Advances in Atomic and Molecular Physics, Volume 3

Advances in ATOMIC AND MOLECULAR PHYSICS VOLUME 3

CONTRIBUTORS TO THIS VOLUME E. CHANOCH BEDER B. BUDICK

H. G . DEHMELT A. L. STEWART ROBERT E. STICKNEY HENRY WISE H. C. WOLF BERNARD J. WOOD

ADVANCES IN

ATOMIC AND MOLECULAR PHYSICS Edited by

D.R. Bates DEPARTMENT OF APPLIED MATHEMATICS

THE QUEEN’S UNIVERSITY OF BELFAST BELFAST, NORTHERN IRELAND

Immanuel Estermann DEPARTMENT OF PHYSICS THE TECHNION ISRAEL INSTITUTE OF TECHNOLOGY HAIFA, ISRAEL

VOLUME 3

@) 1967 ACADEMIC PRESS New York London

COPYRIGHT @ 1967, BY ACADEMIC PRESS INC. ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

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United Kingdom Edition published by

ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.1

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 65-18423

PRINTED I N THE UNITED STATES OF AMERICA.

List of Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin.

E. CHANOCH BEDER, Department of Fluid Physics, Aerosciences Laboratory, TRW Systems, Redondo Beach, California (205) B. BUDICK, The Hebrew University, Jerusalem, Israel (73)

H. G. DEHMELT, Department of Physics, University of Washington, Seattle, Washington (53) A. L. STEWART, School of Physics and Applied Mathematics, The Queen’s University of Belfast, Belfast, Northern Ireland (1)

ROBERT E. STICKNEY, Department of Mechanical Engineering and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts (143) HENRY WISE, Stanford Research Institute, Menlo Park, California (291)

H. C. WOLF, Physikalisches Institut der Universitat, Stuttgart, West Germany (1 19) BERNARD J. WOOD, Chemical Dynamics Department, Stanford Research Institute, Menlo Park, California (291)

V

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Foreword This serial publication is intended to occupy an intermediate position between a scientific journal and a monograph. Its main object is to provide survey articles in fields such as the following: atomic and molecular structure and spectra, masers and optical pumping, mass spectroscopy, collisions, transport phenomena, physical and chemical interactions with surfaces, gas kinetic theory. It is the aim of the editors to select articles which combine a rigorous yet understandable introduction to their subject, a summary of past work with emphasis on progress in the last five years, and a sufficiently complete bibliography to enable the reader to locate quickly relevant details in the literature. Suggestions for topics to be covered in the future will be appreciated.

D. R. BATES I. ESTERMANN

Belfast, Northern Ireland Haifa, Israel November, 1967

vii

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Contents LISTOF CONTRIBUTORS FOREWORD CONTENTS OF PREVIOUS VOLUMES

V

vii xi

The Quanta1 Calculation of Photoionization Cross Sections A . L. Stewart I. Introduction 11. Basic Concepts and Formulas 111. Resonance in the Continuum

IV. Oscillator Strength Sum Rules V. Comparison of Experimental and Theoretical Results VI. Conclusion References

1 2 16 26 30 41 48

Radiofrequency Spectroscopy of Stored Ions. I : Storage H . G . Dehmelt 53 55

1. Introduction 2. Containment of Isolated Ions

12

References

Optical Pumping Methods in Atomic Spectroscopy B. Budick I. Introduction 11. Experimental Techniques 111. Results of Double Resonance and Level-Crossing Experiments IV. Optical Pumping Experiments in Ground States References

13 74 83 108 114

Energy Transfer in Organic Molecular Crystals: A Survey of Experiments

H . C. Worf I. Introduction 11. The Crystals 111. The Optical Spectra

IV. Experiments on Energy Migration in Mixed Crystals V. Summary References ix

119 121 122 123 140 141

Contents

X

Atomic and Molecular Scattering from Solid Surfaces Robert E. Stickney I. Introduction 11. Preliminary Considerations 111. Experimental Apparatus for Gas-Solid Scattering Studies IV. Experimental Data on the Scattering of Atomic and Molecular Beams from Solid Surfaces V. Classical Theories of Atomic and Molecular Scattering from Solid Surfaces VI. Concluding Remarks Appendix. Classical Mechanics of Hard-Sphere Collisions Refcrences

143 144 154 161 137 198 200 20 1

Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering

E. Chanoch Beder I. Introduction 11. The Interaction Potential 111. The Hamiltonian of the Point-Mass System

IV. General Scattering Theory V. Elastic Scattering VI. Inelastic Scattering List of Symbols References

206 214 234 238 243 261 285 281

Reactive Collisions between Gas and Surface Atoms Henry Wise and Bernard J. Wood I. Introduction 11. Energetic, Mechanistic, and Kinematic Considerations

111. Experimental Methods and Results

IV. Concluding Remarks List of Symbols References AUTHORINDEX SUBJECT INDEX

29 1 292 31 1 344 347 348 355 365

Contents of Previous Volumes Volume 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A . T. Amos Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch Atomic Rearrangement Collisions, 3.H. 3ransden The Production of Rotational and Vibrational Transitions in Encounters between Molecules, Kazuo Takayanagi The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H . Pauly and J. P . Toennies High Intensity and High Energy Molecular Beams, J. 3.Anderson, R. P. Andres, and J . 3.Fenn Amnou INDEX-SUBJECT INDEX

Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W. D . Davidson Thermal Diffusion in Gases, E. A. Mason, R . J. Munn, and Francis J. Smith Spectroscopy in the Vacuum Ultraviolet, W. R . S. Carton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A. R . Samson The Theory of Electron-Atom Collisions, R . Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J. de Heer Mass Spectrometry of Free Radicals, S. N. Foner AUTnoR INDEX-SUBJECT INDEX

xi

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THE QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS A . L . STEWART School of Physics and Applied Mathematics. Tlre Queen’s University of Belfast. Belfast. Northern Ireland

I . Introduction ...................................................... i I1 . Basic Concepts and Formulas . . . . . . . . . . . . . . .................... 2 A . The Velocity and Acceleration Formulas . . .................... 6 B. Accuracy of the Calculations .................................... 8 C . The Continuum Wavefunctions ....... ....................... 9 D . The Normalization Process ...................................... 14 E The Solution of Continuum Differential Equations . . . . . . . . . . . . . . . . . 15 111. Resonance in the Continuum ...................................... 16 A . Resonances and Configuration Interaction ........................ 20 B. Resonances and Photoionization ....... ................. 23 IV Oscillator Strength Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 A . The Generalized Oscillator Strength . . . . . . . . . . . . . . . . . . . . . 28 B. General Formulas for Photoionization C ns . . . . . . . . . . . . . . 28 C . Extrapolation at the Spectral Head .............................. 29 V . Comparison of Experimental and Theoretical Results .................. 30 A. Photoionization of Helium ...................................... 30 B. Photoionization of Neon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 C . Autoionizing Levels in Neon . . . . . . . . . .................. 39 D . Photoionization of Lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 E . Photoionization of Sodium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 F . Photoionization of Potassium .................................... 43 G . Photoionization of H - ......................................... 44 H . Photoionization of C- and Other Negative Ions . . . . . . . . . . . . . . . . . .45 I . Photoionization of Atomic Nitrogen .............................. 47 VI Conclusion ................................. . . . . . . . . . . . . . . . . . . . 47 References ........................... . . . . . . . . . . . . . . . . . . . 48

.

.

.

.

I Introduction Photoionization or continuous absorption is the process in which an atomic or molecular system. neutral or ionic. normal or excited. absorbs a photon and is raised to a state of the electron continuum so that one or more 1

2

A . L. Stewart

electrons are observed to be emitted. In the particular case of negative ions this process is referred to as photodetachment. Electron detachment can also occur through many-electron excitation to a short-lived quasi-discrete state with subsequent decay through a radiationless transition, autoionization, to an adjacent state of the continuum. This effect is observed as a resonant distortion of the photoionization cross section. Obviously the cross sections for these reactions and their inverse, continuous emission or radiative attachment and dielectronic recombination, are of fundamental importance for a quantitative understanding of the many fields of physics that involve the interaction of radiation and matter. To take a recent and important example, such processes can be shown to control the temperature of the solar corona (cf. Burgess, 1964, 1965). The subject has been recently reviewed (Ditchburn and t)pik, 1962; Branscomb, 1962, 1964; MiDaniel, 1964) and the special case of resonances in photon absorption has been reviewed by Burke (1965). The experimental situation at the present time is emphasized in the article by Samson (1966) in the previous volume of this series. Thus in this article only the most immediate theoretical developments, or those topics considered outside the scope of the recent reviews, need to be discussed. At the same time it seems desirable to bring together the many quantal formulas and relationships which are used in the study of photoionization, to facilitate the entry of new research workers to the field. Reference should be made to Condon and Shortley (1935) and to Heitler (1954) for detailed descriptions of spectroscopic terminology and for the basic theory of dipole radiation.

11. Basic Concepts and Formulas The quantal formula for the cross section for ejection of photoelectrons in any direction from an N-electron system (atom, positive or negative ion) by absorption of photons of energy hv and wavelength L from an unpolarized beam is

where

is the " line " strength of the optical dipole transition involved and N

P=-eCri i=l

(3)

QUANTAL CALCULAlION OF PHOTOIONIZATION CROSS SECTIONS

3

is the electric dipole moment of the N-particle system. In these formulas

X

= ( x i , x2

. . . ,x N )

=

{(rl, 011, ( r 2 3 021, .

--

3

(rN , g N ) >

(4)

represents the set of space, ri , and spin, oi, coordinates of all the electrons and A and B represent collectively the complete set of configuration and quantum numbers defining the initial bound and final continuum states, respectively. The sum in (2) is over all the wA degenerate levels with energy EA and all the w B degenerate levels with energy EB such that hV = EB - EA = I ,

8 ~ ,

(5)

where IA is the threshold ionization energy and 8, is the kinetic energy of the electron in the continuum state B. The initial wavefunction is normalized to unity and the final wavefunction to

( y ( B I x) I Y ( B ' I x))= S ( 8 B - &B,).

(6) Expressing energy in rydbergs, wavelength in angstroms, and all other quantities in atomic units, ( 1 ) becomes

= 2.690 x

where c1 is the fine-structure constant, a, is the Bohr radius, S ( A ; B ) is in atomic units, and

Alternatively the rate of the reaction can be expressed in terms of the oscillator strength per rydberg energy interval which, in the unit!: of (7) above, is

so that

dfAB = 1.239 x 1017 oAB(v). dbB It should be pointed out that Eq. (6), in absolute units, gives the free electron radial function wave amplitude to be (l/r)(8m/h28B)"4,if the angular function is normalized to unity. The identical relationship to Eq. (6) in atomic units and rydbergs gives the free electron radial function wave amplitude to be ( l / r ) ( l / ~ ~ 8 ~Accordingly, )''~. in Eqs. (7) and (9), S ( A ; B) contains Y ( B I X)

4

A . L. Stewart

normalized with this latter asymptotic amplitude, that is, according to the equivalent of (6) in rydbergs and atomic units. The numerical multiplying factors in (7) and (9) must of course be adjusted if Y ( B l X ) is differently normalized; for example, with asymptotic amplitudes (l/r)(l/&E)1/4 or (l/t-)(l/&E)l/’, which are also commonly used, the numerical factor in (7) becomes 8.56 x and, in the second case, the factor &k/2 appears. Some scientists prefer to normalize the final wavefunction with the free electron in the form of a plane or Coulomb wave, so that the numerical factor in Eq. (7) becomes aaO2/3= 6.814 x lo-” and is again multiplied by &;I2. Since the final wavefunction in this case is a superposition of degenerate states of the free electron, care should be taken with the B summation in Eq. (2). It is usually assumed that non-Coulomb interactions are never so large as to invalidate the L-S coupling approximation, and so the quantum numbers for the initial and final states are taken to be A = (aSLM,M,)

B = (PS’L‘M,M.),

and

(11)

where c1 and p represent collectively the configuration quantum numbers, or other parameters, necessary to completely specify the states. The degeneracies are WA

= (2s

+ 1)(2L + l),

wg

= (2s’

+ 1)(2L’ + l),

and since P is independent of spin we have S = S’, M , I

S ( A ; B) = (2s

(12)

= Msj, I

+ 1)M L2M L ’ (aLM, IPIPL‘M,?) ,

which can be reduced with the aid of the L-S coupling selection rules

L= L ’ , L ‘ + 1 ; M , = M L , , M,,+I. (14) To proceed it is necessary to specify the form of the wavefunctions. Except for simple systems with two or three electrons the only wavefunctions available, or easily obtainable, for the calculation of photoionization cross sections take the form of linear combinations of Slater determinants whose elements are orthonormalized one-electron orbitals. The final state is usually formed by adding an electron to the core of N - 1 electrons, the core orbitals being unperturbed by the presence of the continuum electron with no coupling between the possible core parent states taking place. In this approximation the core orbitals in the final state can be taken as equal to those in the free core (cf. Sewell, 1965a) or equal to those in the core of the initial state (cf. Cooper, 1962). With the latter choice the orbital for the continuum electron is necessarily orthogonal to the core orbitals of the initial state and the selection rule that only one-electron transitions occur is obtained. With the former choice this rule is violated because with differences between the core orbitals in the

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

5

initial and final states the continuum orbital may no longer be exactly orthogonal to all core orbitals in the initial wavefunction. [See however the recent discussion by Henry and Lipsky (1967).] In the usual notation the one-electron orbitals are written

where x = (r, a) is the composite of space and spin coordinates. Then S ( A ;B ) =

n:j1 nl

P:(r) P$(r) dr Y ( A )CT'

where Y ( A )is the relative multiplet strength and

where I, is the greater of fi and 1,. In Eq. (16) the product is over all orbitals nI of the core of the initial and final states. Since the two sets of core orbitals are not necessarily identical the integrals in each term of the product are not necessarily unity. It has been assumed that in the transition only one electron changes its orbital state, the initial quantum numbers being ni Ii and the final quantum numbers &I,. The omission of double-electron jump terms from (16) is unlikely to be important since, with 1, = f and a compact core, j;P:(r)

P&,(r) dr *

J:

P$(r) P&(r) d r = 0.

(18)

The fact that the product of core integrals in (16) is approximately unity is the basis of the one-electron approximation in which the line strength is reduced to S ( A ; B) = Y ( A )02.

(1 6 4

However, it should not be forgotten that its validity depends on either the first or second integral in (18) being close to zero. The relative multiplet strengths may be obtained by a straightforward application of Racah algebra to the angular integrals in (1 3), after expressing the initial wavefunction in terms of its fractional parentage coefficients and using the selection rule that transitions between states of different core parentage do not occur. Formulas and values for multiplet strengths of aeronomical and astrophysical interest are given by Rohrlich (1959a), Burgess and Seaton (1960), and Dalgarno et af. (1964), who give references to tables of Racah and fractional parentage coefficients.

6

A. L. Stewart

It often happens that the radial function P i l , is independent or nearly independent of L’ (as it is of A4L1)so that (i2 is independent of L‘. It is then convenient to sum S(A ;B ) over L’ and use the sum rules for multiplet strengths (cf. Rohrlich, 1959b). A.

THEVELOCITY AND ACCELERATION FORMULAS

The “line” strength in (2) is written in terms of the dipole length operator, which in atomic units is N

c

pL=-

(19)

Ti.

i= 1

Assuming that the wavefunctions in (2) are exact and, by appeal to the exact Schrodinger equation (Chandrasekhar, 1945a), (19) can be effectively replaced by the dipole velocity operator, energy being expressed in rydbergs, -I

N

or by the dipole acceleration operator, N

where V is the Schrodinger potential

giving

Written in these alternative forms the matrix element in (17),

F , = j:p:lrpi,l*

is replaced by the dipole velocity form

or by the dipole acceleration form

dr,

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

7

The differences between the three formulas become apparent if PAI and are solutions of equations of the form 1(1

d2

+ 1)

( g- GnI - r2

(27)

and

respectively. The potential functions G in a real situation would contain both the direct and exchange Hartree-Fock potentials with the possible addition of polarization terms (see Section V,B). Identically then

and

+ (8-

EnJ2

jomPiI(G8,1*l - Gnl)(1 f (21 2r + 1 ) pi!

*1

"

+-&Pi, t l d r . (30)

Comparing (29) with (25) it is clear that in an approximation in which it is assumed I = - enl and G8,1+ = Gnl the length and velocity formulas give the same results. However, in the same approximation, comparing (30) and (26) and writing G8.f

*

1

-2(Z - N r

+ 1) 3

there results Z F A * Z - N + l F"

*

Major differences between the values obtained using the velocity and acceleration formulas are accordingly to be expected in approximations based on

8

A . L. Stewart

one-electron orbitals (Stewart, 1954). This arises because the acceleration operator (23) has the peculiar property of being independent of electronelectron repulsion and it would appear to be of value only when used in conjunction with wavefunctions which take full account of electron-electron repulsion, that is of angular correlation.

B. ACCURACY OF THE CALCULATIONS In general the accuracy of photoionization cross-section calculations based on one-electron orbital approximations cannot be expected to be better than 20 %, although in particular cases comparison with experiment reveals surprising agreement. Obviously it is important that the transition matrix element should not be unduly sensitive to changes in the wavefunctions. The quantity 1 - D introduced by Bates (1947), which is the ratio of the value of the radial integral to whichever is the larger, the positive or negative part of the integral, is an important guide to the sensitivity of the matrix element. The relative merits of the length and velocity matrix elements are difficult to assess. However, the conclusions arising from the discussion of Dalgarno and Lewis (1956) seem to be borne out in practice. If the photon energy (5) appropriate to the transition of interest is greater than that associated with other possible strong transitions from either of the states then the dipole velocity form is to be preferred, and if the photon energy is smaller then the dipole length is to be preferred. The argument of Dalgarno and Lewis further suggests that preference should be given to the acceleration matrix element for high-energy transitions and accordingly that the acceleration matrix element should be relatively insensitive to changes in the wavefunctions at high energies. The asymptotic form for high energies of the line strength can be obtained by allowing the continuum electron to have plane wave form and expanding the resulting line strength as a power series in €-’/’ (Dalgarno and Stewart, 1960). Alternatively the matrix element and Schrodinger equation can be transformed‘ to momentum space to give a formal expression for the leading term in the high-energy expansion (Kabir and Salpeter, 1957). For example, the oscillator strength for photoionization of a two-electron system is in atomic units and rydbergs and, using the acceleration matrix element,

For large values of d = k 2 one can replace t+h8(r1,rt) in the lsdl configuration by

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

9

where uO(rI)is the wavefunction of the one-electron core. The leading term in the expansion gives

where

Applied to 1s’ ‘S-ls&p ‘ P transitions in helium and using 1-, 2-, 6-, and 20-parameter wavefunctions for the ground state the values of C are 256, 297,288, and 286, respectively, demonstrating the insensitivity of the acceleration matrix element at high energies to changes in the wavefunctions. C. THE CONTINUUM WAVEFUNCTIONS To evaluate the line strength (2) it is necessary to obtain solutions of the equation ( H - E ) Y(X) = 0, (35) where

H

=

-C

Vi2 + V,

i

with the boundary condition lim,,,, Y = 0 for bound states. The boundary condition for the one-electron continuum states is lim Y ( B I X ) =

1

N-’”(-I)~-~

C.l,rnr.m,

rj-+ m

x Y(C I Xi) @(C&1m,m,I X j ) ( C l r n , r n , I B),

(37)

where the core states Y(C I Xi) are normalized and

with the Coulomb phase shift given by (39)

.

and the phase shifts due to non-Coulomb fields represented by D,, In Eq. (37)

Xj = (XI,

x2 3

* * * 9

xj- 1,

xj+

19

.*-,

X~V)

(40)

A . L. Stewart

10

and the sum contains the vector-coupled superposition of the states formed from the states C = (y, S", L", M L r rMS.,) , of the isolated N - 1 electron core and the states lm, m, of the continuum electron, to give the continuum state with quantum numbers B. Defining the functional

s

L = Y*(B 1 X)(H - E ) Y ( B I X) dX

(41)

one has the variation principle

s

6L = 2 R.P. 6Y*(H - E)Y dX

6DC, + C ( B I Cl)(CI I B ) 71, c.1

(42)

in which the sum over the magnetic quantum numbers M,-M,..m, and m, has been carried out, using the fact that the phase shift is independent of these quantum numbers. For the exact solution of (35) these give, to first order,

and one can force a trial solution with undetermined parameters to imitate the exact solution in satisfying (43) with respect to first-order variations in the parameters. 1. The Hartree-Fock and Close-Coupling Approximations

The boundary conditions are most easily satisfied by taking the fully symmetrized trial solution Y ( B I X) =

i

N-'"( - l)N-j

1 Y(C I Xi) $(CBlm,m, I xj)(Clmlm,I B ) ,

Clmlm,

(44) where $ has, asymptotically, the Coulomb form (38), and carrying out independent variations with respect to the radial parts of $. Using (43) this gives

R.P.

1

(BIClrn,m,)

M~~~Ms~~mnlrn.

x k * ( C I Xi)

Y z l (Pi) 6(m, I o j ) ( H - E ) Y ( B I X) dXj dPj doj = 0, (45)

which reduces to a set of coupled differential equations for the radial parts of $, there being an equation for each core term, specified by yS"L", and each angular momentum quantum number 1 of the continuum electron.

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

11

It is a simple matter to show that (44) and (45) are in accord with the orthonormalization condition (6). However, it does not follow, as some assume, that necessarily ($(Cdlm,m,)l $(CS'Zm, m,)) = S(S' - S),

(46)

i.e., using (15), that

j:P8.1(r) Ptl(r) d r = S(S' - 8).

(47)

For example, for the lsdl states of the two-electron systems with only a single 1s core included in (44),Eq. (45) requires that the radial function P81(r)be a solution of

r 8Z3 e - z r [ r - f a,+'( r ) 21+ 1

=k-

where

and the plus and minus symbols refer to the singlet and triplet states, respectively. The usual proof of the orthogonality theorem leads to

[ [;Pt,,P8,

m

dr 5

a,,

jOm(4Z3)'/'r e-"

P8,, dr Jo (4Z3)'/'r e-"P,,

= 6(S - S').

dr]

(50)

Written in terms of the complete core and continuum orbitals $(ls Ir) and $(Sl Ir) this equation is (B'l

I a,)f (S'l I

ls)(ls I Sf) = 6(6 - 8').

(51)

Thus (46) and (47) are correct only when the continuum and core orbitals are mutually orthogonal. Similar considerations apply to more complex systems. Thus in the case of the (lS;&?p)state of the three-electron systems (44) and (45) ensure that the wavefunction is properly normalized even when correlation terms are included in the ' S core.

A . L. Stewart

12

In most applications to photoionization the sum over the core terms in (44) is restricted to a single term to give only one equation of the form (45). Secondly the core wavefunction Y(C I Xi)is usually expressed in terms of one-electron orbitals in which case an equation of the Hartree-Fock type, of which (48) is an example, is obtained for the radial part of the continuum orbital. Since the core functions are assumed known and fixed, no Hartree-Fock equations for the core orbitals arise out of (45). These require to be determined by a separate variation principle for the ( N - 1)-electron core system or taken equal to the core orbitals of the initial bound state in the phototransition, as mentioned earlier. The theory outlined is discussed in detail by Seaton (1953) and appIied to the two-electron systems by Percival and Seaton (1957). In collision theory (45) corresponds to the close-coupling approximation, the equations for which have been solved in the cases of e + H and e + He' scattering (cf. Burke and Smith, 1962; McCarroll, 1964; Burke et al., 1964a,b). For example, in the case of the two-electron systems, with only the ground u(1sIr) and first excited u(2s I r) and u(2p I r) one-electron core terms included in (44),Eqs. (45) are, omitting spin,

s s

R.P. u(ls I rl) Y;Cm,(P2)( H - E ) Y ( B I rl, rz) dr, dPz = 0 ,

R.P.

4 2 s I rl) Y,*,(P,)

( H - E ) Y ( B I rl, r,) dr, dPz = 0,

x Y ( B I r,, rz) dr, dPz = 0,

(52)

(53)

(54)

with

In Eq. (55) P,,is the permutation operator and the spin quantum number S is such that S = 0 and 1 for singlet and triplet states, respectively. Equation (52) is the differential equation for the radial part of i,b(ls€lm, I r) and is coupled to (53) and (54) which are the differential equations for the radial parts of $(2sblm, I r) and J/(2p61miI r). The coupling takes partial account of

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

13

the perturbation of the core by the continuum electron and its removal reduces (52) to the Hartree-Fock equation P R.P. u( 1s I r , ) Yfz,(P2)(H- E )

J

1

x - (1

fi

+ (-

1)'PI2} u(ls I rl) $(lsblmf I r2) drl dP2 = 0 ,

(56)

which reduces to the simple form (48).

