / enre showii

iit

Fig. 28.

71

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

It is possible to make a comparison of these calculations with experimental data in the following way: The evaporation theory so far presented predicts unambiguously the relative heights of the peaks of all the ionization probability curves and the heights of the intersections of each pair of curves, but does not predict the energies at which they occur, only the “horizontal” order in which they are found. It can be seen from Table I1 that the theoretical values taken from Fig. 28 agree well with experimental values for A+ on A at 25, 50, and 100 kev, listed in the correct order of decreasing scattering deflection el. This is so because E is a monotonic increasing function of Q. I

1

E+

FIQ.28. Values of

P, calculated by Russek and Thomas.

This impressive agreement between experiment and theoretical analysis is a justification for the assumptions mentioned above. For example, variations in the statistical weights of the cells in the bound and free electron range are found to disturb the agreement, and an assumption of a “staggered” series of ionization energies, as is the case for electron or photon collisions, makes the agreement very much worse. It is also a justification for the rather doubtful assumption that the single electron deficiency will be associated with either atom upon separation; light may be thrown upon this by the use of the same calculations for interpreting data on multiply charged ion-atom scattering, 20nm eto. It retilaitis to obtain a measure of the energy which is nssunied to I J ~ made up from a large number of two-body el~ctroii-elec.troiic~ollisioiis, iiicreasing thr orhitnl eiitlrgy of the t :irgt*t t k t r o i i :

Ii&,

1.”) = f J T ( E ) *

V(lS,

To)

JOHN B. HASTED:

72

TABLE11. EXPERIMENTAL AND THEORETICAL VALUES OF IONIZATION CURVEINTERSECTIONS Intersection or peak

P, x P, Pa x Po

P, P, x Pl

t3;, x Po

A x P2

P, x PI Pa P, x P2 P, x Po P, x P, PS x P2 p6 x Pl P, P, x Pa

Experimental Values 25 kev

50 kev

0.42 0.09 0.42 0.24 0.07 0.34 0.15 0.41 0.23

0.07 0.51 0.22 0.05 0.33 0.13 0.38 0.24

0.03 0.33 0.13 0.02 0.38

100 kev

Theoretical values

0.36 0.13 0.36 0.24

0.42 0.09 0.46 0.25 0.02 0.39 0.11 0.42 0.25

0.32 0.14 0.02 0.37 0.24 0.06 0.32 0.14 0.04 0.35 0.24

0.35 0.13 0.01 0.37 0.24 0.07 0.32 0.14 0.03 0.35 0.24

0.00

P6 x P2

P6 x P4

9 6 x P8 P7-x 4

PS

A x P,

where e T is the average energy transferred in one such colIision, v is the total number of collisions, and TO the distance of closest approach of the two nuclei. It is assumed in this that the electrons adjust adiabatically to the nuclear motions (v, >> v), and that energy is transferred only when the impact parameter of the electron-electron collision is less than some fixed length L which is small compared to the atomic radius. Now

where vo and eo are the average orbital veIocity and energy. Moreover, Y

=

ifL2g(E) M(ro),

where M(r0) = Iu*(z,Y> a&, ’

YMXdY,

and 6 2 being the “squashed densities” of the two atoms, obtained by numerical integration of p, the number density which follows from a Hartree self-consistent calculation for neutral argon; g(E)is a multiplica-

u1

73

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

tive function representing the number of collisions between the same pair of electrons during the collision. It is reasonable to assume that g(E) has the form 1 BE-%, so that, substituting in the above equations and putting eo?rL2= A we have:

+

with A and B undetermined parameters. To obtain the best fit to all the differential cross-section data, however, some empirical adjustment must also be made to the function M(rO),which is different when only the M-shell or a sum of M L shells are considered. A function is taken which is composed of M plus a fraction of the L contribution. With the correct adjustment of these parameters, admittedly fairly numerous, the relative differential cross sections for scattering into eight different states of charge a t three representative energies can be fitted to the experimental data, as is shown in Fig. 29. For multiple charges the agreement is fairly good, and refinements can be made only by the consideration of other atomic collisions, such as those of alkali ions. The relative importance of M and L electrons in contributing to M ( r o ) ,and elimination of the parameter B are both of importance. The scattering into a state of zero charge (charge transfer) is difficult to fit using these assumptions, since there is interference arising from the proximity of the potential-energy curves, that is. the perturbation of the electron densities during the collision. The success of this statistical method of analyzing complex collisions raises the question of whether such methods could be used for nuclear collisions, for dissociation collision of such complex molecules as hydrocarbons, or even for collisions between galaxies. A most interesting energy variation of differential charge transfer a t comparatively large scattering angles of primary ion beam has been reported by Ziemba and Everhart (138). These are the first multiple resonance phenomena to be found in this field, and are shown in Fig. 30, in the form of eiiergy variation of probability Po that a helium ion scattered to el = 5' in laboratory coordinates will emerge as a neutral. Since the preparation of this manuscript some of the most significant advances in the field have in fact been made in the study of multiple ionization. Firsov is reported to have formulated a statistical theory capable of

+

explaining the energy dependence of total ionization cross section

1

10untn,

m,n

and also the value of the mean energy transferred in the collision, which may be deduced from Afrosimov and Fedorenko's measurements of target kinetic energy and angular distribution. From these measurements we may also deduce the over-all transferred energy distributions for collisions a t a

74

JOHN B. HASTED

0.5t

t

t

t

f

t

0.5

02

0 1 L 4 e L

00

%:/ ;- j 03 02

.

%

1 ;-

.

01

00 100

200

300

e

100

200 300

e

100 200

30-

e

FIG.29. Comparison of angular distributions of multiply ionized atoms, measured by Everhart et nl., with those calculated by Russek and Thomas.

75

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

particular impact energy resulting in the removal of a certain number of electrons from the target atom. These distributions are of forms very similar to the Russek-Thomas calculations of P,. Moreover, some of Everhart’s \ a€ 0.7 -

INCIDENT ENERGY, KILOVOLTS

g 48:: R

p

0

2!

m

;:

u

250kv 43kv 175kk96kv 58kv 40kv

I

1

I

I

I

2

?

27kv

I

I

06 0.5 aN n z a 0.4 no Lo

g- 0.3 I0

a

,“ 0 2 I

01 -

I

I

I

I

8Okv 26kv 135kv 72kv 48kv 1 1 1

I

33kv

I

22kv

p2

7

Ir

I

I

I

01 02 03 04 05 06 07 COLLISION TIME, UNITS OF (KILOVOLTS)‘~’~

0

FIG.30. Multiple resonance phenomena.

recent measurements of projectile inelastic scattering through 5 O , as a function of impact energy, and hence of energy transferred, have the same general form. It would seem that the statistical distribution of energy among outer shell electrons is fundamental to the fast inelastic collision, although resonance effects may be superposed over the general picture.

ACKNOWLEDGMENTS The author wishes to express his gratitude to many workers who have made unpublished results available to him. He is indebted to the following for most helpful criticisms and discussion: Profs. D. R. Bates, E. Everhart, J. L. Magee, Drs. J. L. Franklin and H. B. Gilbody. The manuscript was prepared in large part by the author’s wife, to whom a debt of gratitude is owed. L1sr Oh’ S Y h r R o L s A Projectile ion or atom Constant in the Arrhenius equation A

JOHN B. HASTED

rlo

j

K,K’

Distance of collision interaction in adiabatic theory Radius of the first Bohr orbit of the hydrogen atom Target ion or atom Ion-molecule reaction constant Constant in empirical highenergy formula for cross section energy variation Another constant in empirical high-energy formula for cross section energy variation The distance from electron beam to extraction slit in a mass-spectrometer source Relative kinetic energy of impact Ionization energy Kinetic energy at which the cross section is a maximum Kinetic energy transferred to internal energy Charge on the electron Average energy transferred in a single electronelectron collision Average orbital energy of the electron The fraction of singly charged ions in an ion beam passing through a gas The equilibrium fraction of singly charged ions A fragment of the j t h type produced from a complex Efficiency factor of Field, Franklin, and Lampe Ratio of statistical weights Hamiltonian operators Planck’s constant Current of a primary beam Initial primary beam current Final primary beam current Primary beam current after traveling a path length 1. Types of secondary ions and fragments produced from complexes Adiabatic constants Boltzmann constant; also constant in asymptotic form of wave function Rate constant for complex formation Rate constant for production of secondary ions of the j t h type from a complex Length of a collision chamber Number of molecules or atoms of a target gas Masses of the projectile and target Number of energy cells in the transferred energy Mass of the electron Avogadro’s number Initial and final unit vectors in the direction r. Pressure in dyne/cm2 Pressure in mm-of-Hg, also the probability that particles approach along a specific potential-energy curve Probability of n electrons being removed by m energy cells Probability of emergence as a neutral Probability of charge exchange Modified ionization probability Probability of transition at a potentialenergy crossing Probability of emergence as a neutral atom Number of collisions made by the primary ion in a mass-spectrometer source

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

P R RZ r.,rd

77

A parameter defined in Fan’s classical equations The gas constant; also internuclear separation Internuclear separation of pseudocrossing potential-energy curves. Internal coordinate of the final systems Distance of closest approach of two nuclei ro A secondary positive ion of the j t h type produced from a complex Sj+ Overlap integral between initial and final states Soi T Absolute temperature 1 Time variable Defined in terms of V(r) U(r) V Potential gradient perpendicular to the magnetic field in a Nier nws-spectrometer source V(rjd,e) Interaction energy between two atomic systems Relative velocity of colliding particles Mean relative velocity of collision Velocity of scattered projectile Velocity of scattered target Final relative velocity of colliding particles Electron orbital velocity Electric field strength Cartesian coordinate Cartesian coordinate Atomic number Effective perturbation Cartesian coordinate; also “effective nuclear charge” Polarizibility A constant, relating phase to impact energy in the Landau-Zener formula Internal energy defect Energy of activation Energy separation at pseudocrossing Energy cell width in statistical theory of ionization Parameter defined in the Landau-Zener formula Angle between unit vectors parallel to the initial and final velocities Angle between the 5- and y-axes in three-dimenpional potential-energy diagrams Angle of scattering of the incident particle in center of mass coordinates Angle of scattering of incident and target particles in laboratory coordinates Fan’s shadow cone angle Transmission coefficient See Bransden, Dalgarno, and King Reduced mass of a system of colliding particles Total number of electron-electron collisions in an ion-atom collision. Relative coordinates of centers of mass of final systems Collision cross section; also a constant in the Landau-Zener formula Maximum cross section Experimental cross section Time spent by an ion in the collision region of a mass-spectrometer source Wave functions Number of atoms in a molecule

JOHN B. HASTED

78

Siibwript,s and superscript^

A,B

f a+ b+ m+ n+

Projectile and target Final Initial charge on the projectile Initial charge on the target Final charge on the projectile Final charge on the target

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2. Fite, W. L., Brackmann, R. T., and Snow, W. R., Phus. Rev. 112, 1161 (1958). 3. Donahue, T.M., University of Pittsburgh, private communication (1958). 4. Fedorenko, N. V., and Afrosimov, V. V., J. Tech Phys. (U.S.S.R.) 26, 1872 (1957);J. Ezptl. Theorct. Phys. (U.S.S.R.) 34, 1398 (1958). 6. Lindholm, E., Arkiv. Fysik 8, 257, 433 (1954);Proc. Phys. SOC.(London) 866, 1068 (1953). 6. Gilbody, H. B. and Hasted, J. B., PTOC. Roy. SOC.(London)U38, 334 (1956); A240,382 (1957);and unpublished data. 7. Fogel, I. M., Ankudinov, V. A., Pilipenko, D. V., and Topolia, N. V., J. Exptl. Theorct. Phys. (U.S.S.R.) 34,579 (1958). 8. Kaminker, D. M. and Fedorenko, N. V., J. Tech Phys. (U.S.S.R.) 26, 1843,2239 (1956). 9. Flaka, I. P. and Soloviev, E. S., J. Tech Phys. (U.S.S.R.) 28, 599, 612 (1958). 10. Ribe, F., Phys. Rev. 83, 217 (1951). 11. Allison, S.K., Phys. Rev. 109, 76 (1958). 12. Montague, J. H., Phys. Rev. 81, 1026 (1951). 13. Whittier, A. C., Can. J. Phys. 32, 275 (1954). 14. Fu~N, E. N., Jones, P. R., Ziemba, F. P., and Everhart, E., Phys. REV.107, 704 (1957);Jones, P.R., Ziemba, F. P., MOW,H. A., and Everhart, E., ibid. 113, 182 (1959). 16. Bydin, I. F. andBukhteev, A. M., Doklady Akad. NaukS.S.S.R. 119,1131 (1958). 16. Keene, J. P., Phil. Mag. [7]40,369 (1949). 17. Hasted, J. B., Proc. Roy. SOC,(London) 8205, 421 (1951); 8212, 235 (1952). 18. Moe, D., Phys. Rev. 104, 694 (1956). 19. Dukelskii, V. M., Afrosimov, V. V., and Fedorenko, N. V., J. Exptl. Theoret. Phys. ({J.S.S.R.) 30, 792 (1956). 20. Field, F. € Franklin, I., J. L., and Lampe, F. W., J . Am. Chem. SOC.79,2419 (1957). 21. Stevenson, D. P. and Schissler, D. O., J. Chem. Phys. 23, 1353 (1955);24, 926 (1956). 22. Sayers, J. and Kerr, L. W., private communication (1959). 23. Fan, C.Y. and Meinel, A. B., Astrophys. J. 113, 50 (1951); 116, 330 (1952);116, 205 (1953). 24. Sluyters, T. J. M. and Kistemaker, J., Physica 26, 182 (1959). 26. Boyd, R. L. F. and Morris, D., Proc. Phys. SOC.(London) 668,11 (1955). 26. Kerr, L.W., J . Eleetronics 2, 179 (1956). 87. Hasted, J. B., Rept. Gatlinberg Conf. on Penetration of Charged Particles in Matter (S.K. Allison, ed.) University of Chicago (1958). 28. Wien, W., Ann. Physik. [4]39, 528 (1912). 89. Stier, P. M. and Barnett, C. F., Phys. Rev. 109, 385 (1958);103, 896 (1956).

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JOHN B. HASTED

61. Bates, D. R. and Dalgarno, A., Proc. Phys. SOC.(London) AM, 919 (1952). 62. Jackson, J. D., and Schiff, H., Phys. Rev. 89, 359 (1953). 63. Bransden, B. H., Dalgarno, A., and King, H. M., Proc. Phys. Sqc. (London) A66, 1097 (1953). 64. Gerasimenko, B. I., and Rosentwcig, L. N., Proc. Phys. Dept. Univ. Kharkov 6, 87 (1955); Soviet Phys. J E T P 4, 508 (1957). 65. Bates, D. R., Proc. Roy. Sac. (London) A243, 15 (1957). 66. Bates, D. R. and Dalgarno, A., Proc. Phys. Soc. (London) A66, 972 (1953). 67. Dalgarno, A. and Yadav, H. N., Proc. Phys. SOC.(London) A66, 173 (1953). 68. Massey, H. S. W. and Burhop, E. H. S., “Electronic and Ionic Impact Phenomena,” p. 441. Oxford Univ. Press, London and New York, 1949. 69. Jackson, J. D., Can. J. Pliys. 32, 60 (1954). 70. Moiseiwitsch, B. L., Proc. Phys. Sor. (London) A69, 653 (1956). 71. Bates, D. R., Proc. Roy. Sac. (London) A243, 15 (1957); 8246, 00 (1958). 72. Haywood, C. A., Proc. Phys. Sac. (London) 73, 201 (1959). 73. Dalgarno, A. and McDowell, M. R. C., Proc. Phys. SOC.(London) A69, 615 (1956). 74. Fite, W. L., private communication (1959). 76. Bates, D. R., Proc. Phys. SOC.(London) 73, 227 (1959). 76. Frame, J. W., Proc. Cambridge Phil. Sac. 27, 511 (1931). 77. Firsov, 0. B., Exptl. Theoret. Phys. (U.S.S.R.) 21, 1001 (1951). 78. Hasted, J. B., J. Appl. Phys. 30, 25 (1959). 79, Fogel, I. M., Ankudinov, B. A., and Pilipenko, D. P., Exptl. Thcoret. Phys. (U.S.S.R.) 36, 868 (1958); Fogel, I. M., Mitin, R. V., Koslov, B. F., and Romashko, N. D., ibid., 86, 5G5 (1958); Fogel, I. M., Mitin, R. V., and Koval, A. G., ibid., 31, 397 (1956); Fogel, I. M. and Mitin, R. V., zbid. 30, 450 (1956). 80. Hasted, J. B., Proc. Roy. SOC.(London) 8212, 235 (1952). 82. Stedeford, J. B. H. and Hasted, J. B., Proc. Roy. SOC.(London) A227, 466 (1955). 82. Sida, D., private communication (1954). 83. Sherwin, C. W., Phys. Rev. 67, 814 (1940). 84. Dalgarno, A., Phil. Trans. Roy. SOC.260, 426 (1958). 85. Buckingham, R. A. and Dalgarno, A., Proc. Roy. SOC. (London) A213, 506 (1952). 86. Stebbings, R. F., Proc. Roy. SOC.(London) A241, 270 (1957). 87. Massey, H. S. W. and Burhop, E. H. S., “Electronic and Ionic Impact Phcnomena,” p. 230. Oxford Univ. Press, London and New York, 1949. 88. Massey, H. S. W. and Burhop, E. H. S., “Electronic and Ionic Impact Phenomena,” p. 231. Oxford, Univ. Press, London and New York, 1949). 89. Herzberg, G., “Spectra of Diatomic Molecules.” Van Nostrand, New York, 1950. 90. Landau, L., Physik. 2.Solujetunion 2, 46 (1932). 91. Zener, C., Proc. Roy. SOC.(London) A137, 696 (1932). 92. Bates, D. R. and Massey, H. S. W., Phil. Mag. [7] 46, 111 (1954). 93. Hasted, J. B. and Smith, R. A., Pvoc. Roy. SOC.(London) A236, 354 (1950). 94. Boyd, T. J. M. and Moiseiwitsch, B. L., Proc. Phys. Sac. (London) A70,809 (1957). 95. Moiseiwitch, B. L., J. Atmospheric and Terrest. Phys. 23,Spec. Suppl. Vol. 2, p. 23 (1955). 96, Bates, D. R. and Boyd, J. M., Proc. Phys. Sac. (London) A69, 910 (1956). 97. Magee, J. L., J . Chem. Phys. 8, 087 (1940). 98. Bates, D. R. and Moiseiwitsch, B. I., Proc. Phys. Sac. (London) A67, 805 (1954). 99. Dalgarno, A., Proc. Phys. Sac. (London) A67, 1010 (1954). 100. Bates, D. R. and Lewis, J. T., Proc. Phyn. SOC.(London) AM, 173 (1955). 101. Massey, H. 9. W. and Burhop, E. H. S., “Electronic and Ionic Impact Phenomena,” p. 532. Oxford Univ. Press, New York, 1949.

INELASTIC COLLISIONS BETWEEN ATOMIC SYSTEMS

81

102. Weizel, W., Z . Physik 76, 250 (1932). 103. Gilbody, H. B. and Hasted, J. B., Proc. Roy. SOC.(London) A240, 382 (1957). 104. Opik, E. J., “Physics of Meteor Flight in the Atmosphere.” Interscience, New

York, 1958. 106. Evans, S. and Hall, J. E., J . Atmospheric and Terrest. Phys. Spec. Suppl. Vol. 2,

18 (1955). Greenhow, J. S. and Hawkins, G. S., Nature 170, 355 (1952). Massey, H. S. W. and Sida, D. W., Phil. Mag. [7] 46, 190 (1955). Hasted, J. B., Proc. Roy. SOC.(London) 8322, 74 (1954). Hydin, I. F. and Dukelskii, V. M., J. Exptl. Theoret. Phys. (U.S.S.R.) 31, 474 (1957). 110. Gilbody, H. B. and Hasted, J. B., Proc. Phys. SOC.(London) 872, 293 (1958). 111. Fogel, I. M., Ankudinov, B. A., and Slabospitskii, P., J. Exptl. Theoret. Phys. (U.S.S.R.) 32, 453 (1957); Souict Phys. J E T P 6, 382 (1957). 118. Fogel, I. M., Mitin, R. V., and Koval, A. G., Soviet Phys. J E T P 4, 359 (1957). 113. Magee, J. L., Discussions Faraday Soc. 12, 33 (1952). 114. Simons, J. H., Francis, H. T., Fontana, C. M., Jackson, S. R., Muschlitz, E. E., and Bailey, T. E., J . Chem. Phys. 13,216 (1945); 21, 689 (1953); 20, 1431 (1952); 18, 473 (1950); 13, 221 (1945); 11, 316 (1943). 116. Bates, D. R., Proc. Phys. SOC.(London) A68, 344 (1955). 116. Eyring, H., Hirschfelder, J. O., and Taylor, H. S., J. Chem. Phys. 4, 479 (1936). 117. Langevin, P., Ann. chim. el phys. 6, 245 (1905). 118. Stearn, A. E. and Eyring, H., J . Chem. Phys. 3, 778 (1935). 119. Coulson, C. A. and Duncanson, W. E., Proc. Roy. Sac. (London) 8166,90 (1938). 1.20. Boyd, R. L. F., Nature 166, 228 (1950). 12Oa. Bohm, D., “The Characterjatics of Electrical Discharges in Magnetic Fields (Guthrie and Wakerling, eds.), p. 77. McGraw-Hill, New York, 1949. 121. Dear, H. D. and Emeleus, K. G., Phil. Mag. [7] 40, 460 (1949). 122. Townsend, J. S., “Electrons in Gases.” Hutchinson, London, 1947. 183. Fan, C. Y., private communication (1958). 124. Fedorenko, N. V., SOC.Ital. Fis. Terzo Congr. Intern. Sui Fenoneni d’ionizzazione nei gas, Milano p. 295 (1957). 125. Blauth, E., Z. Physik 147, 228 (1957). 1.26. Gryzinski, M., Phys. Rev. 107, 1471 (1957); Polish Acad. Sci. Inst. Nuclear Research Rept. No. 59/I-A (1958). 127. Reynolds, H. K., el al. Phys. Rev. 92, 742 (1953). 188. Fite, W. L. and Brackmann, R. T., Phys. REV.112, 1141 (1958). 189. Takayanagi, K., Sci. Repts. Saitama Univ. I, 9 (1952); MI, 33 (1955). 130. Mott, N. F. and Massey, H. S. W., “The Theory of Atomic Collisions,” 2nd ed., p. 271. Oxford Univ. Press, London and New York, 1952. 131. Rutherford, J. J., Phil. Mag. [6] 47, 277 (1924). 132. Barnett, C. F. and Reynolds, H. K., Phys. Rev. 100, 385 (1958). 133. Stier, P. M., Rept. Gatlinberg Conf. on Ptnetration of Charged Particles in Matter (S.I<. Alliso ed.), University of Chicago, (1958). 134. Lassen, N. Kgl. Danske Videnskah. Selskab, Mat.-fys. Medd. 26, No. 5 (1951); 26, No. 12 (1951); SO, No. 8 (1955). 135. Bohr, N and Lindhard, J., Kgl. Danske Videnskab. Selskah, Mat.-fys. Me& 28, No. 7 (1954). 196. Bell, G.I., Phys. Rev. 90, 548 (1953). 137. Russek, A. and Thomas, M. T., Phys. Rev. 109, 2015 (1955). 138. Ziemba, F. P. and Everhart, E., Phys. Rev. Letters 2, 299 (1959). 106. 107. 108. 109.

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Field Ionization and Field Ion Microscopy ERWIN W. MULLER Field Emission Laboratory, The Pennsylvania State University, University Park, Pennsylvania Page I. Introduction. . . . . 11. Field Ionization of Free Atoms.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

. . . . . . . . . 8-1

................................. 86 111. Field Ionization near a Metal Surface.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 A. Theoretical Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 B. Experimental Investigations . . . . . . . . . . . . . . . . 90 IV. Field Ion Emission from a Metal Surface.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 ......... V. Field Ion Microscopy.. . . . . . . . . A. Theory of the Field Ion Mic B. Experimental Procedures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

F. Surface Effects.. . . . . . . . . Acknowledgments. . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

...........................

Ii7

I. INTRODUCTION The ionization of a free atom by the tunnel effect in a high electric field is called field ionization. This process was visualized by Oppenheimer (I), when he used the then new concepts of wave mechanics to show that the electron of a hydrogen atom in a field has a finite ionization time. In a field of 1 v/cm this time was found to be extremely long, about 101o'o sec. On the other hand, the field necessary for ionization within a reasonable observation time, was estimated to be of the order of several hundred Mv/cm, forbiddingly high for an observation with the experimental techniques known 30 years ago. However, some experimental evidence for field ionization from high lying excited states of the H atom was found in the quenching of the lines in the Stark effect, as observed by Hausch von Traubenberg (2) and calculated by Lanczos (3). For instance, the reddest component of H,, originating from the level n = 5 , disappears at a field of 0.7 Mv/cm. Because of the complex nature of the problem no further basic progress 83

84

ERWIN

w.

MULLER

has been made in the theoretical study of field ionization, and our knowledge today is based almost entirely on empirical results, which became possible with our improving ability to handle extremely high fields. High electric fields became accessible for quantitative work with the introduction of the needle-shaped, surface migration-polished field emitter (4) and the invention of the field emission microscope (6). With a negative emitter the maximum field strength is, of course limited by the large electron emission and subsequent heating of the tip. The limit for tungsten is approximately 80 Mv/cm which can only be maintained for microseconds because of the electron current density of lo8 amp/cmZ according to Dyke and Dolan (6). Much higher fields could be handled by reversing the voltage of a field emission microscope to make the tip positive. This led first to the discovery of a new effect, field desorption (7). Later, the field emission microscope was operated with hydrogen ions (8) and field ionization in space was observed to occur above 290 Mv/cm. The main objective of operating a field ion microscope, however, was the prospect of increased resolution. In the course of the development, field ionization of helium a t approximately 450 Mv/cm at the surface of a liquid hydrogen-cooled emitter tip proved to be most effective (9). The field ion microscope, now the most powerful smicrocope, is capable of showing individual atoms as they form the crystal lattics of the tip metal with full resolution of high index net planes and epareating atoms with spacings down to 2.3 A. With these possibilities, the microscope seems to be a very promising tool for basic studies in solid state physics and in some of the more practical lines of physical metallurgy.'.At the time of this writing most of the work with the low-temperature field ion microscope has been done in the author's laboratory with its very limited capacity, and it is hoped that this review may stimulate and encourage other workers to utilize the new tool more effectively. Besides field ion microscopy there are several other aspects of field ionization that make the physics of this effect attractive because one can now control electric fields that are comparable with the ones acting between the electronic charges at atomic distances. All metals can be evaporated in the form of ions a t temperatures close to absolute zero, and unusual low-temperature chemical reactions between metals and gases may occur. Some practical aspects may also be of importance, such as the use of field ionization in mass Epectroscopy and perhaps in ionic propulsion systems. 11. FIELDIONIZATION OF FREEATOMS A . Theory of Field Ionization A free atom can be considered as a potential trough in which the valence electron is trapped a t a depth equal to the ionization energy V I (Fig. 1).

FIELD IONIZATION AND FIELD ION MICROSCOPY

85

In an electric field the potential toward the anode is lower than in the atom, and according to wave mechanics ionization is possible by penetration of the electron through the potential barrier on the anode side of the trough. A rigorous treatment of this problem has been carried out by Lanczos,

(b)

(a)

FIG.1. Potential trough representing a free atom of ionization energy V , (a), and the same atom in high electric field (b).

and a more simplified derivation for the case of a hydrogen atom can be obtained by using the WKB method [see for instance Bethe and Salpeter (10) or Good and Muller (1111. As a one-dimensional problem, the barrier penetration probability can be writt,en as

where m is the electron mass, A is Planck's constant divided by 2 r , V the potential energy, and E the total energy of the elect,ron. The integral is to be extended over the barrier width. Numerically the linear barrier penetration probability is

if potential energy V and total energy E are expressed in electron volts, and the barrier width is measured in centimeters. Except in the case of a hydrogen atom, the potential energy cannot be written down in a simple expression, but in some eases it may be sufficient for an approximation to assume a Coulomb potential, so that in the field F the potential is --U

=

e

- -r

+ Fx.

(3)

The time r it will take to ionize an atom depends also on the frequency v with which the electron inside the atom strikes the barrier,

86

ERWIN

w. M ~ ~ L L E R

where v call be calculated for hydrogeii for Bohr's t,heory aiid for other atoms by maintaining the Bohr orbit picture and using for instance Finkelnhurg's (12) values for the effective nuclear charge. Typical data are Y = 4.1 X 1OI6 seo-1 for H, 2.4 X 10'6 sec-1 for He, 1.5 X 10l6 sec-I for A. Table I, adapted from Inghram and Gomer (IS), gives ionization times for a free hydrogen atom in typical fields. TABLEI. IONIZATION TIMEOF

F (Mvjcm) 50 100 150 200 250

7

A

HYDROGEN ATOM

(sec)

1.3 x 1.6 X 1.6 x 1.7 X 2.0 x

10-1 10-13 10-14 10-16

For a crude estimate of the ionization time of an atom in a field one can replace the rounded off potential hump in Fig. l b by a triangle of height V I and a base Vr/F, and insert this function in the integral of Eq. (2). Combined with Eq. (4) and-an average v one obtains

where Vr is measured in electron volts and F this time in volts per angstrom. Actually the lifetime will be somewhat shorter because the tapering of the potential funnel reduces the exponent in Eq. ( 5 ) by some 10-20%, but Eq. ( 5 ) is quite useful for the comparison of equivalent field strength for gases with different ionization energies.

B. Observation of Field Ionization The first observation of field ionization of free atoms was made by Muller (8) when he conceived the idea of operating a gas-filled field emission microscope with reversed polarity in order to depict surface details of the emitter tip with field desorbed ions instead of electrons. The more important ionization process near the metal surface which permits the image formation shall be considered later. However, i t was noticed that when the field a t the tip was raised above 290 Mv/cm, the ion image on the screen became quite suddenly blurred. It was concluded that above this field strength the ionization of the arriving hydrogen molecules took place in free space before the tip surface was reached. The observed field strength for the onset of ionization of free molecules is in good quantitative agreement with the previous theoretical considerations. No further measurements were made at that time, and there also seems to be no other work

FIELD IONIZATION AND FIELD ION MICROSCOPY

87

since then dealing particularly with free autoionization. A number of ohservations have been made in connection with studies of field ionization near metal surfaces, which will be discussed later. I t may be only mentioned here that the blurring of a helium ion image occurs a t about 500 Mv/cm. The ratio of this field strength and the one given for the hydrogen molecule equals closely the ratio of the 95 powers of the respective ionization potentials a s is predicted by Eq. (5).

111. FIELDIONIZATION NEAR

A

METALSURFACE

A . Theoretical Considerations If field ionization takes place very close to a positively charged metal surface the conditions become somewhat more complicated because of the electric interaction with the metal. Barrier penetration from atoms near a surface in positive and negative fields was first suggested by Kirchner ( l 4 ) , and the considerable increase of penetration probability due to the image force was then noticed by Inghram and Gomer (15).If the ion, represented by a Coulomb potential, is a t a distance d from the surface, the total effective potential of the electron at a distance x from the surface is

e e TI(=) = - ___ + F x - - + + * Id

-4

4x

e

d+x

The third term is due to the attraction of the electron by its image in the metal, and the fourth term is due to its repulsion from the image of the ion in the metal. The Pauli principle imposes a condition for the closest distance d = xc from the surface a t which field ionization can occur. Since all states below the Fermi level inside the metal are occupied, the tunneling electron can be accommodated only above the Fermi level. This condit,ion can be written as

eFxc 2 Vz

-

e2

C$

-4x

+ -21 F?(w

- ffz)

where VZ is the ionization energy, 4 the work function, C Y A and CYI the polarizabilities of the atom and the ion, respectively. The last term represents the difference in polarization energies before and after ionization, and, although not accurately known except in the case of hydrogen, it is prohably not larger than 0.1 ev. I n a given case the probability of ionization during a time elenieiit 1 can now be calculated by using Eqs. (4)and (6) to be

P(t) = 1 - e-lIr = t

/ for ~ t / < ~< 1.

(8)

This equation must be written in integral form when the position of the

88

ERWIN

w.

M~LLER

considered atom changes during this time with points X A and XB

P(t) = 1 - exp

(-

B

velocity v between the

L%)*

(9)

For some considerations it may be more practical to calculate a probability P(d) which describes the ionization probability while the atom travels within a certain distance between d and d-dx a t a velocity v towards the surface. Figure 2 shows curves obtained by numerical integration (16) for DISTANCE FROM SURFACE (A) f

-4

8

l,O

;1

;4

'p

1,s

4 0 0 MV/CM

FIG.2. Logarithm of probability for an argon atom (a) and a helium atom (b) to be ionized by tunneling while traveling the distance dx = 1 A towards the surface with a velocity determined by dipole attraction.

argon and helium. The velocity of the atoms approaching the metal surface was assumed to be determined by the dipole attraction in the inhomogeneous field, which is I---

v = F dms for molecules without a permanent dipole moment and, under the conditions of the diagram, is much larger than the gas kinetic velocity a t room temperature. Field ionizat,ion will usually be measured in terms of the ion current emerging in space more or less near the positive point electrode. This cur-

2r

FIELD IONIZATION AND FIELD ION MICROSCOPY

89

rent is the product of the number of gas molecules arriving a t the emitter per unit of time, which we will call the supply function 2, and the efficiency of field ionization as expressed in Eq. (9). When field ionization of hydrogen at the tip of a field emission microscope was discovered by the author (8), it was noticed that the supply function was considerably larger than the number of z of gas kinetic impacts on a surface of 1 cm2,

P

=

d%izT1

and this increase was recognized to be due to the dipole attraction of the polarized molecules in the inhomogeneous field around the emitter tip. One can define a radius of capture r, by the condition that the actual supply to the tip surface Z equals the number of molecules which strike the sphere of capture without a field,

Z

=z

*

4m-2.

(12)

Since the gas molecules move in a central field, their radial velocity can be written as

dr dt

=

&(I32 m

f

-I- a P

+ p F - &), mLr2

where E = kT is the mean thermal energy of the molecule in the plane, &bF2 p F is the energy of the molecule in the field due to its polarizability a and its permanent dipole moment pl and j = mva its angular momentum at velocity v and for the impact parameter a. At the instant of closest approach of molecule and tip the motion is tangential] for example dr/dt = 0, and the condition of striking the tip is r 5 ro, the tip radius. By setting r = ro in Eq. (13) and realizing that the maximum value of the impact parameter is re, we obtain

+

which describes the supply increment due to the attraction of the induced dipole. The same result was first obtained by Inghram and Gomer (IS)in a slightly different way. As a typical example one finds for helium (a = 2X cm3,p = 0) a t FO = 450 Mv/cm and T = 300"K, an increase of the supply by a factor of 6.5. For a crude calculation of the ion current that can be expected a t a voltage sufficient for ionizing all arriving molecules we consider only the second term of Eq. (14) and obtain

90

ERWIN W. MULLER

By expressing the field as a function of tip radius and voltage with the empirical formula

V Fo = 5r0 which is useful for the conventional tip shape, we find the emitted ion current

Here p is measured in dynes/cm2, V in esu, and e is 1.6 X

coul.

B. Experimental Investigations 1. Ion Currents in Various Gases. Measurements of field ion currents in hydrogen and some rare gases have been reported by Drechsler and Pankow ( l 7 ) , but the current range covered little more than one order of magnitude, and no interpretation was presented. More detailed measurement was carried out by Miiller and Bahadur (16) who studied field ion emission from tungsten tips of different radii in gases such as Hz, He, Nz, 0 2 , A, Kr, Xe, and Hg, in fields u p to 500 Mv/cm and in the pressure range from 0.1 to 6fi in Hf and to 25p in He. Figure 3 represents some of the results of current measurements in simple field emission microscopes. In order to compare the features of these curves with the theory preaented in the preceding sections it is necessary to consider a few more details. The theoretical ion current can be computed by numerical integration of the ionization probability P(d) from the minimum distance to infinity and by multiplying the result with the supply function Z and the ion charge e. The actual ion current is larger because the molecules that have not been ionized while approaching the tip will be reflected from its surface and pass again in the reverse direction through the zone of high ionization probability close to the surface and beyond the Fermi level barrier. This time the ionization probability will be larger than on the approach. The arriving molecules come in almost perpendicular to the surface, since the dipole attraction energy component +&F2 is much larger than kT, but the reflected molecules may have any random direction. They may also have a lower average velocit,y, corresponding more or less to the tip temperature, depending on the accommodation coefficient of the gas a t the surface. Assuming a small accommodation coefficient and random reflection, the average normal rebound velocity is half the incident velocity, and the ionization twice as much. Therefore, the total ion current will be

I

=

3eZ

L:in

P(d) dx.

91

FIELD IONIZATION AND FIELD ION MICROSCOPY r,

1he total cwrrciits :icatually measured iii a field ion microscope as b' rive11 in Fig. 3 must he expected t o hc larger by a factor y 1 hecause of y secondary electrons released by eac*hion a t the screen and other negative electrodes. Literature data about the secondary emission factor of not extremely outgassed surfaces vary widely, and particularly nothing definite could be found for the screen material. An empirical formula is y = 4 X lW4 X V for helium (V in volts) for the number of secondary electrons

+

-7

-8

-

TIP RADlUS

450 A

c

-:

-9-

u

"?-lo

s

-I I

-

I

-12

41 0

-7

-8

-

TIP

RADIUS

1000 A

-z - 4 -9

I

"?-lo.

s

-11

-12

.

0

FIG.3. Logarithm of current versus field strength measured in a field emission micromope filled with different gases of lrr pressure. Tip radius 450 A (a) and 1000 A (b).

released by one ion. There is also some increase in current by the collision of ions with gas molecules, but in most gases this effect is still negligible a t 1p pressure. This follows from the experimental fact of a linear pressure dependence of the ion current as expected from Eq. (17). The author does not share Becker's (18) conclusion that the multiplying factor due to volume effects is as large as 7 in ljt of hydrogen, and the volume effect has been neglected in the calculation of field ion current as presented in Fig. 4. The

92

ERWIN W. MULLER

comparison of theoretical and experimental results is as perfect as could be expected with the large number of fairly crude assumptions that had to be made, and it supports fully the mechanism that has been described here. 1here is no sharply defined minimum field strength for field ion emission, although the current rises rather steeply in the lowest current range. For different gases one finds experimentally the field for the onset of the current to be proportional to the 35 power of the ionization potential, as suggested by Eq. (5). Linearity with pressure as suggested by the supply function Eq. (15) has already been mentioned. In the conventional size -6,

-8

-

I

n

-

L -10. c

-

0

$12

.

A

-14

’

I I

I I I

I

I -WO

100

HELIUM

I

200

I

I

300

400

I

I0

FIG. 4. Theoretical current-field characteristics (dashed line) compared to experimental curves (solid lines).

field ion microscope with the circuit protected by a high ohmic resistor a glow discharge and corresponding voltage breakdown occurs at about 8 p in hydrogen, and up to 50p in helium, while current multiplication through of this pressure. At high volume effects becomes noticeable a t about fields where the total supply is ionized the current should increase with V 2 according to Eq. (17), which is found to be true. Also the dependence of , current upon tip radius in Eq. (15) expected to be proportional to T ~ agrees sufficiently with the experiments since the radii were not determined with the accuracy that is possible today by using the low-temperature field ion microscope. The temperature dependence predicted by Eq. (17) has not yet been accurately measured. At elevated temperatures, up to 5OO0C, the current appears to decrease much faster then predicted, because the rebounding molecules which are not ionized are less efficiently trapped in the sphere of capture. Below room temperature down to liquid nitrogen temperature there is a great increase in ion current, as observed by the brightness in the ion image, which may correspond closely to the predicted

FIELD IONIZATION AND FIELD ION MICROSCOPY

93

' T

dependency. At still lower temperatures, however, the additional gain in current is much smaller. This may be because the sphere of capture becomes comparable with the dimension of the tip cone, and many of the captured molecules will go to the cone rather than to the cap of the tip. 2. Energy Distribution by Retarding Potential Method. An analysis of the energy distribution can be used to localize the origin of the ions, and the best resolution in energy can be obtained by applying the retarding potential method with spherical symmetry. This arrangement has proved its usefulness in the measurements of energy distribution of field-emitted electrons (19), and Muller and Bahadur (16) adapted it to field ion emis-

FIQ.5. Retarding potential tube and circuit.

sion. The tube as shown in Fig. 5 is first operated at a positive accelerator potential in order to determine the work function of the collector and to check the resolution with the known energy distribution of fieldemitted electrons; &,u equals the voltage of the collector at which electrons start to enter this electrode according to the potential diagram Fig. 6 . If now the of gas and the voltages are reversed, the tube is filled with about 0 . 1 ~ field ions produced near the emitter surface can reach the collector only if it is negative so that

94

ERWIN W. MULLER

EMITTER

COLLECTOR

ACCELERATOR

E

f

FIQ.6. Potential diagram for retarding potential tube, above for electron emission from negative emitter, below for ion emission from positive emitter. Ionized atom is at minimum distance z5 from surface.

ELECTRON

EMISSION

w 0

I

9eou

RETARDING

POTENTIAL

(VOL1

I6 POTENTIAL

PO (VOLTS)

ARQON ION EMISSION 0

W

WI

0

5

10 RETARDINQ

FIQ.7. Recorded collector ciirrerit8, approximately 10-" amp.

FIELD IONIZATION AND FIELD ION MICROSCOPY

95

The recorded collector current,.s are presented in Fig. 7, and it is a t a first glance surprising to find the collector current to have the direction of entering electrons again. At low collector voltages these electrons are mostly secondaries released at the outside surface of the accelerator electrode when the ions cannot enter the collector and fall back to the accelerator. Some of the electrons originate from collisions in the gas, but this is only a small fraction since the free path at 0 . 1 is ~ about 10 times the tube radius. The collector current begins to drop when the field ions start contacting the collector. The onset of ion current is a t 11 v, exactly where it should happen according to the potential diagram, with Vr = 15.7 ev, q&11 = 5.7 ev, and e2/4x2 = 0.8 ev. The onset for Xe was found a t 6.5 v , and for He at 19 v. By graphical differentiation of the voltage current curve one obtains the energy distribution, Fig. 8, which shows as expected the origin of the ions beyond the minimum distance x, from the tip, and the increase in distribution width when at higher fields the ionization space becomes effective. The rounding off a t the onset of the distribution curves is partly real, since the theoretical treatment has not considered local variations of 4 and F on the emitter surface, and partly owing to the limited resolution of the apparatus because of misalignment and a too large time constant of the current recorder. With improved techniques quite more detailed results appear possible with this method, since Young and Muller (20) have shown that the resolution of the apparatus when operated with field electrons can be as good as 0.005 ev. 3. Mass Spectroscopy with Field Ion Sources. The fact that one can expect fairly monoenergetic ion8 from a field emitter immediately suggests its use as a source for mass spectroscopic analysis. Miiller (21) expected the following particular advantages compared to the conventional electron impact source: (1) The absence of fragmentation of organic molecules since the gas molecule need not even touch any surface before becoming ionized ; (2) The absence of a hot electrode; (3) The small dimension of the point source. Inghram and Gomer ( I S , 15) were the first ones to publish experiDz, N2, mental results on the mass spectroscopy of field ionization of HZ, Oz, some hydrocarbons, and methyl alcohol. For hydrogen they found that Hz+ ions predominate a t lower fields, while H+ ions are more abundant a t higher fields, where eventually pure autoionization occurs a t distances up to 100 A from the tip and the effect of the image potential becomes less important. This was concluded from the increasing peak width, which began to surpass the resolution of the mass spectrometer of 20 v a t an apparent field strength of 580 Mv/cm. The significance of this paper for the understanding of the mechanism of field ionization is not reduced by the fact that the given field strengths are too high by a factor of two, which was caused

96

ERWIN

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MULLER

THEQiTlCAL VELOCITY DISTRIBUTION

WB€R

or

IONS

ARBITRARY UNITS

10

12

I4

I6

10

I2

14

I6

I8

VOLT1

FIQ.8. Theoretical and experimental energy distribution of field emitted argon ions. by a n erroneous assumption about the presence of an oxygen film of high work function on the tip during the field calibration with field electron emission. With this correction the beginning of pure autoionization at about 290 Mv/cm agrees with the earlier observation in the field ion microscope by Muller (8). Muller and Bahadur (16) have also made a more qualitative analysis of field ions in a simple mass spectroscope (22),Fig. 9, in which an ion beam of approximately 20" aperture is focused by the electrostatic lens onto the screen. With the magnetic deflection field turned off and the voltage reversed, focusing with the voltage divider can be done easily with the much more intense electron beam. The relatively large lens aperture makes i t difficult to maintain a low pressure behind the lens when the gas is being let in on the emitter side. Therefore, some charge exchange is observed on

FIELD IONIZATION AND FIELD ION MICROSCOPY

97

the screen recognizable by an undeflected or partially deflected beam of neutral atoms. Sufficient resolution to just barely resolve the krypton isotopes of mass 82, 83,84, and 86 was achieved. With this apparatus and a measurement of the spot intensities with a photomultiplier the marked difference in abundance of the two hydrogen ions according to Inghram and Gomer could not be confirmed. The ions H2+and K+ appeared rather to have equal abundance within about 10% in the entire range of the previous

FIQ.9. Mass spectrometer tube.

measurements. Since it is hard to understand how the only variable factor, the cleanliness of the tip surface, could influence the abundance in the case of pure autoionization, further studies either with a high-resolution mass spectrometer or preferably combined with accurate retarding potential measurements seem to be desirable. Exact localization of the origin would permit detection of a possible contribution of ions supplied by a surface migration from the shank of the tip, as was suggested by Muller (22) and by Becker (18), although a large flow of surface diffusing hydrogen atoms or ions from thd shank to the central region of the tip is hard to imagine in the presence of the high desorbiiig field. With a resolution of 20 v and later 5 v in the mass spectrometers used by Inghrnm and Gomer (13) the line details could be measured well enough to locate the origin of the ions. Some of the lines do not show broadening with increasing field, indicating their origin a t the tip surface itself. Figure 10 shows the line profiles in the field

98

ERWIN

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MijLLER

ion spectrum of methyl alcohol. The parent ion CH30H+exhibits the typical field broadening, while the CH30+ion line remains narrow. The latter must therefore be a product of the dissociation at the surface, and since a t high fields the supply to the central part of the tip vanishes, there must be also a surface migration from the shank. Field desorption of the species CH30+orcurs only at a higher field than the ionization of the methyl alcohol molecule in space before the tip. The failure to detect hydrogeii ions in the spectrum remains unexplained. If the field is applied only in microsecond pulses the number of gas molecules that are available in the ioniza-

33

32

31

30

29

MASS NUMBER

FIG.10. Line profiles in the ion spectrum of methyl alcohol recorded by Irighram and Gomer.

tion space before the tip becomes very small, while all the CH30+ions that have accumulated in the off period of the cycle are being desorbed almost in the same way as with dc operation (except that in the latter case the surface migration occurs in the presence of the field). I n contrast to the case of hydrogen the gases oxygen and nitrogen appear only a s molecular ions. The field strength for desorption a t least of the oxygen atom is very high; also it is doubtful that the adsorbed atom is capable of surface migration at or below room temperature (23, 24) to allow for a supply from the shank. The mass spectrum of hydrocarbons is particularly simple, with the parent i ns predominating the dissociation products by three to four orders of magnitude. In conventional electron collision ion sources the impact causes severe fractioning, so that for instance in the case of acetone there appear 19 fragments of approximately equal abundance. The use of a field ion source for mass spectroscopic analysis appears therefore promising, although there are some difficdties connected with the differences in

s

FIELD IONIZATION A N D FIELD ION MICROSCOPY

99

ionization poteiitials aiid, consequent,ly, ionization fields for thc different species. The abundance of the secondary ions rises usually with the square of the pressure, and the masses seem not to be integers. This indicates their origin in collisions after acceleration. Sometimes there are ions with an abundaiice increasing linearly with pressure, and again noninteger masses. These are metastable ions which were fragmented by vibrational excitation before the acceleration. The behavior of molecules with permanent dipoles particularly water, is also quite complex and is characterized by the occurrence of associations. Inghram and Gomer report the ions H20+, (H20)2+, (HzO)a+, and (HPO)~+ with the abundances 0.47, 0.51, 0.02, and 0.003. However, Beckey (25) arrived at the conclusion that their assignment of masses to the observed water molecules was not correct. His observations are quite illustrative of the difficulties that may be encountered in field ionization mass spectroscopy. Beckey originally used HzO, NP,Ar, and C02 for the calibration of his mass scale, assigning to them the masses 18, 28, 40, and 44. However, when NO was checked with this scale, its ion appeared a t mass 29 instead of 30. In a following investigation of other molecules with unpaired electrons such as 02,NO,, C10, C102,and Sz the main maxima appeared again one mass number too low. Since this finding was independent of acceleration voltage and pressure it was concluded that the mass scale was too low by one unit. The originally used test molecules HzO, N2, Ar, and COP had actually produced the ions H30+, N2H+, ArH+, and COzH+. This surprising result could be confirmed in the case of water by isotope substitution. If, as suggested by Inghram and Gomer, the water molecule would produce the ion H20+, then one would expect in a mixture of light and heavy water the ions HzO+, HDO+, and D2O+, i.e., three different masses. If, however, field ionization produces by a secondary process an ion &0+, then the mixture with heavy water should contain the ions H30+, H2DO+, HD20+, and D30+. The experiment actually showed four maxima. Field ionization of light water produces therefore predominantly H30+ ions. The contribution of H20+ is only 0.1%. Beckey could not observe the ions of 0 2 molecules or OH radicals that must be formed to gether with the H30+ ions, probably because the field used for the ionization of water was too low for the ionization of the by-products. More than half of the water molecule ions were associated to form H30+ * H,O, H3O+ * (H2O)2 and H30+(H20)3,and since their abundance is approximately proportional to the pressure, the association cannot have taken place in the gas phase. Because of the high polarization energy of water, the smallest traces of water vapor, even in the baked vacuum system used by Beckey, are sufficient to build up a condensed film on the tip, in which the association is going on. Also other molecules with paired electrons such a s N2, Ar, and C02 react exothermally with the condensed water film to form the

100

ERWIN

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MULLER

hydrogen association. On the other hand the ions of molecules with unpaired electrons attack the hydrogen atoms to a much lesser degree, so that the oxygen spectrum shows mostly Oz+ ions and only a few percent KOz+ ions. The reaction of the ions with water is so very disturbing because of the extremely small partial pressure that is required in the vacuum system to build up a condensed film on the tip surface. The density of the vapor near the tip surface is increased because of the polarization to P = poe

E,/kT

,

+

where the polarization energy is E, = p F >GaF2. The dipole moment of water is p = 1.84 X lo-'* dynes% cm2, and the polarizability is (Y = 1.57 X cm3. With a typical field of 1.5 X lo8 v/cm one obtains a t room temperature a pressure increase by a factor of lo".'. That means that the partial pressure of water vapor must be below lo-'* mm in order to prevent condensation a t the tip surface.

IV. FIELDION EMISSION FROM

A

METALSURFACE

A . Theory of Field Desorption and Field Evaporation In the foregoing sections the ionization by tunneling of an electron was considered for an atom in free space or a t a certain distance from a metal surface. We consider now the case of field emission of positive ions that originate from a bound state a t the metal surface, either in the form of adsorbed atoms and ions, which we call field desorption, or from the metal surface itself, in which case the term field evaporation seems appropriate. For the mechanism of these effects it is practical to distinguish between strong covalent binding on the one hand, which may be characterized by a large value for V I - 4, and weak covalent or essentially ionic binding, occurring when V I - 4 is small. I n the latter case, which applies to field desorption of electropositive adsorbates and to field evaporation as well, the mechanism (26)is a direct ionic evaporation over a field reduced energy hump, with a time constant 7

=

.,eQ/kT

(20)

where TO = l/v is the reciprocal vibrational frequency of the bound particle. The desorption energy QO without field is derived from the thermionic cycle by considering first the evaporation of a neutral atom, which requires the vaporization energy A, then the ionization which needs the energy Vr, and finally the energy 4 which is gained when the electron is returned to the metal; &o =

A

4-Vr - 4.

(21)

FIELD IONIZATION AND FIELD ION MICROSCOPY

101

Since the process of ion evaporation is actually done in a field, we have to add the polarization energy >8F"(ai- ao),where a; is the polarizability of the free ion, and a. the polarizability of the particle in the bound state before the evaporation (27). Both figures are not known quantitatively, so that not even the sign of the polarization contribution can be given. Experimental evidence will show that the polarization term is usually small, and in some cases definitely positive. Presently we disregard this term entirely. The energy hump Q that has to be overcome by vibrational excitation is reduced from QO by the effect of the field on the image force corresponding to the Schottky effect for an ion with charge n e :

-

Q = Q o - W .

By combining Eqs. (21) and (22), the desorption field is then FD = r3e-l(A V I - 4 - kT In r / r ~ ) ~ ,

+

(22) (23)

if the desorption time r is considered as the independent parameter. For field ion microscopy it is essential that field evaporation be negligible a t the field strength required for producing the image by field ionization of the gas. In Table I1 the evaporation field, as calculated from Eq. (23) for T = O"K, has been listed for the number of metals (and graphite a t the top of the table) that are most promising objects in field ion microscopy, The arrangement is made in the sequence of decreasing FD down to 288 Mv/cm only, since the more easily field evaporated metals are of lesser interest. The temperature dependent last term in Eq. (23) is fairly small sec and compared to Q. With ro lying probably somewhere between sec the temperature contribution amounts to between 0.78 and 0.48 ev a t 300°K. Data for the heat of vaporization A, for the ionization energy V I , and for the work function 9 were taken from the literature (28, 29). While the first two quantities are fairly well known for most metals, the basic difficulty lies in the large variation of work function with crystallographic orientation of the surface. Only in the case of tungsten was it possible to consider two extremes, the (111) plane with I#I = 4.35 ev, and the (011) plane with 4 = 5.99 ev (SO). An interesting difference between thermal evaporation and field evaporation stems from the fact that in the latter case the activation energy contains the sum of A and V I .As a consequence, the platinum metals, with their relatively large ionization energy rank high in the list, and even easily evaporating metals such as gold, zinc, and mercury appear not to be out of the range of field ion microscopy. In a recent paper Gomer (91) has illustrated the scheme for field desorption from the ionic ground state as give11 above by elucidating potential diagrams, of which some have been adapted here in Figs. 1 1 , 12,and 13. If

102

ERWIN

w.

MULLER

TABLE11. FIELn EVAPORATION OF IONS AT 0°K

C W

7.40 8.67

11.26 7.98

Ta Re

8.10 8.10

Tr Nb Mo Pt

7.7 7.87

4.34 4.35 5.99 4.20 5.1

14.32 12.30 10.66 11.60 10.87

1428 1052 786 930 819

6.50 6.87 6.82 5.84

9.2 6.88 7.13 8.96

5.0 4.01 4.30 5.32

10.70 9.74 9.65 9.48

792 659 648 626

Zr Be Rh Ru

6.33 3.38 5.77 5.52

6.84 9.32 7.46 7.36

4.12 3.92 4.8 4.52

9.05 8.78 8.43 8.36

569 536 495 485

Si Au Fe co

4.90 3.68 4.30 4.40

8.15 9.22 7.90 7.86

4.80 4.82 4.17 4.40

8.25 8.08 8.03 7.86

473 454 448 429

U Th V Ti

5.09 6.30 5.32 4.92

6.0 4.74 6.74 6.83

3.27 3.35 4.4 4.17

7.82 7.69 7.66 7.58

424 410 408 399

Pd Ni Ge

cu

4.08 4.38 3.99 3.52

8.33 7.63 7.88 7.72

4.99 5.01 4.80 4.55

7.42 7.10 7.07 6.69

383 351 347 312

La Hg Zn

4.33 0.64 1.36

6.51 10.39 9.39

3.3 4.52 4.31

6.64 6.51 6.44

307 294 288

+

Vr - is small, the particle will be adsorbed in ionic form, although t,he evaporation in the absence of a field will predominantly occur in the form of neutral atoms (Fig. Ila). If a field is applied, the potential curve for the ion is bent down so that desorption can occur by vibrational excitation over the Schottky saddle of height Q (Fig. l l b ) . The case of an intermediate value of V , - 9 is shown in Fig. 12. Here the ground state is atomic, but if a high field is applied the two potential curves for the atom and the ion cross each other at the critical distance r,. The desorption can be considered as a result of field ionization of the adsorbed, vibrationally excited atom at zc,followed by evaporation of the

FIELD IONIZATION IONIZATION AND AND FIELD FIELDION ION MICROSCOPY MICBOSCOPY FIELD

-6

-

-6

I

-8

I

I

I

I

-8

-

103

FXnh

I

I

I

I

I

FIG.11. (a) Energy level diagram for the adsorption of a n ion and an atom a t a metal Rurface when V I- 4 is small. Ground state is ionic, but evaporation occurs predominantly as a neutral atom (adapted from Comer). (b) The same system with a field applied. Reld deeorption of the ion occurs by overcoming the Schottky saddle Q by vibrational excitation.

-6

-

,I:: *C

I

I

1

1

'

I

M-+A+

.,

I

Pic;. 12. ( 8 ) Energy levels for arlsorptioii a t u metal surf:wc r h r n 1’1 - qi is iiitermediate. Adsorption and evaporatioll occur in the neutral crtute. (b) In a11applied field the ionic curve crosses the neutral onor field dcsorption of the ion occursover the Schottky mddle of height Q.

104 E (ev)

ERWIN

w. MULLER E

(ev) ' ,

10

10 -

8

8-

6

6-

'\\

\,Fx \

\ \

F=200 M v / m

'\\ \

\ \

4

2

a -2

-4

-f

-f

Fro. 13. (a) Energy levels when V J - 4 is large. Evaporation of adsorbate in neutral form only. (b) In an intermediate field desorption requires approximately full evaporation energy from vibrational excitation, then field ionization of the atom is possible beyond critical distance zc. PF is the field-bond interaction energy. (c) In a very high field, desorption of a covalently bound atom requires a reduced activation energy Q, followed by field ionization beyond 2.

FIELD IONIZATION AND FIELD ION MICROSCOPY

105

ion again over the Schottky hump with energy Q a s calculated from Eqs. (21) and (22). A modification of the desorption mechanism is to be made for the case of large values of V I - 4 which applies for the desorption of gases such as oxygen. Here Comer proposes the almost complete desorption of the neutral particle by thermal excitation, followed by field ionization at the critical distance xe. In Fig. 13a a hypothetical case is shown where the ionic potential curve lies way above the atomic curve. In Fig. 13b the same system is shown with a field of intermediate strength applied. The ionic curve “crosses” the atomic curve at x,. In the absence of symmetry or spin differences permitting degeneracy the curves will repel1 rather than cross, and the desorption from the atomic ground state will result either in the emission of an ion by adiabatic transition, e.g., by field ionization at or beyond x,, or the transition is nonadiabatic and the particle remains neutral following the atomic curve. The probabilities of both events can be calculated in principle by using Eqs. (1) and (9), but in practice the potential curve for the atom, depending on the interaction with the surface, is not known in sufficient detail. In a very high field as shown in Fig. 13c the intersection point x, comes so close to the surface that the binding energy Q is greatly reduced, and the separation of the two states is so great that desorption can only be expected in ionic form. Comer suggests that since Q depends on the shape of the atomic potential curve, the latter could be investigated in principle over a limited range of x by measuring the activation energy of field desorption as a function of F . As will be seen from the following description of experimental work in field desorption there are quite a few practical difficulties encountered in such a n attempt.

B. Experimental Results of Field Desorption Field desorption was discovered experimentally by Muller (7) when with the method of temporarily reversing the polarity of the field emission microscope and subsequent observation of the electron image the field range at and beyond 100 Mv/cm became accessible for measurements. An adsorbed film of barium was found to be stripped off the tungsten tip a t a well-defined field strength. The field calibration was obtained from the known emission of the clean tungsten surface, and the state of the barium film was characterized by the average degree of coverage as derived via the work function from the slope of the Fowler-Nordheim plot. The desorptioii was found to depend on the crystallographic orientation of the substrate, beginning at the high work function region near the (011) plane. A t room temperature the desorption field a t other crystal planes was considerably higher. However, when a t elevated temperatures surface migration sets in, the entire film is torn off a t the instant when desorption started a t

106

ERWIN

w.

MULLER

(011). The experimental data derived from a large number of measurements (19) are given in Fig. 14. The desorption field was applied for 3 sec. If the field exposure was extended to 300 sec a t room temperature, the desorption field was about 2% lowcr, and if the desorption was made with a single 1-psec pulse, the field was approximately 5% higher. For the interpretation of the data Muller (26) suggested evaporation over the Schottky hump of doubly charged ions. Using Eq. (23) with a value

u

'0

2

A .6 .a 1.0 DEGREE OF COVERAGE ( 8 )

FIG.14.Desorption field of a barium film from the vicinity of the (011) plane of tungsten.

of A = 5.1 ev and assuming a work function of 4 = 5.0 ev for the vicinity of ( O l l ) , to which the desorption data refer, a desorption field of F = 78 Mv/cm at room temperature, and of 44 Mv/cm a t 1200°K was calculated, in excellent agreement with the measurement a t low degree of coverage. For evaporation times of 7 = 300 sec and T = 10-6 sec, the room temperature desorption field was calculated by Eq. (23) to be 76.5 Mv/cm and 85 Mv/cm, respectively, again matching the observed data. Of course, there is ambiguity in the values of A and 4, and it is not too surprising that Kirchner

FIELD IONIZATION AND FIELD ION MICROSCOPY

107

and Kirchner (32) found agreement with Miiller's previously published data by assuming the desorption of only singly charged ions. I n his recent reconsideration of field desorption as ior ic evaporation Gomer (31) makes the interesting suggestion that Muller', data may be interpreted a s indicating desorption in the form of doubly charged ions below about 700"K, and as singly charged ions at higher t,empcrature. One can write Eq. (23) in the form

and plot the measured data as F D s versus T . From this plot, o w would expect a straight line, the intercept of which would immediately give Qo, while the slope would measure the vibration time 7 0 . In the case of barium, however, the plot seems clearly to consist of two straight sections, representing singly and doubly charged ions, respectively. The intercepts yield Q O ~ * + = 4.26 ev, and Q O ~ * + + = 10.0 ev, whose difference should be theoretically equal to YI++- 4 = 9.95 ev - 4. The correct value for 4 is again uncertain, 4w = 4.5 ev would fit better than the previously assigned value of 5.0 ev for the vicinity of (011). The heat of vaporization is A = 3.57 or 4.07 ev depending on the value of 4 used. Particularly the latter value is in good agreement with the accurately measured data of Moore and Allison (33), who find he=o.a= 3.7 ev and d A / d 0 negative (0 = degree of coverage). The vibration times 7 0 derived from the slope of the Gomer plot are sec for unexpectedly large, namely 7 X lo-* sec for Ba+ and 2.5 X Ba++. Gomer suggests temporary trapping due to overlapping of the potential curves a s a possible explanation. A general feature of field desorption of electropositive adsorbates is the increase of desorption field with increasing degree of coverage, as seen for barium in Fig. 14. Obviously the applied field must additionally balance the opposite field which is produced a t the considered adsorption site by the neighboring ions. These ions and their electric images form dipoles, the field of which at the considered site can be computed by summing up the contribution of a square array of dipoles with moments p and a lattice constant ae-%

Fdlp= 9.05,d%/n3.

(25)

The dipole moment of a single adsorbed ion as a function of e can be calculated from the experimental decrease of thermioriic work fuuctioii A 4 as measured by Becker (34), which may be approximated by the empirical formula A+

=

nBq

(26)

108

ERWIN

w.

MULLEB

(with B = 2.9 ev for barium). A sheet of 0a-* dipoles per square centimeter reduces the work function by AC#J= 2~peOa-~.

(27)

Only one-half of the dipole moment effective in producing the field contributes to A$. This gives the desorption field as a function of 0:

With an estimated density of barium ions of N = a t 0 = 1 the calculated desorption field is

FD = 78

+ 66 X 0% Mv/cm

=

2.5 X 1014 cm-2 (29)

in perfect agreement with the experimental 300°K desorption curve of Fig. 14. A thorium film on tungsten is in some respects more convenient for field desorption experiments than barium because of the better vacuum conditions that can be obtained. However, the much higher binding energy requires fields up to 350 Mv/cm, that is, 10 times the field strength used for field electron emission. As a consequence, the film cannot be produced with the usual activation process with tungsten wire containing about 1 % thorium oxide. The high temperatures required for the reduction of the oxide would cause an excessive blunting of the emitter tip so that voltages in the 50-100 kv range would be necessary for the desorption. The experiments reported by Muller were made with a thorium film condensed from the vapor from a nearby evaporation source. The field electron emission patterns were identical with the ones obtained by thermal activation of thoriated tungsten wire. Figure 15 gives the lowest desorption fields again a t the (011) vicinity of the tungsten crystal for various temperatures and as a function of the degree of coverage (0 = 1 for minimum work function). Desorption fields a t the other crystal planes are considerably higher, increasing in the same sequence (011)) (112), (OOl), ( l l l ) , and (116) as the work function of the tungsten substrate decreases. For desorption a t room temperature the dependence on degree of coverage 0 in Fig. 15 can again be perfectly matched by Eq. (28) as

F D = 243

+ 43 X 0% Mv/cm

(30)

with 13 = 1.9 ev and N = u-* = 2.5 X 1014 cm-2. The mechanism of ionic evaporation over the Schottky saddle is again proved best by plotting with Comer FD% versus T according to Eq. (24). From Fig. 16 one obtains 70 = 2 x sec by assuming singly charged ions, and Qa = 6.45 ev. Unfortunately the ionization energy Vr for thorium is not known. If

FIELD IONIZATION AND FIELD ION MICROSCOPY

109

Langmuir's value of A = 7.7 ev is correvt8,our field desorption data would give Vr = QO 4 - A = 4.74 ev if = 5.99 ev is assumed for the (011) plane of tungsten.

+

"ic m 260.

5 eoo

300'U 440 110

!!!

u.

2

2 I60 0 I-

620

a

0

ID

XI00

6ot-00

.2

A

.s

.a

1.0

1.2

DEQREE OF COVERAQE (8)

FIG.15. Desorption field of a thorium film from the vicinity of the (011) plane of tungsten.

With the field desorption of Ba and T h from tungsten, as described above, the list of quantitative work that is suitable for comparison with theory is exhausted. There are a number of qualitative observations of a

1

500

T PK)

1000

FIG.16. Plot of square root of desorption field versus temperature for thorium on tungsten.

more exploratory nature that may be added here, whiIe the experimental work on the closely related field evaporation can be discussed in detail only in Sec. V, after an introduction to the techniques of field ion microscopy. Field desorption experiments were made by Muller with Li, Na, Te, 02, and Nz films, all on tungsten, and also with individual phthalocyanine

110

ERWIN W. MeLLER

molecules, always usiiig the field electron microscope. The alkali metals are most easy to desorb in the range of 60 to 80 Mv/cm, but no well-defined data could be taken because of the impossibility of eliminating gaseous contaminations at the tip surface. A vacuum deposited tellurium film that was spread out evenly by surface migration begins to be desorbed at 280 Mv/cm. A nitrogen film formed by adsorption a t room temperature and about lop6mm field desorbs in two steps. A top layer disappears a t about 300 Mv/cm, and the strongly bound chemisorbed layer, which is not very clearly recognizable in the field emission pattern, seems to come off only at fields near 450 Mv/cm. Some more detailed, but far from conclusive studies were made with oxygen films (24). If the clean tungsten emitter of a field electron microscope is exposed to about lo-'' mm-of-oxygen for a period of some minutes at room temperature, the effective over-all work function of the practically saturated film as measured by the slope of the Fowler-Nordheim plot is 6.6 ev. The surface appears grainy because of some corrosion of the tungsten surface. The first slight change in the pattern and a decrease of work function by one tenth of an electron volt occurs if a t least 300 Mv/cm are applied for a few seconds. The grainy structure in the pattern disappears above 400 Mv/cm, and it takes 430-450 Mv/cm to remove the chemisorbed oxygen film almost completely, so that the original slope of the Fowler-Nordheim plot, corresponding to a work function of 4.5 ev, is re-established. I n this field the tungsten surface itself begins to field evaporate a t room temperature, and the resulting tip geometry differs from the previous one that was obtained by thermal surface migration, so that accurate measurements are difficult to make. Field desorption experiments were also made with individual phthalocyanine molecules on tungsten and platinum tips. These molecules, and a number of other organic compounds, form very strange doublet and quadruplet patterns in a field electron microscope (35). In spite of the efforts of a number of investigators the mechanism of the image formation is still mysterious (11). Melmed and Muller (36) observed the field desorption in order to find out if the molecules would be bound tightly enough to be depicted with higher resolution in a field ion microscope. The desorption experiments were made with the substrate cooled to liquid nitrogen temperature. On tungsten some molecular patterns of Cu-phthalocyanine survived a field of 200 Mv/cm. The average desorption field for 43 doublets which were individually adsorbed was 154 Mv/cm, and the average for 15 quadruplets was 126 Mv/cm. On platinum, a few molecules survived a field of 210 Mv/cm, and most of them 160 Mv/cm. A most interesting system was studied by Kirchner and Ritter (37) when they investigated qualitatively the desorption of KC1 in a field emission microscope with a tungsten tip. A presumably monomolecular KC1

FIELD IONIZATION AND FIELD ION MICROSCOPY

111

film emits a t a voltage about 20% lower than does the clean tungsten tip. If now a positive voltage of only 65% of the tungsten emission voltage is applied (about 21 Mv/cm), potassium ions are stripped off, so that the average work function of the remaining film is higher than on a clean W surface, probably because of an excess of chlorine. From a negative tip, on the other hand, it is possible to strip chlorine ions, provided the KCl film is two to three monolayers thick. Such layers have a higher intrinsic work function, so that initially they do not give any field electron emission if the voltage is raised to 80% of the voltage for clean tungsten emission. Then suddenly a very bright emission pattern appears, and the necessrtry voltage is only 34% of the clean tungsten voltage, indicating a very low work function. This change in effective work function is certainly due to field desorption of negative chlorine ions, leaving behind a film with a n excess of potassium. From such a film potassium ions can again be desorbed by applying a positive field of 64 Mv/cm. While field desorption is usually detected by changes in the electron emission pattern or the total current of the tip, a new and quite sensitive method of detection was introduced by Herron (38)with the use of a radioactive adsorbate. a-Active curium (Cm2d2)on a tungsten tip was found to begin to evaporate at 1600"K, if no field was applied, while in the presence of a high positive field (of unreported magnitude) the evaporation rate was 5-10 times larger as measured by the activity appearing on a collector plate opposite the tip. A technique particularly suited for exploratory work on field desorption consists of the operation of a field electron microscope with ac in such a way, that the positive voltage peak is used for field desorption, while the negative peak produces the electron image, the intensity of which may be limited by a resistor (7). An improvement of this method was developed by Cooper and Miiller (39) by using dc-biased ac according to the schematic diagram in Fig. 17, so that the amplitude of the desorption voltage and the image voltage can be adjusted independently. The high voltage transformer and the dc power supply are fed from the same hand-operated variable transformer A, while the auxiliary transformer B is used to adjust the dc bias so as to keep the image intensity essentially constant while the desorption peak voltage is slowly and continuously raised. The gradual progress of field desorption by successively peeling off the adsorption film from the different crystal planes can be observed visually, photographically, or by taking motion pictures. The desorption peak voltage is continuously nieasured with an electrostatic voltmeter in connection with a diode. The image producing electron emission as well as the desorption take place during very small fractions of the cyale only. A typical example has been calcrilntcd for the fic.lt1 cv:~p~r:iti~ii of a tmigstcn tip giving :L dc electron emis-

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ERWIN W. MULLER

sion of 10pa a t 2500 v. Such a tip has a radius of about 1000 A and emits electrons at a field of 41.5 Mv/cm. In the example, the ac voltage is assumed to be 11,500 v rms, and the dc bias 13,750 v. This voltage will then produce a peak of 500 Mv/cm over-all field strength, sufficient for field evaporation

A

FIG. 17. Schematic diagram of dc-biased ac circuit for pulsed field desorption and observation of the electron image of the tip.

of tungsten at room temperature. Actually the local field at the evaporation site at a lattice step is larger by a factor of approximately 1.4 and a peak field of 700 Mv/cm is therefore used for calculating the evaporation rate. The number of tungsten ions coming off the tip area 2rr2 during the time eIement dt is 2nrzNdt n =, 7

where N is the number of evaporation sites per square centimeter, and 7 the evaporation time of a single surface atom, which is according to Eqs. (20) and (22) r = rOexp

kT

In Fig. 18 the resulting number n of field evaporating ions per degree phase angle (dt = 4.44 X lovs sec) is given. The evaporation rate drops to 1% of the peak value when the voltage is only 2.2% below the peak, therefore a quantitative study of field desorption or field evaporation at relatively low temperatures can be made by simply measuring the peak voltage only. More accurate data may be obtained by using square wave rather than sine wave ac.

FIELD IONIZATION AND FIELD ION MICROSCOPY

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This ac field desorption technique seems to offer considerable advantages for field electron microscopy. Previously the cleaning of a field emitter tip rould only be done by heating, with the undesired result of excessive tip blunting due to surface migration. Quite often a surface contaminant can not be removed a t all by heating the metal tip, or the concentration of bulk impurities such as carbon or silicon may be increased at the surface. Using controlled field desorption the contaminating surface layers may simply be peeled off. For observing a clean metal surface the vacuum conditions KV W

-I 0

z

4 v)

-10

-

1

J

- 10 w

FIQ.18. Tip potential and calculated ion and electron pulses a8 a function of phase angle.

are not very stringent since the surface is being cleaned in each ac cycle, so that at a given pressure only a very low degree of coverage will be built up during the short time of a fraction of the cycle. Field electron emission patterns of difficult to handle metals such as nickel, iron, copper, and silver were produced, and it was also possible to obtain electron images of tips that were made from cold-worked metals without being subject to annealing. A difficulky of this method when applied to the very soft metals may be encountered in the fatiguing of the specimen by the continuous cycling of the mechanical stress of magnitude P/& acting at the surface under observation, The ac technique is also promising for the investigation of the dynamic equilibrium of adsorption films in an external field. Cooper and Muller

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~ i i t ~ ~ ~ i ~ lsl~i ilt lii csv ~ l ~iitrogc~ii ~ t ~ ~ l ) O I I iritliiini t i p h :it, 78°K. ' l ' h ~ tlesorplioii field WLS ahou t 370 hilv/cwi. 'I'hc nitrogcn films forxntd a sharp i~oiiiidary w h i d i c*oiildtie n i o v r d from t,hc rrrst of thc tip towards the shaiik when the field was increased. Upon lowering the field, the boundary moved back to the center as the nitrogen film was replenished by surface migration from the shank. The operation of an ion microscope using field desorption by mierosecond pulses was suggested by Muller (16, 40). The field emission mieroscope is filled with an easily adsorbed gas, such as nitrogen, and the repetition rate is chosen so that in the time between two pulses the tip surface is covered up to an appreciable fraction of a monolayer. The pulse peak must be high enough to desorb this layer during the short pulse time. The ratio of pulse width to repetition time must be very small in order to keep low the background which is produced by autoionization far above the tip surface during the pulse time. With nitrogen of 1p one can build up and tear off some 1000 monolayers per second, which should give a recognizable image on the screen. The practical results were not encouraging. Besides the low intensity there are other difficulties arising from surface migration of the adsorbate, desorption during rise time, and fatiguing of the tip nietal by the cycling stress of the field force. From all the work on field desorption described above it appears that only a few simple cases have been investigated so far. However, the results obtainable from field desorption measurements promise to present a valuable supplement for the investigation of adsorption phenomena by providing a new range of data beyond the conventional measurements of activation energies for surface migration and thermal desorption. Results about and the significance of the closely related effect of field evaporation shall be discussed later after the technique of field ion microscopy has been described.

V. FIELDIONMICROSCOPY

A . Theory of the Field I o n Microscope 1. Introduction. When the conditions affecting the resolution of the field electron microscope (Fig. 19) were considered (41), it was evident that diffraction due to the wave nature of the electron and particularly the fairly large (some tenth of an electron volt) tangential velocity component of the emitted electrons placed an insurmountable limit on the resolution of this appealingly simple device. The disturbing tangential velocity component of the field-emitted electrons cannot be reduced significantly by external means, such as cooling the emitter. The electrons inside the metal have a Fermi-Dirac velocity

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distribution, and although the barrier penetration prefers the emission of electrons with a large velocity normal to the surface, the tangential velocity components will be conserved in the emission act. Under most favorable conditions the diameter of the scattering disk in the image corresponds to 10 A on the object, so that there is no chance of seeing the atomic lattice. This limitation does not exclude the possibility that individual atoms or small molecules projecting from the tip surface in sufficiently wide spaced

FIG.19. Schematic diagram of field emission microscope showing exaggerated tan gential velocity component zit of emitted particles causing wide scattering disk.

positions may be visible, a claim originally made by Miiller (5, $5) and recently becoming more generally accepted by other workers in field emission microscopy [see review articles by Dyke and Dolan (6) and by Good and Muller (11)l. In order to improve the resolution it seems then quite an obvious idea to look for a process of image formation with the help of positive ions. The shorter de Broglie wavelength associated with ions would alleviate the diffraction difficulties, and disturbing tangential velocity components may possibly be reduced by cooling since the particles obey Boltzmann statistirs. The basic problem is the supply of suitable ions, as they cannot travel through the lattice of the emitter in the way electrons do. A process such as the diffusion of protons throtigh palladium is murh too slow. With the typical tip area not much larger than 10-lo cm2, a current density of th(b order of 1 amp/cm2 is necessary for producing a faint image on the scrcen. The desorption for instance of a barium film seems not to be very promisiiig either, since the stripping of a single monolayer would yield a cnrrent pulse

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of only about coul, many orders of magnitude too small for visual observation. Looking for a practical way of replacing an adsorption layer very quickly in order to be able to repeat the desorption act many times in a second, the author came upon the idea of supplying the adsorbate from a gas filled into the microscope. The ionizing field at the tip is kept on all the time, thereby also enhancing the gas kinetic supply by the attraction of the polarized molecules. The ions travel to the screen through the gas of such a low pressure that the free path is of the same magnitude as the tipscreen distance. The first successful experiments were made with hydrogen (8), and although the occurrence of field ionization in free space far above the tip surface at the highest voltages was realized, the origin of the ions in the sharp image field range of 200 to 280 Mv/cm was thought to be directly in the adsorption layer a t the surface. Three years later, by the independent and simultaneous work of Kirchner (14), Inghram and Gomer (15),and Muller (22) the tunneling mechanism of field ionization a t the surface and the condition of a minimum distance to lift the ground state of the ionizing molecule above the Fermi level became evident. The crucial experiment of operating the microscope with helium ions from a tip cooled to very low temperature, which was already suggested in the first paper on the field ion microscope (a), and which established the full resolution of the atomic lattice, could be performed not earlier than in fall 1955 (4.2). The real reason for the great improvement to be expected by the cooling, however, had been realized only shortly before (22).It is connected with the temperature dependence of the accommodation coefficient between the helium gas and the tip metal. 2. Resolution of the Field I o n Microscope. At the present time no complete theory of the resolution of the field ion microscope can be given. Usually the resolution 6 is defined as the smallest distance between two object points that can be separated in the image. One may also define 6 as the object-side diameter corresponding to the smallest scattering disk on the screen. In our case the object points are atoms of a diameter comparable to the resolution, and these lie more or less deeply imbedded between other surface atoms, thus making the local field distribution in the space immediately around and above the test atoms dependent upon the specific arrangement. It is practical to distinguish between the resolution of two adjacent atoms projecting from a smooth surface or from a lattice step, which we may call point resolution; between the resolution of the members of an atom row arranged in the form of a pearl chain, which may be named chain resolution; and finally the resolution of individual atoms forming a square or triangular array within a complete net plane, which we call plane resolution. The difficulty of resolving adjacent atoms increases in the given sequence of the arrangements.

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The resolution that may theoretically be obtained with a point projection microscope using ions is easily calculated by finding the image size Ay. on the screen of an object of diameter Ayt on the tip and minimizing Ayl with regard to Ayt. The resolution 6 is then the minimum image size Ayamin divided by the magnification M = R/@ ro. Here R is the tip-screen distance and B the geometrical image compression factor taking into account that the emitter is not a freely suspended sphere of radius To, but a spherical cap on a cone-shaped shank. For some simple geometries /3 could be calculated, but it is easier to measure it in the image of a crystal tip as the angular demagnification factor that determines the ratio of the true angle between two crystallographic directions and the angle between the image of these two poles on the screen as seen from the tip. Usually B lies between 1.5 and 1.8. There are three contributions to the broadening of the ion beam through a spot on the screen. Straight forward projection magnifies the spot to

-

Al/rmin = MAY:.