2. The Method of Polarized Orbitals A theory which takes account of the effect of core polarization on the continuum orbital has recently been developed by Temkin and Lamkin (1961) and shown to bederivable from a variation principle (Sloan, 1964). This method of polarized orbitals has been applied to e + H and e + He' scattering and shown to give scattering phases which are superior to those given by the closecoupling approximation when compared with the phases obtained by the quantum defect method (see Section IV,B). The method is presently being applied to photoionization of the two-electron systems by Bell and Kingston (1967). Essentially the method consists in replacing the core function Y(C I X j ) in (44)by Y(C I X)

= Y(C

I X j ) + Y'PO'(CI X),

(57)

where Ypo'(CI X) is an asymmetric function of Xj and xj which represents the perturbation of the core in the dipole approximation by the continuum electron located at x j . If the variational functional (41) is used the method further neglects second-order terms arising from the polarized part of the wavefunction. Thus the energy functional L is not stationary in this method. In the case of the two-electron systems Sloan (1964) writes

where the unperturbed core is represented by u(ls I rl) and

v Thus YPo'(rl, rz) is the dipole correction for the polarized one-electron core (cf. Dalgarno and Stewart, 1956) suitably modified by &(TI, r z ) so that the effect vanishes when the continuum electron is within the core. The variation principle used by Sloan leads to the differential equation

x +(lsBlrn, I r2) dr,, dP2 = 0.

(61)

which is similar to (56). Further analysis reduces (61) to the form (48) with the addition of terms arising from the polarization of the core, one of which is asymptotically the dipole polarization potential c(,/r4, the strength of which depends on the dipole polarizability q.

3. The Method of Correlated Orbitals The full effect of the interaction between the continuum electron and the core can be obtained only if the trial function Y ( B 1 X) depends explicitly on the distance r i j between the core and continuum electrons. Such wavefunctions have been obtained for two-electron systems by a number of authors but only in one case (Geltman, 1962) has the method been used to calculate a photoionization cross section. A method that includes correlation terms with the minimum of additional labor has recently been developed by Burke and Taylor (1966) and promises to yield interesting photoionization results, particularly in the resonance region. This method retains the close-coupling expansion, which has the desirable feature that it includes the correct long-range forces coupling the core and the continuum electron, and adds correlation terms, r:j, n 2 1, to make allowance for the short-range forces. The Burke and Taylor calculations on the 'S and 3S,e H and e + He+, continua show that the resonance widths differ considerably from those found by the Is-2s-2p close-coupling approximation. The correlation terms also effect the resonance positions, particularly the 2pz ' S resonance which was poorly given by the close-coupling approximation.

+

D. THENORMALIZATION PROCESS Equation ( 3 9 , or the variational or Hartree-Fock systems of equations derived from it, is usually solved without regard to the amplitude of the solution in the infinite asymptotic region. The procedure is then to examine the amplitude and adjust it so that (37) and (38) are satisfied. This is conveniently achieved by means of the Stromgren method (cf. Bates and Seaton, 1949). For

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

15

in the asymptotic region the radial part P of the continuum orbital satisfies an equation of the general form d2P dr2

-+ A(r) P ( r ) = 0, which has the solution p ( r ) = 71- Wz-112 sin r z dr,

(63)

where

Since A-6+

2(Z - N

+ 1) -~ t!(l + 1) r2

’

the boundary condition (38) is satisfied by (63). Thus solutions with the required asymptotic amplitude can be obtained by solving (64). This can be achieved by iteration, and often a single iteration is sufficient, giving

In the discussion given by Cooper (1962), Eq. (64) is in error and accordingly his version of (66) is also in error. If the unnormalized solutions initially achieved in the asymptotic region, so that (62) holds, are written P‘ such that (1IN)P‘ = P and if we write 12

a, = n1/2z:/2P’(r),

u 2 = n1/2z:/2P‘(r2),

a =f

J rl

z dr,

(67)

then it can be shown that N =+[(a,

+ a2)’ sec2 a + (al - a2)’ csc2

(68)

An alternative normalization procedure is discussed by Temkin and Lamkin (1961).

E. THESOLUTION OF CONTINUUM DIFFERENTIAL EQUATIONS Most of the methods of determining continuum wavefunctions require, finally, the solution of a set of coupled integrodifferential equations. A convenient technique for numerical integration of such equations is that of Fox and Goodwin (1949). Because of the coupling, and of the integral form, the

A . L. Stewart

16

straightforward use of the integration procedure involves the necessity to iterate and the use of valuable memory space for the storage of intermediate solutions. Iteration can also lead to difficulties with convergence, particularly at an energy nearly coincident with a resonance or a degeneracy. For example, Eq. (48) for the 3 S continuum of H - has the bound solution Pio = re-‘ at 8 = 0.25, and at this energy completely damps out the oscillatory solution which is required and with which it is degenerate. The number of iterations required for a solution can be reduced by adopting for a particular stage of the iteration the arithmetic average of the two previous solutions (cf. Saraph and Seaton, 1962). The necessity to iterate can be avoided by manipulation of the equations in the manner developed by Percival (cf. Temkin and Lamkin, 1961), a method which has been extended to a set of coupled integrodifferential equations by Marriott (I 958). For continuum solutions it is only necessary to integrate outwards from the origin, the two starting values being calculated by means of series expansions at the origin. Otherwise the starting values can be taken as P,,(O) = 0 and P,,(w) = h, where o is the mesh width, and h is an arbitrary small number which is finally determined by the chosen normalization amplitude for the solution. For the investigation of resonances, closed channel, 8 < 0, solutions are necessary and a method of inward and outward integration with subsequent matching must be adopted. The problems which occur in the solution of coupled integrodifferential equations are detailed in the paper by Smith et al. (1966).

111. Resonance in the Continuum It is clear that if all possible core terms are included in (44)an exact solution of (35) can be obtained by solving exactly the complete set of differential equations (45). In each of these equations the energy Ecan be taken as consisting of two parts, one part associated with the core, E c , , and the other associated with the separated electron moving in the field of the core, 8,. . Thus we have the following relation for the different core terms :

E

= Ec,

+ bC,= Ec,. + &,.

=*

* *

(69)

With I C>.0 it is possible to have, for example, Ic

= 8Cr

- ( E p - E C , ) < 0,

(70)

so that it is necessary to solve (45) (with C replaced by C”) in the region of negative values of bc..yielding a solution $(C”8,,,/) corresponding to an electron bound to the core C”. The exact solution (44) needs necessarily to contain terms which are such that they satisfy the boundary condition for

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

17

bound states $(C’’&c,fl) 0. With the inclusion of these terms the curve for the probability of finding the system with energy E plotted against E can be shown to have the form of a resonance curve, and it follows that physical properties of the system will exhibit resonance effects at the resonance continuum energy which, in the case considered above, is N

Considered in isolation the resonance terms in (44) are “closed” eigenfunction representations of multiply excited states of the N-electron system and since in (44) they are coupled with the “open” continuum states they decay with a lifetime that is inversely related to the strength of the coupling. Physically the decay takes place by deexcitation of the core and the radiationless emission of an electron into the continuum. The atom is said to autoionize. However, it is clear that this is never a separate ionization process, either theoretically or practically. For example, in ionization by photoabsorption it is the state of energy E, satisfying (69) and represented by (44), which is created. Only approximately is it possible to associate separate cross sections with the individual terms of (44) and so determine separate cross sections for the processes of photoionization and photoexcitation followed by autoionization. The nonseparability of the resonance and continuum terms in the wavefunction leads to formal difficulties of definition and interpretation if a method of “configuration interaction ” is used to solve (35). By this method configurations which are solutions of equations approximating to (35) are superposed and coupled by substitution into (35). Ideally, in the case of autoionizing states, the separate configurations would consist of closed ” configurations representing multiply exicted states with no continuum components and “ open ” configurations with no resonance components, the former being solutions of an approximate Schrodinger equation with no open channels in the energy region of interest and the latter being solutions of a Schrodinger equation with no closed channels in the energy region of interest. Although quite unnecessary (Burke and McVicar, 1965), separate configurations could be achieved in calculations based on (45) by removing the interactiors coupling different core parents so that the separate eqLations generate separate systems of states, each consisting of a bound series and an associated continuum with mutual orthogonality between the separate systems. Coupling could then be reintroduced by a theory of configuration mixing. A method of achieving this separation of systems of states has been described by Feshbach (1958, 1962) and applied to two-electron systems by O’Malley and Geltman (1965) and other workers. Formally, projection operators satisfying P+Q=1 (72) “

18

A . L. Stewart

are introduced so that the Hamiltonian can be written H =(P

+ Q)H(P + Q ) = P H P + Q H Q + P H Q + Q H P

(73)

and partitioned into an approximate Hamiltonian H , = PHP

+ QHQ

(74)

HI = P H Q

+ QHP.

(75)

and a correction Since for projection operators P 2 = P and Q2 = Q one has PQ

= QP = 0 ,

(76)

so that P and Q operate on a general wavefunction to project out orthogonal parts. Thus the solutions of the equations P H P Y p = EpYp

(77)

are orthogonal since (YQ1 Y

p )

= EP '(YQJ P H P JUp) = E; 'EP '(YQIQHQPHPI Yp) = 0 ,

and the matrix of the approximate Hamiltonian based on a representation in terms of Y pand YQis diagonal. It remains then to define the projection operator P so that (77) and (78) generate separate systems of states in the approximation H = H,, and which resonate under the perturbation H I in the energy region E p = EQ . Specifically for a resonance in the region above the first ionization potential Ic. = Ec, - EA of an atomic system and below the second ionization potential Ic.. = Ec- - EA it is desirable for Y pto embrace the first ionization continuum and for YQ to embrace the doubly excited bound states lying above the first ionization potential and below the second. Thus Y pmust have core parentage C' whilst YQcan contain all core parents except the ground core parent C'. This makes it possible to use the variational principle for bound states to obtain bound solutions of (78); for, if it exists, the lowest-lying solution of (78) must represent a doubly excited bound system. In the application to the two-electron systems O'Malley and Geltman (1 965) set, in the Dirac notation,

P =P,

+ P , - P,P,,

p i = lu(1s 1 ri)>(u(ls I rill

(79)

where u(ls I ri) is the wavefunction for the ground state of the one-electron parent ion, and use the variational method for bound states to solve (78) with Q = 1 - PI

- P2 + P I P , .

(80)

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

19

As desired Y, has no components in wavefunction space belonging to the u(1s)r) ground state parentage. On the other hand Y , is constrained to the form of the first term of (55) and the continuum orbital is the solution of the Hartree-Fock equation (56). In the continuum region 8 > 0, Y , has thus the simple form, which is usually assumed in calculations of the photoionization of a two-electron system (cf. Stewart and Webb, 1963). In contrast YQ may be a wavefunction of considerable complexity as it is in the treatment of O’Malley and Geltman. The resonance phenomenon in the two-electron systems occurs when the Y, functions obtained from (78), such as those of O’Malley and Geltman, and the Y continuum functions obtained from (77), such as those of Stewart and Webb, are combined in a theory of configuration interaction (Fano, 1961). This calculation has not been carried out, but the results should be comparable with those of Burke and McVicar (1965) who instead solve (45) for the twoelectron systems with a finite number of one-electron core terms, Is, 2s, and 2p. This close-coupling calculation requires the solution of the differential equations (52), (53), and (54) in the region below the second ionization potential, that is, in the region where the 2s and 2p channels are closed and the 1s channel open. Applied to the photoionization of normal helium this yields resonances corresponding to the photoabsorption anomalies observed by Madden and Codling (1965). With coupling between the Is channel and the pair of channels 2s and 2p removed, Eq. (52) reduces to (56) or (77), and (53) and (54) together form an approximation to (78) in the closedchannel region. Burke and McVicar (1965) also solve this pair of equations, uncoupled from the 1s channel, and so obtain representations of the separated resonance states which are approximations to those of O’Malley and Geltman. The separation of resonance and continuum states achieved through (79) and (80) is only applicable in the energy range between the first and second ionization potentials of the two-electron systems. For example, the doubly excited 3sns configurations interact with both the LsGs and 2sbs continua and the separate V, for this case must not contain any Components in wavefunction space belonging to either the u( 1s J r) or 4 2 s I r) one-electron parentage. The appropriate projection operators for this and higher-energy cases are given by O’Malley and Geltman (1965). The difficulties of applying the projection operator technique to more complex systems have not yet been solved. Only in the two-electron systems are the core terms known exactly but this may not be critical. Greater inaccuracies may arise from the necessity of expressing the core wavefunctions in terms of one-electron orbitals to ensure that the resonance states contain no components belonging to the wavefunction space of states with ground core parentage.

,

A. L. Stewart

20

A. RESONANCES AND CONFIGURATION INTERACTION To illustrate the nature of resonances in the continuum it is of interest to discuss the theory of Fano (1961) which uses the configuration interaction approach. Consider an atomic system which can be represented approximately by the wavefunctions (P, ,n = 1 , 2, . . . ,k belonging to discrete configurations and by $ E , belonging to a continuum configuration, all of which are nondegenerate. It will be assumed that this basic set of functions has been derived so that (4, IHI 4rn) = Ern6nm (81) 7

!HI$ E )

= E’ 6(E” - E’),

(82) where H is the Hamiltonian for the system. These are realistic assumptions. For example, 9, might be the solution of (78) obtained using a variation principle and @ E , the solution of (77). Allowing for configuration interaction the system with energy E is more accurately represented by ($E”

and a E sso that The energy matrix is diagonalized by choosing (I, a,(E,

-E) + k

s

n = 1,

a , g , * , ,dE‘ = 0,

C a , gEpn(E)+ a,.(E’ - E ) = 0,

n= 1

...

k,

(84)

(85)

where gE’n = ($E’

IH - E l

In what follows it need not be assumed that the case studied by Fano. Equation (85) gives

which on substitution into (84) yields k

where

$E,

and

(P”

(86) are orthogonal as in

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

21

and

s

Fmn(E) = P g;sm(E) g E Z n ( E ) ( E - E')- dE',

(90)

the principal part of the integral being indicated by P. Thus there is a nontrivial solution when I(Em - Q6mn

+ Fmn(E) + z Gmn(E)I = 0.

(91)

This determinantal equation is linear in z and gives

where lxmnl = I ( E m

- E P m n + Fmn(E)l

(93)

and lymn(r)1is the determinant lxmnlwith the rth row replaced by the rth row lymn(r)lare real and of the determinant IGmnl.The determinants lxmnl and so z is real. lxmnlis just the secular determinant for the energies of the bound states coupled through the continuum. Some manipulation of determinants shows that

cr

and the normalization condition

(Y(E)I Y ( E ) ) = 6 ( E - E )

(95)

gives I n

Thus the normalized solution, obtained by comparing (94) with (88) and using (96), is

which on substitution into (87) gives

only gE,,,(E)and 6 ( E - E') on the right being functions of E'.

22

A . L. Stewart

The resonant form of the function (83) is apparent. For example, with a single bound state p , Eq. (97) gives

Thus the probability of finding the system in the bound state p varies with energy in the form of a resonance curve with half-width

tr, = nGpp, so that if the system were prepared in the state q, it would autoionize with the mean life

assuming that G,, and Fppvary little with energy E. The resonance is centered at

E - E , - F,, = 0,

( 102)

i.e., at the bound state energy E, shifted by interaction with the continuum through Fpp. In the general case the resonance is centered at the zeros of the secular determinant lxm,,l when z = 0 with the half-width

again assuming that the elements of the determinants IFmnIand lGmnlvary slowly with energy E. The resonant structure is also apparent in the shift of the phase of (83) when one of the electrons is removed to infinity. Thus if the asymptotic form of q E ,is given by (37) and (38) the asymptotic form of Y ( E ) using (98) can be shown to involve the shift in phase due to interaction with the bound configurations which is (Fano, 1961)

This can be used to give a precise meaning to the autoionization lifetime of the bound configurations, as the inverse of the time delay in the elastic scattering. The delay time due to configuration interaction is (Eisenbud, 1948; Smith, 1960)

h dA 271 dE’

At=--

23

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

which, on introduction of (104), gives for the mean life for autoionization

At resonance, where Jx,,,,J = 0, this becomes

which is in accord with (103) since together they satisfy

zr

= h,

(108)

the uncertainty principle relation.

B. RESONANCESAND PHOTOIONIZATION Consider the matrix element ni, E ) =

(Wi) IT/W E ) )

(109)

controlling the rate of transitions under the perturbation T between a bound state of an atomic system Y(i) and a resonating continuum state Y ( E )given by (83). In the case of photoionization (109) would be the matrix element for the line strength (2). Introducing (83) and (97) and (98) we obtain

where E-E’

’

which can be written in the form

= 0, the second term vanishes, so that at the resoAt resonance, where lxrnnl nance energies transitions to the bound configurations are dominant. The variation of T(i, E ) with energy near the resonances is rapid and this causes the ratio of the transition matrix element to the pure photoionization transition matrix to be an important physical quantity. We have

24

A . L. Stewart

where q, is referred to as the line profile index. If q is approximately independent of energy it can be seen that the cross section is zero when E = - q and has its maximum when E = q-'. The curve (1 13) for various values of q is displayed in Fig. 1, giving an absorption line shape of the traditional kind when 141 is large but an absorption window when q is zero.

c

FIG.1. Natural line shapes [Eq. (113)l.

It is evident from (116) that if the resonance states 'p, constitute a Rydberg series then q, varies little from series member to series member, for (Y(i) IT1 0,) and gEp(E)are both proportional to, and controlled in magnitude

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

25

by, the normalizing constant of qn, so that their ratio is nearly constant (Fano, 1961). Furthermore, since (1 15) can be written

with

q is in reality an averaged value of the separate line profile indices. It can be shown that for a Rydberg series the normalization constant of the resonance states is proportional to v - ~ ' ' , where v is the effective quantum number of the resonance level. Accordingly the modified width of the level y p = +v3TP= nv3G,,

(118)

is, using (loo), only a slowly varying function of energy along a Rydberg series. When two or more continua interact with the resonant states the simplicity of the preceding theory is lost (Fano, 1961 ;Fano and Cooper, 1965). However, with a single resonant state Fano has shown that in this case the total continuum can be divided into two orthogonal parts, one of which interacts with the resonant state and one of which does not. Thus the equivalent of (113), written in terms of cross sections, becomes

where crO is the cross section due to the resonant continuum and ob is the cross section arising from the orthogonal nonresonant continuum. Corresponding to (1 16) one has

where

rp= 2n

G;, i

and the elements of the j sums are the interactions of the separate continua $ j E with the resonant state 'pp and the initial state Y ( i ) taking part in the transition.

26

A . L. Stewart

Whether or not more than one continuum channel is open can be determined from (119) since, with E = -4, the cross section is only zero if there is but one open channel. This can also be determined from the correlation coefficient defined by Fano,

which is unity when only a single continuum is operative. We shall be interested later in physical situations in which there is only a single continuum and where the resonances due to the quasi-discrete states are well separated. In the vicinity of the resonance p then (1 19) becomes, in the oscillator strength form,

where (df/d&), represents the continuum background and is just the limit of df/dS as E + co.The effective oscillator strength of the quasi-discrete level is obtained by integrating (123) and subtracting off the continuum background giving (Fano, 1961)

where all energy quantities are in rydbergs. Equation (123) shows that a net decrease in the absorption is possible if lqpl < 1. Introducing the modified width yp which is approximately constant for a Rydberg series, Eq. (1 18), we have

where q p is also approximately constant. Thus the theory predicts that the oscillator strengths for transitions to a Rydberg series of resonant levels should fall off as v - ~ ,as is the case for transitions to a Rydberg series of real bound levels.

IV. Oscillator Strength Sum Rules A knowledge of the oscillator strengths for transitions from a particular atomic level can be used to determine many of the properties of that level which may be difficult to obtain otherwise. Transition probabilities for bound-bound

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

27

as well as bound-free transitions are necessary for this technique which employs the sum rules (Vinti, 1932a,b; Dalgarno and Lynn, 1957)

Conversely the sum rules can be used to check the accuracy of available oscillator strengths. These equations, which are expressed in atomic units and rydbergs, refer to an atomic system of nuclear charge Z containing N electrons, with position vectors r, and momenta ps, and sum the oscillator strengths for transitions from the state m with energy Em to all final states n, including the continuum, with energy E n . The quantity PI can be determined from experimental data on the refractive index and the Verdet constant of the system in the level m, P2 can be determined by a variation principle (Schwartz, 1961), a1 is the dipole polarizability of the system in level m, and, if angular correlation is not an important effect in level m, S(-1) can be related to the diamagnetic susceptibility. Further S( 1) gives information about the mass polarization isotope effect and S(2) is an important term in the relativistic shift of the level. These sums are obviously related to the mean energies d -1 dk log ~ ( k =) f m n ( E n - Em>' log(En - ~ m ) ) frnn(En - Em)'] * (133)

(c

(c

With k = 0 and k = 1 this is the expression for the logarithm of the mean excitation energies, log I and log I,, which control the stopping power and straggling, respectively, of atoms in collision with fast charged particles

A . L. Stewart

28

(Livingston and Bethe, 1937; Fano, 1963), and with k = 2 it is related to the mean excitation energy which enters the expression for the Lamb shift (Dalgarno and Stewart, 1960). Except near the singularity, which for an s - p transition is at k = 5/2 [see Eq. (33)], the function S(k) varies slowly with k and so can be well defined knowing only a few points. Accurate interpolation and extrapolation is possible to find values for sums and associated atomic properties which are unknown (see Section V).

A. THEGENERALIZED OSCILLATOR STRENGTH The probability of excitation or ionization of an atomic system in collision with electrons or other charged particles is clearly related to its optical properties. For example, at high energies of impact, when the first Born approximation is valid, the cross section for an electron- or proton-induced transition from a nondegenerate state tjo to a nondegenerate state of the continuum $# depends on the generalized oscillator strength of Bethe (Mott and Massey, I 949),

df(K)= ) ( I db

+ 8)

(134)

where K is the momentum change in the collision. The similarity of (134) and (10) and the fact that they are equal for an optically allowed transition in the limit K --t 0 is clear. In general it is a simpler task to calculate the optical oscillator strength than the generalized oscillator strength so that often it is necessary to use wavefunctions in (134) which are inferior to those used in (10). In such a case the ratio of the generalized oscillator strength at K = 0 to a well-determined optical value can be used as a correction factor over the whole range of K to derive superior electron collision cross sections (cf. Miller and Platzman, 1957). On the other hand experimental differential cross sections for electron collisions can be used to determine optical oscillator strengths at the limit K + 0 (Silverman and Lassettre, 1964; Kuyatt and Simpson, 1964) both for the case of excitation and photoionization.

B. GENERAL FORMULAS FOR PHOTOIONIZATION CROSSSECTIONS The theoretical analysis of stellar and laboratory spectra requires a knowledge of photoionization cross sections for a wide variety of transitions over a comprehensive range of energies. This is a forbidding task using ab initio calculations starting from formulas based on a detailed knowledge of wavefunctions. It is desirable then to have available formulas and tables from which

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

29

photoionization cross sections can be readily obtained, based on a knowledge of a small number of atomic parameters. Bates (1 946a) gives formulas which express the photoionization cross sections in terms of the Slater orbital expansion coefficients of the initial bound wavefunction, assuming that the state of the ejected electron can be adequately represented by a regular Coulomb function. The formulas of Burgess and Seaton (1960) are of wider validity. The approximation is equivalent to that of Bates and Damgaard (1949) who have published tables which allow rapid evaluation of bound-bound transition probabilities. The method assumes that the radial orbitals of the optical electron used in (17) are solutions of (27) and (28), respectively, where the potentials G,,, and G,,,*, are Coulombic, and the energy E , , ~ , which determines the long-range asymptotic form, is taken from the known energy levels of the initial atomic system. The asymptotic form of the continuum orbital is fixed by taking the phase shift 6 to be that given by the quantum defect method (Seaton, 1958)

6 = 7cp,

(135)

where p is the bound energy level quantum defect extrapolated into the relevant region of the continuum. Some further, but important, approximations are made in the solution of (27) and (28). In nearly all the cases studied by Burgess and Seaton their general formulas give results at least as accurate as those obtained using the best alternative methods of calculation. The theory has also been applied to neon (McGuire, 1965) and the rare gases (Schluter, 1965). Recently a more comprehensive set of tables from which photoionization cross sections can be obtained using the theory of Burgess and Seaton has been compiled by Peach (1966). Further, Seaton and co-workers have considered the general treatment of the many-electron problem using the quantum defect theory (Bely et al., 1963; Seaton, 1966a) to account successfully for series perturbations and resonances (Seaton, 1966b). The theory has been applied to e - He' collisions (Bely, 1966), and to photoionization and autoionization in calcium (Moores, 1966).

c. EXTRAPOLATION AT THE SPECTRAL HEAD The principle of continuity at the spectral head implied in (135) applies to other atomic properties. In this particular connection (Hargreaves, 1929) it can be shown that

where v is the effective quantum number of the upper level in transitions to

30

A. L. Stewart

the members of a Rydberg series. Thus the curves versus & ' I 2 and fv3fv versus -z/v should be smoothly continuous across the spectral head for a Rydberg series with effective Coulomb charge z (Burgess and Seaton, 1960). Alternatively (136) allows the bound-bound oscillator strengths to be determined from the bound-free oscillator strengths using the relation

f, = C/v3,

C = 2 lim df/d&. 8+0

(137)

V. Comparison of Experimental and Theoretical Results The recent review by Samson published in the previous volume of this series contains a comprehensive comparison between the best experimental and theoretical determinations of photoionization cross sections for a large number of atomic systems. The following discussion is an appendix to the Samson review, illustrating the theory of the previous chapters.

A. PHOTOIONIZATION OF HELIUM With only two electrons, helium and its isoelectronic ions are sufficiently simple so that almost complete allowance for correlation can be made in the discrete wavefunctions (cf. Stewart, 1963). Although similar accuracy has not been achieved for the free states, there are available heavily correlated wavefunctions for the continuum, which have however only been used for the purpose of photoionization in the case of H - (Geltman, 1962; Doughty and Fraser, 1964). Thus the best available calculations of the photoionization cross section of helium are based on Hartree-Fock or close-coupling wavefunctions for the continuum. Cross sections obtained using the method of polarized orbitals should be available soon (Bell and Kingston, 1967), and will be presented in an appendix to Table I. The most recent calculations are due to Burke and McVicar (1965) who use the 20-parameter ground state wavefunction of helium (Hart and Herzberg, 1957) together with 1s-2s-2p close-coupling wavefunctions for the continuum. The coupling takes account of the resonances lying below the second ionization potential. It is of interest to compare these calculations with those of Stewart and Webb (1 963) in which Hartree-Fock representations of the continuum are taken together with a six-parameter ground state wavefunction but which do not include the resonance effects. The results using both the length and velocity formulas are given in Table I, from which it is clear that from 0-2 Ry the change in complexity of the wavefunctions used has changed the oscillator strength by less than 2%. As the tail of the resonance region is reached the values of Burke and McVicar fall off less rapidly than those of

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

31

Stewart and Webb, as they should, so that at 2.5 Ry they are some 10 %larger, a difference which is in large part due to the first resonance which lies at 2.623 Ry. Included in Table I are the most recent experimental measurements. In contrast to the radiation absorption experiments of Samson (1964) and Lowry et al. (1965), the Silverman and Lassettre (1964) and Kuyatt and Simpson (1964) values are derived from high-energy electron collision experiments from which a generalized oscillator strength and accordingly an optical oscillator strength can be obtained as explained in Section IV,A. The agreement between theory and experiment at the spectral head is excellent but becomes less and less good as the energy increases, particularly when the comparison is made with the results of Samson (1964). However, this may not be significant since there is considerable scatter in the values obtained in this experiment in the region beyond 1 Ry. On the other hand the agreement between theory and the experimental values of Silverman and Lassettre (1 964) and Kuyatt and Simpson (1 964) in the region between 1 and 2 Ry is still good. In the region of the resonance tail, 2.0-2.5 Ry, the scatter of points about a smooth curve in both the experiments of Silverman and Lassettre and of Lowry and collaborators makes comparison with the theory of Burke and McVicar difficult. However, the essential agreement in this region is encouraging (see the graph published by Kuyatt and Simpson). Altogether nine resonances appear in the theoretical oscillator strength curve obtained by the close-coupling calculation of Burke and McVicar (1965). These are too narrow to display in a table or a comprehensive graph. Where comparison is possible the positions and widths of the resonances agree with the calculations of O’Malley and Geltman (1965) and the absorption experiments of Madden and Codling (1963, 1965) and the electron transmission experiments of Simpson et af. (1964). For the purpose of illustration attention will be concentrated here on the 2s2p Presonance and on the (2s3p 3s2p) ‘P, (2s4p + 4s2p) ‘P,and (2s5p + 5 . ~ 2 ~‘P ) resonances which form with it a Rydberg series. Burke and McVicar have analyzed these resonances using the Fano theory given in Section III,B, assuming that the resonances are sufficiently sharp and sufficiently separated, so that the single resonancesingle open-channel theory is applicable. The results are given in Tables I1 and 111 for both the dipole length and dipole velocity calculations of (d’/d~?)~, each curve being analyzed using (123). Since the background oscillator strengths (dfld€), are strictly comparable with nonresonant calculations the values obtained by Stewart and Webb at the same energies are also given in Table 11. The differences are seen to be in the region of 4 to 6 %. The fact that the oscillator strength curves computed by Burke and McVicar can be accurately fitted to the analytic form (123) is striking confirmation of Fano’s theory. The absorption experiments of Madden and Codling (1963, 1965) provide further confirmation in the case of the 2s2p ‘P resonance, the

+

w

N

TABLE I

THEOSCILLATOR

STRENGTH

dfld8 OF THE TRANSITION IS2 *%1dp 'PIN HELIUM (RYDBERG UNITS)

Experiment

Theory Dipoie length

Silverman and Lassettre ( I 964) 0.00 0.25 0.50 0.15

I .oo 1.25 1S O 1.75 2.00 2.25 2.50 2.75 3.oo

0.91 0.75 0.64 0.54 0.43 0.34 0.28 0.24 0.19 0.16

0.23 0.14

Lowry ef a/. (1 965)

Dipole velocity

Samson (1964)

Stewart and Webb (1963)

Burke and McVicar (1965)

Stewart and Webb (1963)

Burke and McVicar (1965)

0.91 0.75 0.60 0.50 0.40 0.32 0.25 0.18

0.15

-

0.159

0.14

-

0.139

0.909 0.749 0.620 0.517 0.435 0.369 0.317 0.274 0.240 0.212 0.194 -

0.886 0.729 0.601 0.499 0.41 8 0.353 0.302 0.259 0.224 0.196 0.172 0. I52 0.133

0.895 0.734 0.606

-

0.916 0.763 0.632 0.525 0.440 0.371 0.316 0.271 0.234 0.205 0.180

-

-

0.22 0.20 0.18 0.19

0.505

0.425 0.361 0.309 0.268 0.234 0.207 0.190 -

b

P

APPENDIX TO TABLE I THEOSCILLATOR STRENGTH dfid8 OF THE TRANSITION, 1s' 'S-Is8p 'P IN HELIUM OF POLARIZED ORBITALS (RYDBERG UNITS): THEMETHOD ~

~

~

~~

Dipole length

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

¶meter ground state

20-parameter ground state

0.947 0.784 0.648 0.539 0.451 0.378 0.325 0.281 0.242 0.212 0.186 0.164 0.145

0.937 0.772 0.639 0.532 0.447 0.378 0.325 0.281 0.243 0.213 0.1 88 0.166 0.147

~~

~

~

~~

Dipole velocity ¶meter ground state 0.972 0.798 0.658 0.547 0.459 0.387 0.331 0.285 0.247 0.216 0.189 0.167 0.147

20-parameter ground state 0.972 0.797 0.657 0.547 0.459 0.388 0.332 0.286 0.247 0.217 0.190 0.168 0.147

c)

7J 0

v v) l v)

rn

2 0

z

r A

w w

34

A . L. Stewart TABLE I1

THEBACKGROUND OSCILLATOR STRENGTH (dfld&), OF THE TRANSITION Isz 'S-Is@ 'P IN HELIUM(RYDBERG UNITS)CALCULATED AT THE CENTERS OF THE (2snp 2pns) 'P RESONANCES [See Eq. (123)]

+

Dipole length

Dipole velocity

n

8

Stewart and Webb (1963)

Burke and McVicar (1965)

Stewart and Webb (1963)

Burke and McVicar (1965)

2 3 4

2.6227 2.8742 2.9323 2.9575

0.168 0.148 0.144 0.142

0.1754 0.1550 0.1505 0.1485

0.161 0.142 0.138 0.136

0.1710 0.1510 0.1466 0.1446

5

TABLE 111 THEWIDTHS,MODIFIED WIDTHS,AND LINEPROFILE INDICES FOR THE (2snpf2pns) 'P RESONANCES APPEARING IN THE CALCULATIONS OF BURKE AND MCVICAR (1965) (RYDBERG UNITS)" Eq. (117)

n En 2 3 4 5

Eq. (103)

+ F,,-E(lsZ IS)

v

4.4298 4.6813 4.7394 4.7646

1.6279 2.8191 3.8426 4.8514

Eq. (118)

r 0.003216 0.0006408 0.0002712 0.0001390

Y 0.006937 0.007178 0.007694 0.007937

9L

-2.59 -2.44 -2.42 -2.41

4v

-2.65 -2.51 -2.49 -2.48

The quantities tabulated here when substituted into (123) reproduce the actual oscillator strengths calculated by Burke and McVicar (1965). (I

measured resonance width and line profile index, r = 0.038 eV and q = 2.80, comparing favorably with the Burke and McVicar calculated values, r = 0.04375 eV, q,, = -2.65, qL = -2.59. The dipole velocity oscillator strength curve plotted against E = ( E - Ep - Fpp)/+rpfor the 2s2p 'P resonance is displayed in Fig. 2. For comparison the experimental points of Lowry et al. (1965) and Silverman and Lassettre (1964) in the neighborhood of the resonance are given numerically in Table 1V. None of the experimental points is sufficiently close to the resonance center to provide a severe test of the theory. Although the resonance region is minutely covered by the data of Kuyatt and Simpson, their graph is too small to allow points to be read off with sufficient accuracy. Their resonance maximum for the 2s2p excitation seems to lie at df/d& = 0.27 which is very much lower than the peak height dAd8 = 1.4

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

35

calculated by Burke and McVicar. The peak is narrow and it is possible that its apex has been missed in the Kuyatt and Simpson experiment. This is supported by the Silverman and Lassettre result at 60.1 eV which lies at dfld8 = 0.38. The point of maximum absorption lies at E = q-' and is therefore displaced from the resonance center at E = 0 by r/2q.This must be taken account of when comparing the results of absorption experiments with calculations on the positions of the resonances.

I

~

-20

-30

-10

0 E

20

10

2

FIG.2. The oscillator strength dfldg of the l s 2 1 S - 1 ~ 81P~ transition of helium near the 2s2p lP resonance. This curve is calculated using the dipole velocity formula and the theory of Burke and McVicar (1965). The units are rydbergs. TABLE IV

THEOSCILLATOR STRENGTH d f l W OF THE TRANSITION1s' l S - l s ~ p'PINHELIUM(RYDBERG UNITS) NEAR THE 2S2p 'P RESONANCE &

- 62.57 -27.04 - 19.84

-7.6 25.49 46.69 47.15

Experiment 0.23" 0.185' 0.192b 0.38" 0.1 S o b 0.151' 0.19"

Theory' 0.186 0.206 0.219 0.305 0.137 0.152 0.152

Silverman and Lassettre (1964). Lowry et al. (1965). Burke and McVicar (1965), dipole velocity.

A . L. Stewart

36

The effective oscillator strengths of the resonance lines can be obtained from the calculations of Burke and McVicar using (124). The results obtained using the dipole velocity values of the line profile index and background continuum oscillator strength are given in Table V. There is little comparison data available. For excitation to the 2s2p state the calculation of Vinti (1932a,b) gives TABLE V EFFECTIVE OSCILLATOR STRENGTHS FOR THE ISz 'S-(2Snp f IN HELIUM

n 2 3 4 5

Burke and McVicar (1965), dipole velocity

nS2p) 'P TRANSITIONS

Direct calculation dipole velocity Eqs. (138), (139)

0.00520 0.000806 0.000325 0.000163

0.00232 0.000614 0.000252 0.000128

0.01 8/v3

0.0042 0.00080 0.00032 0.00016

f= 0.01 1 and on the basis of the experiments of Silverman and Lassettre an oscillator strength lying between 0.002 and 0.004 is favored by Fano (1961). It is interesting to note, however, that similar values to those of Burke and McVicar are obtained using the dipole velocity approximation together with modified hydrogenic wavefunctions for the resonance states and the wavefunctions of Green et al. (1954) for the ground state. These are of the form 1

W s n P 'P)= - {4@s

J2

Irl) 4 m I r2> + w

s I r2) IC/(np1 r l ) } ,

(138)

where the 2s orbital is hydrogenic with 2 = 2 and the np orbitals are hydrogenic with Z = 1, and

These values are also given in Table V. In agreement with (125) the results of Burke and McVicar can be fitted to

f" = O.O18/V,

(140) as is done in Table V. Thus if the continuity rule (136) applies at the threshold for double excitation and ionization, as it does at the first ionization threshold (Samson, 1964), we obtain the value

"f (1s' db

' S 4(2sdp + ds2p) 'P)= 0.0090

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

37

at the second ionization potential, due to this process. This is 0.0099 of the oscillator strength at the first spectral head and is somewhat less than the ratio 0.062 assumed initially by Dalgarno and Stewart in their calculation of the Lamb shift. B. PHOTOIONIZATION OF NEON The experimental situation on the measurement of the photoionization cross section of neon has been summarized by Samson (1966). The results are sufficiently extensive to allow the use of the oscillator strength rules to check the values and to determine the atomic properties of neon (see Section IV). Such an assessment based on the cross sections of Ederer and Tomboulian (1964), which are in excellent agreement with the electron collision values of Kuyatt and Simpson (1964), has been made by Piech and Levinger (1964). The energy range covered was extended beyond photon energies of 13 Ry using the experimental data of Dershem and Shein (1931), Bearden (1963), and Allen (1935). At energies above 7353 Ry, Piech and Levinger assume that absorption takes place through the K-shell electrons, which are nearly hydrogenic, and take a - 7/2 power law for the oscillator strength in this region, the constants being determined using the Born approximation (see Section II,B) (Heitler, 1954). To obtain an estimate of the contribution from discrete transitions Piech and Levinger force the sum rule S( -2) to fit the measured polarizability c1 = 0.398 x cm3 (Cuthbertson and Cuthbertson, 1911). With an average excitation energy of 1.4 Ry, midway between the discrete excitation limits 1.22 and 1.58 Ry, the contribution of discrete transitions to the other sums is obtained. The results are displayed in Table VI where they are compared with other TABLE VI

OSCILLATOR STRENGTH SUMSS(k) FOR NEON Experimental

S(k)

S(-2) S(- 1 ) S(0) S(1)

S(2) a

Piech and Levinger (1964) 0.67 1.96 10.32 318.5 11.16 x

Direct estimates.

Samson (1966) -

9.7

lo4

Theoretical

Dalgarno and Kingston (1960)"

Sewel (196Sa)

Bell and Dalgarno (1965)"

0.6658 1.925

0.670 1.922 9.86

0.66575 2.0225 10 302.85 10.383 x lo4

-

-

-

-

-

38

A . L. Stewart

experimental and theoretical estimates. The difference between the value for S(0) and that obtained in the experiments of Samson (1966) reflects the difference between the experimental oscillator strengths of Ederer and Tomboulian and those of Samson which separate by some 10% in the 200-100A region. The values attributed to Sewell (1965a) are based on oscillator strengths calculated using the Hartree-Fock approximation and are accordingly strictly comparable with the direct estimates of Bell and Dalgarno (1965) which are obtained using the right-hand sides of the sum rule expressions and HartreeFock wavefunctions. The direct experimental estimates are due to Dalgarno and Kingston (1960) who carefully analyzed the published data on the refractive indices and Verdet constants of neon [see references in Dalgarno and Kingston (1960)]. The agreement between the experimental and theoretical sum values given in Table VI is seen to be excellent and is a striking verification of the accuracy of the Hartree-Fock approximation to the oscillator strengths of neon. Certain inaccuracies and possible errors in the papers referred to in this section should be pointed out here. Thus the sum rule for S(l), Eq. (131), is different from that quoted and used by Piech and Levinger who appear to be in error. Further, certain of the values of atomic quantities calculated by Piech and Levinger in their analysis of the sum rules are questionable. These authors obtain

s+t compared with the value - 29.95 calculated by Bell and Dalgarno (1965) using their Hartree-Fock wavefunctions. A possible defense of the Bell and Dalgarno estimate of the mass polarization operator is that with the experimental value S(1) = 318.5 it gives, using (131), Eo = -269 Ry compared with Eo = -314 Ry, which is obtained using the Piech and Levinger estimate, and Eo = - 258 Ry, which is the nonrelativistic binding energy of neon calculated by Scherr ef al. (1962). In view of the success of the Hartree-Fock approximation for neon a table of values of atomic quantities, which occur in the sum rule expressions, calculated by Bell and Dalgarno, is provided in Tab!e VII. To calculate the stopping power and straggling mean excitation energies Z and Z,, using (133), it is assumed that S(k) may be written

S ( k ) = { a + bk

+ C k 2 + d Iog(5 - 2k)}k S(0)

and the constants a, b, c, and d a r e determined by requiring this expression to reproduce the Hartree-Fock values for the sums (128) to (132). It would be of interest to compare the similar work on beryllium, argon, and krypton carried out by Bell and Dalgarno using the Hartree-Fock

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

39

approximation with the most recent experimental oscillator strength data (Samson, 1966). This has not yet been done. The Hartree-Fock results for beryllium, argon, and krypton, as well as neon, are given in the report by Bell (1965). TABLE VII VALUES OF ATOMIC QUANTITIES FOR NEON (ATOMIC UNITSAND RYDBERGS)

Bell and Dalgarno (1965)

Piech and Levinger ( 1964)

Experiment

~~

C C r,. S

r,

-3.2965

- 3.01

9.364

-

t

S f t

C r,'

9.66"

I

-257.09

Eo

C CP,*P, S

-29.952

~

75.2

-257.88b -

t

S t t

10.383 x lo4 I 11

150 1200

10.4 x lo4 -

-

140' -

Havens (1933). Scherr ef al. (1962). Bichsel (1961).

C. AUTOIONIZING LEVELSI N NEON Madden and Codling (1963) have observed, in absorption measurements in neon, a series of lines corresponding to the transitions ls22s22p6' S o + ls22s2p6np 'PI0 and these have also been observed by means of an electron impact technique (Simpson ef a/., 1964). Although these lines have not been investigated theoretically in the detail of the helium autoionizing lines, the recent work of Sewell (1965b) indicates that calculations based on the HartreeFock approximation should be of value. In the neon continuum the Hartree-Fock approximation consists in solving (44) and (45) with a single fixed orbital core term Y(C 1 X j ) corresponding to the ls22s2p6 2S1,2 state, which with the addition of an np electron yields the ls22s2p6np'PI0state. A comparison of the orbital energies obtained by Sewell with the observed energies is given in Table VIII. The agreement is excellent, especially for the higher members of the series. The values of

TABLE VIII WAVELENGTHS OF AUTOIONIZING LINES IN NEON,ls22s22p6--z ls22s2p6np

(A) n

Experiment"

Estimatedb

Theory without exchange'

Theory with exchange'

3 4 5 6 7

272.21 263.1 1 259.96 258.48 257.68

272.9 263.3 260.0 258.5 257.7

268.55 262.03 259.48 258.22 257.52

270.66 262.74 259.80 258.40 257.63

a

Madden and Codling (1963). Estimated using binding energies of alkali metals (Samson, 1966). Hartree-Fock theory of Sewell (1965b).

Ionization energy (Ry)

0.19617 0.094610 0.055339 0.036289 0.025733

u2 x lo3 (a.u.) [see Eq. (17)]

10.02 3.466 1.512 0.07955 0.04299

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

41

the line strength o’, given by Eq. (17) with li = 0, If = 1, are also given in Table VIII. It would be of interest to use the Hartree-Fock theory together with the Fano theory of configuration mixing to calculate the shift and width of these autoionizing lines in neon.

D. PHOTOIONIZATION OF LITHIUM There are now available a number of highly accurate correlated wavefunctions for the bound states of three-electron systems (cf. Weiss, 1961, 1963) and these have been used for the calculation of oscillator strengths and other atomic properties. However, apart from the work of Tait (1964), the photoionization cross sections have not been calculated in a manner taking account of the correlation effect, despite the simplicity of the system. The purpose of the Tait (1964) calculation was to investigate, without undue labor, the effect of correlation in the initial bound state and so use was made of published wavefunctions for both the initial and final states. Accordingly the calculation contains a number of defects considered of negligible importance by the author. The initial state (James and Coolidge, 1936) contains an admixture of 4S symmetry and the final state consists of the HartreeFock continuum orbital of Stewart (1954) with non-Hartree-Fock core orbitals. A comparison of the results with experimental (Hudson and Carter, 1965a) and previous theoretical values is given in the companion review by Samson (1966). In view of the rather large separation of the dipole length and dipole velocity curves obtained by Tait a further calculation using more complex wavefunctions would be of interest. The difficulty envisaged by Tait in doing this seems only apparent (see Section I1,C).

E. PHOTOIONIZATION OF SODIUM It is a considerable defect in the Hartree-Fock theory of the motion of a valence electron that the central potential which it allows the core to provide is essentially Coulombic except for terms which fall off exponentially at long range. In reality the core is polarized by the presence of the valence electron and reacts by providing a polarization potential which may tend to control the photoionization cross section in cases where the cross section is small. The influence of the polarizability of the core on the photoionization cross section of atomic potassium demonstrated by Bates (1947) is an important example. That the inclusion of polarization has a critical effect on the oscillator strength for the ls22s22p63s’S-ls22s22p66?p ‘P transition in sodium is apparent from the calculations of Boyd (1964). The radial functions for the initial and final states are assumed to be solutions of equations of the form

R

TABLE IX THE

ABWRPITON CROSS SECTION (10-'8 CmZ)FOR THE 3S-@

TRANSITION IN SODIUM AT THE

SPECTRAL HEAD U3s--tlJ

Reference Boyd (1964) Boyd ( I 964) Seaton (1951) Seaton (1951) Burgess and Seaton (1960) Hudson (1964) Ditchburn et al. (1953)

Theory Dipole length Dipole velocity Dipole length, Hartree-Fock Dipole velocity, Hartree-Fock Quantum defects

Theory

Experiment

0.136 0.126 0.096 0.0725 0.18 0.130%0.018 0.116* 0.013

b

P

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

43

(27) and (28) with a central potential which takes account of exchange and polarization in the manner described by Biermann and Lubeck (1948). Thus G is taken to be

G = -2VH{1

+ A/.? r e~p[-(r/r,)~]} - (c(/r4)(l- e~p[-(r/r,)~]}, (143)

where VH is the Hartree self-consistent field potential, ro is the effective radius of the Na' core, and a is the polarizability of the core. The term involving Afl allows for exchange effects, being assumed constant for the bound and continuum states of each I series and chosen to give the correct energy for the bound states. The similarity of the method of polarized orbitals (see Section II,C,2) and the theory used by Boyd is clear. The calculations of Boyd at the spectral head are compared with the experimental values obtained using the most up-to-date techniques (Hudson, 1964, 1965) and with previous theoretical and experimental values in Table IX. Considering the sensitivity of the calculated values to changes in the wavefunctions and the 11-14% error estimates in the experimental values the agreement is excellent, the latest theoretical and experimental results being specially close. Further, both the theory of Boyd and the experiment of Hudson predict a sharp drop in the cross section beyond the spectral head falling to a zero minimum at electron ejection energies of E = 0.094 and 0.090 Ry, respectively. The zero minimum arises theoretically from exact cancellation of the positive and negative portions of the radial integrand (17) or (25) and is predicted by all the theoretical calculations but is in conflict with experimental determinations previous to those of Hudson.

F. PHOTOIONIZATION OF POTASSIUM It has been shown by Bates (1947) that the degree of cancellation in the matrix element involved in the photoionization transition matrix elements for potassium is such that the final results of any conventional calculation must be unreliable. The problem as it arises in the quantum defect method has been discussed by Burgess and Seaton (1960) who demonstrate the sensitivity of the matrix element by showing that, with a Hartree-Fock bound wavefunction, the cross section varies by an order of magnitude as the spectral head quantum defect for the final state varies from 1.711, the experimental value, to 1.690. Accordingly it is not surprising that the most recent theoretical estimates (Boyd, 1965) depart considerably from the current experimental curve (Hudson and Carter, 1965b). The method used by Boyd is the same as that employed in the case of sodium, the polarizability of the core being taken as 5 . 8 ~ 7 , ~The . results are

44

A . L. Stewart

given in Fig. 3. The dipole velocity calculation is in better accord with experiment, giving a threshold cross section of 2.3 x cm2 and a zero minimum at 2550 A compared with the experimental points 1.0 x lo-” cm2 and 2670 A, respectively. Perhaps the most depressing feature of the comparison is the very high ratio of the experimental and theoretical cross sections at their maxima. I

I

I

E(eW

FIG.3. Photoabsorption cross section for the 4 s 4 p transition of potassium. (1) Dipole velocity, Boyd (1966). (2) Dipole length, Boyd (1966). (3) Experiment, Hudson and Carter (1965b).