(33)

The diffraction effect can be described by the Heisenberg uncertainty principle. As the ion of mass m passes through the object region Ay,, the uncertainty of its y momentum is a t least h/2Ayt, and its corresponding lateral velocity component is h/2mAyl. Because of the concentration of the field near the tip the ion travels most of its way to the screen with essentially the final velocity as determined by the total applied voltage V , so that the travel time is

=R

d m m .

(34) The contribution to the image width of the lateral uncertainty velocity or diffraction is then 1

In the act of ionization the lateral motion of the original gas molecule is preserved. The kinetic energy of the gas molecule arriving a t the tip is Jdmv' = 3.jkT

4->$xP,

(36)

and the corresponding velocity vector is usually close to normal t o the surface since a t room temperature or below the dipole attraction term in Eq. (36) is much larger than the thermal energy. However, since a detailed image can only be expected if the ionization is made near to the surface, e.g. just beyond the critical minimum distance x, set by the condition that the ground state of the gas molecule lies above the Fermi level, the optimum field strength is so low that only a minor part of the arriving molecules will

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MVLLER

he ioiiizcd at, their approach. Most of the molecules will pass the eonc of forbidden ionization within z,, hit the surface and bounce off as neutrals. Because of the random reflection at the atomically rough surface the rebounding molecules will have on the average a large tangential velocity component vt, which is determined by provided the accommodation coefficient of the gas to the substrate is small, as is true for helium on tungsten a t room temperature. The random direction of the rebounding molecules, compared to the almost normal incidence at the arrival, makes the rebounding molecules stay twice as long in the zone of high ionization probability just beyond xC. Therefore, the image consists of two thirds of ions from rebounding molecules, and their large tangential velocity component vt determines the resolution, by contributing the amount Ayv

=

2tijt.

(38)

The square of the total diameter of the scattering disk on the screen is now given by summing up the squares of the contributions in Eqs. (33, 35, 38) :

The minimum of this expression with respect to yt is

and the resolution 6 is obtained by dividing by the magnification

-

2rph

+ aF2) + n2p2r2(3kT 8e V

6 = J m .

(42)

At not too low temperatures, where the condition of a small accommodation coefficient is fulfilled in all practical cases, the first term, based on the uncertainty principle, can always be neglected. Using Eq. (16) for the relation of field and applied voltage, we obtain then 6 = 40.6r(3kT

+ aF2)/eF.

(43)

The only independent variable in this equation is the tip radius, since the best image field has a definite value for each gas. Noticing that for all gases aF2 is much larger than the gas kinetic energy a t room temperature, no

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improvement can be expected from working at lower temperatures, e.g. as long as no accommodation is taking place. I n Fig. 20 the resolution as a function of tip radius is plotted for He, H2, A, Xe and, for comparison, also for the operation as a field electron microscope according to the resolution formula derived by Good and Muller (11).Experimental results shall be discussed later, but it may be stated here that they are in agreement with the calculation above. Considering the polarization energy at the best

.2

.I

I

too

moo

.

.

,

,

K)ooo

TIP RADIUSA

FIG.20. Theoretical resolution of field emission microscope as a function of tip radius when operated with electrons, various gas ions a t room temperature, and with helium ions a t liquid hydrogen temperature.

image field as a given value for each gas, the resolution is proportional to F-35. The improvement of resolutiori a t smaller tip radii cannot be much utilized since the depicted area decreases with r2, and the image intensity with a t least the third power of the radius. Equation (43) describes the potential resolution by which the field distribution and the possibly inhomogeneous density of the rebounding atoms in the ionization zoiie can be depicted. The field profile produced a t the edge of the ioniziltion zone of distance zcis still a fairly true replica of the geometrical profile of the real surface, but the details fade out within a few angstroms further away. The ions originating in this more distant region will therefore not contribute very much to the image detail. Most favorable conditions will prevail where the local field and with it the ionization probability drop more rapidly with the distance from the surface. This will be true especially

I20

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MULLER

al)ove small projechg object,s such as iiidividual atoms git,ting on top of a densely packed net plane. The field drops off by a lesser degree above chains of atoms on such planes, or above the steps formed by net plane edges. Thus it is necessary to define the object on which the resolution is measured as mentioned at the beginning of this section. 3. The Low-Temperature Field Ion Microscope. As soon as the large lateral velocity component of the rebounding gas molecules was recognized to be the main cause of limitation of the resolution it was easy to remedy the situation by cooling the tip. If the temperature is low enough to accom+$aF2 modate the impinging gas molecules of average energy 3$kT,,. completely to the tip temperature, the molecules will re-evaporate with an energy of ?$kTti,. The ions made from rebounding molecules will then have a tangential velocity vt = (2kTtip/m)" only, permitting a much better resolution. From Eq. (41) we obtain

+

and by putting in numbers (with F in esu) and using Eq. (16)

=

1.5, r in centimeters, m in grams,

+

0.45 lo-'' ,sr~sc " 0.52

-.Tr F

(45)

Numerical values are plotted in Fig. 20 for the most important case of helium a t 450 Mv/cm and the temperature of liquid hydrogen, 21°K. The contribution of the uncertainty principle term increases from M a t 10,000-A tip radius to 35 a t 100-A tip radius, but it seems that this effect cannot be observed directly because the experimentally obtainable resolution does not go down to these small dimensions. There are several reasons for this, the most important one being that the edge of the ionization zone is located a t xc 2 4 A above the real surface. Moreover, the efficiency of ionization fades out only slowly with increasing distance, and we have also to consider that the helium atom itself has a diameter of about 2 A. The situation with the much smaller hydrogen atom is not quite clear since i t is not known to what extent the dissociation of the Hz molecule occurs before the ionization. Mass spectroscopic analysis of the field ions from a tungsten tip a t liquid nitrogen temperature showed equal amounts of protons and molecule ions. The accommodation of the impinging molecule to the emitter surface temperature has an interesting consequence. If at a relatively low field strength the rebounding molecule has not lost its electron while it passed

FIELD IONIZATION AND FIELD ION MICROSCOPY

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through the ionization zone it cannot escape entirely from the surface. The molecule is rather pulled back by the inhomogeneous field acting upon the induced dipole. At a low field strength where the probability of ionization during a single pass through the ionization zone is small, the molecules will perform a hopping motion until they are ionized (43). A t very low fields the trapped molecules will eventually escape by diffusing towards the shank, where the field drops fast. The average hopping height can be calculated as follows: The force on the moleculur dipole in the inhomogenous field above the tip surface is I’ = cYF(dF/dr). Sett,ing the energy in the vertical direction of the re-evaporating molecule equal to kT,;,, and assuming the field gradient above the cap of the cone-shaped tip to be smaller than around a free sphere, for instaiicc by assuming the empirical relation

F = Fo ( ) % J then the average hopping height is

Numerically one obtains for a typical case with helium a t T = 2I0K, cm3,and F = 450 nlv/cm, an average hopping cm, a = 2 X ro = height of 4.8 A, e.g., just about the distance of the edge of the ionization zone, which is according to Eq. (7) zc = 4.25 A. It may be favorable for the resolution to adjust the tip tcmperature to a given tip radius in such a way that the crest of the hopping orbit coincides approximately with the inner edge of the ionization zone, since the not yet ionized molecule will stay relatively long in the region where the surface details are most pronouiiced in the local field profile (44). The hopping height of the individual gas molecules varies of course in a wide Maxwellian distribution, and an image can also be formed if, for instance, a t liquid helium temperature the average hopping height is much smaller than xc. Local variations of the field gradient due to projections on the tip such as atom clusters or lattice steps will not only change the hopping height but also influence the diffusion flow of the hopping gas molecules and give rise to intensity variations as a result of the inhomogenous supply. The confinement of the hopping gas molecules in a layer near the edge of the ionization zone is particularly favorable for the image contrast. Above a close-packed net plane the differentiation in the field profile fades out very quickly with increasing distance from the surface. Model measuremerits in an electrolytic trough show the field ripple 156 atom diameters above a net plane of a close-packed sqiiare array to be only 1%. By using Eq. ( 5 ) one can estimate that, for 1% field diflerence the ionization prob-

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ability in this region will be different by 30% in the case of helium. Less densely packed arrays of atoms will allow much more contrast. The thermal accommodation of the incoming gas molecules probably requires the time of only some 10 to 100 vibrations, during which the energy of the molecule is transferred to the substrate. When a t lower temperatures the sticking time increases further, the degree of coverage in the resulting adsorption layer may become appreciable. If such a layer is highly mobile, as may be the case when a helium ion microscope is operated with the tip a t liquid helium temperature, one might expect a blurring of the image due to the smearing out of the local field profile by the ever changing dipole fields of the adsorbate. When the microscope is operated with a gas that is strongly adsorbed such as hydrogen, the adsorbate will tend to fill in the lattice steps and the recessed gaps between the atoms on low-density net planes, where it will find sites with a maximum number of nearest neighbors. The result will be a flattening out of the field profile in the ionization zone and a loss in contrast and resolution. It is necessary to consider briefly the possible effects of space charge on the resolution of the microscope. Space-charge limitation of the ion current itself is not a problem because of the very high field at the emitter. Any influence of space-charge limitation should be most noticeable near the emitter, where the ions are still fairly slow. Dyke and Dolan (6) have shown that space-charge effects at field electron emitters become noticeable at current densities of the order of magnitude lo8 amp/cm2. For ions, spacecharge current density will be smaller by the ratio of the square root of ion mass and electron mass, which is a factor of 86 in the case of helium. Space charge is therefore expected to influence field ion emission at a current density of lo8 amp/cm2. However, with the presently used gas supply to the tip the current density actually cannot he made much larger than 100 amp/cm2, if the resolution should not be reduced by gas scattering. The next question would be whether there is not, at least, a small spacecharge effect on the resolution due to the mutual repulsion of the ions on their way to the screen. An estimate can be made as follows: A t a current density of 100 amp/cm2 from a typical tip of ro = cm there will be one ion leaving the tip surface every lo-" sec on the average. During this short time element the helium ion will travel to a distance of 65 tip radii until the next ion originates near the surface. This distance between the two ions will continuously increase because the one that has started earlier is always faster. The Coulomb field of an ion a t a distance of 6.5 X 10-4 cm is only 0.35 v/cm, and since this field is also acting in essentially radial directions with' respect to the tip center, the mutual lateral repulsion of the ions can be completely neglected at the presently used current density levcl.

FIELD IONIZATION AND FIELD ION MICROSCOPY

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The current, density could be increased by providing a greater supply through increased gas pressure. The limitation is set by the collision of the gas molecules with the ions on their way to thc screen. This interaction can be calculated as single scattering by considering the lateral deflection of an ion due to its attraction to the dipole which is induced during the short time of near approach by the Coulomb field of the ion. This attraction drops with T - ~ , and there is also an r6term due to a charge-induced quadrupole Contrary . to the usual defienergy and to London dispersion energy (14) nition of a collision we are here only interested in very small deflection angles, since the angular aperture of a beam coming from an individually depicted surface atom is sometimes as small as two minutes of arc. Unfortunately, classical mechanics is not applicable for the description of very small angle scattering. Practically, because of its low polarizability helium is again the most suitable gas. The influence of scattering in helium becomes noticeable above a pressure of 1 . 5 a~t a tip-screen distance of 10 cm, while in hydrogen with its larger polarizability the loss of resolution already begins below 0 . 5 ~ .

B. Experimental Procedures 1. Microscope Design and Operation. Field ion microscopes without emitter cooling are obsolete. However, some of the early observations are still of interest. The first instrument (8)was built like a conventional field electron microscope and had a polished doughnut-shaped accelerating electrode near the tip in order to reduce the field in the screen region, to prevent a gas discharge breakdown and field electron emission from the negative electrodes. Hydrogen was used at a fairly high pressure of 3 to 6 p in order to get more image brightness. The diameter of the finest image spots on the screen was estimated to correspond to 2 to 4 A. However, from the improved calculation of the relation between tip radius and voltage for field electron emission by Drechsler and Henkel (46) it must be assumed that the magnification used was actually smaller by a factor of 1.5 and the image spot diameter amounted accordingly to 3 to 6 A. Atomic distances could be resolved a t protrusions such as the central (011) net plane edge of an annealed tungsten tip, or the built-up ridges on a tungsten tip which was annealed with adsorbed carbon. Another way to produce resolvable surface details was found when the tip was heated to 500 to 600°C in the presence of a field of 200 Mv/cm. Under this condition the tip crystal dissolves slowly by field evaporation, whereby net plane edges several atomic diameters high collapse as essentially concentric rings of some 20-A width. This method was later used extensively by the author’s former co-workers Drechsler, Pankow, and Vanselow (47)a t the Max-Planck Institute in Berlin. Without this artificial build-up of the surface the hydrogen ion microscope cannot

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show mrivh atomic detail at, room t,empernt,urc, in agreemciit, with the (hcorctically expected resolution [Eq. (43) aiid Fig. 201. The same limitations on the resolution owur for the operation of the microscope with other gases. Drechslcr and Pankow (17) rcportcd the use of hydrogen, oxygen, mercury, cesium, argon, neon, and helium, claiming that they succeeded in getting pictures with a resolving power of 3 A, as a result of their attempts to increase the resolution. However, the authors show only a diagram of Muller's 1951 hydrogen ion microscope, without giving any indication as to what their improvements in the technique were. Since the presented photographs do not support their claim, there seems to be no reason for questioning the theoretical resolution limit as given by our diagram Fig. 20. Drechsler and Pankow also showed an ion image of a tungsten surface covered with the decomposition products of phthalocyanine, which they thought was made by cesium ions. However, the tip radius of 550 A according to their scale and the voltage of 12 kv suggest that the over-all field strength was as high as 250 Mv/cm. This means that the image was actually produced by ions from a gaseous contamination having an ionization energy of about 15 ev. Using Eq. (5) one can easily compare the field strengths necessary for approximately equivalent ionization conditions of different gases, since V z w / F should be a constant. Assuming for helium an image field of 450 Mv/cm, one obtains the image field for other gases as

F , = 3.7

VIw

(48)

(Fi in Mv/cm, Vr in ev). Applying this formula to cesium one would except an ion image a t 27 Mv/cm. It appears that the authors failed to observe this low field image, which was probably very weak because of the low cesium pressure, low voltage, and low screen efficiency. A high resolution is not to be expected anyway. Mercury gives also an extremely weak image. While Drechsler and Pankow claimed a very good resolution in this case, it is this author's experience that the resolution a t a pressure of 2j.4, corresponding to a saturation temperature of the vapor of 35"C, is very poor. about 8 A for a tip of radius 1000 A, as can be expected theoretically from Eq. (43). The first traces of an image representing small protrusions on the tip appeared a t about 80 Mv/cm, and the best image field was 140 Mv/cm. The picture was all blurred due to ionization in space at 150 Mv/cm. These fields were, as usual, calibrated by measuring the voltage for field electron emission of the same tip before the mercury was distilled into the microscope tube, and by using the current density versus field strength relation from the Fowler-Nordheim equation (11). The measured image field of 140 Mv/cm for mercury agrees well enough with Eq. (48) from which 125 Mv/cm is expected. The pump is usually shut off during the operation of the microscope,

FIELD IONIZATION AND FIELD ION MICROSCOPY

125

and this sometimes results in poor vacuum conditions. Drechsler and Pankow (17) observed occasionally argon ion emission that came from narrow bands and rings which were oriented with respect to the crystallographic structure of the tip and increased in diameter, when the voltage was raised. These rings can be interpreted as zones of locally enhanced field strength as a result of the piling up of adsorbed material by fieldinduced surface migration. Similar adsorption rings were observed later by this author with mercury ions under poor vacuum conditions. Since the field that was necessary to move these rings was as high as 400 Mv/cm, it may be concluded that the adsorbate was a tightly bound oxygen film. Helium ion images of tungsten surfaces cleaned by field evaporation at room temperature were first obtained by Muller and Bahadur (16).Up to 10 concentric rings could be counted around the (011) plane, and from the geometry and the approximately known tip radius it was recognized that these lattice steps of equal height (2.24 A) form a perfect topographic map of the emitter cap with the steps as contours I t was suggested that this map be used for obtaining a more accurate field calibration of the tip in order to check more closely the Fowler-Wordheim theory of field clcctron emission. Although the operation of a microscope with helium ions from a strongly cooled tip was suggested early (a), the actual experiment was only carried out in 1955 when the effect of the velocity of the rebounding gas molecules on the resolution was realized (9, 40, 42). The technique used in field electron microscopy of immersing the entire microscope into the cooling liquid is not practical because of the more stringent optical requirements for the undisturbed observation of the finely detailed image, which would not be easy through the double glass walls of dewars or through the bubbling surface of the coolant. Therefore, all the designs used so far are based on the principle introduced by the author of supporting the tip on metal leads that are sealed in the bottom of cold finger inside the microscope envelope, while the screen is a t room temperature. If the cold finger is cooled with liquid hydrogen or liquid helium, it is quite economical to have a doublewalled cooling mantle with liquid nitrogen around the cold finger, and to have it extended as far down below the tip as is possible without vignetting the image on the screen (Fig. 21). This mantle will also cool the major part of the gas molecules traveling towards thc tip, thereby increasing the supply function and the image intensity. For simplicity microscopes were also used that had only a eheet metal cylinder of silver, copper, or aluminum wrapped tightly around the cold finger of glass (48).With liquid nitrogen cooling this sheet-metal mantle increased the image intensity by a factor of three compared to the operation of the same tip a t the same temperntiire with the cold finger aIoii(~.Another tiihe was desigiied with two out-

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side dewars (44), so that the hydrogen-cooled finger was surrounded first by the hydrogen-cooled part of the vessel, and this again was surrounded by a liquid nitrogen mantle. If the temperature of the gas supply was raised by replacing the liquid hydrogen in the intermediate vessel by liquid nitrogen (with the cold finger still filled with liquid hydrogen), the image intensity dropped to 60% of its previous value. All these tubes were completely sealed and were baked out and processed with the conventional procedure

-------+

HIGH VOLTAGE

FIG.21. Low-temperature field ion microscope.

of high-vacuum techniques. This is particularly necessary for work with all gases other than helium, and special precautions have to be taken t o insure that no contaminations are introduced when the tube is filled with the gas. Actually, this is quite a difficult task if one wants the emitter tip to stay perfectly clean even for a rclatively short time. The partial prcssure of undesired gases must be in the ultra-high vacuum range, which can be achieved in some cases by using a molybdenum, zirconium, or titanium getter, or by providing a cooled charcoal or alumina trap. The situation is entirely different if one wants to work only with helium ions, which give the best resolution, but also restrict the specimen to the more refractory metals. Helium has the highest ionization energy of all elements (V, = 24.47 ev), and if we disregard neon (VI = 21.45 ev) which caiinot be considered as a contaminant because of its small adsorption

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energy and the low abundance, all the other contamination gases which may occur in a vacuum system have ionization energies near or below 15 ev. According to Eq. (5) these gases will be ionized at about one half of the field that is necessary for helium. The contamination gases will therefore ionize on their approach to the tip in the space above the surface approximately where the field is between 200 and 300 Mv/cm. Assuming a field distribution according to Eq. (46) this ionization zone for the contaminations will extend from about 2ro to 1.570above the center of the tip of radius ro. After ionization has occurred the contamination ions will be carried away towards the screen, where they give a negligibly weak background light, while the tip surface after an initial cleaning by field evaporation will remain untouched by any contaminant as long as the high field is kept on. This unique feature lessens the conditions for vacuum cleanliness so much that no-bakeout demountable vacuum systems with conventional gaskets or greased ground joints can be used as microscope tubes. A typical design that is being used for most of the present work is shown in Fig. 22. The simple cold finger has a conically ground part over which a sheet-metal cone is slipped with good thermal contact to cool the tip environment. This cone may have a slit or a hole a t the level of the tip for observation or for allowing the deposit of material from an evaporation source in a side arm of the tube, or for irradiating the tip from an ion source or a-particle source outside the cone. The specimen tip is mounted on a metal wire loop and this loop is plugged into receptacle leads which are sealed into the bottom of the cold finger. This allows quite convenient replacement of a specimen. The lower part of the microscope tube including the screen bottom has a conductive coating of transparent tin oxide, which is kept at ground potential, together with the metal cone, while the positive high voltage is connected with the tip. The glass wall coating and the screen may also be made more negative than the cone for the purpose of postaccelerating the ions for greater image brightness. Although neither exact calculations nor accurate measurements have been made it can be estimated that under normal operating conditions the tip temperature is not more than one degree above the temperature of the coolant in the cold finger because of the good heat conduction of the tungsten loop at low temperature. If, for instance, the loop is flashed to 2000°K while the cold finger is filled with liquid hydrogen, it takes not more than about 10 to 20 sec to reestablish full image sharpness, i.e., a tip temperature below perhaps 40°K.Measurements of the tip temperature can be made by determining the resistivity of the tungsten loop. For this purpose, two more leads are necessary in order to measure a t a given heater current the potential drop along the central, fairly uniformly heated part of the loop. Under certain conditions the true tip temperature in the very low-temperature region can

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be determined by measuring the rate of field evaporation, as shall be discussed in the next section. For a finer temperature control it may then be better to mount the tip on a nichrome loop rather than on tungsten with its extremely small resistivity a t very low temperature. Little scientific background can be given for the mostly empirical art of producing fine emitter tips. Compared to field electron emitters it is usually desirable to have smaller radii, because for a given tip the voltage for a helium ion image is 10 to 12 times larger than for electron emission. A small radius also gives enough magnification so that the potential resolu-

FIG 22. Field ion microscope preferably for helium ions. Ground joint and plug-in emitter loop permit eaay tip replacement.

tion is not lost in the grain of the screen and the photographic material. Usually the tips are made from wires preferably 2 to 6 thousandths of an inch thick, or from rods of the same dimensions, by a suitable chemical or electrochemical etching processes. Tungsten or carbon filaments can be etched in a hot gas-oxygen flame. After this procedure the carbon tips are often found covered with microscopic transparent beads, probably SiO,, which can be removed by hydrofluoric acid. Table 111 of metals and suitable etches is not complete, and there may be some better methods in different cases. Thcre is a largc variation in the treatment possible and sometimes nccesmry t,o t,ake care of the different behavior of a specific

FIELD IONIZATION AND FlELD ION MICROSCOPY

Metal

W Ta

Re Ir

Nb Mo Pt

Zr Be Rh Si

Au Fe co Ti Pd Ni cu Zn TiN

Eti!hnnt.

hlolten N:tN02 Molten NaNO, Aqu. sol. NaOH Molten NaN0, or KOH Conc. HNOa Molten NaCl Aqu. Sol. KCN Molten NaNO, Same as W Molten NaCl Aqu. Sol. KCN 10% H F Conc. Hap01 Aqu. sol. KCN 45 parts HF (40%) Sol: 60 parts HNOs (conc.) 20 parts acetic acid 3 parts bromine 50% HCI, 50% HNOJ 1% HCl 10% HCl 40% H F 30% HCI, 70% HNOI 40% HCl Conc. HIPOl Conc. KOH 40% HF

{

129

Ilr~narks Dip wire into melt 0.5-1 v ac 1-5 v ac Up to 8 v ac 10 v dc 2-3 v ac 2-3 v ac 6 v ac 5.5-6 v dc 1 v ac Dip into solution 30-50 v dc 1 v ac Dip into fresh solution

10 v ac 0.5-1 v ac 4-6 v dc 4-12 v dc 3 v ac 1-2 v ac 1-5 v ac 10-15 v dc 1-6 v ac

sample which depends upon the degree of cold work, partial or full recrystallization, the orientation of a single crystal wire, the presence of contaminations, the exact composition of the etching bath, the applied voltages and currents, and some other conditions. The progress of the etching process is checked by microscopic inspection. Although the actual cap of the tip with a typical radius below 1000 A can only be seen in an electron microscope, experience shows that if a tip appears perfectly smooth and sharp in an optical microscope at 500 times magnification, chances are good that the tip will be useful in the field ion microscope a t a magnification of a million diameters. The phokphorescent screen should have a good efficiency in order to amp. The give maximum intensity at a total image current of 10-lo to grain must be much finer than the smallest image detail. In a tube such a s that shown in Fig. 22 the tip-screen distance is R = 75 mm, and if a tip radius of 2000 A is used (at approximately 40,000 v in helium) the magnifi-

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(&:itionM = R/pro is oiily about 250,000 diameters, which makes the easily rcsolvd)lc distmw of 4.5 A appear slightly more than one-tenth of a millimctcr. l’hr phosphor layer should be quit,e thin, since the ions do not perictrate very dcep anyway. A typical coating which can be obtained by dusting the fine powder onto a suitable binder such as phosphoric acid has a density of 0.5 mg/cm2. Unfortunately the most efficient phosphor materials such as ZnS or CdS-ZnS suffer badly from the ion bombardment. Under typical operating conditions with helium ions a bright spot on the screen amp/cm2, where a zinc sulfide screen may carry a current density of looses half of its efficiency within a fraction of a n hour. Zinc silicate phosphors deteriorate about 10 times more slowly (49),so that they are usually preferred in spite of their lower efficiency. According to Ha n k and Rau (60) the efficiency of binder-free ZnS for 25-kv helium ions is 23%, and of ZnzSiOl it is only 6%. Heavier ions are much less effective, namely, 0.9% for Ne+ and 0.65% for Xe+ on zinc silicate phosphor. The intensity of the total image increases rapidly with the tip radius and the corresponding voltage. For tips with different radii the total current as calculated from Eq. (17) increases with V 2 ,and as the brightness of the screen goes up linearly with voltage, the total image brightness rises with the third power of the best image voltage. It is essential that the screen be deposited on a flat glass plate. The low image intensity makes the use of high-aperture photographic objectives mandatory, and an F:l objective has not enough depth of focus to permit a curved screen. 1t may also be mentioned that a n objective of such a large aperture must be used a t the object distance for which it is corrected in order to keep the spherical aberration within permissible limits. The focal length of the objective must be properly chosen to keep the finest details on the screen above the photographic resolution which is determined by the quality of the lens and the grain size and thickness of the photographic emulsion. Finally, the spectral sensitization of the negative material should match the spectral distribution of the screen. For a blue-green silver activated zinc orthosilicate screen the Kodak Spectroscopic Film 103 a-G was found to be most sensitive when processed with a highly active developer (Ethol) and an extended development time. Typical exposure times vary from a few seconds for tip radii above 2000 A to 1 hr for tip radii below 100 A. It is amazing to note that the human eye well adapted to the dark can see changes in the faintest images almost immediately, while it takes u p to 1 h r to pbotograph it with the best available material. The use of a n image amplifier could be very desirable, however, it is not easy to develop a system that is not impaired by the necessity of opening the microscope to air pressure quite frequently. The high resolution of a t least 0.1 mm which is desirable over a screen diameter of 100 mm is

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another condition that is difficult to match with the presently available systems. Von Ardenne (51)mentions the operation of a hydrogen ion microscope in which the ion image is converted into an electron image by the secondaries released on a fine metal mesh. The brightness is reported to have been equal t,o a field electron microscope image, which is unfortuiiately not too well defined. After the description of the microscope design the mode of operation needs to be considered further. N o accurate experimental determination of the optimum tip temperature has been carried out so far. The general observation with a helium ion microscope and a typical tip of 1000-A radius is that cooling with liquid nitrogen increases the image intensity by a factor of about 5 compared to room temperature, and also the resolution is better. A considerable improvement in resolution occurs in the small temperature range between 78°K and about 53"K, which can be reached easily by reducing the pressure above the nitrogen in the cold finger with a mechanical pump. Apparently the accommodation coefficient of helium on tungsten increases quite suddenly in this temperature region. Whether the improvement in resolution occurs also with other metals such as platinum in the same temperature range has not yet been checked. When pure nitrogen is pumped in the cold finger, it solidifies a t a tcmperature of 63°K and a pressure of about 100 mm-of-Hg. With continued pumping the temperature of the solid nitrogen can be reduced by another 10"K, but the heat transfer from the leads and the metal cone is impeded by a gas layer. It appears to be more efficient to use liquid air in the cold finger. The oxygen content maintains the liquid phase to a lower temperature, $0 that a more efficient cooling of the tip and the cone is possible because of convection and better thermal contact. The observation of the author (9) that no further advantage in the resolution is obtained if one goes down to liquid hydrogen temperature was probably the result of too high heiium gas pressure in the microscope. With only inadequate photographic equipment available most of the pictures were then taken a t 8 to 20p pressure, where the scattering is quite considerable. Only for very small tip radii may solid nitrogen cooling be fully adequate, since the hopping height, for instance, for ro = 250 A equals the width of the zone of forbidden ionization (4A) already at T = 85" according to Eq. (47). For a radius of 1000 A the theoretical optimum temperature is 21"K, and this seems to agree with the experience. The best plane resolution observed so far was obtaiiied a t 21°K with a silicon tip, where the triangularly arranged atonis of 2.35-A spacing on the (111) plane could be separated visually (52). Because of the high rate of field evaporation going on during observation it was impossible to make a photographic record. Photographically the square arrangement of the

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2.77-A spaced atoms on the (001) plane of a platinum tip of 200-A radius could be fully resolved. A chain resolution of the rows of atoms on the (112) plane of tungsten, with 2.74-A spacing, can be obtained fairly easily on tips with radii up to 1300 A a t 21°K (44) and on very fine tips even with liquid nitrogen cooling only (53).For a tip-screen distance of 75 mm it is found that the helium pressure may be as high as 1 . 5 without ~ any noticeable loss in resolution due to scattering. For most exploratory observations a pressure of 3 to 5 p is adequate in order to double or triple the image intensity without too much loss of image quality. An adjustment of the tip temperature to the given tip radius is not necessary, since the average hopping height varies locally with the atomic surface structure, and there is also the wide Maxwellian distribution of the re-evaporating molecules. At least an increase of resolution was never observed when a tip of very small radius was heated from liquid hydrogen temperature to the temperature of optimum hopping height. Because of the limited availability of liquid helium in the author's laboratory not too many experiments have been made a t very low temperatures, after a number of preliminary runs showed that there was not only no improvement compared to liquid hydrogen cooling, but rather a loss in resolution and no gain in intensity. It is assumed that a mobile adsorption layer of helium is present a t the tip surface, where it fills in the recessing lattice steps and thereby reduces the contrast of the field profile in the ionization zone. However, more experiments with liquid helium cooling are necessary in order to evaluate the performance of the microscope a t the very low temperatures. By pumping on a liquid hydrogen-filled cold finger the temperature can easily be lowered to 12"K, and it seems that there is an improvement in resolution for large tip radii. Although helium ions yield the best resolution, it is still necessary to explore the feasibility of other gases for the image formation, in order to make accessible the nonrefractory metals which would field evaporate a t the helium image field. A number of experiments were made by producing first the helium ion image of a tungsten tip, and then introducing the other gases for comparing the best image field strength and image quality a t the same specimen (44). The results are not too encouraging. Neon requires about 85% of the voltage for helium, and the resolution in some parts of the image may he even dightly better than with helium. However, the contrast is greatly reduced, probably due to adsorption, and a t the same gas pressure the image intensity is only ?&of the helium image, mostly because of reduced screen efficiency a t larger ion masses. Moreover, the cm3) seems to cause more larger polarizability of neon (a = 4 X scattering of the ions in the gas on their way to the screen, so that the pressure must he reduced a t least by a factor of two compared to helium.

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There is dso a diffiise light. glow iir the gas space lwt,wreii the tip assemhly and the metal cone, which shows up if the screen is photographed from below. 111 spit8eof these disadvantages i t may sometimes he iiecessary to resort to the use of ~ieonions. The only other gases remaining for an operation a t 21°K are hydrogen and deuterium. There is a marked improvement in resolution a t this temperature compared to the liquid-nitrogen temperature, but the contrast is very poor compared to helium, probably again because of adsorption. Deuterium gives definitely sharper images than hydrogen, as the scattering disk of single atoms is about 3070 smaller and the resolution correspondingly better. This isotopic mass effect may be explained with a slower motion of the heavier molecules through the ionization zone. The heavier rebounding molecules stay near the inner edge of the ionization zone longer, where the ionization probability is so much larger. On the average the heavier ions will then originate closer to the surface, where the field profile is more differentiated. According to the mass spectroscopic determination, the hydrogen ion image is made up of equal parts of molecule ions and atomic ions. However, from the following observation it may be concluded that the atomic ions are more responsible for the best resolution: The field of best image quality is 40 to 42% of the best image field for helium ions, e.g. 180 to 190 Mv/cm, which corresponds according to Eq. (48) to an ionization energy of 13.3 to 13.7 ev. This points to the hydrogen atom with Vr = 13.56 ev, rather than to the molecule with Vr = 15.4 ev. The scattering of the ions in the gas is larger than in helium because of the higher polarizability of the molecule (a = 8 X cmS), so that the pressure should not be increased above 0 . 5 ~This . is not very disturbing since owidg to the better screen efficiency for the lighter ions the image is brighter than with helium ions a t the same voltage and current density. For maximum intensity, a t the cost of resolution, the hydrogen pressure can be increased to 6 or 8p at a tip voltage of 10 to 20 kv before a gaseous discharge breakdown occurs. If the impedance of the power supply is high (100 meg) the discharge will not damage the tip. At 0.lp hydrogen pressure an extremely fine detailed ion image could be obtained with very blunt tips at voltages up to 50 kv without a discharge. I n the case of helium a microscope was operated at 50p pressure and 20 kv without a discharge, and the ring-shaped net plane edges could still be seen well enough to observe the onset of field evaporation. Gases other than helium, neon, hydrogen, and deuterium are riot suitable for operation a t liquid hydrogen temperature because of their low vapor pressure. With liquid-nitrogen cooling a number of gases such as methane, oxygen, nitrogen, argon, and krypton have been used to produce ion pictures of tips of tungsten and other metals, and the resolution is still

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very good compared to a field electron microscope image. However, the adsorption of the image gas or contaminations is quite disturbing for the reproducibility of the state of the tip surface. The use of gas mixtures has some interesting implications. If, for instance, helium is used with 10% hydrogen, one sees first the perfect hydrogen image when the field is raised to about 200 Mv/cm. The image turns blurred and then disappears entirely in a faint background illumination from the hydrogen ions autoionized in space when the voltage is further increased. Then, a t about 300 Mv/cm the helium ion image of individual

FIQ.23. Schematic diagram showing (cross hatched) 200-300 Mv/cm zone where ionization of gases with V I near 15 ev occurs while the surface field is 500 Mv/cm. Electrons falling back to the tip arrive with an energy of 2000 to 3000 ev in the center of the tip. r = 1000 A, V , = 20,000 volts.

protrusions begins to appear, and eventually the entire surface shows up in its best resolution a t 450 Mv/cm. When the ionization of the hydrogen molecules is now taking place some 500-1000 A above the tip surface (Fig. 23), the electrons are being accelerated towards the tip surface, where they impinge with an energy of some 2000-3000 ev. When the hydrogen partial pressure is l p , and the tip radius 1000 A, then the current density of impinging electrons is about 10 amp/cm2, and the energy density is 20-30 kw/cm2. Under those conditions the temperature at the cap of the tip may rise by perhaps 30-50" when the tip is cooled with liquid hydrogen, as can be measured by the rate of field evaporation. The tip should also emit

FIELD IONIZATION AND FIELD ION MICROSCOPY

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X-rays (64) of a wavelength above 4-6 A, representing a n X-ray point source of extremely small dimensions (1000 A). Attempts to photograph the X-ray emission from the small cap region of the tip with the help of a pinhole camera inside the microscope have not yet been successful. The photographs taken through a pinhole which was covered by a 5-mil beryllium window gave only an image of the entire tip and loop assembly with an evenly distributed X-ray density, which may be due to the impact of highenergy secondary electrons from the screen. Another attempt was made with 2p of xenon added to the helium, which because of its low ionization energy will ionize further away from the surface and yield higher energy for the impinging electrons. In order to avoid condensation of xenon, the cold finger was filled with liquid oxygen. Emission currents up to 10-7 amp were measured, but no especially strong X-ray emission was found a t the crest of the tip. The role of the helium gas in this experiment is only to monitor the field strength so that one can go close to the onset of field evaporation. The intensity of the field ion image is usually limited only by the maximum gas pressure that will not cause loss of resolution due to scattering on the path from the tip to the acreen. Preliminary experiments have been successful in which the high pressure was only supplied near the tip. To this purpose the metal cone is entirely closed except for a fine hole in the center of which the tip is showing towards the screen. Helium gas is fed into the cone, so that the tip is in a high-pressure region. The large volume of the microscope between the cone and the screen is pumped by a faEt oil diffusion pump, so that the prewure there is less than 1p. In such a dynamic system, in which the helium gas was circulated, an increase in brightness by a factor of 50 was obtained without any loss of resolution, and higher gains seem very likely possible. 2. Field Evaporation. The most important experimental procedure in field ion microscopy and an interesting researrh subject in itself is the process of field evaporation of the tip metal. It was first noted as a dissolution of the surface of the emitter tip at :in elevated temperature (8), where field-enhawed surface migration plays a predominant role in building up broad lattice steps with a height of many atomic distances (47, 55). Drechsler (56) confirmed the fact that tungsten field evaporates i n the lOCO'K range in the form of positive ions by letting them impinge on another field emitter tip a t negative potential and detecting the impact spots in the electron image of t,his tip. The high-temperature field evaporation is further complieat(4 by the appearance of slip b:ittd\ a t thc siirfaw (.57). Morc iiitcrestiiig as :ti1 cllrmenttary physical c4Tcc.f is, 1hcrcfow, field c>\~~por:itioii at, rciom tcrnperature or evc'ii at very low temprraturcs, whcrc~the wmov:il of atoms occurs without surface migration directly from the edge of a lattice

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step of atomic height (26). At low temperatures, where surface migration from the shank of adsorbed contaminations, such as oxygen, is brought to a standstill, field evaporation of the refractory metals produces the cleanest possible surfaces. After removal of surface layers the number of foreign atoms in the surface is determined only by their bulk concentration. As long as a field above 300 Mv/cm is maintained, no contamination atoms from the gas can reach the surface, either. The field-evaporated surface is also very regular in structure if the crystal is faultless. Whenever some protruding atoms cause a local field enhancement, they will be compelled to evaporate, Fig. 24. As a result a crystal surface is obtained with perfect atomic smoothness. However, the end form established, after removal of

FIG.24. Typical changes of tip profile (a) by thcrmal surface migration (b and c) and by continued field evaporation (d and e ) .

a sufficient number of surface layers, is quite different from the form obtained by prolonged annealing. The latter is almost hemispherical, except for the fairly large flats developed by the densely packed low-index net planes, such as (011) on the b.c.c. crystals, and (111) and (001) on the f.c.c. crystals. No high index planes are properly developed, as the difference in their free surface energy per single atom is comparable with kT at the surface migration temperature. The field evaporation end form on the other hand is determined by the fact that the evaporation energy for ions varies a t the different crystal planes of the surface with their work function, while the evaporation energy for atoms is the same a t the evaporation sites along the net plane edges of all crystal planes (58). It may be noted that dc field evaporation of a bulb-shaped tip surh as shown in Fig. 24c will lead to an instability and to tip destruction because once the evaporation sets in, the radius will decrease. With a c~mstant~ voltage applied this will lead to a catastrophic increase of the field mid the evaporation rate. A cone-shaped tip, on the other hand, will simply increase its radius until the field hwomes too small for further evaporatioii. I n order to find the end form of a field evaporating crystal it is sufficient

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to consider it at zero temperature and for singly charged ions. Equation (23) is then reduced to

FevBp ec3(A 1 :

+ V I - +)*.

(49)

If on a stable tip configuration a sufficiently high voltage is applied, field evaporation will progress further a t regions with a higher work function until their radii of curvature have become so much larger than those of the regions with lower work functions that the local field is reduced t o give a homogeneous rate of evaporation. I n Table I1 the evaporation field at T = 0 is calculated to be Foil = 786 Mv/cm for the rhombododecahedron plane with q5 = 5.99 ev, and FIl, = 1052 Mv/cm for the octahedron plane with = 4.35 ev. The same low work function exists along the [Oll] zone line, e.g. around planes (116) to (113). A measurement of absolute field strength a t the various crystal planes of the mic*rosropically small tip is not possible. Even the over-all field of 450 Mv/cm given in this paper as best image field for helium ions is uncertain by a t least f 1 5 % , since the field calibration is based on field electron emission of the same tip. According to Dyke and Dolan (6) the FowlerNordheim theory of field electron emission has been proved only within this range of accuracy in field. The additional uncertainty here is that field electron emission has only beeii measured for heat-polished tips, which, as the field ion microecope has revealed, display a great randomness of atomic arraitgenient in the strongly emitting planes owing to the frozen in thermal disorder. I t is also only of such imperfect, surfaces that the thermionic work fuiwtioii is known. A difficult experimental task is presently being undertaketi in the author’s laboratory (wit,h It. D. Young) to determine field electron emission current densities from individual planes of a tungsten cry>tal with field evaporated and consequently perfect surfaces. However, there is no way known to determine accurately the ratio of field strength and applied voltage. Relative field strengths can he measured quite acciirat,ely in the heliumoperated field ion microscope. Experience shows that the best resolution caii only be obtained within a narrow range of about 1% of the applied voltage, and for a given tip this best image voltage is different for each set of crystal planes. Assuming that the mechanism of ion formation i i i the ioiiization zone above the surface is the sailie above a11 cqrystsl platies, except perhaps for minor effects due to variations in siipply density by the hopping atomh, we w i i expwt that the local field strwgth will be the s:mc at :ill places of equal image quality. Figure 25:i hhows ;I tiiiigsten tip takcii a t 9000 v, and Fig. 25b the same tip depicted a t 12,000 v, both after the tip had beeii hhapcd by field cbixporatioii a t 21°K and 13,500 volts (57). The low work fuiiction region:, on the [Ol1]-2011~~, partirdarly (1 11) protrude,

+

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FIG.25a. Helium ion image of a tungsten tip eftrr field evaporation at 13,500 v, 21' I<. Picture taken at 9000 v to depict the more protruding low work function rrgioiis.

because they have assumed a smaller radius of curvature, aiid therefore the best image field is reached a t a low applied voltage. In order to depict with best resolution the high work function-low raLdiLis-of-ci1rvature region around (011) a much higher voltage is necessary. The ratio of evaporation field strengths, F h k l , for a pair of planes should be inverse to the best image voltages V M ,for example,

In the example of Figs. 25a and 25b the voltage ratio is 1.33 in best agreemriit with the ratio 1.32 of the calculated field strengths above. Obscrvations on other tips of widely different average radii gave ratios between

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E'ic. 251). Same tip photographod at 12,000 v. Best irnngv sti:irpiwss r i w r t h v high ~ o r l fiinetion i are34 around the rhomlmdoder.al~cdion plnncs.

1.26 a d 1.36 for this w t of planes. If in our example lrevap = 13,500 v produces the theoretical Fll, = 1052 Mv/cm, the best image field hhoiild be

and the same value is found from thc data 0 1 1 the (011) plane. This figure is 1.57 tirncs larger then the over-all value of 450 Mv/cm for best image field. This factor may be due to the local field enhancement above the lattice steps, which with its short range does not contribute much to field electroii emission used for calibration. Prom model measurements of the field distribution near lattice steps, however, it may be concluded that a typical field enhancement factor should not be larger than 1.4. The small remaining

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differelice between the measured and the calculated evaporation field may be due to the inadequacy of the simple image force scheme, or it may point to the neglected polarization term f.5F2(a,- a,). Assuming a field enhancement factor of 1.4, a difference in polarizability of the tungsten atom and the tungsten ion of only 2.3 X cm3 would suffice for: an explanation. All the other data in Table I1 could only be calculated with literature values of work functions which do not consider the dependence upon crystallographic orientation. These thermionically or photoelectrically measured effective work functions emphasize the lowest values, while for field evaporation the high work function regions are the more significant ones. Most of the 27 elements in Table I1 have been checked qualitatively in the field ion microscope to determine whether their rate of field evaporation corresponds to their rank in this list, with the exception of Ru, U, Th, V, Ge, La, and Hg. Carbon tips, made from carbon incandescent lamp filaments proved to stmid a field a t least 30% beyond the helium ion image field, but the small cryst,allite size of 20 to 30 A and the obviously great difference in work function of t,he basal plane from the other crystal planes prevented the development of a perfect surface. The next seven elements from W down to Pt have theoretically an evaporation field sufficiently high to allow a good helium ion image, that is more than 1.4 times 450 Mv/cm, and this is fully confirmed experimentally. The following elements behave less regularly. Rhodium still gives a very good helium ion image, while zirconium field evaporates a little too easily to give a good helium ion image. Steady images can be obtained by using neon ions. Be, Si, and AU rank further clown, only the more protruding low work function net planes (*anbe depicted with helium for visual observation, since the dissolution of the surface is continually going on. Surprisingly the much lower ranked metals Fe, Co, Pd, and Ni give fairly satisfactory helium ion images a t least of their low work function regions. Some details such as atom chains and completely resolved high index net planes could even be seen around the (1150) plane of zinc, or on and around the (102) plane of gold. On the other hand metals such as Ti and Cu definitely do not stand higher fields than calculated theoretically, as can be observed by imaging them with argon a t lower fields. This different behavior of the metals may be explained as an effect of the polarizability term $@(az - a,) which is contributing to the evaporation energy of the metal ion. None of the polarizabilities are known quantitatively, so we are forced to hypothesize with the assumption that in the cases where the observed evaporation field is higher than calculated in Table I1 the term is appreciably positive. This assumption may be justified when the ion is alkali-like, with one single electron in the outer shell. Looking over the electron grouping in the periodic table of the elements one

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will fiiid thih to hr tme for tlhe following ions of the nictals in Table 11: Re+, Si+, Fe+, Nit, Zn+, Ge+, Hg+ ant1 maybe some more of the heavy elements. On the other hand Cu+ and Ti+ have a noble gas structure in the outer shell, with a presumably tmall at,so that the polarizability term may even subtract from the evaporation energy. The good agreement of the theoreticd evaporation field based on the measured work functions of the (011) aiid the (111) plane of tungsten with the experimental results suggests the use of field evaporation for determining the extreme work functions of other metals if only their thermionic, that is approximately their lowest work function, is known. In the case of Pt one finds in the field evaporation end form an extreme curvature in the region around (102), while the vicinity of (001) and (111) is more flat. On a (001) centered Pt tip the best image voltage ratio Vlo2/VOolwas found to be 0.76, which would be according to Eq. (50) also the ratio of FWI/F1Oz. Ascribing the thermionic work function of 5.32 ev to the (102) plane, we can calculate trhe unknown work function for the cube plane +0ol

=A

Vl,, = 6.48 electron volts. 4- V I - Je3F10z VO,,

(52)

In the same way the work function of the octahedron plane of Pt is found to be +ill = 6.60 ev. With Re only one set of data is available. Here the vicinity of (1120) and (1011) are the regions of lowest work function, for which the thermionic value is 5.1 ev. For the high work function regionF (except for the basal plane, which was never found anyway near the tip axis) field evaporation data give 9 = 7.3 ev, a very high value which cannot be discussed further as long as 110 other data are available. I n the practice of field ion microscopy, field evaporation is not only used to remove the surface roughness of chemically etched tips and to develop the high crystalline perfection of the end form, but also for the controlled removal of surface layers for the purpose of investigating the interior of the crystal. The evaporation can be controlled best when the margin between best image field and evaporation field is narrow. This makes platinum one of the most easily handled metals, since one can see the progress of evaporation in almost best image sharpness. One can concentrate attention on an atom a t a specific site such as a t a dislocation, and by gradually increasing the voltage see it evaporate either earlier or later than its neighbors, thus obtaining an indication of the specific binding energy. Also, one full net plane after the other can be renioved a t a time to inspect each for vacancies. If the tip crystal does not contain imperfections that lead to destruction by rupture under the surface stress F2/% of the field, many thousand atomic layers corresponding to 5 or 10 tip radii may be removed by field evaporat,ion, as sketched in Fig. 24e, until the increasing radius becomes so

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large that the availalh volLage is ilisufficirnf~to produce Lhe evaporatioii field. In principle it should be possible to continuously field evaporate a fine wire which is being fed through a nozzle, but difficulties arising from disruptures caused by crystal defects make it a not too practical ion murce. On the other hand such a eource of heavy metal ions might be quite useful for ion propulsion systems, the more as the final product of the evaporation is doubly charged ions (57). The removal from the surface occurs certainly, according to our previous theoretical picture, in the form of a singly charged ion. However, as the accelerating ion travels first quite slowly, another electron will tunnel out before the ion has escaped from the high field zone near the surface. The second ionization energies of all the refractory metals in the upper part of Table I1 lie between 15 and 20 ev, well below the first ionization energy of helium. The ratio of the lifetime before ionization of a helium atom or a tungsten ion, for example, can be calculated from Eq. (5) to be

which amounts to 186 a t F = 6 v/A and with Vwt = 17.7 ev. A helium atom approaches the tip surface by the dipole attraction with such a speed [Eq. (lo)], that it travels the last 10 A within 2.9 X sec and a t the given field it will certainly be ionized during this time. A tungsten ion, being accelerated from the surface, will travel 13.3 A during the same time, and with its 186 times larger ionization probability will definitely loose a second electron. The occurrence of doubly charged ions when tungsten was field evaporated a t room temperature could be confirmed experimentally with the mass spectrometer shown in Fig. 9. No singly charged ions were seen at all. When a t elevated temperatures field evaporation of tungsten occurs at much lower field strengths an increasing number of eingly charged ions should appear. Also the metals further down in Table I1 should give an increasing ratio of singly charged to doubly charged ions as the field strength goes down. No quantitative observations have been made so far. Measuring the rate of field evaporation provides a means of determining the tip temperature by assuming the validity of Eq. (23). In one experiment the liquid nitrogen cooled emitter was operated a t 40 N of helium a t 122% of best image voltage, .ie., a t 550 Mv/cm. The current density was 100 amp/cm2. The equivalent amount of electrons falling back into the surface from the ionization zone estimated to be at least 20 A away heats up the tip with an energy density of 90 v X 100 amp/cm2 = 9 kw/cm2. The tip temperature rose from 80°K to 136"K, which is in agreement with an estimate from the heat conduction of tungsten a t that temperature.

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FIG.26a. Field electron microscope image of a tungsten tip of 800 A radius, 10- amp a t 2000 v.

FIO.2Gb. Helium ion image of B tungstw tip of average rsclius GOO A, after partid field evaporation; l p helium, 21"K, 3 X 10-0 amp, 12,500 v.

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FIG.27a. Cork ball model of the rap of a tungsten tip, built up of (011) net plaiies of increasing diameter; (011) plane in conter, (001) and (010) on top and bottom edge, respectively; equivalent tip radius 135 A.

c. Observation of

the Atomic Lattice in Perfect Crystals

The atomic lattice of all metals suitable for field ion microscopy is already known from X-ray structure analysis. The study of perfect or nearly perfect tip crystals is therefore centered on the better uiiderstanding of the mechanism of image formation, so that it can be applied to the interpretation of the complex structure of the more interesting crystal imperfections. Some of the properties of perfect crystals can also be investigated in a new or sometimes unique way, such as surface migration, work function of individual planes, or field evaporation. The progress made by converting the field electron microscope into an

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FIG.271). Same cork Iiall mocld with most Irotrriding arid serond nest protruding atoms marked with fluorescent paint, illuminated with dtraviolct light.

ion microscope is demonstrated in Figs. 26a and b. In the electron image of a heat-polished tungsten tip of approximately 800 A radius we find the familiar pattern of the low emitting (011) plane surrounded by the foiir { 112) planes. No finer details (’an he expected with the resolntion of 20 A. In the ion image of a similar tungsten tip, with the surface perfected by I field evaporation, we recognize the pearl chain-like rows of individual atoms aloiig the net plane edges arid the fully resolved two dimcnsional atomic. arrays on the high index planes. A cork ball model Fig. 27a, representing a hemispherical cap built up in a b.c.c. arrangement of the “atoms” to an equivalent radius of 135 A shows the close-packed (011) plane in the wntcr. This model may appear soniewhat confusing with its grcat iirim1)c.r of siir-

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FIG.28. Helium ion image of a tungsten tip of approximately 300 A radius. Rcsolution in chains across the [ill] zoncs up to 2.74 A. The (111) plane at t,he estreme left is fully resolved.

face atoms all looking alike, and me may be fortunate that the actual ion image does not show all the surface atoms either. Only the protruding atoms which produce a locally enhaiieed field appear in the ion image. In order to match the cork ball model, the most exposed atoms at the corners of the lattice steps, essentially those with only four next ncarest neighbors, were painted with bright fluorescent paint. The atoms in the chains next to them on slightly less protruding sites were marked more dimly. Viewing the model ih the dark, Fig. 27b, gives now a quite accurate picture of what one sees on the fluorescent screen of the ion microscope. This is particularly evident if a tip of approximately equivalent radius is compared in Fig. 28. The triniigrilar arraiigcmcii t of Iho iirdivitliinl atoms o i l thv (11 1) plnirc,

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FIQ.29. Orthographic projectioii of hemispherical cubic crystal with (011) in the axis. Emphasis is on the planes predominating on b.c.c. crystals; [110] zone lines solid, [ I l l ] zones long dashed, [loo] zones short dashed.

4.47A apart, and the atom chains of the same spacing across t,he [ O l l ] zone which connects the (111)-(011)-(111) planes are easily resolved. The chains of adjacent atoms, 2.74 A apart, across the [ l l l ] zone lines between the { 112) planes via (Oll), are just barely resolved in some places and provide the best test sites to look for with respect to resolution. For the orientation in the crystal patterns it is useful to have a rrystallographic map. We represent here in Figs. 29-31 the orthographic projections of the basic planes on hemispheres of the b.c.c., f.c.c., and the h.c.p. lattice. The first two cubic lattices contain, of course, the same planes, but the relative size of the planes differs greatly, (011) and (112) being predominant in the b.c.c. lattice, and (OOl), (111) followed by (Oll), (013), (113) being most pronounced in the f.c.c. lattice. The maps can easily be extended to planes of higher Miller indices. As an illustration for

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FIG.30. Orthographic projection of hemispherical cubic crystal in (1101) position, with planes appearing preferentially on f.c.c. crystals; [110] zones solid, [100]zo~icsshort dashed.

the typical lattices, images of a face-centered platinum crystal and a hexagonal rhenium crystal are shown in Figs. 32 and 33. While in conventional electron microscopy the exact determination of magnification is difficult, we can use here the known lattice distances to find the magnification and also the exact geometrical shape of the crystal. The edges of the net planes may be considered as topographic lines in an elevation map of the tip cap, and from the known apex angle between a pole and another crystallographic direction one finds immediately the tip radius (16,52, 53). Counting the number n of net, plane rings of known step heights gives first the height of the cap, ns, within the considered angle y, and then the local radius of curvature

r=

n8

1 - cosy

.

(54)

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FIG. 31. Orthographic projection of hemisphere of hexagonal close-packed crystal; [2IiO] zones solid, [OOOl] zone long dashed.

[ioio] zones short dashed,

The angle y between the normals of two net planes with known Miller indices (h.llcll1) and (hzkzh) is calculated by using the well-known formula of crystal geometry

Table I V gives the angles y , the corresponding 1/(1 - cos y), and the step heights for the most significant planes of the cubic lattices. These data allow a quick det)errnination of the local tip radii as shown in the example of a platinum tip in Fig. 34. One notices the surprisingly large variation of local radius up to a factor of three even in the more central part of the tip (58). This is typical for a field-evaporated crystal, the shape of which is determined by the condition of equal field evaporation rate on the regions of different work function. This results in an end form where the local radius

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TABLEIV. ANULE y 1/(1 - COB y )

Zone

(hlklll)

BETWEEN

w. MULLER

CRYSTAL PLANES(hlklZl) AND (hakzlz), A N D LOCALTIP RADIUS

FOR THE CALCULATION OF

(hzkzh)

Y

1/(1

- 0 0 8 y)

Step heights

b.c.c. 8"56' 10"54' 13'54' 19'6' 30'0' lO"2' 13"16' 19'28' 25'14' 35'16' 8'8' 11'19' 18'26' 26"34' 45'0' 9'27' ll"18' 14"2' 18'26' 26"34' 33'41' 45'0' 11'25' 13'15' 15'48' 19'28' 25'14' 35'16' 43'19' 46041' 54"44' lO"2' 15'48' 19'28' 22"O' 23'50' 25'14' 35'16' 6'12' 8"3' 11'25' 14'25' 15'48' 19'28' 29'30'

82.4 55.4 34.2 18.2 7.46 65.5 37.5 17.5 10.5 5.45 99.4 51.5 19.5 9.47 3.41 73.5 51.5 33.5 19.5 9.47 5.96 3.41 50.5 37.6 26.5 10.5 5.45 3.67 3.18 2.37 65.5 26.5 17.5 13.7 11.7 10.5 5.45 170.6 101.5 50.5 31.7 26.5 17.5 7.71

f.c.c.

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FIG.32a. Platinum crystal after annealing with frozen in thermal disorder. Picture t,akenat 21°K with helium ions and 24,000 v.

FIG.321). Sam(. platinum tip after removal at 21"I.i of about 3 monoatomic layers by field evaporation with 26,200 v. Picture taken at 25,200 v.

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FIG.33. Rhenium crystal (h.c.p. lattice) photographed with 21,000 v. Average tip radius 1200 A. Orientation corresponds to crystallographic map Fig. 31.

of curvature changes in the same sense as the work function of that particular region. As a consequence of the stronger curvature in the low work function regions around 1012) and { 113} the field strength there is too high for best image sharpness, as the voltage chosen for the photograph had to be a compromise between the best image voltages of the different tip regions. The large variation of radius occurs only over narrow regions, so that the entire cap of the tip still resembles quite cloeely a hemisphere. Another result of the variation of local tip radius is an inverse change of magnification over the field of view. As an example one finds in Fig. 28 that the 4.47 A spacing on the protruding (111) plane appears much wider than the same distance between the at.oms in the chains across the [ O l l ] zone. Local variation of magnification is also responsible for the

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FIG.34. Platinum crystal in (001) orientation with local radii in angstrom units determined by counting number of net plane edges within a given angle.

curvature of actually straight atom rows on essentially flat low index net planes, such as on (113) in Fig. 32b. Besides the distortion, we have the additional effect of variation in apparent atom diameter and intensity, depending again upon the local field enhancement and caused by the locally inhomogeneous supply of hopping and surface-migrat ing heliiim at,oms.

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The platinum crystal of Fig. 32b contains more than 1000 discernible facets of individual high index net planes. On the [130]zone line for instance, coiit4ainingthe major planes (313) and (315), indices as high as (3,1,15), (3,1,17), (3,1,19),and (3,1,21) can be distinguished, while in the [lTO] zone the higher indices are (115), (117), (119) and barely developed ( l , l , l l ) . Geometry requires for the appearance of such high index net planes a sufficiently large radius of the crystal sphere or more accurately of the polyhedron. The minimum radius to permit geometrically the occurrence of a plane with index (101) or (111) on an f.c.c. crystal sphere can be calculated and is approximately T = 6 a / 2 ( a = lattice constant) (54, 59). Whether or not the geometrically possible high index planes do appear depends upon the continuity of the local conditions that determine field evaporation. The heat of vaporization of the neutral atom, the work function, and the local field strength must change only gradually if one moves from one high index plane to the next one. Usually these quantities will be influenced by localized crystal imperfections due to impurity atoms, interstitials, and dislocations. Probably one impurity atom with properties sufficieiitly different from the matrix will suffice to change the field evaporation energy of a small high index net plane of step width a 1 in such a way as to bring it out of the order of the neighbor planes, so that this particular plane cannot appear individually. Assuming an essentially homogenous distribution of the impurity atoms, their coilcentration should not be larger than k3in order to permit the appearance of the plane with Miller indices (hkl), where h and k are smaller than 1. This lP3 rule for the concentration of impurities (59) is in agreement with the experience that only very pure metals, such as thermocouple quality 99.99yo Pt or highly outgassed pure W develop the very high index planes.

-

D. Crystal Imperfections I. Vacancies. The most promising application of the field ion microscope is the study of' imperfections in metal crystals. All other methods known for this increasingly important subject of research are indirect ones in so far as they use secondary or gross effects caused by the imperfections, such as the mechanical or electrochemical behavior, electric conductivity, and X-ray or electron diffraction of crystal areas large compared to atomic dimensions. With the advent of the field ion microscope it, is now possible to see the imperfections directly. In some instances the defects can be generated while the specimen is kept in the microscope, or even during the observation. This work is still in its earliest stage, and only 811 outline of the exploratory work can be given at the present time. The simplest defect of a lattice is an unoccupied lattice site, a vacancy. An equilibrium concentration of vacancies can be produced a t elevated

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temperature, if one allows the xpecinien to assume the state of minimum free energy, F , = IT - TS. The internal energy I J increases with the introduction of lattice defects, but so does the entropy S as well. The contribution of the vacancies to U is proportional to their concentration, while their contribution to S is most significant a t lower concentrations and rises more slowly with higher concentrations. The resulting vacancy concentration nvac/na t which the free energy has a minimum follows a Boltzmann law, %ac

= ne

-Ef/kT 1

(56)

where E, is the energy of formation of a vacancy. This concentration becomes appreciable near the melting point of the metal, and by rapid quenching it can be frozen in and observed at a convenient temperature. Other ways to produce vacancies are plastic deformation and irradiation, which, however, are both connected with the simultaneous introduction of other lattice imperfections. For the separate observation of vacancies it is therefore better to use annealed and quenched specimens, for can be expected. which a concentration of order of magnitude lop5 to As most of the visible atoms form the edges of closely packed net planes, it is seldom possible to distinguish with certainty a vacant site from a kink. However, a vacancy can be unequivocally identified when it is found in the interior of a completely depicted net plane (69). The best planes to search for vacancies are the more protruding low work function planes such as (111) in the b.c.c. lattice, and (012) in the f.c.c. lattice, which can be resolved even when they are large enough to contain more than 100 atoms. Higher indexed planes are more easily resolved, but they contain too few atoms to provide a good chance for encountering a vacancy. Figure 35a shows the (012) area of a platinum crystal which after quenching from 1500°C had been etched electrolytically at room temperature to preclude annealing. With slowly continued field evaporation at 21°K the 15 following net planes were examined individually and found vacancy-free. A vacancy was then found in the 16th plane (Fig. 35b), and two vacancies in the 21st plane (Fig. 35c). In order to find out whether there ie a large lattice strain around the vacancies, field evaporation was continued very slowly so that one atom after another was seen to disappear until the net plane edge had been moved to the site of one of the vacancies (Fig. 35d). From the smooth progression of field evaporation it can be concluded that there was no particularly high strain acting around the vacancy. When the removal of complete (012) net planes was continued] a new single vacancy was found in the 12th following plane, and another one after removal of 22 more net planes. In total the 71 removed net planes of an average size of 120 atoms contained 5 vacancies] e.g., the vacancy con-

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FIG.35. (a) Complete (012) net plane on a platinum crystal which was quenched from 1500°C. (b) After removal of 15 layers from the (012) plane a vacancy appears. (c) After continued field evaporation of 20 complete net planes the next contains two vacancies. (d) A few more atoms have been removed from the (012) plane. One vacancy is left forming an indentation a t the net plane edge.

centration was 5.9 X 10-4. This corresponds according to Eq. (56) to a formation temperature of T = 1800°K and to an energy of formation E f = 1.84 X erg = 1.15 ev. This figure, based on a statistic of only five vacancies, is in good agreement with a value of B f = 1.2 ev found by other investigators (60,61) when they interpreted the additional resistivity which is introduced by quenching pure platinum. It may be remarked that the choice of the convenient (012) plane for the observation does not influence the result since one simply inspects a prismatic piece of the interior

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3 57

of the crystal of about 11 by 11 atoms cross section arid 71 layers depth. In order to avoid anilealing a t room temperature quenching can also be done in s i t u by heating the t5ipinside the liquid hydrogen-cooled microscope and turning off the heater current. The quenching rate is increased by using extra fine loop wire and by filling the tube with some l o o p of hydrogen. So far only one divacancy and no small vacancy clusters have been observed with the field ion microscope, but the number of specimens studied is still too small for the conclusion that such oftendiscussed agglomerations do not exist. Probably one will have to look for them near a dislocation line, from which they may be emitted, and this dislocation line will have to be near a crystal region which is suitable for vacancy visibility. Large agglomerations of vacancies as they may occur in cold-worked metals are difficult to recognize as such because of the local image distortion around an irregularly shaped “hole” in the heavily strained tip surface. 2. Interstitials. Vacancies are not the only possible “point defects” in a lattice. There also may be extra atoms squeezed in between the regular lattice points, which are then called interstitials. Their thermodynamic equilibrium concentration is very small as their energy of formation is quite high because of the large strain around them. Larger nonequilibrium concentration may emerge under certain conditions from dislocation lines, and interstitials are, together with vacancies, the predominant defects in ion-bombarded or irradiated specimens. Although the displacement of atoms a t the site of a n interstitial is only less than one atomic diameter such places can be found occasionally when a good and sufficiently magnified micrograph is carefully scanned. The atom rows are sometimes bulging out sidewards, to make room for the inserted extra atom. On close-packed net planes of which only the edges are depicted as rings one can also find some spots that appear brighter than individual metal atoms that have been deposited on this plane by evaporation from a metal coil near the tip. The brighter spots can often not be removed by field evaporation without taking off the entire net plane, beginning from the edge, while the individual atom is more easily field evaporated. The interpretations of the bright spot may be as given in Fig. 36. The bulging out of the surface layer above the interstitial a t B increases locally the field strength over a wider range than does a single atom A, thus increasing the supply from the hopping helium atoms and the ionization above it. The atoms on the bulge are more strongly bound than the individual atom A on top of the net plane. Vacancy-interstitial pairs have not yet been found in the limited amount of ion micrographs that have been searched for them. It may be that the interstitial atom is too far away from the vacancy site to permit an unequivocal association of the two point defects. The activation energy

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for the migration of interstitials should he much lower than the one for vacaiicics, which are usually responsible for volume diffusion. It should therefore be possible to find a mobility of the interstitials at fairly low temperatures, perhaps between 50 and 200°K in Pt or W, but no experiments have been made as yet with the ion micromope. Foreign atoms in interstitial positions, especially oxygen in platinum and in rhodium shall be discussed in Sec. V.D.4 as impurities. A defect related to interstitials A

B

FIG.36. Schematic diagram of a single atom A on top of a close-packed net plane, and of an interstitial atom B just below the surface.

is the crowdion, which is visualized as a straight row of n atoms approximately equally ?paced along n - 1 lattice sites. The number of atoms in a crowdion may be perhaps 4 to 8, and the defect is considered to be important because of its dynamic properties when it moves through the lattice without much energy loss. Atom rows that may be interpreted as crowdions have been found occasionally on pictures of platinum, but no attempts were made to investigate their motion as they were found only after searching the photographs when the tip was no longer available. 3. Dislocations. The most important crystal defects are dislocaticns, which allow crystal growth a t low degrees of supersaturation and which cause plastic deformation when they begin to move through the lattice under a stress exceeding the yield stress. An even approximately adequate introduction into the basic features of dislocation theory is beyond the scope of this section, and we refer to the many excellent reviews on the subject (62, 63). No systematic investigations in field ion microscopy of dislocations have been made so far, and only preliminary observations can be reported. Drechsler, Pankow, and Vanselow (47, 64) noticed that the well-known dissolution of the tip crystals a t temperatures around 1000°C in the presence of the image field, which goes on in the form of broad steps, occurs predominately in the form of collapsing open loops or sections of spirals rather than closed rings. They ascribed this to the intersection of screw dislocations with the tip surface and compared the patterns with the well-known schematic diagrams of screw dislocations of atomic step height by Frank and Read (62). This may not be quite permissible since the observed steps were of larger than atomic dimensions, as the images were obtained either with the field electron microscope or the hydrogen ion

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microscope at room temperature with their limited resolution. Also, from the voltages given with the hydrogen ion pictures we know now that the tip radii must have been much larger than reported, and the magnification correspondingly smaller. While there is no doubt that dislocations are involved in the observed process of crystal dissolution through high-temperature field evaporation, it appears now that the main feature in that temperature-field range is the occurrence of slip as a result of the field stress (57);and what had been observed were the protruding edges of slip bands, and in some cases pseudo spirals created by closed low index net planes that were stacked off center. Pseudo spirals may appear when the two upper net planes form a double height step along a certain length of their circumference, which will dominate in the ion image because of its locally enhanced field, while the field is too low for depicting the parting single steps ahJllg the rest of the cirri1mferenc.e.

FIG.37. Spirul edge on (013) plane of a. {)latinurn crystd whirh was produced bv ,4ip a t 500°C u n t l t ~the stress of a fic~ldof 200 hlv Icni.

Svrew disloc.atioiis intersecting the surface in one of the major net plancs so th:i.t il perfectly spiraling net plaiie edge tan be observed are quite rare. On platinum they have been found occasionally on the (013) plane (Fig. 37). I t may be that under most c.ircumstanc,esscrew dislocations are not stable at the large external field stress. Either they move out of the crystal before the field is high enough for a perfect image, or they cause thc crystal to rupture by 4ip. In most instances the intersection of a single dislocation line with thc surface is not very spectacular, and one has to scan the image carefully to find the imperfection. Well-annealed tip specimens of pure metals such as W, Ir, Pt appear often to be entirely free of dislocations, which can be expected because of the small field of view of some cm2. Into such a seemingly perfect crystal a large number of dislocations can be introduced i n sitir by plnst ic deformntion at, elev:it,ed temperatiirf~( ~ ~ i i s ehy d slip I I I I ~ C

~ ~

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the stress of the applied field, by strain release, fatigue, or radiation damage as shall be discussed in Section V.E.1-E.5. Occasionally one finds a horseshoe-shaped incomplete net plane edge suggesting the presence of two parallel screw dislocations of opposite sign. Such loops have been found on the (011) plane of tungsten, and particularly on the (001) plane of iron whiskers, Fig. 38. In the latter case the observation of the double screw (65) agrees well with the necessity of having an axial dislocation for the growth mechanism and yet no axial Eshelby twist (66)which would result from the presence of only one screw and which could not be found by X-ray diffraction. Iron whisker tips give just barely stable helium ion images of the low work function vicinity of the ( O O l } planes, so that the double screws can be seen on (001) oriented whiskers. On 811

FIG.38. Loop shaped (001) net plane edge on a n iron whisker. Picture taken with helium ions at 21"K, represents only central part of the entire tip pattern; tip radius 430 A.

(011) oriented whisker the (001} planes appear as complete, concentric net plane rings. It should be mentioned that a still unidentified pinning mechanism is necessary to prevent the mutual attraction and annihilation of the two opposite screw dislocations which are only about 100 A apart. Pure edge dislocations seem to be more easy to see in the field ion image. Since the Burgers vector is normal to the slip direction and in the slip plane, one finds these dislocations particularly near the zone lines of the slip planes, which are the [lll]zones on the f.c.c. lattice, and the [ O l l ] and the 11121 zones on the b.c.c. lattice. However, Fig. 39 shows an edge dislocation on the (021) plane of tungsten which lies on the [001] zone. The surface geometry that would indicate in the field ion image the presence of stacking faults in f.c.c. and h.c.p. lattices has not yet been studied. It appears that some of the jogs and off sets in the zone lines on the rhenium pattern Fig. 33 may be explained as stacking faults. Lowtemperature hydrogen ion images of a cobalt tip of 2000-A radius showed

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161

an extended system of 20- to 30-A wide stripes probably caused by stacking faults formed during the incomplete phase transition a t about 430°C. Most of the dislocations in the tip crystals seem to be immobile. If they were not pinned somehow, the tip would not be capable of standing the enormous radial stress F2/8n-of the field, which amounts to 10" dyne/cm2 or about 1100 kg/mm2 a t 500 Mv/cm. Sometimes, however, one observes a sudden glide motion of a part of the tip over a limited distance, when the field is slowly increased, and a t rare occasions in tungsten as well as in platinum a more complex dislocation system could be moved smoothly and controllably through the lattice by simply raising the voltage by a few percent, which probably resulted in t,winning. 1he possibilities of the field ion microscope for studying the motion of dislocatioris under the stress of the applied

FIG. 39. Edge dislocation on (021) plane of tungsten.

field h a w iiot yet h e n utilized. The difficulty i?, as with all other firicly detailed imperfectionp, that one finds them in most cases only by careful scanning of the photographs when the tip is no longer available for further experiments with that specific defect. When the present work on the improvement of the microscope in the direction of larger image brightness succeeds, then it will be easier to follow the events directly on the screen and to manipulate more effectively the crystal imperfections. 4. Inipuiities. The effect of impurities on the development of high index planes by field evaporation was already touched in See. V.C. Indeed the order of magnitude of impurity concentration c can he estimated by the k3rule which states that only planes with a highest Miller index I = c-% can appear. This of course applies oiily to impurity atoms that are sufficiently different from the bulk metal, e.g. either nonmetallic or metal atoms of a considerably different diameter and electron configuration. Commercial nickel wire with approximately 1% impurities, mostly man-

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ganese and silicon, develops only planes with indices up to 5, while on tungsten tips that were made from highly outgassed NS-wire or from single crystal wire, planes with indices up to 10 were found. The material of highest purity used was 99.999% platinum, which showed all planes with indices up to 21 that were compatible with the geometrical condition of sufficient polyhedron radius. No high index planes could be observed on zirconium tips, probably because of the high solubility of oxygen, nitrogen, carbon, and other contaminants which get into the interior when the surface-contaminated material is annealed in vacuum. Nonconforming individual impurity atoms on normal lattice sites can be seen in most metals when they lie within a completely depicted net plane or atom chain. I n such two- or onedimensional arrays one atom occasionally appears much weaker than the other atoms of more uniform brightness. Presumably this is an impurity atom of smaller diamet,er bound in a more recessed position (Fig. 40).

FIG.40. Impurity atoms on a rhenium crystal, indicated by reduced intensity (less protrusion) in an otherwise homogeneous atom chain. Dot on upper right dark (lOi0) plane is a single atom (position A of Fig. 36).

Impurities in iiiterstitial positions may become visible by bulging out the surface as suggested in Fig. 36. It ip assumed that the great number of bright, randomly distributed spots in Fig. 41 that appear on a tip of commercial better than 99.9% pure rhodium (67) are caused by interstitial oxygen, the concentration of which can be estimated from the photograph to be 3 X This number agrees also with the appearance of the { 3,1,15] planes as the highest indexed ones. The metal could not be depleted of these inclusions by heating in vacuum up to 900°K. The interpretation as interstitials is supported by the following t,hree other observations of such bright spots. They appear on high-purity platinum if t,he specimen was annealed in air. In contrast to rhodium the spots can be removed by annealing the platinum tip ill the 10-6-mni high vacuum of the pumped out, microscope. In well-outgnssed Pt! tips, 011 the other hand, these bright

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spots can :~IsoI)e iiitroclwxi Ijy 1ic:ttiilg tho t i p in a11 atmosphere of l W 4 mm of oxygen. The third niethod is a mild rltthode sputtering of the plutiiium tip with slow helium ions hy simply operating the gas-fillcd t ul)c with a negative tip to draw a field electron currelit of some lo-* amp for the time of a few seconds. If an oxygen adsorption film is adsorbed 011 the

FIG.41. Rhodium crystal with a number of bright spots indicating interstitials (oxygen) just below the surface (position I3 in Fig. 36).

platinum during the bombardment, some oxygen atoms are knocked into the surface, and when the helium ion image is inspected after field desorption of the surface contaminations, the bright oxygen interstitial spots appear. Controlled field evaporation reveals that they have been knocked in to a depth of 4 to 5 atom layers. Such oxygen interstitiale can also be produced by mild ion bombardment of an oxygen-covered tungsten surface. However, here the spots are not so pronounced as on platinum, prob-

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ably because of t hc leaser lat,f,ic.e distortion and hulging iiecessary for arcommodating the sniall oxygen atoms in the more spacious b.c.c. lattice. This idea about the relation between relative spot intensity and space requirement is supported by the finding that interstitials of platinum atoms in the platinum lattice as generated by radiation damage give much brighter spots than do interstitials of the smaller oxygen atoms. 5. Alloys. Very little work can be reported on the promising subject of field ion microscopy of alloys or compounds. No alloys of refractory metals were tried so far. A platinum 10% iridium alloy gave an image that can not be distinguished from pure platinum with its many high index planes. However, field evaporation occurred a t a much higher field, corresponding more to the evaporation field of pure iridium. The opposite behavior was found with platinum 10% rhodium alloy, the conventional thermocouple material. The field for evaporation a t 21°K was lower than for the pure constituents, so that photographs had to be made slightly below the best image voltage in order to prevent evaporation during exposure. Planes were only developed up to Miller index 5, in spite of the high purity of the material. There may be an unknown miscibility gap in the alloy responsible for this irregular behavior, or it may simply be an effect of a short-range order. Some exploratory work was done with tips of stainless steel, but the structure was very irregular, perhaps because of insufficient annealing of the heavily cold-worked wire for imaging with helium ions. Coming to conipounds, the attempt to use tips of titanium nitride whiskers may be mentioned. Field evaporation rate is almost too high even for neon, and besides a number of concentric net plane edges around the axial (111) plane no further details could be depicted from the apparently quite faulty crystals. The most promising alloys and compounds for the high-resolution field ion microscope are, of course, those containing a t least one of the highly refractory metals, and also their nitrides, silicides, borides, and carbides. In the past, several of these compounds have been formed in situ on emitter tips for the purpose of studying their field electron emission, and some should be strong enough to stand the high ion image field.

E. Disturbed Crystal Struclures I . Slip. One of the earliest observations with the field ion microscope was the formation of broad steps when the tip was exposed to a high field a t elevated temperature (8). It was thought that surface migration in the presence of the field builds up sharp ridges as the polarized surface atoms tend to move to places of higher field strength (19).Drechsler (55) discussed this mechanism in detail and calculated the size of the ridges as limited by the onset of field evaporation a t the crest. While this seems convincing, it cannot explain the irregularity of the ridge structure. If the ridges were

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oidy a surface effect, the perfect surface structure of the original tip should also be restored by removing the ridges with low-temperature field evaporution. Actually it turns out tjhat a profusion of defects of various types is being created throughout the entire crystal (Fig. 42). The main feature of thc high temperature-high field stress treatment of a W tip is the production

FIG.42. Slip bands on a tungsten tip produced by glide under the stress of a field of 200 Mv/cm applied at 600°C, followed by removal of protruding large steps by field evaporation a t 21" K.

of slip bands (57),lying some 50-200 A apart in the { 011) and ( 112) planes, which are known as the preferred slip planes in the b.c.c. lattice. This explanation is supported by experiments of the same kind with plabinum (Fig. 43), iridium, and nickel. The f.c.c. metals have only one set of glide planes, 1111, arid therefore the slip bands are clearly centered around the ( 1111 poles. No Drechsler-type step formation due to field-enhanced surface migration is found in the slip-free region near the (111) pole, or near the (001) plane, but a fairly large concentration of nonoperative dislocations

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caii be seen t,here. The interpretatioii of the steps as slip bands is also suggested by the observation of Drechsler et al. (47) that the onset in a given field owiirs suddenly when the temperature is raised beyond a critical value, and that it is not possible to slow down gradually the dissolution of

FIG.43. Platinum crystal with slip hands in (111) oricntntion.

the tip below a niiiiinium rate of 2 layers per second. Viewed as a slip mechanism this is iri agreement with Schmid’s law of a critical yield stress. Slip hand formation on field emitter tips certainly deserves further study t l s it niay give a clue about the mechanism by which the ordinarily pitiiietl dislocations bcrome active sources of new dislocations. The typical field stress at 200 Mv/cm and 1000°K for tungsten is equivalent to 180

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kg/mm2, which is still a very high stress compared to conventional slip experiments with macroscopic specimens. 6. Cold Working. A great advantage of the field evaporation technique is that the specimen need not be heated in order to remove surface contaminations. The cold-worked Btructure, for instance, of hard drawn wires can be seen without any annealing after etching the tip electrolytically a t low current density. Gradual field evaporation removes the surface layers and exposes the severely disordered interior structure. If an NS tungsten

FIG.44. Cold working structure in a p1:ttinum tip, ctched near the broken end in a tensile strrrigth experiment.

wirc is sharply bent and straightened out again several times and theii used for etching a tip, a number of grain boundaries roughly parallel to the wire axis are found after annealing. Such grain bouiidaries were studied by Wolf (68) with a liquid nitrogen-vooled helium ion microscope. As an interesting example of cold work we show here (Fig. 44) a platinum tip which was etched electrolytically at rooni temperature within 0.01 nini of the point where the original wire of 0.1-mm diameter broke under ft load of 0.35 kg. It is typical thnt besides the severely disturbed stmcture there is also a region of approximately 150-A diameter around a (111) pole that represents an apparently perfect crystal. No vacancies were found in this crystal using the controlled field evaporation technique to a depth of 25 atom layers.