G. PHOTOIONIZATION OF HBecause of its importance in the analysis of stellar atmospheres the photoionization cross section of the negative hydrogen ion has been the subject of increasingly accurate calculations (cf. Chandrasekhar, 1958). In the most recent work of Geltman (1962) a ground state function with 70 parameters (Schwartz, 1961) depending explicitly on the electron-electron distance was used. The continuum wavefunction used by Geltman also allows explicitly for angular correlation, the ten independent parameters being determined using the Hulthen-Kohn variational procedure. This is the most sophisticated continuum wavefunction used to date for the purpose of a photoionization calculation. Despite the complexity of the wavefunction used there are still considerable differences in the values obtained using the different possible formulations. However, theoretical arguments based on the sensitivity of the formulas to wavefunction changes favor the velocity formula. There is also

QUANTAL CALCULATION OF PIIOTOIONIZATION CROSS SECTIONS

45

the fact pointed out by Chandrasekhar (1945a,b) that the major contribution to the velocity integral comes from a region of wavefunction space where the wavefunctions are likely to be well determined by variational procedures. Lastly the velocity curve provides the best over-all agreement with the experimental measurements (Smith and Burch, 1959). The accuracy of the velocity curve can be assessed using the oscillator strength sums which are given in Table X. The best available direct calculations of S ( k ) are also given in the table, S(2) and S(l) being taken from TABLE X VALUESOF S(k) FOR H ~~

k

Accurate values

2 1 0

-1 -2 -3

5.512 1.495 2 7.55 0.1 53k 2 500 40

~

Geltman (1962) 4.72 1.22 1.76 7.09 50.6 468

Pekeris (1958) and S(-l), S(-2), and S ( - 3 ) from Schwartz (1961). Since the velocity curve of Geltman ends at 500A the computed sums for S(l) and S(2) require a careful extrapolation of cifld8 to shorter wavelengths (Dalgarno and Ewart, 1962). Although the agreement is good, Table X indicates that the Geltman oscillator strength is less accurate at high energies than at low energies, or that the extrapolation beyond 500 A requires modification. The comparison does not support the view of Dalgarno and Ewart (1962) that the contribution to the total oscillator strength from two-electron transitions is unusually large in H - . The computations of S(1)and S(2) from the Pekeris data quoted in Dalgarno and Ewart appear to be in error. Although the calculations demonstrate the sensitivity of the H - photodetachment cross section to wavefunction changes it has been shown by Geltman (1956) and Tietz (1961) that a very simple approach can be reliable. These authors use the one-electron approximation and determine the central potential from the criterion that the binding energy be correctly reproduced. No polarization potential is included. A similar technique has been applied to the case of Li-. H. PHOTOIONIZATION OF C -

AND

OTHERNEGATIVE IONS

Continuous absorption of radiation by C- in stellar photospheres may be an important source of opacity, especially in stars where the effect is not obscured by an abundance of H - (cf. Branscomb and Pagel, 1958). For this

46

A . L. Stewart

reason a number of theoretical studies of the photodetachment cross section of C- have been carried out (Breene, 1959; Cooper and Martin, 1962; Myerscough and McDowell, 1963). The more recent calculations of Myerscough and McDowell (1964) and Henry (1966) employ formally identical Hartree-Fock wavefunctions for both initial and final states but obtain somewhat different results. Thus the peak of the dipole length cross section lies at 25 x lo-'' cm2 according to Myerscough and McDowell whereas a peak height of 18 x lo-'' cm2 is predicted by Henry. The dipole velocity results are in closer accord and, except near the threshold, lie close to the experimental measurements of Seman and Branscomb (1962). The paper by Myerscough and McDowell should be referred to with care since the terms in the exchange potential for the 6 d electron-electron coupling with the 2p core electrons appear to be too small by a factor of two (see also Seaton, 1965). Myerscough and McDowell also consider the effect of including a polarization potential but neglect the exchange potential for these cases. Although there is improved agreement with experiment near threshold in the dipole velocity cross section the difference between the dipole velocity and dipole length cross sections is increased. The results of the Hartree-Fock approximation seem preferable. Calculations of photodetachment cross sections for the negative ions C-, 0 - , Ar-, and F- have been published by Cooper and Martin (1962). The wavefunctions are determined in the Hartree-Fock approximation with the exclusion of the exchange potentials and the inclusion of a polarization potential of the form (cf. Bates, 1947; Klein and Bruckner, 1958)

where CI is the polarizability of the neutral atom and rp is the average distance of the valence electron from the nucleus in the neutral atom. For C- the results are in essential agreement with the polarization results of Myerscough and McDowell, who took care to ensure the validity of the one-electron approximation by orthogonalizing the continuum orbitals with respect to the orbitals of the neutral atom core. Rather better agreemeet with experiment (Smith, 1960) is obtained in the case of 0 - in the range up to 2 eV but discrepancies appear at higher energies. The extent of the agreement obtained, however, indicates that the value of the polarizability of atomic oxygen assumed by Cooper and Martin is close to the correct value, since Bates (1946b) has shown that the low-energy cross section is rather sensitive to the polarizability parameter. It is clear than many of the difficulties encountered in predicting accurate photodetachment cross sections in negative ions and pointed out by Bates (1946~)still remain.

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

I.

PHOTOIONIZATlON OF

47

ATOMIC NITROGEN

The photoionization cross sections of atomic oxygen and nitrogen are basic atomic parameters of fundamental importance in the quantitative understanding of the ionosphere. The study of atomic oxygen using the Hartree-Fock approximation by Dalgarno et a/. (1964)has been followed by an application to nitrogen (and its isoelectronic ions) by Henry (1966). The results agree with the earlier calculations of Bates and Seaton (1949)at the spectral head. The measurements of Ehler and Weissler (1955) show a maximum of 14.4 x cm2 at 650 A which compares favorably with the dipole length cross section 12.4 x cm2 at tht same wavelength. The general shape of the measured cross-section curve is reproduced by the calculations. It is of interest to note that the measurements of Samson and Cairns (1965) on the photoionization cross section of molecular nitrogen are very close to twice the dipole length calculations in the L , absorption region.

VI. Conclusion During the last few years considerable emphasis has been given to developing the formalism of electron-atom scattering and photoionization to make use of the speed of digital computers to solve not only the coupled integrodifferential equations but to aid in the preliminary analysis. The results of these endeavors should soon be available. Thus the equations for the elastic scattering of electrons by atomic systems with 2pq and 3p4, q = 0 , . . . , 6, configurations have been solved taking account of coupling between the ground state terms (Smith et a/., 1966). Such studies may lead to improved photoionization calculations for the atoms and positive and negative ions of carbon, nitrogen, and oxygen, among others. Calculations making allowance for correlation in such systems may also be tractable (Myerscough, 1965). More sophisticated calculations on photoionization of systems with three electrons now seem possible, to take their place alongside the bound-bound oscillator strength results of Weiss (1963) who used wavefunctions with 45 parameters. Although photoionization experiments on lithium do not reveal a resonant structure, this is nevertheless a system of great theoretical interest to investigate the problems connected with the application of the closecoupling theory of Burke et al. (1964a,b) to more complex systems. The considerable body of experimental data on calcium makes it a system worthy of intensive theoretical investigation. Being essentially a two-electron system the technique employed by Burke et al. is applicable and it would appear possible to achieve a solution to the five-channel, ' P I , e + Ca' scattering equations including coupling to the Ca' 4s, 3d, and 4p states

48

A . L.Stewart

(Burke, 1965). These results, when available, shouId make revealing comparison with the calculations of Moores (1966) who uses the many-channel quantum defect theory.

ACKNOWLEDGMENT The research reported has been sponsored by the U.S. Office of Naval Research for the Advanced Research Projects Agency, Department of Defense, under Contract N62558-4297.

REFERENCES Allen, S. J. M. (1935). “X-rays in Theory and Experiment” (A. H. Compton and S. K. Allison, eds.). Van Nostrand, Princeton, New Jersey. Bates, D. R. (1946a). Monthly Notices Roy. Astron. SOC.106,423. Bates, D. R. (1946b). Monthly Notices Roy. Astron. Soc. 106, 128. Bates, D. R. (1946~).Monthly Notices Roy. Astron. Soc. 106,432. Bates, D. R. (1947). Proc. Roy. Soc. A188, 350. Bates, D. R., and Damgaard, A. (1949). Phil. Trans. Roy. SOC.London A242, 101. Bates, D. R., and Seaton, M. J. (1949). Monthly Notices Roy. Astron. SOC.109, 699. Bearden, A. J. (1963). Bull. Am. Phys. Soc. 8, 312. Bell, K., and Kingston, A. E. (1967). Proc. Phys. SOC.(London) 90, 31. Bell, R. J. (1965). Nat. Phys. Lab. Gt. Brit. Rept. Ma 52. Bell, R. J., and Dalgarno, A. (1965). Proc. Phys. SOC.(London) 86, 375. Bely, 0. (1966). Proc. Phys. Soc. (London) 88, 833. Bely, O., Moores, D., and Seaton, M. J. (1963). In “Atomic Collision Processes” (M. R. C. McDowell, ed.). North-Holland Publ., Amsterdam. Bichsel, H. (1961). Linear Acceleration Group, Tech. Rept. No. 3, Univ. Southern California. Biermann, L., and Lubeck, K. (1948). 2. Astrophys. 25,325. Boyd, A. H. (1964). Planetary Space Sci. 12,129. Boyd, A. H. (1965). Private communication. Branscomb, L. M. (1962). I n “Atomic and Molecular Processes” (D. R. Bates, ed.). Academic Press, New York. Branscomb, L. M. (1964). Ann. Geophys. 20, 88. Branscomb, L. M., and Pagel, B. E. J. (1958). Monrhly Notices Roy. Astron. SOC.118,258. Breene, R. G. (1959). Planetary Space Sci. 2, 10. Burgess, A. (1964). Astrophys. J. 139,776. Burgess, A. (1965). Astrophys. J. 141, 1588. Burgess, A,, and Seaton, M. J. (1960). Monthly Notices Roy. Astron. SOC.120, 121. Burke, P. G. (1965). AERE Harwell Rept. T.P. 183. Burke, P. G., and McVicar, D. D. (1965). Proc. Phys. Soc. (London) 86,989. Burke, P. G., and Smith, K. (1962). Rev. Mod. Phys. 34,458. Burke, P. G., and Taylor, A. J. (1966). Proc. Phys. SOC.(London) 88, 549.

QUANTAL CALCULATION OF PHOTOIONIZATION CROSS SECTIONS

49

Burke, P. G., McVicar, D. D., and Smith, K. (1964a). Proc. Phys. SOC.(London) 84, 749. Burke, P. G., McVicar, D. D., and Smith, K. (1964b). Phys. Letters 12,215. Chandrasekhar, S. (1945a). Astrophys. J. 102,223. Chandrasekhar, S. (1945b). Astrophys, J. 102, 395. Chandrasekhar, S.(1958). Astrophys. J. 128, 114. Condon, E. U., and Shortley, G. H. (1935). The Theory of Atomic Spectra.” Cambridge Univ. Press, London and New York. Cooper, J. W. (1962). Phys. Rev. 128,681. Cooper, J. W., and Martin, J. B. (1962). Phys. Rev. 126, 1482. Cuthbertson, C.,and Cuthbertson, M. (1911). Proc. Roy. SOC. A84, 13. Dalgarno, A., and Ewart, R. W. (1962). Proc. Phys. SOC.(London) 80,616. Dalgarno, A., and Kingston, A. E. (1960). Proc. Roy. SOC.A259, 424. Dalgarno, A., and Lewis, J. T. (1956). Proc. Phys. SOC.(London) A69, 285. Dalgarno, A., and Lynn, N. (1957). Proc. Phys. SOC.(London) A70, 802. Dalgarno, A., and Stewart, A. L. (1956). Proc. Roy. SOC.A238,276. Dalgarno, A., and Stewart, A. L. (1960). Proc. Phys. SOC.(London) 76,49. Dalgarno, A,, Henry, R. J. W., and Stewart, A. L. (1964). Planetary Space Sci. 12,235. Dershem, E.,and Shein, M. (1931). Phys. Rev. 37, 1238. Ditchburn, R. W., and Opik, U. (1962). In “Atomic and Molecular Processes” (D. R. Bates, ed.). Academic Press, New York. Ditchburn, R. W., Jutsum, P. J., and Marr, G. V. (1953). Proc. Roy. SOC.A219, 89. Doughty, N. A,, and Fraser, P. A. (1964). In “Atomic Collision Processes” (M. R. C. McDowell, ed.). North-Holland Publ., Amsterdam. Ederer, D. L., and Tomboulian, D. H. (1964). Phys. Rev. 133,A1525. Ehler, A. W., and Weissler, G. L. (1955). J. Opt. SOC.Am. 45, 1035. Eisenbud, L. (1948). Dissertation, Princeton Univ. Fano, U. (1961). Phys. Rev. 124, 1866. Fano, U.(1963). Ann. Rev. Nucl. Sci. 13, 1. Fano, U.,and Cooper, J. W. (1965). Phys. Rev. 137,A1364. Feshbach, H. (1958). Ann. Phys. (N. Y.) 5,357. Feshbach, H. (1962). Ann. Phys. ( N . Y.) 19,287. Fox, L., and Goodwin, E. T. (1949). Proc. Cambridge Phil. SOC.45, 373. Geltman, S. (1956). Phys. Rev. 104,346. Geltman, S. (1962). Astrophys. J. 136,935. Green, L.C., Lewis, M. N., Mulder, M. M., Wyeth, C. W., and Woll, L. W. (1954). Phys. Rev. 93,273. Hargreaves, J. (1929): Proc. Cambridge Phil. Sor. 25, 75. Hart, J., and Herzberg, G. (1957). Phys. Rev. 106,79. Havens, G. G. (1933). Phys. Rev. 43, 992. Heitler, W. (1954). “The Quantum Theory of Radiation.” Oxford Univ. Press, London and New York. Henry, R. J. W. (1966). J . Chem. Phys. 44,4357. Henry, R.J. W., and Lipsky, L. (1967). Phys. Rev. 153,51. Hudson, R. D. (1964). Phys. Rev. 135,A1212. Hudson, R. D. (1965). Aerospace Corp. Rept. TDR-469 (9260-01)-8. Hudson, R. D., and Carter, V. L. (1965a). Phys. Rev. 137, A1648. Hudson, R. D., and Carter, V. L. (1965b). Phys, Rev. 139,A1426. James, H.M., and Coolidge, A. S. (1936). Phys. Rev. 49, 688. Kabir, P. K., and Salpeter, E. E. (1957). Phys. Rev. 108,1256. Klein, M., and Bruckner, K. (1958). Phys. Rev. 111, 1115. ‘I

50

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Kuyatt, C. E., and Simpson, .I. A. (1964). In “Atomic Collision Processes” (M. R. C. McDowell, ed.). North-Holland PubI., Amsterdam. Livingston, M. S., and Bethe, H. A. (1937). Rev. Mod. Phys. 9,245. Lowry, J. F., Tomboulian, D. H., and Ederer, D. L. (1965). Phys. Rev. 137,A1054. Madden, R. P., and Codling, K. (1963). Phys. Rev. Letters 10, 516. Madden, R. P., and Codling, K. (1965). Asrrophys. J. 141,364. Marriott, R. (1958). Proc. Phys. SOC.(London) 72, 121. McCarroll, R. (1964). Proc. Phys. SOC.(London) 83,409. McDaniel, E. W. (1964). ‘‘Collision Phenomena in Ionized Gases.” Wiley, New York. McGuire, E. J. (1965). Reported in Lowry e t a / . (1965). Miller, W . F., and Platzman, R. L. (1957). Proc. Phys. SOC.(London) A70, 299. Moores, D. (1966). Proc. Phys. SOC.(London) 88, 843. Mott, N. F., and Massey, H. S. W. (1949). “The Theory of Atomic Collisions.” Oxford Univ. Press, London and New York. Myerscough, V. P. (1965). Proc. Phys. SOC.(London) 85, 33. Myerscough, V. P., and McDowell, M. R. C. (1963). Proc. Sixth Intern. Conf. Ionization Phenomena Gases, Univ. of Paris 1, 135. Myerscough, V . P., and McDowell, M. R. C. (1964). Monthly Notices Roy. Astron. SOC. 128,288. O’Malley, T. F., and Geltman, S. (1965). Phys. Rev. 137,A1344. Peach, G. (1966). Private communication. Pekeris, C. L. (1958). Phys. Rev. 112,1649. Percival, I. C., and Seaton, M. J. (1957). Proc. Cambridge Phil. SOC.53,654. Piech, K. R., and Levinger, J. S. (1964). Phys. Rev. 135,A332. Rohrlich, F.(1959a). Astrophys. J. 129,441. Rohrlich, F. (1959b). Astrophys. J. 129,449. Samson, J. A. R. (1964). J. Opt. SOC.Am. 54, 876. Samson, J. A. R. (1966). Advan. Atomic Mol. Phys. 2, 178. Samson, J. A. R., and Cairns, R. B. (1965). J. Opt. SOC.Am. 55, 1035. Saraph, H., and Seaton, M. J. (1962). Proc. Phys. SOC.(London) 72, 121. Scherr, C. W., Silverman, J. N., and Matsen, F. A. (1962). Phys. Rev. 127,830. Schliiter, D. (1965). J Quant. Spectr. Radiative Transfer 5, 87. Schwartz, C. (1961). Phys. Rev. 123,1700. Seaton, M. J. (1951). Proc. Roy. SOC.A208,408. Seaton, M. J. (1953). Phil. Trans. Roy. SOC.(London) A245, 469. Seaton, M. J. (1958). Monthly Notices Roy. Astron. SOC.118,504. Seaton, M.J. (1965). Proc. Phys. SOC.(London) 85, 197. Seaton, M. J. (1966a). Proc. Phys. SOC.(London) 88, 801. Seaton, M. J. (1966b). Proc. Phys. SOC.(London) 88, 815. Seman, M., and Branscomb, L. M. (1962). Phys. Rev. 125, 1602. Sewell, K. G. (1965a). Phys. Rev. 138,418. Sewell, K.G. (1965b). J. Opt. SOC.Am. 55, 739. Silverman, S. M., and Lassettre, E. N. (1964). J. Chem. Phys. 40, 1265. Simpson, J. A., Mielczarek, S. R., and Cooper, J. (1964). J. Opi. SOC.Am. 54, 269. Sloan, I. H. (1964). Proc. Roy. SOC.A281, 151. Smith, F.T. (1960). Phys. Rev. 118,349. Smith, S. J., and Burch, D. S. (1959). Phys. Rev. 116, 1125. Smith, K.,Henry, R. J. W., and Burke, P. G. (1966). Phys. Rev. 147,21. Stewart, A. L. (1954). Proc. Phys. SOC.(London) A67, 917. Stewart, A. L. (1963). Advan. Phys. 12,299.

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51

Stewart, A. L., and Webb, T. G. (1963). Proc. Phys. Sac. (London) 82, 532. Tait, J. H. (1964). In “Atomic Collision Processes” (M. R. C. McDowell, ed.). NorthHolland Publ., Amsterdam. Temkin, A., and Lamkin, J. C. (1961). Phys. Rev. 121, 788. Tietz, T. (1961). Phys. Rev. 124, 493. Vinti, J. P. (1932a). Phys. Rev. 41, 432. Vinti, J. P. (1932b). Phys. Rev. 42, 632. Weiss, A. W. (1961). Phys. Rev. 122, 1826. Weiss, A. W. (1963). Astrophys. J. 138, 1262.

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RADIOFREQ UENCY SPECTROSCOPY OF STORED IONS I: STORAGE* H . G . DEHMELT Department of Physics, University of Washington Seattle, Washington

1. 2.

Introduction. .................................................... Containment of Isolated I o n s . . .................................... 2.1 Limitations.. ................................................ 2.2. Penning Trap ................................................ 2.3. Average Force in Inhomogeneous rf Field. ...................... 2.4. Single Ion Motion in Quadrupole rf Trap ....................... 2.5. Ion Cloud in Temperature Equilibrium ......................... 2.6. Collision Heating ............................................ 2.7. Ion Creation ................................................ 2.8. Sample Trap Data ........................................... References ......................................................

53 55 55 56 58 61 63 67 68 69 72

1. Introduction Experimental attempts to approach the ideal of isolated atomic systems floating at rest in free space for unlimited periods and free from any undesired outside perturbations appear to be worthwhile for a variety of reasons. At the same time such experiments are of limited value unless one also devises means for first preparing the atomic systems in certain selected states and for later observing the development of these states in time due to internal or controlled external interactions, as in a hfs or magnetic resonance experiment. The first experimental goal therefore is to develop techniques to isolate, contain in a trap, thermalize, and possibly refrigerate the atomic systems under study. Since these problems appear to be most easily solved for charged particles, we restrict ourselves in the following to ions. Next, collision reactions with suitable, state-selected projectiles may serve to create oriented, aligned, or somehow state-selected ions from atoms or cause ions already present in the trap to undergo transitions to selected states. A second reaction, not

* Part 11: Spectroscopy is now scheduled to appear in Volume V of this series. 53

H . G. Dehmelt

54

necessarily of the same type, may be used for analysis or interrogation of the stored ions. A whole arsenal of suitable projectiles becomes available once ion containment times of sufficient duration, of the order of seconds, are realized, bringing down the necessary projectile flux densities to feasible values. General discussions of the subject have been given by the author who also previously suggested rf spectroscopy of stored ions (Dehmelt, 1956a, 1962, 1963). An experiment (Dehmelt and Major, 1962; Major, 1962; Major and Dehmelt, 1967) carried out on He' ions with the intent of observing their magnetic resonance may serve as an illustrative example; He' ions contained under ultrahigh vacuum conditions in an electric quadrupole rf trap (Paul etal., 1958; Fischer, 1959; Wuerkeretal., 1959)were bombarded with polarized Cs atoms. In the ensuing spin-exchange collisions, the He' electron spins became oriented according to the reactions and

Cst+He+t+Csf+He+t

Csf+He+l +Csl+He+T

(1.1)

while the Cs atoms lost orientation. This experiment is closely related to an earlier experiment (Dehmelt, 1956b, 1958a,b;Balling and Pipkin, 1965)inwhich free electrons contained by a positive ion cloud were oriented by spinexchange collision with oriented sodium atoms, Nat

+ e t + Nat + e t

and

NaT

+ e i -+

Nal

+ et.

(1 -2)

In both instances, magnetic resonance disorientation of the ions He' and e, the process of interest, was monitored by a second collision reaction. In general, information about the magnetic resonance is contained in the final states of all the reaction products. In the electron experiment the monitoring focused on the orientation loss of the sodium atoms or of the " projectiles." Since spin-exchange reactions between oriented electrons e t and N a t atoms do not produce N a l atoms while el, Nat collisions do, the appearance of N a l did serve as an indicator for the magnetic resonance disorientation of the electrons. In the He' experiments the orientation monitoring phase focused upon the He' targets and not the Cst projectiles. By employing the nearly resonant charge-exchange reaction leading to excited He states,

+ He+J Cs+ + He* Cst + He+?-%Cs+ + He* Cst

rast

-+

(singlet), (triplet),

(1.3)

it was possible to translate the magnetic resonance disorientation of the initially oriented He't ions to He'l into a reduction of their number, which in turn could be conveniently measured. The appearance of a new reaction product, namely, Cs', might have provided an even more effective indicator of the He' resonance. In fact, in a third experiment on H2+ ions (Dehmelt and Jefferts, 1962; Jefferts, 1962; Richardson et al., 1967), using linearly polarized photons as projectiles in the dissociation reaction

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

+

h~ 1 Hz+ + H+

+ H + K.E.,

h~ 5

55

+ HZ+*-* tr H + + H + K.E.,

(1.4) the appearance of H + was used to monitor the magnetic resonance of H2+. This again is possible because the photodissociation rate depends on the angle between electric light vector 5: and the internuclear axis 1. As illustrated by the three experiments just described, our studies seem to indicate that a wide variety of target-projectile combinations of interest may be imagined. The state selection of the projectiles necessary may be purely by energy eigenstate, as low as that found in an unpolarized monoenergetic paralleI beam of electrons (alignment by electron impact), or as high as that in a beam of circular polarized optical resonance radiation (optical pumping), or in a beam of polarized alkali atoms (polarization by spin exchange). The ion-storage collision technique outlined has now been applied successfully to a precise determination of the hfs separation of the hydrogen-like (He3)+ ion in the 8-GHz region (Fortson et al., 1966) and to the first study of the Zeeman effect in H2+ yielding a value for the spin-rotational coupling constant.