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3. Strain Release. The immobility of dislocations that causes the extraordinary strength of the tip specimen must be connected with the small dimensions of the tip end. There is certainly some yield to the much smaller stress further down the shank where the dimensions of the crystal are much larger. This can be concluded from tJhefollowing experiment : Aft,er taking

FIG.45n. Nearly prrfcrt platinum t i p aftrr field evaporation.

the picture of a nearly perfect plstiriuni crystal (Fig. 452) 011 which only t\vo dislocations cwiild be found, the high voltage \vas turiied off and then t hr tip was w r m d up to 1~0o111ternperatiirc. The helium i v a h pumped oiit niid the tip renmined iu high v:wiuni for s e i ~ r ahours. l The11the tip w s cool~tl agaiii with liquid hydrogen, helium \vas filled into the miwoscwpe, : ~ n dt h r tip voltage slonly applied to remove the c.oiitaiiiiiitltioiis by field evapor~tion of the :dsorption film aiitl ahout two additional layers o f the tip metal

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itself. Figure 45b shows the resulting image which is characterized by a large number of lattice imperfections including dislocation spirals on the (315) and (221) planes, a subgrain boundary beginning at the lower edge of the picture near (101), passing between (113) and (221) via (203) to the (135) region, and many other individual dislocations ail over the field of

FIG. 45b. Same platiniim tip after strain release at room temperature. A copious numbcr of dislocations appeared a t the surface by creep.

view. As an explanation of the appearan~eof this profusion of imperfections we assittne that while the tip cap with its originally low dislocation density was only elasticdly strained by the iniage field, some yield occurred further down i i i the shaiik, and when the field was removed, the tip found itself under a stress from the yielded region below. A part of this strain is then released when the tip cap has time to creep at room temperature.

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Even a time of 1 hr a t room temperature is enough for platinum to bring a large number of imperfections to the previously perfect surface. Creep during a week produces a still larger abundance of defects. As a result of this experience it is not possible to design a field ion microscope experiment in which an individual specimen such as a platinum tip is first inspected for perfection, and then taken out of the microscope for some treatment, for instance neutron irradiation in a reactor, and eventually inserted again for the study of the effect of the special treatment. With a metal as soft as platinum such experiments can only be made by evaluating statistically the results obtained with a number of specimens, some of them studied without the special treatment, and others with it. However, it appears that the more refractory metals such as iridium, rhenium, and tungsten do not exhibit this disturbing effect of strain release by creep a t -room temperature. 4. Fatigue. The motion of dislocations in the tip crystal can also be activated by applying the field stress periodically (69).A simple circuit for doing this is presented in Fig. 46. The image is continuously observed during the flat top part of the voltage-time curve, while the specimen is undcr the radial stress S = F2/8n dyne/cmP. With the 60-cps half-wave amplitude as shown in the diagram the stress is periodically released to about J$.~oof its maximum value. I n the diagram the maximum strehs vectors acting radially over the surface of three typical tip shapeF, are shown to produce a shear component across the slip planes which may cause the dislocations to move. The only experiments PO far were made with platinum tips of various orientations. Changes in the origiiially perfect surface appear already aftcr a few thousand load cyclea. The degree of disorder increa gradually, while field evaporation is slowly going on, perhaps as a rcsult of a temperature rise due to energy released hy the motion of dislocations to the surface. The dynamic ficld rvnporation end form changcs i n so far as apparently more material i h removed around the (135) regions. After several hundred thouwid cycles suhgrain boundaries develop, which may be the beginning of the eventual fatigue failure (70).Fatigued platinum tips show a very strong strain release effert when left without field at room teniper:it,iire for a fern hourh, ofteii in the form of extrusion, so that after slight field cvaporutioii for cleaning up the surface the effective radiuh of the no\\ cpitc irregular tip is caonsiderably snialler than before the strain release. In order t o speed up thc numbcr of stress cycles pcr unit of time the circuit of Fig. 46 was o p r i ~ t :it ~ l higher frequencies lip t o 5000 cps ( 5 2 ) . The tip buspension has of twurse sonie rcsoiiaiiw v i h t ioiis, whirh ( Y ~ I I S C the iinngc to bc hlurrcd. Somt. w ~ o i i a ~ ~freqiicnries co w r t l f o i i n t l at \vhicah appureiitly niosi of the image rmiaiiicd i n a fisrd posit i o n , ~ h i the h nioi’~ protrudiiig parts shifted by as much as two atom dinmetcrs. This was first,

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thought to be a n indication of the specific audiof requency resonances that have been discovered by Fitzgerald (71)in a number of metals, but it seems now that the field ion microscope observations have a simpler explanation : Small, more protruding parts on the emitter appear brighter when the roltage is so much lowered that the surrounding low field region.. can no Q

0

I0

20

30

rnilli seconds

STRESS AT F ~ ~ x ' 5 0M0V K M

P

1011

DYNESICM~

106PSI

FIG.46. Schematic diagram of circuit for pulsed stress release, pulsed voltage at 60-cps operation, and distribution of stress vectors about tip surface for three typical tip shapes. Sets of (111) glide planes are shown for a (111) and a (001) oriented tip.

longer ionize the helium atoms and send their entire supply of hopping helium atoms to the protrusion. In the image of the vibrating tip the protrusions are then depicted a t a different time than the rest of the surface, so that the image of the protrusions appears to shift its position with respect to the substrate when the ac amplitude is increased.

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5 . Rudiation Duinugc. In spite of the large amount of work done IIY many investigators in the study of radiation damage, the nature of the defects is by no meaiis agreed upon at the prcseiit time. It appears that the application of the field ion microhc.opc should shed some light on the unsolved qiiest,ione. Radiation damage of perfect crystal structures is experimentally done either with electrons in the 1-Mev range, or better with heavier particles such as neutrons, protons, deuterons, and a-particles. While there can be little doubt that electrons produce mostIy vacancies and interstitials, the knock on of a metal atom by a heavy particle can result in a localized release of a h g e amount of energy of the order of lo4to lo5ev, which is assumed to cause a large disturbed region referred to as a thermal and a displacement spike ( 7 2 ) .Since some of the defects anneal a t very low temperatures, probably beginning as low as 50°K in platinum, the original plan to investigate neutron-irradiated specimens was postponed, and the present experiments were made with a-particles. A silvercoated Po2l0source of 2-mC strength, 1 cm2 in size is mounted about 1 cm away from the tip inside the microscope, so that several hits on the small tip area can be expected every hour. The 5.4-Mev a-particles are capable of penetrating any tungsten tip. The experiments show that every hit of an a-particle ie visible. Although the collimation is poor with the unfavorable geometry necessary because of the weak source, it appears that thc damage is visible only on the exit side. There a displacement of about 15 to 30 atoms occurs within an area of approximately 50 A diameter. Two thirds of the displaced spots look like intcrbticials right below the surface, and about one third of the displi~cedatoms have disappeared from the surface. This may represent the surface end of a disphcement spike. In about half of the impact events ohsrrved so far there appeared Eimultaneoiisly with the large exit disturbanre a smaller defect, such ah a singlc or up to three rlo.;ely clustered interstitials as far away from the main disturtxinre as one-half t>ipradius. The sinall defcrt may mark the exit of a high energetic knock on atom branching off the main track, or it may be the result of the transfer of energy by focuring cwllisions along close-packed atom rows (67). These firstJ experiments were made with tungsten tips, and more information can be expected when platinum tips are used which allow a better controlled field evaporation for “digging out” the displacement spike inside the crystal. Preliminary experiments with Pt have shown thc damage on the exit side of the trark to be similar to the rase of tungsten. There is no formation of a crater as could be expected as a result of enhanced field evaporation if a “thermal spike” would have raised the track temperature. Another radiation damage experiment is presently being done (with M. K. Sinha) ill which the tip is bombarded with a well-collimated beam of

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neutral helium atoms of about 20-kev energy. This beam is obtained by producing an ion beam in a sidearm of the microscope with a Penning discharge and accelerating the ions in a two-lens system. If the tip is operated a t a higher voltage than the acceleration voltage, the helium ions of the beam are deflected from the tip, while the high-energy neutral atoms in the beam, produced by charge exchange, will bombard the tip. The beam density is about 1O’O atoms/sec-cm2, so that there is about one impact per second in the field of view. However, only about 1% of the hits become visible. Contrary to the case of a-particles the impact site is then marked by an interstitial. The range of the 20-kev helium atoms appears to be about 500 A in tungsten. When the particles penetrate the cap section of the tip, of 300 to 600 A radius, the exit side shows a cluster of 3 to 5 interstitials, and a few surface atoms are missing. These metal atoms were knocked off the surface so that they rose higher than 5 A, where they were ionized and removed by the field. I n such experiments removal or displacement of a few or maybe one atom among the many thousands on the entire image can be detected by comparing the two photographs taken before and after the event. This is done most conveniently by viewing simultaneously the first picture through a red filter, and the second one through a green filter with the help of a beam splitter (4.3).With the images coinciding the details common to both photographs appear white or yellow, while the new atoms present only on the second photograph appear green. Correspondingly the atoms that are only on the first picture and have disappeared in the experiment will show up red. For scanning purposes the observation is done visually. The composite image can also be photographed on color film. Such photographs have appeared on the covers of several journals and in textbooks (48, 73). The greatest damage to the lattice seems to be caused by t,he interaction with fairly slow particles, as is the case in cathode sputtering when the gasfilled microscope is operated for a short time with a negative tip and at such a low voltage that the electron current emitted is only some to amps. Since only those ions that, are produced by collisions near the tip have a good chance to hit the cap of the tip, they will have a fairly low, but rather inhomogeneous energy, probably rarely more than 300 ev. This is a condition that often prevails in conventional cathode sputtering experiments. Ion microscopic inspection reveals severe damage to the lattice structure. Point defects, single dislocations, and complex dislocation systems are seen not only directly at the surfaces, but also to considerable depth, as revealed by field evaporation. The profuse abundance of mobile dislocations is probably responsible for the easy destruction of a field electron emitter tip which is subject to cathode sputtering in a poor vacuum. The high defect density may also be the reason for the difficulties in finding

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clear-cut relations between the physical constants of the lattice and the cathode sputtering effects (74).

F . Surface Eflects 1. Adsorption and Corrosion. The field ion microscope cannot match its electron-operated predecessor in its suitability for the investigation of the adsorption phenomena. Under the best conditions of resolution the field is so high that all surface layers will be desorbed. Oxygen films can be seen quite well a t a reduced field by using neon ions. With neon ions, or a t least with hydrogen ions it should be possible to decide the old question of whether adsorbed atoms may be seen individually as blurred spots in the field electron microwope (6, 11, 35). Thorium or zirconium with their fairly high desorption field should be the most favorable adsorbates, as they produce in the field electron pattern a t low degrees of coverage the granulation which is assumed to represent the individual atoms. Successful observat,ion of adsorbed phthalocyanine molecules on tungsten alternately with electrons and hydrogen ions were made by Melmed and Miiller (%), but the hydrogen image did not show dctails of the molecule, as the desorption field is below the best image field. Mulson and Muller (75) found a strange etching occurring when the niicroscope was cooled oiily with liquid nitrogen. I t mas first noted when the recording of t,he image directly on a photographic p1a‘r.einside the vacuum system was attempted. As soon as the cmuision faced the tip, the image began to change by moving and collapsing of the concentric iiet plane edges siniilar to the process of field evaporation. The same attack on the tip stiriwe ocwirs in an unbaked microscope operated with liquid or solid nitrogen cooling after an induction period of 1 to 15 min, during which undisturbed observation is possible. No etching of the surface is observed when the sealed tube of a design similar to Fig. 21 has been baked. In a ground joint microscope according to Fig. 22 it suffices to bake only the screen witth the help of a hot plate to prevent the etch. The etching of the surface, observed so far on tuiigsten and platinum, is definitely due to the attack by water vapor, as could be shown by deliberately introducing water vapor of about mm pressure into a baked tube. TJnder this condition the dissolution of the surface occurred very rapidly. Oxygen or hydrogen alone do not attack the tip when added to the helium gas. The amount of water vapor necessary to cause the “water etch” must be extremely small, considering the presence of the cold finger in the tube, but there is, of course, always the large dipole attraction which according to Eq. (19) increases the density near the tip. Because of its low ionization energy water cannot reach the tip cap directly while the high image field is on. The water molecules will rather approach the tip further down the shank

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where the field is too low for ionization. The induction period often observed is then due to the necessity of building up a sufficiently high.degree of coverage which will allow surface migration and chemical reaction with a metal, followed by field evaporation of the metal-oxygen or hydroxide complex. Observation confirms that the attack always starts a t the circumference of the tip cap, and moves then inwards towards the center. By properly manipulating the applied voltage it is possible to use this etching process to sharpen the tip by field evaporating more material from the side of the cap than from the center. Tips of a high degree of perfection have been obtained this way with radii down to 50 A, which were stable when the tip temperature was lowered to stop the dissolution process. Water etch may present a problem to the operation of a ground joint microscope tube with only liquid nitrogen cooling. This, however, can be solved by a proper tube design which prevents evolution of water vapor from the screen and by improving the vacuum with a getter. Adsorbed oxygen atoms have been seen as individuals on a platinum crystal a t 21°K. Since at that temperature the saturation pressure of oxygen is of the order of lo-'" mm, a continuous supply at about mm was provided besides the 1p of helium. An oxygen film can then be established on the tip by lowering the voltage for a few seconds in order to allow the approach of the oxygen. Best observation is possible on the protruding and easily resolvable (120) plane. In one experiment the best image voltage was 14,300 v. At 9000 v, where (120) is already well depicted, one sees a complete oxygen film covering this plane. At 11,500 v only a few individual oxygen atoms are seen hopping around on (120). They appear weaker than the platinum atoms on that plane, and are too mobile to be photographed. If the voltage is slightly lowered, the film on (120) is replenished from the more recessed surroundings, while at 12,000 v the plane is completely free of oxygen. At still higher fields the gas also desorbs from other areas of the tip, and at the best image field reached a t 14,300 v all the oxygen has disappeared. Hydrogen atoms are too small to protrude enough from a metal surface to be depicted. They only fill in the spacings between the metal atoms and reduce the contrast and resolution either when added as a small percentage to the helium gas, or when the microscope is operated with hydrogen ions. When an oxygen film is allowed to form on a tungsten or a platinum surface that had been made perfect by field evaporation, and the film is then removed by field evaporation slightly below the best image voltage, it turns out that the new metal surface is not exactly identical with the one before the adsorption. A few per cent of the surface atoms have been removed with the oxygen film, even if the tip is kept a t liquid hydrogen

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temperatuie all the tinic. If :HI oxygcii film-rovcrcd t m g allowed to warm up to room temperature for a short time, the c.orrosion of the hiirface is fouiid to have gone as deep a h 3 to 4 a t o m layers (67).Up to the present time it was always assrimed that, low-pressure adsorption forms a closed monoatomic film on a surface without doing aiiy harm to the structure of the underlying metal sui face. 2 . Field-Induced Chemical Reactions. Copper field evaporates too fast to give a picture with helium ions. When an attempt was made to use other gases, it was fouiid that hydrogen behaves anomaloudy. The field evaporation rate as easily measured by the rate of collapsing net plane rings around the (111) or the (001) plane is very large, while nitrogen at the same field strength gives a steady picture. The field evaporation rate a t a given field strength is proportional to the hydrogen prefsure. I t must therefore be coiioludcd that a field-induced chemical reaction is taking place, and that the reactioii product, probably copper hydride is more easily field dcsorbed thaii the metal atom itself (27). Presently, a mass spectroscopic aiialysis of the field evaporation product is being prepared to find out its nature. Although the formation of copper hydride ions is iiot yet definitely established, oiie might well speculate that, some other metals might also form hydrides under the polarization conditions in a field of some 200 Mv/cm. Under ordinary conditions the most easily formed hydrides are the ones of the alkali metals in which hydrogen acts as the electronegative part. One might suspect that in the field-induced reaction the alkali-like metal ions listed in the section on field evaporation on p. 141 may be able to form positively charged metal hydride ions. Iiideed, the observation showed the same features for beryllium, silicon, iron, and zinc a s for copper, namely, the anomalously high and pressure-proportional field evaporation rate in hydrogen. Nickcl, gCrmanium, and mercury were not checked. On these metals the effect is least pronouiiced on iron, and very strong on zinc, silicon, and beryllium. The ring collapsing rate of iron was found to be one layer per second a t 2 p of hydrogen a t about 200 Mv/cm. When the same iron tip was viewed with iicon ions a t a field of 380 Mv/cm, the surface was perfectly steady. When 5 7 , hydrogen was added to the neon, there was a fast rate of field evaporation noticeable a t fields below about, 280 Mv/cm, and the neon ion-depicted rings stopped collapsing when the field was raised so that no hydrogen could approach the tip any more. The effect of hydrogen on cobalt is about the same as on iron. 1he idea that these field-induced reactions are quite diff erelit from ordinary chemical activity is underlined by the observation that there is no increase of field evaporation rate by hydrogen in the case of zirconium or tungsten. Since all these experiments were made in a nonbaked microscope, there exists a possibility that the field-induced reaction with hydrogen involves oxidation or water

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formation followed by “water etch,” but the failure to observe a hydrogen reaction on zircon or tungsten favors the explanation of a hydride-like ion. This question will be resolved when the mass spectroscopic analysie is available. It should not be surprising if in the future, field ion microscopy reveals some more field-induced chemical reactions since by using this tool we can now control, by the turn of a knob, fields that are equal in magnitude to the ones acting between the ions in chemical compounds.

ACKNOWLEDGMENTS The author wishes to acknowledge thankfully the help of the past and present co-workers of the Field Emission Laboratory a t the Pennsylvania State University. Particularly Dr. K. Bahadur, Mr. J. F. Mulson, and Dr. R. D. Young have contributed much to the results reported in this paper. The author is also obliged to several other colleagues and to publishing agencies for allowing the use of some of their material. The continuous and encouraging support of most of the work in this laboratory by the Officeof Scientific Research of the TJ. S. Air Force is gratefully recognized. REFERENCES 1. Oppenheimer, J. R., Phys. Rev. 81, 67 (1928). 2. Rausch von Traubenberg, H., 2. Physik 66,254 (1929). 3. Lanczos, C., 2. Physik 68, 204 (1931). 4. Muller, E. W., 2. Physik 106, 132 (1937). 5. Muller, E. W., 2. Physik 106, 541 (1937). 6. Dyke, W. P. and Dolan, W., Advances in Electronics and Ekctron Phys. 8,89 (1956). 7. Muller, E. W., Natunoissenschaften 29, 533 (1941). 8. Muller, E. W., 2. Physik 136, 131 (1951). 8. Muller, E. W., J . Appl. Phys. 27, 474 (1956). 10. Bethe, H. A. and Salpeter, E., in “Handbuch der Physik,” 2nd ed., Vol. 35, P. 321. Springer, Berlin, 1956. 11. Good, R. H., Jr., and Miiller, E. W., in “Handbuch der Physik,” 2nded., Vol. 21. p. 185. Springer, Berlin, 1956. 1%. Finkelnburg, W., “Einfiihrung in die Atomphysik,” 3rd ed., p. 145. Springer, Berlin, 1954. 13. Inghram, M. G. and Gomer, R., 2. Naturforsch. 10a,863 (1956). 14. Kirchner, F., Natunoissenschaften 41, 136 (1954). 15. Inghram, M. G. and Gomer, R., J . Chem. Phys. 22, 1279 (1954). 16. Muller, E. W. and Bahadur, K., Phys. Rev. 102, 624 (1956). 17. Drechsler, M. and Pankow, G., Proc. Intern. Conf. on Electron Microscopy, London, 1954 (1956). This paper is a 1956 revised form of the 1954 lecture (private com-

munication). 18. Becker, J. A., Solid Stale Phys. 7 , 416 (1958). 19. Miiller, E. W., Ergeb. exakt. Naturw. 27, 290 (1953). 20. Young, R. D. and Muller, E. W., Phys. Rev. 113, 115 (1959). 21. Miiller, E. W., unpublished data (1953). %2. Muller, E. W., 2nd Field Emission Symposium, Pittsburgh (1954). 93. Gomer, R. and Hulm, J. K., J. Am. Chem. SOC.76, 4114 (1953). 94. Miiller, E. W., 2. Elektrochern. 69, 372 (1955). %<5. Beckey, H. D., 2. Naturforsch. lCa, 712 (1959).

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26. Muller, E. W., Phys. Rev. 102, 618 (1956). 27. Muller, E. W., 5th Field Emission Symposium, Chicago (1958). 28. Honig, R. E., RCA Rev. 18, 195 (1957). 29. “Handbook of Chemistry and Physics,” 34th ed., p. 2180. Chemicitl Rubber Co., Cleveland, Ohio, 1953. YO. Muller, E. W., J . Appl. Phys. 26, 732 (1955). 81. Gomer, R., J . Chem. Phys. 31, 341 (1959). 52. Kirchner, R. and Kirchner, H., 2. Naturforsch. 10a, 394 (1955). 33. Moore, G. E. and Allison, H. W., J . Chem. Pliys. 23, 1609 (1955). 34. Becker, J. A., Trans. Faraday SOC.28, 151 (1932). 36. Muller, E. W., 2. Nuturjorsch. Sa, 473 (1950). 36. Melmed, A. J. and Muller, E. W., J . Chem. Phys. 29, 1037 (1058). 37. Kirchner, F. and Ritter, H. A.,Z. A’aturforsch. l l a , 35 (1956). 88. Herron, R. G., University of California Radiation Laboratory Report UCRL-3001 (1955). 89. Cooper, E. C. and Muller, E. W., Rev.Sci. Instr. 29, 309 (1958). 40. hfuller, E. W., 2. Naturjorsch. l l a , 88 (1956). 4 1 . Miiller, E. W., Z. Physik 120, 270 (1943). 43. hliiller, I?. W., Ann. Meeting Electron Microscope SOC.of America, Penn. Stole Uniu., 1065. 48. Miiller, E. W., J . Appl. Phys. 28, 1 (1957). 44. Miiller, E. W., Ann. Physik [6] 20, 316 (1957). 46. Mason, E. A. and Vanderslice, J. T., J . Chem. Phys. 31, 594 (1959). 46. Drechsler, M. and Henkel, E. 2. ungew. Physik 6, 341 (1954). 47. Drechsler, M., Pankow, G., and Vanselow, R., 2. physik. Chem. (Frankfurl) [N.S.] 4, 249 (1955). 48. Muller, E. W., Umachuu 67, 579 (1957). 49. Young, J. R., J . Appl. Phys. 26, 1302 (1955). 60. Hanle, W. and Rau, K. H., 2. Physik 133, 297 (1952). 61. von Ardenne, M., “Tabellen der Elektronenphysik, Ionenphysik und Ubcrmikroskopie,” Vol. 1, p. 581. Deut. Verlag Wissenschaften, Berlin, 1956. 62. Muller, E. W., I V . Intern. Kongr. Elektronenmikroskopie, Berlin, 1968, Vol. 1, p. 820 (1960). Springer Verlag, Berlin. 53. Drechsler, M., ZV. Intern. Kongr. Elektronenmikroskopie, Berlin 19668 Vol. 1. p. 835 (1960). Springer Verlag, Berlin. 54. Muller, E. W., 6th Field Emission Symposium, Washington, D. C. (1959). 56. Drechsler, M., 2.Elektrochem. 61, 48 (1957). 66. Drechsler, M., Bull. Am. Phys. SOC.[2]3, 265 (1958). 57. Miillcr, E. W., Acta Met. 6, 620 (1958). 68. Miiller, E. W., Bull. Am. Phys. SOC.[2] 3, 69 (1958). 69. hluller, E. W., 2. Physik 166, 399 (1959). 60. Lazarcv, B. G. and Ovcharenko, 0. N., Doklady Akad. Naufi 8.S.S.R. 100, 875 (1955). 61. Ascoli, A., Asdente, M., Germagnoli, E., and Manara, A., J. Phys. Chem. Solids 6, 59 (1958). 62. Read, W. ‘l‘ Jr., ., “Dislocations In Cry8taI8.” McGraw-Hill, New York, 1953. 68. “Growth and Perfection of Crystals,” Rept,. Intern. Conf. (Doremus, Roberts, and Turnbull, eds.). Wiley, New York, 1958. 04. Drc~hsler,M., 2. MdallL. 47, 305 (195(i). 6‘5. Muller, F,. W., J . Appl. Phys. 30, 1843 (1!159).

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66. Eshelby, J. D., J . Appl. Phys. 24, 176 (1953). 67. Muller, E. W., in “Structure and Properties of Thin Films” (C. A. Neugebauer, J. B. Newkirk, and D. A. Vermilyea, eds.), p. 476. Wiley, New York, 1969. 68. Wolf, P., 6th Field Emission Symposium, Washington, D. C. (1959). 69. Miiller, E. W., Bull. Am. Phys. Soc. [2] 8, 265 (1958). 70. Muller, E. W., Pimbly, W. T., and Mulson, J. F., in “Internal Stresses and Fatigue in Metals” (G. M. Rassweiler and W. L. Grube, eds.), p. 189. Elsevier, Amsterdam,

1959. 71. Fitzgerald, E. R., Phys. Reu. 108, 690 (1957). 72. Brinkman, J., J . Appl Phys. 26, 961 (1954). 73. Muller, E. W., Sci. American 197, 113 (1957). 74. Wehner, G., Advance3 in Electronics and Electron Phys. 7 , 239 (1955). 76. Mulson, J. F. and Miiller, E. W., 6th Field Emission Symposium, Washington, D. C. (1959).

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Velocity Distribution in Electron Streams P. A. LINDSAY Research Labordories of The General Electric Company Ltd., Wetnbley, England

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 A. The Physical Basis of the Problem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 B. Discussion of Some Simplifying Assumptions. . . . . . . . . . . . . . . . C. Brief Summary of References on the General Problem of Electro D. Some Recent Advances Excluding the Effect of Velocity Distribution.. . 184 11. General Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 A. Particle Mechanics.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Liouville's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Maxwell-Boltzmann Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 111. Probability Considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 A. Sample Point, Sample Space, and Random Variables.. . . . . . . . . . . . . . . . . 190 B. Probability and Probability Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 C. Distribution Function and Probability Density Function. . . . . D. Current Density and Its Probability Density Function., . . . . . . . . . . . . . . 196 IV. Velocity Distrihution of the Electrons Emitted by a Thermionic Cathode. . 197 A. Velocity Distribution inside the Emitter.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 B. The Current Density of the Emitted Electrons. C. The Electron Density a t the Surface of a Plane D. The Distribution of Velocities a t the Surface of a Plane Emitter. . . . . . . . 202 E. Electron Density and Velocity Distribution a t the Surface of a Curved Emitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Generalizations. . . . . . . . . . . . V. Velocity Distribution in Plane A. General Considerations. ... 13. A Plane Diode. . . . . . . . . . . C. A Plane System with Two Emitting Cathodes.. . . . . . . . . . . . . . . . . . . . . . . 236 D. A Plane Triode or Tetrode.. . . . . . VI. Velocity Distribution in Cylindrical Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 250 A. General Considerations . . . . . . . . . 250 B. The Cylindrical Diode.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 VII. Velocity Distribution in the Presence of a Magnetic Field.. . . . . . . . . . . . . . . . 294 A. General Considerations. . . . . . . . . . . . . . . . . . . . . . . ....... 294 B. The Plane Magnetron.. . . . . . . . . . . . . . . . . . . . . . . 295 VIII. Experimental Support for the Th ....... 308 A. Electron Scattering or Interaction between Individual Electrons.. . . . . . . 308 H. The Half-Maxwellian Velocity Distribution of the Emitted Electrons. . . 30!1 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 References. . . . . . . . . . . .... ....... . . . . . . . . 311 181

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I. INTHODUCTION

A . The Physical Basis of the Problem In electronic engineering it is often necessary to calculate the paths of free electrons moving in an external field of force. Such a field may be either electric or magnetic in nature or, as is often the case, the electric and magnetic components of the field are both present, their directions being perpendicular. (In general the electric and magnetic fields could be either stationary or varying in time, but in what follows only stationary fields will be considered.) When the external field of force is known, for example from the shape and potential of the electrodes, it is possible, in principle, to calculate the trajectories of electrons entering the field. As a rule, such a general problem is sufficiently complicated to require some simplifying assumptions. These assumptions will be discussed in Sec. I,B. At this point it is only necessary to point out that one of the difficulties encountered in the calculation of the electron trajectories concerns the velocity spread which the electrons acquire on their emission by a thermionic cathode. The main object of this article is to review the work which has been done in the past in calculating the velocity spread in the flow of an electron stream. Furthermore, the article aims a t clarifying the general problem of velocity distribution by presenting a concise method for deriving the effects which a velocity spread must have both on the surrounding field and on the electron stream itself. The method of approact adopted in this article is based exclusively on the powerful concept of electron density in a sixdimensional space composed of configuration and momentum spaces joined together. It will be assumed in this that the electrons can be represented by a smooth cloud of charge-thus the local interactions among individual electrons will be entirely neglected in the treatment adopted here. This simplifying assumption seems to be fully justified on purely physical grounds, the interaction between individual electrons and their time of flight being generally small.

B. Discussion of Some Simplifying Assumptions In considering the flow of electrons in an external field of force it is usual to make various simplifying assumptions. When the fields are stationary tjhese assumptions affect mainly: (1) the influence which the free electrons exert on the surrounding field and (2) the velocities with which they enter the field. Here the initial velocities may either possess a natural spread due to the emission of the electrons from a thermionic cathode or their spread may be very small when the electrons enter the field through a positively charged electrode (grid).

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183

The problems mentioned under (1) are usually referred to as spacecharge effects. being associated with the presence of charged particles (electrons) in the interelectrode space. The space-charge effects are notoriously difficult to calculate since they represent a sort of equilibrium between the free charges and the external field. For a stable flow this equilibrium must extend over the whole of the interelectrode space. For obvious reasons the space-charge effects are ignored, whenever possible, the paths of the electrons being calculated on the assumption that the presence of the free charges does not affect the structure of the external field at all. Such an assumption is justified whenever the volume density of the electrons is small. The calculated electron trajectories then give a good approximation to the actual paths of the electrons, always provided, of course, that the initial velocity and position of entry of the electrons are known. The problems mentioned under (2) refer exclusively to the initial conditions. It is usual, except in discussing the problem of noise, or when the space-charge effects are neglected, to assume that the electrons enter the tield in a continuous fashion. Furthermore, their initial velocity is assumed to be either: (1) nonexistent, (2) single-valued, or (3) distributed according to some well-defined probability law. The first assumption is normally made either when calculating the path of a single electron, or, more often, in calculating the potential distribution when the space-charge effects are taken into account, for example in the well-known case of potential distribution in a space charge-limited diode. This assumption may lead, however, to solutions which are singular, as is probably the case in some well-known examples of potential distribution in a magnetron. The assumption of zero initial velocity should therefore be avoided whenever possible. The second assumption, of a single initial velocity, is usually more sstisfactory. It is quite accurate, in fact, when the electrons enter the interelectrode space through a grid which is at a high potential relative to the cathode. Then the initial velocity spread due to thermionic emission becomes negligible compared with the high velocity of entry governed by the positive potential of the grid. This assumption is less satisfactory, however, when the electrons are emitted from a thermionic cathode directly into the interelectrode space. Yet, even then, it is still possible to assume with reasonable accuracy that they possess a single initial velocity given, for example, by the root-mean-square value of the velocity distribution. This approximation, however, breaks down completely when it is necessary to consider the sorting out process introduced by a potential minimum, or when the initial velocity distribution is fundamental to the problem, as is the case for example in the discussion of a “thermionic electron engine” (1, 2 ) .

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The third assumption is, of course, the most realistic of them all. Here, it is fully recognized that the electrons are emitted with varying velocities ranging from very small to very large, and the only simplifications refer to the neglect of the corpuscular nature of the flow and to the details of the cathode surface which is assumed to be equipotential and perfectly smooth. When the space-charge effects are neglected, it is possible in this case to calculate a number of trajectories corresponding to different initial velocities and positions of emission and to obtain in this way some idea of the electron current density. When the space-charge effects are taken into account, however, such a procedure becomes extremely laborious. It is then best to adopt a different approach altogether. Instead of calculating the trajectories of small groups of electrons endowed with a given mean initial velocity it is preferable to look at the electron stream as a whole. Suppressing all discontinuities in the stream, it is possible to accommodate the variations in the initial velocity of the electrons by considering their flow not in the three dimensions of configuration space but in the six dimensions of phase space consisting of the configuration and momentum spaces joined together. Variations in the initial velocities of the electrons then appear as a probability distribution in the velocity space. This approach to the calculation of velocity distribution in electron streams will be adopted here.

C. Brief Summary of References on the General Problem of Electron Flow For economy of space it is not possible to include in this article a detailed discussion of the laws of electron ballistics. However, it seemed desirable, for the convenience of the reader, to provide a list of references where the necessary information could be found. They may be subdivided roughly into three groups: those which cover in a fairly general manner the whole field of electron ballistics (3-16'), those which consider in particular axially symmetric systems (17-24), and finally those which are largely concerned with electron flow in the presence of space charge (25-29). Any three of these references, one from each group, should be sufficient as an introduction to this article.

D. Some Recent Advances Excluding the E$ect of Velocity Distribution All problems of electron flow are strongly interrelated, irrespective of the simplifying assumptions. Although this article is primarily concerned with the actual velocity distribution in electron streams, it is of interest to review here very briefly some recent advances in calculating electron flow when the initial spread of velocities is neglected. Since with the advent, of digital computers the problem of electron

VELOCITY DISTRIBUTION I N ELECTRON STREAMS

185

tracing in the absence of space charge has become almost a matter of routine, most recent efforts have been concentrated on the much more difficult problem of electron flow in the presence of space charge. The general problem of space-charge flow was reviewed some time ago by Ivey (27) and, in the case of highdensity electron beams, more recently by Siisskind (28). Since then, notable advances have been achieved on several fronts. Starting with numerical solutions of the problem the work on electrolytic tanks (30, 31), digital computers (Sd), and other iteration processes has been sufficiently advanced to tackle the problem of electron trajectories in crossed fields (33) and in the presence of space charge (34-37). Another method of approach, based on the theory of analogs (38),mostly in the form of simple resistance networks (7,39), has been recently extended to include space-charge effects (40-42);a t the same time it was shown that the Cauchy type of boundary problem was basically unsuitable for this type of approach (43-46). This brings to mind the extensive field of high-perveance electron beams. Here further experimental and theoretical work has been done on periodic focusing using magnets (47-50) and some new methods of electrostatic focusing (51-56). Instabilities (57-60) and spurious oscillations have been investigated both in beams (61-63) and other devices (64). Some general aspects of electron beams (65-73) and in particular the details of their structure (74-79) have been considered by several authors and some striking advances have been made recently in the difficult field of general two-dimensional, space-charge flows (8G87).Further, new advances have been made in the discussion of plane, cylindrical, and spherical diodes (88, 89) and the problem of the steady state in magnetrons has been extensively reviewed (90-102). Finally, the effects of relativistic velocities (103) and the associated magnetic fields have been discussed by some authors (104,105).It should be added here that this brief survey makes no mention (with the exception of spurious oscillations) of any time-dependent phenomena such as space-charge waves, noise, electron plasma oscillations, etc. The vast field of neutral plasmas has not been considered here either.

11. GENERALCONSIDERATIONS

A . Particle Mechanics Consider a stream of electrons moving in a stationary field which is partly electric and partly magnetic in nature. The Hamiltonian, which in this case is equal to the total energy of an individual electron, is then given by (3, 106, 107)

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P. A. LINDSAY

The corresponding Lagrangiaii can be writteii iii the form

L

=

+

f%mgklQkq' e ( I$

- qkAk).

(2)

Here m and - e represent the mass and the electric charge of an electron, g = (ql, q2, 9") and p = (pl, p2, p s ) being the generalized coordinates and momenta. The latter are defined by the usual expression

where the metric of the configuration space is given by the metric tensors gk' and g k l . The remaining two quantities appearing in Eqs. (1) and (2) are, respectively, the electrostatic potential 4 and the magnetic vector potential A = ( A 1 ,A2, A3).Here, as usual, the subscripts and superscripts denote, respectively, the covariant and contravarian t vectors, the summation extending over all quantities in which the same subscripts and superscripts appear twice. The introduction of the generalized coordinates and momenta is necessary for the discussion of systems with cylindrical geometries, Sec. VI, but a note of warning on the use of curvilinear coordinates should be added at this point. In Eqs. (1) and (2) the quantities A4 are the components of the generalized vector potential A = (A1, A 2 , A 3 ) and not merely the projections of the ordinary vector potential A = (Azl A,, A , ) in the directions of the generalized coordinates Q = (q1, q2,a*). For example, in the case of cylindrical coordinates the covariant components of the generalized vector potential B a r e ( A , = A,, A2 = rAB,A 3 = A,) whereas the projections of the ordinary vector potential in the direction of the cylindrical coordinates are: A , = A , sin 0 A , cos 8, A B = -A, sin e A , cos 0, A , = A,. The contravariant components of a vector can be obtained from its covariant components by using the relationship Ak = gk'Al. In all cases the square of the magnitude of a vector is always defined as AkAk. This difference between the covariant and contravariant components of a vector must always be born in mind since in particle mechanics some quantities transform as covariant and other quantities as contravarian t components. I n Cartesian coordinates Eqs. (1-3) reduce to a more familiar form

+

1 x={ p + eAI2 - eI$, 2m p = -aL =

av

+

(4)

mv - eA.

Here the poeition and velocity vectors are respectively defined by r = (z, y, z ) and v = (v,, v,, s,) and the differential a/av = (a/avz, a/&,, a/av,),

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

187

as i t 1 reference 108. Equatiotis (1-3) :tiid (4-6) differ merely hy the fact that

in Cartesian coordinates the metric tensors gkL and gk’ become unit tensors. ItJ should he pointed out that Eqs. (1-6) will be subject to several simplif’yitig assumptions. First of all it, will he assumed that the function 4 = +(q) represents the effect of the external electrostatic field and the smoothed-out effect of the moving charges (the “space-charge” effect) but that it does not account for the local variations due to short-range forces of interaction between individual electrons. Furthermore, the magnetic forces introduced by the movement of the electrons (e.g. “pinch effect”) will not be allowed for, A representing the vector potential of the external magnetic field only. In view of these simplifying assumptions i t can be seen that Eqs. (1-6) now represent in fact the movement of a small element of volume of the electron stream rather than the movement of a single electron, but for brevity the latter description seems justified whenever it does not lead to contradictions. The electron trajectories, in the sense of the previous paragraph, can be calculated from the well-known canonical equations

These equations, together with the appropriate boundary conditions, will be used extensively in the following chapters for the calculation of the so-called “limiting trajectories” of the electron stream. At this stage, however, they are required for the derivation of an important theorem of particle mechanics.

B. Liouville’s Theorem Partial differentiation of Eq. (7) with respect to qk and of Eq. (8) with respect to p k shows that, as long as the order of differentiation can be reversed, the sum of the differentiated equations is identically equal to zero, or, using the summation convention

a _ dqk _ + - -a= od .p k apk

dt

apk clt

The six terms of Eq. (9) can be compared to the divergence of a generalized vector in the six-dimensional phase space, defined by the three generalized configuration coordinates q l , q2, q3 and the three generalized momentum coordinates pl, p2, p3 all a t right angles to each other. I n such a phase space Eq. (9) can be written in the form

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P. A. LINDSAY

where e = ( q l , q2, q3, pl, p2, p3) has the properties of a sixdimensional velocity. Since, when there are no sudden jumps either in position or velocity of the particles, the continuity equation for the flow in such a phase space requires that

Eqs. (10) and (11) can be satisfied for a steady flow only when the phasespace volume density (12)

n(q,p,t) = n(q,p> = const

along any given electron trajectory or line of flow of the electron stream. Thus, as the stream moves through the phase space, a given volume defined by a large number of individual electrons or points in the electron stream remains constant, although its shape may change. This, in brief, is the substance of the Liouville's Theorem of classical mechanics. Since it is based on the general properties of canonical equations it embraces all the laws of motion of particles.

C. Maxwell-Boltzmann Equation

It is of interest to note that Eq. (12) can be derived also from the Maxwell-Boltzmann equation of statistical mechanics (108). Consider a gas in which each molecule (electron) is moving in a field of force. Further assume that there exists a density function n(q,p,t) so defined that n(q,p,t)dqdp gives the number of molecules which at time t occupy the volume element (9, q dq) and have momenta in the range (p, p dp), where q and p are the generalized coordinates and momenta and d q = dq1dq?dq3,d p = dpldp2dpl. At some later time t dt the position vectors and momenta of the molecules which did not collide will have changed to q &it and p pdt, so that the same number of molecules is now given by n(q Qdt, p pdl, t dt)dqdp. However, due to collisions, some electrons will leave the set and other electrons will join it, so that eventually

+

+

+

+

+

+ +

+

The term on the right-hand side of Eq. (13) represents the change in the function n(q,p,t) caused by the collisions. Division of both sides of Eq. (13)by dqdpdt leads to the following expression

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

189

which is the celebrated Maxwell-Boltzmann equation. It provides an expression for the total variation of the phase-space volume density n(q,p,t). If Eq. (10) of the previous section is now multiplied by n(q,p,t) and then added to Eq. (14) it can be found after a simple rearrangement of terms that

an at

a a a + aq7 (nq') + (np*)+ a!l aq2 8-

a a a +(nSl) + - (npz) + -(nP,) aPz 8P3

(n4j3)

aP1

This, except for the collision term a,n/at, is identical with Eq. (11). Thus, the Maxwell-Boltzmann equation can be interpreted in a descriptive way by saying that intermolecular collisions act as sources (or sinks) for the flow of particles in the six-dimensional phase space. When the collisions, or, in the case of electrons, the short-range interactions, are neglected, the flow in the sixdimensional phase space becomes continuous. In view of Eq. (10) the flow also becomes incompressible when the system is in a steady state. In other words the change in the shape of electron trajectories with time is now represented by the corresponding variations in the density function n(q,p,t). I t should be noted that although for a steady state the function n(q,p) remains constant along any particular stream line, its value for a given stream line depends on the abundance of the clectrons in the surrounding element of phase space (q,q dq,p,p dp). Finally it should be noted that the condition of flow specified by Eq. (12) is equivalent to saying that the lines of flow in the sixdimensional phase space can never cross, although they may close on themselves. This is of particular importance in the treatment of electron motion in the preEence of a magnetic field, where the frequent crossing of the electron paths in the configuration space adds greatly to the mathematical difficulties of the problem. This statement can be understood by remembering that two trajectories which may cross in the z,y-plane, for example, will be separated in the phase space defined by the x, y, vz, and v, coordinates because of their different velocity components. This is similar to the case of two straight lines which although in fact skew, may appear to cross when projected on a plane.

+

+

111. PROBABILITY CONSIDERATIONS

It is now necessary to consider briefly those ideas borrowed from the theory of probability which will be useful in the discussion of velocity (or momentum) distribution in electron streams. For a more detailed exposition of the theory of probability the reader is referred to suitable textbooks on the subject (109-116).

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P. A. LINDSAY

A . Sample Points, Sample Space, and Random Variables Before introducing the concept of probability it is necessary to discuss first the idea of a sample space and then the associated idea of a random variable. Both concepts are fundamental in the discussion of most probability problems. Consider the classical experiment of tossing a coin, which then falls to the ground with either its “head” or its “tail” facing up. I n the theory of probability it is customary to call the result of such an experiment an event. I n general, consecutive events can be either independent or they may depend on each other, further they may be simple, when their decomposition into simpler events is impossible, or they may be compound. However, when the events are simple, it is usual to refer to them as the sample points, since they are then capable of defining all the theoretically possible outcomes of an idealized experiment. In the case of tossing the coin there are two such sample points, called “heads” and “tails.” I n the theory of probability an experiment is fully defined when all the sample points referring to that experiment are known. The collection of all the sample points is called the sample space. In considering velocity distribution of a n electron stream the idealized experiment takes the form of a n observation of the position and velocity (or momentum) of a minute element of the stream. The corresponding sample point is the result of such an observation, the sample space consisting of all the possible results of such observations, which, in this case, will be m 6 in number. The next concept to be considered is that of the random variable. In principle the introduction of a random variable is equivalent to the process of labeling the sample points in the sample space. I n the simple case of tossing the coin such a labeling is quite arbitrary-the two points of the sample space (“heads” and “tails”) could be called, for example, 0 and 1 and plotted on the real axis. In the case of velocity distribution however it is more natural to w e the actual numerical result of any given experiment as the label for the corresponding point in the sample epace. Then to each point in the sample epace will correspond one and only one point in the six-dimensional phase space (q,p) defined in Sec. 11. At this stage it is important to note the difference between the function called the random variable which performs the labeling operation and its value a t a given point which fixes this point with respect to all the other points of the sample space.

I3. Probability and Probability Distribution It, is now necessary to discuss very briefly the actual concept of probability. The exact meaning of this concept is still liable to arouse strong passions

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

191

among niaiiy, :wd little would lie gained Iiy laking sides a t this point. Most, people however seem to agree that onre the concept of probslility has lwei) ncwpt8ed,a ~iumherof mathematlical operations involving probubility c m be performed. Since this article has a strong eiigirieeritig bias it would be most appropriate to define the probability of a n event by considering its relative frequency of occurrence. This approach, although somewhat crude, seems to work whenever the distributions are discrete, that is, whenever the number of sample points is finite. It breaks down, however, when distributions become continuous as, for example in the case of velocity distribution in an electron stream. Then it is necessary to define the probability of an event in terms of a function built on the sample points which are infinite in number so that the theory of probability becomes to some extent a branch of the theory of measure (109, 111).

C. Distribution Function and Probability Density Function 1. A Single Random Variable. In the case of continuous distributions the probability distribution can be specified by means of a distribution ]unction, given by

F(x) = Pr(X

6 x).

(16)

The right-hand side of Eq. (16) expresses the probability of an event in which the random variable X 6 x. By definition the distribution function is a monotonically increasing function which extends from F ( - w ) = 0 to F(+ a ) = 1, where unity represents a probability amounting to certainty. The derivative of the distribution function with respect to 5 is called the probabzf7ty density (or frequency) function; it is defined by

so that,

F ( W ) = J f ( x ) d x = 1, where the limits of integration are (- w , m). These limits of integration apply to all the other integrals in this section-they will be omitted for convenience in printing. Probability distributions are often characterized by their moments. A moment of order n is defined by a,>=

F:(X*) = J.n](x)d.

= (xn),

(19)

where E { X r 1 }is the expectation value of the nth power of the random variable X . The most important among these definitions are the first moment of the distribution, which represents the mean (x)or the expectation value of the variable itself, and the second moment which is often

P. A. LINDSAY

192

called the mean-squaw value or (&'). Often it is also useful to consider the so-called centrnl (moments of the distribution. They are defined by p, =

E ( (X - a,),} = J(2 - a,)nf(s)dz = ((z

- ,,In>

(20)

where aIstands for the mean value of the distribution. By definition PO = 1 Pl = 0 p2 = a2

- a?.

(21)

The most important among these momenta is the second central moment which is often called the variance of the distribution, p2 =

D"X]

=

E ( X 2 ) - E2JXI = (22) - (x>2 = d.

(22)

The quantity D ( X ) = u is called the standard deviation. As a rule the larger the value of the standard deviation the flatter is the peak of the corresponding probability density curve. 2. Two Random Variables. Equations (16-22) can be generalized to cover the case of several random variables. Consider for brevity the case of two random variables. It is then customary to define the joint distribution function, given by F(z,y)

=

Pr(X


and

Y

< y}

(23)

where X and Y are the two random variables. The joint probability density function is now given by

and F ( - m ,- a ) = P(z,- w ) = F ( - cu ,y) = 0. The joint distribution function, Eq. (23), leads to the so-called marginal distributions, i.e. to the distribution of each of the random variables separately,

Fl(z) = F ( z , m ) = P r ( X 6 z and Y < w ) , Fz(y) = F ( m , y ) = P r ( X < and Y 6 y]. Q)

The corresponding marginal probability denszfy functions are given by and

(26) (27)

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

193

In the case of two random variables it is also possible to define two conditional distributions,

F(xly) = lim P r ( X 6 sly du-0

and

6 Y6

y

+ dy)

< ylx 6 X 6 x + dx) Pr(x < X 6 z + dx & Y 6 y). = lim P r ( x 6 X 6 x + dx)

F(ylz) = lim Pr{Y lIZ-10

dz-0

(31)

Here, Eq. (30) for example, expresses the probability of an event X 6 x on the hypothesis Y = y. The conditional probability density functions corresponding to Eqs. (30) and (31) are given by

and

As in the case of a single random variable it is now possible to define the moments of the distributions. The moments a,,, and the central moments ,g of the joint distribution, Eq. (23), are given respectively by amn=

E {XmYn}= JJzrnynf(x,y)dx& = (~"y")

(34)

and gmn

=

E { (X - ~ r i o ) ' ~ ( Y

( ~ 0 1 ) ~= )

JJ(. - alo)m(~ - dnJ(x,Y)dxdY = ((5

- a d m ( y - aoJn). (35)

Here a , and ~ aOn are respectively the nth moments of X and Y taken separately, that is, they are the nth moments of the respective marginal distributions (fl(z)and f 2 ( y ) .The central moments c(20

=

E [ (X - a 1 0 ) Z j

= a20 -

a102

= (x2yO) - (zy">"

(Yo?

=

(36)

and poz =

E / ( Y - aoJ2)= an.. 1

(x"y?) -

(."2/)2

(37)

194

P. A. LINDSAY

are respectively the variances of X and Y , whereas fill

=

E { (X- a101 ( Y - ao1) 1 = a11 - a10ao1 = (ZY> -

(Z>(Y>

(38)

is normally called the covariance of X and Y . The function

is referred to as the correlation coefficient of X and Y . The function p is equal to 0 when the two variables are independent and it is equal to 1 when a functional relationship exists between them. It is still possible to define the moments of the conditional probability distributions, Eqs. (32) and (33). These moments, called regressions, are given by a,(y) = E ( X m ( Y= y j = JZmf(Zly)dZ= ( ~ ( y )

(40)

an(z) = E'{Y"JX = 2) = Jy"j(y(z)dy= (yn1z).

(41)

and For m = 1, for example, Eq. (40) represents the regression or fuactional dependence of E ( X }on the value of the variable Y = y. 3. Position and Momenturn as the Random Variable. In the rase of the velocity distribution in an electron stream it is convenient to adopt the notation which was originally developed by Chapman and Cowling (108). This notation is based on the concept of a six-dimensional phase space consisting of configuration and momentum subspaces (see Sec. 11). Consider a density function

N ( q , p ) being the number of electrons in the phase space. The density function given by Eq. (42) is in all respects similar to the function n ( q , p ) already defined in Sec. I1 of this article. (For convenience of printing the elements of the subspaces are written in the form d q = dq1dq2dq3and d p = dpldp2dpa.) I n terms of the density function the number of electrons in an element of volume around a point ( q , p ) is given by d,,N(q,p) = n(q,p)dqdp.

(43)

Here an element of volume d q d p is chosen sufficiently small to permit only a small variation of the denpity functioii and yet sufficiently large to coiltaiii eiiqugh electronr for the density function to be virtually smooth. Integration of Eq. (43) with respect to p gives the number of electrons of all possible velocities contained in a n elrment of configemtion subsp:K!e rlq r/,N(q.p) = 4 q ) m (44)

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

195

whcre It (9) is iiow the generalized voliinie density of the part,irles. Similwrly an integration of Eq. (43) with respect to q gives the total number of electrons in the momentum range (p,p dp), irrespective of their position in the configuration subspace,

+

dd\r(q,p) = 4P)dP.

(45)

Here n(p) is the generalized momentum density of the electrons. It is now possible to define a joint probability density junction given by

where Ntot is the total number of electrons. Integration of Eq. (46) either with respect to p or q leads to two marginal distributions which are characterized by the following probability density functions

and

The conditional probability density junctions based on the joint probability, Eq. (46), are now given by the following expressions

and

Here the conditional hypotheses are expressed reFpectively by X, = q and = p, X, and X, being the position and momentum random variables. It is worth noting that in this case the distribution function F(q,p) can be defined as [see Eqs. (23, 24, 42, 46)]

X,

F(q,p) = P r ( X , 6 q and X,

< p} = N(q P) ’ Ntot

(51)

196

P. A. LINDSAY

that is, the probability is expressed in terms of the relative frequency of occurrence of individual electrons. This definition would suggest a discrete probability distribution and a discontinuous probability density function. However any reference to the “number of electrons” in the definition of N(q,p) is more symbolic than real since the function itself must be subjected to the normal smoothing process before it can be differentiated a t all [see the qualifying remarks following Eqs. ( 6 ) and (43)].

D. Current Density and its Probabi1,ity Density Function Since the electrons are invariably in a state of motion it is necessary to consider the contributions of various velocity groups to the total current density at different points in the stream. In general an element of the generalized current density is given by

Here Jk(q) and Jk(q,a) are respective components of the generalized current density and of the generalized velocity-current-density vectors. These components are in the directions of the generalized coordinates ql, q2, qs, the corresponding element of ordinary current density being Jk(S) = l/gkk/(gkJ(Jk(q),the argument s indicating that Jk(S) gives the rate at which electrons pass an element of area dsidsj = &$qi &dqi (in technical literature gll, gzz,ga3are often called hl, hz, ha). In Cartesian coordinates Eq. (52) reduces to

d,J(r) = vn(r,p)dp = vn(r,v)dv = J(r,v)dv,

(53)

the vectors J(r) and J(r,v) being in the same direction. Integration of Eq. (52) with respect to q gives the k-component of the generalized current density at q. Jk(q>= SQk49,Q)dil (54)

A similar integration of Eq. (53) gives J(r) = Jvn(r,v)dv = iJv,n(r,v)dv

+ jJu,n(r,v)dv + kJv.n(r,v)

where i, j, and k are the unit vectors in the directions of z, y, and z. It is now convenient to define a velocity contribution function

(55)

197

VELOCITY DISTRIBUTION I N ELECTRON STREAMS

where (J(q)j2is the sum of the squares {Jk(q)}z. For Cartesian coordinates Eq. (55) reduces to

fdplr) =

+

WrlP) - ~24r,P)Jz(r) v,n(r,P)J,(r) ~

J (r)

J2W

+ vzn(r,p>Jz(r).

(57)

In generalfJ(p1q) is not a probability density function since it may become negative. The function is useful however in analyzing the positive and negative contributions of various velocity groups to the total current density at a given point q. It will be shown in Sec. V that the function fJ(plq) can be defined in such a way that it acquires the properties of a probability density function of the current density J(q). Then fJ(plq)dp represents the conditional probability of occurrence of a momentum range (p,p dp) in the current density at q; that is, on the hypothesis that the electrons had crossed an element of area at q in the configuration subspace (113). Writers on thermionic emission have of ten overlooked the fundamental difference between the functions f(p]q) and f ~ ( p ] q[(8) ) and (9) are two notable exceptions to this rule]. From the physical point of view the difference becomes clear when the electrons with zero velocity are considered. These electrons can play a prominent role in the composition of the velocity distribution function, the function often having a maximum at the origin, but they contribute nothing to t,he current density function. By similar reasoning one can 6nd that in symmetrical distributions, due to the cancellation of contributions from equal but oppositely directed velocities, the total current density at a given point may be zero in spite of a very large local electron density n(q). Equations (50,52, and 56) lead to the following relationship which holds between the two functions f(plq) and fJ(plq)

+

Here t.heexpreesions in angular brackets are the usual first moments defined by Eq. (41) and taken with respect to the probability density function f(p1q). For Cartesian coordinates Eq. (58) reduces to

IV. VELOCITY DISTRIBUTION OF THE ELECTRONS BY A THERMIONIC CATHODE EMITTED For the full discussion of tharmionic emission the reader is referred either to textbooks (8, 9, 11, 13, 114-11 6) or to special review articles (14, 117-

198

P. A. LINDSAY

120)on the subject. Here it is only necessary to derive the basic expressions for the velocity distribution of the electrons emitted by a thermionic cathode. For this purpose consider a metal, such as tungsten, which is known to be a good thermionic emitter. As is normally the case in solids the electrons in such a metal can possess only certain energies, which are grouped on the energy scale in bands, as shown in Fig. 1. The energy bands in tungsten are all fully occupied except for the topmost or conduction band which is half full. There the electrons can change their energy by small amounts and, if the changes are coherent, give rise to an electron current. (No such current would be possible if all the energy levels in the band were

D(7CONDUCTION

BAND

'

'

Ky\)(

FORBIDDEN BAND

FORBIDDEN BAND

FIG.1. Energy bands in a metallic emitter. El = Fermi level, Eg = work function, E, = total potential barrier.

occupied.) At absolute zero the electrons occupy all the lowest energy levels of the conduction band, shown hatched in Fig. 1. When the temperature of the metal is above zero, some electrons gain enough energy to cross the broken line of Fig. 1 and move from the hatched to the dotted area. The number of such electrons increases with temperature and a t some sufficiently high temperature the electrons gain enough energy to surmount the potential barrier existing on the surface of the metal and leave the interior of the metal altogether. Since in this case the electrons leave the metal entirely under the influence of heat, the process is generally called thermionic emission.

199

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

A . Velocity Distribulion. i i d e the Emitter Assuming that the electrons in the conduction band are virtually free, that is, that the influence of the individual atoms in the interior of the metal can be neglected when considering the energy of these electrons, it is possible to show that the phase-space density of the electrons, when they are in a state of equilibrium, is given by n,(r,v)

=

G

(F)~ {1 + exp (2) exp

= 'G(3"{1 2

(%)]-I

+ tan h a($

-

g)}.

Here r = (z,y,z) is the position vector, v = (u,,uy,v,) the velocity of the electrons, T the absolute temperature of the emitter, m the mass of a single electron, Er the energy of the Fermi level (in metals this energy is usually measured from t,he bottom of the conduction band), h the Planck constant, k the Boltzmann constant, and G the so-called occupation number, which in metallic emitters is usually equal to 2. Equation (60) can be derived from quantum mechanical considerations, assuming that the changes in the energy of the electrons in the conduction band are purely kinetic. Since, in general, ne(rlv) = d2N(r,v)/drdv, for homogeneous distributions n,(r,v) = n,(V)/(volume of metal), where n,(v) is the density function normally quoted (11, 115) and defined by the statement that n,(v)dv gives the number of electrons dN with velocities in the interval (v, v dv). From Heisenberg's Principle ApAq 2 h3 so that the quantity h3 appearing in the denominator of Eq. (60) limits the final accuracy with which the function n,(r,v) can ever be known. Figure 2 shows n.(r,v) as a function of v2. For T = O'K, ne(rlv) = 0 for h' > E I ; that is, all electrons have energies E' 6 h'r and are situated in the hatched area of Fig. 1. When T > O'K, ne(r,v) > 0 for E > E'r so that some electrons will occupy the energy levels in the area shown dotted in Fig. 1. However, due to the nature of the distribution given by Eq. (60), the number of such electrons will be relatively small. The number of electrons which in addition could then overcome the potential barrier a t the surface of the metal would be even smaller. Furthermore, if the surface of the emitter is a plane perpendicular to the x-axis it is not enough for the electrons to have a total energy sufficient to surmount the potential barrier, but their 5irnv2 component of kinetic energy alone must be sufficiently large to overcome the barrier. Thus, even at temperatures of the order of 2000°K only a very small percentage of the free electrons will escape from the interior of the metal, the condition for their escape being given by > ~ r n u z 2>> Er. Substitution in

+

a00

P. A. LINDSAY

ne mox,

inemax

..-- - - ---- - _ _--

Eq. (60) shows that the phase-space density of the electrons which are likely to take part in the emission is given by

the unity in the denominator of Eq. (60) becoming negligible compared to the rapidly increasing exponential term in v2 (for metals G = 2).

B. The Current Density of the Emitted Electrons Calling the minimum energy necessary for an electron to overcome the total potential barrier of the emitter E, = $$rnvzr2 and using Eq. (55) it can be shown that the current density of the electrons emitted from a plane cathode perpendicular to the x-axis is given by

J,

=

/

vn,(r,v)dv

the remaining integrals in the directions j and k being zero. Introduction of a new variable v,? = vz2 - vm2 reduces this equation to a more convenient form

VELOCITY DISTRIBUTION I N ELECTRON STREAMS

201

where Eg = E, - Er is the so-called work function of the emitter. When multiplied by -e, Eq. (61) is identical with that given by Dushman, Richardson, et al. (121-125) for the maximum emission current density. Here it has been derived in a comparatively simple manner using the powerful phase-space density method and neglecting the reflections which may occur at the boundary of the emitter [for metals the mean coefficient of transmissivity is approximately equal to unity (126, 127)l. It should be noted here that the quantities E+ and El. are not true constants but depend on temperature. This dependence, which in metals is slight, can be explained by the fact that the integration of Eq. (60) with respect to r and v must give the same total number of electrons whatever the temperature of the metal. Thus, as can be seen from Fig. 2, the constant El must be slightly different for different temperatures in order to give in each case the same value for the area under the curve. For $.jmvz = El, the value of the phase-space density n.(r,v) always drops to one-half of its maximum value. This property is often used as a definition of Er. Equation (61) has been derived on the assumption that the electron gas inside the emitter remains in a state of equilibrium in spite of the continuous loss of electrons caused by the thermionic emission. This aesumption is quite justified, however, since, as has been already mentioned, the number of electrons actually emitted from the cathode is very small compared to the total number of “free” electrons inside the emitter. In most practical cases conditions are further improved by the fact that a potential minimum is allowed to develop in front of the cathode. This may cause the return of something like 95% of the emitted electrons, and virtually creates nearequilibrium conditions in the spa(-c bctwcen the surface of the emitter and the potential minimum. C. The Electron Density at the Surface of a Plane Emitter

From Eq. (55) and from the integralid of Eq. (61), it can be eeeu that the phase density at a point roimmediately outside the surface of an emitter must be given by

202

P. A. LINDSAY

where the subscripts e and 1 have now been dropped. Integration of Eq. (62) wit>hrespect to v gives the volume density at the surface of the emitter nT

=

1 Im lm(7) (y)’ (2) dv,

-a

dv,

n(ro,v)dv, =

2akT

exp

(63)

-m

where the subscript T has been introduced to indicate that this particular value corresponds to the volume density at the so-called temperaturelimited cathode, that is, when all electrons are drawn away from the vicinity of the cathode after they have been emitted. If all electrons return to the cathode, for example, when acted upon by an infinitely strong retarding field, the volume density of the electrons at the surface of the cathode becomes equal to

(64)

which is of course twice the value of nT. Subscript eq indicates that this particular volume density prevails outside the emitter when a state of equilibrium exists between the interior of the emitter and the surrounding vacuum. The thermionic emission current and its density are then, of course, equal to zero. For “temperature-limitsd” emission, 0 6 v, < m at the surface of the emitter. In the state of equilibrium, that is, when all electrons return to the emitter with the velocities they had on leaving it, - m < v2 < w . I n both cases - 00 < v, < w and - w < v, < a.

D . The Distribution of Velocities at the Surface of a Plane Emitter From Eq. (50) the probability density function of velocity distribution on the hypothesis X, = rois given by f(vlro) =

=

(E)‘ akT (A) 2akT exp (- 2 2k T (vz2 + vy2 + v?).

(65)

Equation (65) refers to the “temperature-limited” operation of an emitter. For the state of perfect equilibrium between the interior and the exterior of an emitter, the corresponding probability function is given by

As was to be expected, Eq. (66) gives the usual Maxwellinn distribution of velocities typical of a state of equilibrium of a nondegenerate electron gas. Ihcept for the constants w1~it.h are indicative of thc c.orrespoiiding i’:~iigcsof iiitcgratioii ntid, i t t tririi, tlcpctid ott thc r:~tigcof thc nv:dnl)lc

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

203

velocities, Eqs. (65) and (66) represent the same type of function. This function is often referred to as the Maxwellian distribution of velocities. When all velocity ranges are present this distribution is typical of the state of equilibrium, and its graph is shown in Fig. 3. In the case of Eq. (65),

FIG.3. Probability density function for the Maxwellian distribution of velocities.

when there are only positive directed velocities it would be more correct to call such a distribution half-Maxwellian, in order to avoid confusion with Eq. (66) The graph of a half-Maxwellian distribution of velocities is shown in Fig. 4.

I

t FIG.4. Probability density function for the half-Maxwellian distribution of velocities.

E. Electron Density and Velocity Distribution at the Surface of a Curved Emitter

It is now necessary to express E ~ F(62-66) . in terms of the generalized coordinates q and the generalized monierita p. From the definition of the phase-space density functions n(r,v) and n ( q , p ) , it is possible to write the following idcntit,y for the total number of electrons ill ail element of volume tlN = 11 (r,v)tlrrlv

=

11

(4,p)rlqdp.

((i7)

204

P. A. LINDSAY

Since, in general,

and

where

are the respective Jacobians of the transformation; the expression for dp can be written, with the help of Eq. (3),

or

Thus, combining Eqs. (68) and (70) drdv = ~n-~dqdp

as could have been expected from the fact that an element of phase space remains invariant under a canonical transformation. Substitution of Eq. (71) in Eq. (67) reveals a perfectly general relationship which exists between the two density functions

n(r,v) = m3n(q1p).

(72)

Bearing this in mind it is possible to see that immediately outside the surface of a cathode the phase-space density of the electrons is given by [see Eq. (62)]

+

where gk[CjkQ' = m-2gk1(pk-I-eAk)(pr eA1) has been substituted in place of v2. An integration of Eq. (73) with respect to p gives the generalized volume density of the electrons immediately outside the emitter. For a

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

205

cathode limited by ql = const. and a rectangular system of generalized coordinates ( g k l # 0 for k = 1, g k l = 0 for k # l),

where the subscript zero indicates the value of the determinant at the cathode. Equation (74) is a generahed expression for w. It reduces to Eq. (63) for the Cartesian coordinat.es when the Jacobian IgklIO' = 1. If all electrons return to the cathode with the velocity a t which they left it, the generalized volume density immediately outside the emitter becomes equal to

For Cartesian coordinates [g&'i = 1 and Eq. (75) rzduces to Eq. (64). By definition, Eq. (50), the conditional probability density function for the case of temperature-limited emission is given by

Similarly, in the case of perfect equilibrium outside the emitter,

=

{ &

(271%kT)-'l~kl10-f*eXp -

gk'(pk

+ eAk)(pz + e A d } . (77)

Both functions, Eqs. (76) and (77), are probability density functions and are equal to unity when integrated with respect to p .

F . Generalizations Equations (60) and (66) have been derived for a simple thermionic emitter, such as pure tungsten, They are also valid, however, for the more elaborate cathodes, for example, composite surface cathodes, (W-Th) , oxide cathodes (BaO with excess of barium) or even the more recent impregnated dispenser cathodes. In the case of composite surface cathodes,

206

P. A. LINDSAY

Eqs. (60-66) are valid since the adsorhed layer, be it ions or dipoles, only affeck t8hevalue of the work function E+ and leaves the interior of the emitker undisturbed. In the case of oxide cathodes, whirh are really semiconductors by nature, the internal structure of the emitter differs, of course, from that of a metal, but the equation obtained for the emission current density is still similar to Eq. (61) (115). There are two reasons for this. First of all, although in semiconductors the distribution of energy among electrons in the conduction band is Maxwellian rather than Fermi-Dirac, this merely cuts out some steps in the derivation of Eq. (61) and does not affect its algebraic form. A more serious difficulty arises in connection with the fact that in semiconductor@the work function E6 depends strongly on the temperature T of the emitter. This, however, only affects the temperature dependence of the emission current and leaves the velocity distribution of the electrons as before. (Slight departures from the Maxwellian distribution of velocities have been reported, however, when pore conduction takes place.) There is one more alteration of Eq. (61) which has to be remembered in the case of composite cathodes. The coefficient 2 in the expression for ne(r,v)is merely a specific value for the product of the occupation number and the transmissivity coefficient and although this value is approximately correct for tungsten it has to be altered in the case of oxide cathodes. All these changes depending on the physical structure of the emitter do not affect, however, the reeults obtained in the following sections of this article since these results are based entirely on the general form of the velocity distribution which is half-Maxwellian in nature.

V. VELOCITYDISTRIBUTION IN PLANE SYSTEMS A . General Considerations Plane systems represent the simplest configuration both from physical and mathematical points of view. Consider for example two plane electrodes perpendicular to the x-axis and situated a certain distance apart. Assuming that the electrodes are sufficiently close together for the fringing effects to be safely neglected, the electric field in such a system will depend on one coordinate only. It is now possible to impose different boundary conditions on the individual electrodes of the system. Assume that one electrode is a thermionic cathode, the other being an anode. The electrons emitted by the cathode can either reach the anode and be absorbed by it [for the discussion of reflections see (128)]or they can return to the cathode. Such a system represents an idealized diode and it will be considered in Sec. V,B. Then assume that both electrodes are thermionic cathodes and are facing each other. If the temperatures of the two cathodes are different, such a system represents

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

207

a simple thermionic engine (1, 2 ) and it will be discussed in Section V,C. Finally assume that the first electrode is an idealized grid which is situated between a cathode and an anode. If the grid is kept a t a potential which is positive with respect to the cathode, the velocity distribution between the grid and the anode will differ from that in a plane diode (129). Such a system is typical of a power triode or a tetrode and will be discussed in See. V , D .

B. A Plane Diode

A plane diode was the first system in which the effect of the Maxwellian distribution of electron velocities was investigated (130-159). Although both Richardson (140) and von Laue (122) had calculated a t a n earlier date the effect of the Maxwellian distribution of elertron velocities on the potential distribution outside the cathode, their results were limited to systems in a state of equilibrium, or in the case of Laue, near such a state (that is, when the currents between the electrodes were vanishingly small). The general solution of the problem of potential distribution between two plane electrodes allowing for the Maxwellian distribution of velocities was given independently first by Epstein (131) and then more fully by Fry (132) and Gans (133). The problem was subsequently reviewed and discussed by Langmuir (134), who also enlarged some of the numerical tables. Later Rakshit (135) recalculated the volume density of the electrons--obviously unaware that a table of this function was already available in (133).A discussion of the whole problem again appeared some time later in connection with the calculation of the electron transit time (1%). Except for the tables in (133) all these references are now of historical interest only. More recently the subject has been revived by Kleynen (137) who published very accurate table? of the potential distribution function and by van der Ziel (138) and later Ferris (139), who presented an up-to-date discussion of the whole problem and provided it with a large number of valuable graphs. Quite recently Ivey reviewed the problem as part of his article on space charge-limited currents (27). The most recent contribution in this field came from Poritsky (141) who again derived the volume density function and the potential distribution between the electrodes in connection with his discussion of the concept of temperature and pressure as applied to an electron stream; and from Auer and Hurwitz (I,$la) who consider the effect of space-charge neutralization by positive ions. In all these references with the exception of (131 and 141), the algebra of the problem is made unnecessarily complicated by the insistence on expressing the right-hand side of the Poisson’s equation in terms of thv current density rather than the volume density of the electrons. This is a curious aberration-it occurs right a t the beginning of a long line of argu-

208

P. A. LINDSAY

ment and adds greatly to the complexity of an otherwise straightforward problem. It is hoped to show here that this pitfall can be avoided and that the subsequent argument can be made simple and clear. 1. Equations of Motion-Simpli$ed Notation. In the case of a one-dimensional field of force which is defined by E = (E,,O,O),B = 0, 4 = 4(x) and A = 0, one can obtain, from Eqs. (4-6) and (8),

p = mv and

_ dP# - 0. dt

Multiplying Eq. (79) by 2p,, integrating with respect to t, substituting from Eq. (78), and then dividing both sides by m, gives

where the integration of Fqs. (80) and (81) presents no difficulty. Here the subscript nought represents the values of the various quantities at t = t,that is, a t the cathode. Equations (82-84) are the first integrals of motion in a one-dimensional electrostatic field of force. As a rule they are sufficient for the solution of the problem of velocity distribution since the main difficulties always arise with the determination of the right limits of integration in the velocity subspace, the limits of integration in the configuration subspace being clearly defined by the position of the electrodes. If it were also neceesary to obtain the second integrals of motion in order to find the potential distribution between the electrodes, the chances of success using the phase-space density function would be indeed slender. The addition of Eqs. (82-84) gives the following well-known energy relationship between the velocity components of an electron as it moves from the cathode to a point x in the interelectrode space

Substitution of Eq. (85) in Eq. (62) leads to the following general expression for the phase-space density of the electrons

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

=

(

2% 2rk T

exp

{& (4 - do)} exp { - $$ ( u 2 4- + UP)}. v,2

2M

(86)

Since Eq. (86) occurs throughout this chapter it is worth-while to simplify its notation. P ut

e

(4

-

dm)

= Ill

where $,I is a constant which will be defined later. An inspection of Eqs. (87-90) shows that the energy has been expressed everywhere in terms of kT, 3BkT being the average kinetic energy of a molecule of gas in a state of equilibrium a t temperature T. (For T = 1000°K the coefficient e/kT = 11.605 per volt and m/kT = 0.3299.10-'O sec2/meter2).The notation indicated by Eq. (87) is already well known (134, 137, 158) and Eqs. (88-90) are merely further logical extensions of it. Equations (82-84), when expressed in terms of the new variables, acquire the following simple form

To complete the transformation it is now necessary to redefine the phase-space density function. Since a small increase in the number of electrons is given by dN

=

n(r,v)drdv = n(rlw)drdwl

(94)

and since

the following general relationship holds between the two phase-space density functions

210

P. A. LINDSAY

Su1)stit~ritiugEqs. (87-90) aiid (98) ill Eq. (86) gives the following simplified expression for the phase-space density of the electrons

+ +

n(r,w) = nrr27r-96 cxp (q - 7”) exp { - (wa wy” w12)1.

(97)

This function has the following intereeting properties: (1) it is independent of y and z , that is, it remains constant in the yand z-directions, and (2) it can be separated into a multiple product n ( z , w z ) n ( y , w u ) n ( z , w = ~) n(w,o)n(wuo)?z(wzO).Since the probability density function f(r,w) = n(r,w)/NtOt[see Eq. (46)], the property of statistical independence (109112) of motion in the three directions of space is preserved under the influence of a one-dimensional electrostatic field of force.

--d---

B

-

2. Density and Probability Density Functions. In the presence of space charge the potential distribution between the cathode and the anode has a general form shown in Fig. 5 . Here the subscripts nought and a refer respectively to the conditions a t the cathode and at the anode of the valve. In g@eral, the function = $J(x)has a minimum 4 = +,, at a point x = xm. The plane x = xm divides the interelectrode Epace into a region in which the electrons are retarded and a region in which the electrons are accelerated. These two regions are called reepectively a! and p, as shown in Fig. 5. In order to obtain the volume density of the electrons it is necessary

VELOCITY DISTRIBUTION I N ELECTRON STREAMS

211

to integrate Fq. (97) with respect to W. To do this the limits of integration for the velocity variable w must be found. At the cathode the h i t s of integration for the w,- and utz-componentE of velocity are, from Eqs. (89), (go), and Fig. 3, - ~0 < wuo < CQ and - CQ < wzo < ~ 1 Since . there are no coniponents of the field in the y- amd 2-directions, 20, and w Lwill remain constant and equal to their value a t the cathode for the whole of the interelectrode space, as shown by 12qs. (92) and (03). Thus, in general, for a plane systcm and in the absence of a magnetic field -03

--co

< w, < < w z<

00,

(98)

CQ.

(99)

In the 1-direction the physical conditions are, of course, quite different. Duc to the presence of the z-component of the electric field the 10,-component of the velocity will vary between the cathode and the anode of the

FIG.ti. The z-component of the reduced velocity w 2 as n function of the distance hetween the electrodes, Ench curve corresponds to n different initial velocity w,o,

valve. Figure 6, the so-called phase-space diagram, shows the function w, = wz(x), Eq. (Dl), for different values of the initial velocity 7uz,,. The dingram sliows very clearly that ill t,hc P region of the vnlvc, whole ~ ; z i i g c ~ of the low vclocitics of thc elcatroris are missing. This is simply due to the

212

P. A. LINDSAY

fact that even tho5e electrons which started with an initial velocity w d = (q(xo)) M just sufficient to clear the potential minimum will have acquired a velocity component w, = {&TO) ~ ( s-) ~ ( z o ) ) = % {~(x))!4 by the time they have reached the point x, xm < x 6 x4. Thus, a t that point there will be no electrons with velocities w, < { q ( x ) ) xand the lower limit of integration will be w, = g x rather than w, = 0. In the a! region of the valve the electrons which have started with the initial velocity wz < (~(z0))fS can never pass the potential minimum in front of the cathode and must return to the cathode, giving rise to the so-called "double stream," as shown by the trajectories extending below the z-axis in Fig. 6 . Thus the lower limit of integration is obtained in this case by putting

+

w, = - ( d x o )

+ o(z) -

tl(Z0))S

=

-(Il(x)}~.

It is assumed here that the electrons which leave the cathode with the initial velocity W,O = { v(xo)) % have a fifty-fifty chance of either passing the minimum or returning to the cathode (from the probability point of view these electrons form a set of zero measure and an assumption that they all return to the cathode or that they all reach the anode would make no difference). To summarize, the limits of integration for the w,-components of the velocity are given by

-+ 6 w, < qf*

6 wz <

a, co.

for xo 6 x < sm,a region for zm 6 x 6 z4,p region

(100) (101)

Finally, it should be added that the limiting trajectory w, = F { q ( z ) ) x is the only one which meets the x-axis at an angle which is different from 90") 6 = tan-' a z f i )where a2 is a coefficient in the polynomial expansion q = U ~ ( Z- 2),'

+ aa(z - z,)~ + . . . .

The volume density of the electrons n(r) can be obtained now by integrating the function n(r,w) with respect to W.

n(r) =

/ n(r,w)dw

=

dw, /-mm

dw,

where erf x

= 27r-56

I"

n ~ 2 7 r -exp ~ ~ (rl - 1101 exp { - (wz2

+ w,2 + w.'> Ww,

exp ( - u 2 ) d u , and the upper and lower signs refer

respectively to the expressions for the a and /3 regions of the valve.

213

VELOCITY DISTRIBUTION IN ELECTRON STBEAMS

Since, from Eq. (87),q = 0 at z = x,,,,the volume density of the electrons at that point is given by n(rd

=

(103 1

n~ exp (-70).

Since for q = 0 the function erf qfs meets the q-axis at right angles, the derivative of n(r) with respect to q also tends to infinity as q + O . The function n(r,w) is shown in Fig. 7. Since the function is constant with respect to y, z, w,, and w., a three-dimensional space is sufficient for its complete representation. In Fig. 7 the function is plotted as a surface above the x,w,-plane. The function, that is, the height of t.he surface above the x,w,-plane, remains constant along any of the electron trajectories shown in Fig. 6. In particular it remains constant along the limiting trajectory wz= f q f a . For constant wz the function varies as exp ( q ( x )

FIG.7. The phase-space density of the electrons as a function of

2

and w,.

- q ( x o ) ) . For constant x the function n(r,w) a exp (-wz2). The latter relationship suggests that except for a constant coefficient, the velocity distribution at a given point can be obtained by cutting off the Maxwellian distribution of Fig. 3 a t the point w, = TqM[see Poritsky (141) and p. 64 in (29) for the drawings of the distribution]. The new velocity distribution obtained by this process may be called part-Maxwellian, just as the distribution in Fig. 4 is sometimes referred to as half-Maxwellian. It can be seen from Eqs. (102) and (103) that, starting from z = x,,,, the volume density steadily decreases in the direction of the anode and increases in the direction of the cathode. The function n(r) has been tabulated by Gans (133) and Rakshit (135) and a graph of the function is shown

214

P. A. LINDSAY

in Fig. 8. The difference in the behavior of the function in the two regions of the valve can be explained by recalling the physical properties of these regions. Thus, region (Y contains not oiily all thc electrons which will eventually reach the anode but also the remairiiiig electrons which, due to their low initial velocity, are turned back by the retarding field. Region p on the other hand contains only those electrons which have passed the potential minimum at 2 = xmand are now safely on their way to the anode. Hence, in region Q the reduction in the volume density is caused by the fact that some electrons fail to reach far enough into the interelectrode space,

I

1

FIG.8. The volume density of the electrons a8 a function of the redured potential 7.

their reduced velocity, if anything, helping to increase the volume density. In the /3 region on the other hand the reduction in the volume density is due entirely to the electrons being accelerated there. Putting q = 770 in Eq. (102) gives the volume density of the electrons immeditltely outside a thermionic cathode

+ erf qo3*] = +.in,,( 1 + erf qof*) . (104) Since, by definition, qo is always positive, the function 1 + erf qo>* lies n(ro)=

nT(1

between 1 and 2; it becomes equal to 1 for qo = 0 ; that is, for do = 4m, and it rapidly tends to 2 for large values of qo = ( k T / e ) ( d o- dm). Thus the volume density of the electrons immediately outside the cathode depends strongly on the depth of the potential minimum in front of it and varies between nT and neq.In order to have n(ro)= 0.95ne, it is only necessary to put qo = 1.355 which corresponds to $o - c $ ~ = 0.1 volt a t T = 858°K and 4o - +, = 0.2 volt a t T = 1716'K. Integration of Eq. (97) with respect to r gives the corresponding electron density in the velocity subspace.

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

n(w>=

n(r,w)cir

= nT2r-35 exp

(-w2)

exp (-qo)(y2

- yl)(zp - zl)

for -756

u’,

[exp qdx <0

(-qo)(y2

- yl)(z2 - 2,) for 0 6 wz< 735

exp vdx

exp ( -w2) exp (-qo)(y2 - yl)(zz - zl)

exp qdx.