2. Containment of Isolated Ions 2.1. LIMITATIONS Having stated our experimental goal, we shall briefly touch upon some fundamental and technical limitations before discussing specific containment methods. It is, of course, impossible in principle to have a confined particle totally at rest. Rather, when for reasons of Doppler effect suppression (Dicke, 1953) it is necessary to restrict a particle to a region smaller than a quarter wavelength A14 of the microwave spectrum to be studied, its de Broglie wavelength A, must always be shorter than A/2. Evaluating the velocity of a cm/sec. On proton with A, = 0.1 cm as an illustration, we find u N 4 x a temperature scale this amounts to only about 6 x 10-'20K, a negligible value. We would like to be sure that the relatively long-range forces between the slow ion of interest and the unavoidable background gas atoms do not in practice lead to an excessively short mean time T, between collisions. From Langevin's analysis (Stevenson and Gioumousis, 1958) of orbiting collisions leading to an inward spiraling orbit and close contact, we obtain

T, = (rn,/a,)'/2/(27ren,) (2.1) in the limit of a stationary ion, where n,, a,, and rn, are density, polarizability, and mass of the residual gas atoms and e is the charge on the ion. Taking He as the residual gas at the technically feasible pressure of Torr, we find the very large value T, N lo7 sec independent of the kinetic energy of the gas atom, K,. . The closest approach re for excluded, nonorbiting encounters is here given by re4 = $a,e2/K,. (2.2)

-

56

H . G . Dehmelt

For He at room temperature impinging upon a singly charged stationary ion this gives re N 2.5 x lo-' cm, a value which is somewhat enlarged by cooling. (Throughout the article cgs units have been used unless other units are specified.) After this illustration of the degree of isolation now within the state of the art we will turn to a review of the principal containment methods useful in rf spectroscopy. 2.2. PENNINGTRAP Even though this scheme does not strictly belong to those admissible under our title, we will begin with a discussion of the Penning arrangement which we are using for the containment of electrons as well as cyclotron resonance and thermalization studies of these particles (Dehmelt, 1961). The confinement mechanism is that of the Penning discharge widely used in cold cathode ionization gages and ion-getter pumps. In our embodiment of this principle (Penning, 1936; Pierce, 1949) hyperbolic electrodes (see Fig. 1) create a dc electric potential of the form 4 ( x , y , z ) = A(x2

+ y 2 - 2z2),

A

= A . = const,

(2.3)

while a homogeneous magnetic field H , points in the z direction. The quantity A , is related to D = +(O, 0,O)- d(0, 0, z,), the depth of the potential well in the z direction, by D = 2zO2Ao.We also have U = Uo = D[l

+ + ( r , / ~ , ) ~ ]>O,

U being the applied voltage. The electrons move along bound orbits which are characterized by three frequencies. An electron moving along the z axis sees a parabolic potential and will undergo harmonic oscillations with a frequency w, , wZ2= 4eA,/m.

(2.4)

Another simple type of orbit is a circle in the x-y plane with center at the origin. For these orbits the electric force -eEr almost balances the magnetic force -e(u/c)H,, . The frequency of this magnetron motion is a constant of the trap : w, = u/r = cEr/(H,r) = 2cA,/H,.

(2.5)

The third frequericy is the cyclotron frequency w, given by w,

= eHo/(mc).

(2.6)

For the particular values of parameters of experimental interest to us, the orbits in general are approximately a superposition of a fast, small-diameter cyclotron motion around a magnetic field line, an oscillation along this field

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

,-+

57

= A (x2+y2-2z2)

FIG. 1. Hyperbolic electrode configuration employed in ion storage devices useful in

rf spectroscopy. Application of an alternating voltage U = Vo cos Rr at the terminals shown in (a) creates a three-dimensional harmonic oscillator potential of depth D = D along the z axis as depicted in (b). When a dc voltage U = Uo is applied the potential at the origin has a saddle point only. While a parabolic well is also obtained along the z axis, motion of the charged particles in the r direction has to be restricted in this embodiment of the Penning discharge geometry by a strong axial magnetic field as usual.

line, and a slow circular drift across the field lines around the z axis, with respective frequencies o,% o,s- w, . From the foregoing we see that these quantities are related by

2w,wm = o,2.

(2.7)

A more accurate analysis shows that the frequencies of cyclotron and magnetron motion are slightly shifted to o,- om’ and om’. Relation (2.7) is replaced by 2wcwmf- 2 4 = o,*.

(2.8)

The trap is filled by shooting through it, along the z axis, a narrow electron beam of an energy sufficient to ionize the residual gas. In ultrahigh vacuums a beam of low energy eiectrons is reflected upon itself. Temporary trapping

58

H. G. Dehmelt

occurs by transformation of longitudinal kinetic energy into transverse due to e-e collisions. By providing simultaneously a radiative damping mechanism the trapping is made permanent. In experiments carried out by Walls, trapping times of hours have been realized for about lo4 electrons in the trap at a background pressure of 2 x lo-” Torr. The trap parameters are listed in Table I. Compare also Fig. 1. TABLE I OP6RATING PARAMETERS OF PBNNlNG TRAPFOR CONTAINMENTOF THERMAL ELECTRONS Electrode dimensions: Magnetic field : dc operating voltage: Cyclotron frequency: Axial oscillation frequency: Magnetron (drift) frequency:

22, = 1 cm, ro/zo = ~‘7. Ho = 1.8 k G CJo = 2 0 = 12.8 volts w C / ( 2 n ) = 22.4 GHz wJ(2n) = 58 MHz wm/(2v)= 75 kHz

2.3. AVERAGE FORCE IN INHOMOGENEOUS rf FIELD The obvious shortcoming of the type of trap just described is the presence of the magnetic field, causing a large and often undesired Zeeman effect. This difficulty does not arise in the trap design based on the use of inhomogeneous electric rf fields (Paul et al., 1958; Fischer, 1959; Wuerker et al., 1959). The related mass-filter (Paul and Steinwedel, 1953) is now widely used in the commercial quadrupole residual gas analyzers. To illustrate it we first consider an ion of charge e and mass m moving in the homogeneous electric rf field of a parallel plate capacitor [see Fig. 2(a)]. The equation of motion is mi’ = Fz(t)= eE, cos Rt,

E,

=

V0/(2z0).

(2.9)

We seek a solution of the form z ( t ) = Z + 5(t), Z = const, and obtain

5 = - 5, cos Rt,

5, = eEo(z)/(mfi2).

(2.10)

The time variation of the force Fz(t) = eE, cos Rt and the coordinate z are depicted in Figs. 2(b) and 2(c). Making the field inhomogeneous, as in the modified geometry indicated by broken lines, without changing its value at 5, slightly perturbs the orbit [now T’(r)] of and the force [now F,’(t)] acting on the ion. The interesting net effect is that the time average of the alternating force,
59

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

though small compared to its maximum value. The direction of F(Z) is toward the region of weaker field. For a quantitative evaluation of F(F) in the limit of very large C? we imagine it balanced, so that the center of the sinusoidal micromotion at frequency R remains at rest at 5.We expand Eo(z)around Z, Eo(z) = EO@)

+ [aEo(wzIi9

I[aEo(z)/azliol Q IEo(Z)l.

0

-t

(bl

FIG.2. Principle of rf electric quadrupole ion cage. While the force F,(t) causing the micromotion displacement 5 of an ion from its guiding center i has a vanishing time average for a homogeneous rf field (full lines) this is no longer the case in an inhomogeneous field of the same intensity (broken lines). Here the averaging of the modified instantaneous force Fz'(f) over the essentially unaltered micromotion < ( t ) leads to a small, nonvanishing average force F(;(23,which always points toward the region of weaker rf fields independent of the sign of the ionic charge.

Averaging over one period, we may write (I/e)F(F)= (E,(Z) cos Rt =

+ [aEo(2)/Z]('cos Rt),,

[aEO(z)/az]( i'cos Rt),, .

(2.1 1)

For (' the expression (2.10) for i may be used in excellent approximation:

(i'cos at),, = -tio= - e ~ ~ ( 3 ) / ( 2 r n ~ ~ ) .

60

H . G . Delimelt

Finally, we have (I/e)P(Z) = - e [ d ~ , ( ~ ) / d ~ ] ~ , ( ~ ) / ( 2 m R Z )

(2.12)

which may also be written

( l / e ) F ( z )=

I&)

-d $ ( ~ ) / d ~ ,

=e~,~(~)/(4mn~),

(2.13)

introducing the electric pseudopotential $(z). This result can be generalized (Kapitsa, 1951; Gaponov and Miller, 1958; Landau and Lifschitz, 1960; Weibel and Clark, 1961):

F(Z, 7, Z)

=

II/ = [e/(4rnR2)]Eo2((X,J , Z),

-e grad I ) ( % J , Z),

(2.14)

K, 7, ,7 denoting the coordinates of the guiding center of the three-dimensional particle motion. We now remove the balancing force compensating F(Z) and allow the guiding center to move. It is plausible that expressions (2.11) to (2.13) remain valid as long as the quasi-stationary behavior condition

1.4 4 c o n

(2.15)

is satisfied. In order to check if this condition can be relaxed, we investigate the case of moderately large, constant i so that over a period 2n/R the derivative dE,(Z)/dZ may be considered as constant,

I[~~E0(Z)/a't~(2ni/R)~ 4 laE,(z)/azl. We use a solution of (2.9) in the form z

=Z

r=+n'n

dt [E,(.Z) (l/e)F(z) = 2n: r=-n/n

(2.16)

+ z't + [ and write

+ ( i t + @3E0(5)/Z]cos Rt.

(2.17)

Inspection shows that in addition to the terms in (2.1 l), a term proportional to ( I t cos Qt),, appears. For suitably chosen end points of the interval 2n/R this term vanishes too, leaving the expression for force and pseudopotential unchanged. Again it is plausible that a slow variation of 2, 12nz'/Qt/4 121, may be tolerated. Evaluating the kinetic energy W(z) averaged over one period gives

< Wz)>av = W = <(m/2)iz>av = (m/2)i2

+ ((m/2)t2>av+ (mz"[),, .

(2.18)

In the limit, R + co,of interest here the last termvanishes becauseeither (2.15) or (2.16) applies for a bound particle. The kinetic energy in the micromotion, ((m/2)fz>av = [ e 2 / ( 4 ~ ~ 2 ) 1 ~ = 0 2ell/(% (3

(2.19)

turns out to be the pseudopotential energy e$ previously introduced. Consequently, w i s a constant of the secular motion corresponding to the total

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

61

energy in systems with time-independent conservative forces and will be referred to as “the energy from here on. The total instantaneous kinetic energy W(z)on the other hand does fluctuate widely,

w”

0 I W(Z)I 2 K

(2.20)

rf TRAP 2.4. SINGLEION MOTIONI N QUADRUPOLE As an application of (2.14) we consider the same potential as for the Penning trap, Eq. (2.3), but we set

A

= A,

cos Rt.

(2.21)

The electric field has the components -&$/ax = -2Ax,

-d$/dy

=

-2Ay,

-a$/&

= 4Az. (2.22)

Accordingly, the resulting three-dimensional elliptic potential well is given by $(F, J , Z) =

D[X2 + j 2 + 4Z2]/(4zOz),

D/(4zO2)= e A O 2 / ( m f l 2 ) , (2.23)

b denoting the depth of the pseudopotential well in the z direction. AS is well known, the motion of a particle in such a well is a superposition of three independent harmonic oscillations, along the x and y axes with frequencies Gr and along the z axis with frequency O,, explicitly, W,= 2W,. Restricting us to motion purely along the z axis, we expect the z-motion of the guiding center to be described by, y a phase constant,

z = Z, cos (w,t

+ y),

eB

= mzo2~,2/2,

(2.24)

and the complete motion by

z = Z - lo cos at.

(2.25)

Without serious lack of generality y = 0 may be assumed in the following. With E,(Z) = - 4A,Z from (2.3), Eq. (2.25) may be written

z

=

[I

+ (J2w,/R)

cos ~ t l z cos , w,t.

(2.26)

The pseudopotential approach is justified because in the region z N 0 condition (2.16) applies, and in the region z N Zo condition (2.15) applies. Provided the “stability parameter” QjG, is large compared to unity, it can also be shown by direct substitution that in this limit (2.26) is a solution of the Mathieu differential equation associated with the rigorous treatment of the problem (Paul et af.,1958; Fischer, 1959). Multiplying (2.26) out, terms proportional to cos 6,t and cos W,t cos Rt occur. The last one may be decomposed to

cos (R + a,)r + cos (R - w,)t.

(2.27)

H. G . Dehmelt

62

This shows that the motion contains spectral components at Ez and R & 4. The rigorous solution for finite values of R/G, also contains components at SRL-G,,

s = 2 , 3 , 4 ,..., 00.

It is now interesting to compare Gz with w, for operation of the same trap in the Penning mode with A = A , . We may write (for a negative ion) using (2.3) and (2.23) G z 2 / ~= , 2 DID = +oZ2/R2

(2.28)

or JZG, R = oz2,

D/D = 2-'/2Ez/R.

(2.29)

We see that for any specific geometry (e.g., as given in Fig. 1) operation at U = V , cos Rt leads to an effective well depth in the z direction which is by the factor 2-'/20,/R smaller than for operation at U = U , = V , . Since U , is related to D by (2.3) we may write

V, = JZ [I

+~(ro/zo)2~(~/~z~b.

(2.30)

These results have been obtained in the limit R/Cz9 1. The question as to how far this restriction may be relaxed arises. It is found (Paul et al., 1958; Wuerker et al., 1959) that trapping can be achieved down to R > 2Cz. For R 5 2Ez, a rapidly growing instability similar to parametric excitation (Landau and Lifschitz, 1960) occurs. Similar instabilities are observed when R coincides with certain higher order (Busch and Paul, 1961) harmonic combinations of Ez and Gr and the rf field is sufficiently inhomogeneous of a related order. It is a general property of a harmonic oscillator to respond to a harmonically related inhomogeneous alternating force field. Because small contributions from higher order multipole fields cannot be avoided in a practical trap it seems advisable to make the stability parameter R/Ezlarge as the contributions from such fields may be expected to be the weaker the higher their order. Avoiding integer ratios we have found values n/w, N 10 satisfactory. Choosing size, depth, shape, and stability parameter of a trap determines the operating voltage by (2.30). The frequency Gz and therefore R are then determined by the energy relation (2.24), (m/2)(zO2 W z 2 ) = eb. Once zo , r,, G z , and R have been fixed for a given apparatus it is also of interest to know at which values of V, other ion masses will come into resonance. We note that under these conditions, according to (2.24), it follows that

Dccm

(2.31)

V,(m) cc m

(2.32)

and with (2.30), we have

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

63

indicating proportionality between ion mass and rf voltage necessary to cause oscillation at Gz . In other experimental situations we wish to know at what frequencies other ions will oscillate in a given trap for fixed R and V, and what their well depth will be. Here we find from (2.30) the relation D/Wz= const, which with (2.24) gives 6,oc l/m

and

Doc Ilm.

(2.33)

This demonstrates the difficulties encountered when attempting to trap ions of very different masses simultaneously.

2.5.

ION CLOUD IN

TEMPERATURE EQUILIBRIUM

The motion of the stored ions is of course three dimensional. Contemplating extended storage it seems reasonable to allow for cumulative effects of small conservative interactions due to various causes which, while leaving constant, may still change the character of the ion orbit. By making the well depth in the axial and the radial directions the same, D = Dz = Dr, we avoid the possibility that an ion initially carrying out a large amplitude oscillation in, for example, the z direction will later hit the ring electrode, as would be the case for Dr < D. Inspection of (2.33) shows that equal depths result for ro = 22,. We also note that W,= 213,. By adding a dc component U, to U, U = U, + Vo cos Rt, the potential due to U , alone simply adds to as given in (2.23). This is to be expected since, in our limiting case G24 n, the small amplitude micromotion responsible for the trapping force is not affected by the presence of the weak dc field. Judicious choice of U , provides another means of symmetrizing a general well, ro # 22,. In both cases it is reasonable to define as the active trap volume an ellipsoid of rotation closely fitting into the electrode structure,

w

v --43nzoro2 . I

(2.34)

As our next task we attempt to estimate the storage capacity of the zero dc trap, with ro = 22,. Filling it with a number of ions at rest (except for the micromotion) we would expect that the electrostatically interacting ions would arrange themselves in such a fashion at the bottom of the well that a test ion would experience no force; that is, we would expect a flat net potential in the ellipsoidal ion-filled region,

$ + 4i= const,

(2.35)

where 4 i denotes the electrostatic potential due to the ions. Taking the Laplacian on both sides, we have - Abi = A$ = 4np,,,

,

pm,,

= 3D/(4nzO2)= enmax,

(2.36)

64

H . G . Dehmelt

indicating a constant charge density. Expression (2.36) may be used to obtain the distance between the stationary ions rnlin,which here equals that of closest approach,

riin = 4nezO2/(3D)= l/nnlax, nlnax= 1.66 x

1O6D/zO2,

[z,]

(2.37) = cm,

[ D ] = volt.

In a numerical example we find, for singly charged ions and for e = 4.8 x lo-’’ cgs, zo = 2.5 cm, and D = 6.2 volts N 1/50 cgs, the value rnlin= 0.84 x lo-’ cm, demonstrating the excellent isolation between the ions. At room temperature this would decrease to rmin= 2 x cm, as evaluated from rnlin= e2/3kTi.

(2.38)

Even at this value the average overlap of the electronic wave functions would be orders of magnitude smaller than for a neutral gas of the same density. After this digression we return to the original question of storage capacity. From (2.34) and (2.36) we get, for the maximum storable charge qmaxand the maximum total ion number N,,,, ,

-

Nmaxe = qmax= 402, ; = 2.8

N,,,

x IO’Dz,

(2.39)

,

[D] = volt,

[z,] = cm.

The important question, if the supposed zero secular motion ion configuration would be stable in actuality, has so far not been discussed. The synchronous breathing of the ion cloud at the frequency Q only modifies the rf field in the trap by a constant factor very close to unity. Ion-ion collisions will of course occur in a cloud of ions at the finite temperature T i , 3KTi = ( . As the appropriate collision interval we take in an approximation that given in the literature for a uniform plasma (Spitzer, 1956; McDonald et al., 1957):

w),,

t, = 0.78 x 107A”2(kTi/e)3/2n-’Z-4(lnA)-’,

[t,]

= sec,

(2.40)

[kTi/e] = volt,

where A and Z are atomic weight and ionic charge and where A = r,,ax/rmin is the ratio of the cutoff distance for Coulomb interaction of an ion with its neighbors to that of closest approach. Since the choice of r,,,, is not very critical it seems reasonable to set rmax z= zo which gives In A N 20 in the region of interest. A numerical estimate for ions at room temperature and density n = nmaxfrom (2.37) with D = 6 volts and zo = 2.5 cm, gives for protons, A = Z = 1 , t, = 1 msec, showing that such collisions can be quite important. However, at least for the case of a cloud of identical ions in a homogeneous

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

65

rf field, these collisions cannot lead to energy absorption from the field. Here the rf field simply causes the center of mass of the cloud of ions to oscillate with amplitude lo [see (2.10)] while the individual ions in the center-of-mass frame carry out their random rectilinear motions. Since ion-ion collisions cannot transform energy of the motion of the center of mass into kinetic energy of the relative motion of the ions, no energy absorption from the field can take place. This is not strictly true anymore in an inhomogeneous rf field; e.g., hard spheres of a radius which is an appreciable fraction of zo and bearing a charge at the center when moving along the z axis of a quadrupole trap undergoing head-on collisions will absorb energy from the rf field. Assuming that analogous long-range effects are unimportant in an ion trap we expect that in a perfect vacuum and in a trap geometrically and electrically perfect a cloud of identical ions would not be subject to any rf heating. Consequently, we propose to treat the ion cloud as if it were contained in an ordinary three-dimensional harmonic oscillator potential. Because of the long available experimental storage times, T 9 t , , the ions may be considered to be in thermal equilibrium with themselves and assigned a temperature T i . When cold ions kTi 4 e 6 , of total charge q < qlnaxare confined in a trap, they fill it up to a level 6, like a fluid occupying at constant density n,,, an ellipsoidal region with half-axes zq , rq = 2z,, ( Z , / Z , ) ~ = 6,/6 and volume V, = (16/3)zq3.For the “fluid” level 6, the relation (2.41) holds. If the temperature rises the ion cloud expands, and its boundaries wash out. Also, loss of ions due to collisions of the Boltzmann tail, W > eD, with the electrodes becomes important. A self-collision parameter T,* for the trap may be defined as the t , from (2.40) for 3kTi = eD and n = nmaxtaking 1,

z=

T,*

E 0 . 0 4 6 ~ , ~ 6 ’ / ~ A ’ ~[zo] ~, =

cm,

[D]= volt.

(2.42)

For ions only partially filling a trap, as will always be assumed now, q 4 qmax, at the temperature 3kTi = eD, where every ion-ion collision may be expected to lead to loss at the electrodes, Tc E (qmax/q)Tc*

(2.43)

may be interpreted as the (hot) ion lifetime. In the intermediate region 0 < 3kT, < eD, we approximate the ion density of the ion cloud roughly extending over an ellipsoidal region with half-axes zqT, rqT by

n N nma,[eD,/(eB4+ 3kTi)I3” ( Z , ~ / Z- ~( e>6 ,~

+ 3kTi)/(e6).

(2.44)

66

H. G. Dehmelt

For the average energy 3kT, $- eD, ,we use this to obtain a convenient expression for the self-collision time I,

N

(3kTi/e4)3T,.

(2.45)

At very low temperatures, 3kTi -geD,, t, becomes independent of q, as seen from (2.37) and (2.40),

t, N 0.24A'/2(kTi/e)3/2z02/D, [zO]= cm,

[kT,/e] = [4] = volt. (2.46)

We now attempt to obtain an expression for the rate of ion loss at the electrodes in the intermediate temperature region by a detailed balancing argument. For a parabolic well extending far beyond zo, ro , the fraction f B of the N ions in the well with W 2 eD, where 4 is now much less than its depth, in temperature equilibrium is f B

N

$(eD/kTi)2exp( - eD/kT,).

(2.47)

An ion in this Boltzmann distribution tail undergoes collisions with the majority of energies around 3kTi and thereby returns to the majority at a rate roughly equal to TC-' as defined in (2.43).On the other hand, tail ions are created from the majority at the rate NT;'. At equilibrium this rate must be given by

NT;'

N

fBNT,-'

or

(2.48) TB N 2T,(kTi/e4)' exp(eD/kTi).

For the loss rate of fast ions in our trap of depth D,we now take the creation rate TB-' of tail ions just developed. Simultaneously, the loss of energetic ions results in a heat loss of the ion cloud

d(3kTi/e)/dtN - B / T B ,

(2.49)

which already in time intervals short compared to TBcan be appreciable on a percentage basis. The equilibrium temperature established will be that at which the heat input due to various dissipative mechanism balances the heat loss of the ion cloud. By using the experimental ion lifetime Tin place of TB, Eq. (2.48) may be used to establish an upper limit for the equilibrium temperature of the cloud for the region eD/3kTi > 2 :

(e4/3kTi)2 3 ln(T/T,)

+ 3 ln[0.1 ln(T/Tc)] + 1.3.

(2.50)

In a numerical example with T = 600 sec, zo = 0.16 cm, and qm,,/q = 30, which is similar to analogous experimentally realized conditions, one would find e4/3kTi 2 4.3 for [He3]+ in a trap of depth 4 = 6 volts.

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

67

2.6. COLLISION HEATING We now turn to specific dissipative mechanisms such as rf heating of the stored ion cloud due to the presence of other atoms and ions in the trap region. As a model of these processes occurring at intervals T, , we choose first purely elastic, totally randomizing collisions of ions in a homogeneous rf field with a “forest ” of uniformly distributed fixed scattering centers. The center of mass of an ensemble of ions, all of which have just undergone a collision with a center near 2, will then be at rest at the instant t l . Interaction with the electric rf field will, according to (2.9) and (2.10), cause the center of mass to move with a translational velocity 2 = - [ ( t l ) ,on which of course the micromotion velocity [ is superposed. The average total energy per ion of the center-of-mass motion, that is available for degradation into heat of amount A W in subsequent randomizing collisions, is, consequently,

AV= 2(m/2)((2),,

= 2e$(Z).

(2.51)

If this is further averaged over the secular oscillation, assuming a constant rate of collisions, we have A W = Ti, (2.52) where Viis the initial ion energy. The generality of this result is not affected by focusing on scattering centers located near the z axis. Reasonably heavy atoms and other ions of mass m, may be expected to cause similar strong heating though reduced by a factor a(rn/m,) 5 1, which may also depend on other variables. The corresponding heat input to the ion cloud is then 8(3kTi)/8t= a(m/m,) 3kTi/T,.

(2.53)

Conversely, collisions with much lighter cold partners, which essentially create viscous drag, result in a lowering of the average total ion energy implying a negative a. In this limit the micromotion is not interrupted by the collisions but only slightly modified in phase and amplitude, while any secular motion gradually damps out exponentially. This effect was first demonstrated for metal particles in air (Wuerker et al., 1959) and later for Hg’ in helium gas at about Torr (Huggett and Menasian, 1965). In order to shine some light on the interesting question at which value m/m, the factor u(m/m,) vanishes, as the above suggests, we now discuss collisions with scattering centers of mass m, = m which are initially at rest, restricting us to head-on collisions. Immediately after such a collision an ion of initial energy Viwill be at rest, and the subsequent average energy absorption from the rf field will again be W i , as described by formulas (2.51) and (2.52), corresponding to zero net energy gain during the collision. This may be taken to indicate that for general, not only head-on, collisions l41)l 4 1.