= nT2n-35 exp

= nT2a-35

6

(-w2> exp

for q35

< w.

<

215

03

(105)

Here x’ and x” are the two roots of the equation w,2 = q, the electrons extending in t,he y- and zdirections from y1 to yz and from z1 to ZZ. The integration can be performed numerically using the tables of the function 9 = rim, (1S7, 1 4 w . Further integration of either Eq. (104) or Eq. (105) gives the total nurnber of electrons in the interelectrode space.

where the electrodes are again assumed to extend between y1, y 2 and 21, 2 2 . For convenience of notation the function G(xO,x,) has been substituted for the sum of the two integrals with respect to x. The probability density function f(r,w) and the two marginal probability density functions f(r) and S(W) can be now expressed, as shown by Eqs. (46-48), in terms of the various electron density functions. Dividing Eq. (97) by Eq. (106) gives the joint probability density function

Similarly, dividing Eqe. (102) and (105) by Eq. (106) gives the two niarginal probability density functions

-

esp 711 - erf q H J (yz - yd(zz - z1)c;(xo,x,)

in the p regioii

(108)

P. A. LINDSAY

226

and

f ( w )= / f ( r , w ) d r

=

n(w> NtOt

The two conditional probability density junctions can now be obtained, following the definitions given by Eqs. (49)and (50).

and

+ + wZ2)1

- 27r-95' exp { - (wZ2 wu2 (1 - erf q%)

for the /3 region

(111)

Equations (110) and (111) have the following physical significance. Assuming that the electron pospesses a velocity w = (m/2kT)Nv, the quantity f ( r ( w ) d rgives the probability that it could be found in an element of volume (rrr d r ) . Similarly, assuming that the electron is at r , f ( w 1 r ) d w gives the probability that its velocity will be in the range (W,W d w ) . In thermionic emission, Eq. (111) is the most commonly used probability density function. Unfortunately it is seldom made clear that this function, in fact,

+

+

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

217

refers to a conditional probability, the condition imposed on the random variable X, being X, = r. It should be added that for q = 0, that is, a t the point x = G,,, the function f(wlr), when multiplied by (m/2kT)36 [see Eq. (96)], is the same as the corresponding probability density function of the emitted electrons, Eq. (65). Furthermore, comparing Eqs. (62) and (86) it becomes clear that, as far as the B region of the valve is concerned, the potential dip - (40 - 4,,,) in front of the cathode is equivalent to an increase in the cathode work e(4& &) [see also (14) p. 36ff., and (115),Secfunction from E, to E, tion 4.21. For this reason the plane x = xmis often referred to as the "virtual cathode," to suggest the physical conditions a t that point. 3. Current Density Function and i t s Probability Distribution. It is of importance to consider the contribution of different velocity groups to the total electron current density at a point r. From the definition of the current density, Eq. (55), one can obtain

+

=

+

(z)' 1

wn(r,w)dw

+ wy2 + wZ2)] dw,

exp 1 - (wZ2

(112)

Here the integrals in the j- and k-directions are both zero. This is entirely due to the fact that in the y- and z-directions the velocity distributions are symmetrical, Fig. 3. Thus, for a given value of wuor w,,the positively or negatively directed velocities are equally probable, the over-all drift of the electrons in either the y- or z-directions being zero. The only contribution to the current density can now come from the nonsymmetrical distribution in the x-direction, that is, in the direction perpendicular to the cathode, Fig. 7. From Eq. (112) the only nonzero component of the current density is given by

= nT2,+4

(5)"exp (q

qo) exp ( - q >

Here the contributions from the electrons in the velocity range - q H 6 w z qx cancel out and the two limits of integration -q34 and qx give the

<

21 8

P. A. LINDSAY

mine vnlw for t8heintegral with respect to w,. This fact makes ICqs. (113) and (113) valid for both the a and the @ regions of the valve. When 70 = 0, that, is, 40 = 4, t,he potential minimum occurs right at the cathode ancl a11 emittcd electrons reach the anode. If the correspondiiig value of the current density is called the saturation current J,, Kq. (113) can be expressed in the more familiar form J z ( r ) = J , exp (-70) = J , exp { - ( e / k T ) ( & - &J) = J..

(114)

Equation (114) shows that the current density in a plane diode decreases rapidly as the potential dip qo = (e/kT)(4o - @J, increases. However, for a given potential distribution the current density J ( r ) is independent of r. This is a well-known feature of plane systems, and it can also be derived from the general properties of the continuity equation, div J = 0. For J , = J , = 0 the continuity equation reduces to dJ,/dx = 0, which, when integrated, gives J , = constant, as shown in Eq. (113). In Fig. 5 the anode is shown positive with respect to the cathode. For a negative anode, z, could be in the CY region of the valve and the ,L? region of the valve would then be missing altogether. If this is the case, the only electrons which can reach the anode are those which have started with an initial velocity w,o(~,~- so)%. When this is substituted in Eq. (91), the lower limit of integration for the x-component of velocity changes to The integration of Eqs. (112) and (113) then gives w, = (q -

J,(r) = J , exp

(71.

- 710)

= J , exp { -(e/lcT)(+o

- 4.))

=

J,.

(115)

In these circumstances the anode current depends exponentially on the anode voltage of the valve. It can be shown that this dependence of the anode current on the anode voltage remains true only as long as the velocity distribution of the emitted electrons is half-Maxwellian. This, so-called exponential range of operation of the diode can therefore be used for an experimental investigation of the velocity distribution Itw for the elec.trons. For s~ifficieiit~ly large anode voltages the potential minimum betweeii the cathode arid the anode may disappear altogether. Point xo of Fig. 5 then moves to the /3 region of the valve and all electrons, even those which started with a zero iiiitial velocity, wZo = 0 , reach the anode. The lower limit of integration in Eqs. (112) and (113) is now w, = (7 v,,), and the electron current density

+

J,(r)

=

J,

=

J,.

(116) This range of operation of the valve is the temperature-limited one since, the definition of the saturation current the anode current then depends on the temperature of the cathode only. When Eq. (114) holds, the valve is space charge-limited.

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

219

Using the defiiiition of Eq. (57) and remembering that fJ(vlr)dv

=

jJ(wlr)dw,it is possible to derive the function which represents the contribcltion of different velocity groups to the current density. Substituting Eqs. (97) and (113) in Eq. (57) gives

The function fJ(wlr)is shown by a broken line in Fig. 9. Although the integral of Eq. (117) with respect to w is equal to unity, fJ(w\r) does not represent a probability density function since it becomes negative for all

FIG.9. Current distribution of the electrons. Full line shows the probability density function of the electron current as a function of wz. The dashed line gives the complete current distribution function. The dotted line gives the continuation of this curve beyond the limits of integration. -736

< w, < C. However, according to Fry (113),it can be made to repre-

sent a conditional probability density function if wt is changed to lwzl in both Eq. (117) and (113). The probability condition in this case is the passage of the electrons through an element of area dydz in either a positive or a negative direction. Such a change in the definition of the function jJ(wlr)does not agree, however, with the usual definition of the current

220

P. A. LINDSAY

density, Eq. (112), in which the contributions of the electrons traveling in opposite directions cancel out. Yet the probability condition used by Fry can be readjusted if one considers the passage of only those electrons which actually contribute to the current density J(r). Then the function fJ(wlr) must remain zero in the interval - 00 < w. < +fj. This interval covers not only those velocities which are altogether missing but also all the other velocities which are situated symmetrically round the origin and which thus fail to contribute to the current density J(r). The newly defined functionfJ(w1r) has now all the required properties of a probability density function-it is shown in full in Fig. 9. When the gas is in a state of equilibrium, the function fJ(w\r) is zero everywhere in accordance with the fact that in equilibrium the current density J ( r ) = 0. At a potential minimum or at a temperature-limited cathode the function fJ(wir) follows the full line of Fig. 9 for all negative w. and then remains positive for all w z > 0. Since the limits of integration in the xdirection now extend over the complete interval 0 wz< 00 the vertical lines of Fig. 9 are shifted to the origin and disappear altogether. It is now convenient to call this function the half-Maxwellian current distribution; it should be compared with the corresponding half-Maxwellian velocity distribution shown in Fig. 4. The current distribution curve has a maximum a t w z = l / d = 0.707. The shape of the curve can be understood when one remembers that the current density depends both on the velocity and on the number of electrons passing through an element of area. As the velocity increases the current density grows a t first, but then a drop in the number of the available electrons becomes increasingly noticeable until eventually fJ(wlr) -+ 0 as w z4 0 0 . The change in the definition of fJ(wlr) which, in Fig. 9, corresponds to the change from a broken to a full line does nbt affect any odd moments of the curve but it does alter the values of all its even moments, for example, the mean kinetic energy of the electrons. This is not surprising since the number of electrons which are liable to be counted in each case is different. There is no harm in adopting either of the two definitions of the function fJ(w1r) provided the neceesary assumptions are clearly stated. In this chapter the second definition of fJ(wlr) will be adopted throughout since it not only accords better with the generally accepted definition of the current density but it also provides the function with the properties of a real probability density function. 4. Potential Distribution in the Presence of Space Charge. It has been mentioned a t the beginning of this section that the problem of the potential distribution between the electrodes had been treated in the past by several authors (131-159, 241). Here the equation for the potential distribution will be derived very briefly by a direct integration of the volume density function n(r).

<

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

221

For a plane diode, Poisson’s equation is given by

where p = -en@) is the space-charge density of the electrons and tois the dielectric constant of free space. Multiplying both sides of Eq. (118) by 2dt+/dx and integrating, one can obtain

Here it is assumed that the lower limit of integration coincides with the potential minimum t+ = $, [see (13‘7-139) for the discussion of the potential distribution in the saturated and exponential regions of the diode]. Introducing the variable q = (e/kT)(t+ - t+,,) and substituting from Eq. (102) gives

It is usual to simplify this expression by putting

where the minus sign is used in the a region and the plus sign in the j3 region of the valve. In terms of the new variables, Eq. (120) reduces to

in the a region (121)

in the j3 region (122) where h-(q) and h+(q) are the functions introduced by van der Ziel (138). The integration of Eqs. (121) and (122) gives a general curve for the potential distribution between the electrodes of a diode. There is no need to discuss this problem any further here since it already has been adequately treated in some recent publications [ ( l c ) “Thermionic Emission”; (d?’), Sec. 7.2; 13’7-139; (13), Chap. 5.2 and (29), Chap. 1.41. However, the actual derivation based on the use of the volume density function is very simple and seemed worth quoting. It may be of interest to mention

P. A. LINDSAY

222

+

here that in the majority of cases J,,, J,, and x,, - 20 (or [,,+ [O-) are known, and the problem reduces to the calculation of &, xm and +a - 40 from the definition of [, Eq. (114), and from the tables of ( = [ ( q ) [see (197,199)].This procedure gives the anode characteristic of a diode when its saturation current and the separation of the electrodes are known. In some cases, however, the potential difference 4. - 40 and J,, are both available and the value of J , is not known. The position and depth of the potential minimum can then be found by solving a transcendental equation involving the function 5 = [(TI). 6. The Mean Kinetic Energy of the Electrons. The calculation of the mean kinetic energy of the electrons emitted by the cathode seems to have been bedeviled in the past by statements which can be readily misinterpreted. Since these statements were made by recognized authorities in the field of thermionic emission (194, p. 430; 142) their meaning must have been quite clear to their authors, but many lesser men have been baffled since. The object of this section is to shed some light on the question whether the mean kinetic energy of an electron emitted from a thermionic cathode is 9.jkT or 2kT. a. Definitions of the mean kinetic energy. The kinetic energy of an electron is usually defined as

E , = >Bmv2 E,, = >.jmvs2,Ev, = >5mv,2,E,, = >imv,2

(123) (124)

where m is the mass of the electron. From the definition of v2

Ev = Em

+ + Evm Ev,

(125)

The so-called “peculiar” kinetic energy of an electron [see Chapman and Cowling (IOS)] is given by

Ev = >dmV2 EvZ = >dmV*2,Ev, = >$mV,2,Ev. = 4dmV:

(126) (127)

where, again,

EV = E v ~ iEv, 4- E v ~ .

(128)

The velocity V is the “peculiar velocity,” that is the difference between the actual velocity of an electron and the mean velocity of the electron stream at that point. Thus, if the conditional probability density function f(vlr), Eq. (50), is known, the peculiar velocity of an electron at the point v can be defined as

V

=

v - Jvf(v1r)dv

=

v

- (vlr).

( 129)

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

223

Here the angular twackets represent as usual trhe mean or the expectalioii value of the random variahle, which in this case is X,. I t will be convenient to adopt again the notation introduced in Fibs. (88-90). Dividing both sides of Eqs. (123-128) by kT,the reduced kinetic energy, measured in terms of kT (for 7' = 1000°K, kT = 0.086167 ev = 0.1 ev) is given by

Ew = w2 Ewz= wS2,Ewy = wU2,Ew,= w.1 E w = E w z Ewy Ewsj

+ +

(130) (131) (132)

and, eimilarly, for the reduced peculiar kinetic energy,

Ew = W2 Ews = Ws2,Ewv = Wu2,Ews = Wz2 Ew = Ews Ewv Ewz.

+

+

(133) (134) (135)

Here again,

W

w - Jwf(w1r)dw = w - (wlr), =

the function f(wlr) being given by Eq. (1 11). It can be seen now from the definition of (wlr), Eq. (136), and J(r), Eq. (112), that the mean velocity a t a point r has the following important property

J(r)

= =

YTY

(wlr)n(r.>

(vir)n(r).

(137)

In other words the current density a t a point r is obtained by multiplying the mean velocity of the electrons a t r by their volume density at that point. In order to find the mean kinetic energy of the electrons relative to a given distribution, it is necessary to write, following Eqs. (41) and (130)

(Ewlr) = JEwf(w/r)dw

Jw2f(wlr)dw = (w2/r). (138) Thus the mean energy of the electrons a t a point r, when measured in terms =

of kT, is given by the second moment of the conditional probability density function f(w1r). It is easy to see from Eq. (41) that in terms of the joint probability density function f(r,w) Eq. (138) represents the regression or

P. A. LINDSAY

224

functional dependence of w2on r. The mean kinetic energy of the electrons at the point r, Eq. (138), should be clearly distinguished from the mean kinetic energy of the electrons which are actually flowing past the point r (see Sec. V.B.5.c).These two mean kinetic energies have different physical origin and as a rule are not numerically equal. Equation (138) explains why in calculating the mean kinetic energy it is usual to consider the meansquare velocity rather than the square of the mean velocity of the electrons. The following definition, which is similar to that given in Eq. (138), holds for the mean reduced peculiar kinetic energy of thse electrons,

A comparison of Eqs. (130-135) and Eqs. (138, 139) shows that

(Ewlr)= (&Jr) (Ewlr) = (Ew.lr)

+ (E&) + (EWN

+ (Ett-,,lr)+ (Edr).

(140) (141)

Furthermore, from Eq. (136),

+

(Wlr) = (w21r)- 2 ( ~ l r ) ~(wlr)2 = (w21r) - (w]r)2.

(142)

It should be added that although Eqs. (140-142) have been derived for one particular probability density function, their validity is quite general. The substitution of Eq. (142) in Eqs. (138) and (139) gives (Ewlr) =.(E,lr)

- (wlr)2= (Ewlr)- E d @ )

(143)

where Ed@)= (w\r)2 is the drift kinetic energy, that is, the kinetic energy of a particle moving with the mean or drift velocity (wlr). With the exception of Eqs. (140-142), which are quite general, the angular brackets are used here to signify the mean or expectation value with respect to the distribution given by Eqs. (111). This restriction is quite important since it is possible to calculate the mean kinetic energy of the electrons with respect to the probability density function given by Eq. (117). As a matter of fact the failure to distinguish between the two mean values lies a t the root of the confusion concerning 3.@T versus 2kT. As has been already pointed out in Sec. 111, the two distributions have an entirely different physical meaning, one giving the velocity, the other the current distribution. The mean of w2when calculated with respect to f(wir), gives the mean reduced kinetic energy of the electrons contained in an element of volume dr at the point r. The mean of w2when calculated with respect to fJ(wir)gives the mean reduced kinetic energy

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

225

of the electron current density J(r) a t the point r. Naturally the two mean values are numerically different as a rule. The definitions given by Eqs. (138, 139) can be further extended to give an additional insight into the properties of the electron flow.Taking the mean or the first moments of the joint probability density function f(r,w), Eq. (107), one can obtain

+

+ kaoolm

+

+

(r>= JdwJrf(r,w)dr= ialom ja0,, = Jrf(r)dr = rm,

(144)

and

(w) = JdrJwf(r,w)dw = iamIoo jarnolo k a w l = Jwf(w)dw = wmc.

(145)

Equations (144) and (145) give respectively, the position and the reduced velocity of the center of mass of the electron stream. It is interesting to note that r,, and w,, together give the six first moments of the joint distribution f(r,w). I t should be noted that in general and Thus the mass center of all thc electrons does not coincide with the mass center of the electrons in a given velocity range (w, w dw). Similarly, the reduced velocity of the mass center differs in general from the mean velocity of the electrons at a point r. Finally, considering the second moments of the joint probability density function f(r,w), one can obtain, first of all

+

(r2)= JdwJr*f(r,w)dr= ~ = Jr”fr)dr = I, + I ,

+

z ~ o wQ O Z W ~ O

+ I,.

+

~ W ~ W O

(148)

Here the integrals are the components of the moment of inertia of the electron stream. For example, the moment of inertia of the stream with respect to the z-axis is given by NMtmkT(I, IJ,similar expressions holding for the moments of inertia with respect to the y- and z-axes. Further,

+

Cp*)= JdrJwW,w)dw = amoO+ = JW*f(W)dW =

(E,)

+

( 149)

gives the mean reduced kinetic energy of the electron stream. Again comparing Eqs. (138) and (149) one can see that (EW)

#

(EPDlr),

(150)

226

P. A. LINDSAY

that is, the mean kinetic energy of the whole stream is a constant of the system and differs, in general, from the kinetic energy of the electrons at a given point r. It is now possible to defme a new mean peculiar kinetic energy given by

(Ew) = JdrJEW.f(r,w)dW = JdrWzf(r,w)dw = JdrJ(w - {w))z.f(r,w)dw= PO~OZW = { E w ) - Wm2 = ( E w ) - E m c

+

+

PWZD

PWIOOZ

(151)

where W is the difference between the actual velocity of an electron and the velocity of the mass center of the stream. The new mean peculiar energy {Ew)is equal to the sum of three second central moments of the distribution. The mean energy (ETv)is a constant of the system and differs, in general, from the mean peculiar energy (Ewlr), Eqs. (139) and (143). b. Expressions jor the mean kinetic energy. I t is now possible to calculate the mean reduced kinetic energy of the electrons a t a point r. Substituting Eqs. (111) in Eq. (138) one obtains

where the upper and lower signs refer to the a and B regions of the valve respectively. It has been shown previously [Eqs. (66) and ( l l l ) ] that the velocity distribution is the same both at the potential minimum and a t the surfa e of a temperature-limited cathode. Putting q = 0 in Eq. (152) gives F

{Ewlrm) = 95

(153)

{Evlrm)= 96k T.

(154)

or

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

Furthermore,

ail

227

inrpect
=

(Z<wulrm) = (&Jr)

=

pi,

(155)

exactly as in the case of equilibrium. Thus the law of equipartition of energy holds both immediately outside the surface of a temperature-limited cathode and at the virtual cathode of a space charge-limited diode. This is not surprising when the physical side of the picture is considered. The velocity distributions for the state of equilibrium and for a temperaturelimited cathode are shown respectively in Figs. 3 and 4. They differ in only one feature: to each component of velocity v. a t a temperature-limited cathode there correspond two components vz and - v z in the state of equilibrium, the shape of the curve for vz > 0 being identical in both cases. Thus the even moments of the two distributions must be the same, although the odd moments (for example, the mean velocity) will be different in each case . For some values of t] it is convenient to express Eq. (152) in terms of series. When q is small, that is, in the immediaie vicinity of the virtual cathode

where the upper and lower signs refer respectively to the a and L3 regions of the valve. For large values of t],

(E,Jr)

-

d>'"

35 - >4 3

e-q

in the a region

(157)

in the 8 region (158) The function (E&) is shown in Fig. 10. I n the region of the valve the function increaees with q, until for large q it approaches asymptotically the straight line 2 q. Here the rise in the mean kinetic energy of the electrons is due entirely to the accelerating effect of the field between the virtual cathode and the anode. The higher the anode voltage the smallex the effect of the initial velocities, until for sufficiently high t], the mean energy is virtually proportional to 1. In the (Y region the conditions are quite different. The function (E&) is equal to 3.5 both for 7 = 0 and for r] --t 00, the latter corresponding to an infinite retarding voltage and a state of equilibrium

+

228

P. A. LINDSAY -2.5

1 /

/

/ /

F

-2.0

+ 7-I8n

I

-d6

-0.4

4

0

<

2

I

FIG.10. The mean kinetic energy of the electrons (E,lr) as a function of the reduced potential q.

a t the cathode. For the remaining values of 7 the second term in Eq. (152) always remains positive so that the mean energy must be less than 95 This is not surprising in view of the fact that in the a region the electrons lose their energy to the field. The mean kinetic energy in fact reaches a minimum when

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

229

that is, when q = 0.352. For this value of q the mean energy differs from 95 only by 9.7%, so that the minimum is a very shallow one. Physically the minimum can be understood if one remembers that, although the mean kinetic energy of the electrons is continuously reduced by the action of the retarding field, their relative composition, as one approaches the virtual cathode, changes in favor of the fast electrons. The two opposing effects are as usual responsible for the appearance of a minimum. The mean kinetic energy is of course independent of the y and z variables. The function shown in Fig. 10 is quite general in character. The conditions in a given valve can be obtained by calculating 4" (27, 18?'-1.99,1.9) and cutting off the curve a t the points q = 90and q = t)4, which points then correspond to the cathode and anode potentials of the tube measured with respect to the minimum (brn and multiplied by e / k T . Finally the mean reduced energy (E&) can be obtained in terms of the interelectrode distance x by replotting it with the help of the (t),[) tables of either Kleynen (1.97), Ferris (1.99) or Lindsay and Parker (146a). Such a curve, however, could no longer be expressed in terms of tabulated functions. The graphs of this and other similar curves are given in ref. (141). The mean or drift reduced velocity of the electrons a t a point r is obtained by calculating the first moment of the distribution given by Eq. (111).

Here the y- and z-components of the mean reduced velocity are zero. For = 0, that is, either a t the potential minimum or a t a temperaturelimited cathode, the reduced drift velocity is given by

t)

(w[rrn}= i c x .

(161)

This of course differs from the reduced drift velocity in a state of equilibrium, for which (wlr) = 0. The drift energy of the electrons a t a point r is given by

-Wr) = (wIrY = ?r-'(exp

q ( l =t

erf q%)}+.

(162)

230

P. A. LINDSAY

0.;

0.r

Oi6

0.1~

FIG.11. The drift kinetic energy of the electrons Ed@) as a function of the reduced potential q.

The function Ed@)is shown in Fig. 11. For 7 = 0 the drift kinetic energy 1

Ed(r)= ;

(163)

For small values of 7, that is, near the virtual cathode, the drift energy is given by the following series

where the two signs again refer t o the two different regioiis of the valve. For large values of 7

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

&(r) &(r)

-+ I

7

t'-v

1

in t,he a region 3 -+ 11 47

2-

4rl

-

231 (165)

- .-

in the B region

(166)

Thus, in the 0 region, the function approaches asymptotically the value 1 q . Here again the initial drift energy loses its importance as the potential 7 increases. In the (Y region on the other hand the function Ed@)4 0 as 7 4 m. This is hardly surprising since, for an infinite retarding field, the conditions at the cathode are those of equilibrium and the corresponding drift kinetic energy must be equal to zero. The mean reduced peculiar energy of the electrons can be calculated by subtracting Eq. (162) from Eq. (152), as indicated in Eq. (143). Thus

+

(Ewlr) = @,lr) - Ed@)

=3r 2

(rlh)?4 exp q(1 erf 7%)

-

1

7riexp v(1

erf rls))' (167)

For 7 = 0, that is, a t the potential minimum or at a temperature-limited cathode 3 1 (Ewlr,) = - - -. (168) 2 7 r This differs from the corresponding value of the mean peculiar energy for an electron cloud in a etate of equilibrium, which is 95. Since in equilibrium Ed@) = 0, the functions (Ewlr) and (E&) become identical and the law of equipartition of energy holds for both of ttem. This is no longer true a t the surface of a temperature-limited cathode, since, although the law of equipartition still holds for (Ewlr),Eqs. (153) and (154), it is no longer true for (Ewlr). By analogy with Eq. (154) one obtains

For small values of q the function (Ewlr) can be expressed as a series

232

P. A. LINDSAY

For large values of q

-

(Ewlr)

(Ewlr)

-

-+ 1

("> e-

-

1

3

47

2T2

+---

H

-- -

1

e"

in the cy region in the B region

(172) (173)

The function (Ewlr) is shown in Fig. 12. It varies rather slowly between 00 in the a region and 1for q + 00 in the 8 region. The function represents the difference between the mean kinetic energy and the drift

95 for q 4

Fro. 12. The mean peculiar energy of the electrons (Ewlr) as a function of the reduced potential q.

kinetic energy of the electrons a t a given point r. When the system is near a state of equilibrium the function 3$(Ewlr)T is taken to represent the temperature of the system a t the point r [see (108, 141)l. c. The mean kinetic energy of the electron current. Substituting of Eq. (117) in the expression for the mean reduced kinetic energy, Eq. (138), gives

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

{ -(wz2 1 1 - 4- 2 2 1 +7

+ 2 exp 7

+ 1 = 2 +tl.

233

+ wy2 + w,2))dw,

w 2 exp (-wz2)dwz

(174)

The function fJ(w[r)is d e h e d in such a way that it applies equally to either the a or the B regions of the valve. The expression (Ewlr)Jrepresents the mean kinetic energy of the electron current passing an element of surface dydz situated a t r. The function is shown in Fig. 13. It depends linearly on the reduced potential 7 = ( e / k T ) ( + - $,) a property which seems peculiar to the Maxwellian distribution of the current density. The function (Ewlr)Jis quite different of course from that given by Eq. (152). For 7 5 0, that is, either a t the potential minimum or at the surface of a tempwature-limited cathode \

\ .

di' Here, at last, is the source of misunderstanding in the discussion of the mean kinetic energy of the electrons emitted by a thermionic cathode. Equation (176) is sometimes referred to, rather loosely, as the mean kinetic energy of the emitted electrons, which of course it ie not, the corresponding expression being that of Eq. (154). What Eq. (176) represents is actually the mean kinetic energy of the electron current emitted by a temperature-limited cathode. An inspection of the integrals in Eq. (174) shows that

Thus, h e corresponding contributions from the movement of the electrons in the y- and zdirectioiie are still given by 36, as is the case in Eq. (154), but the contribution from the movement in the xdirection is now given by 1. One might ask why the contributions from the y- and zdirections are present a t all, since these velocity components do not contribute to the current density. The answer is that those electrons which do contribute to the current density possess, as a rule, the w, and w,components of velocity aE well as the UL component, and the corresponding contributions to thc mean kinetic energy of the electron current must be included. The mean reduced velocity of the electron current is obtained by calculating the first moments of the current distribution, Eq. (117).

234

P. A. LINDSAY

FIQ.13. The mean kinetic energy of the electron current (Ewlr)J and the drift kinetic energy of the current EJ(r), both as a function of the reduced potential v.

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

235

As in Eq. (160), the integrals in the j- and kdirections are again equal to zero. Apart from this, Eqs. (160) and (179) are quite different. In particular for q = 0 the mean velocity of the electron current is given by

(wlr,)J = i$5+, (180) which should be compared with Eq. (161). It should be added that unlike (wlr) [see Eq. (137)] the mean velocity (wlr)= does not give the current density J(r) when multiplied by (2kT/m)w n(r). The reduced kinetic energy associated with the velocity ( w 1 r ) J is given by

E l @ ) = (wIr)J2 = (954 -l$+rs exp q(1 - erf

qs))2.

(181)

For q = 0 this function reduces to

EJ(rm)= %&.

(182)

For small values of q, that is, neax the potential minimum,

For larger values of q , the function EJ(r)can be expressed by an asymptotic series

The function EJ(r)is shown in Fig. 13. For large q the function approaches asymptotically the straight line 1 q. It is now possible to calculate the mean reduced energy associated with the new peculiar veIocity W J= w - (wlr)$. From the usual properties of the mean velocities, Eq. (142), one can obtain

+

(&VJlr)J

For

q =

-

= (WJ21r)J = (w21r)J (wlr)J2 = (E,lr)J Edr) =2+qIq exp d l erf q w ) 1.

-

+ >++

-

(185)

0, this function reduces to

(EWJ]r,,JJ= 2 For small values of

q,

- ?&r.

(186)

P. A. LINDSAY

236

For large values of 7,

-+

(EwJ[r)J 1

1

--

47

1

2T2

+ ..- .

The function (EWJlr)=is shown in Fig. 14. Just like (Ewlr), the function (EwJlr)Jtends to 1 for large values of q but it is great.er than (Ewlr) everywhere.

FIG.14. The mean peculiar energy of the electron current (EwJlr)J as a function of the reduced potential q.

C. A Plane System with Two Emitting Cathodes There are usually two reasons for considering systems with more than one emitting cathode. They are (1) generation of current and (2) noise investigations. Recently simple systems comprising two emitting electrodes kept a t differenttemperatures seem to have attracted considerable interest (I, 2, 143). Such systems are, in principle, capable of direct conversion of heat into electricity since, in general, the difference in temperature will cause an over-all drift of electrons from the electrode a t a higher to the electrode a t a lower temperature. Since in this case no potential difference will be required for the flow of electricity the whole system will act as a generator of electric current. The operation of such a “thermionic engine” is governed t o a large extent by the fact that the distribution of the initial velocity of the electrons depends on the temperature of the emitting cathode. The other reason for considering systems with two thermionic cathodes is associated with the investigations of noise in diodes. In order to check the validity of certain theories it has been necessary to consider the

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

237

space-charge and potential distribution between two emitting electrodes taking into account the Maxwellian distribution of the initial velocities of the electrons (144-1 46). A brief description of the velocity distribution in systems comprising two plane emitting electrodes will be given in this section (see aIso 146~). Consider a system comprising two cathodes which have work functions E+l,E+zand are kept a t temperatures T 1 ,Tz.The cathodes are situated a t zcl and xc2 along the x-axis and they are perpendicular to it. For the sake of generality a potential difference &2 exists between the cathodes. This potential difference is equal to the potential difference between the Fermi levels of the cathodes (which may be due to an external battery or load) plus or minus the contact potential, that is, the difference in the work functions of the two cathodes. To avoid fringing effects the cathodes are assumed to be relatively close together, a condition which is usually satisfied in practice ( 1 , 2 ) . The potential distribution in such a system is given by the general curve of Fig. 5, although for a given separation and potential difference between the electrodes both position and depth of the potential minimum will be different in systems with one or two cathodes. In the presence of a second cathode a shift in xm and d,,, will be caused by the additional space charge contributed by the second emitter (see 146a). 1 . Equations of Motion. An extension of Eqs. (82-84) which are, in general, valid for any plane electrostatic system, gives

and

where the subscripts cl and c2 refer to the initial velocities and potentials a t the corresponding cathodes. It is now usual to assume that the electrons emitted by both cathodes have their velocity distribution given by Eq. (62),with appropriate values T I , T2 and E.+l,Eg2 substituted for the temperatures and work functions of the corresponding cathodes. Equations (189-194) show that in the y- and z-directions the velocity distribution of the electrons remains Maxwellian and extends from - CQ to m. In the xdirection the velocity dis-

238

P. A. LINDSAY

FIQ. 15. The z-component of the velocity .u as a function of the distance z.Each curve corresponds to a different initial velocity uZsl or v z a .

tribution is of course altered by the presence of the electric field. Figure 15 shows the plot of the electron trajectories in the x., v,-plane aesuming that the electrons are absorbed completely by either cathode. The trajectory of the electrons which have just enough energy to pass the potential minimum between the electrodes is obtained by putting

or

Both values, when substituted in the appropriate equations give the same expression for the limiting trajectory, namely,

VELOCITY DISTRIBUTION

239

IN ELECTRON STREAMS

Equation (197) shows that the whole strip of the x, v,-plane between x = xcland x = xCz can be subdivided into two areas which are marked A1 and A2 in Fig. 15. All electrons emitted by the first cathode are permanently in the area Al, whereas all electrons emitted by the second cathode are permanently in the area AIL. A t this stage it is again convenient to introduce the following simplifying notation e

k ~ (6 , - 6m)

m VU2

m 2kTz v,2

= 771

=

(198)

WZY2

= w2,2.

Similarly to Eq. (96) the following expressions hold for the phase-space density n(r,v) in terms of the new variables w1 and w2

2. Density and Probability Density Functions. The phase density function, Eq. (86), has been derived for a general potential distribution of the form 6 = $(x) and it is still valid for the region between two emitting cathodes. Substituting the appropriate values for the temperature and work function of the corresponding cathodes one can obtain, in terms of the new variables, Eqe. (19&207),

n(r,wl) = m,2*-5* exp n(r,wd

= n~,27r-96exp

(771 (772

+ + in the regioii A l - m)exp { - (w2+ + 1. - m)exp { - ( ~ 1 2 2 wIu2 wlr2)1 wZy2

~

2

~

(208) ~

)

in the region A2 (209)

P. A. LINDSAY

240

Here, from Eq. (63),

and

The functions nTI and nTI are the volume densities of the electrons a t the cathodes cl and c2 when they are temperature limited. It is interesting to note that in general the phase-space density n(r,w) is discontinuous along the curve separating the two areas A1 and AS,the discontinuity disappearing only when the temperatures and work functions of the two cathodes are similar. (In practice a slight amount of electron interaction will cause a blur in the steplike transition between AQand A1.) The volume density of the electrons is obtained by the integration of Eqs. (208) and (209) with respect to W.

n(r> = =

1n(r,w)dw nT1 exp

(01

- rllC1)27r-x

+ nT2 exp

exp (-wlL.2)dwII (72

':/

- r/2&)2?r-x

+

exp

(-w2z2)dwZz

exp (71 - vlC1>(1 f erf ~ 1 % ) nT2 exp (92 - q2cz)(l T erf 92%) Sr (92), (212) where the upper sign referg to the range xE1 x < x, and the lower sign to the range zm 6 x 6 zC2.The functions g* (vl) and gr (92) have been in. troduced for convenience of notation and their meaning is self-explanatory. From Eq. (212) the following expressions can be obtained for the volume density of the electrons at the surface of either of the two cathodes =

~1

= g*

(91)

+

n(rcl) = nT,(1

<

+ erf meP)+ nT%exp

(712~1

and

n(rcd = n ~exp ,

(rllCz

- rllcJ(l - erf rllcZq)

- ~2~2)(1 - erf qSClx), (213)

+ nr,(l + erf q2e~x).(214)

These values are always greater than the correrponding expressions for an ordinary diode, obtained by putting 7 = 90 or q = q,, in Eq. (102). For two identical cathodes E+l = E+z= E+ and 711= 92 = 7 if T1 = T2 = T. Also, from Eqs. (210) and (211), n ~ = , nT, = *. The expression for the volume density of the electrons, Eq. (212), then reduces to

n(r) = n~ exp vjexp (-qC1)(1 f erf 9%)

+ exp (-rlCz)(l

f erf ?%)I.

When there is no potential difference between the two cathodes qcl and Eq. (215) reduces to

(215)

= qcS =

VELOCITY DISTRIBUTION IN ELECTRON

n(r>= 2 n exp ~

(7

STREAMS

24 1

- rlc)

(216) =k lexp i(e/kT)(9 - 4Jc) I Equation (216) gives the volume density of the electrons when the system is in a state of equilibrium. This expression was fist derived by von Laue (122) from purely statistical considerations. Recently Loosjes and Vink (145) and Knol and Diemer (146) have given another derivation based on the concept of the current density. The approach adopted in this article is based on the concept of the phase-space density and extends the results previously obtained to the more general case of two cathodes kept at different temperatures and potentials (see 1 4 6 ~ ) . Integrating Eqs. (213) and (214) with respect to r gives the density of the electrons in the velocity subspace.

n(w) =

n(r,w)dw

= 2r-$*(yz

- y1)(z2 - 211%~ exp (-m) exp

(-w22)

exp v2dxfor - co =

2 r - s l ( ~ ,-

Y ~ ) ( Z Z - Z I ) ( ~ Texp ,

+ n~~exp

< wZz <

Lcl w2) I."z 2'

(-m) exp (-w,2> (-17~~2)

exp (for

-T,+

- 21) ( mexp (-m)exp (-wlz>

+

~

T

- yl)(zz

< wlz < 0 < <0

Lr

w20

exp vldx

exp Z (-wed exp (- wz2)/zcz exp q~dx1 2"

for 0 0 = 2r-?*(yz

exp mdz exp &I?

-72%

= 2r-3*(~2- yl)(zZ

-17234

- z~nT1exp (-mCl)

exp

(-w12)

6 wlZ < vlY 6 wzz < 9zf6

Jzz

exp Tldx.

for qlj* 6 wlZ < co (217) Finally, integrating either Eq. (212) with respect to r or Eq. (217) with respect to w one can obtain an expression for the total number of electrons in the interelectrode space

P. A. LINDSAY

242

As iii the c u e of Fqs. (107-1 11) t.he prohability density fuiictioiie now can he obtained by a suitable combination of the expressions for n(r,w), n,(r), n(w), and NtOt. These operations are quite straightforward and for reasons of economy of space the corresponding expressions will not be quoted here. 3. Current Density of the Electrons. In the analysis of the two cathode systems it is of some importance to consider the current density of the electrons. By definition, Eq. (55), the expression for the current density gives the surplus of the eIectrons which flow in the direction of one of the two electrodes. I n the case of a “thermionic engine” this surplus is equivalent to the generated current which can be used in an external load. Substituting Eqs. (208) and (209) in Eq. (55) gives

the integrals in the j- and k-directions being zero. Carrying out of the integrations indicated in Eq. (219) gives

=

J,I exp

(-qlc1)

- Ja2 exp

= const’.

where Jsl and Jez stand respectively for the saturation currents of the two cathodes. Equation (220) is quite general and shows that the current density in the presence of a second cathode is always less tha,n the corresponding current density in a simple diode, Eq. (113). This is hardly surprising in view of the fact that from the definition of the current density Eq. (55) the contributions of the electrons traveling in opposite directions cancel out. Thus in Eq. (220) the current density is largest when the second cathode is a t the temperature T = 0°K or in practice, when its emission becomes negligible. ,Since 9 = (e/kT)(t$ - t$m) the current density depends in each case exponentially on the depth of the potential minimum (pm. For two identical cathodes kept a t the same temperature T 1= TI = T, Eq. (220) reduces to

Jdr)

- exp (-

= J , {exp (-- vC1)

rlC2)

1

(221)

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

243

[see (1467, Eq. (23)l. When the potential difference between the cathodes is zero, qcl = qc2 = qc, the current density

Jz(r) = 0

(222)

as it should for the state of equilibrium. 4. Potential Distribution between the Cathodes. As in the case of a plane diode it is possible to derive the potential distribution between the electrodes by substituting the right expression for the volume density of the electrons in Eq. (119). Thus from Eq. (212)

4-n ~exp , (-q2c2>

.$

i m e x pq2(1 f erf qzs)d+}

(223)

Putting

gives

{exp q z - 1 f 2 =

hr(~i)

=F exp q 2

erf ??} (224)

+ Ah*(~]p)

where

and h-, h+ are again the functions introduced by van der Ziel (138). Iiearranging of Eq. (224) and integration give

244

P. A. LINDSiW'

where 72 = (T2/Tl)71. Equations (225) and (226) give a family of curves which are the generalized characteristics of the valve and should be compared with the single generalized characteristic which is obtained in the case of a simple diode (137, 139). For two identical cathodes kept at the same temperature, 71 = 2 2 = g and A = exp (vcl - qC2).Equations (225) and (226) then reduce to

Equations (227) and (228), when integrated, give a complete solution of the potential distribution probIem. Numerical values of the integraIs . approximate solugiven by Eqs. (225-228) are collected in ( 1 4 6 ~ )An tion of the problem, valid only for small departures from the state of equilibrium has been given by Furth (147). For small values of 7, that is, when the potential minimum is relatively small, the functions h-(.rl) and h+(q) can be expressed in the form of a series (138). A term by term integration of Eqs. (227) and (228) then gives

tf

=

(1

(H)'") $6 - 1+A

37r

l - A 2 + A ) - > $ [27%+ - (' - - A)' * } $6 + . l+A %

.] a ]

for xcl 6 x

< xm

for xm 6 x

< x c 2 (230)

(229)

These expressions do not suffer from the assumptions about the shape of the potential distribution which Fiirth (147)found it necessary to make. When the two cathodes are kept at the same potential rpc, = rpcz = $=, the constant A = 1 and Eqs. (227) and (228) reduce to

VELOCITY DISTRIBUTION IN ELElCTRON STREAMS

245

The function is now symmetrica1 with respect to the potential minimum and can be integrated, (1CS), giving

5 = 1Tz tan-' (exp 1 - I)%,

(231)

or for 7 in terms of f 7 =

In sect

((/di$.

(232)

This is the familiar expression which has been discussed by several authors (14 p. 20; 122; 144-147). Substituting Eq. (231) in Eq. (216) gives the volume density of the electrons in terms of the reduced distance f ,

In Eqs. (232) and (233) the distance can be measured in the direction of either electrode, the origin f' = 0 being at x = xm.

D. A Plane Triode or Tetrode In high-power triodes which are opeiated as class C amplifiers it is usual for the control grid to stay positive with respect to the cathode for an appreciable part of the rf cycle. Similarly, tetrodes or screen grid valves have a second grid which is permanently at a potential higher tha$ that of the cathode. When the current density is sufficient, it is necessary to consider the velocity distribution of the electrons in order to be able to calculate the potential distribution between the grid and the anode (129). 1. Equations o j Motion. Consider a plane triode with the cathode, the grid, and the anode situated at the respective points x,, x, and x,, along the x-axis and perpendicular to it. Figure 16 shows the general curve for the potential distribution in such a valve. Between the cathode and the grid the potential distribution is the same as in a space charge-limited diode. (In the case of a screen grid valve there will be a slight change in the shape of the curve caused by the presence of a negative control grid. However, unless an additional potential minimum is allowed to develop between the control and the screen grid, the following arguments will not be affected by the presence of the control grid.) Assuming an idealized grid equivalent to a plane at a potential $, a second potential minimum can now exist between the grid and the anode of the valve, provided the current density due to the electrons passing the grid is sufficiently high. The second potential minimum a t x = xm2 divides the grid anode space into two regions a2 and /32 which have properties similar to the corresponding regions in an ordinary diode. It is convenient, however, in this case to subdivide the

P. A. LINDSAY

246

FIO.16. PotentiaI distribution in a triode with a positive grid. region a2 into atz and attzby a'plane situated at x = xl, where XI is the point at which the potential C#I = C#Iml. Introducing the reduced potential

where &,2 is the potential of the second minimum a t x = %,a and using the reduced velocit,y w, Eqs. (88-go), the equations of motion of the electrons reduce to wz2 = wz2 (s - s c ) (235) wy2 = wy,2 (236) W.2 = w ,:. (237)

+

Except for a slight change in the definition of the reduced potential q, Eqs. (235-237) and (91-93) are identical. It is necessary now to calculate the limiting trajectories of the electrons in order to be able to integrate the function n(r,w) in the velocity subspace. Since in the system considered there is no field in the y- and z-directions, the limits of integration for the reduced velocity components w, and w, are the same as a t the cathode, namely, co and co . In the x-direction, however, the limits of integration must be found again from a phase-space diagram, Fig. 17. There are two types of limiting trajectory which give respectively the outer and the inner boundary curves. Consider first those electrons which have enough energy to pass the second potential minimum at x = xm2.Their initial velocity is given by w,,2 = qo. Substituting this in Eq. (235) one can obtain the fol-

-

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

247

FIG.17. The z-cornponent of the reduced velocity w r as a function of the distsnce z. Each curve corresponds to a differentinitial velocity wSc.

+

lowing expression for the limiting trajectory w2 = qc q - qo = q. This trajectory is represented in Fig. 17 by the outer boundary curve. When rp,,z < &,lr there is however another limiting trajectory associated with the electrons which have enough initial energy only to pass the first minimum at x = zml. The initial energy of these electrons is given by to2> = qc - vml. Substituting this particular value of the initial velocity in Eq. (235) gives an expression for the second limiting trajectory namely w22 = qe - qrnl ’I - qc = q - qrnl. This trajectory corresponds to the inner boundary shown in Fig. 17. (If Qrnl < drn2all electrons which pass the first minimum at x = Xm1 reach the anode and the inner boundary curve of Fig. 17 disappears altogether.) 2. Electron Density Functions. Substituting Eqs. (235-237) in Eq. (62) and bearing in mind the transformation of velocities Eqs. (88-90) and Eq. (96) one can obtain the phase-space density of the electrons

+

n(r,w) = ~ 2 7 J*- exp (T

- qc) exp {

- (w2+ wy2 + w,”)}.(238)

Here nT is the volume density of the electrons at a temperature-limited cathode, Eq. (63). Integrating Eq. (238) with respect to w gives the volume density of the electrons anywhere in the interelectrode space

P. A. LINDSAY

248

= %27r-x exp (7 - 70) =

n~2u-Nexp (7

- 70)

On carrying out the indicated integration this reduces to the simpler form

3. Potential Distribution between a Positive Grid and an Anode, As in the case of a plane diode it is now possible to obtain the potential distribution between the grid and the anode by substituting Eq. (239) in Eq. (119). This gives

+ s,"u exp 7{ 1 + erf 7% - 2 erf (7 -

7mJ

1d+

in the

region

CY'~

Substituting 9 for 6, putting tr = F {2e2wexp ( - q O ) / k T e O ) ~-( zzm2) and carrying out the indicated integration gives

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

in the a"t region and

($>'exp =

q

-1

+2

(n?->" -

exp q erf

756 =

249

(241)

h+(T>, in the & region

(242)

where the functions h-(q) and h+(q) have been defined previously. Equations (241) and (242) correspond to Eqs. (10a-c) of Ramberg and Malter (129). In the derivation presented here the problem treated in (129) has been somewhat generalized by introducing a potential minimum between the cathode and the grid in addition to the potential minimum between the grid and the anode of the valve. This requires changing - V of (I29) to 4ml everywhere except in Eq. (8), where it is necessary to put V' - 7 = 4c - dlnZin order to obtain the customary expression for tF. An integration of Eqs. (241) and (242) gives

5-

=

dq '1-

+

dh-(q)

- exp (-qrnl){h-(q - qml) - h+(q - vrnl>+j =

11-

+ xl(q,qml)

in the Q'Z region

(243)

in the CY"Z region

(244)

in the 8 2 region

(245)

Equations (244) and (245) give the function tabulated by Kleynen (I37),Ferris (139) and Lindsay and Parker (I46a). A graph of the function xl(q,q,l), Eq. (243), is given in Fig. 2 of ref. (I29), where is used in place of qml. I t appears from Eqs. (243-245) that the general effect of the second

potential minimum 4%2 being deeper than the potential minimum a t x = xml is to reduce the slope of the potential distribution curve in the region a'2. This can be understood easily by comparing Figs. 6 and 17. The existence of the inner boundary in Fig. 17 shows that certain low-velocity electrons are completely missing from the region ar2.As a result of this, the space charge there will be somewhat less than it would have been in the corresponding region of an ordinary plane diode.

P. A. LINDSAY

250

VI.

VELOCITY

DISTRIBUTION I N CYLINDRICAL

SYSTEMS

A . General Considerations For experimental reasons it is often necessary to consider electrode arrangements which are in the form of concentric cylinders and possess axial rather than plane symmetry, Although such sygtems are convenient from the experimenter’s point of view they are something of an embarrassment to the mathematician. This can be understood when one recalls that the equations of motion of an electron traveling in a field of force between two concentric cylinders are much more complicated than for an electron traveling between two parallel planes. In the cylindrical case an electron leaving the inner cylinder or the cathode tangentially acquires a radial velocity component on its way to the outer cylinder or the anode. This transfer of energy from the tangential to the radial direction must be allowed for in the equations governing the motion of the electron and thus makes the algebra of the problem that much more complicated. The other reason for the mathematical difficulties associated with cylindrical systems is the fact that the potential distribution between two coaxial cylinders, in general, has a curvature of opposite sign to that introduced by the electron space charge. Thus, the corresponding solutions of Poisson’s equation must be sufficiently complex to allow for the fact that the function may have different signs of curvature a t different distances from the cathode. [This point has been fully confirmed by the difficulties encountered by Page and Adams (148) in their efforts to obtain series solutions for the potential distribution in the case of a single valued initial velocity.] In plane systems the potential distribution in the absence of space charge is a straight line and the only curvature possible is that due to space chargenaturally the sign of the curvature will then be the same everywhere. In spite of these mathematical difficulties the problem of electron flow in cylindrical systems has been of sufficient practical importance to attract considerable attention. However, in general, it has only been possible to consider the simple two-electrode case of a cylindrical diode.

B. The Cylindrical Diode Consider two coaxial cylinders of radii ro and r., respectively, the inner cylinder forming the cathode and the outer cylinder the anode. A,qsume further that the cylinders are sufficiently long for the end effects to be neglected. As was the case in plane diodes it will be assumed here that the distribution of the electrons a t the surface of the cathode is half-Maxwellian in the direction perpendicular to the cathode and fully Maxwellian in the

VELOCITY DISTRIBUTION I N ELECTRON STREAMS

25 1

tlirectioiis which arc tangcntia! to it. Further, it will be assumed that no reflections owur at, either of the two electrodes. The number of papers which discuss the flow of electrons in a cylindrical diode, including the initial distribution of velocities, is relatively small. Schottky seems to have been the first person to realize that because of the geometry of the valve the anode current in a cylindrical diode, in general, would differ from that in a plane diode of similar dimensions. In two classic papers Schottky obt,ained in the first case an expression for the anode currelit (14.9) in a retarding-field diode and then an approximate expression for the potential distribution between the cathode and the anode (150), assuming in each case that the potential distribution is a monotonically decreasing function. In a subsequent paper (151) an attempt was made to extend these results to the case of a space charge-limited flow, that is, a flow past a potential minimum in front of the cathode. Some years later Langmuir (IS,$), Davisson (152), and Gehrts (155) discussed the problem of current flow in a cylindrical diode again. Davisson discussed the conditions which the potential function must satisfy in order to ensure the validity of Schottky’s expression for the anode current in a retarding-field diode (149), and Langmuir tried to develop the argument advanced in (151) and suggested a new expression for the anode current applicable to the ease of a space charge-limited diode. Since then the most notable contribution to the solution of the problem has come from Wheatcroft (154). In his paper he derives not only an approximate expression for the magnitude of the anode current, but also provides graphs of the approximate position and depth of the potential minimum expressed in terms of different parameters of the valve. [These graphs are also available a t the end of Chapter 5 of Rothe and Kleen ( I S ) . ] Finally it should be added that Bell and Berktay recently published a critical review of the various approximations involved in the calculation of the anode current (155). All t,heee investigations have been severely hampered by the difficulties associated with the choice of the right limits of integration in the velocity subspace, which is particularly true in the case of space charge-limited flow. The usual insistence on treating the current density rather than the volume density of the electrons seems to have added further to the mathematical difficulties. In this chapter an attempt will be made to free the problem from a t least some of these difficulties and to present it in a form which is perhaps more suitable for further computational advances. 1. Equations of Motion. In the analysis of systems possessing axial symmetry it is often convenient to use cylindrical coordinates T , 8, z. The following expressions give the corresponding covariant and contravariant metric tensors

P. A. LINDSAY

252

9

(246)

and

[

1

gk‘ = 0

0

0 T-2

0

0 0 1

(247)

Substituting these values in Eqs. (1) and (2) and assuming the existence of an electrostatic potential t$ = $ ( T ) and a zero magnetic field, gives the following expressions for the Hamiltonian and Lagrangian of the system

and

L

= >5m(i2

+ r282 + i 2 ]+ et$(r).

(249)

The differentiation of Eq. (249) with respect to the generalized velocities qklas indicated in Eq. ( 3 ) ,gives the following expressionsfor the generalized momenta p , = mf, p 2 = mr28, p3 = m i .

(250) (251) (252)

The differentiation of Eq. (248) with respect to the generalized coordinates pk, as indicated in Eq. (8), gives the following three equations of motion

Multiplying both sides of Eq. (253) by 2p1, substituting from Eq. (250), and integrating, gives (256) pz2 = p3* =

P202, p302j

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

253

the integration of Eqs. (254) and (255) presenting no difficulties. Here the subscript nought refers to the values of the corresponding quantities at t = lo, that is, a t the cathode. Equations (256-258) describe completely the motion of an electron in an axially symmetric onedimensional field of force. Equations (257) and (258) respectively represent the conservation of the moment of momentum p2 and of the linear momentum in the z-direction pa. Equation (256) contains both p l and p 2 and expresses the coupling which exists between the generalized velocities i and 8. This can be understood better if one remembers that even in a field-free space an electron which leaves the cathode tangentially, that is, with i = 0, will arrive a t the anode with a radial component of velocity i # 0. The coupling bet.ween the 1' and d velocities is typical of systems which can be expressed easily in terms of cylindrical coordinates. It is one of the main sources of mathematical difficulties which one has to face in attempting to solve the problem of electron flow in such systems. The addition of Eqs. (256-268) leads to the well-known expression for the conservation of energy

The phase-space density of the electrons a t the surface of a curvilinear cathode has been given already in Sec. IV, Eq. (73). Substituting the appropriate values for the components of the metric tensor, Eqs. (246) and (247), one can obtain the appropriate expression for the phase-space density of the electrons a t the surface of a cylindrical cathode:

Here a suitable expression for w has been introduced from Eq. (74). The phaee-space density of the electrons anywhere between the cathode and the anode can be obtained by substituting Eq. (259) in Eq. (260): 1 ro

n(q,p) = 2 w ( h k T ) -% - exp

By Liouville's Theorem the function n(q,p) again remaim constant along any given electron trajectory (see Sec. 11,B). Equation (261) occurs so frequently that it calls for some simplification of notation. Put

P. A. LINDSAY

254

where 41, is a constant to be determined later. ‘l’he transformation Eqs. (262-265) is equivalent to expressing all energy in terms of IcT, as was done before in Sec. V, Eqs. (88-90). Since the sets of quantities in the two sections are identical in character the same symbol w can be used for both. A comparison of Eqs. (263-265) with Eqs. (250-252) leads to the following relationships

(&) (&) (&)

H

w,=

i.1

9i

w,= w, =

4

9i 2.

Since i., 8, i are the covariant components of the generalized velocity vector 4, the reduced velocities w,, w,, w,are neither covariant nor contravariant, but represent the components of w in the directions which coincide with the axes T , B, z at a given point. Substituting Eqs. (262-265) in Eqs. (256-258) gives the following simplified expression for the equations of motion of an electron

where R = T / T O is the reduced radius. Equations (269-271) when added together give a simplified expression for the conservation of energy, W d

+ + w,02

wzo2 =

w,2

+ w,2 + w,2 - (71 -

70).

(272)

It is now necessary to express the phase-space density n(q,p) in terms of the new variables q and W.Since the number of electrons in an element of volume of the phase space is given by dN

=

n(q,p)dqdp = n(q,w)dq&

(273)

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

255

and further since

= (2mkT)%dw,dw,dw,,

(274)

the following relationship must hold between phase-space density functions expressed in terms of the two sets of variables q,p and q,w. 4 9 , P ) = (2mkT)-+'n(q,w)

(275)

Substituting Eqs. (262-265) and (275) in Eq. (261) gives the following convenient expression for the phase-space density of the electrons

+ + w , ~ ) } . (276)

n(q,w) = n~R27r-35exp (7 - 70) exp [ -(q2 w 2

It should be noted here that although, by Liouville's Theorem, the function n(q,p) remains constant along any given electron trajectory, the same certainly does not apply to the new function n(q,w). The following properties however do apply equally to both functions: (1) the functions are independent of 0 and z, (2) they can be separated into products containing w, and wg together and w. alone which shows, from Eq. (46), that the statistical independence of the movement in the z-direction is preserved under the influence of an axially symmetric electrostatic field of force. Since there is no field of force in the zdirection, the latter statement indicates that the velocity distribution of electrons in the z-direction is everywhere the same as at the cathode, that is, Maxwellian, the limits of integration in the velocity subspace being - 00 < wa < Q).This general property will apply to all systems considered in this section. 2. The Temperature-Limited Diode. It seems easiest to consider first of all the case of a temperature-limited cylindrical diode. By definition there exists now an accelerating field everywhere between the cathode and the anode of the valve so that all electrons emitted by the cathode will reach the anode. Since the cathode is now the point of lowest potential it is convenient to put = & in Eq. (262), so that the value of the reduced potential a t the cathode is qo = 0. a. Limits of integration in w,-and w,-directions. Equations (269) and (270) represent the motion of the electrons in the R,w,,w,-space. Substituting for w,2 from Eq. (270) in Eq. (269) gives the projections of the corresponding electron trajectories on the R,w,-plane, W? = w,02

+ (1 - l/Rz)>w,oz+

70

(277)

A set of such projections is shown in Fig. 18. Figure 19 shows projections of the same electron trajectories onto the R,w'-plane. These projections remain the same whatever the value of w,. as shown in Eq. (270).

256

P. A. LINDSAY

n

2

We

4

W

0

Ba

< IV

0

I

2=-----

__c

Ra.

R

FIQ.19. The 0-component of the reduced velocity, we, as a function of the reduced distance R. Each curve corresponds t o a different initial velocity wh. The same curves are obtained for all values of W,Q.

It can be seen from Fig. 18 and Eq. (277) that the limiting electron trajectory, that is, the trajectory of those electrons which leave the cathode tangentially (wd = 0) is given by wr2

= (1

- l/RZ)W,(12 + r],

(278)

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

257

and is a function of w,o. Thus the surface limiting the range of integration in the velocity subspace is no longer a cylinder, as was the case in plane diodes, but has a more complicated shape. The expression for this limiting surface can be obtained by eliminating w,o between Eqs. (278) and (270). This gives w,z

=i

(R*- 1 ) w t

+

?#I.

(279)

This surface is shown schematically in Fig. 20.

FIG.20. The limiting surface generated by all possible trajectories of those electrons which leave the cathode tangentially. The dashed lines drawn on the surface represent typical electron trajectories.

b. The generalized volume density of the electrons. The generalized volume density of the electrons, Eq. (44),can be obtained by integrating the phasespace density, Eq. (276), with respect to the reduced velocity W. In order to find the appropriate limits of integration for the w,and w,variables it is necessary tp cut the surface given by Eq. (279) by a plane perpendicular to the R-axis. Figure 21 shows such a limiting curve and indicates the preferred order of integration. The limiting curve is a hyperbola

P. A. LINDSAY

258

FIG.21. The curve obtained by cutting the limiting surface of Fig, 20 by a plane perpendicular to the R-axis. The shaded area indicates the range of integration.

It degenerates into a straight vertical line through the origin for R = 1 and at the cathode of the valve. The integration of Eq. (276)

q = 0, that is,

over the shaded area of Fig. 21 gives

=

1

dw,

--DD

=

~ T exp R 75

dw,

\

n~R27r-35exp q exp { - (wr2

+ w? + wL2))dw,

1- 1:

+ w,"))dw,

+

-

- 0

dw,

exp (-(wr"

where u'(12= (R2 - 1)w,2 q and erfc x = 1 erf 2. This exprewion should be compared with Eq. (102) (lower sign) which gives the volume density of electrons in the p region of a plane space charge-limited diode. Apart from the factor R, which is neces.sary sincen(q) is a generalized volume density (that is, the density in an element of volume d q = drdedz and not

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

259

in an clenictit of physiral volume rdrdMz), the function n(q) is everywherc less than the corresponding function for the plane diode. This is fairly obvious since erf[ (R2- 1)w? v ) s 2 erf vx for R 1. In physical terms one can explain the difference by recalling that in a cylindrical system the electrons continuously increase their radial component of velocity at the expense of their tangential component so that for the same potential distribution and separation of the electrodes they stay in the interelectrode space for a shorter time in a cylindrical than in a plane diode. This means that for the same cathode emission the volume density of the electrons must be less in the cylindrical than in the plane case, except of course at the cathode ( R = 1) where the two densities must be equal. Equation (281) shows that for cylindrical systems the volume density depends on two variables 7 and R. This differs from the conditions in a plane diode, Fig. 8, where the function depends on a single variable 7. There is a very good physical reason for this. In plane systems the conditions at a given point are fully specified by stating the value of the reduced potential q which in turn gives the position of that point with respect to some reference plane, for example, the cathode, the distance of the reference plane from the origin being immaterial. In cylindrical systems, however, the situation is quite different. In order to specify the conditions a t a given point it is necewry to know not only the value of the reduced potential q but also the distance between that point and the origin. Because of the cylindrical geometry of the whole system it matters now a great deal whether the point considered is situated for example 0.1 cm in front of a cathode of radius T~ = 1 cm or ro = 1 meter. In the first case the effect of the natural curvature of the system will be much greater than in the second one. In general the function n ( q ) can be represented by a surface over the R,v-plane. A cut of the surface along R = 1 then gives Eq. (102) (lower sign). For any given cylindrical diode, n ( q ) will be uniquely determined by the usual relationship = v(R), which can be traced as a curve in the R,v-plane, the actual electron density a t a given point R being given by the value of n ( q ) at [R,v(R)I. The important conditional probability density function for the velocity distribution of the electrons a t a point q, Eq. (50), can now be obtained by dividing n ( q , w ) by the generalized volume density n ( q ) . This step presents no special difficulties and will be omitted for economy of space. c. The generalized current density of the electrons. The expressions for the components of the generalized current density of the electrons in terms of the curvilinear coordinates q have been derived in Sec. II1,D. From Eq. (54) one obtains the following expressions for the components of the geiieralized current density in the directions r, 6, z

+

>

260

P. A. LINDSAY

=

=

=

(g)" 1

wrn(q,w)dw

(zy 1

w,ncq,w)dw

(F)'1w.n(q,w)dw.

Now consider the limits of integration. Since the function n(q,w) is symmetrical, the value of the last two integrals can be obtained by inspection. In the case of Eq. (284) the situation is very simple since the limits of integration are - and w and the integral reduces to Jdq) = 0. (285) In the case of Eq. (283) the situation is still fairly simple since wo makes the integrand skew symmetric so that the integral over the shaded area of Fig. 21 is zero, giving JJs)= 01 (286) as would be expected from the symmetry properties of the system. The only integral which has actually to be calculated is that given by Eq. (282). Substituting Eq. (276) in Eq. (282) and putting qo = 0, gives

where wrl has the same meaning as in Eq. (281). Defining the total electronic current per unit length of the cathode as

11 = fhreJr(q),

(288)

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

261

one can show from Eq. (287) that the total saturation current becomes equal to

It is now possible to write Eq. (287) in a simple form It = I18 = It,.

(290)

This expression is similar to Eq. (116) which was derived for a temperaturelimited plane diode. Comparing Eqs. (287) and (113) it becomes clear that, because of the symmetry of the system, the generalized current density does not vary with T just as the ordinary current density does not depend on z in a plane system. This is fairly obvious since the generalized current density J,(q) gives the total current flowing across an element dodz, which does not depend on r. In the case of a cylindrical system the ordinary current density can be obtained by dividing Jl(q) by r. This gives the total flow of current acroea an element of surface rdedz which is a function of T . 3. The Retarding-Field Diode. Assume that the field between the cathode and the anode is a retarding field, that is, that the reduced potential q is a monotonically decreasing function of the distance R. As was the case with a temperature-limited diode it is convenient to put $1 = $0 in Eq. (262) so that 7 = 0 a t the cathode. For a retarding-field diode the anode potential q, 6 0, the case qa = 0 being possible only when the space-charge density and the electron current are zero. On the other hand for large current densities, a limiting case is reached when the slope of q = q ( R ) becomes zero a t the anode. Any further increase of the current density necessarily introduces a potential minimum and changes the conditions from retarding-field to space charge-limited flow. The case of a retarding-field diode has been discussed in the past on 15.4-165), , but the results of these invesmore than one occasion (lo-162 tigations are somewhat scattered. Here an attempt will be made to discuss the whole problem perhaps a little more fully and to clarify the reasons for the limited validity of some of the previous results. a. Discussion of the electron trajectories. It has been shown in the previous section that Eq. (277) gives the projection of the electron trajectories on the R,w,-plane. In the case of an accelerating-field diode q 2 0 for all 1 R 6 R,, and the right-hand side of Eq. (277) remains positive whatever the value of either wd or w,o. Thus irrespective of their initial velocities all electrons leaving the cathode will reach the anode. This is by no means true in the case of a retarding-field diode. Here the reduced potential q 0 and the right-hand side of Eq. (277) may become negative. The physical significance of this is simply that certain electrons, depending on

<

<

P. A. LJNDSAY

262

their initial velocities wd and w , ~ will , not be able to penetrate the field beyond a certain point given by w,2 = 0. It is best to consider fist those electrons which leave the cathode with W,Q = 0 and W ~ # O 0. An electron which possesses no radial component of its initial velocity w,.~can escape from the cathode only when its centrifugal force is greater than the retarding force exerted on it by the electric field a t the cathode. Equating the two forces gives an expression for the critical velocity of the electrons

mrodo=

=

-e -

fz)

0

e(g) kTero -1

o

the subscript nought referring as usual to the conditions a t the cathode. Multiplying both sides of Eq. (291) by ro/kT gives w,02 = ,1/ / 2 7 0'1 (292) where the prime indicates differentiation with respect to R. As long as wtQ = 0, the electrons can escape the cathode only when their initial tangential velocity we0 > { ->4$0)% (in the case of a retarding field the slope 7' < 0 by definition). It is now convenient to rewrite Eq. (277) in the following form: w,2 =

= wro2 =

+ (1 -); + + Y*(R) + d R )

%I2

Y(R)

+7 m 1

w,02

7

(293)

where the definitions of y and yt are selfevident. For the electrons leaving the cathode tangentially, the right-hand side of Eq. (293) becomes zero whenever yt = -7. Figure 22 shows a schematic diagram of yc and -7. Both functions are zero for R = 1, and the function yt is defined over the whole of the range, but apart from its end points, the function 7 = q(R) is not known in advance. It is necessary to consider the following two possible cases.

Case I Assume that -7' 3 yltEreverywhere, as shown in Fig. 22, where the primes represent differentiation with respect to R, and ytcris that particular yt which corresponds to the electrons eatisfying Eq. (292); that is, for which y t is tangential to - q at the cathode. The condition -9' 3 ytceris certainly

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

263

satisfied for a vanishingly small space charge, since theri q = qa In R/ln R. and -q co whereas yIa Weo2 < 00 for R ---f 00 ,both being monotonically increasing functions for R > 0. As the effect of space charge begins to be --f

---f

FIG.22. The reduced potential - q and the functions yt = (1 - l/Rz)weoz plotted against the reduced distance R. Case I of the retarding-field diode.

felt the condition -q' 3 y'ta grows weaker until it will cease to hold altogether. However as long as the condition -7' 2 ylIW is satisfied, there must be one y t for which yt = - q both a t the cathode and at the anode. The value of wm2 corresponding to this y c is given by

The tangential electron trajectories (wd2 = 0) now can be divided into three distinct groups: (1) When yt < tIcr the corresponding electrons can never leave the cathode. (2) When yIcr< y t < yta the corresponding electrons can leave the cathode, but they can never reach the anode and must come back to the cathode. (3) When y I > y14 the corresponding electrons can always reach the anode. It is now necessary to consider the effect of the radial initial velocity WIO. It can be seen from Eq. (293) that the introduction of wro is equivalent to putting y in place of y;. In Fig. 22 this corresponds to an upward shift of the yl curves so that each curve now cuts the vertical line R = 1 a t y = wd instead of y = 0. Considering the three different groups of electron trajectories, one can now draw the following conclusions.

264

P. A. LINDSAY

(1) As w,O iiicreaEes the electroils in the first group penetrate further and further into the interelectrode space, always turning back at a point wheri the two curves y and - q cross, that is, when y 7 = 0. When wIo is sufficiently large for y 2 -)I at the anode, the electrons reach the anode. (2) As wroincreases the electrons in the second group penetrate further and further into the interelectrode space, each time reaching a point which is further away from the cathode than if they left the cathode tangentially. O sufficiently large for y 2 -7 at the anode, the electrons reach When W ~ is the anode. (3) All electrons in the third group reach the anode whatever their value of WN. Figures 23a-e show the projections of the electron trajectories on the R,w,-plane, each set of curves corresponding to a different value of w,o. In Figs. 23a-d the limiting trajectory A is obtained by putting W,.O = 0 for R = Re in Eq. (293). This gives

+

w: =

- ;( - &)

w,o2

+q -

70.

(295)

However, this is not the only limiting trajectory. In a retarding-field diode there exists another limiting trajectory which is shown in Figs. 23c-e. It corresponds to those electrons which leave the cathode tangentially and acquire the wr-component of velocity entirely by virtue of the cylindrical geometry of the system. In a plane diode these electrons would graze the cathode all along their path and could never penetrate into the interelectrode space. The limiting trajectory of the tangential electrons, marked B in Fig. 23, can be calculated from Eq. (293) by putting w ~ o= 0. This gives

It can be seen from Eq. (270) that the projections of the electron trajectories on the R,w,-plane are independent of q and will be the same as ior a temperature-limited diode. These projections were shown in Fig. 19. Case I I

>

In Case I it was assumed that -q’ yltcr everywhere in the interelectrode space. Since the reduced potential -)I depends on the space charge, its curvature will increase with current until the sign of the inequality is reversed, a t least in places. It is instructive to consider first the case for which -1’ Y’tcr everywhere. The functions -7 and ytcrare both shown in Fig. 24. It can be seen from Fig. 24 that whenever the slope of y1 at the cathode is greater than that of ytcrthe function yt q > 0. Thus whenever

<

+

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

265

electrons have enough tangential velocity W,O to leave the cathode, they will invariably reach the anode. The rate at which the electrons convert their tangential into radial velocity is then greater than the retarding effect of the field and the electrons possess a surplus of kinetic energy which

FIG.23. The reduced radial velocity wra8 a function of R and we. Case I of the retarding-field diode.

will carry them on to the anode. This is a radical departure from the conditions which applied in Case I. However, as before, there now exists a functioii yl, for which yfo tl = 0 a t the anode. The value of w,a which correspotids to this function is again givcii by 13;q. (294). The tangential electron trajectories (,wfl = 0) now can IE subdivided intfo two different groups. (1) When u1 < yfer t,he correspoittliiig electroiis m i I I C V O ~leave the cathode because of the retarding field there.

+

P. A. LINDSAY

266

(2) When y t > ytErthe corresponding electrons leave the cathode and always reach the anode. The introduction of the radial component of the initial velocity WrO changes yt into y. This corresponds to a vertical shift of the yt curves in Fig. 24. Considering the two different groups of the electron trajectories one can draw the followingzconclusions.

-*

-

FIG.24. The reduced potential and the functions yt = (1 l / l P ) w ~ Splotted against the reduced distance R. Case I1 of the retarding-field diode.

(1) As w,o increases the electrons belonging to the first group will advance further and further into the interelectrode space, always turning back a t the point where y q = 0. As long as yf < -q' at the cathode, the two curves y and - q must become tangential a t one point, say R = R1. For any W,O greater than this value the electrons can alwaye reach the anode. As U'ao increases the point R1 will recede from the anode to the cathode. R1 = R, when y' = q' at the anode, R1 = 1 when y' = -?'at the cathode. The dependence of R1on WOOis shown schematically in Fig. 26. (2) All electrons in the second group, whatever their initial radial velocity W,O, must reach the anode. Figures 25a-e show the projections of the electron trajectories on the R,w,-plane. They look quite unlike the corresponding projections of Fig. 23. Although they may appear to be simpler, most calculations in this case are, in fact, much harder because of the variable radius R1,at which the electrons have SL choice of either going forward or turning back to the cathode. Since at the point R = R 1 the two functions y and - q are tangential, one has y = -7 and y' = -7'. Thus, from Eq. (293), the radius RI is given by the equat,ion

+

2weo2

+ Rl3q? = 0,

(297)

VELOCITY DISTBIBUTION IN ELECTRON STREAMS

(4

w:o=

-*dl

(e)

267

wzo>-~

2 ’lo

FIG.25. The reduced radial velocity w, ae a function of R and we. Caee I1 of the retarding-field diode.

where q’l stands for the value of dV/dR at R = R I . Equation (297) can be solved only when the potential function q = q ( R ) is known. The limiting curves A and B are the same as the corresponding curves in Case I-they are given respectively by Eqs. (295) and (296). The new curve C is obtained by putting W,O = 0 a t R = RI which, when substituted in Eq. (293), gives

The fact that RI is not a constant but varies with weo or wg is very awkward. This probably explains why Case I1 does not seem to have been discussed

268

P. A. LINDSAY

FIG.26. The point RI at which the electrons have a choice of either going forward or turning back, as a function of the tangential component of the initial velocity w ~ .

in the past. The dependence of R1 on woo is shown schematically in Fig. 26. It is worth noting that for each woo the trajectory which goes through the point R1 meets the R-axis at an angle

where q" and yftl are the second derivatives of the functions q and y with respect to R, taken a t the point R = R1. This result should be compared with the conditions in a plane diode, where the angle 6 = tan-' (>$q",,,]%, where q",,, is the second derivative of the function 11 = ( e / K T ) ( +- +,J at the potential minimum z = xm (see Sec. V,B,2).In a plane system there is only one electron trajectory which cuts the axis a t an angle different from the right angles, that trajectory always passing through the point of the potential minimum. In a cylindrical system no potential minimum is required for a trajectory to meet the R-axis a t an angle which is different from 90". All that is required is a retarding field q' which is sufficiently powerful to overcome the rate of transfer of energy from the tangential to the radial directions, given by y'. The potential distribution which separates Case I and I1 is given by either of the two equations --TI

dlc,

(300)

= 1Jtcr.

(301)

=

or -71

269

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

Now the tangeiitial velocity sufficient for the electrons with leave the cathode is given by w,o2

=

U',O

=

0 to

- 217 0, = 1 - -(l/Ru2)' I10

In this case the electron trajectories are still as shown in Fig. 25 but Fig. 25b would consist of a series of straight horizontal lines, whatever the , Figs. 25c and d would not be present at all. The limiting value of W ~ Oand potential distribution curve can be calculated from Eqs. (301) and (302)

This curve has been derived by Davisson (15.2) as a limit of validity of Schottky's expression for the anode current in a retarding-field diode (see Sec. VI,B,S,d). It is now possible to answer the following question: What happens when the potential distribution is of the form shown, for example, in Fig. 27, so that neither of the two conditions characterizing Cases I and I1 is satisfied? By analogy with Figs. 20-25 it can be seen that when - q has the form shown on Fig. 27a the trajectories will look somewhat like those shown in Fig. 25, except that instead of Fig. 25d there will be a combination of Figs. 25c and 23c, R1of Fig. 25c becoming R, of Fig. 23c. This will then be followed by a combination of Figs. 25c and 23d ending eventually with the electron trajectories shown in Fig. 25e. All these trajectories will occur again in the case of a space charge-limited diode and are shown in Figs. 35d-f. If the potential distribution were of the general form shown in Fig. 27b then the corresponding electron trajectories would he as follows: instead of Fig. 25c there would be a figure consisting of Figs. 25c and 23c put together, Fig. 25c now coming fist, the point R = 1 of Fig. 23c coinciding with point R1 of Fig. 25c. This would be followed by a similar combination of Figs. 25c and 23d and finally by Fig. 25e. It is a bit doubtful whether these more complicated potential distributions of Fig. 27b would ever occur in practice in a retarding-field diode. The interesting point to observe is that for all potential distributions, except that of Case I, the limiting trajectories have to be specified in terms of the point R1, which is itself a function of the reduced potential 7. The potential distribution of Fig. 27a could occur easily in the limiting case of a potential minimum at the anode. As a matter of fact it is clear from Fig. 22 that if -9' = 0 at the anode, the condition -q' 3 yrtercharacterizing Case I cannot be fulfilled and at least for a certain range of we0 the limiting trajectory will be of the type C, Fig. 25. Since this involves the critical

270

P. A, LINDSAY

(b)

FIQ. 27. Possible potential distributions in a retarding-field diode, other than Cases I and 11.

radius R1,Schottky’s equation for the anode current in a retarding-field diode (Sec. VI ,B,S,d) cannot possibly apply to all potential distributions. b. Limits of integraiion in w,- and we-directions. As was the case in an accelerating-field diode, the limits of integration in the velocity subspace are given by the limiting trajectories A-C in Figs. 23 and 25. Case I

Substituting Eq. (210) in Eqs. (295) and (296) one can obtain expressions for the following two surfaces

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

271

The two limiting surfaces joined together are shown in Fig. 28. The ELKfaces cut along the trajectory given by w** = 0, weo2 = -qa/(l - Ra2), as shown in Fig. 23d. The crom section of the composite surface by a plane

FIQ.28. The limiting surface generated by all possible trajectories of those electrons which either graze the anode or leave the cathode tangentially. The dashed lines represent typical electron trajectories.

perpendicular to the R-axis is shown in Fig. 29. The two curves are respectively an ellipse and a hyperbola, both centered on the origin. For R = R, the ellipse degenerates into a straight vertical line through the origin; for R = 1 the hyperbola degenerates into a similar straight line. The portion of the plane shown shaded in Fig. 29 represents that part of the phase

272

P. A. LINDSAY

FIG.29. The curve obtained by cutting the limiting surface of Fig. 28 by a plane perpendicular to the R-axis.

space which can be reached by electrons-it is the range of integrat,ion of the phase-space density function. It should be noted that the two curves of Fig. 29 cross at w,p =

-?a

R2(1- 1/R,2)

(306)

and that the hyperbola cuts the we-axis a t a point

Case

ZZ

Substituting Eq. (270) in Eqs. (295), (296), and (298) gives the three limiting surfaces corresponding to the limiting trajectories A, B, and C in Fig. 25. The h s t two surfaces are given, as before, by Eqs. (304) and (305). This time however they do not cross and they are joined by a third surface which crosses the other two and is given by

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

273

FIQ.30. The limiting surface generated by all possible trajectories of those electrons which either graze the anode, pass through the point R I , or leave the cathode tangentially. The dashed lines represent typical electron trajectories.

The composite limiting surface in the R,wr,we-space is shown in Fig. 30. It can be seen from (304) and (305) that the two surfaces become tangential along the trajectory Wr.0 = 0, we? = -~./(1 - 1/Ra2) when 7 is given by Eq. (303). The cross section of the limiting surface by a plane perpendicular to the R-axis is hown in Fig. 31. It is convenient to note that the curves obtained by c tting the surfaces given by Eqs. (304) and (308) cross at

E

Similarly the curves obtained by cutting the surfaces given by Eqs. (308) and (305) cross at

274

P. A. LINDSAY

FIG.31. The curves obtained by cutting the limiting surface of Fig. 30 by a plane perpendicular to the R-axis.

In both cases the values for we2have been derived t y noting the appropriate values for weo2in Figs. 25(b) and (d). Finally, the curve obtained by cutting the surface given by Eq. (308) crosses the we-axis a t a point

c. The generalized volume density of the electrons. The generalized volume density of the electrons can be obtained by integrating Eq. (276) with respect to w over the range shown shaded either in Fig. 29 or 31.

275

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

=

OD

--oo

=

dw,

\ /{Iwo' i:

wR2r-W exp q exp f - (w2

dwe

dwe

nTRexpq-2 ?r

exp f - (w?