(2.54)

H . G. Dehmelt

68

The same formulas (2.51) and (2.52) may be applied in even better approximation now to the important case of collisions with parent gas atoms. Here resonant charge exchange dominates by about an order of magnitude over momentum transfer. The ions, having been formed by fast ions passing on their charge to thermal atoms, may, independent of the collision parameter, be assumed to be initially at rest in good approximation. By the same argument as given before no energy input into the self-regenerating ion cloud occurs due to the collision process. A constant collision rate has been assumed everywhere. This is a fair approximation in general since the average instantaneous total kinetic energy is a constant of the motion and according to (2.18) equal to W.In the region of orbiting collisions the assumption is even rigorous [see (2. l)]. However, at higher relative velocities, about lo6 cm/sec, the charge-exchange rate Tc;’ becomes roughly velocity-proportional (Rapp and Francis, 1962). Now, from ions of initial oscillation amplitudes Z,, zero velocity ones are preferentially created when i is largest, i,,,, = t,,, = J%,,,, which is in the Z N Zo region whereby, in accordance with the expression (2.51), net energy absorption results and the factor a consequently assumes a small positive value. By taking T, = T,, in (2.53) and using the above a value we obtain the appropriate rate of heat input. Charge exchange with other atomic species leads of course to immediate loss of the ion of interest. When predominant, the heating processes sketched before eventually cause the ions to be lost at the electrodes. 2.7. ION

CREATION

The principal way in which traps have been filled has been electron bombardment of background gas. It is essential here to realize that the maximum current density of electrons at energy eU, that can flow through the space between two highly transparent grids at equal potential separated by a distance d is limited to (2.55) j,,,,, = 1.9 x 10-5U3/2/d2,

Lie,,,,]

= amp/cm2,

[u,]= volt,

[d] = cm,

provided no virtual cathode is to appear (Rothe and Kleen, 1951). In the axially symmetric design it is not practical to make the beam cross section larger than about d Z ,resulting in a maximum electron current available of 5 312 i,, N 2 x 10- U , [i,,,,,] = amp. (2.56) This is of course only a rough estimate since (2.55) is not strictly applicable to the complex trap structure. The total electron charge qe necessary to produce an ion charge qnlaxalso does not depend on the trap size zo , qe N 2.2 x 10-l’ D/(n,Qi), [qe]= coulomb, [ B]= volt. (2.57)

69

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

Here Q iand n, are ionization cross section and density of the atomic species to be ionized. Such qe values can usually be realized within milliseconds. At ultrahigh vacuums it is of course still possible to pulse the gas pressure or even use pulsed atomic beams. The release of bursts of adsorbed parent gas from metal electrodes under electron bombardment acts in a similar way. An experimental value for the stored fractional charge q/qma,or filling factor may be obtained by measuring the dc bias voltage necessary to restore the oscillation frequency of a small sample of indicator ions, of a mass not too different from that of those under study, to its old value after it has been shifted by the space-charge potential of the ion sample of interest of total charge q. Assuming constant ion density over the whole trap, we have from (2.36) that q/qmax = ADID,

(2.58)

where A D is the well-depth change corresponding to the compensating dc voltage applied. Of course, Vo has been adjusted to bring the indicator ions into resonance. Ideally, Vo should be held constant in this measurement. However, if some change of Vo can be tolerated subsequent to filling of the trap, the constant Gz operation assumed here is much more convenient experimentally. Because of the averaging effect of the indicator ion motion, this procedure should give good approximate values even for inhomogeneous ion distributions, when large oscillation amplitudes of the indicator ions are used.

2.8. SAMPLE TRAPDATA In conclusion, as an illustration of the various points raised in the preceding discussion, we describe the trap used in Major’s experiments in numerical detail. Its relevant parameters are listed in Table I1 and a scale drawing of its mechanical construction is shown in Fig. 3. The trap depth was derived from

FIG.3. Scale drawing of illustrative rf quadrupole trap described in Table 11.

70

H . G. Dehmelt TABLE I1

DIMENSIONS, OPERATING PARAMETERS, OBSERVED AND DERIVED ION DATAFOR ILLUSTRATIVE TRAP Axial dimension: Radial dimension: Effective trap volume: dc bias : ac drive amplitude: Frequency: Vacuum: Background: Electron current: Electron acceleration voltage: Electron pulse duration: Ionic species: Axial oscillation frequency: Maximum velocity in trap center: Axial depth : Radial depth : Maximum instantaneous energy : Maximum experimental stored charge: Maximum charge: Ion lifetime: (maximum attained): Self-collision parameter : He+-He charge exchange:

zo = 2.5 cm ro = 3.5 cm V, = 128 cm3 uo = 7 volt v a = 175 volts Q=27r x 1 MHz p N_ 3 x Torr Mostly He4 i, = 1 mA

U,= 400 volts N

0.1 sec

[He4]

+

8,=27rx

llOkHz

zo8, = 1.73 x 106cm/sec

b = 6.2 volts b. = 8.2 volts W,,, 4~

2:

12.4 eV

107~

2: 3 x 10% T = 8 sec T = 50 sec T,* = I .5 sec Tce2 0.3 sec

relation (2.24), which applies also in the presence of a dc bias, as D = 6.2 volts. Since ro/zo N J2 the dc bias voltage Uo = 7 voIts contributed - 3.5 volts to D = B, . Therefore, we have Dz (ac only) = 9.7 volts and 0, (ac only) = 4.9 volts giving Dr = 8.4 volts. In order to check Eq. (2.30), we evaluate W,(ac only) = 27c x 138 kHz using the relation

W,(ac only)/WS,= [D, (ac ~ n l y ) / D ] ' / ~ and obtain Vo = 200 volts, in only fair agreement with the experimental value. The electron acceleration voltage was chosen as U , N 2V0 in a compromise effort to retain reasonably straight electron trajectories inside the trap, in the face of the electron space-charge potential depression and the strong rf fields

RADIOFREQUENCY SPECTROSCOPY OF STORED IONS

71

there, without causing excessive heating of the electrode structure The electron current was pulsed by applying a positive voltage pulse, +45 volts with respect to the cathode, to the Wehnelt cylinder, ordinarily held at a small negative voltage. Comparison of the experimental lifetimes T = 8 sec and T = 50 sec (the latter attained at best vacuum, background not analyzed) with the hot ion self-collision time T, N 50 sec defined in (2.43) shows that ion-ion collisions are not very important for this trap and that the average ion energy must be near eD. We attribute the ion loss primarily to the repeated small energy increments furnished by charge-exchange collisions with the parent gas according to the arguments presented, which suggest a small positive a and which are in agreement with the experimental observation that about T/T,, N 30 such collisions are necessary to drive an ion against an electrode. The maximum charge density in a symmetrized ro/zo =J2 trap is, for equal D and zo , given by F ? ~ , , ,=+nmax. ,~~ Applying this to the Major trap gives ns,maxN 2.2 x 106/cm3,and with the listed volume qmaxN 2.8 x 108e. The maximally storable experimental charge listed was determined for a (He3)' sample according to the procedure leading to Eq. (2.58), using H 2 + created from the background gas as the test ions. Application of a dc voltage of -0.3 volt compensating the (He3)+ space charge was required. With D(He3) = 2D(He4) = 4.6 volts this is equivalent to a filling factor q/qmaxN 1/30. Besides the axially symmetric quadrupole trap discussed here, other configurations have also been operated successfully. These include circular traps for electrons and ions simultaneously contained by the electronic space charge (Drees and Paul, 1964), circular (Church, 1966) and race-track-shaped traps for hydrogen and helium ions (Church, 1965) and for mercury ions (Huggett and Menasian, 1965), all obtained by bending a 4-wire mass filter structure (Paul and Steinwedel, 1953) into these shapes, and also a cubic trap (Wuerker et al., 1959). In the circular trap, lifetimes of 600 sec were observed for He' ions at a background pressure of 3 x lo-'' Torr of helium. It had a depth = 6 volts and separation between electrodes was 2x0 = 3.2 mm. Much shorter self-collision times t, obtain here than in the Major trap. Using in a rough approximation Eqs. (2.42), (2.43), and (2.45) developed for the axially symmetric configuration, setting z,, = x,, , and assuming qma,/q= 30, we find T,* = 6 msec, T, = 200 msec, and t, = 3 msec. Comparable t, values have been experimentally observed for room temperature electrons stored in the Penning trap (described earlier), to which system much of our analysis is applicable. By observing the delayed temperature increase in the z motion following pulse heating of the cyclotron motion in the x-y plane we have estimated (Walls, 1966) t, 5 5 msec. All this demonstrates the importance of the concept of a stored ion cloud in thermodynamical equilibrium with itself in future experiments.

72

H . G . Dehmelt

REFERENCES Balling, L. C., and Pipkin, F. M. (1965). Phys. Rev. 139, A19. Busch, F. V., and Paul, W. (1961). Z . Physik 164, 588. Church, D. (1965). Private communication. Church, D. (1966). Private communication. Dehmelt, H. G . (1956a). Phys. Rev. 103, 1125. Dehmelt, H. G. (1956b). Optical Pumping Symposium, 123rd AAAS Meeting, New York, Dec. 26, 1956. Dehmelt, H. G. (1958a). Phys. Rev. 109, 381. Dehmelt, H. G. (1958b). J . Phys. Radium 19, 866. Dehmelt, H. G. (1961). Unpublished. Dehmelt, H. G . (1962). Bull. Am. Phys. SOC.7, 470. Dehmelt, H. G. (1963). Bull. Am. Phys. SOC.8, 23. Dehmelt, H. G., and Jefferts, K. B. (1962). Phys. Rev. 125, I3 18. Dehmelt, H. G., and Major, F. G. (1962). Phys. Rev. Letters 8, 213. Dicke, R. H. (1953). Phys. Rev. 89, 472. Drees, T., and Paul, W. (1964). Z . Physik 180, 340. Fischer, E. (1959). Z . Physik 156, 1. Fortson, E. N., Major, F. G . , and Dehmelt, H. G. (1966). Phys. Rev. Letters 16, 221. Gaponov, A. V., and Miller, M. A. (1958). Zh. Eksperim. i Teor. Fir. 34, 242. Huggett, G. R., and Menasian, S. (1965). Private communication. Jefferts, K. B. (1962). Thesis, University of Washington, University Microfilms, Ann Arbor, Michigan. Kapitsa, P. L. (1951). Zh. Eksperim. i Teor. Fir. 21, 588. Landau, L. D., and Lifschitz, E. M. (1960). “ Mechanics. ” Pergamon, Oxford. Major, F. G. (1962). Thesis, University of Washington, University Microfilms, Ann Arbor, Michigan. Major, F. G., and Dehmelt, H. G. (1967). Phys. Rev. To be published. McDonald, W. M., Rosenbluth, M. N., and Chuck, W. (1957). Phys. Rev. 107, 350. Paul, W., Osberghaus, O., and Fischer, E. (1958). Forschungsber. Wirtsch. Yerkehrsministeriums Nordrhein- Westfalen No. 41 5 . Paul, W., and Steinwedel, H. (1953). Z . Naturforsch. 8a, 448. Penning, F. M. (1936). Physica 3, 873. Pierce, T. R. (1949). “Theory and Design of Electron Beams,” Chap. 3. Van Nostrand, Princeton, New Jersey. Rapp, D., and Francis, W. E. (1962). J . Chem. Phys. 37, 2631. Richardson, C. B., Jefferts, K. B., and Dehmelt, H. G. (1967). Phys. Rev. To be published. Rothe, H., and Kleen, W. (1951). “Grundlagen und Kennlinien der Elektronenrohren,” Ch. 6. Akademische Verlagsgesellschaft, Leipzig. Spitzer, L. (1956). ‘‘ Physics of Fully Ionized Gases.” Wiley (Interscience), New York. Stevenson, D. P., and Gioumousis, G . (1958). J . Chem. Phys. 29, 294. Walls, F. (1966). Private ccmmunication. Weibel, E. E., and Clark, G. L. (1961) Pfusma Phys. 2 , 112. Wuerker, R. F., Goldenberg, H. M., and Langmuir, R. V. (1959). J . Appl. Phys. 30,441.

OPTICAL PUMPING METHODS IN A TOMIC SPECTROSCOPY B. BUDICK The Hebrew University Jerusalem, Israel

.................... ........................................ A. Optical Double Resonance ....................................

I. Introduction

13 74 74 B. Level-Crossing Spectroscopy ... 80 111. Results of Double Resonance and Level-Crossing Experiments ........ 83 A. Atomic Lifetimes and Oscillator Strengths ...................... 83 B. Atomic Fine Structure . . . . . . . . . . . . . . . . . . . . . . . . 86 C. Atomic Hyperfine Structure . . . ................. 88 D. Polarization of the Atomic Core ................. 103 IV. Optical Pumping Experiments in G ................. 108 A. Optical Orientation of Atomic Ground States ............... B. Spin-Exchange Experiments on Light Systems .................... 112 References ...................................................... 114 11. Experimental Techniques

I. Introduction Experimental techniques that involve the use of optical pumping have been applied to a wide variety of phenomena. The earlier work was primarily concerned with lifetimes of excited states and with the effect of collisions in producing broadening, shifts, and asymmetries in spectral lines [I]. Resonance fluorescence from an atomic beam is still used for high-resolution optical spectroscopy as a means of reducing the Doppler width [2]. Procedures utilizing magnetic resonance are clearly more suited for precision work. Radio-frequency spectroscopy of excited atomic states was first suggested by Bitter [3] and by Brossel and Kastler [4]. The natural lifetime of the excited state is in principle the only limit on the precision of such an experiment. With the extension of the technique to ground states, linewidths comparable to atomic-beam magnetic-resonance experiments have been obtained. All optical pumping experiments utilizing magnetic resonance have in common the achievement of a population inequality in the ground and/or excited state. This inequality is partially or completely eliminated by application of the radio frequency. The population change is detected 73

74

B. Budick

by the change in the polarization or intensity of the transmitted or scattered light. An extremely fruitful technique has recently been discovered by Franken [ 5 ] which does not involve a redistribution of the atomic population. It relies rather on interference effects in the scattered light produced when two radiating excited-state Zeeman levels are made degenerate. This levelcrossing technique avoids complications of radio-frequency power requirements, which may be severe for short-lived excited states. Its precision is also limited by the natural linewidth. Most of the recent work has been devoted to the study of atomic fine and hyperfine structures (hfs). Many magnetic dipole and electric quadrupole moments of nuclei have been measured. Lifetimes and oscillator strengths of excited levels have been accurately determined. Work in excited states has both demanded and encouraged a study of configuration interaction in its various forms, particularly core polarization. By comparing hfs results in a number of configurations for two isotopes of the same element it has been possible to evaluate the Sternheimer corrections to nuclear quadrupole moments. This article will deal with those physical phenomena whose study has been facilitated by optical pumping techniques. The techniques themselves will be treated in some detail in Section 11. Section 111 will treat each of the subjects mentioned in the previous paragraph in turn. In Section IV we will discuss orientation of atomic ground states and spin-exchange optical pumping. A compilation of the data that has been obtained is included. In view of the large number of experiments that have been performed in excited states of atoms, this article will be limited to a consideration of those experiments in which resonance radiation is used both as a means of excitation and as a means of detection. Consequently, the many elegant experiments, particularly on light atoms employing electron beam excitation, will not be treated here. The reader is referred to the review article by Series [6].

II. Experimental Techniques A. OPTICAL DOUBLE RESONANCE The optical double resonance method and its applications have been the subjects of a number of review articles [6-81. The method was first successfully applied by Brossel and Bitter 191 in a study of the mercury isotopes. A consideration of the even isotopes will serve to illustrate the method. With regard to Fig. 1, a beam of resonance radiation (2537 A) is incident along the x axis on a cell containing the even mercury isotopes, situated in a magnetic field which is pointed in the z direction. The light can excite the 6 3 P , level

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

75

FIG. 1. Schematic representation of an optical pumping experiment. Incident light is polarized along the magnetic field direction.

shown in Fig. 2, but because it is linearly polarized in the incident channel it manages to excite only the m = 0 Zeeman level. Thus the emitted resonance light also contains only II radiation. Since the intensity of the emitted 7~ light follows a sin28 distribution (8 is the angle formed by the direction of the magnetic field and the direction of the observer), a photomultiplier placed along the magnetic field direction will not receive any light. Alternatively, the photomultiplier can be placed in a direction perpendicular to both the incident light and the magnetic field, and a polarizer used to isolate the B light whose intensity follows a 1 + cos2 8 distribution. In the absence of magnetic resonance this photomultiplier will also register zero intensity. If now a radiofrequency (or microwave; we shall use the two interchangeably where not important) field is applied in a direction perpendicular to the static field so that transitions of the type Am = 1 can be induced, atoms will be transferred to the other magnetic sublevels and the emitted light will now contain B components which will be detected by the photomultipliers. The frequency of the oscillating field must equal the Larmor precession frequency, vL = gJ(po/h)/H, m

6 9

{

.''I'I !

c

A '

',' f

I

I

:

*1

O -1

I I I

I I

'I

6'5,

L

1 1 ;

FIG. 2. Effect of a magnetic field on the polarization of resonance radiation from the 6 3P1level of mercury. T light is used selectively in excitation.

76

B. Budick

where gJ is the atomic g factor, po is the Bohr magneton, h is Planck's constant, and H is the static field. In the original experiment of Brossel and Bitter 191 an optical bridge technique was used to eliminate noise due to fluctuations in lamp intensity. n light was viewed in a direction perpendicular to the magnetic field and u light was viewed along the field direction. As a result of magnetic resonance, the D light increased at the expense of the n light. The output currents of the two photomultipliers were subtracted from one another so that the fluctuations canceled and the magnetic resonance signals added. The above detection geometries indicate how one may use changes in polarization to advantage. A technique due to Kohler [lo] utilizing intensity differences before and during rf resonance can be orders of magnitude more sensitive. The scattering cell is placed in a high magnetic field which separates the Zeeman levels by more than the Doppler width of the reemitted light. The wavelength of the reemitted n radiation has not been changed by the field since both ground and excited levels have m = 0. An absorption cell containing mercury in zero magnetic field placed in the detection arm will absorb any scattered light due to atoms or to the apparatus. Magnetic resonance now produces light of a different wavelength which can pass through the absorption cell and reach a photomultiplier.

I . Methodology 4. Light Sources. The success of optical double resonance and levelcrossing spectroscopy depends critically on the light source. Both of these techniques involve the optical excitation of an atomic resonance level. The requirements on the light source can be briefly summarized as follows:

(i) The total optical output in the resonance line must be in the range 1015-10'8photons/sec (I mW to 1 W for ultraviolet lines). The smaller output would be suitable for fully allowed transitions, while the larger would be required for partially forbidden transitions. (ii) The spectral line must be free of self-reversal. Only the light within the Doppler width (typically 1500 Mc/sec) of the center of the line is useful for producing fluorescence. Light in the wings of the line contributes to the background but not to the useful signal. (iii) The light source should require a minimum of sample material. This is to facilitate construction of lamps of rare or radioactive isotopes for the excitation of particular isotopes. In some cases it is possible to excite selectively hyperfine levels in rare and radioactive species with lamps containing stable, abundant isotopes. The success of the method depends on favorable combinations of isotope shift and hyperfine structure that produce coincidences.

OPTICAL PUMPlNG METHODS IN ATOMIC SPECTROSCOPY

77

(iv) The lamp must be free of fluctuations. Short-term noise should be less than one part in lo5 and long-term drifts less than one part in lo3. (v) The lamp should be adaptable to a wide variety of chemical elements. Details of design and construction of three types of resonance lamps (electrodeless discharge, Cario-Lochte Holtgreven flow lamp and hollow cathode) have recently been published [l 11. The success with which each of these light sources meets the above requirements is also treated by Budick et al. [ll]. b. Scattering Cells. For elements which have a sufficient vapor pressure at low temperature, and which do not attack quartz, such as the group IIb elements, the resonance vessel can be a simple quartz bulb. Details of construction and filling procedures of such bulbs have been published [12]. An atomic beam has proved a more advantageous source of scattered radiation for a variety of purposes. (i) Materials as refractory as palladium have been successfully studied using an oven heated by electron bombardment [131. (ii) Elements which react with the cell material such as lithium and sodium can be kept away from the lenses that are used to admit and observe the light. A geometry that is most effective in avoiding such attack is shown in Fig. 3. In general, no attempt is made to collimate the atomic beam. Where collimation is necessary or desirable, the crinkle foil plugs first introduced by Zacharias and described by Giordmaine and Wang [ 141 are capable of producing a directed beam with a density of l O I 3 atoms/cm3.

MODULATION

HEATER LEADS

CIRCULATION

FIG.3. Schematic diagram of an atomic beam optical pumping apparatus.

78

B. Budick

(iii) In an experiment in which very low densities of absorbing material must be studied, an atomic beam presents the possibility of eliminating most of the instrumental scattering [151. c. Radio-Frequency Requirements. In order to reorient the excited atom before it decays the radio-frequency field ( H , ) must be of sufficient amplitude to produce a rotation in a time comparable to the atomic lifetime. If y is the gyromagnetic ratio of the excited level the requirement becomes

y H , x l/z. For T = lo-' sec and y x lo7 sec-' G-' we must have radio-frequency fields of the order of a gauss. Shorter lifetimes require correspondingly larger fields. d. Vacuum Requirements. Holtsmark and Lorentz broadening due to collisions between atoms being studied with one another or with foreign gas atoms, respectively, become important for vapor pressures of lo-' Torr. Detailed studies of their effects have been carried out both experimentally [16-181 and theoretically [19]. These requirements are easily met except in cases where weak resonance lines are being studied and the vapor pressure is raised in order to increase the signal. e. Magnetic Field Requirements. If the magnetic field that is responsible for the Zeeman splitting varies over the sample volume, the resonance condition will be fulfilled for a broad band of radio frequencies. The magnetic field inhomogeneity must be reduced until the bandwidth is small compared to the natural linewidth.

2. Signal Size and Shape The intensity of a double resonance signal will be proportional to I, the intensity of the incident light, to 0,the cross section for absorption by the atom, to N , , the number of absorbing ground state atoms, to P, the probability that a radio-frequency transition has occurred, to e, the efficiency of the photocathode, and to a geometrical factor (D to account for solid angle losses. Thus the number of photons arriving at the photomultiplier is h N , P @ . For Z we may take 1015per sec per cm' and, for N , , 10" atoms/cm3 in accordance with the discussion above. For the cross section 0 we may write*

where AvA is the Doppler width for the absorbing atoms, Avs the Doppler width of the source atoms, ro the classical radius of the electron, c the speed of light, andfthe oscillator strength for the transition. The oscillator strength

* This result can be derived from Mitchell and Zemansky [l, Chapter 31.

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

79

is a measure of the ability of an atom to absorb light relative to that of a classical dipole oscillator. For resonance transitions in the near uv with source and absorber at 200°C, f = 1, AvA = Avs 2 1500 Mc/sec, and 0x cm’. In the experiment of Brossel and Bitter [9] the probability for a transition to the m = & 1 levels is that given by the Majorana formula and is time dependent. The initial population of the m = 0 level possesses an exponential time factor characteristic of spontaneous decay. An effective transition probability can be arrived at by integrating over time. One finds

(YHd2 (l/Te>’ 4(yHJ2 4(0 - WJ2

1 (YHA2 2 ( y H 1 ) 2 (w -

Perf = -

+

+

+

where y is the gyromagnetic ratio, T, is the mean life of the atom in the excited state, and wo is the resonance frequency. In the limit yH,T, %- 1 this approaches 0.125 at resonance. A plot of this function which determines the intensity as a function of the radio frkquency is shown in Fig. 4 together with the experimental points for various values of yH,T,. The full width at halfmaximum may be derived from Eq. (3) by putting w = w o and taking the first two terms in a power series expansion. We find (Ao)’

):(

=

2

[l

+ 5.8(yH,Te)2].

(4)

From a plot of the square of the half-width against the square of the radiofrequency field extrapolated to zero field one can determine the lifetime of the excited state. In any particular experiment the radio-frequency matrix elements that determine the probability of a transition must be evaluated. However, a crude estimate of the change in x light at resonance can be made by assuming saturation, i.e., that the rf is sufficient to equalize the populations of the resonating levels. Of course, this may be a poor approximation in the case of short-lived states. This method leads to effective transition probabilities of the order of 0.1. Solid angle losses may amount to a factor of low3.The power arriving at the photocathode is the product of the number of photons per second and the energy of a single photon: Power = laN,P@hv

(5)

for a uv photon ( v = and the values we have assigned this is a power of 66 pW. A photocathode of sensitivity 3 x l o p 6 A/W will convert this to a

80

B. Budick

Ho (gauss)

FIG.4. Effect of increasing the amplitude of the radio-frequency field on the shape of a double resonance signal.

current of 2 x lo-'' A. This is much larger than the dark current or the shot noise due to random fluctuations in the photocurrent. One of the reasons for the sensitivity of double resonance experiments as pointed out by McDermott and Novick 1201 is " that although resonances are effected by low-energy photons (hv,, 4 kT), they are detected optically (hv,,,, % k T ) so that thermal noise is of no concern."