+ + w$))dw, wg2

+ we2))dwr

+ we2)Idw,)

exp f - (w? =

2 exp q _ _

n&

Ir,

+I

e-u'oz[l- erf { (P- l)we2

+ q)!+'j]dwe

Wtl

where

+

wrz2 = (R2 - l)we2 7, and the values of we1 and w62 are given by Eqs. (306) and (307). As in the case of an accelerating-field diode the generalized electron density n(q) is now a function of both q and R and could be represented by the height of a surface over the q,R-plane. For a given diode, q = q ( R ) is fixed, and the dependence of the electron density on either R or q is then given by the height of the surface over the q = q(R) curve. Case II n(q)

=

/n(q,w)dw

=*Rexpq-2

?r

+

dw,

{ Jo

w8a

1;

dtue

exp ( - (

+ we') 1 ~ +2 we2) )dwr + I,dwe exp

- (w?

dulr

L+

exp { - (w?

we2)) dwr

P. A. LINDSAY

276 =

2 wR exp 7 T <

where

and the other quantities wrl, w,z, we3, we4, and We6 have been defined in connection with Eq. (312) and by Eqs. (309), (310), and (311). In this case the generalized volume density of the electrons n(q) ie again a function of the two variables R and q. However this time the "constant" R1 is itself a function of we2, so that the integration can be carried out only when the functions = q(R) and thus R1 = Rl(we2) [see Eq. (297)l are known. For this reason Case I1 is much harder to treat than Case I. It might be added that the integrals appearing in Eqs. (312) and (313) are somewhat involved [see (156, 157)], but they are certainly not beyond the powers of an electronic computer. d. The generalized current density of the electrons. As was the case with an accelerating-field diode it is now possible to calculate the generalized current density of the electrons. Using Eqs. (282-284), which are quite general, and noting the symmetry of the shaded areas of Figs. 29 and 31, it is possible to show that Eqs. (285) and (286) are again valid in the case of a retarding-field diode. Thus J,(q) is the only component of the current density which is different from zero. It is again convenient to coneider the two cases I and I1 separately. Case I

Substituting Eq. (276) in Eq. (282) and introducing the correct limits of integration shown in Fig. 29 one obtains

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

{ eqa /d"" exp (-

we2) dwe

+

277

I:

exp (- R2w2)dwa}

(3 14) where wrl, wr2, and wel have been defined before. Substituting from Eq. (289) for the electronic saturation current per unit length of the cathode, Eq. (314) reduces to

I I = It,

=

It,

{

?4

erf

(A)+ 1 - erf (1 - (1/Ra2) )"}. (315)

Equation (315) is the well-known Schottky expression for the anode current in a retarding-field cylindrical diode [Eqs. (1) and ( 2 ) in (149)].Equation (115) is its counterpart for a retarding-field plane diode. Equation (315) reduces to Eq. (115) when R, -+ 1. In the case of a plane diode the shaded area equivalent to that of Fig. 29 would consist of the whole of the wr,wpplane to the right of a vertical line through wr = - (7- qa}H. For very large values of -qa/(Ra2 - l), Eq. (315) reduces to

It = It,

Il.Rae*a. (316) For small values of --fa/(Ra' - l), for example, when R, > 30 (single wire cathodes) and I], < 10 (small anode voltages), Eq. (315) reduces to

It

=

11, = It8 ( 2

" + erfc ( (2) eqn

-~

~ ~ R a 2 ) ) " }(317) ,

where again erfc z = 1 - erf 2. Both expressions, Eqs. (316) and (317), were derived by Schottky in (149).

P. A. LINDSAY

278

Case I I Substituting Illq. (276) in Eq. (282) and introducing the correct limits of integration, one obtains, with the help of Fig. 31,

=

(--) 2lcT " m R e q -2 H

Introducing I I , from Eq. (289) this reduces to

where, since toea2 = R2we2, it proved convenient to introduce weo as the variable of integration. Here both 711 and R1are functions of WEO,v1 = @,), R1 is defined by Eq. (297). The function R1 = Rl(we3) is shown schematically in Fig. 26. In view of this functional dependence Eq. (319) is much more complicated than the somewhat similar Schottky expression, Eq. (315).

VELOCITY DISTRIBUTION I N ELECTRON STREAMS

279

Comparing Eqs. (317) and (319) and the shaded areas in Figs. 29 and 31, it is fairly obvious that for a given R, and va the current density in Case I1 is larger than in Case I. This is hardly surprising since in practice Case I1 would be reached by increasing the cathode emission and depressing the potential beyond the limit allowed in Case I. [See Eq. (303).] When the emission is sufficiently large -7’ can become less than y’tcFeverywhere in the interelectrode space, as shown in Fig. 24. As long as there is no actual potential minimum between the electrodes, an increase of the cathode emission is invariably associated with a larger anode current. 4. The Space Charge-Limited Diode. It is now possible to extend the investigations to the case of a space charge-limited diode, where a potential minimum exists between the cathode and the anode. In practice this is the most common arrangement, but unfortunately it is also the most difficult to treat. As mentioned above, Wheatcroft (154) is probably the only writer to have considered it in any detail, apart from some brief references by Langmuir [Eqs. (28) in (134)].In this chapter the discussion will depart from that given by Wheatcroft by considering more thoroughly the actual trajectories of the electrons in the velocity subspace. a. Electron trajectories i n the space charge-limited diode. Putting (61 = (6m in Eq. (262) and eliminating up2between Eqe. (269) and (270) gives the projections of the electron trajectories on the R,w,-plane

w12 = w,02

+ (1 - $) + 81 + -

= %I2

9

W802 70

+

r]

- 70 (320)

= y+q-vo.

Most of the remarks concerning the pecularities of the electron motion in axially symmetric retarding fields apply equally to the electron motion in the retarding portion of a space charge-limited diode. The differences are largely due to the fact that in the retarding-field diode the anode is placed a t a point where the potential curve has a negative slope, whereas in a space charge-limited diode the potential distribution is much more complicated, the anode being placed beyond the potential minimum and a t a point where the potential curve has a positive slope. However, due t80 the transfer of energy from the tangential to the radial direction, a feature of the systems possessing axial symmetry, the difference between the electron trajectories in retarding-field and space charge-limited diodes is actually less than would be expected. In the analysis of the retarding-field diode it proved convenient to distinguish between Case I where -7’ 2 yltcrand Case I1 where - q’ 6 y’lcr. A somewhat similar distinction is also convenient in the case of the space charge-limited diode.

280

P. A. LINDSAY

I

-==I

(b>

-

FIG.32. (a) The reduced potential - ( q - VO) and the function y: = (1 l/R*)w&S plotted against R. (b) The point RI as a function of the initial velocity wm. Case I of the space charge-limited diode. Case I

<

Assume that -)I’ ytrcrin the a region, that is, between the cathode R = 1 and the potential minimum R = Rm. (In view of the negative slope of -7 past the potential minimum, the above condition will be satisfied automatically in the P region of the valve, that is, for all €2, R 6 R,,.) It appears from inspection of Fig. 32 that once WOOis sufficiently large for the tangential electrons to leave the cathode, they will automatically

<

VELOCITY DISTRIBUTION I N ELECTRON STREAMS

28 1

reach the anode whatever the value of the radial conipoiieiit of their initial velocity wro. Furthermore, it appears from Fig. 32 and Eq. (297) that the range of values for which -q' and g'# can be equal extends from R 1= 1 t o RI = R,. Figure 33 shows the usual projections of the electron trajectories on the R,w,-plane. They are obtained with the help of Eq. (320) and Figs. 32a and b, Fig. 32b giving the approximate functional dependence of R1

"'t t

R

Wrl

"'t

CURVE

wgo.

c : w', '7 -7,-

(I/$-

I/p;)W&

FIG.33. The reduced radial velocity wr as Case I of the space charge-limited diode.

CURVE D : w: '(l-'/R*)We,%+?-l~

a

function of R for different values of

on wgc. It is important to note that, except when Woo = 0, the electrons do not turn back a t the potential minimum R = R,, but at some point R1 which is nearer to the cathode than R,. Thus, from Eq. (320), the initial radial energy which the electrons require in order to reach the anode is only wd2 = 70 - - (1- 1/R1*)weo2for a given w&. That this remarkably low value of w , ~is sufficient for an electron to reach the anode, is a result of the transfer of energy from the tangential to the radial directions, a process which is quite impossible in a plane diode. Thus, in a cylindrical diode, the only electrons which still require wd2 = 70 for reaching the anode

282

P. A. LINDSAY

are those which leave the cathode with weo = 0, t!hat is, at right angles to its surface. This is t,he reason for the commonly expressed opinion that iii cylindrical diodes the effect of the potential minimum is much less marked than in the casc of a plane diode. Similarly, since fewer elect,rons can stay between the electrodes in a cylindrical than in a plane diode, one would expect that for the same values of the emission current, potential difference, and electrode separation the depth of the potential minimum would be less in a cylindrical than in a plane diode. This is confirmed by Fig. 3 of Wheatcroft (154), where it is shown that for the same depth of the potential minimum the anode current is higher in a cylindrical than in a plane diode [in terms of Fig. 3 in (154) the potential minimum in a plane diode is given by wm = log. (10/1)1. It should be noted that, as was the case in a retarding-field diode, Eq(299) gives the angle at which the limiting trajectory, curve C in Fig. 33, meets the R-axis. Substituting Wro = 0 at R = R1 in Eq. (320) it can be shown that these trajectories are still given by Eq. (308). The other limiting trajectories are formed by the electrons leaving the cathode tangentially. They are marked D in Fig. 33 and are obtained by putting wro = 0 in Eq. (3201, wr2

= (1

- +j)

we02

+7-

70.

(321)

Case I1 I n practice Case I is reached by increasing the cathode emission in a temperature-limited diode until a potential minimum appears first a t the cathode and then in front of it. As the cathode emission increases even further the potential minimum deepens until a state is reached when the conditions imposed on -q are no longer satisfied. Case I may then go over to Case 11, shown schematically in Fig. 34a. The requirements which now have to be fulfilled by the potential distribution -q are -9’ 3 yllcr for 16 R RI,, where RI, is some point between the cathode and the potential minimum. I t can be seen from Fig. 34a that a t the limiting point R1, one of the yt-curves, in fact yt = ytm,becomes tangential to -7. Having established the main features of the potential distribution curve it is now possible to consider the new shape of the electron trajectories. The electrons with no radial component of velocity cannot leave the cathode until their tangential velocity we0 is equal or greater than the value given by Eq. (292), (that is, until the effect of the centrifugal force outweighs that of the retarding field a t the cathode). Second, for one particular value of WOO, y t becomes tangential to - q . This occurs a t a point R1 = R,,, R 1 again being given by the general Eq. (297). For any w#o equal or greater than this the electrons will reach the anode, whatever their initial radial

<

VELOCITY DISTRIBUTION I N ELECTRON STREAMS

283

t

04 Fro. 34. (a) The reduced potential - (7 - V O ) and the functions gr = (1 - 1/R2)weo2 plotted against R. (b) The point R1 as a function of the initial velocity WBO.Case I1 of the space charge-limited diode.

velocity wd.The dependence of R I on weo2 is shown schematically in Fig. 34b. Consider the projections of the electron trajectories on the R,w,-plane. Figures 35a-f show the electron trajectories for different values of the initial velocities wd and weo. It is interesting to note that as a consequence of the different conditions imposed on the function -1, new types of electron trajectories appear in Figs. 35d and e. They have no counterpart in Fig. 33 and are related to the electron trajectories shown in Figs. 23c and d which refer to the retarding-field diode.

FIG.35. The reduced radial velocity w, as a function of R for different values of woo. Case I1 of the space charge-limited diode.

b. L i m i t s of integration in w,-and zoo-directions. The surface which limits the range of integration in the R,w,,wo-space can be obtained by eliminating woo between Eqs. (270), (298), and (321). This process gives two limit.jng surfaces. One surface, Eq. (308), represents the trajectories of those electrons which have enough energy to reach a point where they have a choice of either going forward or turning back to the cathode. The other surface, given by

+ 7 - 70,

~ 1= 2 (Rz - l ) ~ $

(322)

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

285

is formed by the trajectories of those electrons which leave the cathode tangentially. Since the limits of integration differ somewhat in Cases I and I1 it is convenient to consider them separately. Case I

The corresponding limiting surface is shown in Fig. 36 and its cross sections by planes perpendicular to the R-axis are given in Fig. 37. The dashed lines in Fig. 36 represent limiting electron trajectories, the arrows

FIQ.36. The limiting surface generated by all possible trajectories of those electrons which either pass through the point RI or leave the cathode tangentially. The dashed lines represent typical electron trajectories.

indicating the direction of motion. In Fig. 37 each curve corresponds to a different value of R, the dashed line representing the projection on the w,,we-plane of the electron trajectory characterized by wTo2= 0, woo2 = - >&‘o. It can be seen from Fig. 33c that along this trajectory the two surfaces given by Eqs. (308) and (322) interpenetrate. Case 11

The corresponding limiting surface is shown in Fig. 38 and its cross sections in Fig. 39. The surface is similar to that of Fig. 36 except for the depression near the cathode. This depression occurs because it is now possible for the tangential electrons to leave the cathode but fail to reach the anode, the transfer of energy from the tangential to the radial directions being too slow to overcome the retarding effect of the field. In Fig. 39 each

286

P. A. LINDSAY

FIQ.37. The curves obtained by cutting the limiting surface of Fig. 36 by a plane perpendicular to the R-axis. The dashed lines represent the projection of the electron trajectory C = D.

curve again represents a cross section of the surface for a different value of R. The dashed lines C = D represent the trajectory which separates the two surfaces, Eqs. (308) and (322). This trajectory now corresponds to those electrons which leave the cathode tangentially (wN2 = 0) but with an initial velocity given by wooZ = ->5(qo - m,J/(l - l/Rlm2),RI, being shown in Figs. 34b and 35e.

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

287

FIG.38. The limiting surface generated by all possible trajectories of those electrons which either pass through the point R1 or leave the cathode tangentially. The dashed lines represent typical elect,ron trajectories.

c. The generalized volume density of the electrons. The generalized volume density of the electrons can be obtained by integrating Eq. (276) with respect to w introducing the limits of integration shown in Figs. 37 and 39.

288

P. A. LINDSAY

FIG.39. The curves obtained by cutting the limiting surface of Fig. 38 by a plane perpendicular to the R-axis. The dashed lines represent the projection of the electron trajectory C = L).

289

VELOCITY DISTRIBUTION IN ELECTRON STREAM9

for 0

6 R 6 R,,

n(q)

= =

that is, for the a region of the valve. Similarly,

"/-

4q1w)dw

~ T exp R

(7

- 70) ;2

{Jd"" dwe 1:exp { - (wr2 + wez)) dw, + JI.: dwe

L:

+ we') ]dwr}

exp ( - (w,2

for R, 6 R 6 R,; that is, iii the j3 regioii of the valve. Here wr2, wr3, we3, we(, and We6 have the same meaning as before. As in Case I1 of the retarding-field diode the integrals defining the generalized density function n(q) contain the quantity R , which in itself is a function of woo, as shown in Fig. 32b. This makes the evaluation of these integrals much harder and probably explains why the space charge-limited cylindrical diode has received so little attention in the past. Case 11

In this case again, the generalized volume density of the electrons can be obtained by integrating Eq. (276), except that the limits of integration have to be slightly altered. Abbreviating somewhat the algebra and taking into account the new limits of integration shown in Fig. 39, one can obtain for 0 R Rim, or the a' region of the valve,

< <

n(q) = ~ T exp R (7 - 70);2

+

l: l.1 dwe

{Lw"dwe JI,: exp +

[ - (w,2

exp [- ( ~ , 2 ws2)ldwr - 2

+ we2)1dwr

Sw0' I

wr:

UjI

dwe

[-

(10:

exp

+ we2)ldwr

290

P. A. LINDSAY

For the other two ranges, that is, for R1, ,< R ,< R,, region a" and for R, 6 R 6 R,, region 8, the expressions for the function n(q) are formally the same as Eqs. (323) and (324). This does not mean that the actual values of the function are the same, but the integration pattern, for example, the subroutine on a digital computer, will be the same in each case. Equations (323-325) are still somewhat formal in nature and they are of value only when a fast computer is accessible. However, they seemed worth quoting in view of their somewhat involved limits of integration. It is worth noting that, from Eq. (50), the ratio n(q,w)/n(q) gives the important conditional probability density function for the velocity distribution of the electrons j(wlq). At the potential minimum, that is, at R = R, this function is still proportional to exp { - (w: We2 w.")) , but the range over which it is different from zero is now given by the R = R, curves of Figs. 37 and 39. These curves show clearly that at the potential minimum of a cylindrical diode certain ranges of low radial velocities are missing altogether, This is quite unlike the conditions in a plane diode where all velocities are present. The missing velocities are caused by the fact that for any WOO# 0 the electrons do not come to rest a t R = R, but at a point R1 < R,, and by the time they reach the potential minimum they have already acquired some velocity. Similarly, some electrons do not stop at all, although they may start with wtd = 0. d. The generalized current density o j the electrons. As in the case of a retarding-field diode it is now possible to calculate the generalized current density of the electrons. Since the contours of Figs. 37 and 39 are symmetrical with respect to the wr-axis and since the limits of integration in the zc*-direction are - a, a, it can be seen that again J,(q) is the only component of the current density which is different from zero. From the symmetry properties of the integrand wrn(q,w) it appears that there is no formal difference in the limits of integration bet.ween Cases I and I1 and that the same equations apply in both cases. Substituting Eq. (276) in Eq. (282) one obtains

+ +

Jr(q) =

2kT ' (m) 2n~Rr-?6cxp ( q - qo)

1 1 \ *

-01

dw, dwe

wr exp

VELOCITY DISTRIBUTION IN ELECTRON STREAMS .

291

After substituting from Eq. (289) this reduces to

Thus in a space charge-limited cylindrical diode the anode current is not proportional to exp (-qo) but is a more complicated function of it. This is unlike the conditions in a plane diode, where, from Eq. (114)) the anode current depends linearly on exp (-’I,,). In a cylindrical diode the expression for the anode current is modified by: (1) a coefficient in the form of an integral which expresses the fact that due to the exchange of energy between the radial and tangential velocities some electrons can pass the potential barrier although their initial radial energy is lower than the corresponding initial energy required when woo = 0, and (2) additional terms which express the fact that in a cylindrical diode tangential electrons can reach the anode. These electrons contribute to the anode current in the cylindrical case, although in a plane diode they would move parallel to the anode and could never reach it. Equation (327)can be looked upon as an extension of Schottky’s equation, Eq. (315))to the more general case of a space charge-limited diode. It reduces to Eq. (114) when one assumes that, as is the case in plane

292

P. A. LINDSAY

diodes, H = R,,, is the only point of unetoble cquilibriurn so that R , = Zi,,,, q 1 = 0, and the int8egralbccomes equal to R,, which in turn +l. Furthermore, since in n plane diode no tangential electrons can contribute to the anode current, -q’o must be put equal to infinity, causing a cancellation of the last two terms in the brackets. Some authors [Eq. (16) in Wheatcroft (154)] use an approximation to Eq. (327) by putting q o and R,,, in place of q, and R, in Schottky’s expression, Eq. (315). This gives an overestimate for the anode current since Schottky’s formula is no longer valid when -17’ 6 0 a t the anode, the limit of validity being given by Eq. (303). In fact, the approximation amounts to taking R1 = R,, 71 = 0 and ~ 0 0 4 ’ = qo/[l - (l/Rmz)] instead of ~ ~ = 0 ->4q’o, Eq. (310). This is equivalent to neglecting the fact that Rl itself is a function of we and as a rule decreases as w o increases. 5. Potential Distribution in a Cylindrical Diode. It has been mentioned elsewhere in this chapter that in a cylindrical diode the potential function can be much more complicated than in a plane diode, since for positive anode voltages it has a natural curvature in the opposite direction to that required by the space charge. Thus the appropriate solution of Poisson’s equation must allow for this. I n cylindrical coordinates r, 0, z the Poisson equation has the following form

where p = -en(q)/r is the space-charge density of the electrons. Substituting this in Eq. (328), multiplying both sides by r and introducing the reduced potential q and the reduced distance R gives

I t is again necessary to consider the two cases mentioned in Sec. VI,B,Q separately.

Case I Substituting from Eqs. (323) and (324) and using the expression for the saturated current per unit length of the cathode, Eq. (289), one obtains

4

~

VELOCITY DISTRIBUTION I N ELECTRON STREAMS

293

where f(R,q) stands for the expressions in brackets multiplied by 2/&. Putting $ = q - qo and changing the variable of integration in f(R,q) from we to WOO= we/R one obtains

where A is a constant and R has been conveniently incorporated in F(R,q), since dwe = Rdweo. No further normalization of Eq. (330) seems possible. The constant A depends on the cathode radius ro and on the saturation current per unit length It,. Equation (330) is of the integrodifferential type and its solution should be possible with the help of a digital computer. Instead of the single curve which would be obtained in the case of a plane diode, the solution of Eq. (330) leads to a family of curves, each corresponding to a different value of A . Case 11 ' It is reasonable to assume that for sufficiently large I I , or ro a situation will be reached when the conditions of Case I1 will apply. The substitution of Eqs. (323-325) in Eq. (329) leads to an expression which is formally identical with Eq. (330). Now the range of integration with respect to R has to be split however into three intervals, 16 R 6 R1,, Eq. (325), R1, 6 R 6 R,, Eq. (323), and R, 6 R 6 R,, Eq. (324). This additional complication makes the solution of the equation one stage more difficult to obtain. Wheatcroft (164) obtained an approximate solution of Eq. (330) by dividing the interelectrode space into retarding-field and accelerating-field diodes. By a judicious under- and overestimation of the space charge he obtained approximate expressions which could be integrated graphically. His results are collected in a series of drawings giving the position and depth of the potential minimum as a function of In I z , / I z . and In I~J-O.Curves for calculating the anode potential are also included. An interesting discussion of the actual accuracy of the anode current calculations, together with the discussion of Davieson's work (15d), can be found in a recent paper by Bell and Berktay (155). From the analysis presented in Sec. VI,B,S of this chapter there is no doubt, however, that whenever the potential in a retarding-field diode is lower than that given by Eq. (303), the anode current will be space charge-limited, although there is no potential minimum between the cathode and the anode. This is a consequence of Eq. (319) where it is shown that in the above circumstances, the anode current does not depend exclusively on the potential of the anode but is a function of the potential distribution in the whole of the interelectrode space.

294

P. A. LINDSAY

VII. VELOCITY DISTRIBUTION IN THE PRESENCE OF A MAGNETIC FIELD A . General Considerations In many technical applications it is necessary to consider the flow of electrons under the joint influence of electric and magnetic fields. In spite of its great practical importance the problem of velocity distribution in such systems has not been considered in sufficient detail, largely because in the presence of a magnetic field the trajectories of individual electrons tend to crosa. This introduces great mathematical difficulties since the solutions of the trajectory equations now have to be expressed in terms of multivalued functions. It seems possible to overcome these difficulties only by considering the motion of the electrons in a configuration-cum-velocity phase space (see Sec. 11).In such a space the point at which two electron trajectories normally cross splits into two separate points EO that the inherent singularities of the electron motion introduced by the presence of the magnetic field are artificially removed. The price which has to be paid for this improvement appears in the form of additional independent variables, the three velocity qk (or momentum p k ) components now being treated on the same footing as the three configuration coordinates qk. Gabor (168) was the first to realize the advantages of the phase-space approach in considering the flow of electrons through crossed electric and magnetic fields. Assuming zero anode currents (i.e. conditions below cutoff) and a narrow range of initial electron velocities he succeeded in calcu lating the space-charge distribution outside the cathode of a cylindrical magnetron. In particular his Eq. (12) gives the limiting surface formed by the trajectories of the electrons leaving the cathode surface tangentially. Since the magnetron is assumed to be below cutoff this is the only limiting surface to be considered. Gabor's approach has been further developed by Twiss (169) who considered both a linear and a cylindrical magnetron, the former in some considerable detail. Assuming zero initial velocities tangential to the cathode and a half-Maxwellian distribution of the initial velocities perpendicular to it, Twiss was able to calculate the space-charge distribution in a linear magnetron which could be either below or above cutoff. Furthermore, assuming zero initial velocities perpendicular to the cathode Twiss found that a spread in tangential velocities of the electrons has a much greater effect on the space-charge and potential distribution between the electrodes than a corresponding spread in the radial velocities. Almost a t the same time Fechner (160)calculated space-charge distribution in a cylindrical magnetron below cutoff. In doing this he assumed a half-Maxwellian distribution of the velocity components perpendicular to the cathode but completely neglected any spread in the tangential velocity

VELOCITY DISTRIBUTION IN ELECTBON STREAMS

295

coniponciits. In view of Twiss's work this seems somewhat unfortunate siiiae it probably invalidates some of the conclusions drawn in Fechner's paper. Finally Hok (161) discussed the problem of space-charge distribution when thermal equilibrium exists in the interelectrode space. He stressed the fact that in the presence of a magnetic field some electrons may be trapped in the electron cloud and stay there long enough for the electron scattering effects to play their part. When this happens the usual assumptions of negligible electron interaction ceasa to hold and the problem must be considered from the point of view of statistical mechanics. Since the purpose of this article is to consider the problem of velocity distribution in an electron stream rather than the space-charge density no simplifying assumptions will be placed on the velocity distribution of the emitted electrons. In order to make the calculations of the electron trajectories a t all manageable it will be necessary instead to assume that the region between cathode and anode is free from electric fields. Thus this chapter is complementary in nature to the work described in references (158-161) where the space-charge effects play a prominent part under various simplifying assumptions placed on the distribution of initial velocities. It should be added that quite recently the work has been further extended to cover the conditions inside linear and cylindrical magnetrons in the presence of space charge. The results of these investigations show the importance of the tangential and azimuthal components of the current density and the importance of the tangential components of the emission velocities in calculating the space-charge distribution between the electrodes (161a).

B. The Plane Magnetron The equations of motion can be derived without difficulty for the general case of a plane magnetron. 1 . Equations of Motion. Consider a plane diode with its cathode and anode situated a t xo and x , along the x-axis and perpendicular to it. Further, assume an electrostatic potential cp = 4(x)and a constant magnetic field B = (O,O,B,) or A = (O,A,,O) where A is the vector potential and A , = xB. = xB. Substitution of these values in Eqs. (4) and (5) gives the Hamiltonian

x = 2m -1 (P,"

+ (P, + eAJ2 + P*zl - e $ ( t > ,

(331)

and the Lagrangian

L = %mIv,2

+ v 2 + vxz) + e t # ( x > - u,A,I.

(332)

296

P. A. LINDSAY

Differentiating Eq. (332) with respect to V, as indicated in Eq. (6), gives the following expression for the momenta (333) (334) (335)

P , = mv,, p , = mv, - eA,, ps = mu,.

Similarly the differentiation of Eq. (331) with respect to r, as indicated by Eq. (€9,gives the following three equations of motion

+ e B r ) e B + e-9d$dx

1 (py @= --

dt

m

Comparison of Eqs. (336) and (79) shows that the magnetic field has introduced a degree of coupling between the x- and ydirected momenta. Multiplying both sides of Eq. (336) by 2p,, adding dp,/dt = 0 to v g B and integrating gives

+ (P, + e W 2

pZ2

+ + e B s d a + 2em($ - $01,

pZo2 (P,O P,2 = P,02, pZ2 = pZo2J =

(339) (340) (341)

where the subscript nought refers to the conditions a t the cathode. Equations (339-341) can be expressed in terms of velocities rather than momenta. Substituting from Eqs. (333-335) and dividing by m one obtains

(vv

-

gx)' ( =

21.2

Vyo

- eB 2 0>' ,

(343) (344)

= VJ.

The quantity eB/m has the dimensions of sec-I and is often referred to as the cyclotron frequency we. Adding Eqs. (342-344) gives v202

+ v,02 +

VZO'

=

vz2

+ + v.2 - 2e-m (4 V!?

40).

(345)

This shows that the magnetic field merely changes the direction of motion of the electrons and does not add to their energy, which is a function of

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

297

the electrostatic field only. In fact Eq. (345) is valid even when 4 = t#~(z,y,z) and not merely when 6 = #(z). Equation (345) also shows that the phasespace density of the electrons, Eq. (86), is still valid in the presence of a magnetic field. Introducing the variables given by Eqs. (87-90) and, in addition,

where R = ( X , Y , Z )is the reduced position vector, Eqs. (342-344) simplify to

Here the coupling between the x- and y-directions is again clearly noticeable. The conservation of energy, Eq. (345), now can be expressed in terms of the new coordinates w,"'

+

U),o2

+ u),o2=

UI,'

+ w,' + w,'

-

(7

- 70).

(350)

Since :m increment in the number of electrons is given by

dN

=

n(r,v)drdv = n(R,w)dRdw,

(351)

it is possible to show, taking into account the Jacobian

Substituting Eqs. (347-349) and (353) in Eq. (86) gives an esprrssion f o r the phase-space density in terms of the new coordinates

n(R,w)= nT2n-5 exp (v - vo) exp - (wZ2+ wu2 + w?)}, (354) where, from Eq. (74),

P. A. LINDSAY

298

wx

t

C W E A.

CURVE

w,'.

0

x(2w~o-x)

c

(4-+x,<w.p


(d)

wYe-

- L2 x a

(e)

-

-

< wvo <-fx,

FIG.40. The z-component of the reduced velocity, w,, as a function of X and wy. Magnetic field is in the z-direction.

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

299

Since the coefficientin Eq. (353) is a constant, the function n(R,w) retains the property of being constant along any given electron trajectory. In addition the function n(R,w) is (1) independent of y and z, and (2) separable into a product n(X,w,w,)n(w,) = n(wd,w&z(wd). Thus the statistical independence of motion in the zdirection is preserved under the influence of the magnetic field, and the limits of integration with respect to the reduced velocity w. will be everywhere the same as a t the cathode, that is, - ca , m . 2. Electron Trajectories. It is now necessary to assume that the cathode and the anode are a t the =me potential $0 = $. and that the electron space charge is sufficiently small for its effect to be neglected. Thus, except for the presence of the magnetic field, the region between the cathode and the anode is now field-free. Putting 7 - qo = 0 in Eq. (347) gives wz2

+ wy" =

WdZ

+ wl/o2.

(355)

Equations (355), (348) define the electron trajectories. Figures 40a-e show the projections of these trajectories on the X,w,-plane. The curves are obtained by eliminating w, between Eqs. (355) and (348)

+ ( X + wfly + (356) Equation (356) represents a family of circles with the center a t (wyo,O) and radius equal to (wJ + wJ)N. It can be seen from Figs. 40a and b that when wyo > 0, the electrons with low w d return to the cathode, whereas w2

= Wd2

W,I'

those with a sufficiently high w+o strike the anode before their xdirected component of reduced velocity wz becomes zero. The limiting trajectory Corresponding to those electrons which just graze the anode can be obtained by putting w+:= 0 a t X = X . in Eq. (356),

w,2 = ( X ,

+ wl/o)2 - ( X + wyo)2.

(357) This curve is marked A in Fig. 40. For negative w,o the situation is more complicated, as shown in Figs. 40c-e. If - x X , , < wu0 < 0, in addition to curve A there is another limiting trajectory corresponding to the tangential electrons. Putting wzo= 0 in Eq. (356) gives

wz2 = wyo2 - ( X

+

w,o)2.

(358)

For W,O 2 0 the tangential electrons can never leave the surface of the cathode, but for wyo < 0 they enter the interelectrode space and, provided lwyol < x X a , are turned back to the cathode before reaching the anode (see, for example, Fig. 40c). When W,O = - W X 0 the tangential electrons are just grazing thc mode, as shown in Fig. 40d. For W~ < ->$Xaall tlangentid electrons, whatever their initial velocity wzo, will be carried to the

300

P. A. LINDSAY

anode as shown in Fig. 40e. The projections of the electron trajectories on the X,w,-plane are shown in Fig. 41. The curves are given directly by Eq. (348)and they are all straight lines inclined to the X-axis a t 45". '

FIG.41. The y-component of the reduced velocity, w,,,as a function of the reduced distance

X.Each line corresponds to a different initial velocity wYo.

3. Limits of integration in w,- and w,-directions. The expression for the surface generated by the limiting trajectories A can be obtained by , between Eqs. (348) and (357) eliminating wO

+ +

w,2 = ( X , - X W,)Z - w;, = ( X , - X)Z 2W,(X, - X ) , = ( X u - X ) ( ( X u- X ) 2%}.

+

(359)

Similarly the other limiting surface generated by curves B can be obtained by eliminating ww,obetween Eqs. (348) and (358) wz2

(wy- X)2 - w,2, = X ( X - 2w,). =

The surfaces given by Eqs. (359) and (360) penetrate along the ellipse

w,' = X ( X , - X ) w, = x - >5x,,

VELOCITY DISTRIBUTION IN ELECTROH STREAMS

301

which lies in a plane through the w,-axis and a t 45' to the X-axis. The projection of the ellipse on the X,w,-plane appears as a circle and is shown in Fig. 40d. The composite surface consisting partly of Eq. (359) and partly of Eq. (360) is shown in Fig. 42. This surface limits the volume which contains all the physically realizable electron trajectories. The cross section of the surface by a plane perpendicular to the X-axis is shown in Fig. 43. 4. Volume Density of the Electrons. The volume density can be obtained by integrating Eq. (354) over the area shown shaded in Fig. 43:

n(R)

=

s

n(R,w)dw

where %I2

Wz2*

= X(X = (XQ

w,1=

wu-2 =

- 2wy)

- X) ( (X, - X) + 2w,)

x - >$X, >.ix.

It can be seen from Eq. (362) that n(R)is a function of X only, X, giving the position of the anode. Since X is a function of B , it is obvious that the space-charge density depends not only on the distance from the cathode x - 20 but also on the magnetic field B. Note that n(R)dR is the number of electrons in an element of volume dR = dXdYdZ.) At the cathode the lower parabola reduces to a vertical line through the origin, as shown in Fig. @a, the upper parabola cutting the w,-axis at w, = $.iX,. Thus the electron velocities present at the cathode comprise not only the velocities of emission represented by the points in the right-

302

P. A. LINDSAY

h i d half-plane but also the velocities belonging to tlic electrons returiiing to the cathode. The latter are represented by points situated between the wraxis and the left-hand branch of the upper parabola. The velocity distribution of the electrons at the anode is shown in Fig. 44b. In the absence of the magnetic field the points corresponding to the velocities of individual electrons would all be situated in the half-plane to the right of the toraxis. (In a field-free space the velocity distribution

"4

Fro. 42. The limiting surface generated by all possible trajectories of those electrons which either graze the anode or leave the cathode tangentially. The dashed lines represent typical electron trajectories.

of the electrons is everywhere the same.) The presence of the magnetic field introduces a gap in the velocity distribution which appears between the w,-axis and the right-hand branch of the lower parabola, as shown in Fig. 44b. This gap occurs because the action of the magnetic field permits the electrons which start with wd as low as wd = 0 to increase their x-component of velocity W~ a t the expense of w,. At an arbitrary point between the cathode and the anode the conditions are slightly more complicated. (See Fig. 43). The area between the upper parabola and the vertical axis again corresponds to those electrons which turn back to the cathode, but the whole shaded area is now limited from below by the lower parabola. The corresponding gap at low values of to., caused by the gain in w. a t the expense of w,,now extends to the left of the vertical axis. This indicates that some electrons pass the plane X E const. twice having acquired enough xdirected energy to

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

303

FIG.43. The curve obtained by cutting the limiting surface of Fig. 42 by a plane perpendicular to the X-axis.

pass this plane but not enough to reach the anode. This could not happen a t the anode since there the electrons are assumed to be caught a t their first contact. The volume density of the electrons a t X = 0, that is, a t the cathode is given by the following expressions:

n(R)

=

/

n(&,W)dw

For X = X., that is, a t the anode, the volume density of the electrons is given by

304

P. A. LINDSAY

FIG.44. The curves of Fig. 43 for (a) X = 0 and (b) X = X , .

n(R)

=

s

n(R,w)dw

-

1 1' - wy2 w

dw,

exp - (wzz

+ w,2) } dw,}

VELOCITY DISTRIBUTION I N ELECTRON STREAMS

305

Equations (363) and (364) sliow that in the presence of the magnetic field the volume density of thc electrons is greater than ?tT at the cathode and less than n~ a t the anode, r t being ~ the volume density a t the cathode when no electrons return to it. Thus somewhere between the cathode and the anode there must be a t least one point a t which the volume density of the electrons will be exactly equal to nT. However, it can be seen from Fig. 43 that even there the velocity distribution of the electrons will differ from that which would exist a t the surface of a temperature-limited cathode. Thus even in the absence of the electric field there is no plane in the interelectrode region which could be said to possess the properties of a virtual cathode, as defined for example for x = x, in the case of a plane space charge-limited diode, Sec. V. The important conditional probability density function specifying the velocity distribution of the electrons a t a given point X can be obtained, following Eq. (50), by dividing n ( R , w ) , Eq. (354) by n ( R ) , Eq. (362). For economy of space the necessary algebraic transformations will not be given here. 5. Current Density of the Electrons. The current density of the electrons can be calculated by integrating Eq. (53) over the area shown shaded in Fig. 43. Substituting Eq. (354) in Eq. (55) one can find first of all that the z-component of the current density is zero,

JdR)

= 0,

(365)

the integrand being an odd function of w,and the integration extending from - 03 to co. It can be seen, however, from Fig. 43 that the same does not apply to the remaining two components of the current density, J,(R) and J,(R). (In the absence of the magnetic field J , ( R ) would be zero of course.) The fact that J , ( R ) # 0 in spite of the geometrical symmetry of the system represents the usual type of anisotropy which is invariably introduced by the presence of a magnetic field. Considering the x-component of the current density first one obtains

/

dw, --m

I =

W,~T

7r

{

/w"' -a

/ dw,/ r

f m

= ~~nT27r-95

dw,

r

lw; w,exp +

L:

306

P. A. LINDSAY

r-

r-

Putting O C

J*=% , %,

(367)

where J , is the current density at the surface of a temperature-limited cathode expressed in the reduced units R,w, Eq. (366) becomes

J,(R)

= JJl

- erf (sXa)].

(368) Equation (368) shows rather well the effect of the magnetic field.* First, it can be seen that the x-component of the current density remains constant in spite of the fact that J,(R) Z 0. Second, for zero magnetic field Xa = 0, and the error function term disappears altogether. Thus the presence of the magnetic field reduces J,(R) to a value which is less than J,(R), where J,(R) gives the electron current in the absence of the magnetic field. For very large magnetic fields X , 3 co as B 3 00 , and the term in brackets rapidly approaches zero. This is fairly understandable since for very large B the electrons are turned back to the cathode and fail to reach the anode even when their initial velocities are very large. It is now necessary to calculate the y-component of the electron current density. Substituting Eq. (354)in the second term of Eq. (55) and integrating over the area shown shaded in Fig. 43 gives J,(R)

7

wo

\

w,n(Rjw)dw

* A generalized form of this equation has been recently suggested by Fulop (108) and Lindsay (1614.

VELOCITY DISTRIBUTION IN ELECTRON STREAMS

{1- exp

307

wia

=

oo

WcnT A

(-wzz) exp

{ - t ( X - $)’} dw,)

where

W d =

X(X,

- X).

Equation (369) shows that J,,(R)is a function of X. At the cathode ( X = 0) and a t the anode ( X = Xa) the tangential component of the current density is equal and given by

J,,(Ro)= w , n ~_2 n-

/” 1.,

w,, exp { - (wzz

dw,

-OD

= Jy(Ra) =

1 Ivy

;

w c n ~

OD

dwz

+ wy2)>)dw,,

w, exp { - (wz2

+ wY’)1dw,

Equations (369) and (370) show that the tangential component of the current density J,,(R)is independent of the y-coordinate but is a function of X . This is in agreement with the current continuity equation which says that

aJz divJ = -

ax

aJ. + aJ, - 4-zay

(371)

308

P. A. LINDSAY

This, of course, does not prevent J, from being a function of x or X . For zero magnetic field X = 0 and X, = 0, and Eqs. (369) and (370) reduce to zero. When the magnetic field is very large, X + 00 and X , 3 a. Somewhat surprisingly this also makes the integrands in Eqs. (369) and (370) tend to zero. The physical explanation of this lies in the fact that the tangential component of the current density differs from zero merely because some electrons are turned back to the cathode whereas other electrons are caught by the anode. When the magnetic field is exceedingly strong all electrons, whatever their initial velocity, turn back to the cathode, and the symmetry of the Fystem which was previously upset by the magnetic field is restored.

VIII. EXPERIMENTAL SUPIWltT

FOR THE

THEoltlCTICA4LRESULTS

At this stage it is only natural to ask whether the analytical results obtained in Secs. V-VII can be supported by some kind of experimental evidence. It seems that on the whoIe they can, although the experimental evidence is perhaps not quite as unequivocal a s one might wish. The analysis of the previous chapters rests on two basic assumptions: (1) that the velocity distribution of the emitted electrons is Maxwellian for the tangential components and half-Maxwellian for the normal component, and (2) that electron scattering effects or the electronelectron interactions are negligible. It is rather unfortunate that no known experiment can show directly whether the first of these assumptions is justified and the experimental evidence must be obtained indirectly by comparing the results of measurements with some of the mathematical expressions derived on the assumption that the velocity distribution is half-Maxwellian. As far as the second assumption is concerned the situation is somewhat more satisfactory, since electron scattering effects can be investigated on their own and a fairly reliable estimate of their magnitude can be obtained. This is quite important since the indirect, experimental evidence concerning the velocity distribution in itself depends on the assumption that the scattering effects are negligible in the circumstances in which the experiments are carried out.

A . Electron Scattering or Interaction between Individual Electrons

It was assumed in the derivation of Eqs. (1-6) that the presence of free electrons affects the poteiitial distributioii betweeii the electrodes in a macroscopic or smeared out fashion. This means that the identity of individual electrons is completely submerged in a vast and smooth electron cloud and the possibility of in teractioii betweeii individual electrons is completely ignored. Was such an assumption justified? It seems that it was,

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a t least in the circumstances in which the potential distribution was analyzed, since the electron densities which occur in ordinary thermionic valves are of the order of 10'" per cm3, the distances which the electrons have to travel are of the order of 0.01-10 em, and the transit times are of the order of a millisecond or less. There is good evidence, both theoretical (16%')and experimental (163,164, that in such circumstances the electron scattering effects are in fact negligible. This certainly seems to apply as long as the electrons remain in a purely electric field of force, but the conditions may not be quite so simple when a magnetic field is also present. It is a wellknown fact that noise generation, which is not fully understood, may occur especially for a beam in crossed electric and magnetic fields. This suggests that in the presence of a magnetic field the whole problem of the exact limits of validity of this assumption well merits further theoretical and experimental research effort.

B. The Half-Maxwellian Velocity Distribution of the Emitted Electrons This problem is probably more difficult to treat than the previous one since it is impossible to carry out an idealized experiment in which all electrons would be arrested at the surface of the emitter and their velocities measured. One can only hope to find some kind of indirect experimental evidence which would give a clue to the actual velocity distribution of the emitted electrons. It has been shown in Sec. I V that there is overwhelming theoretical evidence suggesting a half-Msxwellian velocity distribution for the electrons emitted from an idealized cathode. The question now is: Do the physical cathodes approach the idealized one sufficiently closely for this to be true in each case? Or, possibly: Is the idealized cathode of Sec. I V too simple to account for all the subtle quantum mechanical processes which are lumped together under the name of thermionic emission? Both questions have been discussed in detail by Nottingham in his monumental work on thermionic emission (14) and by Fan (165), Hung (166), Hadley (167), and Sparks and Philips (168). It seems that there are in existence two schools of thought, one represented by Nottingham and the other by Fan, Hung, and Hadley. Nottingham is of the opinion that the differences between the experimental and theoretical results are due to the deficiency of low-velocity electrons caused by reflections occurring at the potential barrier on the surface of the emitter. Fan, Hung, and Hadley are of the opinion that since, among other things, the discrepancies are particularly noticeable in the case of oxide-coated cathodes, they are largely apparent and are caused by the voltage drop in the ohmic resistance of the coating. When the effect of this resistance is allowed for, the calculated veIocity distribution of the emitted electrons becomes almost exactly half-Maxwel-

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lian in inost cases. This differeiicc of opinioii has not, yet been finally resolved. However, even if it is resolved in favor of Nottingham, the analytical results of the previous chapters will still give a very good approximation to the actual state of affairs. Furthermore, in view of the great analytical difficulties in trying to incorporate any departures from the half-Maxwellian law of velocity distribution, they can be used, a t least as a first approximation for comparison with experiment. It might be added here that in the history of thermionic emission the problem of the velocity distribution of the emitted electrons was one of the first to be subjected to &nexhaustive experimental investigation. However, this is neither the time nor place to give an extensive summary of these efforts, especially since some excellent reviews covering the whole of this field are already available (14, 116, 117-120).It is enough to mention here that in principle there are two basic experiments which can be used in investigating the law of velocity distribution. In the first place one can use accelerating anode potentials, the velocity distribution being deduced from the discussion of Richardson’s equation, Eq. (61). I n the other case the anode potentials are suitably adjusted to produce a retarding field in the electrode space, the law of velocity distribution being deduced from the discussion of Schottky’s equation, Eq. (315), the geometry of the system being almost invariably cylindrical. In each case it is possible to predict theoretically what the anode current ought to be if the electrons in fact were emitted with a half-Maxwellian distribution of velocities. Any discrepancies between experiment and theory must then be caused either by some departures from this law of velocity distribution or by some extraneous effects which have nothing to do with the actual velocity distribution of the emitted electrons. Some fairly recent work throws further light on this difficult problem (169-175). It may be appropriate to end this article by quoting Prof. Nottingham on the whole subject of velocity distribution [p. 35 of ref. ( I d ) ] . “Even though experiment has established that the distribution of the electrons emitted from smooth surfaces constituting a measurable emission current is deficient in the lowenergy group, it would complicate the analysis to attempt to bring this fact into theory.” We are faced here with the old dilemma of choosing between the idealized representation which although only approximate can be analyzed and the measurable reality which, by its sheer complexity, defeats analysis. ACKNOWLEDGMENTS This work could never have been completed without generous support from The Research Laboratories of the General Electric Company, Ltd., Wembley, England and from Columbia University, New York, who very kindly extended to me the facilities of a

vEu)CITY DISTRIBUTION IN ELECTRON STREAMS

31 1

Quincy Ward Boese Post-Doctoral Research Fellowship. Among members of the scientific staf€of the Wembley Laboratories I am particularly indebted to Mr. W. E. Willshaw for his unfailing support and encouragement of my efforta, to Mr. Eric Kettlewell for his support in the more recent stages of my work, and to my colleague, Mr. Alan Reddish, for his patient and time-consuming advice. I am also grateful to Prof. John R. Ragazzini, the acting head of the Department of Electrical Engineering at the time of tenure of my fellowship, for placing all the necessary facilities at my disposal. It gives me great pleasure also to thank The California Institute of Technology, Paaadena, California, for their kind invitation to spend there the summer of 1955 and in particular to express my indebtedness to Prof. R. V. Langmuir, who was kind enough to read and correct the first draft of the manuscript. REFERENCES 1. Moss, H., J . Electronics 2, 305 (1957); Feaster, G. R., J . Eleclronics and Control 6, 142 (1958). 2. Hatsopoulos, G. N., and Kaye, J., J . Appl. Phys. 29, 1124 (1958); Proc. I . R. E.46, 1574 (1958); Nottingham, W. B., J. Appl. Phys. SO, 413 (1959); Nottingham, W. B., Hatsopoulos, G. N., and Kaye, J., ibid. p. 440; Wilson, V. C., ibid. p. 475; Houston, J. M., ibid. p. 482; Webster, H. F., ibid. p. 488. 3. MacColl, L. A., Bell System Tech. 22, 153 (1943). 4. Page, L., and Adams, N. I., Jr., “Electrodynamics.” Van Nostrand, New York, 1940. 6. Gabor, D., Proc. I . R. E . 3S, 792 (1945). 6. Spangenberg, K. R., “Vacuum Tubes.” McGraw-Hill, New York, 1948. 7. Liebmann, G., Advances in Eledronics 2, 102 (1950). 8. Millmann, J., and Seeley, S., “Electronics.” McGraw-Hill, New York, 1951. 9. Dow, W. G., “Fundamentals of Engineering Electronics.” Wiley, New York, 1952. 10. Harman, W. W., “Fundamentals of Electronic Motion.” McGraw-Hill, New York, 1953. 11. Beck, A. H. W., “Thermionic Valves.” Cambridge Univ. Press, London and New York, 1953. 1%. Gray, T. S., ed., “Applied Electronics.” Wiley, New York, 1954. 19. Rothe, H., and Kleen, W., “Hochvakuum-Elektronenrbhren,” Vol. 1 : Physikalische Grundlagen. Akad. Verl&gsges.,Frankfurt am Main, 1955. 14. Fliigge, S., ed., “Handbuch der Physik,” Vol. 21: Electron Emission. Discharge in Gases I. Springer, Berlin, 1957. (See articles by W. B. Nottingham, pp. 1-175, and W. P. Allis, pp. 383-444.) 16. Ollendorff, F., “Technische Elektrodynamik,” Vol. 2: Innere Elektronik, Part I, Elektronik des Einaelelektrons. Springer, Vienna, 1957. 16. Kleen, W., “Electronics of Microwave Tubes.” Academic Press, New York, 1958. 17. Mdoff, I. G., and Epstein, D. W., “Electron Optics in Television.” McGraw-Hill, New York, 1938. 18. Zworykin, V. K., Morton, G. A., Ramberg, E. G., Hillier, J., and Vance, A. W., “Electron Optics and the Electron Microscope.” Wiley, New York, 1945. 19. Cosslett, V. E., “Introduction to Electron Optics.” Clarendon Press, Oxford, 1946. 20. Glaser, W., “Grundlagen der Elektronenoptik.” Springer, Vienna, 1952. 21. Klemperer, O., “Electron Optics.” Cambridge Univ. Press, London and New York, 1953. W .Sturrock, P. A., “Static and Dynamic Electron Optics.” Cambridge Univ. Press, London and New York, 1955.

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Electron Probe Microanalysis RAYMOND CASTAING Ddpartement de Physique Gdndrab, FacultS des Sciences, UnivcrsitS de Paris, Orsay, France Page . . . . . . . . . . . . . . . . . . . 317 A. Old Methods of Quantitative Determination.. ......................... 318 B. An “Absolute” Method Using Pure Elements as Standards.. . . . . . . . . . . . . 319 C. Emission-Concentration Proportionality Law. . . . . . . . . . . . . . . . . . . . . . . . . . 320 11. General Structure of the Microanalyzer.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 A. The Electron Probe.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 B. Thermal Conditions of the Analysis.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 C. Contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 D. X-ray Recording.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 340 E. Localization of the Analysis.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 111. The Fundamentals of Quantitative Analysis by X-Ray Emission.. . . . . . . . . . . 360 A. Absorption Correction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 B. Distribution in Depth of the Characteristic Emission.. . . . . . . . . . . . . . . . . . 362 C. The Physical Basis of the Emission-Concentration Relation. . . . . . . . . . . . . 366 D. Experimental Absorption Correction Curves. . . . . . . . . . . . . . . . . . . . . 370 E. Fluorescence Correction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 F. Fixed-Time versus Fixed-Charge Measurements. . . . . . . . . . . . . . . . . . . . . . . 376 IV. The Contribution of Microanalysis to Scientific Research.. . . . . . . . . . . . . . . . . . 379 A. Metallurgy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 B. Mineralogy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 C. Technical Studies.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

I. Introduction. ...

........

I. INTRODUCTION The spot analysis technique known as “electron probe microanalysis” or “X-ray microanalysis” was developed ten years ago by the author in his thesis prepared under the direction of Prof. A. Guinier (1-3). The principle of the method is as follows: a finely focused electron beam (electron probe), of a diameter less than 1 p, is directed onto a particular point of the surface of a sample whose chemical composition is to be examined. The very small volume of material irradiated by the electron beam (about one cubic micron) then emits a complex X-ray spectrum which includes the characteristic radiations of the various elements present at the point, of impact of the probe. Spectrographic analysis of this X-ray spectrum permits the respective concentrations of these elements to be determined. Such a principle was hardly novel: it can be observed that the apparatus (Fig. 1) used by Moseley (4) in his historical work on the frequencies of 317

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RAYMOND CASTAING

characteristic X lines contains, with its trolley carrying various targets under an impinging electron beam, all the main elements of an electron probe microanalyzer. Ten years after the epoch-making experiments of Moseley, the first applications of X-ray spectrography to chemical analysis were beginning to bear fruit; these mainly concerned the analysis of powders. In this field, X-ray emission had from the very first some remarkable successes, in particular, the discovery of new elements. In this connection

PUMP

FIQ.1. Moseley’s apparatus (by courtesy of Philosophical Magazine).

one may mention the discovery of element 72, whose La1 and L/3z lines were first identified by Urbain and Dauvillier (6) and whose entire L spectrum was obtained a little later by Coster and von Hevesy (6). The extreme simplicity of X-ray spectra where, contrary to the case of light spectra, the characteristic lines are few and practically insensitive to chemical bond?, led physicists to consider the problem of quantitative analysis. The wavelengths of the various characteristic radiations made it possible to define the nature of the elements present in the sample. There remained to deduce from the intensities of these various radiations the relative proportions of the constituent elements; and then many difficulties began to appear. These difficulties will be reviewed briefly; it will be shown that the main advance brought about by the electron probe microanalyzer results more from the principle involved in quantitative analysis than from the spot character of this analysis.

A . Old Methods of Quantitative Determination For many years, quantitative analysis by means of X-ray spectrography proceeded on the essentially empirical method used by the early workers.

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319

Tho principlc was as follow,.::Let us suppose we have to determine the concentration of an element, A in a mixture (generally a mixture of oxides in powder form). The mixture is deposited on the anticathode of an X-ray tube and is subjected to bombardment by the beam. A measurement is made (generally by simple photographic recording of the spectrum) of the intensity with which the mixture emits, under definite bombardment conditions, the most important characteristic line of the element A. There is then added to the mixture increasing quantities of a reference element B, with an atomic number close to that of element A, until the homologous characteristic lines of elements A and B are emitted with the same intensity by the mixture. It is then taken that the concentration of element A is equal to that of the reference element B. This method of determination has the double disadvantage of being highly complex and rather inaccurate. Even when precautions are taken to avoid selective volatilization of one of the constituents under the impact of the electron beam, many other sources of error make the results very uncertain. For example, the method is based on the comparison of the intensities of two radiations of different wavelength. These two radiations are adsorbed differently in the sample, in the output window of the tube, and in air; they are reflected with a different efficiency by the crystal and recorded with different sensitivity by the receiver unit of the spectrometer; so the ratio of the measured intensities differs appreciably from the ratio of the intensities actually emitted by the sample. It becomes necessary to apply rat her uncertain corrections, which depend essentially on the experimental arrangement. If an attempt is made to reduce these causes of error by choosing lines of extremely close wavelengths, one is led to comparing lines of different kinds (for example, the Lal line of element A and the Lp, line of element B) and the ratio of line intensities for equal concentrations has to be empirically estimated. And moreover, the X-ray lines of two distinct elements, even of very close wavelengths, generally have distinctly different excitation thresholds, with regard to both direct excitation by the electron beam and secondary fluorescence excitation. Under such conditions it is necessary for the determination to proceed in an empirical manner, calibration from mixtures of known composition being essential in practice.

B. .4n “Absolute” Method Using Pure Elements as Standards It has just been stated that the main source of inaccuracy in old methods of X-ray spectrographic analysis rests on the fact that they are based on the comparison of the intensities of two radiations of different wavelengths or kinds. For this reason, the author ( 2 ) has proposed a method of analysis based on a new principle in which comparisons of intensity are made on

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RAYMOND CASTAING

idoiltical rdiatioils. Mczlsureinents thus have ail iiitriiisic: sigiiificwwe aiid are not affected by properties characteristic of the apparatus used. In this new method, the concentration of an element A in a given alloy is deduced from a comparison between the intensity I A of an important characteristic line of element A emitted by the alloy under given conditions of electron bombardment and the intensity I(A) of the same charackristic radiation when emitted by the pure element A under the same electron bombardment conditions. The operation consists of taking two readings on the spectrometer, by successively bringing under the impact of the electron beam the region of the sample to be analyzed and an element for comparison in the form of the pure element A. The main advantage of this method lies in the fact that the simple ratio of two readings supplies with good accuracy the mass concentration of element A in the region analyzed, as was shown by the author (d) and later by Castaing and Descamps (7) in a more rigorous treatment .

C . Emission-ConcentrationProportionality Law It is assumed that the sample, homogeneous over an extensive region, contains n elements Ai of respective mass concentrations ci; it is desired to determine the mass concentration CA of one of these constituents, such as A. For this purpose, the emission I A of the sample in the Kal line of element A is compared with the emission I(A), in the same line A Kal and under the same electron bombardment conditions, of a reference target consisting of the pure element A. Let us show that the concentration of element A in the region analyzed is supplied, as a first approximation, by the simple relation IA/I(A) = cA. Consider the trajectory of an electron within the sample. Along this trajectory, the energy E of the electron passes from EO= eV (V being the accelerating voltage of the beam) to zero, this energy E being a function of the path x followed from the point of impact. Let EK = eVK be the critical excitation energy of the K level of element A, and nK the number of electrons K(A) present per cubic centimeter in the sample. The number of K(A) ionizations produced along the path dx is then dn = +(E,EK,nK)dx; the ionization function CP contains nK as a factor, and designating p as the density of the analyzed region, A the atomic weight of element A, and #A a function depending only on the characteristics of element A, we can write (1)

As a first approximation let us assume that Williams’ law (8) applies to the deceleration of the electron

ELECTRON PROBE MICROANALYSIS

dE -= dx

321

kpp-1.4

where k is a constant and P = v / c is a function only of the energy E of the electron; we have A a(E)dE,

(3)

the function j A depending only on the characteristics of element A. The total number of K ( A ) ionizations produced by the electron along its trajectory is then

The same electron would have produced in the pure element A a number of ionizations

We have assumed that the characteristics of the electron beam and thc adjustment of the spectrometer remain the same for the two succcssive measurements. Neglecting for the present the X-ray absorption in the sample itself and the secondary fluorescence emission, points which will be taken up again later, it is clear that the ratio l A / l ( A ) of the intensities seen on the spectrometer is equal to the ratio n/n' of the number of ionizations produced by an electron in the region analyzed and in the pure element A. From this the required relation is derived which is valid as a first approximation

IA

I0 =

(5) cA.

In a second approximation we can take for the deceleration of the electrons the law proposed by Webster (9) which, for a pure element A of atomic number Za, is

and which becomes, for the complex anticathode,

The same calculation then leads to the relation

-IA -

~AZA/A. - ,&&/Ai

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RAYMOND CASTAING

If we designate by ni the number of atoms Ai per unit volume in the sample, the two approximations considered may be written IA n d , --I(A) - CniAi

_ I A_ -nAZA I(A) -

Cn,Zi’

It is clear that the validity of these approximations is not tied to the exact form of the deceleration laws of Williams or Webster, since, in order for the first approximation to be valid, it is only necessary that the deceleration law of the electron along its path be written d E / d x = pcp(E),whereas the second approximation assumes that the deceleration law can be written d E / d x = n+(E), n being the number of electrons per unit volume of the sample and the functions (o and $ being any functions of the argument E. Neither of these two deceleration laws, of course, is rigorously obeyed and a still better approximation could be obtained by applying to each of the elements Ai a coefficient ai representing its “specific deceleration power,” so that the deceleration law of the electron in the pure element Ai of specific weight p i may be written d E / d x = a;pif(E).where f is a universal function of any form. The emission-concentration relation would then take the form

It is the latter form which would have to be adopted as a second approximation, the coefficients ai being empirically adjusted by means of measurements made on alloys of known composition. The validity of this last approximation holds only on the assumption that the deceleration curves of the electrons in the various elements can be deduced from each other by expansion or contraction of the coordinates. It should now be noted that, in order to simplify the calculations, we made two assumptions the validity of which requires examination. (1) We have neglected the X-ray absorption in the sample itself and the secondary fluorescence emission; in fact, such a simplification is generally not admissible. Actually, the foregoing relations are valid only for the intensities emitted in the sample (i.e. corrected for their absorption in the sample itself) from atoms directly ionized by the electrons of the beam. It will therefore be necessary, after each measurement, to deduct from the measured intensity the fraction of this intensity corresponding to a secoiidary fluorescence emission. It will be seen later that these various corrections can be sufficiently well estimated so they do not affect, the accuracy of the measurement. (2) It has been implicitly assumed that the trajectory of the electron,

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323

or a t least that part of the trajectory where the electron has an energy greater than the critical excitation energy of the X-ray level, is entirely within the sample. This assumption is incorrect because of back-scattering. We shall see further that back-scattering tends to raise the apparent concentration of the heavy elements and so acts in the opposite direction to the factor Z / A of the Webster equation [Eq. (S)]. This effect produces a kind of compensation with the result that the validity of the fist approximation, which assumes strict proportionality between emission and eoncentration, is often much better than might have been supposed. In any case, back-scattering can be taken into account by a suitable choice of the empirical coefficients ai,and the following basic relations can be written between the intensities and the concentrations

A -I -

I(A)

- "'

-.

IA = ~ A C A __ I(A)

(first approximation) (second approximation)

2aici

It is understood that these relations are with respect to the intensities corrected for their absorption in the anticathode and for the fraction due to a secondary fluorescence excitation. We shall return later to the validity of these relations, which has been verified experimentally for a considerable number of analyses carried out on homogeneous samples of known composition. Stress should be laid on the fact that the reason for the simplicity of the relations rests mainly on the absolute character of the measurements: the radiations whose intensities are being compared have the same wavelength, and all the difficulties which might arise from the absorption of the radiation-apart from that in the sample itself-or from the efficiency of the spectrometer are automatically eliminated since they occur equally for both terms of the ratio. The quotient of the two readings (after correction for fluorescence and self-absorption) very accurately gives the ratio of the over-all emissions, in the line analyzed, of the sample and of the pure element; this ratio has an intrinsic significance which is independent of the experimental equipment. The absolute character of this method of analysis makes superfluous the use of reference samples with a composition close to that of the sample to be examined, whereas such practice is essential in light spectrography, for instance. This circumstance is a particularly happy one, and the method would hardly be usable as a quantitative analytical method did it not possess this absolute character. It would be impossible to prepare a whole series of reference samples with sufficient small-scale homogeneity. In the

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RAYMOND CASTAING

field of nietdlrirgy, for iiistmce, dcfinite phascs or solid soltithis (which, however, may con taiu submicroscopic precipitates) alone are uwhle as staiidards for local analysis. Even in the case of binary alloys, definite phases, if such there be, are few in number; the concentration of solid solutions can be varied continuously, but over a range which is generally narrow in the neighborhood of pure elements. Last, it is generally impossible to obtain phases containing more than two elements and of sufficiently well-known composition to be used as standards; in fact, the precise role of electron probe microanalysis will be to determine the composition of these multiple phases which generally appear as small precipitates.

XI. GENERALSTRUCTURE OF THE MICROANALYZER The principal elements of a conventional electron probe microanalyzer are four in number: (1) An electron optics system, consisting of an electron gun followed by reducing lenses, whose role is to produce at the level of the sample an electron probe with a diameter approximately between the limits of 0.1 and 2 p. It will be seen that, in the present state of technique, diffuse penetra-

T

ELECTRON GUN

MAGNETIC CONDENSER

R

REFL ECTiNG OBJECTIVE-

SPECiUEN

FIG.2. Schematic diagram of the French microanalyzer (Castaing and Descamps).

ELECTRON PROBE MICROANALYSIS

325

FIG.3. The French microanalyzer (Castaing and Descamps). Parts of the housing have been removed and the right-hand spectrometer has been opened to give an inside view.

tion of electrons in matter makes it unnecessary, a t least for the analysis of massive samples, to use probes of very small diameter. (2) A mechanical arrangement for bringing successively under the probe the point of the sample to be analyzed and the reference targets consisting of the pure elements or of compounds of known composition. (3) A viewing device (generally a metallographic microscope) for accurately choosing the point to be analyzed. (4) A set of spectrometers for analyzing the X-ray radiation emitted. The first experimental model built by the author in 1949 by converting

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RAYMOND CASTAING

a C.S.F. electron microscope (It?), used electrostatic reducing lenses and a single spectrometer in air. The latter practically prohibited the analysis of light elements. Since then, many more or lees improved models have appeared in various countries. We shall describe here in detail the apparatus built by the author in the laboratories of the Office National d’Etudes et de Recherches ABronautiques. This instrument, shown a t the French Physical Society’s Exhibition in June 1955, is now available commercially and it will be referred to here a p “the French model.’’ Figure 2 shows diagrammatically the principle of the apparatus, a general view of which is given in Fig. 3.

A . The Electron Probe The probe is obtained by forming, by means of electron lenses, a much reduced image of the crossover produced by an electron gun. The gun is of the conventional hot cathode triode type; the three models which are most frequently used are shown in Fig. 4. The French model uses a type A gun

A B C FIQ.4. Three usual models of electron guns.

which was designed by Bricka and Bruck (10) for the C.S.F. electron microscope. This gun concentrates in an electron beam of very small aperture almost the total current of the HT source. Its efficiency is excellent and it has the advantage of producing a crossover located immediately below the anode level, in a region where it is easy to place a fixed aperture. The type B gun, used in particular by Mulvey (11)’ has similar properties. As regards type C (RCA gun) used by Birks and Brooks (12)its efficiency is lower but its emission seems to be less sensitive to accidental decentering of the filament (warping may’shift the filament position). All these guns use a dropping resistor for self-bias operation; Marton and Simpson (13) on the contrary, use a long-focus gun with fixed battery bias. In The French model, the tungsten filament is heated at high frequency and the HT, stabilized by a conventional electronic process to a few parts in a hundred thousand, is adjustable from 5 kv to 35 kv; for it is essential that the HT used be adapted to the critical excitation energy of the X-ray line being measured.’ The electron gun is followed by two magnetic reducing lenses (Fig. 2); 1

See Sec. II,E,R

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327

the first of these lenses has a focal length which is variable between 2 mm and infinity; it acts as a condenser and the adjustment of its excitation makes it possible to obtain probes with a diameter of 0.1 to 3 p. The second reducing lens (probe-forming lens) accurately focuses the electron beam on the surface of the sample. Its focal length is about 0.9 cm and its spherical aberration coefficient C, = 3.6 cm. The point of formation

FIG.5. Probe-forming lens and viewing device (French microanalyzer) (Castaing and Descamps). l.-Mirrors, 2-semitransparent mirror, 3--coil, &reflecting objective, 5-X-ray spectrometer (operating in air, in low vacuum or in high vacuum), 6-main frame, 7F-outlet window (can be turned down during work), S-specimen, +pole pieces, 10-stigmator (astigmatism correction and probe deflection), ll-electrostatic shield, l2-lens casing (iron circuit).

of the probe is 0.6 cm below the lower face of the pole piece, which gives sufficient clearance to pass the X-ray beams to be analyzed (Fig. 5 ) . An electrostatic device corrects the natural astigmatism of the probe-forming lens; it also makes it possible to apply to the probe small lateral displacements to compensate for slight deviations which sometimes occur in the analysis of magnetic samples. Last, on the extension of the beam and about 30 cm below the object level, a fluorescent screen is inserted for observing the beam, which is most useful for certain adjustments; a photographic chamber also provides

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RAYMOND CASTAINO

nieaiis for obtaiiiiiig electroiidiffract,ioii patkerns (accurate measuremelit of the beam accelerating voltage). Figure 6 shows three types of pole pieces which may be used for the probe-forming lens. Type A (French model) accommodates optical viewing of the sample along the beam axis; type B (Fisher) is designed for about 1:1 demagnification as a transfer lens; it accommodates lateral viewing of the sample, but its spherical aberration is larger than that of type A. With

A

B

C

Fro. 6. Three usual models of probe-forming lenses.

the pinhole type C (Mulvey) optical viewing of the specimen is impossible, but it has the advantage of reducing the magnetic field strength at the sample level, leading to minimum beam deflection with ferromagnetic specimens. 1 . Probe Brightness. Theoretical limitations. The aim is naturally to obtain maximum electron intensity on a probe with the smallest possible diameter, in order that the analysis may add high sensitivity to high resolving power. If electron lenses free of aberrations were available, and more particularly lenses free of spherical aberration, it would be possible to reduce the diameter of the probe by a considerable factor without reducing the current carried by the electron beam, and very high electron densities could be obtained. Unfortunately such is not the case in the present state of technique, and the spherical aberration of the probe-forming lens-the only one working a t relatively large aperture-sets a limit to the current that can be obtained in a probe of a given diameter. This limit can be calculated as follows (2): According to Langmuir (14) the current density in a crossover is a maximum a t the center where it is equal to

In this equation, io is the emissive power of the cathode, expressed in amperes per square centimeter; T is the absolute temperature of the

ELECTRON PROBE MICROANALYSIS

329

cathode, V is the accelerating potential, and B is the half aperture of the beam at the crossover level. This amounts to saying that the crossover may be considered as a source of electrons whose brightness (at least at the center) is

B

=

ioeV nkT

--9

neglecting unity compared to eV/kT. The successive images of the crossover produced by the various reducing lenses are formed in constant potential media, so that the brightness of the probe is again equal to B. If the reducing lenses were perfect, a probe of Gaussian diameter do formed by a beam with an aperture u would have a true diameter do and would carry an electron current

nd2e-V . i = -- zou2

4 kT

(u is supposed to be small), and it would only be necessary to increase u

in order to increase the current without changing the diameter of the probe. We must, however, take into account the spherical aberration of the probeforming lens, which spreads the current i over a probe of true diamet.er d = do C,u31/2.This gives, in a probe of true diameter d formed by a beam of aperture u,an electron current

+

where C, is the coeficient of spherical aberration. This expression has a maximum value for

.-(a M

'

that is, for an aperture such that the diameter of the aberration disk is equal to one-quarter of the true diameter of the probe. The maximum value of the current is then equal to

The procedure for obtaining a probe of true diameter d carrying maximum current (assuming that the only aberration of the probe-forming lens is the spherical aberration) is then as follows: (1) Adjust, the Gaussian diameter of the probe to the value 3 d / 4 (whatever the value of C,), and (2) limit the aperture of the prohe-forming lens to the value u = (d/2CJx. In praetiw, i i i order to avoid cxcwsive wear of the filament, its temperat i i w is set, to the v:~Iue T = 270O"K, :it which i h cmissive powcr is io = 2

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RAYMOND CASTAING

amp/cm2. As a result the theoretical brightness of the crossover, for an accelerating voltage V = 30 kv, is B = 82,000 amp/cm2 steradian. 2. Attempts to Obtain the Theoretical Brightness. It must be noted that the theoretical value of B, which has just been calculated, is the maximum value of the brightness at the center of the crossover; Haine and Einstein (16)have shown that it was possible, under optimum operating conditions, to approach closely this theoretical value. On the other hand, if the probe is obtained by forming the image of the over-all crossover, as was the case in the first experimental instrument built by the author (d), the brightness which has to be inserted in the formulas is an average brightness. This value is much lower (the brightness decreases exponentially a t the edges of the crossover) and it is necessary to introduce the notion of “gun efficiency” (A’, 10). This efficiency is nothing more than the ratio of the mean value of the brightness of the crossover over its “whole surface area” (i.e. over the area where the brightness is noticeable) to the maximum value of this brightness at the center, and is about 7-10%. If it is required to obtain in the probe a current as close as possible to the theoretical maximum, it is necessary to eliminate the peripheral area of the crossover, by means of a limiting aperture, and to retain only its central part where the brightness is a maximum. This amounts to replacing the natural exit pupil, which the crossover constitutes for the electron gun, by a physical exit pupil of smaller diameter. The gun model of Bricka and Bruck is particularly well suited to this operation, as the beam shows a crossover a few millimeters below the anode aperture, in a region of zero field. A11 that is necessary is to place a t that level a platinum aperture whose diameter, of the order of 0.1 mm, is rather less than the half-diameter of the cross-over and which therefore eliminates all those parts of the crossover where brightness is less than about 60% of the maximum. If a second aperture of small diameter (0.1 mm) is then placed in the path of the beam, all that need be done is to measure the electron current passing through the two successive apertures in order to deduce, knowing the distance and the diameters of the aperturee, the brightness of the beam. The author was able to obtain, for a beam accelerating voltage of 30 kv and a cathode emissivity io = 2 amp/cm2, an average brightness B = 58,000 amp/cm2 steradian, or 70% of maximum theoretical value. 3. Measurement of Probe Diameter. Since the diameter of the electron probe is one of the main determining factors for the resolving power of the analysis, it is important to know it accurately. Several procedures are available for the purpose. The simplest arid quickest method consists in measuring on the micro-

-

ELECTRON PROBE MICROANALYSIS

331

scope the diameter of the contamination spot2 which is formed on the sample a t the point of impact of the electron beam. This spot appears more or less rapidly depending on the nature of the object and on the bombardment current (16). For instance, a visible spot is obtained on a copper sample bombarded under normal conditions (V = 20 kv, i about 0.05 pa, probe diameter 0.5 p ) after a 20-sec bombardment, while it is necessary to wait several minutes, under the same bombardment conditions, for the contamination spot to appear on a chromium sample. The contamination spot tends to spread for a prolonged bombardment, with the result that the diameter measured on the microscope is greater than the true diameter of the probe. The relative error may become large with small diameter probes. To take an example, it can be noted that the contamination spot formed on a copper sample with a 1-p probe (even for a short-time bombardment) reaches a diameter of about 1.5 p. This method cannot be considered to be a precision method; in particular it is quite inadequate for verifying that maximum theoretical current has been obtained in the probe. A second method consists in cutting off the probe by means of a sharpedged metal strip (1) and observing on a fluorescent screen the shadow of that edge, considerably distorted by the spherical aberration of the reducing lens. The beam is then moved a t right angles to the metal edge by means of a n electrostatic or magnetic deflecting arrangement; calibration of the deflector allows the probe’s minimum movement between the appearance of the shadow and the complete occultation of the beam to be determined. This arrangement provides means for easy adjustment of the probe on the sharp edge, this adjustment being obtained when the sharp edge cuts the beam in the neighborhood of the circle of least confusion. If the astigmatism of the reducing lens has been corrected previously, the adjustment criterion is very simple. With the metal edge located, say, to the left of the observer, the shadow should appear simultaneously a t the extreme left of the field and a t a spot situated at the right-hand side of the field, at three-quarters of its diameter (Fig. 7); the two shadows should then join up in a perfectly symmetrical manner. This procedure is an excellent test of the correction of astigmatism (2, 17). This method is extremely simple if the instrument is normally provided with a probe deflection arrangement; all it requires is the insertion of a sharpedged metal strip. This strip should previously be sharpened by electropolishing followed by ionic bombardment to elimiiiate impurity layers, for, if the edge is not perfectly olean and conducting, charging up may cause probe deflection. The amount of additional deflection varies during occultn-

* See Sec. I1,C.

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RAYMOND CASTAING

tion, and the exact deflection of the probe is no longer that which is deduced from the calibration of the deflector system. This additional deflection introduces some inaccuracy in the measurement of the probe diameter. For accurate measurements, it seems preferable to use a third method (2) which consists in occulting the probe with a wire of known diameter. Suppose for instance that a probe is required with a diameter of 1 I.C and carrying, for a given accelerating voltage, maximum electron current. First, an aperture diameter is chosen for the probe-forming lens leading to the optimum value u = (d/2CJfs; then a tungsten wire 1 p in diameter is

Jmega of Without astigmatism.

8n

edge W i t h rstigmatism.

E) Image of a wire Wthwt 8stlgmtltm. W i t h 8s:lgmtism.

FIQ.7. Shadow microscopy images (Cashing and Descamps).

placed in the beam at right angles to its axis and the shadow of the wire is observed on the fluorescent screen placed in the lower part of the apparatus. The condenser is then strongly excited, so that the Gaussian diameter of the probe is very small, and the excitation of the probe-forming lens is adjusted so that the shadow of the wire covers the whole field. This complete occultation of the beam is easily obtained for if the astigmatism of the lens has been suitably corrected, the diameter of the probe is of the order of 0.25 p . Then the beam current is gradually increased by reducing the condenser excitation until it is no longer possible to obtain complete occultation of the probe. The criterion for condenser excitation is as follows: With the wire properly centered in the beam, the excitation of the probe-forming lens is slightly modified; if the Gaussian diameter of the probe is too great (Fig. 7) two lwight, spots appear i t 1 the ficld, at oiic-qriartcr mid a t thrcvquiwtcrs of thc tliamctrr, before thr pwiphcry of tbc shadow hns fully

ELECTRON PROBE MICROANALYSIS

333

covrwd the field. When the maximiim ( :aussiaR tliatnctor conipalible wit8h complete occultattion is obtained, a measurement is made, by means of a Faraday cylinder, of the current carried by the beam, which can then be compared with the theoretical value deduced from the emissivity of the filament and from the spherical aberration of the probe-forming lens. In a typical experiment using a probe-forming lens with a spherical aberration coefficient C. = 3.6 cm, an accelerating voltage of 30 kv and a tungsten wire 1.2 p in diameter, the author was able to verify that complete occultation of the probe was still obtainable for a beam current i = 0.77 pa. The theoretical maximum calculated from Eq. (17) is, for a probe 1.2 p in diameter, ith = 1.07 pa. The efficiency of the probe-forming system is thus 72%. Naturally, such an efficiency can be attained only if the central part of the crossover alone is used for forming the probe. In the author’s first, experiments (2) in which the whole of the crossover was used, and in which, in addition, the spherical aberration coefficient of the probe-forming lens was 20 cm (electrostatic lens), the maximum current obtained on n 1-p probe was 0.015 pa, or about 30 times less than in the present instrument (0.47 pa on a 1-p probe). Such a current is much too great for the needs of spot analysis and it is possible to reduce the probe diameter to a value of about 0.4 p while retaining high enough an intensity. When the probe current approaches its maximum value, the tempernture of the tungsten wire used for occultation is raised to a considerable extent and mere radiation loss would not enable it to keep below its melting point. It is therefore necessary to arrange effective cooling by conduction. To this end, instead of a wire of uniform diameter, the end of a 0.1-mm wire is thinned down by electrolytic polishing. It is relatively easy to obtain the shape shown in Fig. 7 by this method. The diameter of the thinneddown part is measured, either with an electron microscope, or by shadow microscopy in the analyzer itself; the conical part is a very effective means for conducting the heat away; in spite of this, the thinneddown part, which can be observed during the experiment by means of the viewing device, is raised to white heat. In order to obtain high efficiencies it may be necessary to effect prior correction of the astigmatism of the probe-forming lens; but correction does not need to be as carefully made as in the case of an electron microscope. It is only necessary that the residual astigmatism after correction be less than 1 p in order that it have no detectable effect on the probe diameter. Simple observation on the fluorescent screen of the shadow of a sharp edge allows the stigmator to be adjusted in a perfectly satisfactory manner. If, for special reasons (for example, in order to obtain a probe of very small diameter) a very carefully made correction for astigmatism is required, use can be made of the method proposed by the author (17). This technique

334

RAYMOND CASTAING

consists in observing 011 the fluorescent screen the shadow of a very fine diameter wire (diameter of the order of 0.01 p ) placed a t right angles to the beam axis at the level of the probe, i.e., u little above the Gaussian focus. If the lens is perfectly corrected for astigmatism, this shadow for any orientation of the wire consists of the combination of a straight line and a circle (the circle corresponding to the cone of rays for which the aperture u is such that the focus F, is on the wire). A similar image is obtained with an astigmatic lens when the direction of the wire coincides with that of one of the focal lines; if such is not the case, the appearance of the shadow is asymmetric (see Fig. 7). I n particular, if the wire is set at 45" to the focal lines, the image obtained provides immediate means for determining the value of the astigmatism by a very simple geometric construction (2, 1'7). In order to obtain perfect correction for astigmatism, the procedure is then as follows: A ring which carries a large number of wires set out in all directions is inserted in the beam. It is convenient to use plexiglass wires prepared from a solution of plexiglass in aniline and then coated with chromium by vacuum deposition. Note is taken of one of the wire orientations which produces the symmetrical circle-and-straight-line image (direction of one of the focal lines) ; a wire is then chosen whose orientation is at 45" to that direction and the stigmator is adjusted to produce a symmetrical image. Correction is complete when the symmetrical image is obtained for any orientation of the wire. This operation takes only a few minutes and ensures a reduction of residual astigmatism to less than 0.5 p. 4. Possibility of Improving the Probe Brightness. Further on it will be seen that probe brightness is one of the main factors which limit the resolving power in electron probe microanalysis. Higher electron densities would be useful for the analysis of very light elements, such as carbon, and it would be extremely valuable in high-resolution analysis of thin films.3 Two processes can then be considered for increasing the brightness of the probe. a. Use of afield emission cathode. Marton (18) has suggested that the use of a field emitter (a tungsten point, for instance) could increase the current density in a probe of a given diameter, and experiments on such cathodes have been carried out a t the National Bureau of Standards. Use of a pulsed field emission has also been investigated by Wittry (19).In good agreement with the calculations of Cosslett and Haine (do), Wittry reaches the conclusion that field emission is preferable to classical thermionic emission as soon as the diameter of the probe required is less than about 0.1 ti, the ordinary hot cathode giving a higher density for probes of a larger diameter. It therefore appears that field emission would be of value only for the high-resolution analysis of thin films. Serious technical difficulties will have to be overcome in order to obtain a sufficiently stable emission for long time operation. I t 8

See Sec. II,E,J.