B. LEVEL-CROSSING SPECTROSCOPY 1. Phenomenology

The first successful level-crossing experiment was reported by Colegrove et al. [21] in the 3P term of He. For purposes of illustration it will be more useful to consider the fine structure of a ' P term. The magnetic field dependence of the two fine structure levels is shown in Fig. 5. For a particular value of the magnetic field the levels J = 3, MJ = -3 and J = 4, MJ = cross.

+

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

81

ti

E

I

ti-

FIG. 5. Zeeman effect on a 2P term. The observable level crossings are indicated by circles.

-+

Both of these levels can scatter cr light from the 2 S , , 2 M , = level of the ground state but will do so independently as long as they are not degenerate in energy. Representing the ground state by u and the excited states by b and c, Franken [ 5 ] has shown that the rate at which photons of polarization and direction fare absorbed (see Fig. 6) and photons of polarization and direction g are reemitted is given by

R(f, 8) = RO = Ifabl’

19ba12

4-

IfacI’ IgcaI’,

(6)

where fab =

If * rl

b).

FIG.6. Energy level diagram pertinent to a level crossing experiment. Incident photons have polarization and direction specified by f, scattered photons by g.

82

B. Budick

If the levels are made degenerate by the magnetic field a single photon can excite both coherently; that is, a definite phase relation exists between the wave functions representing the excited levels. This coherence manifests itself in an interference in the emitted light in a manner similar to that in which a double slit produces interference when a single source is used. The expression for the rate now becomes R(f7

g, = Ro + 1

A + A* + 4n2r2 vyb, c )

+

( A - A*)2niz v(b, c) 1 + 4n222v2(b,c) ’

(7)

where A = f b a ~ c g c o g a bz , is the lifetime of each excited state, and v(b, c) = (Eb

- Ec)h*

None of the matrix elements in the product, A , must vanish if a signal is to be observed. A will in general be a complex quantity. For real A the signal will have a Lorentzian absorption type shape with a full width at half-maximum given by Av@, c ) = l/m. For imaginary A a dispersion-type shape results. In all cases such as that considered above where the crossing levels are excited by o light (Am = 2 crossing) the z axis is chosen along the direction of the magnetic field and the incident and scattered light propagate and are polarized in the xy plane. If the x axis is taken as the direction of the polarization of the incident photon, the scattered photon is uniquely specified by the angle 0 between its direction of polarization and the x axis. Thus the matrix elements fba and fa, have no angular dependence and the matrix elements gac and gob which have P k ij factors [23] will produce a factor coS(28)i sin(20) in the product A . The signal shape is determined by the angle 8. For 0 = O,n{2 a pure absorption curve is expected, while for 8 = nj4 or 3n/4 a pure dispersion curve should result. The crossing of levels with Am = 1, the second circled point in Fig. 5, can also be observed provided the direction of polarization of the incident light is neither parallel nor perpendicular to the magnetic field; i.e., a superposition of n and u light must be used. These and many other results are derived in the paper by Rose and Carovillano 1241. For small quantum numbers the level-crossing signal may amount to 15 of the total scattered light signal. In the general case more than one ground state Zeeman level can excite the two excited-state crossing levels. If the ground levels are denoted by m, m‘ and the excited levels by p, p’, the expression for the rate becomes

where K is a constant that depends on the pumping intensity and on the apparatus geometry. The matrix elements fprn , etc., will be field dependent. It is therefore difficult to predict the intensity of a level crossing a priori.

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

83

2. Hanle Efect A special case of Eq. (8) and one for which no field dependence of the matrix elements exists is the crossing of Zeeman levels as the magnetic field is swept through zero. The resulting depolarization of the scattered light was observed by Hanle [25]and used by Mitchell and Murphy [26] to measure the lifetime of the 7 3 S , state of mercury. A classical discussion of the Hanle effect is given by Mitchell and Zemansky [I]. From Eq. (7) it follows that for real A the lifetime can be determined from the linewidth in gauss by the relation 1

(9) n(dv/dH)AH ’ where v is the separation of the crossing levels in units of frequency and av/aH, the slope of the Zeeman levels, is evaluated at zero field. In the above, it has been assumed that all crossing levels in the excited state have the same lifetime. z=

3. An tileuel- Crossings It has been noted that level-crossing spectroscopy differs essentially from double resonance spectroscopy in that it does not depend upon the establishment of a population inequality. Instead, it requires a coherent excitation of the crossing levels, which in turn implies the selection rule Am = 1, 2. The crossing labeled 6m = 1 in Fig. 5 is in reality a superposition of sixteen closely spaced crossings as a detailed analysis of the hfs shows. It is well known that levels of the same m do not cross, because some interaction, perhaps in nth order, intervenes so that they repel one another. At the point, at which the levels would have crossed in the absence of such an interaction, the wave functions representing these levels are a 50-50 mixture of the wavefunctions of the unperturbed states. As a result of this state mixing, it is possible to observe changes in the resonance fluorescence as the magnetic field is swept through the point of closest approach, provided that the states are populated unequally by the incident resonance radiation. Antilevelcrossing signals have been observed in the 2 ‘P level of lithium by Eck et al. [27]. The experimental conditions required for observation of the effect together with applications are discussed by the authors.

111. Results of Double Resonance and Level-Crossing Experiments A. ATOMIC LIFETIMES AND

OSCILLATOR STRENGTHS

Knowledge of atomic oscillator strengths is valuable in many fields of research. A comparison of measured and calculated values serves as a check on any proposed atomic wavefunctions and helps to determine the breakdown

84

B. Budick

of Russel-Saunders coupling and configuration interaction in complex spectra. They are further useful in the calculation of emission and absorption probabilities, in an analysis of collision broadening experiments, and in a study of the phenomenon of coherence narrowing (discussed below). Finally, they are of interest in astrophysics for purposes of studying stellar structure and in plasma physics for the determination of plasma densities and temperature. Oscillator strengths and lifetimes are inversely proportional to one another (see Mitchell and Zemansky [l], p. 97). Equations (4)and (9) are the starting points for determining the lifetime of the excited state from the linewidth of the double resonance and level-crossing signals. The results of lifetime measurements of both methods are contained in Table I, which includes recent Hanle effect work on the ions of Ca and Mg.

1. Coherence Narrowing In the course of double resonance measurements of lifetimes the phenomenon of coherence narrowing was discovered. Experimentally it was found that the double resonance signal became narrower as the density of scatterers increased. The effect was first noticed in the even isotopes of mercury [49], and then carefully studied in the odd isotopes [50]. Barrat showed that the same phenomenon occurs in the Hanle effect [51] and that the lifetimes as determined by the two methods are in excellent agreement provided an extrapolation is made to zero density. A theory for coherence narrowing using the density matrix formalism has been worked out by Barrat [52]. The coherence-narrowing phenomenon is attributed to resonance trapping which results in a multiple scattering of the photons. Provided the atoms that are responsible for the multiple scattering are contained in the same magnetic field, the time dependence of the wavefunctions representing their excited states is exactly the same. As a result, information contained in the. phase factor, such as the orientation of the atom in the excited state, is transferred by the photon from one atom to another. The time that the light maintains coherence between the excited states is greater than the lifetime of a single atom. Experiments to elucidate this phenomenon in which double scattering was assured by the use of two single scattering bulbs are in excellent agreement with the theory [53]. A maximum narrowing of a factor of 8 has been predicted by Omont for mercury experiments in high magnetic fields [54]. A factor of 4.4 has been achieved by Otten with the discrepancy ascribed to relaxation by collision at the high densities employed (lo-’ Torr) [55]. In the most recent work in which excited-state Zeeman levels are coupled to one another through resonance trapping or collisions different lifetimes have been observed for different Zeeman levels. Physically, this may be

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

85

TABLE I COMPtLATtON OF LIFETtME MEASUREMENTS Element Li Na Mg Mg+ A1 K Ca Ca Cr Cr +

cu

cu cu Zn

Zn Zn Rb

Sr Ag Cd Cd In Xe cs Ba Au Hg

TI

Pb

State

Method

Lifetime

D.R." D.R. H.E.b D.R. H.E. H.E. H.E. D.R.' H.E. H.E. H.E. H.E. H.E. H.E. D.R. D.R. H.E. H.E. D.R. D.R. D.R. D.R. H.E. D.R. H.E. H.E. D.R. H.E. H.E. H.E. H.E. D.R.' D.R. D.R. D.R. H.E. H.E. H.E. H.E. H.E. D.R. H.E. H.E. H.E. H.E. H.E.

2.78(8) x 1.61i7jx 10-8 1.63(5) x 0.9x 10-7 1.99(8) x 3.67(18) x 1.36(14) x lo-' 1.5 x 10-7 4.48(15) x 6.72(20) x 3.34(50) x 6.51(90) x 7.~7) x 10-9 7.0(2) x 3.2(3) x 10-7 2.0(2) x 10-5 i.38(5) x 10-9 1.41(4) x 2.82(9) x 1.0

x

10-7

2.w)x 10-7 4(1)x 10-7 4.97(15) x 2.1(3)x 10-5 6.7x 10-9 7.4(7) x 10-9 2.39(4) x 1.66(5)x 10-9 6.7x 10-9 3.17(19) x lo-' 3.79(12) x lo-' 1.6x 10-7 2.4~) x 10-7 1.21(12) x 10-6 1.04(5)x 10 -' 8.2(2)x 10-9 5.3(10) x 0.7(1) x lo-' 0.81(2)x 1.18 x 10-7 7.4(3) x 10-9 7.6(2) x 10-7 6.2(10) x 5.2(8) x 10-9 5.75(20) x lo-' 1.29(14) x lo-'

D.R. indicates a determination using double resonance. H.E. indicates Hanle effect measurement. This value is a lower limit.

Ref.

86

B. Budick

attributed to different relaxation times for the various multipoles that describe the angular configuration of the excited atoms such as the magnetic moment (the orientation) and the alignment (polarization with zero magnetic moment). Experiments have been performed in mercury [56] where multiple scattering is responsible for the different relaxation rates, and in lead where collision effects are most important [57]. 2. Hanle Effect in Odd A Isotopes

Equation (8) was derived under the assumption that the light source has a uniform power spectrum for exciting all the m -+ p transitions [see Eq. (S)]. This is not the case for isotopes whose ground state hfs is greater than the Doppler width (the group I and group 111elements). In this case the resonance radiation from the lamp has resolved hyperfine components of unequal intensity. The importance of this circumstance lies in the fact that the Hanle effect will generally be a superposition of curves of different widths, corresponding to zero field level crossings in each of the hyperfine levels. This is true because the slope of the Zeeman levels, av/aH in Eq. (S), is in general different for each hyperfine level. In analyzing the superposition, the intensity of each curve depends on the power spectrum of the lamp. Lamp self-reversal must therefore also be considered. These effects have received careful analysis in the case of thallium [44,45] and silver [58].

B. ATOMIC FINESTRUCTURE Several level-crossing experiments have been performed [21,28] and are in progress [59] on the fine structure of simple atomic systems whose principal goal is an improvement in our knowledge of the fine structure constant a. The present value of a is based on the fine structure separation in the 2 2 P term in deuterium [60]. In order to obtain a result accurate to about 10 ppm, the center of the resonance line had to be determined to one thousandth of its natural width ( x 100 Mc/sec). (The fine structure interval in the 2 *Pstate of D is about 10,000 Mc/sec so that the ratio of interval to linewidth is about 100.) The most recent value for a has been deduced from measurements on the hyperfine structure of muonium [61]. It is in excellent agreement with the deuterium result. The first level-crossing experiment [21] yielded values for the fine structure intervals in the 2 3Pterm of helium. The results are listed in Table 11. In the experiment the helium atoms were first excited to the 23S, level by a highvoltage radio-frequency discharge. Excitation to thc 2 3Plevels was accomplished with resonance radiation. A sketch of the helium energy levels is shown

87

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

TABLE I1 FINESTRUCTURE INTERVALSFROM LEVEL-CROSSING EXPERIMENTS Atom

Interval

Measured splitting (Mc/sec)

He

2 3P1 - 2 3Pz 2 3P0 - 2 3P1

Li

2 *PIP - 2 zP3/2 3 2P1/z 3 2P3/2 4 2P,,z 4 2P3/2 ~

~

Corresponding interval in H

Ref.

2291.56(9) 29650(280)

-

[211

10,053.0(7) 2882.9(12) 1199.72(19)

10,971.59(20) 3250.7(20) 1367

[281 [591 [631

in Fig. 7. For the notation (J, rn) as well as for calculations of the fine structure separations and the Zeeman effect see the article by Lamb [62]. The lifetime of the 2 3P level is calculated to be 0.98 x lo-? sec, corresponding to a natural linewidth of 6 Mc/sec. The ratio of interval to linewidth is therefore about 380. For the (ls23p)32P term in lithium the fine structure interval is approximately 2833 Mc/sec and the linewidth about 1.5 Mc/sec with a corresponding

I I I

3?

3p2

+I

600

loo0 1400 MAGNETIC FIELD (GAUSS)

FIG.7. Level crossings between helium fine structure levels. Full circles indicate Am crossings.

=2

88

B. Budick

ratio of 1900. An evaluation of c1 from the measured doublet separation in lithium, however, depends upon the calculation of precise three-electron wavefunctions. In addition, important relativistic and radiative corrections have to be evaluated. Finally, the possibility of three-electron interactions must be considered. Experiments on the 3 'P and 4 2P terms of lithium [59,63] were performed with the dense atomic beam apparatus shown in Fig. 3. A major experimental difficulty in these experiments was the coating of the lenses at the high beam densities necessary to compensate for the low oscillator strengths. A few monolayers severely cut down uv transmission. This effect limited the running time to about 4 hours. At the end of this period the apparatus had to be taken apart, the lenses cleaned, and everything reassembled. The 45" geometry shown in Fig. 3 was chosen to minimize the rate of lens coating. The results of fine structure determinations by level-crossing spectroscopy are also compiled in Table 11. The intervals measured in the successive lithium doublets are 8, 12, and 11 % smaller than the corresponding hydrogen doublets, respectively. This result is surprising for two reasons: first, one would have expected an effective nuclear charge greater than one for lithium and hence a larger fine structure interval; in addition, any deviations from the hydrogen value caused by penetration should decrease for increasing values of n.

C. ATOMICHYPERFINE STRUCTURE

1. The Hyperfine Structure Hamiltonian The Hamiltonian from which the energy of an atom in a magnetic field may be determined can be written in the form

+

B + 21(21- 1)5(25 [3(I - 5)' + $(I - 1) -Z(I + l)J(J + l)] + g.r/.J-Z H +

@ . = Wl,J A1 J

*

*

J) H

(10)

where W,,Jis the multiplet and spin-orbit energy, A the magnetic dipole interaction constant, B the electric quadrupole interaction constant, I the nuclear spin, gJ the electron g factor, gI the nuclear g factor, and po the Bohr magneton [64]. The second and third terms represent the interaction of the nuclear magnetic dipole and electric quadrupole moments with the electron cloud. Higher multipoles have been neglected. The last two terms represent the interaction of the atomic and nuclear magnetism with the external field. For I = 0, 3 or J = 0, 4 the quantity B must vanish. Physically this means that for these

89

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

angular momenta no quadrupole interaction takes place, because a quadrupolar distribution of charge or an electric field gradient, respectively, is lacking. It follows that the nuclear quadrupole moment cannot be measured in the ground levels of the group I, group 11, and group I11 elements. This circumstance provided one of the major incentives for developing radiofrequency spectroscopy of excited states. On the vector model, the coupling between I and J due to an I * J interaction results in a precession of each about the resultant total angular momentum, F = I + J. In the absence of an external field the total angular momentum is a conserved quantity and the energy of the atom is expressible in terms of F :

(1 1) where

C =F(F

+ 1) - 1(1 + 1) - J(J + 1).

For the case of a nuclear spin I = 3 (NaZ3,K39,Rb87,C U ~C~U, ~A~u ,’ ~ the ~) separations between the hyperfine levels is found to be J = 12. .

W(2) - W(1)

= 2A,p

3:

“(3) - W(2)

= 3A,/,

J

=

+B

(12)

W(2) - W(1) = 2A3,, - B

W(1) - W(0)= A312 - B.

+

In the state J = the order of the levels depends critically on the ratio A / B . If a small magnetic field is applied to the atoms so that the coupling between I and J is not broken down, F will precess about the field direction with the Larmor frequency O v = gF PHm,,

h

where m F is the projection of F on the field direction and with gF given by

+ 1) + J(J + 1) - I ( I + 1) 2F(F + 1) F ( F + 1) + Z(I + 1) - J ( J + 1) 2 F ( F + 1)

F(F gF

=g J

+g,[

I

(14)

The second term in the above expression is three orders of magnitude smaller than the first and may usually be neglected.

90

B. Budick

In a magnetic field of sufficient strength to uncouple I from J (535 G ) each angular momentum will precess individually about the field direction. Their projections on the field direction are conserved quantities and may be used to evaluate the energy: B[3mJZ- J(J l ) l [ 3 m r z - I(I l ) ] W ( m , , m l ) = AmIWtJ 41(21 - 1)J(2J - 1)

+

+

+

The last term is usually negligible. 2. Extraction of Nuclear Moments.from the Hyperfne Structure Constants

The nuclear moments are related to the hfs constants by the equations A = -PIHJ

IJ

and where pI is the nuclear magnetic moment, H j is the field at the nucleus produced by the electrons in the state m, = J, e is the electron charge, Q is the nuclear quadrupole moment, and cpzz is the z component of the gradient of the z component of the electric field at the nucleus. Expressions for H j and cpzz are given by Kopfermann for the case of a single valence electron 1651:

Here F, and R, are relativistic corrections; 6 and E are corrections for the finite distribution of nuclear charge and magnetism, respectively. Tables and graphs of these quantities are given by Kopfermann for s, p , and d electrons. The expectation value for l / r 3 in the n, 1 configuration depends on a knowledge of the electronic wavefunction. When this is not available approximate values may be deduced from known values of the fine structure separation which also depends on l/r3:

where 6 W o is the fine structure splitting, H, is another relativistic correction, and Ziis the effective nuclear charge (Zi N 2 - 4 for p electrons and ZiN Z - 11 for d electrons).

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

91

Equation (1 8) implies that the A values in the two levels of a doublet must bear a definite ratio to each other; e.g., A1,2/A3,2= 5[Fr(5)/Fr(3)] for p electrons. Deviations from this conclusion are generally attributed to core polarization and are treated below. If both A and B are measured in an experiment and the nuclear g factor is known, the quadrupole moment can be deduced from the ratio of Eqs. (18) and (19) in which ( l / r 3 ) , , ,does not appear:

The validity of the procedure assumes that the radial parameter is the same for both multipole interactions. Corrections for polarization of the electron distribution by the nuclear quadrupole moment have been treated by Sternheimer [66] and are discussed below.

3. Hyperfine Structure of Isotopes of the Alkali Metals Most of the experiments on the alkali metals have been performed using double resonance in zero magnetic field. A typical setup for such experiments is shown in Fig. 8. Light is incident along the x axis with its electric vector polarized along the z axis. In the absence of an external field, the direction

source

FIG.8. Schematic diagram of an apparatus to observe double resonance in zero magnetic field.

92

3. Budick

of polarization defines an axis of quantization. Extreme care must be taken in keeping Zeeman splittings due to stray fields within the natural linewidth, for if an extraneous field is present it defines a new axis of quantization. The representations of the atomic state with respect to the two axes are related by a linear transformation, but the populations in the two representations are markedly different. This will affect the signal intensities. The light is incident on the resonance vessel from which all traces of foreign gas must be removed. Presence of a foreign gas creates the danger of a radiofrequency discharge at the high rf currents necessary in work on short-lived states. Lorentz broadening by collisions may also be caused by the foreign gas. Temperature fluctuations over the resonance vessel must be avoided. This is usually accomplished by hot-air streams or oil baths. Transitions between the F states can be induced by a radio-frequency field which is perpendicular to the z axis. The intensity of the observed signals is greater for excitation with i~ light than with 0 light and has a strong dependence on the nuclear spin and on the amplitude of the rf field [8]. Since the light signals are monitored as a function of the frequency, it is imperative that the rf amplitude be kept constant over the sweep range. Additional transitions as well as a shift in the position of the resonance line may result if the resonant frequency is comparable with the width of the resonance line. The former are due to multiple quantum transitions of the type AF = 2. The shifts are due to the component of the oscillating field which is rotating in the opposite direction from the component responsible for the transitions (Bloch-Siegert effect [67]). One variation of the zero field technique exploits the resolved power spectrum of the light source mentioned earlier in Section III,A,2. The large hfs splitting in the ground state of the heavier alkalis results in two resolved hyperfine components of unequal intensity. In the case of the 6 2S1,2- 7 transition of C S ' (I= ~ ~ 7/2) the two components have the ratio 9 : 7. The F = 4 level of the excited state will have a greater population than the F = 3 level by a ratio close to this. The actual ratio is 8.68 : 7.32 as calculated by Bucka [68]. Radio-frequency resonance between the two F states will tend to equalize their populations. To detect the resonance, Bucka resorted to the phenomenon of self-absorption. Of the light incident on the cell, more is absorbed by the F = 4 level than by the F = 3 level. A transfer of atoms to the more weakly absorbing level means that more light will be transmitted. The radio-frequency resonance can then be detected as a change in intensity. The results of all double resonance experiments on the alkali metals are contained in Table 111. With the exception of a few additions and modifications this table was compiled by zu Putlitz [8]. The table lists only the quadrupole moments deduced from the measured B values with the aid of Eqs. (1 9) and (21). The magnetic dipole moments are known very precisely from

93

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

TABLE 111 Hfs MEASUREMENTS IN ALKALI ISOTOPES Nucleus Li NaZ3

K39

K40 Rbs5

State

Method"

2 'Pi12 3 2p312

4 2P312 5 'P3l2 5 'Piiz 5 'Pw, 5

6 2P3f2

RbS7

Cs'33

cs'34 Cs'35

Cs'37

7 8 'P3l2 5 'P3i2 6 zP312

7 'P3i2 8 'Pw 7 2P312 7 2P1,2 8 2P31z 7 2P312 7'PW~ 7 'P3/'

b a b b a a, b b a a a a b a a a a a a a a a a a a a a a

A (Mc/sec)

B (Mc/sec)

46.0( 10) 19.5(6) 18.5(6) 18.7(4) 6.20( 12) 1.97(10) 8.99(15) -2.450(46) 25.029( 16) 8.16(6) 8.178(9) 8.25( 10) 3.72(3) 1.99(2) 84.852(30) 27.63(10) 27.707( 15) 27.70(2) 12.58(2) 6.75(3) 16.60(1) 16.609(5) 100.2(3) 7.626(5) 16.849(8) 17.576(6) 18.280(6)

2.4(1.4) 2.25(40) 3.4(4) 1.0(3) 1.7(3) -1.31(33) 26.032(70) 8.40(40) 8.199(40) 8. I6(20) 3.65(10) 1.98(12) 12.61 l(70) 4.06(20) 4.000(39) 3.94(4) 1.72(4) 0.96(6) -0.11(8) -0.16(6) -0.049(42) 18(1) 2.19(9) 2.23(9)

-

Q

Ref.

(barns) +0.10(6) +0.097(13) +0.13(4) +0.11(2) -

-0.093(25) +0.298(1) +0.29(2) 0.286(1) +0.283(8) +0.280(6) 0.316(20) +0.144(1) +0.14(1) 0.140( 1) +0.138(1) 0.133(1) 0.153(9) -0.003(2) -0.0036(13) -0.0024(20) 0.43(4) 0.O49(2) +0.050(2)

+

+

+

a indicates a zero magnetic field determination ; b refers to a measurement in interrnediate or strong field.

nuclear magnetic resonance work. The quadrupole moments have not been corrected for the Sternheimer effect. Where possible all three hyperfine intervals in Eq. (12) were measured yielding an overdetermined set of equations for A and B. The internal consistency was very high in all cases. The Q values are given in barns cm2).

4 . Level-Crossing Experiments in the Noble Metals The same considerations that make optical pumping attractive for investigation of the alkali metals apply in the case of the noble metals. The experimental situation is made more difficult by the more refractory nature

B. Budick

94

of the noble metals and by the fact that their main resonance lines lie in the uv. The 6s-6p transition in gold is at 2428 A. These difficulties are surmounted by an atomic beam spectrometer with quartz optics. A hollow cathode of pure copper, or plated with silver or gold, was an adequate source of resonance radiation. Level-crossing spectroscopy is particularly suited for the noble metals, since the lifetimes of their first excited states are about 20 times shorter than the corresponding states in the alkalis. Experiments on copper were first performed with a sample containing Cu63and Cu65in their natural proportions [48,84]. Subsequent experiments were carried out with enriched samples. Observed level crossings in are shown in Fig. 9. From the measured crossing fields the value of A and B listed in Table IV were deduced with a least-squares computer program [85].