ELECTRON PROBE MICROANALYSIS

335

would seem that best results are obtainable from the use of thermionic field emission (21). This technique consists in heating the emitting point to a temperature slightly below that which corresponds to thermionic emission, and then applying the extractor field in the form of very short pulses (1 psec). This source shows fair stability under the relatively high residual pressures which cannot be avoided in an electron probe microanalyzer; unfortunately, in the present state of technique, the life of such cathodes is hardly more than 1 hr. b. Correction of the spherical aberration of the probe-forming lens. A less revolutionary solution, but probably more practical, consists in reducing or even in trying to correct completely the spherical aberration of the probeforming lens, which enters at the two-thirds power in the expression for maximum electron current attainable in a probe of a given diameter [Eq. (17)]. I t is well known that the reduction of C , in a lens with axial symmetry can be obtained only by reducing the focal length; such an improvement of C, is achieved in Duncumb’s and Melford’s instrument (26) at the expense of the X-ray spectrometer resolution for the probe is formed inside the lens4 and the closest the crystal can be moved to the specimen is about 15 cm, necessitating a spectrometer of the semifocusing type. Archard (23) has proposed the use of spherical aberration correction devices, originally designed for electron microscopy (magnetic quadrupoles and octupoles), in electron probe systems. This does appear to be the best solution, and spherical aberration correction seems likely to find an easier field of application in electron probe systems than in high-resolution electron microecopy. In the present state of technique, the minimum value of the spherical aberration coefficient is about 0.03 cm (24) for an axially symmetrical lens. We shall call “ideal probe” one obtained with such a lens associated with a perfect hot cathode gun, as opposed to the probe-which will be referred to as “long-focus probe”-obtained experimentally by the author by means of a relatively long-focus lens, the gun efficiency being 72%. Assuming for both cases a cathode emissivity of 2 amp/cm2 and a cathode temperature of 2700”K, we obtain from Eq. (17) the following expressions for the current carried by a probe of diameter d , a t an accelerating voltage V :

i i

= =

0.535Vd” 0.0158Vd”

(ideal probe conditions) (long-focue probe conditions)

(18)

(19)

where i is expressed in microamperes, V in kilovolts, and d in microns. 5. Automatic Regulation of Probe Current. The fundamental operation of the analysis consists in comparing the intensities of one characteristic 4

See Fig. 15.

336

RAYMOND CASTAING

radiation in two successive measurements in which the sample and the reference target in turn are subjected to the same electron bombardment conditions. Sometimes many points of the sample are analyzed in succession (this is the case, for instance, in the determination of intermetallic diffusion curves) and the intensities emitted by the various points analyzed are all compared to the intensity emitted by the pure metal, the latter being measured once for all. In this way each analysis requires only one measurement instead of two, which is an appreciable saving of time. It is CONDENSER

R LGULATOR

AMPLiFiER

9, OEJECTiVE

"

'

SPECIMEN

FIG.8. Probe current regulating system (Castaing and Descamps).

then important that the electron current carried by the probe remain rigorously constant over the whole period of the measurements, to the same extent as the beam accelerating voltage and the adjustment of the spectrometer. But it may happen that the gun emission is subject to a slight drift in the course of measurements made over a long period; this drift is due mainly to an off-centering of the tungsten filament. Figure 8 shows the arrangement of an automatic regulation system (25) which makes the probe electron current practically independent of any variation in the total gun emission. The aperture DOof the probe-forming lens is preceded by a coaxial aperture D1 with a diameter k times as great. Since the electron density in the plane of aperture D1 is uniform around the center, the aperture DO receives an electron bombardment of intensity I proportional to the current i carried by the probe, the proportionality coefficient being it2- 1. A constant fraction r of this intensity I is diffused back; as a result, if the aperture DOis connected to earth through a high resistance

ELECTRON PROBE MICROANALYSIS

337

It, f hr ~ i i r r c ~fplirougli n~~ to t l i k rclsis(,:uic*cis I(1 - T ) , aiid hcnc*ct,lie potciitial clifferencc developed at, its termiiials is proportional to the current i, the proportiona1it.y roeffivient being ZZ(1 - r)(k2 - 1). All that is then iiccessary is to control the excitation of the condenEer by this potential difference by means of an electronic arrangement in order to obtain an excellent regulation of the current i carried by the probe. The regulation may be sufficiently effective to ensure that a two-to-one variation in the total gun emission does not bring about a variation greater than 1% in the electron current of the probe. Another method would consist in regulating the electron current absorbed by the sample in accordance with Wittry's suggestion (19). This current may differ considerably from the probe current through backscattering effects and the author feels that such a procedure would present several disadvantages : First, emission concentration proportionality is no longer valid in this case, and this introduces a serious complication a s well as some uncertainty in the interpretation of the results.6 Also, it would be rather difficult, in scanning analysis for instance,b to cause the probe to adjust its current without any delay to the rapid changes which would be involved by a sudden move from a low scattering point on the sample to an adjacent point where back-scattering is strong. In this case, the procedure of coiltrolling by means of the condenser, whose inductance is quite large, would be difficult to apply. B . Thermal Conditions of the Analysis The electron density in the region of the sample bombarded by the probe is distinctly greater than that on the anticathode of a usual X-ray tube; it might therefore be feared a priori that the sample might become seriously overheated at the analyzed point. This is, in fact, not the case since the very small diameter of the bombarded region permits very strong cooling by thermal conduction. The temperature rise in the center of the probe has been calculated by the author (2) for the case of a massive hemispherical sample (the external shape of the sample has practically no effect on the result) with its outer surface kept at room temperature. Electronic bombardment brings a uniform amount of power into a hemisphere whose radius TO is taken a s being equal to that of the probe (it is actually greater, and so the calculated temperature rise is overestimated). Under these conditions maximum temperature rise at the center of the probe is 6

6

See Sec. II1,F. See Sec. II,E,l,c.

338

RAYMOND CASTAING

where R is the radius of the sample ( R >> T o ) , J is the mechanical equivalent of heat, C is the thermal conductivity of the sample, and Wo is the power carried by the beam (which we suppose entirely converted into heat). In the case of a metallic sample, the temperature rise obtained in this way is quite negligible. For example, under the impact of a probe 1 p in diameter carrying a maximum current of 0.47 pa a t 30 kv the temperature rise of a copper sample would be less than 18". Such is not the case, however, if the sample analyzed is a thermal insulator, and, under these conditions, one often observes thermal effects a t the bombarded point when the low proportion of the constituent analyzed requires the use of a high electron current. Such samples can be coated with a thin conducting film (metal layer or carbon coating) with the double purpose of evacuating the heat dissipated in the sample and of holding the surface of the sample a t constant potential by removing electrostatic charges. It is also of interest to examine what goes on when a thin sample is penetrated by the beam electrons (high resolution analysis by transmission); for it is in this case that it becomes necessary to use the highest possible electron density. In the case of a thin layer of thickness el cooled over a circle of radius R, and bombarded a t its center by an electron probe of radius r0, it is found (2)that maximum temperature rise a t the center is

where W is the power absorbed by the sample. As long as e is small compared to the maximum path of the electrons, W is practically given by the relation W = upeWo, where Wo is the power carried by the beam, p is the specific weight of the sample, and u is the Lenard's absorption coefficient. Then 6, = u is

Woap (1 6JC

+ 2 In E),

of the order of los for an accelerating voltage of 40 kv (26) and would be much less at the high accelerating voltages which would have to be used for high-resolution transmission analysis.' Therefore the temperature rise would be considerable in probes of large diameter carrying large electron currents. This is the case of the temperature rise, in the electron microscope, of samples sufficiently thick to ensure that radiation loss is small compared to conduction loss. But analysis of thin layers will necessarily require the 7

See Sec. II,E,3.

ELECTRON PROBE MICROANALYSIS

339

use of probes of very small diameter. Since WOvaries as the 8/3 power of the diameter of the probe and since the logarithmic term varies only slowly, the temperature rise will decrease with the decreasing size of the probe. Let us take for example the case of an “ideal probe” (Eq. 18) 0.1 p i l l diameter, carrying a current of 0.046 pa a t an accelerating voltatge of 40 kv and impinging on a thin copper sample. The rise of temperaure a t the, center of the point of impact is of the order of only 7”. Even if it is assumed that the spherical aberration correction of the probe-forming lens can briiig the diameter of the probe to 0.01 p while retaining its current value (the solid angle of the probe-forming beam would have to be greater than 0.5 steradian in the case of a hot cathode gun!), the temperature rise a t the point of impact would increase from 7 to 9”. It is therefore certain that thermal limitations will in no way hinder the securing of high resolution in the analysis of thin layers by X-ray epectrography.

C. Contamination

It is well known that a sample subjected to electron bombardment in a dynamic vacuum gradually becomes covered with a “contamination” layer due t o polymerization, under the action of the beam, of organic matter adsorbed on the surface. Ennos (27) has shown that organic molecules condense directly on the sample from vapors present in the enclosure. To this phenomenon there is added, in the case of a highly localized bombardment, a superficial migration towards the bombarded point of organic matter condensed on the surface as a whole (16).Ennos found that heating the sample to 250°C or surrounding it with a cold trap effectively reduced the contamination rate. Unfortunately, heating the eample is not always acceptlable; certain alloys, for instance, are liable to suffer a transformation at a relatively low temperature. Also, the installation of a cold trap raises some practical difficulties, and the necessity of providing clearances for the exit path of X-rays lowers its efficacy. Castaing and Descamps (16) have shown that it is possible, not only to lower the contamination rate but also to remove preexisting deposits by means of a low-pressure air jet directed onto the region bombarded by the beam. However, this solution, as pointed out by Wittry (19), has two undesirable effects: as a result of operating the beam a t higher pressures, filament life is reduced and the contamination rate in other parts of the beam system is increased because of greater back diffusion of vapors from the pump. The effect of the contamiliation is not very troublesome so long as the accelerating voltage used for the beam is much greater than the critical excitation voltage of the line analyzed. Such is not the case if, with the object of increasing the resolving power,* a low accelcrating voltage is 8

See Sec. II,E,b.

340

RAYMOND CASTAING

chosen; in this case it is essential to eliminate contamination completely. Consideration of Eq. (23) shows, for example, that, in the case of the analysis of copper by means of an accelerating voltage of 10 kv, the presence of a contamination layer only 200 A thick (in the absence of special arrangements such a layer is formed in a few seconds) suffices to lower the energy of the incident electrons by some 70 ev and to reduce the emission of Cu K q line by more than 10%.

D . X-ray Recording 1 . Conditions lo be Satisfied. X radiation emitted by the sample consists, in addition to the characteristic radiations of the various constituents of the bombarded region, of a continuous background which increases as the mean atomic number of the bombarded point increases. Therefore the analyzing device for this radiation has to satisfy the two essential conditions: (1) High sensitivity, as the intensity of the X-rays is relatively low. The electron current used for the excitation’being of the order of 0.1 pa, the X-ray emission of the sample is about 100,000 times lees than that of an ordinary X-ray tube and the detection technique necessarily uses a counter capable of recording the individual X photons. This detector may be a GM counter, scintillation counter, or proportional counter. The counter is generally preceded by a bent crystal monochromator, the whole making up a conventional X-ray spectrometer. In order to give some idea of the experimental conditions, we shall just indicate that the impact of a 0.33-p probe, carrying an electron current of 0.025 pa a t an accelerating voltage of 30 kv, on a pure copper block gives rise to a Cu Kal emission which, when measured in a bent quartz spectrometer and a GM counter, reaches 1,500 counts/sec. Such a n intensity is adequate for a rapid and accurate measurement of the characteristic emissions. (2) Good discrimination; it would seem a priori that the extreme simplicity of the spectra would not require the use of a highly dispersive instrument for isolating the characteristic lines, and, in fact, quite a coarse discrimination is sufficient for isolating the K spectra of the various elements. This is no longer true when using the L spectra, which is essential in the case of heavy elements for which the excitation of the K levels would demand an electron accelerating voltage incompatible with correct localization of the analysis. The L spectra have many more lines, and it is obviously important that they be perfectly separated by the spectrometer to avoid all risk of ambiguity between separate elements. But the usefulness of high discrimination arises mainly from the fact that the limit of detection of the analysis (minimum d c t 2 c r t d h con(wi-

341

ELECTRON PROBE MICROANALYSIS

tration) is lowered as the discrirnhiatrion of the spectrometer increases as ;I consequence of improved signal-to-noise ratio. 2. Practical Designs. In the French model (25) analysis of X radiation is effected by means of two similar spectrometers, the crystals and the counters rotating over the same focusing circle (Rowland circle) of 25-cm radius (Fig. 9a). A gear system gives the counter a speed of rotation which

a

b

C

FIG.9. Three spectrometer arrangements.

is twice that of the crystal. The radiations reflected by the crystal thus come, whatever the angular position chosen and hence the reflected wavelength, to a focus exactly at the center of the counter imput window. A dial graduated in degrees and minutes shows continually the angle of incidence of the X-ray beam on the crystal and hence the wavelength of the reflected rays. a. Usual wavelengths. One of these spectrometers is used to detect radiation of medium wavelength in the band 0.6 to 4.5 A, and so it permits the analysis by K lines of all elements situated in the periodic table between chlorine and molybdenum, and the analysis by L lines of all elements heavier than molybdenum. The detector is a GM counter with a mica window and the crystal is a quartz plate 0.3 mm thick of the Johannson t,ype (ground and bent). A similar arrangement is used in many other instruments (11, 12), the quartz crystal often being replaced by a lithium fluoride crystal. Lithium fluoride gives a broader reflection, which could be an advantage if the spectrum as a whole were recorded by a scanning method, because the integrated intensity reflected by the lithium fluoride is in this case much greater than that reflected by a quartz crystal under the same conditions. But microanalysis is a case of a method of analysis in which we have to consider peak intensity and not integrated intensity. Peak intensity is practically as great for a quartz crystal (provided the crystal is correctly curved and the spectrometer is well adjusted) as for a lithium fluoride crystal, with the considerable advantage that the discrimination-and consequently the signal-to-background ratio-is much better in the case of the quartz. The bent quartz spectrometer provides means for

342

RAYMOND CASTAING

the convenient separation of the two components of the Ka doublet for elements of medium atomic number. This possibility is of no particular interest in itself for the analysis, but this resolution is related to the ability to achieve a value of the signal-to-continuous background contrast which is of the order of 400 when pure elements are bombarded. This observed signal-to-continuous background contrast is in good agreement with Cambou’s measurements (28) in which he used the method of Ross’s double filters and a beam accelerating voltage of 30 kv. It can be deduced directly from Cambou’s results than the portion of continuous spectrum emitted between the absorption limits Cu K and Ni K (Ah = 0.1065A) by a pure zinc anticathode has an intensity equal to 18% of the intensity emitted in the Zn Ka doublet. As a result, for a spectrometer whose discrimination is just sufficient to separate the two components of the doublet (Ah = 0.00386 A), the signal-to-continuous background contrast is, when the spectrometer is adjusted to the Zn Kal line which carries two-thirds of the Ka intensity, 2 0.1065 = 100, 30.18 X 0.00386

Q = -

neglecting the self-background of the counter, which is mainly due to cosmic rays and amounts to about 30 counts/min. The value of 400 obtained for the line-to-continuous background contrast permits very low concentrations to be detected; an iron concentration of 0.02%, for instance, is very easily detected since the Fe Kal line projects above the continuous spectrum by nearly 10%. Other types of spectrometers are also used. Borovsky (29), in an apparatus designed in 1953 independently of the previous work of the author, uses a bent crystal operating by transmission (Fig. 9c) which naturally is applicable only for relatively hard lines. Figure 9b shows the arrangement used by Fisher (SO), in which the reflecting crystal moves in a straight line from the source and so constantly sees the source in the same direction. Some authors also use a plane crystal whose effectiveness, however, is less than that of the bent crystal. Last, Birks and Brooks (31) set up several crystals and fixed detectors for the simultaneous analysis of several elements (system analogous to that of the quantometer) which has the advantage of reducing the time required for the analysis on a given point of the sample and consequently the degree of its contamination by the beam. b. Light elements. The analysis of light elements is made rather more difficult by the fact that their characteristic radiation is much softer and hence easily absorbed. We shall here describe the soft radiation spectrometer used in the French model (16)which enables the waveband from 4 A to 12 A to be analyzed.

ELECTRON PROBE MICROANALYSIS

343

The reflecting crystal is a mica strip curved on a 50-cm radius; the receiver is a proportional counter with gas flow (Fig. 10). The window of this counter consists of a 6-p Mylar foil whose absorption is only 40% for a radiation as soft as Si Ka (7.11 A). The Mylar is somewhat permeable to water vapor, and this prevents the design of a sealed-off counter; gas flow has therefore t o be used, the rate of flow of the gaseous mixture (90% argon and 10% methane) being about 10 cma/min. The lateral arrangement of the window makes it possible to avoid any blind space; the drop in sensitivity of the counter at the longer wavelengths then arises only from the absorption in the window, which is slight up to 10 A. A bypass arrangement (Fig. 11) enables the counter to be evacuated before the gas is introduced.

P

I

1

. . TO PRE AMPLIFIER

GAS INLET

FIG.10. Proportional counter (soft X-rays) (Castaing and Descamps). 1. Iraldite (a trade name for an epoxy resin); 2. Window (Mylar 6 f i thick); 3. Tungsten wire (20 p diameter); 4. Wire tension spring; 5. Araldite.

This arrangement avoids the long flowing process which would otherwise be necessary to eliminate completely the inside air, and the counter is ready for operation in less than 5 min. The constancy of the gas flow is secured by a capillary; the gas pressure is slightly above 1 atm, avoiding contamination by air. The small diameter (20 p ) of the central tungsten wire enables the operating voltage to be lowered to 1500 v. A preamplifier located in the evacuated space of the spectrometer follows the counter in its motion. To take a n example, it may be mentioned that the Si Ka line emitted by a block of pure silicon under an electron bombardment of 0.1 pa a t 15 kv gives an intensity of 1000 counts/sec, the signal-to-background ratio being about 100, which is sufficient for a quick and accurate measurement,. I t is, of course, necessary to evacuate the spectrometw in order to rword such soft lines (the introduction of atmospheric air in tlhe spectrometer would lower the intensity of the Si Ka line in the ratio of l O I 3 ) . In order to avoid the presence of an absorbent window between the source and the crystal, a secondary vacuum is applied in the spectrometer, and the outlet

344

RAYMOND CASTAING

window of the instrument, which normally isolates the spectrometer from the object space, is turned down. This beryllium window 0.08 mm thick, inserted in the path of the beam, would m a c e to reduce by a factor of 140 the intensity of the Mg KCYradiation. The combination of the two spectrometers thus allows all the elements whose atomic number is not less than 11 (sodium) to be analyzed either by their K lines, or by their L lines.

Fro. 11. Spectrometer arrangement (Castaing and Descamps). Cl, Cz,capillary pipes; El, &, Ra, valves; 1, pressure-reducing valve; 2, gas tank; 3, linear amplifier; 4, pulseheight discriminator; 5, counting device with integrator; 6 , HT supply for counters; 7, gas-flow meter; 8, preamplifier; 9,pressure meter; 10, proportional gas-flow counter; 11, mica crystal; 12, probe; 13, quartz crystal; 14, GM counter.

c. Other arrangements. Certain authors, in particular Mulvey and Campbell (32),have investigated the use of nondispersive systems for measuring characteristic radiation. The use of scintillation counters (unfortunately limited to lines harder than Fe Kor ) and especially of proportional counters makes it possible, with the aid of a pulse analyzer, to avoid reflection on the crystal and so to obtain a considerable gain in sensitivity. This gain is mainly due to the large aperture of the admitted beam; unfortunately, the power of discrimination is much poorer, and special artifices must be used t o separate the characteristic emissions of adjacent elements (33). Cambou (28), in the author’s laboratory, has examined the possibility of applying Ross’s double-filter method, which ensures excellent separation of the lines and which is applicable to wide beams. The conclusion is that the interpretation of the results is much more difficult than in the case of ordinary spectrographic measurements. This difficulty arises mainly because of the complexity of the absorption correction for the continuous

ELECTRON PROBE MICROANALYSIS

345

spectruni. But it shorild IJC Iiotcd tlist oiie of tJir cnses where it, will be essclitial t,n iisc large-apertme h a m s ant1 noiidispersive detector systems is high-rcsolutioti analysis of thin layersg wherc such absorption is negligible; hence this happy consequence that the mail1 disadvantage of the doublefilter method vanishes in one of the cases for which a large collection efficiency becomes essential. d. Special design. Riggs ($4) has designed an arrangement which makes it possible to keep the specimen in air outside the electron optics system. A 10-p hole through a mica sheet allows the electron probe to emerge and strike the surface of the sample. Emitted X-rays which leave the surface a t a take-off angle of 55’ first pass back through the mica sheet and then emerge again through a beryllium window; excessive air leakage through the 10-p hole into the electron optical system is prevented by an auxiliary pumping path near the window. With Riggs’s instrument, specimens weighing several hundred pounds may be examined; this is of particular interest in the analysis of big meteorites. e. Possibility of extension to very light elements. The extension of the analysis to very light elements, such as carbon introduces quite serious difficulties. The characteristic lines are of very long wavelength (44 A for carbon) and therefore easily absorbed. Moreover, the efficiency of the dispersive systems which can be used in this wavelength region (ruled gratings for instance) is very bad (11). Dolby and Cosslett (35) have undertaken a detailed examination of the possibility of using nondispersive systems for the separation of characteristic lines in this region, and the results they have obtained are very encouraging. They start from the observation that “high efficiency is necessary to keep the beam voltage and current as low as possible, both from specimen heating and spatial resolution considerations. The quantum efficiency problem is particularly acute when scanning imnges1° are desired. The best available energy discriminating detecltor having high collection efficiency is the proportional counter. However, proportional counters do not have sufficient energy resolution for separating the K X-ray lines of elements closer than about three in atomic number. But the collection efficiency overrides this disadvantage” and the authors have centered their effort around the direct use of a proportional counter followed by a special pulse analysis method which overcomes many of the problems associated with low-energy resolution. This “matrix method” is illustrated in Fig. 12. The three overlapping 10

See Sec. II,E,S. See Sec. II,E,l,c.

346

RAYMOND CASTAING

pulse height distributioiis from elements A, €3, 2nd C, adjacent in the periodic table, add together to give the composite curve Y ;this last curve is known from the experimental measurement, but the constituent amplitudes a, 8, and y are unknown. These amplitudes can then be found by expressing three ordinate measurements Y A , Y B ,YC in terms of the component amplitudes and solving the three equations for a, b, and y. The coefficients evidently depend on the exact shape of the constituent curves, which may be known from experiments on pure elements. The values chosen

Measurements Solutions (Y = 1.83Y~- 1.52Y Y A = 0.6078 0.1357 0.68Yc Y = 0.607n 8 0.607~ /9 = -1.52Y~ 2.85Y 1.52Yc 7 = 0 . 6 8 Y ~- 1.52Y 1.83Yc Yc = 0 . 1 3 5 ~ ~0.6078 3- y Fro. 12. A composite curve and its constitutent curves A, B, and C (by courtesy of R. M. Dolby and V. E. Cosslett). (y

+

+ + + +

+

+ + +

in the example (Fig. 12) correspond to Gaussian distributions with the same standard deviation, and peaks separated by a distance equal to this standard deviation. Electronic mixing of the outputs with the appropriate signs (Fig. 13) makes it possible to solve the equations automatically; ordinate measurements are replaced by area measurements (vertical strips) performed by three identical pulse analysis channels. The method could be applied to the separation of the characteristic lines of carbon, nitrogen, and oxygen; however, absorption in the inlet window makes it rather difficult, for the moment, to design proportional counters recording efficiently the nitrogen and oxygen lines. Meanwhile the authors have applied the method to the separation of magnesium, aluminum, and silicon lines with excellent results.

347

ELECTRON PROBE MICROANALYSIS

The only disitdvaulage of this niethod lies in the fact that it increases the illfluenre of stutisticd fluctuations on the accuracy of the results; statisticd errors of 5% in the determination of the ordinates Y A , Y B , Yc causc, for instalice, in the fl amplitude (Fig. 12) an error of about 30%. In practice this amounts to lowering the collection efficiency of the counter by a factor of about 40. Nevertheless, this method seems to hold out a real hope for the microanalysis of very light elements, a t least in the absence of heavy constituents whose L spectrum would seriously complicate the situation. ELECTRON GUN

-:!

LENS

7

SCANNING

4

DISPLAY C.R.T.

SUNNING CIRCUITS

m

-

A

MOWLATION

FIG.13. Scanning X-ray microanalyzer incorporating the matrix met,hod of pulse analysis (by courtesy of R. M. Dolby and V. E. Cosslett). But it must not be forgotten that, even if a satisfactory method for recording lines is developed, serious difficulties will remain for the practical application of the method to precision quantitative analysis, mainly as regards absorption correction.1' In order to maintain self-absorption within acceptable limits, it will be necessary to use very low accelerating voltages (a few kilovolts), and the analysis will be extremely sensitive to the presence of surface impurities such as contamination.

E. Localization of the Analysis 1. Various Procedures for Localizing the Point of Impact of the Probe. The accurate determination of the point of impact of the probe is one of the main problems in electron probe microanalysis. This determination has to be effected with an accuracy a t least equal to the discrimination of the analyzer, i.e. better than 1 p , The method most commonly used consists in observing the sampIe during the operation by means of a viewing device consisting of an optsirid mirroscope. 11

See Sec. II1,D.

348

RAYMOND CASTAING

a. Optical viewing of the sample. In the original apparatus developed by the author (d), the viewing device consisted of the object lens of an ordinary microscope, preceded by a mirror placed between the object and the probe-forming lens. The mirror had a hole 0.2 mm in diameter to let the electron beam through, and its orientation was such that the viewer's optical axis, perpendicular to the surface of the sample, was inclined a t about 10" to the electron beam axis. This arrangement makes it possible to reject the shadow generated by the mirror orifice. A similar arrangement is a t present used in several instruments; it has the disadvantage of limiting the numerical aperture of the viewing microscope used to about 0.25 and its resolving power to about 1.5 p. But the experimenter should be able to distinguish easily the structural details of a sample with which he is familiar from observation with excellent metallographic microscopes. A perfectly satisfactory solution consists in observing directly the surface of the sample by means of an objective centered on the electron beam; a reflecting objective (36) was adopted in the French model (Fig. 5). The advantages of such an arrangement are many: The reflecting objective has a large working distance (about 17 mm) while retaining a large numerical aperture (0.48) and a resolving power of 0.7 p . I n addition, the axial part of the objective plays no part in the formation of the image and so can conveniently be provided with a hole for the electron beam; the objective has no nonconducting surface liable to cause charging up effects; last, none of the optical surfaces is opposite and close to the point of impact of the probe. This last feature makes it possible to prevent contamination of the surfaces under the action of back-scattered electrons. This objective is associated with a reticular eyepiece; it is only necessary to make a preliminary adjustment to cause a coincidence of the point of impact of the probe with the cross lines. This adjustment is effected by means of a small fluorescent screen set up permanently on the specimen holder. The position of the probe is therefore fixed in three dimensions because of the very small depth of field of the viewing microscope. The construction of the specimen holder makes it possible to bring in succession under the impact of the probe the point chosen for the analysis and the standard composed of the element to be assayed. A set of 42 standards composed of pure elements or definite compounds is permanently fixed on the specimen holder. The exact position of the point of impact of the probe can be constantly verified by observing the contamination spot produced by a prolonged bombardment. It is thus possible to observe slight displacements of the probe which occur, in particular, when analyzing a highly magnetic sample. These displacements are then compensated for by means of the electrostatic device for deflecting the beam. It is important that the probe

349

ELECTRON I’ROIJE MICROANALYSIS

always be in precisely the same position with respect to the spectrometer, a t least when the latter uses a dispersive crystal of high discrimination. b. Rotating drum. The arrangement developed by Mulvey (11) avoids the use of a reflecting objective and the difficulties involved in placing it on the axis of the probe-forming lens; the specimen is mounted on a drum and is rotated out of the electron beam in front of a regular microscope for viewing. In this way it is possible to observe the sample by means of an excellent metallographic objective with a resolving power close to 0.3 p . Ako, the probe-forming lens does not have to be specially designed to accommodate the objective on its axis, hence a greater structural simplicity. But the difficulties are transferred to the design of the COLUMN

v-

Electron Gun

a+Condensar

Lens

OPTICAL MICROSCOPE

Specimen Table Proportional Counter

DISPLAY

TUBES

Pen Recorder

FIQ. 14. Schematic diagram of the X-ray scanning microanalyzer (by courtesy of P. Duncumb and D. A. Melford).

rotating drum which has to ensure absolute correspondence between the point viewed by the microscope and that bombarded by the probe. It seems that a precision better than 1 B can be held only with difficulty over a long period of routine operation. The scanning technique proposed by Cosslett and Duncumb enables this difficulty to be overcome. c. Scanning analysis. Cosslett and Duncumb (37) have developed a scanning technique which permits rapid study of the surface distribution of the different elements. The electron probe is scanned over the specimen surface in synchronism with the spot of a cathode-ray tube whose brightness is modulated by a signal from the spectrometer detecting the characteristic emission of the selected element (Fig. 14).

350

RAYMOND CASTAING

The value of this method arises from the fact that it makes it possible to detect rapidly local variations of concentration (e.g. segregation) which often fail to appear on the image obtained on the optical microscope. Scanning hardly replaces the optical viewing method when a definite point of the sample (such as a precipitate) has to be rapidly brought under the impact of the probe in order to effect its quantitative analysis. According to Duncumb and Melford (22) “an image of the specimen surface showing the distribution of the element is obtained, and, after stopping the scan, the electron probe can be accurately positioned from the image afterglow for quantitative analysis.” But it should be noted that a really accurate positioning is possible by this method only after the image has been established for a rather long time. If, for instance, a detail of the object (e.g. a precipitate), whose diameter is a t the limit of resolution of the analyzer, i.e., 1 p , has to be centered under the impact of the probe, positioning has to be effected to a fraction of a micron. This centering requires that the resolution of the image be itself of the order of 1 p. Now the intensity recorded on the spectrometer is of the order of 10,000 counts/sec for pure elements and each surface element of one micron square must deliver a t least some 20 counts in order that statistical fluctuations shall not completely mask the concentration fluctuations of the constituent to be assayed. Therefore the period required for establishing a sufficiently clear image of 0.4mm side is of the order of 10 min, which necessitates the use of a very long remanence oscilloscope as a more complicated storage system. Most fortunately, a more rapid view of the surface of the sample can be obtained by using the back-scattered electrons. In the Duncumb and Melford instrument, a scintillation counter is used to collect scattered electrons (Fig. 15) and the signal delivered by this counter is used to modulate the brightness of the oscilloscope. By this method image formation is considerably faster, the number of back-scattered electrons being lo’ times greater than that of the X photons recorded on the spectrometer. But contrast in this case arises mainly from local variations of mean atomic number, and the examples of application given by the authors refer to samples in which these variations are large (e.g. aluminum-tin). It is not yet certain that these two modes of observation are capable of completely replacing visual observation of the sample in the course of analysis by means of the metallographic microscope, a t least if the method is to be capable of application to any type of sample. The ideal would be to combine direct vision on the metallographic microscope with mapping possibilities of the various elements by the scanning method developed by Cosslett and Duncumb. Electrostatic or magnetic scanning of the probe is, however, incompatible with the use of a high-resolution spectrometer. Displacement of the probe by a few hundredths of a millimeter completely

SECTION

OF

OBJECTIVE

LENS

H

r n

M

c3

T! Z cd W

:: M

z

i; zf

PLRSPLX L W T TO PHOTOHULTIP

B

?r

2 v1

SPECIMEN

CHAMBER

SPECTROMETER

FIG.15. The objective lens and X-ray spectrometer of the scanning microanalyzer (by courtesy of P. Duncumb and D. A. hlelford).

352

RAYMOND CASTAING

upsets the adjustmerit of the spectronieter and falsifies the measured intensities. In Duncumb’s and Melford’s apparatus the crystal used (LiF) has a broad reflection, and this makes it possible to depart considerably from the strict focusing conditions; the disadvantage of this crystal is that the line-background contrast is low, which limits the possibilities of quantitative analysis in the region of weak concentrations. Considering that the period required for establishing a good X-ray image is in any case long, it is possible to replace the scanning of the probe by a mechanical scan of the sample itself, the probe remaining fixed. This allows the spectrometer to remain constantly focused on the X-ray source; in this way the advantages of scanning analysis can be combined with those of precision quantitative analysis. 2. Limitation of Discrimination !rg Diffuse Penetration of Electrons. In the course of their progress inside the sample, the electrons deviate from their initial direction; a t the end of a travel which is shorter for a weaker accelerating voltage and for a higher atomic number of the bombarded region, the direction of the electron trajectory is no longer related to its initial direction and the penetration of the electrons in the sample is consequently completely diffuse. Some of the electrons may even be diffused back and leave the sample. As a result, the diameter of the region analyzed, i.e. the diameter of the sample region where the electrons still have sufficient energy to ionize the characteristic X-ray levels, is greater than the diameter of the probe itself. It is quite futile to use probes of very small diameter for the analysis if, a t the same time, the precaution is not taken of lowering the accelerating voltage of the electron beam. It is normal, in the usual X-ray tube technique, to use an accelerating voltage a t least equal to three times the critical excitation voltage of the X level since it is well known that this gives the optimum value of the contrast between the characteristic emission and the continuous spectrum. If this rule is maintained in electron probe microanalysis, the voltage which will have to be used for assaying copper, for instance, is in the neighborhood of 30 kv. Under these conditions, the electrons follow, within the sample, a track of the order of 2 p before their energy falls below the 9 kev which is necessary for the excitation of the K level of copper. Along this track, which obviously is not a straight line, the electrons depart from their initial trajectory by an amount which may reach about half the total path. As a result, the diameter of the region analyzed is greater by 2 p than the diameter of the electron probe, and consequently there would be little point in seeking to improve the discrimination of the analyzer by reducing the diameter of the probe below 0.5 p . But, as was shown by Wittry (38),it is not to be deduced that the resolution of the method is thus subject to a fundamental limitation and that

ELECTRON PROBE MICROANALTSlS

353

efforts to improve the probe brightness are of no interest. It is possible, if one wants to do away with the use of the best accelerating voltage for the excitation of X levels, to improve considerably the discriminating power of the analyzer, the only limit being, in fact, imposed by the performance of the probe-forming system (probe brightness) and the quality of the spectrometer used (line-continuous background contrast). Wittry was able to establish the experimental conditions leading to optimum discrimination, on the basis of the observation that the volume or the diameter of the region analyzed may be made arbitrarily small by a suitable choice of the probe diameter and of the beam accelerating voltage; the only limitation is imposed by the necessity of obtaining a sufficiently accurate measurement of the line intensity without extending the measurements beyond a reasonable time. Without entering into the details of Wittry’s calculations, the various stages of his argument can be summarized -with minor modifications-as follows: First, the depth of the analyzed region can be estimated from Williams’ law of deceleration modified, as suggested by Webster (9), by introducing the factor 2 2 / A . The equation obtained by Williams is applicable to relatively fast electrons > 0.5);here we shall modify slightly the numerical constant of Williams’ equation (8) in order to adapt it to electrons with an energy between 10 and 30 kev; in good agreement with the results obtained by Williams on argon (39)and by Terrill on aluminum (40) we shall adopt the deceleration law

(a

where V is expressed in kilovolts and x in centimeters. It can be assumed that excitation a t the maximum depth is produced by electrons which have suffered practically no deviation along their path. Writing that their deceleration has brought them from their initial energy E = eV to the minimum energy necessary for exciting the K level (for instance) or EK = eT.‘K, we obtain an approximate value of the depth of the analyzed region

A zm = 0.033(V1,7- v K 1 ’ 7 ) - microns, PZ

(24)

where V and V K are expressed in kilovolts; A is the mean atomic mass of t,he bombarded point, 2 is its mean nt,omic number, and p the local density in grams per cubic centimeter. I t can be assumed that the total diameter of the analyzed region is equal to 6

=

d

+ z,,,,

(25)

354

RAYMOND CASTAING

where d is the diameter of the electron probe and Zm the maximum effective range. The total diameter can, of course, be reduced indefinitely by reducing the diameter of the probe and lowering the accelerating voltage V to the immediate neighborhood of the threshold voltage V K .But in this case the intensity of the line emitted by the sample becomes extremely weak and an accurate measurement of this intensity requires a very long time. For example, let us consider attaining an accuracy of 0.4% on the intensity emitted by the pure element by means of a measurement lasting no more than 1 min. If the intensity of the continuous background can be neglected compared to that of the characteristic line (which will be assumed in order to simplify calculations), the number of counts per second recorded by the counter should be n = 500 for the pure element. But the intensity of the characteristic radiation is proportional to (1) the electron current, i.e. to Vd", (2) the efficiency of the electron-photon conversion, which is substantially proportional to (V V&s. To take an example, let us consider the case of a pure copper sample (VK = 9 kv) and a curved quartz spectrometer for which the intensity Cu Kal is equal to 1500 counts/sec for an accelerating voltage V = 30 kv and a beam current of 0.025 Na. These conditions result from Eq. (19) (long-focus probe conditions), for a probe diameter d and a n accelerating voltage V, in an intensity of the Cu Kcq line of

-

~ ( C U=) 7.3V(V - 9)'a8d',

(26)

where V is expressed in kilovolts and d in microns. The intensity of 500 counts/sec is thus obtained for a probe diameter d = 4.74V-"(V

-

9)-"s6

microns.

(27)

The application of Eq. (24) to the case of copper leads to the expression for the depth of penetration zm zm = 0.0081(V1.7- 42) microns. (28) The diameter of the analyzed region, therefore, varies with the accelerating voltage used (the intensity of the Cu Kal line being constantly held a t 500 countx/sec) according to the relation 6 = d zrn = 4.74V-'(V - 9)-O.' 0.0081(V1*7 - 42). (29)

+

+

This expression gives a minimum for V = 14.5 kv; this gives the optimum conditions leading to the best discrimination when the line examined is the Cu Kal line:

V = 14.5 kv; i = 0.07 pa; d = 0.646 p ; Zm

=

0.421 p .

ELECTRON PROBE MICROANALYSIS

355

tinder t,hesc conditions, Ihe diamctm wid f hc volunic of thc analyzcd region arc 6 = 1.07 p

&lid

v = 0.88 p3.

For the ideal probe conditions [Eq. (IS)] we would obtain the optimum values

V = 11.7 kv; 6 = 0.475 p ; v = 0.034 p3. We have assumed that the influence of the continuous spectrum is negligible; this is entirely justified in the case of a high-resolution spectrometer provided the concentration of the element being measured is not too small. I n a more accurate calculation, Wittry (38) takes the continuous spectrum into account and reaches the following conclusions : (1) The diameter of the analyzed region is a minimum for a rate of excitation V / V x between 1.6 (strong concentrations for which the continuous spectrum is negligible) and 1.9 (low concentrations) ; however, this minimum is broad and slightly higher accelerating voltages may be used without reducing the linear discrimination markedly. (2) The volume of the analyzed region has a much more sharply marked minimum for a rate of excitation V/VK between 1.1 (strong concentrations) and 1.5 (low concentrations), minimum volume being in the neighborhood of 0.2 p3 in the case of pure copper. The discrimination power can be even better in the case of the analysis of light elements for which the critical excitation energy of the X levels is in the neighborhood of 1 kev. Duncumb (41) obtains under these conditions a discrimination close to 0.1 p. His estimate is a little optimistic since the diameter of the analyzed region is obtained by a quadratic sum of the Gaussian diameter of the probe, of the diameter of the aberration disk, and of the maximum depth of excitation. This method certainly leads to a n underestimation of the total diameter of the analyzed region. Considering only the first two terms, the intensity distribution in a probe whose Gaussian diameter is infinitely small shows a marked maximum a t the edges, so that the optimum probe has a sharp edge intensity distribution. I n order to improve the discrimination in the case of elements of medium atomic number, Duncumb (41) recommends the use of very soft lines ( L spectrum in the case of copper, M spectrum in the case of heavy elements). However, in addition to increased experimental difficulties, the use of the L or M spectra makes the estimation of fluorescence correction12a more delicate matter, and it seems preferable, in the case of a precision quantitive analysis, to limit ourselves to the K spectrum up to atomic number Z = 35 and to the L spectrum for the heavy elements. fi

See Sec. 1II.E.

356

RAYMOND CASTAING

But it is nevertheless valuable to use for the analysis a relatively weak accelerating voltage exceeding only by 50 to 80% the critical excitation voltage of the element to be analyzed. The resulting gain in discrimination is particularly important when a relatively heavy element has to be measured in a sample of low density. Suppose, for example, that copper is the element to be measured in an aluminum-copper alloy of low copper content and of density around 2.7. With the electron current such that the pure copper emission is equal to 500 counts/sec we find the optimum conditions V = 12.2 kv, d = 0.953 p, zm = 0.712 p or a discrimination 6 = 1.7 p, the volume of the analyzed region being of the order of 0.5 p3. The use of an accelerating voltage of 30 kv would make the diameter of the analyzed region greater than 7 1.1, and its volume greater than 300 p 3 ! The use of low accelerating voltages has the further advantage of reducing the magnitude of the absorption c~rrection,'~ but it has the disadvantage of increasing the thermal load of the sample, which is not too troublesome in the case of metal samples but could cause an unacceptable temperature rise in the analysis of thermal insulators. Another disadvantage arises from the increased relative importance of fluorescence secondary emission excited by the continuous spectrum.14 However, it is possible that the ratio I f / I which begins to grow when the excitation rate V/VK passes from 3 to 2 (this effect is due to the increase in the average absorption of the continuous spectrum in the sample), decreases again when the excitation rate approaches unity. This behavior results because the intensity of the exciting continuous spectrum decreases as (V V K ) ~while the intensity of the line excited by direct ionization decreases as ( V - VK)'.~. Last, a practical disadvantage is the fact that the improvement in the discrimination is obtained mainly by a reduction of the maximum penetration of the electrons in the sample; the analysis thus becomes more superficial in character and the infiuence of the surface state of the sample becomes a dominant feature. Unfortunately, chemical processes such as oxidation may modify this surface state. Also, the influence of a slight variation of the beam energy may become important. Even if the accelerating voltage is perfectly stabilized, a slowing down of the electrons may result from passing through a contamination layer, or from charge effects in the case of insulators. It will therefore be prudent to maintain the rate of excitation between the limiting values 1.5 and 2; the discrimination is then close to its optimum value without the disadvantages noted being too troublesome. 3. Possibility of Improving the Discrimination. From the considerations

-

18

See See. II1,D. Sec. III,E$.

'(See

ELECTRON PROBE MICROANALYSIS

357

which have just been set down it follows that the discrimination power of spot analysis can easily be brought to the neighborhood of 1p ,with a possibility of pushing the limit back to about 0.3 p under the best conditions. The further lowering of this limit requires some considerable effort and it seems clear that the only possibility of an important advance in this line lies in the analysis “by transmission” of previously thinned down samples, as was suggested by the author (1, 42). Suppose for instance that the sample to be analyzed is no longer in the form of a solid sample, but in the form of a thin film, the thickness of which is much less than the maximum range of the electrons. The deviation of the electrons in passing through the sample is then very slight and the diameter of the analyzed region is practically equal to the diameter of the probe. Discrimination is then limited only by current considerations, and this limitation is much less harsh than in the case of solid samples, since the accelerating voltage can be held at a sufficiently high value to ensure good efficiency of the electron-photon conversion. In the case of solid samples, improvement in the discrimination power can be obtained only a t the cost of a catastrophic collapse of this efficiency. It will even be worthwhile to use relatively high voltages for the analysis of thin films, for the following reasons : (1) The mean deviation of the electrons on crossing a layer of a given thickness falls as the initial energy of the electrons increases; this mean deviation depends only on the ratio of the energy of the electron to its initial energy (43) and tends to zero as this ratio tends towards unity. (2) The ratio of the intensity of the line to that of the continuous spectrum recorded simultaneously in the spectrometer (signal-to-background ratio) increases with the electron accelerating voltage. If we consider the emission as a whole in all directions, we find that the intensity of the continuous spectrum over a band of wavelengths of a given width encloeing the line varies as V-I (44a) while the intensity of the characteristic line varies roughly as V-~(VK-’- V-I ) (44b); so that the line-background contrast increases with V and tends to a constant value for very high acclerating voltages. But, in addition, the direction of maximum emission of the continuous spectrum tends to move towards the beam direction a t high energy (44c). As a result of this fact if the X-ray beam analyzed is on the side of the bombarded surface of the sample, it contains practically only the characteristic line as soon as the beam accelerating voltage becomes large (100 kv, for instance). (3) The probe brightness is proportional to V for a given diameter, and this is quite an advantage since the main limitation in the discrimination power arises in this case from current considerations. The discrimination power can be estimated in the following way:

358

RAYMOND CASTAING

In the case of analysis by transmission of a thin layer, the X-ray emission is considerably reduced by the fact that the path of the electrons inside the sample is very short. Suppose for example that the measurement of line intensity is always effected by means of a curved quartz spectrometer and that the sample consists of a thin copper foil E microns thick (e << 1). From the measurements made by Castaing and Descamps,16 it can be deduced that the ratio of the Cu Kal emission of this layer to the Cu Kcrl emission, under the same bombardment conditions, of a solid copper anticathode is

for an accelerating voltage of about 30 kv. Let us measure the intensities of the characteristic radiations in pulses per second in the counter. If i is the beam current in microamperes, we have I(Cu) = 60,OOOi,

or

I,(Cu) = 36,000ie.

(33)

If the examination of the sample is performed by transmission electron microscopy, a reducing lens of very short focal length can be used; we are therefore under “ideal probe conditions” (V = 30 kv) and we obtain from Eq. (18) I,(CU) = 5.77 10%d8’3,

(34)

where d is the diameter of the probe in microns. For a sample 0.2 p thick and a probe diameter of 0.1 p , the Cu Kal intensity is 250 counts/sec, the discriminating power being close to 0.1 p ; such an intensity is quite sufficient for a correct measurement of characteristic emission. However, it will be noted that the intensity of the characteristic lines decreases more or less as the 11/3 power of the resolution, and there is not much hope of improving the latter with the same experimental arrangement. There are then two possibilities: (1) Increasing the probe brightness by correcting the spherical aberration or using a field emission gun. (2) Replacing the spectrometer by a nondispersive system securing high collection efficiency; the solid angle of the beam received by the curved quartz spectrometer is about 1/300 steradian and so it can be hoped to attain, in the limit, a gain of about five in the discrimination power. The ultimate discrimination of transmission microanalysis would then be in the neighborhood of 0.02 p ; we have seen that thermal limitations do not play any part (st least in the case of metal films). 15

See See. II1,B.

ELECTRON PROBE MICROANALYSIS

359

On the other hand, the coiit,amiiiat,ioiiof the sample, which is a fuiwtioii of the total beam current, will be fairly rapid, and it can be expected from practical considerations of available spare around the specimen that it will be even more difficult to eliminate it completely than in the case of solid samples. It might be feared that the contamination layer might cause a serious perturbation; in actual fact this cause of error is not at all serious in the case of transmission analysis. The main effect of crossing a contamination layer is to slow down slightly the incident electrons. This slowing down would cause a large variation of the characteristic emission in the case of a solid sample bombarded by electrons with an energy close to the critical excitation energy of the X level; but it has no appreciable effect on the emission of a thin layer bombarded by highenergy electrons. This behavior can be observed on the experimental curves (variation of the ionization function with the rate of excitation V / V K )obtained by Webster, Hansen, and Duveneck (&d). The only effect of the contamination layer, therefore, is to introduce some diffusion of the electrons which slightly lowers the discrimination. From Bothe's formula, the most probable angular deviation suffered by 50-kv electrons after passing through a contamination layer (mainly carbon) 0.1 p thick would be about 3". The loss of discrimination is negligible, and the same applies to the lengthening of the electron trajectories within the thin sample. The latter point is an important one since the result of the analysis is not to supply directly the concentrations of the element present a t the point analyzed, as is the case for solid samples, but the masses per unit area of these various elements. It is therefore necessary to measure in turn all the elements present. in order to obtain the chemical composition of the sample. A gradual inclination of the electron trajectories within the sample could thus lead, if the various elements are measured in turn, to a slight overestimate of the elements measured last, when the contamination layer becomes rather thick. In actual fact, the effect is a second order one, and the same applies to the error introduced by the formation of a superficial film of any kind (oxidation, for instance) provided of course that it is not the element to be measured which gets deposited on the sample. Analysis by transmission measures directly the superficial masses of the various elements present a t the analyzed point. Any surface phenomenon such as oxidation or deposition of a contamination layer which leaves these superficial masses unvaried (it is assumed that the element measured is neither oxygen nor carbon!) has no effect on the results of the measurements. 4. Experimental Verifications. Many processes can be used for estimating approximately the diameter of the X-ray source constituted by the analyzed region. For example, this source can be used to form the image

360

RAYMOND CASTAING

of a very fine grid by X-ray shadow microscopy. It, is also possible to estimate visually the resolution on the image obtained by X-ray scanning microscopy (37’). For an accurate measurement of the true discrimination power (i.e., of the minimum diameter of the region where an accurate quantitative analysis is possible), the best method consists of analyzing a specimen showing abrupt phase boundaries such as a diffusion couple or better a composite sample obtained by pressing two metals against one another. As an example, one can verify, by scanning the probe across a copper-tantalum boundary, that a movement of 1.2 p of the probe is sufficient to raise the Cu Kal emission from 0 to 100% (after correction for the secondary fluorescence emission) when the accelerating voltage is 14 kv, while the minimum displacement is 3 p for an accelerating voltage of 30 kv. In the analysis of thin samples, one may mention the discrimination of 0.3 p obtained by Duncumb with an accelerating voltage of 25 kv (41). It should be noted that, in what has already been said, absolute discrimination is referred to, i.e. the total diameter of the region excited by the electron beam. Resolution could also be defined in a manner similar to that used in optics, by means of samples with a periodic structure formed by stacking alternate layers of element A and element B. A movement of the probe through the sample then reveals its periodicity, even if the latter is much smaller than the absolute discrimination. The resolution of an image obtained in X-ray scanning microscopy can therefore be, provided the differences of chemical composition are sufficiently marked from one point to the next one, considerably greater than the absolute discrimination of the same instrument for precision quantitative analysis. In regard to the detection of small particles (analogous to ultramicroscopy in the case of light), the limit is even lower; iron inclusions less than 0.1 p in diameter can be detected in a sample of medium atomic number.

111. THEFUNDAMENTALS OF QUANTITATIVE ANALYSIS BY X-RAY EMISSION We have seen in Sec. I that the concentration of an element A in a complex sample is measured by comparing the intensity I A emitted by the sample in a strong characteristic line (A Karl for example) of element A with the intensity I(A) emitted in the same line and under the same bombardment conditions by the pure element A. The intensities I A and I(A) are the intensities actually emitted by the atoms of the anticathode directly ionized by the electron beam. In order to obtain them it is important to apply various corrections to the raw intensities read in the spectrometer (apart from the correction for dead time of the counter). A first correction consists in subtracting from each reading the part due

ELECTRON PROBE MICROANALYSIS

361

to the portion of the continuous spectrum recorded simultaneously by the spectrometer; this correction is readily obtained by taking the mean of the intensities recorded for two settings of the spectrometer on either side of the line; it is generally small as long as the concentration of the element being measured is greater than a few per cent. It is then necessary to subtract from the intensity obtained the fraction of that intensity arising from an excitation of the sample by the X-rays themselves (characteristic radiation and continuous spectrum) ; it is this correction for fluorescence which proves to be the most difficult to evaluate with accuracy. FinalIy, the intensity of the line must be corrected for its absorption in the anticathode itself [the other absorptions in the outlet window, in air, etc. do not enter since they are eliminated when taking the ratio IA/I(A)]. This absorption correction may become particularly large in certain extreme cases, and it is essential to apply it very carefully.

A . Absorption Correction To take an example, let us consider a plane anticathode consisting of the pure element A, of density p , receiving normally on its surface an electron beam with an accelerating voltage V , and let us designate by 0 the angle of emergence of the X-ray beam analyzed. Let I be the intensity which would be recorded by the spectrometer if there were no X-ray absorption in the anticathode, and let dI be the fraction of this intensity which is emitted by an infinitely thin layer with a thickness dz located at a depth z under the surface of the anticathode. Measuring the thicknesses in masses per unit area, we shall write

d

= 'PA(PZ)d(PZ).

(35)

The function (PA, which we shall reconsider later, represents the distribution in depth of the characteristic emission A Kal in an anticathode formed of the element A and subjected to normal electron bombardment with an accelerating voltage V . In the absence of X-ray absorption in the anticathode, the spectrometer would then record an intensity

In actual fact., since the mass absorption coefficient of the radiation A I
362

RAYMOND CASTAING

from which we shall obtain the intensity corrected for absorption by the relation

where F is the Laplace transform for the function

F(x)=

im

'pA

exp (-xu)du.

~ ~ ( 2 1 )

(39)

The curve which represents the variation of the function F(x)/F(O) as a function of the argument x = ( p / p ) cosec 8 has been obtained directly by the author (2) in the case of an iron anticathode, by a method which consisted of examining the variation of the characteristic emission as a function of the angle of emergence 0; it was used as a single absorption correction curve which was supposed to be valid, provided the electron accelerating voltage remained always the same, for all samples and all characteristic radiations, a t least in a first approximation. This amounts to assuming that the function 'p depends broadly only on the beam accelerating voltage, and constitutes a somewhat rough approximation. In actual fact, an exact determination of absorption correction requires a knowledge of the law of distribution in depth of the X-ray emission in the anticathode. We shall give a brief description of the experiments which have enabled Castaing and Descamps (7,46)to determine directly the form of the function 'p for various types of anticathodes.

B. Distribution in Depth of the Characteristic Emission 1. Experimental Determination of the Distribution. In order to determine experimentally the variation of the function ~ ( p z ) it, is necessary to isolate the emission of a thin slice of the anticathode with a constant thickness dz, located a t varying depths below the surface. The intensity read on the spectrometer represents, of course, the total characteristic emission of the slice dz multiplied by some coefficient. This coefficient, which depends on the aperture of the X-ray beam analyzed, on the efficiency of the spectromcter, on the absorption by the windows, etc., is invariable, provided the spectrometer setting and the characteristics of the electron ticam arc unchanged during the whole of the series of measurements and Frovided also that the result of the measurement is each time corrected for abaorption in the anticathode itself; this absorption depending on the drpth of the emitting layer. It is then possible to refer the intensities emitted by the layer dz at the various depths to a common unit. This unit is the intensity of the r h m w twistic radiation cmit tctl iiiider tjhc snmr ronditlions hy nil itlriitical layer

ELECTRON PROBE MICROANALYSIS

363

dz isolated in space and subjected to the same normal electron bombardment conditions. The unit so chosen is directly related to the value of the ionization function and to the thickness of the layer dz. Because the layer is infinitely thin and the beam normal to the surface, the path length for each electron is exactly equal to de. The choice of this unit therefore enables the function cp(pz) to be obtained in absolute terms. I n order to separate from the total emission of the anticathode that of a particular layer, Castaing and Descamps use an artifice which consists in replacing, in the massive anticathode consisting of the element A, the layer dz by a thin layer of an element B close to element A and whose properties are substantially the same as regards diffusion and deceleration of the electrons. Layer B then acts as a “tracer” whose characteristic B Krvl emission

FIG.16. Tracer composite sample for determining p h (Castaing and Descamps).

(say) can be easily separated by the spectrometer from the A Ka, emission of the anticathode as a whole. The choice of the tracer is limited by the following considerations: The element of which it is composed must be as close as possible in the periodic table to element A which makes up the anticathode; its characteristic emission must not be liable to secondary excitation by an intense line of element A, and the line emitted by the tracer must be only slightly absorbed by element A, so as to reduce the correction for absorption in the anticathode. Suppose we have to determine the law of distribution ‘pcu(pz) of the characterietie emission in an anticathode of pure copper. A carefully polished block of copper is covered, by vacuum deposition, with a zinc layer approximately 0.03 mg/cm2 thick. An identical zinc layer is collected on a thin collodion film plxraed in the evaporator ill the immediate neighborhood of thc surface of the copper block. This layer can later be considered as isolated, t,he collodion playing an entirely negligible part. Then , on various regions o f the s1irfac.e of the })lock already cmted with zinc, ropper layers arc tlcposit ctl (Fig. 16) with iticwnsiiig thic*kricsscs,siwh as for iiistxtiw 0.05,

364

RAYMOND CASTAING

0.1, 0.2, 0.5, 1, and 2 mg/cm2. The various layers are deposited simultaneously on large plates, which makes it possible to determine their thickness accurately by weighing, at the cost of a few corrections. This procedure provides the whole of the experimental equipment required, i.e. zinc layers of rigorously equal thickness, one being isolated and the others more or less deeply buried in a copper anticathode. Then, by means of the spectrometer, a measurement is made of the intensity emitted in the Zn Ktvl line when the various parts of the surface, which correspond to various values of depth z, are successively brought under the impact of an electron probe with constant accelerating voltage and current. For the common unit, the intensity is taken which is recorded in the spectrometer when the probe strikes the zinc layer deposited on the collodion. This series of measurements gives the curve of variation of the function cpcU(pz).There is no need to take into account the radiation absorption in the zinc layer since the correction term is eliminated when taking the ratio of the two intensities, but, each intensity measurement is corrected for the absorption suffered by the Zn Kal radiation in the superficial copper layer. Also, it is necessary to take into account the fact that the zinc layer has a finite thickness. This is done by sliding the curve in a direction parallel to the pz-axis by an amount equal to half the superficial mass of this layer. 2. Interpretation of the Results. Figure 17 shows the curves of distribution in depth of the characteristic emission in copper (zinc tracer), aluminum (copper tracer), and gold (bismuth tracer) obtained by Castaing and Descamps for a beam accelerating voltage of 29 kv.'" The abscissas represent the depth in mg/cm2, while the ordinates show log cp. It will be noted that the unit, i.e. ordinate 0, corresponds to the emission of the tracer isolated in space and subjected to bombardment by the same electron beam. These curves suggest the following remarks: (1) The value of (p for zero depth is always greater than unity, the tracer emission being reinforced by the underlying block. This effect is mainly due to back-scattering of the electrons. A slight increase also arises from the excitation of the tracer by the continuous spectrum of the anticathode as a whole. But it can be verified that it is important only in the case of heavy elements (compare Figs. 17 and 18).Back-scattering increases with the value of the atomic number of the anticathode; after correction for fluorescence (Fig. 18) we obtain pAl(0)

= 1.16; cpcU(O) = 1.475; p ~ u ( 0 )= 1.613.

(2) The function cp starts to increase for small depths; this is due to the gradual diffusion of the electrons which lengthens their path within the dz layer (2). The effect of this gradual diffusion is felt down to a depth corla

Further measurements have slightly lowered the tail of the gold curve.

ELECTRON PROBE MICROANALYSIS

365

responding to a coriditrioiiof complete diffusion, from which point the mean angle of incidence of the electrons on the tracer becomes constant. (3) When the complete diffusion is established, cp decreases proportionately to the number of electrons reaching the layer, as shown by Fig. 18 where the tcu curve represents the absorption curve of electrons in copper (7). This proportionality is due to the fact that, in these measurements, the beam accelerating voltage is much higher than the critical excitation voltage of the X-ray levels. This linear behavior would not appear in the case of

FIG.17. Distribution in depth of the total emissions (Castaing and Descamps).

low accelerating voltages leading to optimum discrimination. For instance, in the determination of (bA1, the replacement of the copper tracer by a bismuth tracer for which the rate of excitation is much lower (7) leads to a faster decrease of (PA] for the deeper layers. The depth z d where complete electron diffusion is established increases when the atomic number of the diffusing element decreases. For aluminum, complete diffusion is reached only at a depth corresponding to PZd = 0.6 mg/cm2, it is already reached in the case of copper for pzd = 0.4 mg/cm2, and in gold for pzd = 0.25 mg/cm2 (Fig. 18). (4) For the deeper layers, the decrease of cp (total characteristic emis-

366

RAYMOND CASTAING

sioii of fho tmver, Fig. 17) is slower, this effect being particularly marked ill the case of gold. This behavior is due to the fluorescence secondary emission of the tracer, excited by the general continuous spectrum of the anticathode. It will be seen a little further on17 how this fluorescence emission can be estimated approximately. By deducting it from the experimental values shown in Fig. 17,the curves of Fig. 18 are obtained; these represent

-P "

&-

ik

2.5

FIQ.18. Distribution in depth of the direct (primary) emissions and absorption curve of electrons in copper (to.) (Castaing and Descamps).

the distribution in depth of the direct characteristic emission (i.e. directly excited by the electron beam) of anticathodes of aluminum, copper, and gold for an accelerating voltage of 29 kv. This deduction is based on the assumption that the fluorescence emission of the tracer is approximately the same a t all depths till 1.5 mg/cm2. It is then found that the constant contribution of fluorescence emission is: for aluminum, c p d = 0.025 (selective absorption of the continuous spectrum in the copper tracer), for copper, cpc,,f = 0.03, and for gold, PA,,' = 0.10.

C , The Physical Basis of the Emission-ConcentrationRelation

-. .. . .. . . .* . .. 'lhe distribution curvea 01 r'ig. 18enable us to analyze in greater aetaa m-.

-n

1

.

.*

the various factors entering in the relation connecting the characteristic emission of the various elements of a complex anticathode with their respective concentrations. I7 See

Sec. III,E,%

367

ELECTRON PROBE MICROANALYSIS

Consider a n anticathode formed of an alloy AB in which the mass concentrations of elements A and B are respectively CA and CB. Let c p and ~ (OAB be the respective distribution functions of the radiation A Kal in the pure element A and in the alloy AB. As before, we shall designate IAand I(A) as the intensities emitted in the A Ka, line, under the same conditions of electron bombardment, by the alloy AB and by the pure element A. As regards the X-ray emission, the anticathode AB behaves as if it consisted of a stack of infinitely thin layers of elements A and B. This gives immediately

where S is the integral from zero to infinity of the function cp. Consider the special case for which CA is infinitely small, c p ~ Bis then the same as the function cpg which represents the distribution of the characteristic emission R Karl obtained by means of an A tracer, and we can write

It therefore appears that the approximate emission-concentration proportionality IA/I(A) = cA is nothing else than a n approximate equality of the integrals SAand S B and is obtained in this case with the same relative error. We shall therefore obtain an idea of its degree of accuracy by comparing the S integrals of the various elements. Referring to the distribution curves of Fig. 18 (direct emissions), we find the following numerical values s A ~

=

1.55

x

10-3,

scU = 1.46 x

10-3,

sAU

=

1.43

x

10-3.

The unit is the intensity which would be emitted under the same conditions by a layer of the same element of mass per unit area 1 gm/cm2, normally crossed through by the electron beam, if the electrons of this beam were to retain their energy throughout the path and if there were neither electron diffusion nor fluorescence emission. These values are close to one another; the agreement between S Aand ~ Scuwould even be improved by using the same tracer for the two distribution curves, as is supposed in deriving Eq. (41). Considering the c p ~ curve l obtained by Castaing and Descamps (7) with a zinc tracer and correcting we may conclude that the it for fluorescence, we obtain SAI= 1.45 X integrals are found to be equal to within the acwrary of determining cp(pz)-about 2%. An exact determination of the cp curves would make it possible, by com-

368

RAYMOND CASTAING

paring curves obtained with the same tracer, to calculate the a,coefficients of the second approximation [Eq. (ll)].However, a direct determination of the ai coefficients by the analysis of alloys of known composition is much simpler and more precise than the experimental plotting of the distribution curves. The relation C Y A / ~ B = S B / S A (same tracer) is really only of theoretical interest, but it shows that the variation of the a coefficients from one element to another is slower than that of the Z / A ratios which appear in the theoretical second approximation deduced from Webster’s law of deceleration [Eq. (S)]. This discrepancy is due to back-scattering which acts inversely to the factor Z / A in Webster’s equation. As a result of back-scattering, many beam electrons leave the anticathode with an energy definitely greater than the critical excitation energy of the line analyzed. The intensity I(A) emitted by an anticathode of pure element A is thus less than the intensity Io(A) which the anticathode would emit if there were no back-scattering and if, in consequence, the whole of the electron trajectories were located within the anticathode. Let us define this “back-scattering loss” by introducing the coefficient XA = Io(A)/I(A) ; the back-scattering loss is then (XA - 1) I(A) and the coefficient XA, always greater than unity, is greater the heavier the element. Applying then the theoretical second approximation [Eq. (S)] to the emissions corrected for back-scattering, we obtain the relation

It thus appears that each

ai is the product of two factors, the first one of which increases with atomic number Z while the second decreases with increasing 2. This “compensation” is the reason why the values of ai vary but little from one element to another. It is possible to get a rough estimation of the values of the X coefficients, (back-scattering correction) by an examination of the curves of distribution in depth of the characteristic radiation (Fig. 18). It can be taken roughly that the back-scattered electrons which reaeh the free surface have on the average followed the same path in the anticathode as those which reach the layer at depth z1 = 2zd ( z d being the depth from which the electron trajectories are completely diffuse). For each anticathode it will be assumed that the back-scattering loss is approximately equal to the total emiseion of the anticathode layers situated below the depth zl,or

lO(A) - I(A) = and hence

/“ 21

‘f‘A(pZ)d(pZ),

1,“

(43)

‘f‘A(pZ)d(PZ)

X A = 1 +

/o “

‘f‘A (pZ)d(pZ)

(44)

369

ELECTRON PROBE MICROANALYSIS

This gives: For copper ( Z / A = 0.456): pzl = 0.8 mg/cm2; hcu For aluminum ( Z / A = 0.482): pzl = 1.2 mg/cm2; XAI For gold ( Z / A = 0.400): pzl = 0.5 mg/cm2;

= = =

1.15; 1.05; 1.34.

If we arbitrarily make acU = 1.00, we find from Eq. (42) the a,coefficients ( Y A ~=

0.96, acU = 1.00, a~~ = 1.02,

in excellent agreement with the values which might be deduced from consideration of the S integrals. The method which has been followed for the estimation of the backscattering loss can be supported by the following remarks: (1) The curve of electron absorption in copper (Fig. 18) shows a transmission factor of about 20% for 29-kv electrons passing through a copper layer of 0.8 mg/cm2. This value is not very different from the percentage of back-scattered electrons, i.e. 29%. (2) It will be noted that in each case the following relation is approximately verified : (PA(PZ1)

=

2[(PA(O) - 11.

(45)

But, the bracket in the second term represents the fraction of the emission of the superficial layer due to the back-scattered electrons as this superficial layer is crossed (from bottom to top) by the back-scattered electrons. The number of these electrons is about the same as the number of electrons which pass downward through the layer at depth zl. The factor 2 can be interpreted if it is assumed that, because of the completely diffuse distribution of the trajectories, the layer z1 is crossed by a n equivalent number of electrons traveling upward. (3) If the qcU curve is compared with the tcu curve representing electron transmission in copper for the same accelerating voltage (Fig. 18), it will be seen that, a t medium depths for which complete diffusion is attained without too severe a loss of energy by the electrons, the relation pcu =

4tcu

(46)

is substantially verified, tcUbeing the electron transmission factor. This relation means that the mean path length, in the dz layer, of the electrons which have passed through a thickness large enough for complete diffusion is of the order of 4 dz. If we assume, as before, that each of these electrons passes, on the average, twice through the dz layer, we obtain, calling e the angle of incidence of the electron trajectory on the dz layer: m

=

2

(47)

370

RAYMOND CASTAING

whidi is vsactly the v:tlilc. which CWI I)ccspwted from a completely random distri1)iition of t hc trajectory dircctions (cosine distrilmtion).

D . Experimental Absorption Correction Curves From the experimental curve of dkribution in depth of the characteristic emission of a given element A (for a given accelerating voltage V ) ,it is easy to plot the Laplace transform F ( x ) [Eq. (39)l. The curve which represenh the quantity log [ F ( x ) / F ( O ) as ] a function of the argument x = ( p / p ) cosec 0 constitutes the absorption correction curve. Figure 19 shows the I.( =-asec 0

ioO0

2000

3000

4000

5000

6000

e

'7000 L

-1.5 T

- At -- cu Au

FIG.19. Absorption correction curves (29 kv) (Castaing and Descsmps).

absorption correction curves (direct emission) obtained in this way from the distribution curves of Fig. 18, for an accelerating voltage of 29 kv and anticathodes of aluminum, copper, and gold. The curve previously obtained by the direct method of variation of the X-ray intensity as a function of the angle of emergence ( 2 ) would lie satisfactorily in this family of curves (36). The various curves are much alike, but it will be noted that, for a n accurate determination of absorption correction, it is necessary to take into account the average atomic number of the region analyzed. Absorption correction can become quite large when analyzing light elements whose characteristic lines are easily absorbed. In this case values of x of the order of 5,000 to 10,000 are commonly found. The use of a n accelerating voltage of 30 kv for the beam would then involve an enormous absorption correction, more than 80% of the emitted intensity being absorbed within the sample. It is then necessary to use the lowest possible accelerating voltage in order to limit the absorption correction to 30 or 400j0.It is important to ensure that this correction shall remain fairly small since one can never be sure of its rigorous accuracy. For however carefully the absorption 'curves have been prepared, there always remains some uncertainty in the value of 0 since the angle of emergence may be locally

ELECTRON PROBE MICROANALYSIS

371

modified by an amount of the order of 1". This is the case when analyzing a precipitate which is generally protruding or slightly recessed after mechanical polishing. I n any case it is necessary to perform this polishing with care and to avoid electrolytic polishing, especially in the case of light alloys where it would be, in other respects, particularly convenient.

h

?2

L

B FIG.20. Absorption correction curves for different voltages and average atomic numbers; (a), V = 9.7 kv, 2 = 13; (b), V = 15.1 kv, Z = 13; (c), V = 27.5 kv, 2 = 26 (Cashing and Descamps).

Figure 20 shows the absorption correction curves corresponding to various accelerating voltages and various kinds of anticathodes obtained by the direct method of variation of the X-ray intensity as a function of the emergence angle.

E. Fluorescence Correction Secondary emission caused by fluorescence is a particularly troublesome phenomenon in the exact interpretation of the results obtained from the electron probe microanalyzer. First, the emission-concentration proportionality, which is the basis of this method of analysis, is no longer true, in general, for the secondary emission. It is clear for instance that an element which strongly absorbs thc exciting radiation suffers a selective excitation within the complex niit,icathode and that, as a result, there is an intensification of its apparent concentration. This situation will be clear immediately if we remember that the emission-concentration proportionality in the case of primary emission is directly related to the fact that all the elements possess broadly the same mass absorption coefficient for the beam electrons. Also, i t should be noted that, although the primary emission excited by an clectron probe is st,rictly restricted to the very small volume swept through hy the electrons, such is iiot, the case for the secondary fluorescence

372

RAYMOND CASTAING

radiation, which may have its source in the whole of the region irradiated by the X-rays issuing from the point of impact of the probe. For example, spot analysis by X-ray spectroscopy of a very small precipitate in an alloy necessarily consists in superposing two analyses : (1) X-ray emission analysis of the precipitate itself, (2) Fluorescence analysis of the precipitate with its surrounding region. It is therefore important to determine the value of this secondary fluorescence emission in order to estimate the correction term to be introduced in the results of the measurements, or at least a n order of magnitude of the resulting uncertainty in the measurement of the various concentrations. 1. Exm'tation by Characteristic Radiations. An important source of secondary emission may arise from the excitation of the element analyzed by the characteristic radiations of a heavier element whose concentration is large. For example, in the case of an iron-chromium alloy with 10% chromium, the secondary fluorescence emission of Cr Kal line excited by the K lines of iron is equal to 24% of the Cr Kal intensity due to direct ionization of the chromium by the beam electrons (direct emission). Very fortunately, the case of excitation by characteristic lines lends itself rather well to calculation, and it is possible to estimate the secondary emission with a sufficient degree of accuracy to remove completely the error which might result in the determination of concentrations. The calculations have been made by the author (2) and the correction formula obtained. Although the expression is a rather complicated one, it is convenient to use and leads to an accuracy which is quite sufficient for the needs of the analysis. It should be noted that Wittry (19) has taken up the author's calculations in a more elaborate form, using the curves of distribution in depth of the characteristic emission plotted by Castaing and Descamps ('7) and thus obtained a more rigorous estimate of the fraction of the fluorescence radiation excited by the characteristic radiations directed outside the anticathode. Matters become a little more complicated when an estimate has to be made of the importance of the fluorescence emission excited by the general continuous spectrum of the anticathode, and two methods will be briefly given which permit the experimental determination of this secondary emission. 2. Excitation by the Continuous Spectrum. Castaing and Descamps (7') use two separate methods for determining the secondary fluorescence emission excited by the continuous spectrum. In the first method, analogous to that previously used by Webster (QS),the surface of the anticathode of element A is covered, by vacuum depoeition, with a layer of aluminum of a thickness sufficient to stop the electron beam completely. The emission of the underlying anticathode is then entirely due to the fluorescence excited

ELECTRON PROBE MICROANALYSIS

373

by the contiiiuous spectrum of aluminum. This continuous spectrum has pract,ically the same distribution in wavelengths as that which would be produced by the same electron bombardment on an anticathode of element A (atomic number Z), the intensity of each radiation being modified in the ratio of 13/2. After a few minor corrections (in particular absorption correction for the fluorescence radiation in passing through the aluminum layer) the experimental value of the ratio I f / I is obtained. This ratio represents the proportion of secondary emission due to the continuous spectrum directed inside the anticathode to the total characteristic radiation, at the exit from the anticathode a t an emergence angle of 16" (value eleclrons

FIQ.21. Tracer composite sample for determining the true intensity of fluorescence emission excited by the continuous spectrum (Cashing and Descamps).

used by the authors). For example, the results obtained for an accelerating voltage of 29 kv are: for zinc, I f / I = 0.031 and for bismuth l f / I = 0.059. Further measurements have shown that this last value is probably a little low and that 0.07 would be a better figure. The second method gives a direct determination of the relative value of the fluorescence emission, as produced within the anticathode and before any absorption. It uses the tracer process already used for the determination of the distribution in depth of the characteristic emission. For example, suppose we wish to determine the magnitude of the secondary radiation in a zinc anticathode. A carefully polished block of copper is covered, by vacuum deposition, with a zinc layer of thickness e2 and the whole is covered with a copper layer of thickness el (Fig. 21). If the thickness el of copper is sufficiently great to stop the electrons completely, then the Zn KaI emission of the stratified layer is derived only from the excitation of the zinc layer by the continuous spectrum from the top copper layer. This continuous spectrum contains an infinite number of radiations occupying the whole of the frequency band between the quantum limit and the critical excitation frequency of the Zn K level. These various radiations

374

RAYMOND CASTAING

are very differently absorbed in the top copper layer and in the zinc layer, but it is possible, by a judicious choice of the el and e2 thicknesses, to arrange matters so that the fraction of the total intensity which is absorbed in the zinc layer is about the same for all the components of the exciting continuous spectrum (to within 10%). The experimental conditions are then as follows : A penetrating component of the continuous spectrum is weakly absorbed in the top copper layer and reaches the zinc a t a high intensity; but only a small fraction of this intensity is absorbed in the zinc layer; a soft component is strongly absorbed in the top copper layer, its intensity is small when it reaches the zinc layer, but a large fraction of this intensity is absorbed by the latter. For a beam accelerating voltage of 20 kv, it is necessary to choose a thickness of zinc of the order of 4 mg/cmz (for absorbing a noticeable part of the continuous spectrum in it) and a thickness of copper of the order of 2 mg/cm2. Under these conditions, the fraction of the continuous spectrum which is absorbed in the zinc substantially keeps the constant value of 0.32 from one end of the spectrum to the other. After a few minor corrections, the authors obtain in this way the proportion If"/I of fluorescence emission due to the continuous spectrum, as produced within the anticathode and before any absorption; the value obtained for a zinc anticathode and a 20-kv voltage is (If"/I)z, = 0.068. If this is compared to the value obtained for the same voltage at the exit of the anticathode at an emergence angle of 16", that is (If/I)zn = 0.036 (7), the large absorption suffered by the fluorescence radiation in the anticathode itself, because of its great mean depth of emission, is readily appreciated. It would be easy to predict theoretically the ratio I f / I f " if the fluorescence emission were excited by a monochromatic radiation issuing from the surface of the anticathode. In this case we would have, designating respectively by p and p' the mass absorption coefficientsin the anticathode of this exciting radiation and of the fluorescence radiation Zn Kal

Actually the exciting radiations occupy a band of wavelengths within which the quantity In (1 x ) / x varies from 0.37 to 0.82. It can be taken that the experimental ratio I f / I f o = 0.53 represents the mean value of the quantity In (1 x)/x, averaged over the exciting spectrum taken as a whole. This result enables us to obtain approximately the intensification by fluorescence due to the continuous spectrum in the case of a complex alloy. Since we are concerned only with an approximate calculation, we can replace the exciting spectrum by a single radiation emitted a t the surface

+

+

ELECTRON PROBE MICROANALYSIS

375

of the anticathode. For example, suppose we have to determine the proportion of secondary emission due to continuous spectrum in the A Kal line emitted by a n alloy in which the mass concentration of A is CA. Calling p' the mass absorption coefficient of the A Kal line in the alloy, Gall and (A the mass absorption coefficients of the alloy and of the element A, for the mean exciting radiation, we obtain

where

x=

p'

cosec 8

-

Pal1

It may happen that the variations of the last three terms in the righthand term of Eq. (49) compensate roughly for one another when varying cA. Such is the case for aluminum-copper alloys where Cu Kal is reinforced in about the same ratio a t all concentrations, which in this particular case makes it unnecessary to apply the fluorescence correction. The above calculations suppose that the sample is homogeneous over an extended region around the analyzed point. If we consider the opposite case where a very small precipitate is analyzed, imbedded in a matrix the chemical composition of which is quite different, we reach the conclusion (7) that: (1) I n the extreme case where the concentration of the analyzed element is zero in the matrix, neglecting the fluorescence excited by the continuous spectrum leads to a relative error in the concentration CA in the precipitate

being the apparent concentration. (2) I n the extreme case where the matrix consists entirely of the element which is being analyzed in the precipitate, neglecting the fluorescence leads to an absolute error of c'A

where ZA is the atomic number of element A and 2 the average atomic number of the precipitate. In this last case, the relative error can be enornmis for small concentrations and it would be difficult to estimate it correc~tly. In such a case, determining CA from measurements of the concentrations of the other elements present in the precipitate is highly recommended.

376

RAYMOND CASTAING

In conclusion Table I shows the various stages of the analysis of a goldcopper alloy by means of the Au Lal line; the accelerating voltage used is 29 kv and the absorption correction is fairly large. TABLEI. X-RAYEMISSION ANALYSISOF COPPER-GOLD ALLOYS

1

3

2

4

_ _ _ ~

True concentrations CA,,

0.4035 0.5393

ZA"/I(AU) zhu/z(Au) (-44 Rough values (total After fluorescence cor- After absorption coremerging intensirection (direct emis- rection (direct emission true insion emerging inties) tensities) tensities)

01364 0.491

0,356*

0.484t

0.402 0.534

~~

* Average value from 11 scanning analyses. t Average value from 4 scanning analyses.

The true concentrations (chemical analysis) are shown in column 1. Column 2 gives the rough experimental values of the ratio la,Jl(Au), corrected for counter dead time and continuous spectrum background (the specimen surface was scanned under the probe to average out segregation effects). Column 3 shows the values corrected for fluorescence excited by the general continuous spectrum [Eq. (49)], and column 4 shows the final values after correction for absorption (Fig. 19). The emission-concentration proportionality is perfectly verified here, the agreement between column 1 and column 4 being as good as could be hoped for from the accuracy of measurements (about 0.5%).

F. Fixed-Time versus Fixed-Charge Measurements It will be interesting to consider here the modification proposed by Wittry (19) which consists in referring the measurements of X-ray intensities not to a constant value of the number of electrons received by the sample (fixed-time measurement with constant beam intensity) but to a constant value of the number of electrons absorbed by the sample (fixedcharge measurement). The fact that the proportion of back-scattered electrons is 29% for copper and 49% for gold (as was checked by the author under the same instrumental conditions) means that a value of the ratio aAu/acuof 51/71 = 0.72 has to be introduced in order to interpret correctly the results of a fixed-charge analysis. Thus, the emission-concentration proportionality law, which is extremely practical for a first approximation so long as repeated measurements shall not have definitely established the values of the empiri-

ELECTRON PROBE MICROANALYSIS

377

cal ai coefficients which should be applied to all elements, is deliberately dropped. This complication could be justifiable if it were proved that the hyperbolic law corresponding to the second approximation is better followed in the case of a fixed-charge measurement than in that of a fixed-time measurement; but such is not the case. Wittry’s argument is mainly based on the results obtained in the author’s thesis (2) concerning the analysis of aluminum-copper alloys with various concentrations. Table I1 gives the true concentrations of these alloys (column 1) and the experimental values (column 2) of the ratio IcU/Z(Cu) obtained for these alloys in fixed-time measurements (2). Column 3 shows the respective errors involved by the assumption that strict emission-concentration proportionality is valid in fixed-time measurements. From the values given in Column 2, it is possible to calculate the ratios Z’C~/Z’(CU)which would have been obtained in fixed-charge measurements. Designating by rail and rcU the respective back-scattering coefficients of the alloy and of the pure element copper, one obtains immediately

The best estimate for raI1 should be

which has to be introduced for interpretDesignating by a the ratio CYC~/CYA~ ing the fixed-charge measurements [a = (1 rcU)/(1 - T A ~ ) = 0.8151, one obtains from Eq. (53)

-

a -Z’cu -.Icu Z’(Cu) - 1 - (1 - a)CC,l(Cu)

(54)

Column 4 gives the values so obtained. Applying to these values the hyperbolic emission-concentration relation of the fixed-charge measurement ( a c U / a ~=l 0.815) leads to the concentrations given in Column 5, which are practically identical to the concentrations obtained directly by assuming strict emission-concentration proportionality in the fixed-time measurement (Column 2). Since the percentage errors are practically the same (compare column 6 to column 3)) it should be concluded that no improvement is brought about by fixed-charge measurements. But, Wittry’s calculations are based on the implicit assumption that the back-scattering coefficient T , ~ Iof the alloy is given by the relation

TABLE 11. FIXED-TIME VERSUS FIXED-CHARGE MEASUREMENTS (ALUMINUM-COPPER ALLOYS)

1

2

~~

ccu

0.01 0.04 0.53 0.88

3

4

5

6

7

8

9

~~

ZCU/Z(CU) Fixed-time

0.0099 0.0373 0.504 0.867

Error (%I First approximation, fixedtime acu/ari = 1 1

6.75 4.9 1.48

Z'cJZ'(Cu) ccu Fixed
0.00808 0.0306 0.4555 0.844

0.0099 0.0373 0.506 0.869

Z"Cu/Z"(CU) ccu Fixed-charge From column estimated 7 fromEq. (56) acu/a.u = 0.815

1 6.75 4.5 1.25

0.00808 0.0307* 0.460 0.848

0.0099 0.0374 0.5115 0.6725

Error (76) ~ C U / ~ A= I

0.815

1 6.5 3.5 0.85

* Wittry [Thesis, California Institute of Technology (1957)l obtains here the wrong figure 0.0317,leading to only 3.6% error in the concentration.

ELECTRON PROBE MICROANALYSIS

379

leading to the fixed-charge result

-I"CU It'( CU)

-

[a

+ (1 - a>cc"1-'IC" I K U )

Equation (55) seems to be less reliable than Eq. (53); but, even if one assumes its validity, consideration of Columns 7 , 8 , and 9 (Table 11) makes it evident that the improvement brought about by the fixed-charge measurement is weak and vanishes at low concentrations. Subtracting the fluorescence emission excited by the continuous spectrum would leave the results practically unaltered (the correction factor oscillates between 0.997 and 1.002).1* Moreover, recent measurements have shown that the large dispersion of the values L Y C ~ / C Y A(fixed-time ~ measurements) which appears to result from the author's experiments (2, 19) (Column 2 of Table 11) is probably related to the extensive segregation which appears in these alloys and makes the results of the chemical analyses rather uncertain. New measurements will be necessary to clear up this last point. As a conclusion, the complication involved in fixed-charge measurements is hardly justified by better results. Moreover] fixed-charge measurements depend directly on the electron current reemitted by the sample, but, a not negligible fraction of this intensity is supplied by secondary electrons. This secondary emission is highly sensitive to the surface state of the sample and may suffer variations independent of the chemical composition of the underlying metal. The author feels that until more data are available] it is more prudent to keep fixed-time measurement, in which the X-ray emissions of the sample and of the pure element are compared under identical conditions of electron bombardment.

IV. THECONTRIBUTION OF MICROANALYSIS TO SCIENTIFIC RESEARCH Electron probe microanalysis has been successfully applied to a lot of problems covering an extensive range of research fields; the most important of them is no doubt metallurgy. But stress must be laid on the fact that microanalysis can be carried out even on insulating materials, after coating them with a thin metal or carbon film by vacuum deposition for ensuring good electrical and thermal conductivities. This last possibility opens to the electron probe microanalyzer new fields of application such as mineralogy or even biology. 18

See Sec. III,E,2.

380

RAYMOND CASTAING

A . Metallurgy In the field of metallurgy, electron probe microanalysis has been used extensively for theoretical as well as technological studies. We will limit ourselves to a few examples picked up from the numerous papers which have already been published on this subject. I . Identification of Phases. One of the important possibilities of this method is the identification of complex phases which appear as small precipitates, whose size lies generally between 0.1 and 10 p . An example is given by the grey ternary phase appearing in the AI-Cu 6%-Fe 2% alloy. Various formulas had been proposed by different authors; spot analysis has shown (25, 47) that copper and iron concentrations (respectively 34.8 f 0.8% and 14.2 f 0.6%) were consistent with the composition A17Cu2Fe. More difficult was an extensive study of zinc-rich complex brasses where the aluminum content was to be determined in very small twophased starlike precipitates (48). Nonmetallic inclusions, such as silicates, sulfides, phosphides, etc., which are generally present in steel are subject to identification only by quantitative determination of their medium atomic number components (Si, P, S, . . .) (48). An accurate quantitative analysis of the metallic elements makes it possible to distinguish between various types of carbides (M3C, MZ3C6, M7C3, for instance) (60). New methods of extracting replicas have enabled some workers to identify with the microanalyzer very small precipitates (size <1 p ) which were present in fracture surfaces of steel (51). Some siderurgical processes of steel production were not fully understood. By picking up the globular particles with diffuse boundaries which appear at the beginning of the temperature-steady state (1490°C) during the Thomas conversion of steel and analyzing the phosphorus content of these particles, it was possible to show that the particles were solid in the molten bath (52) (Fig. 22). 2. Segregation. Spot analysis has provided a most valuable tool in the study of various types of segregation which are not revealed by conventional metallographic methods. Researches in this field are quite numerous, mainly concerning technical problems (53-66). As an example, a map showing the segregation of manganese in Hadfield steel (C: 1.88%, Mn: 12.5%) has been obtained (Fig. 23) by plotting numerous point-by-point determinations (56). Curves of equal manganese concentration, different from one another by 0.5% Mn are drawn. This is a good example of what can be done automatically (57) by means of a scanning technique; although it should be noted that obtaining the same accuracy would require approximately the same bombardment time as point by point determination. 3. Superficial Layers. The best way of studying superficial layers con-

381

ELECTRON PROBE MICROANALYSIS

Distence h m center of e glehbr putleh (mlcmm)

FIO.22. Distribution of phosphorus acrosa a globular particle (Thomas conversion of steel) (by courteey of J. Philibert and Mrs. H. Bizouard) (69).

loop M~(o,~)

From 11.5 To 12

From 12 TO 12.5

From 12.5 TO I3

From 13 13.5

To

From 13.5 TO 14

,,4

FIO.23. Map showing segregation of manganese in Hadfield steel (by courtesy of J. Philibert and Mrs. H. Bizouard) (68).

sists in operating on a polished section of the sample previously imbedded in copper or any convenient metal by plating. If waxes or plastics are used, care must be taken to eliminate charging up effects and coating with a conducting layer is highly recommended. In any case, a polishing technique must be used which results in a flat surface, for level differences

382

RAYMOND CASTAING

a t interfaces may introduce serious errors (absorption correction needs an exact knowledge of surface orientation). Many kinds of layers have been investigated already, including corrosion layers as well as plating or chemical deposits. As an example, it was possible to show that in the new process of “sulfinization,” used to improve

A 85

k

8

0

0

1 20

I

I

I

I

80

penetmtion ( microns)

k

80

m

I

I 60

I

40

I

I

I

m

by a large amount the friction properties of mild steel, sulfur remains in the top thick superficial layer where it is located quite heterogeneously (58). 4. Diflusion. Drawing intermetallic diffusion curves was previously quite tedious work involving the preparation of a lot of samples; electron probe microanalysis needs only one sample and gives quickly and accurately

ELECTRON PROBE MICROANALYSIS

383

thc wholo diflusioii curve without, ally preliminary dibratioii. Ail advantage of using this technique is that a special curve is obtained for each componeiit ( 2 ) .The method has heen widely applied for determining diffusion constants, for studying the effect of ordering processes on the mngnitude of mutual diffusion, for studying peculiarities in diffusion processes during the formation of substructure in single crystals and transfer processes between d i d and liquid media (59). Recently, a lot of work has been done on the diffusion of refractory metals such as titanium, molybdenum, and zirconium in uranium; the laws of diffusion have to be known quite well for the purpose of building nuclear reactors (60);variations of the diffusion coefficient and of the heat of activation have been determined, in the y body-centered solution, directly from the diffusion curves obtained by spot analysis. More precise equilibrium diagrams can be drawn from accurate measurements of the concentrations close to the interfaces of neighboring phases (55, 61). Quite recently, accurate spot analyses have made evident an important phenomenon : the influence of external pressure on stoichiometric deviations in the intermetallic compounds prepared by diffusion. Heating a copper-uranium diffusion couple to 700°C for 48 hr gives rise to the upper diffusion curve shown in Fig. 24; the atomic concentration of copper is changed by 3% when going through the diffusion zone, which corresponds to a continuous change from UCu4.7 (uranium side) to UCu6.26 (copper side). The concentration gradient disappears when the diffusion couple is prepared under an external pressure of 500 kg/cm2; the concentration is then perfectly constant throughout the diffusion zone and corresponds to the UCu6 stoichiometric composition (Fig. 24, lower curve) (62).

B. Mineralogy In two main branches of mineralogy and petrography, electron probe microanalysis has already enabled workers to elucidate special questions concerning origin, development, and evolution of ore deposits. The first of these is the analysis in situ of ores and minerals in polished samples of stones coming directly from the pit or the sounding. We may cite here the extensive study of a complex copper-iron sulfide containing arsenic and tin; limits of concentration for the rather rare “orange bornite” have been given (25). Spot analysis has been applied to samples of polyphased ores containing elements whose determination by conventional methods would have been very difficult or even impossible. Maps of germanium and gadolinium concentrations have been drawn on polished sections of Tsumeb (Rhodesia) copper ore. It was also possible to determine the scandium and ytterbium content in Thortveitite (complex silicate), and to distinguish between

384

RAYMOND CASTAING

microinclusions (10-20 p ) of Xerotime (yttrium phosphate) and Zircon (zirconium and hafnium silicate) by measuring separately Y, Zr, and Hf (63).

A study of the French oolitic iron ore known as minette lorraine was taken into account in the planning of a new process of enriching the ore before melting in the high furnace (64). The second area covers the analysis of dusts of various origins (volcanic, cosmic, precipitation, sedimentation, or built by microorganisms as in the sea plankton). Magnetic spherules, about 50 p in diameter, collected in the Pacific red clays showed two types of structure; some of them had a metallic core (30 p ) surrounded by a layer looking like oxides; the others were monophased (oxides). The analysis of many particles of both types supported the theory of their cosmic origin and permitted a study of their further transformation among the deep sea sediments (66). Scanning analysis has been used for determining the composition of fine exsolution intergrowths in natural minerals (66).

C. Technical Studies A great help has been given to radio and TV tube builders by electron probe microanalysis of cathodes, filaments, grids, etc. The size of the analyzed region and the accuracy of the analysis are the main factors in the success of such studies of very small technical parts. REFERENCES 1. Castsing, R., and Guinier, A., Proe. 1st Idern. Con$ on Electron Microscopy, Delft, 1949, pp. 60-63 (1950);Caataing, R.,and Guinier, A,, Congr. intern. de Microacopie Electronique, Paria, 1960 p. 391 (1952). 2. Castaing, R., Thesis, University of Paris (1951),publ. O.N.E.R.A. No. 55. 3. Castaing, R., Recherche d r m u t i . 83, 41 (1951);Natl. Bur. StclndaraP (U.8.)Circ. 627, 305-309 (1954); Castaing, R., and Guinier, A., Anal. Chem. 26, 724 (1953). 4. Moseley, H.,Phil. Mag. [4]26, 1024 (1913);27, 703 (1914). 6. Dauvillier, A., Compt. rend. Acad. Sci. 174, 1347 (1922);Urbain, G., ibid. p. 1349. 6. Coster, D.,and von Hevesy, G., Nalurwissaschaften 11, 133 (1923). 7. Caetaing, R.,and Descamps, J., J . phya. radium 16, 304 (1955). 8. Williams, E.J., Proc. Roy. SOC.(London) 8130, 326 (1932) (read mg/cm* instead of gm/cm*!). 9. Compton, A. H., and Allison, S. K., “X-Rays in Theory and Experiment,” 2nd ed., p. 76. Van Nostrand, New York, 1935 (the numerical coefficient is false by a factor 1,000). 10. Bricka, M., and Bruck, H., Ann. rudidkdricitb [3]14, 339 (1948). 11. Mulvey, T.,Mem. sci. Rev. m&. 66 (2), 163 (1959). 12. Birks, L.S., and Brooks, E. J., Rev. Sci. Instr. 28, 709 (1957). 13. Marton, L.,and Simpson, J. A,, Electron Probe Microanalyser Conference, Washington, D. C. (Feb. 1958),report by Birka, L. S. (U. S. Naval %search Lab.). 14. Langmuir, D. B., Proc. I.R.E. 26, 977 (1937).

ELECTRON PROBE MICROANALYSIS

385

16. Haine, M. E., and Einstein, P. A., Brit. J. Appl. Phys. 3, 40 (1952). 16. Castaing, R., and Descamps, J., Compt. rend. Acad. Sci. 238, 1506 (1954). 17. Castaing, R., Compt. rend. Acad. Sn’. 231, 835-994 (1950). 18. Marton, L., Natl. Bur. Standards ( U . S.) Circ. 627, 265 (1954). 19. Wittry, D. B., Thesis, California Institute of Technology (1957). N. Cosslett, V. E., and Haine, M. E., Proc. Intern. Cmf. on Electron Microscopy, Londun, 1964 p, 639 (1954).

21. Dyke, W. P., Barbour, J. P., Trolan, J. K., and Martin, E. E., Bull. Am. Phys. SOC. 29, 15 (1954); Dyke, W. P., Trolan, J. K., Dolan, W. W., and Grundhauser, F. J., J. Appl. Phya. 26, 106 (1954). 26. Duncumb, P., and Melford, D. A., 2nd Intern. Symposium on X-Ray Mirroscopy and X-Ray Microanalysis, Stockholm, 1969. Paper 52. Elsevier, Amsterdam (to be published). 23. Archard, G. D., 2nd Intern. Symposium on X-Ray Microscopy and X-Ray Microanalysis, Stockholm, 1969 Paper 48. Elsevier, Amsterdam, 1959. 24. Liebmann, G., Proc. Phys. SOC.(London) 64, 972 (1951). 26. Castaing, R., and Descamps, J., Recherche adronaut. 63,41 (1958). 26. Lenard, P., “Quantitatives uber Kathodenstrahlen aller Geschwindigkeiten.” Carl Winters Universitjitsbuchhandlung Heidelberg, 1918. 27. Ennos, A. E., Brit. J. Appl. Phys. 4, 101 (1953); 6, 27 (1954). 28. Cambou, F., Dipl8me d’Etudes Sup&ieures, University of Toulouse (1955). 29. Borovsky, I. B., Collection “Metallurgical Problems” (for the 70th anniversary of Acad. I. P. Bardin). Acad. Sci. U.S.S.R., 1953; Borovsky, I. B., and Win, N. P., Doklady Akad. Nauk S.S.S.R. 106, (4), 655 (1956). 30. Fisher, R. M., Denver Research Symposium (1956). 31. Birks, L. S., and Brooks, E. J., Electron Probe Microanalyzer Conference, Washington, D. C. (Feb. 1958), report by Birks, L. S. (U. S. Naval Research Lab.). 32. Mulvey, T., and Campbell, A, J., Brit. J. Appl. Phys. 9, 406 (1958). 38. Dolby, R. M., Proc. Phys. SOC.(London) 79, 81 (1959). 34. Riggs, F. B., Electron Probe Microanalyzer Conference, Washington, D. C. (Feb. 195&),report by Birks, L. S. (U, S. Naval Research Lab.). 36. Dolby, R. M., and Cosslett, V. E., 2nd Intern. Symposium on X-Ray Microscopy and X-Ray Microanalysis, Stockholm, 1969 Paper 51. Elsevier, Amsterdam (to be published). 36. Castaing, R., Proc. Intern. Conf. on Electron Microscopg London, 1964 Paper 68 (1954). 37. Cosslett, V. E., and Duncumb, Cambridge University (1958).

38. 39. 40. 41. 46.

43. 44. 46.

P., Nature 177, 1172 (1956); Duncumb, P., Thesis,

Wittry, D. B., J. Appl. Phys. 29, 1543 (1958). Williams, E. J., Proc. Roy. Soc. (London) AlSO, 320 (1932). Terrill, H. M., Phys. Rev. 22, 106 (1923). Duncumb, P., 2nd I n t a . Symposium on X-Ray Microscopy and X-Ray Microanalysis, Stockholm, 1969 Paper 53. Elsevier, Amsterdam (to be putllished). Castaing, R., Rev. d t . 62 (9) 675 (1955); Castaing, R., Compt. rend. call. C.N.R.S. de Toulouse “Les Techniques RQentes en Microscopie Electronique et Corpusculaire ” p. 123 (1955). Blanchard, C. H., and Fano, U., Phys. Rev. 82, 767 (1951). Compton, A. H., and Allison, S. K., “X-Raye in Theory and Experiment:” (a) p. 105; (b) p. 71; (c) p. 99; (d)p. 79. Castaing, R., and Descsmps, J., C O T Prend. ~ . Acad. Sn’. 297, 1220 (1953).

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46. Webster, D., Proc. Nall. Acad. Sci. U . S. 14, 330 (1928). 47. Descamps, J., and Philibert, J. Proc. G.A.M.S. Meeting on Non-destructiveAnalytical Methods, Paris, 1957, p. 275, (1958). 48. Weill, Mrs. A. R., Rev. mdt. 66 (4), 371 (1959). 49. Bizouard, Mrs. H., and Philibert, J., Rev. mkt. 66 (7), 123 (1959). 50. Pomey, G., Mem. sci. Rev. mdt. 66 (5), 471 (1959). 51. Plateau, J., Henry, G., and Philibert, J., Compt. rend. Acad. Sci. 246, 2753 (1958). 62. Philibert, J., and Birouard, Mrs. H., Mem. sci. Rev. mdt. 66 (2), 187 (1959). 53. Crussard, C., Kohn, A., de Beaulieu, C., and Philibert, J., Rev. m6t 66 (4) 395 (1959). 54. Collettc, G., Crussard, C., Kohn, A., Plateau, J., Pomey, G., and Weisr, M., Rev. m6t. 64, 433 (1957). 65. Austin, A. E., Richard, N. A., and Schwartz, C. M., 8nd Intern. Svmposium on X-Ray Microscopy and X-Ray Micrwnalysis, Stockholm, 1959 Paper 58. Elsevier, Ameter-

dam (to be published).

56. Philibert, J., and de Beaulieu, C., Rev. d t . 66 (Z),171 (1959). 67. Melford, D. A., and Duncumb, P., Metullurgia 67, 159 (1958); Melford, D. A., 2nd, Intern. Symposium m X-Ray Microscopy and X-Ray Microanalysis, Stockholm, 1959 Paper 59. Elsevier, Amsterdam (to be published). 58. Pons, M., Discussion of Philibert-Bizouard paper (62). 59. Borovsky, I. B., 2nd Intern. Symposium on X-Ray Microscopy and X-Ray Microanalysis, Stockholm, 1969 Paper 49. Elsevier, Amsterdam, 1959. 60. Adda, Y., and Philibert, J., Proc. h d U.N. Intern. Conf. on Peaceful Uses of Atomic Energy, Geneva 1958. Vol. 6, pp. 72-90 (1968). 61. Philibert, J., and Adda, Y., Compt. rend. Acad. Sci. 246, 2507 (1957). 6.92. Adda, Y., Beyeler, M., and Kirianenko, A., Compt. rend. Acad. Sci. 260, 115 (1960). 83. Guillemin, C. and Capitant, M., unpublished; will be presented at 9lst Intern. Congr. f o r Geology, Copenhagen (1960). 64. Castaing, R., Philibert, J , and Crussard, C., J . Metals 9, 389 (1957); Castaing, R., Philibert, J., and Cruseard, C., Trans. A.I.M.E. 209, 5 (1957). 65. Castaing, R., and Fredriksson, K., Geochim. et C o s m him . Acta 14, 114 (1958). 66. Agrell, S. O., and Long, J. V. P., 8nd. Intern. Symposium on X-Ray Microscopy and X-Ray Microanalysis, Stockholm, 1959 Paper 57. Elsevier, Amsterdam (to be

published).

Television Camera Tubes : A Research Review PAUL K. WEIMER RCA Laboratories, Princeton, New Jersey Page I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 . . . . . . . . . . . . . . . . . 389 11. Television Pickup with Nonstorage Devices. 111. The Concept of Storage.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 A. The Iconoscope ........................................... 390 B. The Orthicon. . . ........................................... 392 IV. The Image Orthicon ........................................... 3!)4 ............................................

V, VI.

VII. VIII.

IX.

X.

XI. XII.

394

B. The Two-sided Target.. . . . . . . . . . . . . . . . . . . . . . . . . . . . Camera Tubes Based on Photoconductivity, . . . . . . . . . . . . . A. The Vidicon-A General Description.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 B. Thin Film Photoconductors for the Vidicon.. . . . . . . . . . . . . . . . . . . . . . . . . 402 Electron Optical Considerations in Camera Tubes. . . . . . . . . . . A. Discharge Properties of a Low-Velocity Beam.. . . . . . . . . . . . . . . . . B. Scanning of a Low-Velocity Beam in a Uniform Magnetic Field. .................. C. Resolution of Low-Velocity Beams Image Section.. . . . . . . . . . . . . . . . 413 D. Electron Optics of the Image Orth Signal-To-Noise Considerations in Camera Tubes. . . . . . . . . . . . . . . . . . . . . . . . 414 . . . . . . . . . . . . . . . 419 Image Intensifier Camera Tubes.. . . . . . . . . . . . . . . . A. Phosphor-Photoemitter Intensifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 B. Bombardment-Induced Conductivity Target. . . . . . . . . . . . . . . . . . . . . . . . . 421 C. Secondary Emission Image Intensifiers, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 D. Solid State Image Intensifiers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4‘23 The Search for More Efficient Methods of Video Signal Generation.. . . . . . . 423 A. The Isocon Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 B. “Grid-Control” Targets. . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Flying Light Spot Scanning of a Charge Pattern.. . . . . . . . . . . . . . . . . . . . . 426 Camera Tubes for Special Applications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 A. Infrared, Ultraviolet, and X-ray Pickup. . . . . . . . . . . . . . . . . . . . . . . . . . 426 B. Camera Tubes for Color Television.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 C. Camera Tubes for Slow Scan and Storage Applic 11s. . . . . . . . . . . . . . . . 429 Fundamental Limitations on Camera Tube Performance. . . . . . . . . . . . . . . . . 430 Image Pickup Devices of the Future.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 References . . . . . . . . . . . . . . . . . . . . ............................ 436

I. INTRODUCTION Most familiar of all television cameras is the heavy tripod-mounted box with a cluster of lenses on its front now used for broadcasting. Less 387

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PAUL K. WEIMER

well-known are the television cairieras for industrial, scientific, or military applications, where unusual requirements are often made on performance or size. In each case, an optical image of the scene is focused on the lightsensitive surface of the “pickup” or “camera” tube. This tube is the counterpart of the film in the photographic camera, or of the retina in the eye. Its function is to generate a time varying electrical signal, corresponding to the light and shade in the scene, from which the original image can be reconstructed on the screen of the picture tube in the receiver. Although a sensitive camera tube has many characteristics in common with a photocell or photomultiplier tube, the step from simple light detection to image pickup has represented many man-years of research. Proposals for image pickup devices appeared soon after the discovery of the photoconductive effect in selenium in 1873. For many years the principal objective of this work was to develop tubes suitable for broadcasting, first from brightly lit studios, then from scenes with normal illumination. As the applications of television widened, camera tubes were required to satisfy entirely new specifications on performance, size, and ease of operation. Tubes for particular purposes, such as infrared imaging, color television, and extremely low illumination were also sought. A sensitivity equal to that of the human eye was no longer a sufficient objective. Some applications required tubes whose performance was much higher. In an ideal tube, the sensitivity, resolution, and contrast discrimination would be limited only by the statistical fluctuations in the number of photons comprising the image. The tube itself should introduce no limitations in the process of generating the video signal. The achievement of such ideal performance in a tube which is simple, compact, and easy to operate is an objective of much of the pickup tube research now being carried on. Although encouraging progress has been made, the demands of the space age are constantly being extended. Pickup devices of the future may even be called upon to carry out a certain amount of information processing, much as the eye does in its function as an input device to the brain. The present review discusses pickup tube design from a research point of view, indicating what has been accomplished and what remairis to be done. The primary emphasis is on the principles of operation, with no attempts to cover details of manufacturing or operating procedures which are of interest only to the specialists. Sections I1 and I11 include some brief historical development for the purpose of introducing the present day problems. The Image Orthicon, because of its widespread usage today and because its principles of operation are basic to much of the work in progress, is discussed in greater detail than other types of tubes. Recent developments are reviewed, and the performance of various tubes compared with that of an ideal pickup device.

TELEVISION CAMERA TUBES : A RESEARCH RKVIEW

389

11. TELEVISION PICEUP WITH NONSTORAGE DEVICES

A photomultiplier tube as a simple detector of low-intensity light is much to be admired by the camera tube designer. Equipped with a lightsensitive photocathode, which can yield as much as one electron for every three photons ( I ) falling on it, and a high-gain electron multiplier (2) which permits every electron to be counted, it approaches the realm of an “ideal” detector. If an engineer who had no prior knowledge of modern camera tubes were asked to design a sensitive television pickup device, it is highly likely that his first thoughts would turn to the photomultiplier. Figure 1shows a simple way of using a photomultiplier for image pickup,

FAST DECAY PHOSPHOR

W

UNMOOULATEO KINESCOPE

‘..

\

TELEVISED \

\

Fro. 1. TeleVieion pickup with a flying spot scanner (nonstorage s p t e ~ ~ ~ ) .

based on one of the earliest proposals for a television system-the so-called flying light spot method. The total illumination consists of a tiny spot of light from an unmodulated kinescope which sweeps across the scene. Since the light reflected by the scene varies with t h e , it can be picked up directly by the photomultiplier tube to yield the video signal. This system works well for slides (3) or motion picture film but is unsuitable for large area scenes lighted in a normal fashion. The flying spot scanner operates a t an extremely low average level of illumination and provides a graphic indication of the performance one might expect of a camera tube if it were a8 efficient a photon counter as the photomultiplier, The Image Dissector camera tube (4), shown in Fig. 2, is so similar in design to the photomultiplier tube that at first glance one might expect nearly “ideal” performance. Indeed, its performance is excellent except that it lacks the very important item of storage as will be shown below. Comparison with the flying spot system in Fig. 1 shows a lamp replacing the photomultiplier tube and the light paths going toward the scanned tube

390

PAUL K. WEIMER

rather than away from it. Light reflected from the scene is imaged on to the semitransparent photocathode of the Image Dissector tube. Electrons emitted in proportion to the light and shade of the scene are brought to an electron optical focus in the plane P. A small aperture A admits to the electron multiplier only those electrons emitted from a single picture element. When the scanning coils are energized the electron image is swept across the aperture so that emission from each element is utilized in turn. The output of the electron multiplier is the time varying video signal. INCANDESCENT LIGHT SOURCE

I MULTl

1'

VIDEO SIGNAL

FOCUSlNGCOlL

,DEFL€LFCZION 1

U

\

'.'.

\

Fro. 2. Television pickup with an Image Dissector camera tube (nonstorage system).

Although the Image Dissector is extremely insensitive as a pickup tube, it is still used for certain applications where high illumination levels are available and simple, reliable operation is of prime importance. Its low sensitivity comes from the fact that only a small fraction of the electrons emitted by the photocathode a t any one time are able to enter the electron multiplier. If a picture resolution of 500 X 500 lines were required, the light level on the scene would have to be 250,000 times greater than if all the emitted electrons were used. Obviously, a method of integrating the total effect of all the light falling on the photosurface is a prerequisite for high sensitivity. 111. THE CONCEPT OF STORAGE

A . The Iconoscope The importance of &orage in a camera tube was recognized very early by Zworykin, who incorporated the idea into the Iconoscope ( 5 ) . Ideally, one would like to accumulate a11 the electrons emitted by the photoeathodc of a camera tube during the period between scans, retaining all the electrons from each picture element separately until at the proper moment they would be admitt>ed into the multiplier. Unfortunately, this somewhat

TELEVISION CAMERA TUBES: A RESEARCH REVIEW

39 1

idealistic way of building a storage-type pickup tube has never hecli mxmiplished. Instead, charge storage by an elemental array of capacitors has been used. In the Iconoscope, and later in the orthicon, the photoemissive surface was formed on a thin sheet of insulator having a conductive coating called the signal plate on the reverse side (Fig. 3). Emission of electrons from the photosensitive target caused the sensitive surface to charge positively a t each point by an amount proportional to the light intensity at that point. The accumulated charge a t each element, being equal in magnitude t o the total charge of all the electrons emitted from that element, was converted into a video signal by scanning the surface with a n electron NONCONOUCTINQ PMOTOLYISSNE TAROW

WW CATHODE

YOKE

Fro. 3. An Iconoscope camera tube.

beam. As the charge a t each picture element was neutralized by the beam a current was induced in the signal plate proportional to the total charge which had accumulated a t that element since the previous scan. Although revolutionary in its concept the Iconoscope did not bring a s large an increase in sensitivity over the Image Dissector as would be expected from the ratio of television frame time to element time. Its inefficiency came about largely because the high-velocity scanning beam struck the target with sufficient energy to yield a secondary emission ratio greater than unity. The bombarded element was therefore driven positive (the same polarity as produced by the light) while a continuous rain of secondary electrons fell back on the target, charging it negatively and prematurely erasing a part of the stored charge. Moreover, a greater fraction of the redistributed secondary electrons would land in the center of the target producing nonuniform “shading” in the picture. The efficiency of eignal generation in the Iconoscope was only about 5% of what might be expected if all the emitted photoelectrons were converted into video signal. Also contributing to the low efficiency was the weak collection field for the photoelectrons which prevented the attainment of fully saturated emission.

392

PAUL K. WEIMER

The Iconoscope was displaced 20 years ago by more sensitive tubes for studio pickup but is still used in some of the older equipment for telecasting movies. The Image Iconoscope (5),which utilizes an image section in combination with a high-velocity scanning beam, is found in many European studios.

B. The Orthicon A n improved form of storage-type pickup tube which made use of a lowvelocity scanning beam was developed during the 1930's. The Orthicon (6, ?'), shown in Fig. 4, has a photosensitive target similar to that of an Iconoscope except that the conducting signal plate is made transparent, thereby avoiding the awkward geometry of the Iconoscope. As in the Iconoscope the insulated photosensitive elements on the target become charged a few volts positive in the bright portion of the picture. FOCUSING COIL

L E F L E m i O N rOuE

/

CLOSE- SPACED STAEILIZINO MESH

SW OUT !NAL

FIG.4. Diagram of an orthicon [shown here is a CPSEmitron, based on J. D. M c h , Proc. Z.E.E. 97, Part 111,377 (1950)). The low-velocity scanning beam deposits electrons on the surface of the target, driving it negative until it reaches the potential of the gun cathode when no more electrons can land. Since the number of electrons deposited by the beam on each element is equal to the total number of electrons ejected by the light prior to scanning, the current induced in the signal plate a t each instant is proportional to the total light flux falling on that element integrated over the time interval between scans. The use of a low-velocity scanning beam eliminates the spurious shading and loss of efficiency caused by the redistribution of secondary electrons in the Iconoscope. Since the electron beam is decelerated just prior to striking the target a strong electric field in front of the target draws all photoelectrons and reflected beam electrons away from the target. Each picture element can then go through a charge-discharge cycle which is nearly independent of its neighbors permitting a fixed black level to be established.

TELEVISION CAMEM TUBES: A RESEARCH REVIEW

393

The chargedischarge cycle of potential variations of the scanned surface of the target for the Iconoscope and Orthicon are compared in Fig. 5. Second-order effects such as the deflection of the low-velocity beam close to the target by the charge pattern itself will be considered in Section V1,C. The Orthicon, introduced commercially in 1940, was superseded in this country a few years later by the Image Orthicon. However, much further work on the Orthicon has been done in England where this type of tube is called the CPS-Emitron (7). The CPS-Emitron is capable of very high quality pictures and is used in some broadcasting studios in England.

A. ICONOSCOPE

B. ORTHICON

FIG.5. Chargedischarge cycle of the target surface potential of (A) an Iconoscope and (B) an Orthicon.

Recent improvements include the incorporation of a high sensitivity trialkali photosurface (7a) in the target (8) and the use of a fine screen closely spaced from the target on the scanned side to prevent unstable operation for very bright lights. (Without such a screen a sudden burst of light may charge the target so far positive that the secondary emission ratio exceeds unity and the beam drives the target to collector potential. The tube would then be inoperative until various electrode voltages were readjusted). The curves of Fig. 24 show the CPS-Emitron to be considerably less sensitive than the Image Orthicon for low scene brightnesses. However, the larger target capacitance of the CPS-Emitron permits a higher signalto-noise ratio to be obtained at high light levels than with any of the Image Orthicons now available.

394

PAUL K. WEIMER

IV. THEIMAGE ORTHICON

A . General Description The television camera tube now most widely used by the broadcasters for studio and remote pickup is the Image Orthicon ($-lla). Although first developed almost 20 years ago the Image Orthicon and tubes derived from it provide the highest sensitivities yet attained for many advanced applications. The Image Orthicon achieved its high sensitivity by the incorporation of two important features: an electron image section and a signal multiplier. Both were based on well-known principles which had been used in earlier experimental tubes of the Iconoscope type but never as effectively as in the Image Orthicon. ALIGNMENT COIL

ELECTRON IMtsGE SECTION

\

.PHOTOCATHODE I- 5 0 0 V . )

~ D E (t13OOV.)

\-\ 'GUN CATHODE (ZERO VOLTS)

/

THIN SEMICONDUCTING TARGET

\TARGET MESH(+ZV 1

FIQ.6. Diagram of an Image Orthicon.

Figure 6 shows a cross sectional diagram of the Image Orthicon. 1he light from the scene is imaged on the semitransparent conducting photocathode deposited on the inside face of the tube. The resulting photoelectrons are focused by the magnetic field and accelerated to form an electron image on the thin Eemiconducting target. An important feature of the image section is that the surface of the target may be processed to have a secondary emission ratio 6 considerably larger than unity. This permits the actual charge stored on the target to be greater than that of the emitted photoelectrons by a factor (6 - 1). A fine mesh target screen collects the secondary electrons and serves the function of a signal plate in determining the capacitance of the target. The potential on the target becomes a few volts more positive in the lighted area than in the dark area, giving the same polarity charge pattern as if the target itself had been photoemissive. A low-velocity electron beam from a gun whose cathode is held wit,hiri a few volts of the target mesh scans the reverse side of the target depositing elcrtrons in the areas corresponding to the bright, parts of the scene. Thc

TELEVISION CAMERA TUBES : A RESEARCH REVIEW

395

resistivity and thickness of the target film must be chosen so that lateral leakage will not degrade resolution while the transverse leakage through the film is adequate for the electrons deposited by the beam to neutralize the picture charge in one scanning period. The video signal could be derived from the target mesh by connecting it to a high-gain preamplifier as is done with the signal plate of the Iconoscope or Vidicon. However, operation a t low light levels is achieved by passing into an electron multiplier the modulated fraction of the beam which fails to be deposited on the target. This group of electrons constitutes a ‘‘return beam” which retraces approximately the path of the scanning beam back toward the electron gun. The return beam signal modulation is usually less than 35% and is of such a polarity that the return beam current is a maximum in the dark. Although beam noise will be present in the output signal and will provide a limitation in sensitivity at very low light levels, the multiplier is very effective in permitting operation a t signal levels considerably below the noise levels set by the camera amplifier. At higher signal levels the useful multiplier gain rapidly decreases and is given approximately by 1000/R, where R is the signal-to-noise ratio. These considerations are discussed in greater detail in Sec. VII.

B. The Two-sided Target 1. Target Design. The two-sided target has been a major fabrication problem as well as a source of some of the most desirable performance characteristics found in the Image Orthicon. The earliest attempts a t a two-sided target made use of a thin sheet of insulator in which were embedded an array of conducting plugs. Although the plug-type target may be made sufficiently fine for high-resolution pictures, the achievement of satisfactory uniformity is difficult to meet. The target now widely used in the Image Orthicon consists of a thin sheet of semiconducting glass accurately mounted close to the fine mesh target screen. The exact spacing may vary from less than 1 mil to in., depending upon the purpose for which the tube is to be used. The mesh should be of uniform transmission and adequat>elyfine not to limit the resolution desired (750 meshes per 1ine:ir inrh or more). The approximate target potentials during a chargedischarge cycle are shown in Fig. 7 . The potential of the mesh is assumed to be 2-v positive with respect to the dark potential of the glass target. After exposure to light thc secondary emission produced by photoelectrons striking the target causes the potential of the picture side to rise to screen potential. (Emission velocities a i d contact potentials are neglected here.) As showii in step I3 (Fig. 7 ) tfhc scanned side of the target is also brought very near to screen potenti:LI

396

PAUL K. WEIMER

by capacitive coupling. In step C the beam has just scanned the target, bringing the scanned side down to zero volts but leaving the picture side of the target a small fraction of a volt positive. There is now a small potential difference between the two sides of the glass which must disappear by conTARGET MESH

THIN SEMICONDUCTIW

PHOTOELECTRENS

-

LOW-VELOCITY BEAM

A. BEFORE EXPOSURE AND SCANNlNG

B. AFTER EXPOSURE AND BEFORE SCANNING

D. 1/30 SECOND AFTER A AND JUST BEFORE NEXT EXPOSURE

FIG.7. SimpIified drawing of the target potential changes during the charge-discharge cycle of an image orthicon.

duction through the glass within a television frame time for the cycle to be repeated indefinitely. If the resistance of the glass were too high the potential difference across the glass would increase with successive scans until the entire 2-v difference existed across the glass while the picture signal would fade to zero. The resistivity and thickness of the semiconductor sheet must be carefully chosen so that the conductivity through the film is adequate to avoid fading without causing loss of resolution by lateral leakage. The time constant for lateral leakage can be computed by considering a uniform sheet of semiconductor uncharged except for a circular disk of diameter d. This time constant. has the form:

TELEVISION CAMERA TUBES : A RESEARCH REVIEW

397

where d = diameter of a picture element (cm) p = volume resistivity of the glass (ohm-cm) h = thickness of the glass (cm) k' = effective dielectric constant of the glass (1 < k'

< 6).

To avoid loss of resolution by leakage the above time constant should be greater than the television scanning period. If we take d = 5 X 10-8 cm, h=5X k' = 3, we find that p should be greater than 1.4 X loEofor TL to be sec. However, if the resolution or storage time were to be greatly extended, a thinner target and/or a higher resistivity semiconductor would be advantageous. The above equation does not take into account the effects of surface conductivity of the glass which would also degrade resolution. An approximate criterion for avoiding signal fading is that the RC time constant of the semiconducting target material be less than the scanning period. Under these conditions conduction through the target should permit the positive charge on the picture side of the target to be neutralized by the negative charge deposited by the beam. This time constant is given by:

s-50

where

k = dielectric constant of the semiconductor p = resistivity (ohm-cm).

To this degree of approximation one would conclude that the resistivity 7 X 1Olo ohm-cm for 7 d to be less than see. However, a more accurate analysis of the picture fading, taking into account the target thickness shows that a higher resistivity is tolerable for very thin targets. The glass target used in the 5820 Image Orthicon is about 0.00015 in. thick and has a resistivity of approximately 2 X 10" ohm-cm a t an operating temperature of 45°C.Below the operating range of temperature fading is observed, while above it loss of resolution will occur. After several hundred hours of operation Image Orthicons using glass targets often show irreversible changes which lead to increased fading accompanied by afterimages. Although such changes are not surprising, in view of the ionic conductivity of the glass, present indications are that a proper choice of glass constituents and target fabrication may substantially increase the life of the target. Several new target materials which are electronic semiconductors are also being investigated in various laboratories. For special applications requiring exposure time and storage over a p should be less than

so

398

PAUL K. WEIMER

pcriod of t,imcgreatly exceeding the usual f&-sec television scaiiiiiiig period it is advantageous to have a target whose lateral conductivity is considerably less than its transverse conductivity. An insulating sheet filled with conducting plugs in a submicroscopic random or regular array is one approach (12) to such a target. A new Image Orthicon target has recently been described (IS)which employs a semiconducting target material which is reported to have anisotropic conductivity. 2. Target Operation. The ability of the Image Orthicon target to operate with full storage at extremely low light levels and to continue to transmit a picture without saturation when the illumination is increased by several orders of magnitude arises from a fortunate property of secondary electron redistribution on the picture side of the target. Figure 8

FIG.8. Basic light transfer characteristic curves for two Image Orthicons (small area highlights).

is a plot of maximum signal for a spot of light as a function of light intensity for two different values of target spacing. At low illumination the gain of the target is approximately equal to the secondary emission ratio minus one since nearly all secondaries are collected by the mesh. At some light level B depending on the target capacity the signal levels off as the glass approaches an equilibrium potential determined by the velocity distribution of the secondary electrons. As the light level is further increased the contrast of a half-tone picture is maintained by the negatively charging effect of secondary electrons falling back on the target. At bright illuminations the returning secondaries cause the effective storage time of the target to be reduced to less than the scanning period. A moving object which would

TELEVISION CAMERA TUBES : A RESEARCH REVIEW

399

normally appear as a blur with full storage will appear as a succession of sharp discrete images. Although the redistribution effects are useful in stabilizing the target for a wide range of light levels some spurious effects result. For example, a very bright area in the scene will have a dark halo surrounding it, caused by the secondary electrons whose energies are no more than a few volts. Higher energy secondaries on returning to the target will strike it with an increasingly higher secondary emission ratio, tending to produce a somewhat larger white halo, or even a partially focused “ghost image.” For applications which require the most accurate tonal reproduction of the scene, such as in color television, it is desirable to operate with the light level below the knee of the curve. For black and white television it is common practice to operate a t a light level several lens stops above the knee. The enhanced edges occurring a t a transition between areas of different brightness tend to sharpen resolution and give a transition effect similar to that occurring within the eye itself. For operation a t very low light levels the wide-spaced target is advantageous. Its lower capacitance gives a larger potential variation for the same charge on the target, thus yielding improved modulation and lower noise in the return beam. V. CAMERA TUBESBASEDON PHOTOCONDUCTIVITY Early workers on camera tubes and photocell$ had three major classes of photoelectric phenomena a t their disposal: photoemission, photoconductivity, and the photovoltaic effect. Although the last to be discovered and the least sensitive of the three effects (in terms of electrical current per unit, of light flux), photoemission provided the basis for all types of camera tubes commercially available up until 1950, when the Vidicon wits introduced. The potential advantages of a photoconductive camera tube in sensitivity and reduced tube complexity were recognized by many of the early workers (14). Photoconductive and photovoltaic tubes were investigated in several countries during the 1930’s. None of these experiments (15) resulted in a useful tube able to compete with the Iconoscope available a t that time. The principal effects were retention of images, spurious spots on the target, and a lack of sensitivity. Photoconductivity for pickup tubes was set aside until after the war when the great surge of interest in solid state phenomena again revived the problem. The Vidicon photoconductive camera tube (If?), resulting from the more recent work, has fulfilled its promise in respect to reduced size and simplicity of operation. Although much more sensitive than the early photoconductive tubes, the Vidicons built to date have not equalled the sensi-

400

PAUL K. WEIMER

tivity of the best photoemissive tubes for reasons to be discussed in the following sections. Research on photoconductive tubes for many different applications is still in progress.

A . The Vidicon-A General Description The most common form of Vidicon shown in Fig. 9 is a tube 1 in. in diameter by 6 in. long, consisting primarily of a gun and a target. The target is formed by depositing a thin layer of photoconductor upon the inside

FOCUSING COIL GUN,

1

i"' \ f

:TIE

LIGHT FROM SOURCE

I i-30 Y.

T

VIDEO SIGNAL

FIG.9. Disgrsm of the Vidicon photoconductive camera tube,

face of the tube which has been previously covered with a transparent conducting coating such as tin oxide. The transparent conductor serves as the signal plate and is connected to the input stage of the camera preamplifier. Figure 10 shows the target in greater detail. Considerable latitude exists in the choice of operating potentials and electrical characteristics of the target material. The commercial Vidicons ( l 7 ) ,such as the 6198 or the 7038 use a low-velocity beam of the Orthicon type. A photoconductor is chosen which has resistivity sufficiently high to give a time constant for charge leakage much greater than the television scanning period. For the conventional go-sec scan rate a resistivity of 1OI2 ohm-cm is adequate. In operation the signal plate is biased 10-40 v positive with respect to the gun cathode. The beam deposits electrons on the scanned surface charging it down to gun cathode potential. Although a considerable field is developed across the opposite faces of the photoconductor its conductivity is sufficiently low that little current flows in the dark. When a light image is focused on the target its conductivity is increased in the bright areas causing the scanned surface to rise a few volts positive during the so-sec interval between scans. The beam deposits sufficient

TELEVISION CAMERA TUBES : A RESEARCH REVIEW

40 1

electrons to return the scii1111~d surface to ground and in doing SO gei1erat.w the video signal iri the signal plate lead. It will be noted that the target is sensitive to light throughout the entire frame time permitting full storage of charge. When scanned at low velocity the charge-discharge cycle is identical to that of an Orthicon except that the positive charging effect is achieved by photoconduction through the target itself rather than by photoemission from the scanned surface. RING GLASS FACE PLATE

PHOTOCONDUCTOR

IGHT FROM SCENE

SCANNING BEAM

TRANSPARENT CONDUCTING SIGNAL PLATE t30V.

T VIDEO SIGNAL

FIG.10. Enlarged view of the target in the Vidicon.

High-velocity scanning may be accomplished by operating the signal plate a t high potential a few volts above or below the potential of the collector for secondary electrons. Although high-velocity operation is capable of somewhat higher resolution and lower lag the redistribution of the eecondary electrons tends to reduce sensitivity and picture quality unless special precautions are taken. A characteristic of the Vidicon which has affected its usefulness for some applications is the tendency for lag or smearing of moving images. This can arise from either of two causes: photoconductive lag or capacitive lag. The former refers to delay in change of conductivity of the photoconductor with changes of light level. It is a property of the photoconductive material (or its processing) and will be discussed more fully in the section on photoconductive layers. Capacitive lag arises from the inability of the beam to return the

402

PAUL K . WEIMER

scanned surface to a fixed potential in one scan. As discussed in Sec. VI,A this is most objectionable when the target potential falls in the exponential region of the discharge curve (see Fig. 11). Lag can be reduced somewhat by tolerating a fixed amount of dark current which will shift the operating region up into the linear portion of the curve. Capacitance lag will be negligible if the product of the effective beam impedance and the total target capacitance is small compared with the scanning period. A target capacitance of 1500 ppf and a beam impedance of several megohms is adequate for the usual go-sec scanning period. Care must be taken in the preparation of the photoconductive layer that the capacitance is not too large. The area, thickness, and dielectric constant of the layer are the significant parameters in determining capacitance. The Vidicon is now widely used in many applications where its smaller size, lower cost, and simplicity of operation make it preferable to an Image Orthicon. These include closed circuit installations for commercial or scient,ificpurposes as well as some broadcast applications. Its widest use in broadcasting has been for film pickup where its high signal-to-noise ratio under bright illumination is advantageous. For remote pickup its small size is an advantage, but under poor lighting conditions the picture quality is inferior to that produced by an Image Orthicon from the standpoint of lag and noise. A typical camera with an f/2 lens using a 7038 Vidicon will transmit a picture having an excellent signal-to-noise ratio with some lag a t a scene brightness of 32 ft-candles. A. D. Cope (18) has described an experimental Vidicon 45 in. in diameter whose sensitivity was several times higher than that of the standard 1-in. tube. The performance of the Vidicon is set by the characteristics of the photoconductor and the discharge capabilities (19) of the beam. If one could produce a suitable photoconductor free of lag with several times the present sensitivity, the Vidicon would stand a good chance of replacing the Image Orthicon for many broadcast purposes.

B. Thin Film Photoconductors for the Vidicon 1. Present Status. Several important considerations affect the technological difficulty of producing a satisfactory photoconductive layer for the Vidicon. First, the problem of uniformity : the photoconductor must have the same sensitivity and electrical properties over an area of approximately 1 cm2. Second, the dark resistivity of the photoconductor required for charge-storage operation must be that of a fairly good insulator (approximately lo**ohm-cm or more). This value is much higher than that usually found in the best-known photoconductive materials. Finally, for most applications, the sensitivity should cover the visible range of the spectrum. The combination of these requirements has put the Vidicon photoconductor

TELEVISION CAMERA TUBES : A RESEARCH REVIEW

403

problem in a class by itself. Unfortunately, not much direct benefit has yet been derived from the advances made in related photoconductive devices based on germanium, eilicon, or cadmium sulfide. For acceptable uniformity the photoconductive layer must have the same sensitivity and dark current over its entire area on a microscopic a s well as macroscopic basis. The thickness which is set by capacitance considerations is of the order of 3 to lop for a target area of 1 cm2. Although various methods of fabricating such layers have been examined the only method which to date has given the required uniformity is evaporation. Other well-known techniques for depoFiting large area photoconductors such as the sintering process, now used in photocells, or the plastic embedded powders (20), used for light amplifiers (21), have given coarse, grainy pictures in the 1-in. Vidicon. A layer consisting of a thin single crystal would be of interest but techniques are not yet available for producing satisfactory crystals of the required dimensions. Evaporation of compounds for optical filters is, of course, well known but the photoconductive properties of evaporated layers are a more critical function of the materials and processing than are the optical properties. In general, evaporated layers are amorphous or microcrystalline and subject to impurities or decomposition products introduced during evaporation. The control of structure and purity of such layers is at, present much poorer than the controls which can be exercised on the materials used in transistors. The first useful Vidicon targets (16) were made by evaporating the element selenium (92).The deposit is the red amorphous form having a dark resistivity of approximately 1018 ohm-cm. This is more than los times higher than that of the metallic selenium which is placed in the evaporating boat. Amorphous selenium possesses the useful property of permitting a largc fraction of the charge carriers excited in a thin layer near the positive electrode to travel through the unilluminated part of the layer to the other side. A satisfactory sensitivity is therefore obtained even when the light does not penetrate the layer. This is to be contrasted with antimony sulfide (23) and most other materials which require the light to penetrate the layer for maximum sensitivity. Although the absorption edge in amorphous selenium is around 6000 A the sensitivity is highest for blue light and extends far down into the ultraviolet. The photoconduction is primarily by holes, and sensitivities approaching unity quantum yield are obtained for blue light. Although selenium has been useful for special purposes such as ultraviolet and X-ray pickup and for storage applications, it possesses two serious disadvantages: (1) low sensitivity to red light and (2) a relatively short life at the usual camera operating temperatures. The amorphous form gradually reconverts to a metallic allotropic form, producing conductirig

404

PAUL K. WEIMER

spots on the target. Antimony sulfide proved to be a superior material in both respects and is now used in practically all commercial Vidicons. The earliest antimony sulfide targets were prepared by evaporation in a high vacuum as had been done with selenium. Antimony sulfide targets prepared in the manner were quite sensitive if made relatively thin but were subject to capacitive lag. Later S. V. Forgue and R. R. Goodrich discovered that evaporation in a system having a pressure of residual gas gave a porous form of antimony sulfide which had very desirable properties. Capacitive lag was greatly reduced because of the effective reduction in dielectric constant of the layer. The porous layer has an average density of about onetenth that of the high-vacuum deposit which approaches the denrjity of the bulk material. The spectral response which was peaked in the red for the high-vacuum layer is shifted toward the green for the porous form giving a fair approximation to the eye response. The dark resistivity of the porous layer is 10-100 times higher than that of the high-vacuum deposit, but the sensitivity for the same dark current is usually lower by a factor of 3 to 10. Higher sensitivities are now obtained in commercial tubes by adding a highvacuum layer over a porous layer, but in' general the sensitivities are somewhat less than for a high-vacuum deposited layer. The electrical characteristics of a Vidicon target are strongly dependent upon the potential applied to the signal plate. As the field across the photoconductor is increased both the dark current and sensitivity increase. A larger dark current will tend to reduce capacitive lag, but a practical upper limit on dark current is set by the beam discharge capabilities and target uniformity. Photoconductive lag will become progressively worse as the light level is lowered. It is somewhat awkward to quote the sensitivity of antimony sulfide in microamperes per lumen as has been done for photoemitters or photoconductors in which the response is directly proportional to the light intensity. In antimony sulfide the response varies as the (light intensity)? where y is about 0.7. A value of gamma less than unity is preferable to a linear response because it extends the effective dynamic range of the tube and tends to compensate for the high effective gamma in the picture tube. In general, one may say that a sensitive porous antimony sulfide target operated under low light conditions a t high target voltages will have a sensitivity of several hundred microamperesper lumen or more. This corresponds to an average charge flow of about one-tenth of an electron per light quantum. A high-vacuum deposited layer such as used by Cope in a >$-in. Vidicon (18) can approach unity quantum efficiency but excessive lag is often a problem with such materials in a larger target. Other photoconductive materials which have been tested in experimental Vidicons include AszSep,ZnSe, and PbO. The latter material has

TELEVISION CAMERA TUBES : A RESEARCH REVIEW

405

been investigated thoroughly by the Philips Research Laboratories (24) in Holland who report a sensitivity of 100 to 200 pa/lumen, fast response, negligible dark current, and a linear light transfer characteristic. 2. The Search for More Sensitive Vidicon Photoconductors. Although the presently attained photoconductor sensitivities exceed the photocathode sensitivity in the Image Orthicon, the threshold illumination for a conventional Vidicon is several orders of magnitude higher than required for the most sensitive photoemissive tubes. This is due primarily to the absence of an electron multiplier in the Vidicon, thup requiring a relatively high signal level to exceed the amplifier noise. While it is a simple matter to build an Image Orthicon-type multiplier into a Vidicon (229, both photoconductive and capacitive lag become quite objectionable a t very low light levels with the usual photoconductors. A multiplier Vidicon with a sensitive, low-capacitance target should rival the Image Orthicon in low light sensitivity. For many applications it is desirable to achieve very high sensitivity in a tube with the compactness and simplicity of the present Vidicons. A photoconductive target in which 10-100 charges traverse the layer for each photon absorbed, with a photoconductive lag less than 450 sec, and a capacitance adequately low to be discharged in one scan is desired. Several investigators (25, 26) have recently inquired whether such performance is theoretically possible in a photoconductor whose resistivity is sufficiently high to permit the normal charge storage operation. Agreement has been reached that for many if not most materials the photoconductive gain in the conventional Vidicon should theoretically be limited to unity. Rose and Lampert (2'7) have derived the following expression for the gainbandwidth product which applies quite generally to photoconductors having ohmic contacts and operated a t voltages below dielectric breakdown : 1 G . -1 = --; To

Trel

(3)

where

G

=

TO

=

photoconductive gain (charges transported/photon) response time rre1 = dieIectric relaxation time M = factor dependent on the trap distribution.

For most materials one would expect the maximum value of M to be unity, so we write G = T O / T ~ ~InI . the Vidicon T ~must ~ I equal or exceed the Wo-sec scanning period for full charge storage. Since TO should be kept less than sec to avoid lag, G under these conditions cannot exceed unity. For a

so

406

PAUL K. WEIMER

very lnggy photot:onductor where T~ is greater than the scmiiiiig period, C can be greater than unity. A vast, amount of experimental work on the Vidicon target can be cited ill support of the maximum value of M in Eq. (3) being approximately I , unity. The value of M calculated from the measured values of TO^ T ~ ~and G occasionally exceeds unity but is usually lower. While such a limitation, if true, might prevent the simple Vidicon without a multiplier from achieving the ultimate in sensitivity, it still permits an exceedingly useful tube which could be operated a t much lower light levels than the Vidicons now available. Several possibilities remain for improving the photoconductive performance of the Vidicon target. Rose and Lampert (27,28) have shown that M can be much greater than unity in a material in which the density of the recombination states near the Fermi level exceeds the density of trapping states. Although photoconductive lag a t low light levels is, in general, caused by trapping effects the above analysis suggests that improved performance can better be attained by effecting a proper distribution of traps than by simply trying to reduce their density. Alternatively, a target gain greater than unity might be obtained by operation a t such high fields that internal multiplication by avalanche effects can occur. Uniformity would be a problem here, of course. Multiple layer junction-type targets incorporating transistor action could also yield higher gain. Finally, if we specify blocking contacts instead of ohmic contacts for the photoconductor a gainbandwidth product larger than permitted by Eq. (13) is possible even though the gain could not be greater than unity. In this case a faster response might be obtained by the use of higher fields without actually going to the breakdown condition. An alternative approach to the Vidicon photoconductor may be found in more complex target structures (29) or in the use of methods of storage alternative to that of simple electrostatic charge storage. Such methods have been tested in the laboratory but have not as yet yielded results sufficiently good for commercialisation.

VI. ELECTRON OPTICALCONSIDERATIONS IN CAMERA TUBES A . Discharge Properties of a Low-Velocity Beam At a bombarding voltage of less than 5 v the secondary emission ratio (or more properly the reflection coefficient) of most materials is less than unity. An insulated target will therefore be driven negative toward a stable potential near that of the gun cathode. Although it is customary in a general description of pickup tube operation to assume that the beam drives the target to a fixed potential with each scan this is true only to the first

TELEVISION CAMERA TUBES : A RESEARCH REVIEW

407

approximation. In practice, a sudden change in light level may require several scanning periods before the target potential before and after scanning becomes stabilized a t a new set of values. This results in lag or smearing for moving objects, an effect wbich is more or less objectionable depending on the type of tube and its application. The discharge capability of the low-velocity beam is determined by the spread in axial velocities of the electrons and the reflection coefficient (30) of the target surface for low-energy electrons. The latter may vary from 0.2 to 0.8 depending upon the processing of the surface, but has been assumed to be relatively constant for electrons having energies less than a volt or two (31).

FIG.11. Typical beam acceptance curve for current deposited on a metal plate at low voltages.

Figure 11 shows a typical plot of beam current deposited on a metal target plate as a function of target voltage (32).The potential V is measured relative to the gun cathode after correcting for contact potential differences. This curve can be divided into three regions. For V < 0, I versus V has the form expected from a Maxwellian velocity distribution;

where T is the temperature of the cathode and 1 0 is a c.onstant. For V > 0, the curve is approximately linear in the region B. Msltzer (33) haz showti that this portion of the curve should have the form:

408

PAUL K. WEIMER

At V = 0, I = IO and dI/dV = l ~ e / k Tindicating , that the curve and its slope are continuous. For higher target voltages, in region C the curve is no longer linear owing to the combined effects of saturation of beam current and variation of reflection coefficient of the target with voltage. Experimentally, if one plots log I versus V , a straight line is obtained in the exponential region for beam currents less than amp with a slope (34) which is proper for a cathode temperature of 1100'K. At higher currents the slope corresponds to that expected for higher cathode temperatures. The failure of the beam to maintain the velocity distribution set by the cathode temperature a t higher currents and voltages has been variously attributed to focusing effects in the gun, patch effects on the cathode, or space charge oscillation with the beam. Failure of the beam to approach the target in a direction normal to the surface will also result in an effective increase in the velocity spread. The consequence of the velocity spread in the beam is that its effectiveness for discharge decreases as the target is driven negative. During the fraction of a microsecond that the beam is on each element, it may not have sufficient time to reduce the element potential to a fixed equilibrium value. Although the target in the dark would eventually go a volt or more negative with respect to the gun cathode, the presence of ions in the tube or other dark currents will prevent this potential from being reached under normal conditions. From the standpoint of minimizing discharge lag it is desirable to provide a fixed dark current so that the average operating potential will tend to remain in the linear portion of the discharge curve. In this region, where I is approximately proportional to 8,the reciprocal of the slope of the curve will yield an effective beam resistance Rb. It can be shown that to avoid discharge lag the product RbC should be much less than T,, where C is the total capacity of the target and T, is the television scanning period.

B. Scanning of a Low-Velocity Beam in a Uniform Magnetic Field Since a low-velocity beam should approach the target surface normally for most effective discharge, early attempts at low-velocity scanning with electrostatic guns used a concave target to obtain uniform beam landing over the target areas. Later, normal incidence was achieved with flat targets in the Orthicon, the Image Orthicon, and the Vidicon by immersing the entire tube in a uniform magnetic field. This type of focus is particularly advantageous for pickup tubes although for certain specific applications other electron optical arrangements may be preferable. To a first approximation the beam follows the magnetic lines which are bent, as shown in Fig. 12, in the region where the transverse scanning field is superimposed on the focusing field. A closer examination of the beam

409

TELEVISION CAMERA TUBES : A RESEARCH REVIEW DECELERAnN RING(ZER))

\

1

FOCUSlNG~COlL

I

I

\

\ WALL COATINCr(lS0VJ

/ I I

I

THIN G L S S ~ R G E T SCREEN TARGET

(ZERO)

PHOTOCATHODE

FIG.12. Magnetic and electrostatic fields in an Image Orthicon.

shows that individual electrons are following helical paths with a radius given by

R (cm)

fi

= 3.34--.

H ’

where V t is the transverse energy in volts and H is the magnetic field in oersteds. The cyclotron period T of each electron is a function of the magnetic field alone: T (sec) = 3.56 Hx 107 sec. (7)

A divergent electron spray emerging from a defining aperture at the gun will be focused at successive nodal points between the gun and target. The distance between the nodal points is given by

Y< H ’

X (em) = 21.2-*

(8)

where V is the voltage on the wall of the tube relative to the gun cathode and H is the magnetic field. Optimum focus at the target occurs when an integral number of loops lies between the defining aperture of the gun and the target. The Image Orthicon is usually operated with four or five loops but the Vidicon has but one. An advantage of uniform magnetic field focusing is that the return beam very nearly retraces its path through the deflection field and can thus be easily intercepted by the first stage of the electron multiplier. However, for reasons described below, the return beam usually retains some residual scan, thus imposing some severe restrictions on the uniformity of the first stage dynode. The return beam is very nearly in focus on this surface since the dynode itself contains the defining aperture for the gun.

41 0

PAUL K. WEIMER

The problem of uiiiformity of landing of the beam (35) on the target is somewhat more severe in the Image Orthicon than in the Orthicon or Vidicon. Any unbalanced transverse force on the beam which would tend to give it an excess helical energy a t the expense of its longitudinal energy would require the target to rise a few volts above cathode potential in order for the beam to land. While the Orthicon target surface could automatically charge a few volts positive to “meet the beam” this was not permitted in the Image Orthicon where the target surface potential was set by the collection of secondary electrons on the target mesh. Failure of the beam to land on the outer portions of the target produces a “porthole” effect on the transmitted picture. Two radially symmetric sources of helical motion giving rise to the “porthole” effect are: (1) the electrostatic lens effect of the decelerating field in front of the target, (2) the curvature of the magnetic lines within the deflection coil. An eIectron approaching the periphery of the target will experience ;I radial component of force while passing through the decelerating field. (See Fig. 12). In addition to inducing helical motion a t the expense of longitudinal energy the electrostatic field will cause a lateral displacement of the beam in such a direction as to tend to rotate the entire scanning raster. Further displacement occurs on the return from the target, resulting in the residual scan of the return beam at the multiplier. A fine screen a t wall potential will eliminate the radial component of the deceleration field and greatly reduce the amount of residual scan of the return beam. Curvature of the magnetic lines within the deflection yoke will have an effect on the beam similar to the deceleration lens. Helical motion of several volts energy may be acquired a t the expense of the longitudinal energy. Sharper bends in the field and higher wall voltages will result in larger effects. The various sources of helical motion can be balanced against one another to give a minimum transverse motion at the target. In the Image Orthicon, that produced by the deflection yoke can be approximately neutralized by the electrostatic deceleration field. Similar principles were applied recently (36) in a Vidicon camera where tthe flare of the magnetic field over the target was used to cancel the effect of the deflection coil.

C. Resolution of Low-Velocity Beams Since the total beam current required to discharge a target is set by signal-to-noise consideration, the ultimate resolution capability of the beam is strongly dependent on the current density available for discharge. D. B. Langmuir (37) has shown from fundamental considerations that the maximum current density at a target whose potential is the same as the

TELEVISION C.\N ER.4 TUBES: A RESEARCH REVIEW

41 1

cathode is the cathode emission density itself. One can therefore compute that a cathode emission of 200 ma/cm2 could provide a beam with a spot diameter of less than in. for a beam current of lo-* amp. This would yield a resolution of greater than 10,000 lines/in., far rixceeding that of the present day camera tubes. Typical pickup tube guns consist of a flat oxide-coated cathode disk with a control grid aperture and an accelerating aperture mounted between the cathode and the defining aperture. The gun is normally positioned just outside the windings of the magnetic focusing coil so that the field strength is somewhat less than a t the tsarget.A cone of electrons emerging from the defining aperture will be focused a t nodal points whose current density is a maximum in the center and diminishes to zero a t the periphery. An electron optical crossover occurs within the gun itself suggesting that one may actually be imaging the crossover rather than the defining aperture. The velocity spread in the beam will contribute to reduced resolution as well as to reduced effectiveness for discharge. I n the guns now in use the velocity spread for beams exceeding lop8amp is considerably higher than would be expected from the thermal velocities a t the cathode. 15lectrons emerging from a point source with an average energy I' and spread of 6 volts randomly oriented in a uniform magnetic field will form n focused image disk whose maximum diameter is given by (38)

for an image a t potential l',

(I (cm) = 2L2

(+)

for :UI image a t cathode poteiitial where LI is the distance (cm) from the point source to the image disk and Lo is the length of the decelerating regioii. It is noted that these expressions are indepeiident of the niagrietic field strength. ]?quation (9A) indicates the degradation in resolution within the scanning section of the tube. Equation (9B) applies to the decelerating region. It is apparent that the greatest deterioration of spot size due to chromatic aberration occurs in the decelerating field region unless :I very strong electric field exists in front of the target. Two additional effects occur in the iieighborhood of the target which will affect the resolution of tt low-velocity scanning beam. 'Hie potential pattern itself can bend the path of the beam so that, a strong white area will appear larger than i t actually is. Alternatively, the coplanar grid effect of an extended black area will screen the beam from reaching a small weakly

412

PAUL K. WEIMER

charged gray area which it surrounds. Both of these effects can be reduced by the use of a strong electric field on the scanned side of the target. A fine mesh screen operated a t wall potential is a simple way of achieving such a field. Although the above discussion would lead one to expect the resolution of a beam scanning a target near zero potential to be inferior to that obtained if the target potential were raised, there are other considerations which are favorable to low-velocity scanning. The low-potential target may actually provide a “self sharpening” effect 011 the beam in cases where the peripheral electrons possess large transveree eiiergies which do not have sufficient longitudinal energy to reach the target. Furthermore, the lowvelocity scanning beam is free of redistribution effects from secondary electrons which call degrado resolution.

I

-

D _.

’I

0.20

0

LOWVELOCITY V,..-.IV. .-I-. 0.lkA BEAM N 680 LINEWINCH 400

I

I \ I 1

800 ICOO IS00 2000 MOO TELEVISION LINES PER INCH ON TARGET

2 00

Fxa. 13. Sine-wave response curves for a typical Image Orthicon gun operated at “high” and “low” vclocity. These curves were calculated from the measured square-wave peak responses obtained by scanning a metal monoscope pattern and may indicate somewhat higher resolution than would be found in scanning a charge patterri.

Since the current density in any electron beam decreases froin t,lic center to the edge in a somewhat indefinite fashion, it is not customary to quote the spot size as a measure of resolution. A more meaningful evaluation can be obtained by measuring the response of a series of black and white lines as il function of fineness of the pattern. In Fig. 13 is plotted the responseversus-line number curve of a conventional Image Orthicon gun, obtained (39) by scanning a series of metal nionoscope h e patterns a t both low and high velocity. Resolution observed in this manner is probably somewhat higher than would be obtained by scanning an equivalent stored charge

413

TELEVISION CAMERA TUBEb : .4 RESEARCH REVIEW

pattern. The “siiie-wave response” plotted here was computed from the square-wave response actually measured. The sine-wave response is the response which would be observed by scanning a test pattern whose lines varied sinusoidally in density. The equivalent pass band N , for the lowand high-velocity beams are 680 and 1040 lines/inch. The quantity N , was calculated by integrating the square of the sine-wave response curve and taking the equivalent rectangle. The N , concept, introduced by 0. H. Schatle (40), is esccrdingly useful where a iiiimber of imaging processes take place in cascade. The equivalent pass band resolution for the entire system can be closely approximnted by the inclividual N , values of the (*omponent.s as follows:

D. Electron Optics of the Image Orthicon Image SectiorL ‘Lo :I first approximation the photoelectrons follow the magnetic lilies from the photocathode to the target giving a slightly reduced image due to the flare of the field. Single loop focus is ordinarily used. Equation (9B) indicates that resolution should vary inversely as the photoelectron emission energy and directly as the potential difference between the photocathode and target. In practice, an image section resolution considerably higher than suggested by (9B) can be obtained by adjusting H and V for optimum focus of the photoelectrons having the most probable emiseioii energies. Figure 14 gives some computed sine-wave respoiise curves for an

0

TELEVISION LINES PER INCH ON PHOTOCATHODE

FIG.14. Compiited sine-wave response curveB for the image section of the 1-e Orthicon.

414

P.4UL K. WEIMER

image section, bawd oil measurements in a special tube (3.9)having a slotted metal plate in place of the target. The equivalent pass band N , for 300 v and 500 v between photocathode and target are 7’70 and 1250 lines/in. respectively, The values of N , which are quoted here indicate electron optical capabilities and not necessarily the resolutioit which is actually obtained in tubes now available.

VII. SIGNAL-TO-NOISE CONSIDEIL~TIONS IS C . ~ I L I L‘YUBE~ I In view of the complex requirements of storage and generation of video signal it is not surprising that present day camera tubes are somewhat inferior as light detectors to simple photomultiplier c.clls. However, to achieve the ultimate in sensitivity the camera tube should be capable of recording every photon comprising the image and converting this information into a video signal with the least possible degradation in signal-tonoise ratio. It is obvious that the tubes should employ a primary photoprocess whose quantum efficiency is high. The present section will discuss some of the design factors in pickup tubes which affect the signal-to-noise ratio of the video signal. It is interesting to note that the most sensitive pickup tubes available today are based on photoemiseion even though photoconductors may have a higher primary quantum yield. A total photocathode current I , produced by a steady uniform illumination of the entire photocathode with the peak picture brightness will yield a peak video signal I , which equals I , in an ideal tube, assuming full storage with no loss or gain of electrons a t the target. The inherent root-meansquare shot noise I, associated with the video signal will be given by I,,

=

Mu*)%

‘11)

where e

=

electronic charge (1.59 X

lOI9

coul/clectroii)

B = video bandwidth (cps) I n and I , are measured in amperes. Figure 15 shows a plot of the signal current and its associated noise for various values of photocurrent in an ideal tube. The separation of the two curves indicates the maximum signal-to-noise ratio R obtainable for any value of photocurrent. Expressed in terms of I,, the ratio of peak-to-peak signal to root-mean-square noise is given by:

The photocathode current I , is directly related to the input light flux and t,he photocathode sensitivity in microamperes per lumen.

TELEVISION CAMERA TUBES : .4 HESEARCH REYIEIY

415

FIG. 15. Signal and noise currents in an ideal tribe for various values of photocathode current. The dotted lines A and B represent the noise levels in the darker parts of the scene for an Image Orthicon and an Imagc Isocon, respectively, when the beam is pamp. (See Bec. IX,A.) adjusted to diacharge a high light signal of 1.4 X

In tubes having 110 niultiplier, such as the Orthicon, CPS Emitroii, or Vidicon the video signal derived from t.he signal plate must compete with the noise spontaneously generated by the input stage of the video preamplifier. The preamplifier noise can be expressed approximately in terms of 311 equivalent root-mean-square noise current a t its input as follows (41) :

whew

U

=

k T

=

video frequency pass bmid Boltzman constant (1.38 X joules/"K) = absolute temperature (taken as 300°K) C = capacitance of the input lead to ground 12, = resistance of the input lead to ground K t = equivalent noise resistance of the amplifier input tube.

If we take R, = 100 ohms, R, = 50,000 ohms, C = 20 ppf, we find the input noise for a bandwidth of 4 Mc to be 1.5 X lop9amp. Assuming negligible noise contribution from the subsequent stages, the signal level must be 1.5 X lo-' amp to obtain a signal-to-noise ratio of 100 a t the output of the amplifier. At very low light levels with such a tube the video signal is

416

PAUL K. WEIMER

lost in the noise even though the inherent signal-to-noise ratio a t the target is quite high. The two general methods of enhancing pickup tube sensitivity are both used in the Image Orthicon : electron image intensification and video signal multiplication. The secondary emission gain a t the target can raise the signal level by a factor of ten or more. An electron multiplier for the return beam will easily provide a signal gain of several hundred to a thousand. By either method the tube output can be made large enough that amplifier noise is not a limitation. It is apparent by inspection of Fig. 15 that the amount of signal gain or image section gain which can be considered useful for improving sensitivity increases from no more than unity at high sigiial levels t o about one thousand a t threshold levels. dlthough adequate signal multiplication may r i s e the signal level far above amplifier noise this does not guarantee that the resultant signal-tonoise ratio in the output, will equal that of the ideal tube plotted in Fig. 15. Both the beam and the target may introduce additional noise. A basic problem in deriving the video signal from the total return beam is that its amplitude is a maximum in the dark and usually has a poor percentage modulation even in the brightest areas. While there is some hope for increasing the modulation factor by reducing the velocity spread in the beam and the electron reflection coefficient of the target, the inverted signal polarity is inherent hi the normal Image Orthicon method of discharge. In order to keep the noise associated with the unused portion of the beam at a minimum the beam current must be carefully adjusted to be just sufficient to discharge the brightest parts of the picture. If, however, t,he beam is greatly reduced for operation a t the threshold of iIlumination. failure to discharge will occur when the light level is increased uiiless the beam is readjusted. The highest beam modulation and over-all best results are obtaiiied in the Image Orthicoii when the potential swing of the target is approximately 2 v. In order to achieve maximum sensitivity and freedom from lag the target capacitance should be kept small. However, for full storage operation a small target caparitance sets an upper limit on the signal level and the maximurn signal-to-noise ratio obtainable. The signal current I, a t the target is related to the target capacitance CT, the potential swing A I - . the scanning period T F by the expression

I, = C T -AV ' TF

(54)

The optimum capacitance may range from less than 50 ppf for a wide-spaced Image Orthicon designed for low light level operation (where I , = 5 X 10-9 amp, AV = 2 v) up to 3000 ppf for a tube without a multiplier (where

TELEVISION CAMHRA TUBES :

.4 R E S E A R C H REVIEW

417

f. = 2 X lo-’ amp, AI- = 2 v). 1 1 1 some applications the capacitaiicc might have to be even larger if the scmning period or the signal current must be unusually large. Image intensification, or gniii prior to storage call be used (ither independently or iu combination (vith a video signal multiplier. I n the latter case, where practically unlimited gaiii is available in the signal multiplier, the image section gain is useful at, low light levels to iiicrease the potential swing of the target,, thereby improving the percentage modulation. An extremely sensitive pickup tube which has recently been reported, the so-called “Image Intensifier Orthicon” (,$3),i w Sec. ~ Vl11,A) has used profitably a multistage image section gaiii of several hundred, in addition to the usual video signal multiplier. Image Orthicons now widely used obtain a gain of 3 to 5 from the secondary emission on the photocathode side of the glass target. Recently Hannam ( I S ) has reported a new Image Orthicon target which has a gain of 10 to 15 at low light levels and is capable of storage for periods as long as 5 or 10 min. The noise characteristics of the Image Orthicon have bee11 coniputed analytically by Ramberg (43) itnd others (44). Although a completely rigorous treatment becomes quite lengthy the following simplified analysis (46) is sufficiently accurate for most purposes. The peak video signal in the return beam is

s

= (6 -

I)!fl,;

(15)

where 6 = secondary emission ratio of photocathode side of target

T

I,

= =

transmission of the target screen total photocathode current if uniformly illuminated a t peak picture brightness.

The total mean square noise

.2‘ro?

=

B m e

iii

the retiirn beam is given by:

T(6 - 1)?2eT,R

T + T62eZPR + ; (6 - 1)1,2eB:

(16)

bandwidth high light modulation of return beam = electronic charge. = =

The first term in Eq. (16) is the Contribution of the photoelectron shot uoise. The second term is the secondary emission noise generated a t the target by the photoelectrons. The third term is the beam shot noise. The error introdured by neglecting the noise contribution of the signal multi-

418

PAUL K . WEIMER

plicr is less than 15% for a gain per stage of four. The final signal-to-uoise ratio, obtained by dividing peak-to-peak signal by the root-mean-square noise is 2cz2

i.

T(6 - 1) ,

1

d"ts-l+G

,

1'

Combiniiig with Fq. (12), we may writ,e (17) in the form

wherc

T(6 - 1) F=Jc

1

6+s_l+m

1)

may be considered as an efficiency factor, indicating the degree to which ttic Jmage Orthicon would approach the signal-to-noise ratio of an ideal pickup tube wit.h the same photocathode sensitivity. Figure 16 is a plot of F for

TARGET SECONDARY EMISSION FACTOR FOR PHOTOELECfRO&S :

8

FIG.16. Efficiency factor F for computing the high light signal-to-noise ratio (S/b-)Io in the Image Orthicon as a function of target gain and the beam modulat.ion vt. = Fa R, where R is the signal-to-noise ratio for an ideal tube.

various values of ni and 6. These curves show the advantage to be gained by increasing na above the rather low values usually observed in actual tubes (0.1 < m < 0.4). In so far as F can be maintained constant for various light levels the high light SIN ratio can be said to vary approxi-

m:itelg iib tlie quare root of the illumination level. For it particular scent’ with a fixed beam current the noise level is about the same or slightly lower i n the dark part,s of the scene. The signal-to-noise ratio of a Vidicon without a multiplier is determined primarily by the amplifier noise given by Eq. (13). We may write the eign:il-to-aoise ratio for the V i d i m ~ns

‘I’hus the high light ( S I N ) varies directly as the signal current, which ill turn varies as the 0.7 power of thc scene illumination for the porous nntimony sulfide photoconductor. +An accurate expression for t.he sigiial-to-noise ratio of a Vidicon having :t multiplier woiild require a detailed knowledge of t,he photoconductive process. I‘or :III ideal phot,oconduct,or which has a primary quaiitrim efficictivy o f iitiity the signal-to-uoise ratio would have the form

where m is the beam modulation. The relatively high capacity of the conventional Vidicon target restricts nL t,o very small values a t low light levels.

A . Phosphor-l’hotoeN1itter Intensifier Of the various possible ways of providing a gain of several hundred prior to storage the so-called “Image Intensifier Orthicon” is a t present the furthest, developed (42). Figure 17 shows a cross-sectional drawing of such a tube. Idectrons from the primary photocathode are focused by means of a n electrostatic lens onto the first intensifier screen. This screen consists of a glass membrane coated on the image side with an aluminum-backed layer of fluorescent material and 011 the other side with a photocnthodc whose spectral response matches that of the phosphor. ‘The incoming pllotoelectrons penetrate the aluminum film producing fluorescencc in the phosphor which in turn excites photoeniissioii from the sensitized ,side. With an accelerating \voltage of 10,000 v between the cathode and the iiitensifier screen each primary electron causes ail emission of 10 to 20 electrons. Electrons from the first intensifier screen may be accelerated and focused to strike a second intensifier screen with a similar gain. Electrons from the final intensifier are focused on to a two-sided target of the Image Orthicon type. A low-velocity beam scans the target and the modulated return beam hignal is passed into an electron multiplier as in the Image Orthicon.

INTENSIFIER

PMOTO-EMISSIVE COATING

THIN GLASS MEMWANE

PHOTO-EMISSIVE

ORTHICON

THIN MICA

PHOTO-EM1551VE

THIN GLASS TARGET

ELECTRON GUN

I

OPTKAL IMAGE

FMMATION

DEO SIGNAL OUT LFINE

PnosPnoR SCREEN ALUMINUM FILM

MESH SCREEN

MULTIPLIER

PHOSPHOR SCREEN ALUMINUM

FILM

0

I

2

3

4

5

c

.

INCHES

FIO.17. A two-stage Image Intensifier Orthicon [G. A. hlorton and J. E. Ruedy, Advances in Electronics and Electron Plqs. 12, 183 (1960)l.

TELEVISION CAMERA TUBES : A RESEARCH REVIEW

12 1

A total gain of three or four hundred prior to scanning is sufficient to raise the stored signal level to a point where the fundamental noise from the primary photocathode can exceed the beam noise. Although such a picture obtained a t very low light levels is inherently noisy and limited in resolution (see Sec. XI), the Image Intensifier Orthicon approaches the ultimate in performance under these conditions. With adequate optics thc Image Intensifier Orthicon will transmit a picture outdoors at night by starlight illumination alone. At high light levels the signal-to-noise ratio would be comparable to an Image Orthicon having the same target capacitance, but the resolution would bc detcoriated by the successive imaging processes.

Lz. IZombal.dn2ent-Induccd Conductivity Target The change in conductivity of a thin film of insulator when bombarded by electrons with an energy of several tens of kilovolts has already been utilized in certain types of storage tubes (46). More recently Decker and Schneeberger (47) have described an experimental pickup tube in which this method is employed to obtain useful target gains of the order of 500. The tube shown in Fig. 18 might be described a s a Vidicon with an image

-

PHOTOEMISSIVE

BOMBARDMENT CONDUCTING TARGET

-n

-2OKV

4 4

a

VIDEO SIGNAL OUTPUT

V.tS0V

FIG.18. A camera tube with a target employing bombardment-iiiduced conductivitj 6, Part 3, 15(i (1957)l.

IR.W. Decker and R. J. Schneeberger, Z.R.E. Natl. Convention Record

section. Electroils from the photocathode are focused electrostatically 011 to a thin target with an energy of about 20 kv. The target consists essentially of a thin insulating layer backed by a conducting aluminum film which is sufficiently thin to be readily penetrated by the high-velocity electrons. Since the metallic layer is biased several tens of volts positive with respect to the gun cathode, a lorn-velocity beam scanning the exposed surface of the insulator will produce a high field across the layer. The chargedischarge cycle is similar to that of the Vidicon except, that the conductiv-

422

PAUL I(. WEIMER

ity pattern is produced by an electron image rather than the light. ‘ l h video signal may be derived from the metallic signal plate or from the return beam. The perforniance of the experimental tube described above indicates an ultimate sensitivity comparable to the Image Orthicon. With a higher gain target and a return beam multiplier the sensitivity of this tube could probably be increased to reach the limits imposed by the photocathode. The relatively high capacity of the thin dielectric layer permits a high signal-to-noise ratio a t large signal levels but may cause poor modulation and capacitance lag at very low levels. The target thickness cannot be made arbitrarily large since in most materials, the bombarding electrons must penetrate the layer to obtain maximum gain. The possibility for a slow transier;t response of t,he conductivity change slid deterioration of the target with life also present some problems. Early experimental tubes of this type were limited in resolutioii by I screen which supported the thin insulating target. Higher resolution should be possible using a self-supporting target, or one supported by a continuous thin film such as aluminum oxide.

C . Secondary Emission Image Intensifiers Image intensification by secondary emission offers a possibility for a simpler, more compact ultrasensitive tube which does not require such high voltages for operation. One approach has been the use of thin film secondary emitters in which the secondary electrons are ejected from the side opposite the incoming electron stream. Sternglass (48) formed thin composite films consisting of KCl, thin gold, and SiO, which were supported by a nickel mesh. These films gave a gain of more than four per stage a t accelerating voltages of 3.5 kv per stage. The high field between stages tends to prevent spreading of the image, but some loss of resolution is inevitable because of the energy spread of the secondary electrons. .\nother type of secondary emission intensifier coiisists of a series of perforated dynodes which channel the electrons from each element independently to provide gain and at the same time retain the image. Since the secondary electrons are drawn from the surface on which the primary electrons impinge, the applied voltage need be no more than a few hundred volts per stage and the entire photocathode-dynode-target assembly can be made very compact. Work on the design of suitable dynodes has been carried on in this country (49) and in England (50). A major problem is that of achieving suitable mechanical precision in structures which are fine enough for high-resolution pictures. Uniformity of gain is also a problem. If these difficulties can be solved, this type of structure should be extremely valuable for image tubes as well as for ultrasensitive pickup tubes.

TELEVISIOK CAMERA TUBES: A RESEARCH REVIEW

D. Solid

423

State Image Intensifiers

The development of thin “light amplifier” panels (21) in which ail optical image can be intensified by a factor of a hundred or more suggests their possible utility for an ultrasensitive pickup tube. For example, the target of an image Intensifier Vidicon might be composed of a triple layer consisting of a photoconductor, an electroluminescent layer, and another photoconductor. The charge stored on the second photoconductor should be considerably enhanced by virtue of the light gain in the first photoconductor-electroluminescent layer combination. Although attractive in principle the materials now available are not suitable for a significant improvement in sensitivity. Rose and Bube (50a) have shown that the operation of light amplifiers in the extreme low light range requires a new highefficiency low-voltage electroluminescent process and/or photoconductors with “M values” (see Sec. V.B) significantly greater than unity. An entirely new design or panel would be required in order to achieve adequate resolution in a small-size target. Other forms of solid state iiiteiisifier panels, such as ail array of transistor elements are, of course, formally possible. While such an approach could coiiceivably lead to an ultrasensitire tube comparable in size and simplicity of operation to the presrut Vidivon such a dcidopment mould be a major undertaking.

Ix.

‘L’Hl: 8E.UICH FOR AfOl
OF

I-IDEO SIGSAL(;ESER.~TION

‘rhe iiiverted signal polarity and poor percentage modulation of the return beam in the Image Orthicon are serious disadvantages for obtaining optimum signal-to-noise ratio at low light levels. While the noise introduced by the scanning process would be negligible in comparison to t,he inherent signal noise if unlimited image intensification gain were available, such gain is difficult to achieve. This section will describe several basically different methods of generating a video signal from a small stored charge which nffer the possibility for reduced noise along with other useful fentiires.

,4. The Isocoii Scan About 10 years ago a new form of experimental camera tube based oil the scattering of low-velocity electrons was described (51). This tube was called the Image Isocon. The recent interest in achieving the ultimate in performance of pickup tubes has prompted further investigation of this type of tube (52). Although a low-velocity beam similar to that of the Image Orthicon is used, only the fraction of the returning electrons which have been scattered by the target (see Fig. 19) are admitted into the multi-

424

PAUL K. WELAXER

plier. Since the maximum number of the electrons are scattered froni the most positive (or lighted) areas of the target and none from the dark areas the video signal has the desired polarity and can approach 100% modulation. An interesting feature here is that the signal polarity inversion is obtained without interfering with the normal charge pattern polarity. Furthermore, by treatment of the target surface to give a reflection coeficient greater than 0.50 one can achieve n signal gain which becomes arhit,rarily high ns the coefficient approaches unit,y. FOCUSING COIL SCANNING YOKE

I THE A X I S 0

PARATION EDGE SCATTERED ELECTRONS SHOWN DOTTED PRIMARY BEAM AND REFLECTED ELECTRONS SOLID

A X I A L VIEW OF APERTURES

FIG. 19. Diagram of the Image Isocon, a low noise camera tube utilizing electrons srnttered from the target.

A disadvantage of the Isocoii is that the electronuptical conditions for separating tthe diffusely scabtered electrons from those specularly reflected from the target are somewhat critical. In the form of Isocon shown in Fig. 19 the separatioii is varried out, at the edge of an aperture placed a t an antinode in the return beam. Since the return beam must be completely in)mobilized at the separation edge, a fine screen on the scanned side of the target is used t o provide a uniform decelerating field at the target. Care must be taken to avoid loss of resolution and spurious signals due to beam bendiiig or excessive helical motion of the beam near the target. At. high light levels the maximum signal-to-noise ratio obtaiiied with the Isocon is not markedly different from that of an Image Orthicon. In the low light parts of the picture, however, the signal-to-noise ratio observed is several times higher than that produced by a n Image Orthicon beam scanning t>hesitme charge pattern. The dotted lilies in Fig. 15 show the nat,ure of the reduction in noise which has been obtained so far in the IsocoiI. With a return beam modidation of 80 to 90% the principle advantage of the

-

Isocoii is in the nedud mise aiid extmded dymmie range im rtdse low E&k, parts of the AWmgh modulation riiS Itheorebidlily posse, spurious scahtering of ebetrons from oitlwm- s u r f a w m h as decelemih screen have $0 far p w m t e d the m5pkte redueticm of the mi&e levd itm that of the p h o h i m t m n noise s t o d on the t a m However, the Isoawn scan even in its present &ate of dmebpment -cantly k e e n s tihe amount of image section gain r e q u i d for an ultrmemitive tube. Used inn combination with a moderately high+in target it pmvides an attractive (but not yet thoroughly explored) a p p m e h to the pmblem of television pickrip a t low light levels.

lowa

B . “Grid-Contral” Turgets 111 a11 tlic commercial types of pickup tubes the total video signal generated at the target is equal to OT less than the stored charge. Various pickup tubes have been proposed in which a video signal gain is accomplished in the discharge process. In these tubes the charge pattern may control the motion of a fraction of the beam electrons which themselves play no part in discharging the target. If the fraction so controlled is larger than thefraction reqiiirerl to discharge the target, gain results. An example of this THIN SEMCONOUCTOR

J

: ELECTRONS FROM PHOTDCATHOOE

LOW-VELOCITY

*

SZ

S,(MSCHARGE MESW

(COLLECTOR MESH)

FIG.20. A grid-control target. type of video signal gaiii has already been mentioned in the discussioii of the isocon where it was pointed out that the scattered fraction of the beam may exceed the stored charge. Another example is the so-called “grid-control” target. Figurc 20 shows u two-sided target in which the landllig of the lowvelocity beam on the mesh SIis controlled by the coplanar grid action of the potential pattern on the thin semiconductor. If the collector mesh S2 is biased slightly negative with respect to the gun cathode so that the beam cnniiot discharge the target ats all, the effective gain beromes very great.

426

PAUL K . WEIMER

Under these coiiditions the beam can scan the mesh Sl repeatedly, continuing to generate video signals for many seconds or minutes until the charge on the semiconductor target disappears because of target leakage. A targct similar to this was used in the Metrechon half-tone storage tube (53). Redington (26) has computed that a Vidicoii target incorporating a coplanar collector for grid action is capable of a gain of 250. An alternative type of video signal modulation offering the gain of thc grid-control target without the need for complex target structures can be demonstrated with the Isocon tube shown in Fig. 19. If the target mesh is biased a few volts negative with respect to the gun cathode the beam will turn back before reaching the target but may be laterally deflected by the charge pattern (51).By positioning the reflected beam on the separatioii edge of the multiplier aperture a ‘‘differentiated” form of video signal is obtained. The resolution and sensitivity are high, butn spurious “edge effects” in the transmitted picture are objectionable if accurate picture reproduction is required. Here also the charge pattern may hc srannrtl repeatedly with only a very slow decay in signal level.

C. Plying Light Spot Scanning of a Charge Patterrr The patent literature contains many proposals for pickup tubes ill which the electron beam is replaced by a spot of light which scans a photosensitive target. In one scheme which has recentiy been investigated (64) the target is so constructed that the photoelectron current excited by the spot of light carries the video signal which is subsequently amplified by an electron multiplier. To maintain conservation of charge on the target the incoming photoelectrons from the photocathode must charge the target negatively nther thaii positively as is the case for all commercial tubes. Although this scheme offers the apparent possibility of improved low light performance because of the favorable polarity of the output signal the coiwtrwtion of a satisfactory target is likely to be very difficiilt.

S.CAMERA TUBESFOR SPECIAL APPLICATIOXS A . Iiifrared, Ultraviolet, a i d X-ray Pickup The spectral respoiise of both the photoemissive and photocoiiductive type of tube can be extended to wavelengths beyond the visible range with moderate modifi~ations.

Injrared Pickup The Image Orthicons used for broadcast purposes arc not seiieitive at wavelenghts greater t,han 0.7-0.811. Special tubes have been made using an S1surfaw (Cs-CsO-Ag) which has its maximum sensitivity hrtmcen

TELEVISION CAMERA TUBES: A RESEARCH REVIEW

427

0.8 and 0.9~ and some sensitivity extending out to 1.25~.For longer wavelengths it is necessary to go to a photoconductive-type tube. Infrared-sensitive Vidicons (66) have been built using a lead sulfidelead oxide surface whose response peaks in the near infrared and extends out to about 2 . 1 ~ The . sensitivity of these tubes is sufficient to image by their own radiation objects which are at 150°C. Ultraviolet Pickup The glass face plate of the standard tubes limits the ultraviolet response to wavelengths above 3500 A (0.35~).Experimental Image Orthicons which are sensitive to slightly below 2500 A have been built for use in ultraviolet television (@) microscopes. Experimental Vidicons with an ultraviolet transmitting face plate and an amorphous selenium photoconductor are sensitive to below 2500 A. X-ray Pickup Television pickup of an X-ray image can be accomplished by direct excitation of a special pickup tube with X-rays, or by the use of an image converter to produce a light image which is subsequently focused on a conventional pickup tube. The converter might simply be a fluorescent screen or it may include an image intensifier tube (56) or panel (57). Experimental Vidicons for direct excitation with X-rays have been reported. Cope and Rose (58) have described a l-in. Vidicon having a thin face plate and an amorphous selenium layer about 2 5 thick ~ which is capable of relatively high resolution over a small area. Each X-ray photon absorbed in the selenium layer was found to produce 500 carriers on the average. This is sufficient to permit direct observation in the transmitted picture of the photon noise in the X-ray image. Jacobs and Berger (69) have reported the development of a large area X-ray sensitive Vidicon having a 12-in. screen. A lead monoxide layer approximately 150p thick yields somewhat higher sensitivity than the thinner selenium layers but the larger area screen requires a thicker face plate with greater transmission loss. B. Camera Tubes for Color Television Color television cameras now widely used for broadcasting require three separate camera tubes to supply the simultaneous primary color information transmitted. The light from the scene is split by means of dichroic mirrors to form separate red, green, and blue images on the sensitive surfaces of the three tubes. Optical and electrical registry of the images is maintained. The pickup tubes themselves are conventional although in

'be(Etesigned h r tqdmnurn perfmaurae in this apfic&ion. C d m cameras are ,mailable u + g either Emage ( O ~ i e o nor s Vidiams. Many proposah h m been ma& fur (color Ipidhqp tdbes which mauld supllr;43r khee sim-aus sig& hm a single ~omllmredimage. WlMb 1 1 1 ~ tube h s 'been d d q p d to gjim performance aqua1 :to the thne-e;huhe

same umrm may

target b k n camma%,a tricolor ViiIicon (60)bmimg alnmultiple~elmtnrokte inv ed at RCA U b o r a t o r k s m d has been &awn Ito possem m e desimbk features. The %axgetinnqpmtes ?red, g m m , mil blue coboar fHkr strips which atre reg-& with h e iutehcking sets af semitrampanat condueking signal strip (see Fig. 21).The signal strip cmesponding itm a

FIQ.21. A target incorporating color filters and multiple electrode signal leads for use in an experimental tricolor Vidicon. The view of the target is from the electron gun.

given color are insulated from their neighbors but interconnected by means of bus-bars to a common output terminal for that color. The photoconductive layer which covers the target structure is scanned with a conventional low-velocity beam generating the three primary color signals simultaneously in the separate output leads. An advantage of the multiple-electrode design over other possible color tubes is that no special requirements are made on the beam with respect to focus or scanning accurwy. The camera can be made as compact and almost as simple to operate as a black and white camera. Registry of the three signals is inherent in the design.

TELEVISION CAMERA TUBES: A RESEARCH REVIEW

429

The relatively high capacitance between the three sets of signal strips requires video preamplifiers with a low input impedance for separable color signals (61). The amplifier noise is several times larger than if no capacitive coupling existed between the three channels. Where adequate light is available, satisfactory signal-to-noise ratios and color separation has been obtained, but, in general, the sensitivity is somewhat less than the equivalent three-tube camera. The construction of a satisfactory color target with sufficient fineness and uniformity for high quality pictures is a difficult problem. The target shown in Fig. 21 was made in the laboratory by evaporation techniques with 290 strips for each color covering an area suitable for the standard l-in.Vidicon. The signal strips are 0.0005 in. wide with a 0.0002-in. space between. Crossover insulators for connections to the signal strips are provided by the filter strips themselves which were of the multilayer interference type. Fine grill masks were used in evaporating the strip patterns. Many experimental targets have been made which were entirely free of disconnected strips or shorts. A more serious fabrication problem which has not been solved is the achievement of identical electrical and geometrical properties from one strip to the next. Tricolor Vidicons produced in the laboratory have yielded pictures, which, although not of broadcast quality, may be considered potentially useful for many applications. The finite number of strips gives rise to some color moire effects in vertical bar patterns but this was not found to be objectionable in actual scenes. Poor uniformity and lack of ample sensitivity are the most serious limitations at present.

C. Camera Tubes for slow Scan and Storage Applications In some military and industrial applications it is necessary to operate pickup tubes at scanning rates which are quite different from the normal commercial broadcasting standards. Slower scanning, for example, permits decreased transmission band pass and increased sensitivity. A target capable of storage over a period of many seconds or minutes will permit the use of time exposures to obtain pictures of very dim stationary scenes which would be impossible to pick up directly with the standard ?&-see storage period. Since the sensitivity of photoelectric surfaces may exceed that of photographic film, pickup tubes used in this manner offer a significant sensitivity increase over film for astronomical purposes. The final record of the picture is made by photographing the kinescope. Both the Image Orthicon and Vidicon are, in principle, capable of long storage times, although the tubes being sold for standard scan rates may not be the most suitable for these applications. In the Image Orthicon lateral leakage in the target must be minimized for the best performance

430

PAUL K. WEIMER

a t slow scan rates. The new Image Orthicon target described by Hannam (13)is reported to be capable of storage periods up to several minutes. For extended storage times in the Vidicon the resistivity of the photoconductor should be as high as possible since the effect of the dark current increases linearly with the scanning period. Shelton and Stewart (62)have reported on the performance of various commercial Image Orthicons a t scanning rates down to one frame per second and Vidicons down to one-twentieth frame per second. From simple considerations it can be shown that for an Image Orthicon with constant target charge the signal-to-noise ratio should remain approximately constant as the frame scan rate changes, provided, of course, that the bandwidth is changed proportionately. The light level required for a given signalto-noise ratio will decrease proportionately as the scan rate is reduced. In the Vidicon, under conditions where the noise level is set by the amplifier, the signal-to-noise ratio for constant target charge varies inversely as the square root of the scan rate. The light level required will decrease as the scan rate is reduced but will also be affected by the gamma of the photoconductor and by the extent to which the target voltage must be lowered to minimize dark current. Numerous experimental and developmental pickup tubes have been described which were designed specifically for storage applications. These include both photoemissive (63) and photoconductive (64) tubes. In cases where electrostatic charge storage is used it is sometimes necessary to increase the target capacity to many times that required for normal television standards. This is required, for example, if the total number of picture elements is to be greatly increased, or if a charge pattern is to be wanned many times without renewal of charge. Recently, Nicholson (65) has described a new photoconductive-type storage tube which employs a hystereEis effect in the photoconductor to permit repeated scanning of the target giving a continuous television picture for many minutes after the optical picture has been removed. XI. FUNDAMENTAL LIMITATIONS ON CAMERA TUBEPERFORMANCE

It has been clearly established that the performance of an ideal image pickup device would be limited by the statistical fluctuations in the number of photons comprising the image. A. Rose (66) has shown that the following relationship exists between the threshold values of scene brightness, contrast discrimination, and resolution :

TELEVISION CAMERA TURES : A RESEARCH REVIEW

431

where U C a

k D 1

0

acelie brightness (fooL-lambcrts) percent contact (C = ( A B / B ) X 100) = minimum resolvable angle (min) = threshold signal-to-noise ratio = diameter of the lens aperture (in.) = storage time (550sec for commercial TV) = quantum yield of the primary photo process. =

=

This expression is obtained by considering the accuracy to which one can count the total number of photons absorbed by an elemental area of the photosensitive image surface in a time t. Since the absorption of the photons is a random process, the average number N absorbed per element will have associated with its fluctuations whose root-mean-square value is N f i . The difference in intensity of two elements A and B can be distinguished pro, k, the threshold signal-to-noise vided N A - N B 2 k d m B where ratio, has a value ranging between 1 and 5. One lumen of white light is assumed to be equivalent to 1.3 X 1OI6 quanta per second. The quantum yield 0 can also be considered as an absorption coefficient which was restricted to those quanta giving rise to countable events. Two important conclusions can be drawn from Eq. (21).First, it provides a very simple relationship between the threshold values of scene brightness, contrast, and angular resolution which applies to any image pickup device. The statement that BC2aZequals a constant expresses quantitatively the common observation that objects of low contrast or small detail inherently require higher scene brightness to be seen. Second, it provides a direct means for evaluating the performance of image devices. Considered in its most general form the quantum yield 8 becomes a figure of merit which can be applied to all types of imaging devices. This interpretation of tl is discussed in more detail below. For a convenient evaluation of the performance capabilities of pickup tubes as a function of photocathode illumination, Eq. (21) may be rewritten in the following form : EC2 1.03k2* 10-" -12 Ate where

E = photocathode illumination [ft-candles (or lumens/ft)] I = limiting resolution of the transmitted picture in television lines A = picture area of the photocathode (sq. ft) C , k, t, 0 are as defined for Eq. (21).

432

PAUL K. WEIMER

Some degree of approximation is involved here since Eq. (21) was derived for elemental areas taken entirely at random instead of in a line pattern. The elemental area was assumed to be a square whose side h is given by hZ = A/+Z2. Morton and Ruedy (42) have plotted Eq. (22) for an ideal tube having a photosurface with a quantum yield of 0.1 and a storage time of 0.2 sec for k = 1 and k = 5 (see Fig. 22). The contrast of the test pattern is assumed

HA$ INTEGRATED LIGHT FLUX ON PH~,TOCATHOD&LUMENS- i i ~ c . 1 I

I I

I

I

,I

I 10-0

E PHOTOCATHODE ILLUMINI\TION(LUMENS/S9.FT. 1 I I ,1111, I

10-7

10-

10-5

164

I

I

lo-' Am 1.251Nc,t~0.2SEC.

FIQ.22. Limiting resolution for an ideal camera tube as a function of the integrated light flux (EAt) on the photocathode. The scale for photocathode illumination E assumes an area of 1.25 in.' and an effective storage time of 0.2 Bec. The dotted curves are measured resolution values for a two-stage Image Intensifier Orthicon and an Image Orthicon with a high-gain target. Although the scanning period for the dotted curves was only 1/30 sec the eye can be assumed to provide an effective storage time of 0.2 sec.

to be 100%. It is noted that the maximum att,ainable resolution increases as the square root of the integrated light flux (EAt) and directly as the contrast in the image. The only way to exceed this resolution limit without increasing the storage time or the input light flux is to use a photosurface having a quantum yield exceeding 0.1. Present day photoemitters have a maximum yield greater than 0.3 at the spectral peak, while some photoconductors have a primary yield approaching unity. The dotted lines on Fig. 22 give the measured values of resolution obtained with laboratory samples of the Image Intensifier Orthicon and a "wide-spaced" Image Orthicon. Although the Image Intensifier Orthicon approaches the theoretical curves over a limited range at low light levels, all pickup tubes fall short of the maximum resolution at higher light levels largely because of electron optical, or target structure limitations. The

48%

Tl&ES'ISI(XK (FA;MERA TUBES: A RESEAWH REVIEW

justification for cornpazing cexperimiitd daka taken ati the norm1 televit &on scanning rate of ?& yec with theoretical curveaplothd for t = 0.2 see is based on the assumptiam that the observer performan equivalent storage process in viewing the n&y picture on the television screen. I t is expected( of course, that photon-lbihed pictures at such low light levels are inherently much more noisy %ham the usual television p i c t u r s used in broadcasting Equation (22) may & be used to caileulate the signal-to-wise riati@ R of an ideal tube as a function of the Iight falling on the phofocathode. Replacing k (the threshold signal-to-noise ratio9 by R we have after rearranging,

where

R = ratio of the peak-to-peak signal to rms noise (EAt) = integrated light flux (lumen-sec) falling on photocathodb during exposure time t I = television line number corresponding to the resolution of the picture to be transmitted 0 = quantum yield of the photocathode.

It is noted that in the ideal tube R is independent of the bandwidth of BhR video amplifier but is inversely proportional to I , the resolution to be trainsmitted. In Fig. 23, R is plotted as a function of the integrated E&t ffux for

lo- "

10-0

10-1

lo-

10"

'

EA L I ~ H TFLUX ON PHOTOCA~HODE[LUME'NS) t z+o&c, 109

lo- a

i0-7

10'

Id'

10-4

6 '

FIQ.23. Maximum signal-to-noise ratio R for an ideal camera tube as a function of total light flux (EAt) on the photocathode for different values of resolution.

434

PAUL K. WEIMER

scveral cliffereiit hie 11~unl~c~rs. l'hesc- ( : I I ~ V A Rshow f h price which must bc paid in light flux for high signal-to-noise ratio and increased resolution. Figure 24 gives the expected sigikal-to-noise ratios for a number of different types of pickup tubes based partly 011 analysis and partly on measurement, assuming in each case the most sensitive photosurface now available.

LIGHT FLUX ON PHOTOSURFACE €.A (LUMENS)

FIQ.24. Calculated signal-to-noise ratios for various type8 of camera tubee, assuming in each cam the most sensitive photosurface now available. For these curves the scanning period waa taken to be 1/30 sec, although in practice the 0.2-532storage time of the eye would increase the effective signal-to-noise ratios to somewhat larger values. Curve 1: Image Intensifier Orthicon (S = 150 pa/lumen) Curve 2: Wide-spaced Image Orthicon with high-gain target (S = 150 pa/lumen, B = 15, m = 0.25) Curve 3: Close-spaced Image Orthicon with high-gain target (S = 150 pa/lumen, 8 = 15,m = 0.25) Curve 4: Developmental Vidicon (RCA C74008) with nonporous photoconductor Curve 5: Experimental Vidicon with a porous photoconductor Curve 6:CPS Emitron [S = 75&a/lumen,based on data from D. J. Gibbons, Advances in Electronics and Electron Phys. 12,203 (1960). Curve 7: Iconoscope (S = 75 pa/lumen, based on measurements on existing tubes having a sensitivity of approximately 7.5 &lumen) Curve 8: Image Diseector (S = 150 pabumen)

The value of R for an ideal tube was calculated from Eq. (23), which can the expression for maxireadily be shown to reduce to R = mum signa1-to-noise ratio given by Eq. (12). For comparison of the performance of pickup tubes with other image pickup devices, such as the human eye or photographic film, Rose hae used Eq. (21) to calculate an effective quantum yield ff from the known performance specifications. The value of f? so computed for any existing pickup tube is less than the actual quantum efficiency of its photosurface. Used in this manner 6 probably represents the best single index for evaluating the relative performance of various types of image devices. A plot of f? versus

fip/m,

TELEVISION CAMERA TUBES: A RESEARCH REVIEW

435

light intensity for the above-mentioned pickup devices has been given by Rose (66) and later by Jones (67). Figure 25 is taken from Jones’s paper. He has calculated 8, which he calls the “detective quantum efficiency” from the performance data for these detectors. Most significant is the dearee

FIG.25. Detective quantum efficiency B for camera tubes, photographic negatives, and human vision @. Clark Jones, Advances in EledroniCa and Eleclrun Phys. 11,87-183 (1959)). Since the Vidicon noise level is set by the input noise of the preamplifier, it should be noted that a Vidicon with a photoconductor sensitivity only 10 times larger than the 6326 would equal the detective quantum efficiency of the 6849 Image Orthicon.

by which the peak value of 0 for photoemissive camera tubes exceeds that photographic film and the eye. It may be noted, however, that considerable uncertainty exists in the calculation of detective quantum efficiency for the eye. The value of 0 obtained by Jones is lower by a factor of ten than that found by Rose.

XII. IMAGE PICKUP DEVICES OF

THE

FUTURE

As long as camera tubes fall significantly short of the theoretically obtainable performance shown by Figs. 22-25, research and development will continue. Further improvement a t both high light levels and low light levels is required. Not shown on the curves is the need for ideal performance in simple, compact devices which can operate unattended for many years. Also not indicated are such special requirements as color pickup, slow speed scanning or pickup from radiation patterns other than light. To attain all of these objectives may require new principles of design, or the discovery of some basically new physical phenomenil. There is certainly no requirement that future pickup devices should be tubes, or even that they be used in a transmission system similar to television of today. Perhaps thc nost exciting area iiito which future work may lead is indicated by the amount of inforrnntion processing which the eye performs

in i b m l e m a major i w t alimd’tcp~thhebrain. I% is entirely possildktbt the liighly sophieticatealloomputersofi the Wure will contain imagwpiuhp deviaes whose function is far mDre complex than m y camera tube knrprvn today. REFERENCES 1. Spicer, W. E., Phys. R&. Ila,114 (Urn). 2. Zworykin, V. K., and Bmtierg, E. G., “Photoelectri~ity,~’ Chapter 8. Wiley, New

York, 1940.

3. Sziklai, G C., Ballard, R C., and $bhroeder, A. C., Pmc. Z.R.E. 36, 862 (1947). 1. Farnsworth, P.T., J. Franklin Inst. $EX$, 411 (1934). 6. Zworykin, V. K., and Morton, G. A., “Television.” Wiley, New York, 1954. 6. Rose, A.,and Jams, H.,,Proc. Z.R.E. 27, 547 (1939). 7. McGee, J. D., Proc. Z.E.E. 97, Part UT, 377 (1950). 7a. Sommer, A. H., Rev. Sci. Znstr. 26, ns(1955). 8. Gibbons, D. J., Advances in Electronics and Electron f i g s . 12, 203 (1960). 9. Rose, A.,Weimer, P. K., and Law, Et B., Pror. Z.R.E. Sn, 424 (1946). 10. Janes, R. B., Johnson, R E., and Moore, R. S., RCA Rev. 10, 191 (1949). 11. Janes, R. B., Johnson, R. E., and Handel, R. R.,RCA Rev. 10,586 (1949). l l a . Kaseman, P. W., National Electronict3 Conference (195n. 12. Ochs, S. A., I.R.E. Electron Devices Meeting, Washinghn, Oct. 1960. 19. Hannam, H.J., Ft. Belkoir Conf. on: Image Intensihafhon (October 1958). 14. Knoll, M.,and Schroteq F., Physilc Z: 98, 330 (1937). 16. Miller, H.,and Strange, J. W . , Proc. Phys. SOC.60,374 (1938). 16. Weimer, P: K., Forgue, 8. V., and Goodrich, R. R., Electronics 23, 70 (1950); RCA REV.12, 306 (1951). 17. Miller, L. D., and Vine, B. H., J . Roc. Motion P i c t m Television Engrs. 67, 149

(1958).

18. Cope, A. D., RCA Rev. Nr1,465(1956).

19. Redington, a. W., IRE Trans. on Electfon Devices 4, No. 3 (1957). M bNicoll, F. H., and Kazan, B., J . Opi, 8oc. Am. 43, 64T(1955). 81. Kazan, B.,and Nicoll, F. H., PTOC. Z.R.E. 43, 1888 (1955).

22. Weimer, P. K., and Cope; A. D., RCA Rev. 12, 314 (1959. 23. Forgue, S. V., Goodrich, R. R., and Cope, A. D., R C A Rev. 12, 335 (1951). 84. Heijne, L.,Schagen, P., and Rruining, H., Philips Teck..Rev. 16,23 (1954). 26. Rose, A.,Bdv. Phys. Acta 30, 242 (1967). 26. Redington, R.W., J . Appl Phys. 29, 189 (1958). 27. Rose, A., mdLampert, M. A.,Phys. Rev.113, 1236 (1950). 28. Rose, A., and Lampert, M. A., RCA Rev. 18,57 (L96?)): 29. Ochs, S. A., and Weimer, P. K., RCA Reu. 19,49 (1968). 30. Fowler, H.A, and Farmworth, H. E., Phys. Rev. lll, 103 (1958). $1. Gimpel, I., and Richardhn, O., Pkoc. Poy. SOC.8182, L 7 (1943). 33. Meltrer, B., and Holmes, F L., Brit. S. Appl. Phys. Q,)139 (1958). 33. Meltzer, B.,J . Elechtonica and ControZ’9, 355 (1957). 3.4. Cope, A. D., RCA Laboratories, unpublhhed. 36. Weimer, P. K., and Rose, A., PTOC.Z.B.E. 36, 1273 (1947). 38. Castleberry, J., and Vine, B. H., J . Soc. Motion Pictune Television Engrs. 68, 226 (1959). $7. Langmuir, D. B., Proc. T.R.E. 26,977 (1937).

TELEVISION CAMERA TUBES: A RESEARCH REVIEW

a.ROW, A., proc. Z.R.E. as, 311 (1940).

437

S.,RCA Laboratories, unpublished. 40. Schade, 0. H., J. SOC.Motion Picture Television Engrs. 68, 181 (1952). 41. DeVore, H.B., and Iams, H., Proc. I.R.E. 2, 369 (194C). 42. Morton, G. A., and Ruedy, J. E., Advances in Electronics and Electron Phys. 12, 183 (1960). 43. Ramberg, E. G., Z.R.E. Trans. PGME 12, 58 (1958). 44. DeHaan, E. F., Advances in Electronics and Eledron Phys. 12, 291 (1960). 46. Vine, B. H., and Borkan, H., RCA Tube Division and RCA Labs., unpublished. 46. Pensak, L., RCA Rev. 10,59 (1949). 47. Decker, R. W., and Schneeberger, R. J., I.R.E. N d l . Convention Record 6, Part 3, 156 (1957). 48. Sternglass, E. J., Rev. Sn‘. Znsk. 26, 1202 (1955). 49. Burns, J., and Neumann, M. J., Advances in Electronics and Electron Phys. 12, 97 (1960). 60. McGee, J. D., Flinn, E. A., and Evans, H. D., Advances in Electronics and Electron Phys. 12, 87 (196C.). 60a. Rose, A., and Bube, R. H., RCA Rev. 20, 648 (1959). 61. Weimer, P. K., RCA Rev. 10,366 (1949). 62. Cope, A. D., and Borkan, H., to be published. 63. Pensak, L., RCA Rev. 16, 145 (1954). 64. Ward, S. L., Ft. Belvoir Conf. on Image Intensification (October1958). 66. Morton, G.A., and Forgue, 6. V., RCA Engr. 4,52 (1959). 66. Teves, T. C., Philips Tech. Rev. 17, No. 3 (1955). 67. Kazan, B., RCA Rev. 19, 19 (1958). 68. Cope, A. D., and Rose, A., J. Appl. Phys. 26, 240 (1954). 69. Jacobs, J. E., and Berger, H., Winter Meeting, Am. Inst. Elec. Engrs., New York (February 1956). 60. Weimer, P. K., Gray, S., Beadle, C. W., Borkan, H., Ochs, S. A., and Thompson, H. C., Z.R.E. Trans. on Electron Devices July 1960. 61. Borkan, H., RCA Rev. 21, 3 (1960). 69. Shelton, C. T., and Stewart, H. W.,J. SOC. Motion Picture Television Engrs. 67, 441 (1958). 63. Forgue, S. V., RCA Rev. 8, 633 (1947). 64. Webley, R. S., Lubszynski, H. G., and Lodge, J. A., Proe. I.E.E. lOa, Pt. B, 401 (1955). 66. Nicholson, J., The “Permacbon.” Weatinghouse Tube Data Sheet, WL-7383. 66. Rose, A., Advanas in Electronics 1, 131-166 (1948). 67. Jones, R. Clark, Aabanczs in Electronics and Electron Phys. 11, 87-183 (1959). 39. Gray,

This Page Intentionally Left Blank

Author Index Numbers in parentheses are reference numbers and are included to assist in locating references when the authors’ names are not mentioned in the text. Numbers in italics refer to the page on which the reference is listed.

A Acton, E. W. V., 185(103), 313 Adams, N. L., Jr., 184(4), 250, 311, 316 Adda, Y., 383(60, 61, 621, 386 Afrosimov, V. V., 5(19), 8, 12, 24, 26, 40(4), 78, 79 Agrell, S. O., 384(66), 386 Allison, H.W., 107, 178 Allison, S.K., 2, 5, 9(1), 26(53b), 48(1), 78, 79, 357(44a, 44b, 44~1,384, 986 Anderson, J. R., 185(50), 319 Angelov, A., 185(96), 313 Ankudinov, V. A., 10(7), 36(79), 37(79), 42(79), 45(79), 53(111), 78, 80, 81 Archard, G. D., 335, 386 Ascoli, A., 156(61), 178 Asdente, M., 156(61), 178 Ash, E. A., 309(163, 164), 316 Ashkin, A., 185(79), 313 Auer, P.L., 207, 314 Austin, A. E., 380(55), 383(55), 986

B Bahadur, K., 88(16), 90, 93, 96, 114(16), 125, 148(16), 177 Bailey, T.E., 55(114), 81 Bailey, T.R., 14, 79 Baldock, R.,18, 79 Ballard, R. C., 389(3), 436 Barber, M. R., 185(35), 318 Barbour, J. P., 335(21), 386 Barnett, C.F., 9, 41(29), 53, 68,78, 81 Bartels, H., 9, 79 Bates, D.R., 27, 28, 29, 32(65), 33, 34, 40(58, 751, 41(66, 751, 48, 50, 53, 55, 79, 80, 81 Battin, R. H., 189(112), 210(112), 314 Beadle, C. W., 428(60), 4.97 Beck, A. H. W., 184(11), 197(11), 199

(111, 311

439

Becker, J. A., 91, 97, 107, 177, 178 Beckey, H. D., 99, 177 Bell, D. A., 251, 261(155), 293, 316 Bell, G.I., 69, 81 Berger, H.,427, @ .7 Berktay, H.O., 251, 261(155), 293, 316 Bernard, M. Y., 184(23), 31.9 Bethe, H.A., 85, 177 Beyeler, M., 383(62), 386 Birdsall, C. K., 185(69), 313 Birks, L. S., 326, 341(12), 342, 384, 386 Bizouard, H., (Mrs.), 380(49, 521, 381, 386 Blanchard, C. H., 357(43), 386 Blanc-Lapierre, A., 189(111), 191(111), 210(111), 314 Blauth, E., 66, 81 Bobroff, D.L., 185(87), 313 Bohm, D.,61, 81 Bohr, N., 69, 81 Borkan, H.,417(45), 423(52), 428(60), 429(61), 43’7 Borovsky, I. B., 342, 383(59) , 386,386 Boyd, J. M., 50, 80 Boyd, R. L. F., 7, 19, 61, 78, 81 Boyd, T. J. M., 49, 80 Brackrnann, R. T., 5, 63, 67, 78, 81 Bradshaw, J. A., 185(101), 913 Bransden, B. H., 29, 80 Brewer, G.R., 185(37, 44, 66, 70, 79), 312, 313 Bricka, M., 326, 330(10), 984 Brinkman, J., 172(72), 179 Brinkmann, H.C., 29, 79 Brooks, E.J., 326, 341(12), 342, SS4, 386 Brown, I. F., 185(39), 319 Bruck, H.,326, 330(10), 384 Bruining, H., 405(24), 436 Bube, R. H., 423, 43’7 Buck, D.C., 185(49), 319

440

AUTHOR INDEX

Buckingham, R. A., 44(85), 80 Bukhteev, A. M., 5(15), 9, 52, 78 Bulyginskii, D. G., 310(173), 316 Buneman, O.,185(99, 100, la), 313, $14 Burhop, E. H. S., 33, 36, 45, 51(101), 80 Burns, J., 422(49), 437 Bydin, I. F., 5, 9, 52, 53, 78, 81 Bydin, I . M., 14, 79

C Cahen, O., 185(34), 318 Cambou, F.,342, 344, 386 Campbell, A. J., 344, 386 Capitant, M.,384(63), 386 Carleton, N. P., 23, 79 Castaing, R.,317(1, 2, 31, 319(2), 320(2), 326(1, 2, 31, 330(2), 331(1, 2, 16, 171, 33307)' 334(2, 171, 336(25), 337(2), 338(2), 339(16), 341(25), 342(25), 348 (2, 36), 357(1, 421, 362(2), 364(2), 365(7), 367, 370(2, 361, 372(2), 374 (71, 375(7), 377(2), 379(2), 380(25), 383(2, 251, 384(64, 651, 384, 386, 586 Castleberry, J., 410(36), 436 Chang, K.K. N., 185(48, 511, 519 Chapman, S., 187(108), 188(108), 194, 222(108), 232(108), 314 Chen, T. C., 185(67), 313 Chen, T.S., 185(72), 313 Clarke, H. E., 23, 73 Cockburn, R.,207(136), 220(136), 314 Collette, G.,380(54), 386 Cornpaan, K.,310(169), 316 Compton, A. H., 357(44a, 44b, 444, 384, 386 Cook, E. J., 185(73), 313 Cook, J. S., 185(52, a), 311 Cooper, E. C., 111, 178 Cope, A. D., 402, 403(22, 231, 404, 405 (23), 408(34), 423(52), 427, @6, 437 Corben, H. C., 285(106), Sf4 Cosslett, V. E., 184(19), S l l , 334, 345, 349, 360(37), 386 Coster, D., 318, 384 Coulson, C. A , 59, 81 Cowling, T.G., 187(108), 188(108), 194, 222(108), 232(108), 314 Cramer, H., lSS(laS), 191(109), 2lO(l@), 314

Cremosnik, G.,185(41), 312

Crussard, C., 38063, 54), 384(64), 386 Cutler, c. C., 185(61), 310(172), 318, 316

D Dalgarno, A., 29, 32, 34, 41(66), 44(85), 50, 80 Danforth, W. E., 197(119), 310(119), 314 Dauvillier, A., 318, 384 Davies, D. H., 185(33), 318 Davisson, C., 251, 261(152), 269, 293, 316

Dear, H. D., 62, 81 de Beaulieu, C., 380(53, 561, 386 Decker, R. W., 421, 487 De Haan, E. F., 417(44), 437 de Haas, E., 23, 79 de Heer, F . J., 11, 79 Descarnps, J., 320, 331(16), 336(25), 339 (161, 341(25), 342(25), 362, 365(7), 367, 372, 374(7), 375(7), 380(25, 471, 383(25), 384, 386 De Vore, H. B., 415(41), 437 Diemer, G.,237(146), 241, 243(146), 245 (1461, 516 Dillon, J. S., 11, 35, 79 Dinnis, A. R., 185(46), 31.9 Dobretsov, D.N., 310(173), 316 Dolan, W.,84, 115, 122, 137, 174(6), 177 Dolan, W.W., 335(21), 586 Dolby, R. M.,344(33), 345, 386 Donahue, T. M., 5, 7, 78 DOW,W. G.,184(9), 197(9), 311 Drechsler, M.,90, 123, 124, 125, 132, 135 (47, 551, 148(53), 158, 164, 166, 177, 178

Dukelskii, M., 14, 79 Dukelskii, V. M., 5(19), 53, 78, 81 Duncanson, W. E., 59,81 Duncumb, P., 335, 349, 350, 355, 360 (371,380(57), 386, 386 DUM,D.A,, 185(593,319 Dustnman, S.,201, 324 W k e , W. P., 84, 115, 122, 137, 174(6), 17'7, 335(21), 386

E Edwads, H. D, 11(35), 35(3616),79 Einstein, P. A, 330,dCa6

EisePaatein, A. S, 197(117), 310(117), 314

lihekus, K. G,., e$ 81 Eimm, A. E., W.W &@&in, D. W., W17), $11 Bp&&, P. S., Znrr(IBl), 'Z@Kl&I), 314 Epsetein, B., UE4fRI); 3ZZ Eshelby, J. D., IkW66), 179 Evam, H. D., 42!%5D), 437' Evans, S., 5Z, 81 Everhart, E., 5, 66, 69, 73, 78, 81 Eyring, H., 56, 58, 82

F Fan, C. Y., 5, 26, 63, 78, 79, 81 Fan, H. Y., 309, 316 Fano, U., 357(43), 386 Farnsworth, H. E., 407(30), 436 Farnsworth, P. T., 389(4), 436 Fay, C.E., 184(25), Slb Feaster, G. R., 183(1),311 Fechner, P., 294, 295(159), 29MltW), 316 Fedorenko, N. V., 5, 8, 10, 12, 24, 26, 40 (4), 65, 78, 79, 81 Feller, W., 189(110), 210(110], 314 Ferris, W. R., 207(139), 220(139), 221 (139), 222(139), 229(1399, 244(139), 314 Field, F. H., 5, 15, 17, 56, 78 Finkelnburg, W., 86, 177 Firsov, 0. B., 35, 80 Fisher, R. M., 342, 386 Fite, W. L., 5, 34, 63, 67, 78, 80, 81 Fitzgerald, E. R., 171, 179 Flaks, I. P., 5, 10, 35, 46, 50, 78 Flinn, E. A., 422(50), 437 Fogel, I. M., 5, 10, 36(79), 37(79), 42 (79), 45, 53, 78, 80, 81 Fontana, C . M., 25(52), 55(114), X9, 81 Forgue, S. V., 399(16), 403(5), 405(23), 427(55), 430(63), 436, 437 Fortet, R., 189(111), 191(111), 210(111), 314 Fowler, H. A., 407(30), 436 Fowler, R. G., 201(124), 314 Fox, R. E., 25, 79 Frame, J. W., 34(76), 80 Francis, H. T., 25(52), 55(114), 79, 81 Franklin, J. L., 5, 15, 17, 56, 78 Fredriksson, K., 384(65), 386 Frei, A., 185(41), 312 Fry, T. C , 197(113), 207(132), 219, 220

(ma)3rd ,

E i +R., 237(144),

244, 245(144, 147),

3d6

Fwhw W., 185(lOZJ, 308,323 Fuls, E. N., 5(14), 66(14), 63(14), 78

G Gabor, D., 184(5), 185(92, !B, 951, 294, 296(158), 311, 313, 916 Cans, R., 207(133), 213, 220(133), 324 Gehrts, A., 251, 316 Gerasimenko, B. I., 29, 80 Germagnoli, E., 156(61), 178 Ghosh, S. N., 11(36), 79 Giacoletto, L. J., 185(88), 313 Gibbons, D. J., 393(8), 436 Gilbody, H. B., 5, 8(27), 12, 35, 37(6), 41(6), 42(6), 43(6), 52(103), 53(110), 78, 81 Gimpel, I., 407(31), 436 Glaser, W., 184(20), 311 Gold, L., 185(88, 98, 101, 102), 313 Goldmann, F., 11, 79 Goldstein, H., 185(107), 314 Gomer, R.. 86, 87, 89, 95, 97, 98(23), 101, 107, 116, 177, 178 Good, R. H., Jr., 85, 110(11), 115, 119, 174(11), 177 Goodrich, R. R., 399(16), 403(23), 405 (231, 436 Gordon, A. R., 276(157), 316 Goss, T. M., 185(64), 312 Gray, S., 412(39), 414(39), 428(60), 437 Greenberg, J., 310(175), 316 Greenhorn, J. S., 52, 81 Griffing, G. W., 28, 40(58), 79 Grivet, P., 184(23), 318 Grove, D. J., 25(53a), 79 Grundhauser, F. P., 335(21), 386 Gryzinski, M., 67, 81 Guillemin, C., 384(63), 386 Guiner, A,, 317, 326(1, 31, 357(1), 384

H Hadley, C. P., 309, 515 Hagstrum, H. D., 26, 79 Haine, M. E., 330, 334, 386 Hall, J. E., 52, 81 Handel, R. R., 394(11), 436 Hanle, W., 130, 178

442

AUTHOR INDEX

J

Hannam, H. J., 398(13), 417, 430, 436 Harker, I<. J., 185(47, 751, 31.9, 313 Haman, W.W., 184(10), 5fI Harris, J. H. O., 185(38), 516 Harris, L. A., 185(73), 513 Harrison, E. R., 185(71), 313 Hasted, J. B., 5, 8, 11, 12, 14, 35, 36(78, So), 37(6), 41(6, 811, 42(6, 811, 43

Jackson, C. M., 61 Jackson, J. D., 29, 33, 80 Jackson, S. R., 25(52), 55(114), 79 Jacobs, J. E., 427, 437 Jaines, R. B., 394(10, 11), 436 Jansen, C. G. J., 310(169), 316 Jeffreys, B. S., 276(156), 316 (6, 80, 81), 49, 61(78), 62(103), 53 Jeffreys, H., 276(166), 516 (110), 78, 79, 80, 81 Jepsen, R. L., 185(63),31d Hatsopoulos, G. N., 183(2), 207(2), 236 Johnson, C. C.,185(56), 31.9 (2), 237(2), $11 Johnson, R. E., 394(10, 111, 4.96 Hawkins, G. S., 52, 81 Johnston, T. W., 185(72, 791,313 Haywood, C. A., 33, 80 Jones, P. R., 6(14), 66(14), 69(14), 78 Hechtel, R., 185(40), 3 l d Jones, R. Clark, 435, 437 Heffner, H., 1&5(50), 316 Jones, T. J., 197(114), 314 Heijne, L., 406(24), 436 K Henkel, E. Z.,123, 178 Henry, G., 380(51), 386 Kaminker, D. M., 5, 10, 12, 78, 79 Hernqvist, K. G., 236(143), 316 Kanefsky, M., 236(143), 316 Herring, C., 197(118), 310(118), 514 Kaseman, P. W., 394(11d, 436 Herrmann, G., 185(77), 197(115), 199 Kaye, J., 183(2), 207(2), 236(2), 237(2), (1151, 206(115), 217(115), 310(115), 515, 314 Herron, R. G., 111, 178 Herrberg, G., 45, 80 Hickam, W. M., 25(53a), 79 Hillier, J., 184(18), 111 Himchfelder, J. O., 66, 81 Ho, K-C,185(43), 316 Hoch, 0. L., 185(73), 313 Hogg, A. C., 185(54), 516 Bok, G., 295(161), 316 Holland, L. I., 27, 79 Hollway, D. L., 185(31, 331, 51.9 Holmes, P. L., 407(32), 456 Honig, R. E., 101(28), 178 Houston, J. M., 183(2), 207(2), $11 Huienga, W., 11(37), 79 H u h , J. K., 98(23), 177 Hung, C. S.,309, 316 Hurvitz, H., Jr., 207, 314

I

Iams, H., 392(6), 415(41), 436, @?’ I U ’ k N. P.. 342(29). 886 Inghram, M. G., 86,. 87, 89, 95, 97, 116, 177

Ivey, H. F., 184(27), 185, 207(27), 221 (27), 229(27), 318

311

Kazan, B., 403(20, 211, 423(21), 427(57),

4%

437

Keene, J. P., 5, 11, 41(16), 42(16), 78 Kerr, L. W., 5, 7, 19, 78 King, H. M., 29,80 King, P. G. R., 185(78), $13 ICino, G. S., 185(85), 315 Kirchner, F., 87, 110, 116, 177, 178 Kirchner, H., 107, 178 Kirchner, R., 107, 178 Kirianenko, A., 383(62), 386 Kirstein, P. T., 185(45, 84, 85, 861, $16, 513

Kistemaker, J., 5, 11(37), 23, 78, 79 Kjeldaas, T., 25(53a), 79 Kleen, W., 184(13, 16), 197(13), 221(13), 229(13), 251, 311

Klemperer, O., 184(21), 311 Kleynen, P. H. J. A., 207(137), 209(137), 215(137),

220(137),

221(137),

Knoll, M., 399(14), 436 K o h A-1 380(53, 54)t ,986 Kompfner, R., 185(52), 316

222

AUTHOR INDEX

Koslov, B. F., 36(79), 37(79), 42(70), 45(79), 80 Kovach, I,., 185(67), Sf3 Koval, A. G., 36(79), 37(79), 42C791, 45 (79), 53(112), 80, s1 Kramers, H. A,, 29, 79 Kyhl, R. L., 185(57), 312

L Lampe, F. W., 5, 15, 17, 56, 78 Lampert, M. A., 405, 406,436 Lancsos, C., 83, 177 Landau, L., 48, 80 Langevin, P., 56, 81 Langmuir, D. B., 328, 384, 410, 436 Langmuir, I., 207(134), 209(134), 220 (1341, 222(134), 251, 279, 314 Laning, J. H., 189(112), 210(112), 31.4 Lasaen, N. O., 69, 81 Law, H. B., 394(9), 436 Lawrence, T. R., 23, 79 Lawson, J. D., 185(74), 313 Lasarev, B. G., 156(60), 178 Lenard, P., 338(26), 386 Lewis, J. T., 50, 80 Liebmann, G., 184(7), 185(7), 811, 335 (241, 386 Lindhard, J., 69, 81 Lindholm, E., 5, 25, 43, 78,79 Lindsay, P. A,, 185(42, 641, 215(146a), 229, 237(146a), 241(146a), 244(146a), 245(146a), 295(161a), 306, $12, 316 Lodge, J. A., 430(64), 4.97 Loeb, L. B., 14, 79 Lomax, R. J., 185(81, 1051, 313, 314 Long, J. V. P., 384(66), 586 Loosjes, R., 237(145), 241, 245(145), 310 (169), 316 Louisell, W. H., 185(55), 312 Lubsrynski, H. G., 430(64), 437 Lucas, A. R., 185(83, 86, 1051, 313, 314 Luebke, W .R., 185(59), 312 Lyubimova, A. K., 17, 79

M MacColl, L. A,, 184(3), 186(3), 201(127), 511, 314 MacDonald, D. K. C., 237(144), 245 (1441, 516 McDowell, M. R. C., 34, 80

443

McGee, J. D., 39201, 393(7), 422(50), 4% 437 Magee, J. L., 50, 53, 80, 81 Maloff, 1. G.,184(17), 311 Malter, I,., 207(129), 2451129), 314 Manara, A., 156(61), 178 Manley, B. W.,185(65), 313 Martin, E. E., 335(21), 386 Martin, R. J., 185(36), 312 Marton, L., 326, 334, 384, 386 Mason, E. A,, 123(45), 178 Massey, H. S. W., 27, 28, 33, 36, 45, 48, 51(101), 52, 53, 68(130), 79, 80, 81 Mathias, L. E. S., 185(78, 91), 313 Meinel, A. B., 5, 78 Melford, D. A,, 335, 350, 380(57), 386, 386 Melmed, A. J., 110, 174, 178 Melton, C. E.,18(48), 55, 69, 79 Meltzer, B., 185(39, 46, 82, 83, 86, 104, 105), 312, 313, 314, 407(32), 436 Meyer, H., 9, 79 Mihran, T. G., 185(62), 312 Miller, H., 399(15), 436 Miller, L. D., 400(17), 436 Miller, W. L., 276(157), 316 Milmann, J., 184(8), 197(8), 311 Mitin, R. V., 36(79), 37(79), 42(79), 45 (791, 53(112), 80, 81 Moe, D., 5, 13, 51, 78 Moiseiwitsch, B. L., 28, 33, 40(59), 49, 50, 73, 80 Montague, J. H., 5, 9, 78 Moon, R. J., 185(43), S l d Moore, G. E., 107, 178 Moore, R. S., 394(10), 436 Morris, D., 7(25), 19(25), 78 Morton, G. A,, 184(18), 311, 390(5), 392 (5), 417(42), 419(42), 427(55), 432, 436, 437 Moseley, H., 317, 384 Moses, H. A., 5(14), 66(14), 69(14), 78 Moss, H., 183(1), 207(1), 236(1), 237(1), 311 Mott, N. F., 27, 28, 33, 34(56a), 68(130), 79, 81 Mott-Smith, H. M., 309(162), 316 Muller, E. W., 84(4, 5, 7, 8, 9), 85, 86, 88 (161, 89(8), 90, 93(19), 95, 96(22), 97, 98(24), 100(26), 101(27, 301, 105,

444

AUTHOR INDEX

106(19), llO(11, 24, 351, 111(7), 114 (411, 115, 116(8, 42), 119, 121(43, 44), 123(8), 12518, 9, 40, 42, 481, 126 (144), 131(9, 52), 132(44) 135(54, 57), 136(26, 58), 137(57), 142(57), 148(16, 52), 149(58), 154(54, 591, 155 (59), 159(57), 160(65), 162(67), 164 (8,191, 165(57), 170(52, 69, 701, 172 (67), 173(43, 48, 731, 174(11, 35), 176 (27, 67), 177, 178, 179 Miiller, M., 185(68), 313 Mulson, J. F., 170(70), 174, 179 Mulvey, T., 326, 341(11), 344, 349, 384, 386

Muschlita, E. E., 55(114), 81

N Nakamura, S., 310 (174),316 Neumann, M.J., 422(49), 437 Nichols, M. H., 197(118), 310(118), 314 Nicholson, J., 430, f l Nicoll, F. H., 403(20, 21), 423(21), 436 Nordheim, L.,201(125), 314 Nordheim, L. W., 201(126), 314 Norman, F. H., 236(143), 316 Nottingham, W.B., 183(2), 207(2), 311

0 Ochs, S. A., 398(12), 406(29), 428(60), 438

Ollendorff, F., 184(15, 291, 213(29), 221 (29), 311, 312 Opik, E. J., 52, 81 Oppenheimer, J. R., 83, 177 Ovcharenko, 0.N., 156(60), 178

P Page, L.,184(4), 250, Sli, 316 Pankow, G.,90, 123, 124, 125, 158, 166 (47), 177, 178 Pargamanik, L. E., 206(128), 314 Parker, F. W., 215(146a), 229(146a), 237(146a), 241 (146a), 244(146a), 245(146a), 249, 616 Peetscher, O., 310(171), 316 Pensak, L.,421(46), 426(53), 437 Philibert, J., 380(47, 49, 51, 52, 53, 561, 381, 383(60, 61), 384(64), 386, 386 Philips, H.R., 309, 316 Pierce, J., 185(58), 312

Pilipenko, D. V., lo(?), 36(79), 37(79), 42(79), 45(79), 78, 80 Pimbly, W. T., 170(70), 179 Pizer, H.I., 185(32), 312 Plateau, J., 380(51, 541, 386 Pomey, G.,380(50, 541, 386 Pons, M., 382(58), 386 Poritsky, H.,207, 213, 220(141), 229 (1411, 232(141), 314

R Radley, D. E., 185(45), 812 Raev, A., 185(96), 313 Rakshit, H., 207(135), 213, 220(135), 314

Ramberg, E. G., 184(18), 207(129), 245 (129), 911, 314, 389(2), 417, 427(431, 4,% 437 Rau, K. H., 130, 178 Rausch von Traubenberg, H., 83, 177 Read, W.T., Jr., 158(62), 178 Redington, R. W., 402(19), 405(26), 426, 436

Reynolds, H. K., 67, 68, 81 Ribe, F., 5, 9, 78 Richard, N. A., 380(55), 383(55), 386 Richardson, O.,407(31), 436 Richardson, 0.W., 201, 207, 222(142), 314, 316

Riggs, F. B., 345, 986 Ritter, H.A.,110, 178 Romashko, N. D., 36(79), 37(79), 42 (791,45(79), 80 Rose, A., 392(6), 394(9), 405(25), 406, 410(35), 411(38), 423, 427, 430, 435, 436,

4 3

Rosentweig, L. N., 29, 80 Rostagni, A,, 11, 52(34), 79 Rothe, H.,184(13), 197(13), 221(13), 229(13), 251, 311 Rowe, J. E., 185(36), 312 Rudolph, P. S., 18(48), 55(48), 69(48), 79 Ruedy, J. E.,417(42), 419(42), 432, .ls7 Russek, A.,69, 81 Rutherford, J. J., 68, 81

S Salpeter, E., 85, 177 Saloom, J. A.,310(172), 816

445

AUTHOR INDEX

Samuel, A . L., 184(25), 312 Sander, K. F., 185(30, 32, 34, 35), 312 Saporosrhenko, M., 17, 56, 60, 7.3 Sayers, J., 5, 19, 78 Schade, 0. H., 413, 437 Schagen, P., 405(24), 436 SchiR, H., 29, 80 Schissler, D. O., 5, 16, 17, 56, 78 Schneeberger, R. J., 421, @7 Schottky, W., 207(130), 251(150, 1511, 261(149, 150, 1511, 277, 314, $16

Schroeder, A. C., 389(3), 436 Schroter, F., 399(14), 436 Schwartz, C. M., 380(55), 383(55), 386 Schwartz, J. W., 185(73), 313 Seeley, S., 184(8), 197(8), 311 Septier, A., 184(23), 318 Shelton, C. T., 430, @7 Sheridan, W. F., 11(36), 35(36), 79 Sherwin, C. W., 42(83), 80 Shockley, W., 184(25), 318 Sida, D., 41(82), 42(82), 80 Sida, D. W., 52, 81 Simons, J. H., 25, 55, 79, 81 Simpson, J. A., 326, 384 Sims, G. D., 185(92, 93, 95), $13 Slabospitskii, P., 53(111), 81 Sluyters, T. J . M., 5, 23, 78, 79 Smith, R. A., 49, 80 Snow, W. R., 5, 63, 78 Soloviev, E. S., 5, 10, 35, 46, 50, 78 Sommer, A. H., 393(7a), 436 Spangenberg, K. R., 184(6, 261, 197(116), 311, 318, 314

Sparks, I. L., 309, 316 Spicer, W. E., 389(1), 436' Stearn, A. E., 58, 81 Stebbings, R. F., 44, 80 Stedeford, J. B. H., 41(81),

42(81), 43(81), 80 Stehle, P., 185(106), 314 Sternglass, E. J., 422, 487 Sterrett, J. E., 185(50), 51.2 Stevenson, D. P., 5, 16, 17, 56, 78 Stewart, A. L., 27, 28, 33, 40(59), 79 Stewart, H. W., 430, 437 Stier, P. M., 9, 41(29), 68, 78, 81 Strange, J. W., 399(15), 436 Strutt, M. J. O., 185(41), 518 Stuart, G. A., 185(80), 313

Sturrock, P. A., 184(22), 311 Susskind, C . , 184(28), 185, 312 Sziklai, G. C., 389(3), 436

t Takayanagi, K., 67, 81 Talroze, V. L., 17, 79 Taylor, H. S., 56, 81 Teves, T. C., 427(56), 437 Terrill, H. M., 353, 386 Thomas, M. T., 69, 81 Thompson, H. C., 428(60), 437 Topolia, N. V., 10(7), 78 Townsend, J. S., 62, 81 Trolan, J. K., 335(21), 386 Twiss, R. Q., 185(90, 97), 294, 295(159), 313, 316

U Urbain, G., 318, 384 Uzunov, I., 185(96), 313

V Vance, A. W., 184(18), 311 Vanderslice, J. T., 123(45), 178 van der Ziel, A., 207(138), 209(138), %0(138), 221(138), 229(138), 243, 244(138), 314 van Duzer, T., 185(37), Slb Vanselow, R., 123, 158, 166(47), 178 Varney, R., 13, 79 Veith, W., 310(171), 3163 Vine, B. H., 400(17), 410(36), 417(45),

434 @7

Vink, H. J., 237(145), 241, 245(145), 316 von Ardenne, M., 131, 178 von Hevesy, G., 318, 384 von Laue, M., 201(122), 207(122), 241, 245(122), 314

W

S., 197(115), 199(115), 206 (115), 217(115), 310(115), 314 Waraven, P. L., 310(170), 316 Ward, 5. L., 426(54), 437 Waters, W. E., 185(53, 89), 312, 313 Watkins, D. A., 185(73), 313 Webley, R . S., 430(64), 437 Webster, D., 372, 386 Wagener,

446

AUTHOR INDEX

Webster, H. F., 183(2), 185(57, 761, 207(2), 311, 318, 319 Wehner, G., 174(74), 178 Weill, A. R. (Mrs.), 380(48), 386 Weimer, P. K., 394(9), 399(16), 403(16, 221, 406129), 410(35), 1123(51), 428, 428(80), 4% f l W e b , M., 380(54), 386 Weiael, W., 51, 66,81 Wheatcroft, E. L. E., 251, 261(154), 279, 282, 292, 293, 316 Wien, W.,9, 78 Whittier, A. C., 5, 78 Williams, E.J., 320, 353, 384, 386 Wilson, V. C., 183(2), 207(2), 311 Winwood, J. M.,185(105), $14 Wittry, D.B., 334, 337, 339, 352, 355, 372, 376, 379(19), 386 Wolf, F.,11, 41(33), 79

Wolf, P.,167, 179 Wright, D.A., 198(120), 310(120), 314

Y Yrtdav, H. N., 32, SO Yadavalli, 5. V., 185(73), 313 Yates, J. G., 185(30, 321, 313 Yocom, W.H., 185(52, 551, 312 Young, J. R., 178 y o ~ gR , . D., 95, 130(49), 177

Z Zandberg, E. J., 14(42), 79 Zener, C.,48, 80 Ziemba, F. P.,5(14), 66(14), 69(14), 73,

rs, 81

Zworykin, V. K., 184(18), 311, 389(2), 390(5), 392(5), 436

Subject Index A Absorption correction, x-rays, 370-371 in anti-cathode, 361362 Adiabatic theory of collision cross sections, 36-48 Adsorption, 174-176 films, dynamic equilibrium, 113-114 ion, energy levels for, 103, 104 Alloys, field ion microscopy of, I64 Amplifier, image, 130 Antimony trisuE.de photoconductivity, 403-404 Atomic lattice in perfect crystals, 144154 Atomic systems, inelastic collisions between, 1-81 Atoms excitation by ions, 21-23 field ionization of free, 84-87 impurity, 161-164

B Backscattering, x-ray coefficient, 377 Barrier penetration probability, 85 Born approximation, 27-33 Brightness, electron source, 328330 improvement of, 334-335

C Camera tubes, 387437 for color television, 427-429 electron optical considerations, 4 W 414 image dissector, 389-390 image intensifier, 419-423 limiting resolution for ideal, 432 performance, 430-435 photoconductive, 399-406 signal-to-noise ratio, 414-419, 433 for slow scan, 429-430 for storage applications, 429-430 Cathode, virtual, 217 Charge collection, 11-14 Charge equilibrium method, 8-10 447

Charge exchange, 2 collisions, 29ff experiments with crossed beams, 21 of protons in atomic hydrogen, 32-33 Charge leakage in target, 396 Charge transfer, 3, 67 daerential, 73-74 partial, 46, 48 resonance, 35 Chemical reactions, field induced, 176

177

Cold working structure, 167 Collisions charge exchange, 3, 29ff low energy, 14-21 classical treatment, 63-68 cross sections, 2, 27-75 adiabatic theory, 36-48 Born approximation, 27-33 distortion approximation, 34-35 high energy, 68-69 impact parameter treatment, 27 inelastic, 2 low energy, 24-25 perturbed rotating atom method, 33 perturbed stationary states method, 33 quantum mechanical calculation, 2736 wave treatment, 28 inelastic between atomic systems, 1-81 classification, 3-4 experimental methods, 4-27 low energy, 33-36 vector representation, 65 ionizing, 3 non-exchange, 28 rearrangement, 28 single, 10-11 Complex phases, identificatiou, 380 Conductivity, electron bombardment induced, 421-422 Con tsmination due to electron bombardment, 339-340

448

SUBJECT INDEX

specimen, 359 Dzraction, 117 Corrosion, 174-176 Diffusion, intermetallic, 382-383 Coulomb repulsion, 48-51 Diode Counters, proportional for soft x-rays, cylindrical 343 current density of electrons, 259-261 Counting, coincidence, 24 electron trajectories, 255-257 CPS emitron, 392, 393 equations of motion, 251-255 Cross sections phase space density of electrons, collision, 27-75 253, 255 -energy curves, 43 temperature, limited, 255-261 ion formation, 5E velocity distribution of electrons, ion molecule reaction, 15-17 250-294 low energy collision, 24-25 volume density of electrons, 257-259 stripping, 10 plane, 207-236 Crowdion, 158 exponential range of operation, 218 crystals Poisson distribution, 221 cubic, orthographic projections, 147, retarding field, 261-279 148 critical velocity of electrons, 262 disturbed structures, 164-174 current density of electrons, 276-279 hexagonal, orthographic projections, electron trajectories, 261-274 149 volume density of electrons, 274-276 imperfections, 154-164 space charge limited cylindrical, 279Current 294 anode, Schottky expreasion for, 277 current density of electrons, 290-292 density, 196197 electron trajectories, 279-287 cylindrical diode, 259-261 potential distribution, 292-293 electron beam croas over, 32t2-329 volume density of electrons, 287-290 field electron emission, 137 Discrimination, electron probe microfunction, 217-220 analyzer, 352-357 planar diode, 242-243 Dislocations, 15S161 plane magnetron, 305-308 Distribution retarding field diode, 276279 conditional, 193 space charge limited diode, 290-292 functions, 191-196 thermionically emitted electrons, marginal, 192, 195 200-201 Drift kinetic energy, 224, 230 electron beam, 329 Drift velocity, 229 automatic regulation, 335-337 E emitted ion, 90 multiplication, 92 Efficiency saturation, 218, 261 detective quantum, 435 Cyclotron frequency, plane magnetron, electron gun, 330 295-308 factor, image orthicon, 418 Electric field, reduced, 60 D Electron Desorption back scattering, 323, 368, 369 energy, 100 beam field, 100-105 crow over current density, 328-329 pulsed, 111-114 current, 329 Detachment, 51-53 focal point diameter, 411

SUBJECT INDEX

449

low velocity discharge properties, phase space density, 199ff, 208ff 406408 plane magnetron, 297 probe, 326337 a t surface of cylindrical cathodes, resolution of low velocity, 410-413 253, 255 velocity spread, 411 probe bombardment current, automatic regulation, 335contamination, 339-340 337 -induced conductivity, 421-422 diameter, 327, 330-334 temperature rise due to, 337-339 microanalysis, 317-386 capture, double, 45 point of impact, 347-360 critical velocity, 262 reflection coefficient, 407 surrent density, 242-243 scattering, 308309 cylindrical diode, 259-261 source brightness, 329 plane magnetron, 305-308 trajectories, 187, 238 retarding field diode, 276-279 cylindrical diode, 255-257 space charge limited diode, 2W292 plane magnetron, 299-301 tbemnionic, 200-201 retarding field diode, 261-274 cydotran period, 409 space charge limited diode, 279-294 idecdorebion law, 3w velocity distribution, 181-315 density inside an emitter, 199-200 functioq, triodes, 242248 in presence of magnetic field, 294at mrfaae of curved emitter, 2W-205 308 at d a c e of plane ern itter, 201-202 a t surface of emitter, 202-205 in velocity sub space, :314-215 thermionic, 197-206 volume, 23!%!Z42, 245 volume density, 210-217 detachment, 3, 53 genemlized, 257-259 emission plane magnetron, 301-305 by characteristic radiatio n, 372 retadimg field diode, 274-276 by continuous spectrum, 3723733 Emission fluorescence, 371-376 cathode field, 334-335 secondary, 371-376 roncentmhon proportionality, 320324, temperature limited, 202 366-370, 371 velocity distribution, 197-2 06 secondary dectron, 371-376 energy, total, 185 space charge limited, 218 flow, 184-185 Aemperahme limited, 218 double stream, 212 IJbermionic velocity distribution, 197gun, 326 206 efficiency, 330 .MY pickup tube, 411 &arnctcrktir, 362-366 kinetic energy, mean, 222-236 qartntitative analysis by, 360-379 microscopy, field, 114-115 Emitters multipliers, 26 curved, 203-205 number in interelectrode space, 215 plane, 201-203 optics thermbnic, volume density of eleccamera tube, 406-414 trons, 214 image orthicon, 413414 tip penetration fatigue, 170-171 depth, 353, 354 production, 128-129 diffuse, 352-356, 365 profiles, 136

450

SUBJECT INDEX

radius, 119 temperature, 142 water etch, 174 Energy bands in metals, 198 conservation, 253, 254 electron total, 185 field ion distribution, 93-95, 96 ionization onset, 51 kinetic drift, 224, 230 electron mean, 222-236 gas molecules, 117 measurements, 26 Evaporation field, 100-105, 135-144 time, ion, 112 Exchange collisions, 3 a t low energies, 14-21 Excitation of atoms by ions, 21-23

F Fatigue in emitter tips, 170-171 Field desorption, 100-105 experimental results, 105-114 pulsed, 111-114 Field electron emission, current density, 137

Field electron microscopy, resolution, 114-115

Field evaporation, 100-105, 135-144 Field-induced chemical reactions, 176177

Field ion current in various gases, 90-93 e m h i o n from a metal surface, 100-114 energy distribution, 93-95, 96 image intensity, 135 microscopy, 83-179 of alloys, 169 demountable, 127, 128 low temperature, 120-154 magnification, 148 resolu tion, 116120 llieory, 114-123 source, mass spectrosropy with, 95100

Field ionization, 83-179 of free atoms, 83-179 near a metal surface, 87-100

observation, 86-87 theory, 84-86 Fluorescence secondary electron emission, 371-376 Flying spot scanning, 389, 426 Frequency function, 191-196 plane magnetron cyclotron, 295-308

G Gain -bandwidth product, photoconductor, 405

photoconductive, 405

H Hopping height, 121 Hydrogen atoms, ionization time, 86

I Iconoscope, 390-392 Image amplifier, 130 compreasion factor, 117 dissector camera tube, 389-390 field, 139 formation gases for ion, 132-133 positive ion, 115 intensifiers camera tubes, 419-423 orthicon, 419423 phosphor photoemitter, 419-421 secondary emission, 422 solid state, 423 isocon, 423 orthicon, 394-399 efficiency factor, 418 electron optics, 413414 electrostatic fields, 409 guns, 412-413 light transfer characteristics, 398 magnetic! fields, 409 ultruviolet sensitive, 427 tube target potential changes, 395, 396 two-sided, 395-399 voltages, best, 135 Imperfections, crystals, 154-164

SUBJECT INDEX

II1rrILrCti Iiicliup f,ul)cs, 426.427

451

401 ~~holorotitlucl.ivc, Imdaii-Zcnrr forniiila, I9 J,cm tnngnctic reducing, 326-327 probe-f orm ing, 327-328 optical aberrations, 335 Light amplifier panels, 423 transfer characteristice of image orthicon, 398 Liouville’s theorem, 187-188

sensitive vidicon, 427 Intcnsity ion image, 135 x-ray characteristic radiation, 364, 358, 361-362 x-ray measurements, 376-379 Interstitials, 157-158, 162 Ion abundance, 18 adsorption, 103, 104 M beams, ground state, 25-26 current Magnetron, plane, 295-308 emitted, 90 current density of electrons, 30.5-308 field, in various gases, 90-93 cyclotron frequency, 296 evaporation time, 112 electron trajectories, 2 W O 1 field emission, 100-114 equations of motion, 29&299 formation cross sections, 5ff phrtse space density, 297 image volume density of electrons, 301-305 gases for formation, 132-133 Magnification in field ion microscopy, intensity, 135 148 microscopy, field, 83-179 Mass -molecule reactions, 53-63 analysis of ions mean cross section, 15-17 after collision, 8-11 measurement in afterglow, 18-21 formed from target, 5-8 pressure ranges, 17 spectrograph sources, 14-18 rate constants, 16 spectroscopy with a field ion source, recombination, negative, 53ff 95-100 Ionization, 51-53 Maxwell-Boltzmann equation, 188-189 energy onset, 51 Mechanics, particle, 185-187 field, 83-179 Metal meteor, 52 energy bands, 198 multiple, 69-75 surface positive ion, 3 field ion emission from, 100-114 probability, 70, 71, 87 field ionization near, 87-100 statistical theory, 69-75 Microanalysis, electron beam, 317-386 time of hydrogen atom, 86 contributions to research, 379-384 transfer, 3 electron probe, 326-337 Ions general structure, 324-360 doubly charged, 142 limitation of discrimination, 352-356 excitation of atoms by, 21-23 localization of impact point, 347-360 ma= analysis after collision, 8-11 in metallurgy, 380483 mass analysis, from target, M in mineralogy, 383-384 Isocon, 423425 optical viewing of sample, 34-49 optimum discrimination, 353 L pulse analysis, 345-347 scanning, 349-352 Lag capacitive, 401-402 thermal conditions of analysis, 337discharge, 408 339

452

SUBJECT INDEX

usc of field emission cathode, 334-335 for very light elements, 345347 Microscopy field electron, 114-115 field ion, 83-179 Moments, central, 192 Motion, equations of for electrons, 208210, 237-239, 245-247, 295-299, 251256

ultraviolet, 427 x-ray, 427 Pinch effect, 1&3 Poisson equation cylindrical diode, 292 plane diode, 221 Polarizability, 140 Porthole effect, 410 Potential changes in image tube target, 395, 396 N distribution between cathodes, 243-245 Neutralization, mutual, 3, 53ff between grid and anode, 248-250 Noise in cylindrical diode, 292-293 total mean square return beam, 417 in presence of space charge, 220-222 shot, 414 in triode, 246 0 -energy curves, pseudocrossing, 48-55 measurements, retarding, 93-95 Orthicon, image, 392-399 Pressure ranges, ion-molecule reactions, P 17 Probability Particle mechanics, 185-187 considerations, 189-197 Penetration of electrons density function, 191-196, 210-217, depth, 353, 354 239-242 diffuse, 352-356, 365 distribution, 190-191 Phase space density of electrons, 199ff, function, marginal, 192 2086 Protons, charge exchange in atomic plane magnetron, 297 hydrogen, 32-33 a t surface of cylindrical cathode, 253, Pulse analysis, matrix method, 345-347 255 triode, 247 Phase space diagram, 211 Phosphor -photoemitter intensifier, 419-421 screens, 129 Photoconductive lag, 401 Photoconductivity of antimony trisulfide, 403404 Photoconductor camera tubes based on, 399-406 gain-bandwidth product, 405 thin film, 402-406 Photography of field emission images, 130 Photomultipliers, 389 Pickup tubes, 388 electron guns, 411 infrared, 42-27 with non-storage devices, 389-390 sensitivity, 416 signal-to-noise ratio, 434

Q Quantum efficiency, detective, 435

R Radius of capture, 89 Rate constants, 56ff ion-molecule, 16 Reactions endothermic, 49 exothermic, 49 ion-molecule, 53-63 Rearrangement collisions, 28 Recombination, 2 negative ion, 536 Reflection coefficient of low-energy electrons, 407 Regressions, 194 Repulsion, coulomb, 48-51 Resistivity of eelenium, 403 Resolution

453

SUBJECT INDEX

field electron microscopy, 114-115 field ion microscopy, llb120 limiting, for ideal camera tube, 432 of low-velocity electron beams, 410413 plane, 131-132 space charge effects on, 122 x-ray scanning microscope, 360

S Scanning flying spot, 389 of charge pattern, 426 of low velocity electron beam, 408-410 Scattering of electrons, 308-309 back, 323, 368, 369 hard, 66 soft, 66 Schottky expression for anode current, 277 Screw dislocations, 158-159 Secondary electron emission, 371376 image intensifiers, 422 Segregation, 380 Selenium, resistivity, 403 Shot noise, 414 Signal current, target, 416 Signal generation, video, 423-426 Signal-to-noise ratio in camera tubes, 414-419 ideal camera tubes, 433 pickup tubes, 434 video signal, 414-419 vidicon, 419 Signal, peak video return beam, 417 Slip, 164-167 Solid state image intensifiers, 423 Space charge effects, 183, 187 electron flow, 185 limited cylindrical diode, 279-294 limited emission, 218 potential distribution in presence of, 220-222 resolution effect, 122 Spectrography, x-ray, 318-320 Spectroscopy, mass, with a field ion source, 95-100 Spiral edge, 159

Stacking faults, 160 Standard deviation, 192 Strew release, pulsed, 170-171 Stripping, 3 cross section, 10 Superficial layers, 380382 Surface effects, 174-177

T Target aging of glass, 397 bombardment induced conductivity, 421422 grid control, 416 lateral charge leakage, 396 signal current, 416 surface potential, 393 two-sided, image tube, 395-399 vidicon, 400-401 Television camera tubes, 387437 for color, 427429 Temperature emitter tip, 142 optimum tip, 131 rise due to electron bombardment, 337439 Tetrode, plane, 245-250 Time constant for lateral charge leakage, 398 vibration, 107 Transmkivity, mean coefficient, 201 Triode, plane, 245-250 electron density function, 247-248 electron trajectories, 246-247 phase space density, 247 volume density of electrons, 248

U Ultraviolet pickup tubes, 427 sensitive image orthicon, 427

V Vacancies, 154-157 Vacancy concentration, 155 -interstitial pairs, 157 Variables, random, 19Off

454

SUBJECT INDEX

Velocity distribution of electrons, 181-315 cylindrical systems, 250-294 experimental support of theory, 308310

half-Maxwellian, 304310 inside emitter, 199-200 Mamellian, 202-203 p a r t - M a x w e h , 213 plane systems, 202-203, 206-249, in presence of magnetic field, 294-

thermionic emitter. 214 in triode, 247, 248

w Webster’s law, 320-321 Wien equation, 9 William’s law, 320-321 Work function extreme, 140, 141 thermionic, 107ff, 198, 201

X

308

a t surface of curved emitter. 203205

thermionic, 197-208 drift, 229 mean of electrons, 229 spread in electron beams, 411 subspace, electron density, 214-215 Video signal generation, 423-426 return beam peak, 417 signal-to-noise ratio, 414-419 Vidicon, 400406 infrared sensitive, 427 signal-to-noise ratio, 419 target, 400401 thin film photoconductors for, 402-406 tricolor, 428 Voltage breakdown, 92 Volume density of electrons, 210-217 cylindrical diode, 257-259 plane magnetron, 301-305 retarding field diode, 274-276 space charge limited cylindrical diode, 287-290

a t surface of emitter, 202

X-raj absorption correction anticathode, 361-362 curves, 370-371 back scattering coefficient, 377 characteristic emission, depth distribution, 362-366 characteristic radiation intensity, 354, 358, 361362

emission analysis of alloys, 376, 378 from field emission tips, 135 quantitative analysis by, 360-379 intensity measurements, 376-379 microanalysis, 317 pickup tubes, 427 recording analyzer, 340-347 scanning microscope resolution, 360 soft, proportional counter for, 343 spectrographs, 340-347 non-dispersive, 345-347 Ross’s double filter, 344-245 soft radiation, 342-344 spectrography absolute method, 319-320 quantitative analysis, 318-319