~u~~ Lwet Crossing

I 860.0 1 840.0 I

I

DC MAGNETIC FIELD ( PROTON RESONANCE

FREQ.

Cu63 Level

(b)

I

1180

DC

1080 MAGNETIC FIELD (PROTON

Crossings

~

I

98 0 RESONANCE

FREQ)

FIG. 9. (a) Trace of a level crossing in C d 3 at approximately 150 G . (b) Two signals observed at higher fields. The asymmetric line shape stems from departure from 90" geometry.

95

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

FIG.10. Zeernan levels in the excited 5 zP3,2 state of the silver isotopes. Am crossings are circled.

=2

level

A schematic diagram of the Zeeman levels of silver in the 5 zP3,2 state is given in Fig. 10. Two Am = 2 level crossings are seen to occur. These are in fact observed in the wings of the Hank pattern as is evident in Fig. 1 I . Careful measurements of the small maximum on the positive side using separated Ag1O9 isotope and a computer analysis of the superposition of the Hanle effect and level crossings yielded the value listed in Table IV. The value for Ag'" was then deduced using the known ratio of the nuclear

O C magwrtc

field (gauss1

FIG. 11. Trace of the signal observed in the wings of the Hanle pattern in Aglo9. The theoretical curve based on a calculation of intensities and widths is indicated by the solid line.

96

B. Budick

moments. Coherence narrowing of almost a factor of three was observed as the beam density was increased [85]. The resonance line from the first excited state in gold is considerably weaker than the corresponding copper and silver lines, due to branching to the lower (5d)'(6~)~configuration. In addition, coating of the lamp and beam apparatus lenses limited the running time to approximately one hour. It was found necessary to boil all lenses in aqua regia for an hour before each run. Only two level crossings were observed indicating considerable inversion of the hyperfine levels. Both large positive and large negative ratios of B / A could explain the number and approximate positions of the observed signals, but a preliminary calculation of intensities distinguished between these two possibili ties. The Hanle effect and first level crossing were observed in cross fluorescence by monitoring the other branching channel but no improvement in signal to noise was obtained. The final values for the hfs constants are given in Table IV. TABLE IV Hfs MEASUREMENTS IN THE NOBLE METALS Nucleus

State

A (Mcisec)

CU63

4 'P3/2 4 'P3,z

Ag'O' Aglo9 AU'~'

5

195.2(2) 208.57(15) 32.1(16) 35.1(20) 12(2)

'P3/2

5 'P3/, 6 'P3/2

+

B (Mc/sec) -28.8(7) -25.9(6) - 304(5)

Q -0.22(4) -

Ref.

[851 [150] [1501 [85,861

1871

5 . Hyperfine Structure Measurements in the Stable Alkaline Earth Isotopes The elements Ca, Ba, and Sr all have sz ground state configurations. No electronic angular momentum exists with which the nuclear moments can interact. The excited sp configuration on the other hand possesses a shortlived singlet state, 'PI, and a long-lived triplet state, 'PI, which are connected to the ground state by resonance and intercombination lines, respectively. The triplet state has the additional advantage of possessing a large hfs. A disadvantage of the intercombination line, however, is that it lies in the red or infrared portion of the spectrum. It is therefore difficult to distinguish from the heat radiation of the oven. An atomic beam is necessary because the group IIa elements react with quartz and require a high temperature for a sufficient density of scatterers. A final difficulty is provided by the fact that the odd A isotopes that possess hfs are at most 10 % abundant in natural samples. A large isotope shift renders the even isotopes useless for excitation.

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

97

The last-mentioned difficulty was surmounted by scanning the hollow cathode resonance lamp. One of the Zeeman levels m, = - 1 of the even isotopes of barium in the lamp was shifted by the magnetic field until it coincided with that of the F = 3 of Bai3'. The situation is illustrated in Fig. 12. Transitions between F levels in zero field and in a small magnetic field were observed. F

6s6p3T

-1

Even isotopes (light source)

Odd isotopes ( I 3/2 )

(atomic b e a m for absorption)

FIG. 12. The Zeeman effect of the even isotopes of Ba (light source) is shown on the left. On the right appears the hyperfine structure of the odd isotopes in zero magnetic field. The magnetic field on the light source is varied until the level m, = - 1 emits a photon in resonance with the F = 312 level in the atomic beam.

In extracting the nuclear moments from the measured hfs constants the magnetic field and electric field gradient of the electrons must be corrected for the effects of intermediate coupling and configuration interaction. The theory of the hfs of an sl configuration in intermeditae coupling was developed by Breit and Wills [88]. The degree of intermediate coupling can be deduced from the deviation of the 3P, level from its position in pure Russel-Saunders coupling [89], from the ratio of the lifetimes of the singlet and triplet states [go], and from the measured g, factor. Of these the lifetime method appears to be the most reliable. Lurio has shown that configuration interaction leads to different radial wavefunctions for the I electron in the singlet and triplet states [22]. Thus individual electron hfs constants and quadrupole moments deduced from them will depend on which states are relied upon. Finally, levels of the 'P multiplet are mixed by the hyperfine operator [90]. These second-order effects become important when the hyperfine separations

98

B. Budick

are not small compared to the fine structure intervals. Their importance has been demonstrated in the case of zinc [35]. The values listed in Table V have not been corrected for second-order effects or for the Sternheimer effect. They were compiled by zu Putlitz [8]. TABLE V Hfs MEASUREMENTS IN THE GROUPIIa ELEMENTS Isotope

Sr8' Bai3'

State Ss5p 'PI

6s6p 'P1 6 . ~' 6Pi~ 6 ~ 'Pi 6 ~

Method D.R. D.R. D.R. L.C.

A (Mcisec) -260.084(2) 1028.31(2) 1150.59(2) - 113.2(10)

B (Mcisec) -35.658(6) -27.08(2) -41.61(2) -

Q

Ref.

+0.36(3) +0.18(2) +0.28(3) -

191,921 [93,94] [93,94] PSI

6. Hyperfine Structure Studies in the Group IIb Elements The elements in group IIb of the periodic table are particularly suited for nuclear structure studies since each possesses a number of stable or long-lived radioactive odd isotopes. The effect of adding successive pairs of neutrons can then be studied in some detail. The elements zinc, cadmium, and mercury in optical pumping work have played the role of the alkalis in atomic beam research. Their experimentally attractive features are as follows : (a) the vapor pressures for maximum scattering are reached at room temperature or at a few hundred degrees; (b) they show little chemical activity and may be conveniently studied in quartz cells ; (c) their intercombination lines do not lie too deeply in the ultraviolet and are particularly suited for electrodeless discharge lamps [l 11. It has been possible to observe Zeeman resonances in the even isotopes of Cd at a temperature of 13°C corresponding to a total of 2 x lo6 atoms in the vapor phase [20]. In double resonance work the nuclear spin is most easily determined by observing low field transitions of the type AF = 0, AmF = f 1. This leads to values for the individual g F factors from which the spin may be ascertained uniquely with the aid of Eq. (14). These resonances may be followed up in field until the perturbing influence of the other hfs levels is felt. The situation is illustrated in Fig. 13 for normal and inverted hfs. For purposes of convenience, the frequency is held fixed and the magnetic field is swept. If unpolarized light is used for excitation, the transitions (mF- m,') -3 - -3 and - +will be observed in the F = 3 state. For normal order and fixed radio frequency the

+

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

99

d __---

- - _-1/2 2'1

/-L-1/2 1/2

Am =+1

Am =+1 NORMAL HYPERFINE STRUCTURE

INVERTED HYPER FINE STRUCTURE

FIG. 13. Excitation of the 3P1 levels of the stable o d d 4 Cd isotopes with circularly polarized light. For a fixed radio frequency the double resonance transition occurs at a higher value of the magnetic field if the order of the hfs levels is normal.

first of these transitions is in resonance at a lower magnetic field than the second. If circularly polarized light o+ is used in excitaticn, the -3, -4 levels are not populated and the low field transition will not be observed. The opposite is true for inverted order. In this case the low field transition is the - 3 resonance which will continue to be observed if o+ light is used for excitation. From Fig. 14 it is apparent that the ordering of the hyperfine

+

Circularly polarized light

1

+--H MAGNETIC FIELD

(H)

J

Fig. 14. Occurrence of only the low field transition, when circularly polarized light is used, indicates inversion of the hfs levels in Cd1I3.

100

B. Budick

levels in Cd"j is inverted. Note that the field increases from right to left. Since the nuclear spin is 3,the ordering directly determines the sign of the magnetic moment. Higher spins present somewhat greater difficulties but the calculated intensities are in very good agreement with those observed for all the F states of Cd109 ( I = 5) [20]. Second-order perturbation theory may be used to obtain rough estimates of the proximity of the perturbing hyperfine levels from the magnitude of the splitting shown in Fig. 14. The direct A F = f l transitions may then be induced in zero or low magnetic field to determine the hfs separations to high precision. When more than one radioactive species is present in the scattering cell, resonances due to each may be distinguished by observing their intensities as a function of time. The halflives of the observed signals can be matched up with the known isotopic halflives [12]. Level-crossing spectroscopy is particularly suited for the group IIb elements. The potentially high precision inherent in the narrow linewidths can be attained by double resonance only with induction of the direct transitions at microwave frequencies. Level-crossing fields are in the kilogauss range and linewidths may be as narrow as 45 mG (in the case of Cd). In the course of the Cd work an important weakness of the level-crossing technique came to light. The position of a level crossing determines only the ratio of the hfs constants to the gJ factor. The precision with which one may infer the hfs constant is limited by the precision with which gJ is known. In both the double resonance studies and level-crossing investigations of the mercury isotopes, lamp scanning techniques were resorted to. With the aid of the scanning apparatus shown in Fig. 15 both the hfs of the odd isotopes and the isotope shift were studied [96]. The light source contained enriched Hgi9', the scattering cell natural mercury. One of the cr components from the light source was isolated with its direction of polarization parallel to the splitting magnetic field in which the scattering cell was situated. By varying the scanning field the resonance radiation populated the m = 0 level of each even mercury isotope in turn. The population was observed by monitoring the radio-frequency transition Am = k 1 for a frequency of 3054 Mc/sec and a field of 6254 G corresponding to gJ = 1.484 for all the even isotopes. The amplitude of the rf signal is proportional to the population, which in turn depends on the degree of degeneracy between the energy levels in the light source and the scatterer. The width of each scanning curve is of the order of magnitude of the Doppler width, 0.03 cm-', while the isotope shift is approximately 0.15 cm-'. The curves are quite well resolved as shown in Fig. 16. The odd isotopes were observed by populating the m = levels of Hg'99 and the m = + f level of Hgzo' and adjusting the splitting field to the value required for transitions to other Zeeman levels. The position of hyperfine

++

101

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

PHOTOMULTIPLIER

FIG.15. Schematic diagram of apparatus used in mercury scanning experiments.

x)oAmp = 55.4.10-' cm-1

0

HgBB

200

Hgmo

400

600

HgZoz

BOO

1000

1200

Hg2"

1400

Amp

FIG.16. Scanning curve for the even mercury isotopes.

levels with F 2 3 could thus be determined relative to Hgt9'. No double resonance signals are observed in the F = f level because n excitation does not produce a population inequality between the Zeeman levels. Table VI contains a compilation of hfs measurements in the group IIb elements and has been taken from zu Putlitz [8]. Each of the values has been rechecked against the original paper in which it is quoted, to eliminate the propagation of typographical errors. The method used is indicated in column 3: a indicates double resonance in zero or weak field, b refers to double resonance in intermediate or strong field, and c indicates level crossing. The quadrupole moments have not been corrected for the Sternheimer effect. An asterisk indicates

TABLE VI Hfs MEASUREMENTS IN GROUPIIb ELEMENTS c

A

Nucleus

State

Method

4 3P1

a a C

a a a

4 3P1

C

5 3P,

b C

5 3P1

b C C

5'P1

5 3P1 5 3P1 5 3P1 5 3P1 5 3P' 6 3P1 6 'PI 6 3PI 6 3P1 6 3P1 6 3P1

a a b b b a C C C C

b C

6 3PI 6 3P1

b, c a a b, c a b

b b, c a a b a

6 3P1 6 3P1

"Q is obtained from

C Q'O'

(Mc/sec)

I

535.117(2) 535.163(2)* 608.99(5) 609.208(2) 609.086(2)* - 854.2( 10) - 853.543(6)* - 1148.6(20) - 1148.784(7)* 186(4) -4123.81(1) -4313.86(1) - 686.0425(8)* -4484(2) - 657.6(6) 16600(1 100) -61 33( 15) -2399.69(6) 15813.46(23) -2368.O4(8) 15394(30) 15388.9(45) 15387.1(53) 15390.91(1) 15392.66(15)* - 2 3 2 8 3 17) -2328.89(84) 14900(390) 14733.3(150) 14,750.7(50) 14,752.37(1) 14754.04(14)* -5437.1(150) -5454.569(3) 4991.37(1)

+

+

and the ratios of B to B z o l

B (Mclsec)

2.445(4) 2.870(5)* 2.867(12)* - 19.37(9) - 19.331(7) - 18.782(8)* - 18.770(12)* - 166(3) - 163.279(5)* - 167.3(20) - 165.143(5)* -

PI

0.7688(6) -

+131(6) -724.8(9OO) -782.45(86) -

-0.024(2)

-

+0.1-8(2) -

-0.8286(15)

+0.77(10) +0.78(10)

- 1.08885(13)

-0.79(10)

-0.61 62(8) -

-

-

+ 169.047(9)* -

Q

-0.6469(3)

- 1.0437(10)

+0.562(35)

-0.620(2)

+0.538860(16) - 1.04903(13)

-

-

+1.3(3)" -

- 0.61(8)

+ 1.40(12)" -

-901(1 3) -902.9(54)

I .61(13)" -

-

-

-283(19) - 280.107(5) -255(1)

+0.50(4)"

+0.46(4)"

Ref.

E

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

103

that second-order corrections have been applied. Uncorrected values are given for purposes of comparison.

D. POLARIZATION OF THE ATOMICCORE I . Core Polarization by Unpaired Electrons Core polarization is a mechanism whereby configurations containing unpaired s electrons are admixed to configurations in which all s electrons belong to closed shells or subshells. The atomic wave function attains a partial s character which may influence the hfs strongly through the Fermi contact term. Core polarization is alternatively described as the elevation of s electrons from closed shells to unoccupied s orbitals by an exchange interaction with the external unpaired electron. The importance of the phenomenon was recognized early in the hfs of the ground levels of Ga [112] and TI [113]. Calculations have been performed by Goodings for atoms containing valence p electrons, particularly alkali and halogen atoms. The hyperfine interaction constants for the 2 P 3 j 2and 2Pl,2states are given by

where A 3 / 2 ( P )and A I l 2 ( p )are the hyperfine constants in the absence of core polarization and bear the ratio +[F,($)/Fr(9] as pointed out in Section 111,C,2. The reason for the departure from the theoretical ratio is now apparent. The symmetrical manner in which A , enters in Eq. (22) is easily explained. Although the interaction contributing to the hfs is of the form c i s i* I, where the si refer to the s electrons, this may be transformed to the form S * I, whereS refers to the total spin of the atom [114]. We may write the magnetic dipole term in the Hamiltonian as A1 * J

+ A’S

*

I.

Recalling that the projection of S on J is given by gJ - 1, this may be rewritten as [ A A’(gJ - l)]I * J ; (24)

+

for a single-valence electron gJ - 1 is equal and opposite in sign for bothlevels of the doublet. A particularly simple and instructive case of core polarization is provided by the magnetic dipole interaction constants in the first three excited P states of lithium. The fine structure of these states from level-crossing experiments was the subject of an earlier section. The Li7 nucleus has a spin of 4. The magnetic field at which the experiment is performed is sufficient to uncouple I and J so that only their projections on the field direction are good quantum

104

B. Budick

numbers. Consequently, each of the mJcrossing levels consists of four closely spaced levels. Interference signals from only four of the sixteen crossings are observable since the nuclear and electronic motions are uncoupled and the photon interacts with the electronic structure ; hence, the selection rule Aml = 0. A typical signal has the shape shown in Fig. 17.

FIG. 17. Level-crossing signal observed in the Li’ 3P state. The markers and numbers in the figure refer to proton nmr magnetometer readings.

The interval between crossings will be some linear combination of the hfs constants AIl2 and A312 . With the aid of Eq. (22) and the ratio between A 3 / 2 ( ~ ) and A I l 2 ( p )it can be written

It was pointed out [I 141 that the ratio of the intervals between peaks, Eq. (25), for the 2P and 3P states is very closely equal to the ratio of the fine-structure intervals of the 2P and 3P states. Since the fine-structure intervals scale as ( l / r 3 ) , Eq. (20), this strongly suggests that AHc scales as ( l / r 3 ) .A 3 / 2 ( ~is) known to have this scaling factor, Eq. (18). It follows that A , also scales as ( l / r 3 ) . Ritter obtained the value of A1/2 in the 22P,12 state in a double resonance experiment [69]. The value of A , given in Table VII for the 2P

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

105

TABLE VII MAGNETIChfs CONSTANTS IN THE LITHIUM P STATES

2P 3Pd 4Pd

7.67(1)” 2.26(5)b 0.95(1)’

61.4 61.1 60.8

-10.5(3) - 3.05(12)

-1.31(4)

46.17(35) 1332) 5.73

-3.36(30) -0.96(13) -0.43

See Brog [28] and Wieder [29]. See Budick er al. [59]. See Isler et al. [63]. Values of the hfs constants in these states are based on the hypothesis that the constants scale ( l / r 3 ) (see text) and that A l l z is positive. a

state could thus be determined. The first column shows the state; the second column contains the measured intervals ;the third column is the product of the interval and n3 and reveals the consistency with which the intervals scale ; the fourth column gives the measured value of A , for the 2P state and the values deduced from it for the higher states. The last two columns are included to illustrate how far the actual A values depart from the theoretical ratio. It has been known for some time that the mechanism of core polarization can seriously affect the magnetic fields at the nuclei of transition metal ions [115]. More recent work has revealed that each d electron in a divalent ion should contribute a magnetic field at the nucleus of approximately 115,000 G and that each d electron in the (34-(4s)’ ground state of free atoms should produce a field of about 23,000 G [116]. This rather remarkable description of a field per unit spin is in good agreement with experimental results for the divalent ions. The agreement in the case of free atoms is less satisfactory, although certain refinements of the theory promise to bring a higher degree of consistency to the ion and atom results [I 171. Part of the difficulty lies in the presence of the outer 4s subshell in the case of the atoms. The polarization of this outer shell by the valence d electrons produces a near cancellation of the magnetic field due to core polarization as the above figures testify. To test the theory for free atoms while retaining the favorable electronic configuration of the ion, i.e., absence of 4s electrons, an experiment was recently performed in chromium [42]. Unique among the first-row transition elements, its ground state is (34’4s. An excited configuration (3~l)~4p is connected to the ground state by an intense resonance line in the visible range and is therefore susceptible to double resonance techniques. The Hund rule ground multiplet of the excited configuration is 7P2,3,4. In the absence of core polarization the magnetic field at the nucleus arises entirely

106

B. Budick

FIG.18. High-field double resonance in the (3dS4p) 7P3,4 states of CrS3.

from the p electron since the half-filled d shell couples to a spherically symmetric 6S5,2ground state. The magnetic interaction constants in the three J levels are expected to be small and in the ratio 4 4 ) : 4 3 ) : 4 2 ) = 81 : 140 : 101. In a field of 310 G the appropriate expression for the energy levels is given by Eq. (15). The quadrupole moment is small enough so that the last term may be neglected. Transitions of the type AmJ = f 1 and Am, = 0 will occur at frequencies hv = gJpOH Am,. (26)

+

For the even isotopes of chromium ( I = 0), which are 90 % abundant in a natural sample, the second term is absent. The remaining term leads to precise values of gJ . For Cr53( I = 2) the transition frequency is shifted from the central maximum and resonances appear at + $ A and ++A. Thus the magnetic constant can be determined directly. Figure 18 is a trace of the double resonance signal with a sample containing 80 % CrS3and 20 % even isotopes. The J = 3 level shows no hfs splitting. The J = 4 peak is a superposition of equally spaced resonances plus a central maximum. This is evident from Fig. 19. The solid curve is the signal recorded with a 99 % enriched sample. The dashed curve represents the sum of the four equally spaced Lorentzian lines.

FIG.19. High-field double resonance in (3dS4p) 'P4 of CrS3.

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

107

In analyzing the data it was found that an additional form of configuration interaction is important in chromium. An overlapping (3d)44s4pconfiguration also possesses a 'P multiplet which is admixed to the d 5 p term via exchange interactions. Once this possibility was allowed for, measurements performed in the overlapping configuration led to a value of -525,000 G for the magnetic field at the nucleus due to core polarization [1181. This value agrees well in magnitude and in sign with ion values. The method for extraction of the core polarization presented here is based on an analysis of measured A values. When these differ from the theoretical values the difference is ascribed to a net unpaired s electron spin density at the nucleus. At a recent conference on magnetic hfs Bleaney pointed out that the ( l / r 3 )factors associated with the orbital and spin dipolar terms in the hfs Hamiltonian will in general be different [I 18al. When this difference is allowed for the residual contribution to the magnetic hfs A value, heretofore attributed to core polarization, is seriously affected. In addition, it has been shown that relativistic effects in heavy atoms are indistinguishable from core polarization effects [118b]. The data on the excited states of lithium presented in Table V1I offer an opportunity to unravel this puzzle in a particularly simple system. 2. Sternheimer Corrections

The mechanism of core polarization via exchange interactions with unpaired electrons results in a modification of the magnetic dipole interaction constant A . For a p electron, the primary modes of excitation that affect the magnetic field at the nucleus are the excitation of an s electron into a higher s orbit and the spherically symmetric part of the excitation of a core electron in a p orbit into a higher unoccupied p orbit [89]. In addition the nuclear quadrupole moment can induce a quadrupole moment in the atomic core by rearrangements of charge which in turn can interact with the valence electron. These rearrangements are of two sorts : (a) an angular redistribution corresponding to I' # I modes of excitation of the core, ns + d, np +f and (b) a radial redistribution corresponding to I' = 1 modes of excitation, np + p , nd-t d. For heavy atoms the radial modes dominate and result in a net antishielding, i.e., the nuclear quadrupole moment is smaller than the measured value. The correction can be applied by multiplying the measured value by a factor 1/(1 - R) where R is a negative number. In particular Sternheimer has calculated the correction factors for excited states of the alkalis as the principal quantum number n is increased [66]. In Table VIII are presented the quadrupole moments of the rubidium isotopes RbE5and RbE7with and without Sternheimer correction. (a), (b)denote application of different radial wavefunctions of the valence electron in the calculation of the correction factors. This table

B. Budick

108

TABLE VIII

STUDY OF THE STERNHEIMER CORRECTION IN Rb" Isotope

State

Rb85

5P 6P 7P 8P

Rbs7

5P 6P 7P 8P

(I

Qbfs

+0.330( 1) +0.319(2) 0.3 16(8) $0.3 16(20)

+

+0.160( 1) +0.153(2) +0.147(2) +0.153(9)

Quadrupole moments in

QJb)

QN(@

+0.260(1)

+

+0.267(7) +0.270(17)

0.289(1) +0.285(2) +0.285(7) +0.286(18)

+O. 126(1) +O. 127(2) +O. 124(2) 0.13l(8)

+0.140(1) +0.137(2) +0.132(2) +0.138(8)

+0.264(2)

+

cm2.

has been compiled by zu Putlitz [118c]. The corrected values in columns 4 and 5 are more consistent. On deducing the uncorrected moments in column 3 from Eq. (21) the A values were corrected for core polarization alterations of the magnetic field at the nucleus. In the case of Rb these corrections amount to about 5 % . Schwartz has proposed the following modification of Eq. (21) for the determination of Q from the measured values of B, A , and p in ' P 3 / 2 states [89]:

where I is the nuclear spin, C is Sternheimer's correction factor, and p3/2is the fractional contribution of s electrons to the magnetic hfs. In the case of indium the factor (1 - P3,J reduces the given value of Q by 28 %.

IV. Optical Pumping Experiments in Ground States A. OPTICAL ORIENTATION OF ATOMIC GROUND STATES The principle of the method by which the ground state of an atom can be polarized optically can be understood by referring to Fig. 20. The Zeeman effect is shown separately for D, (zSl/z -,'P1/J and D2 ('Sll2+ 'P3/2) lines of sodium, The hfs has been neglected. .n transitions are indicated by vertical lines, a + and a- transitions by lines which slope to the right and to the left, respectively. Suppose optical pumping is effected with o + D, light. Atoms raised to the rn = +* level in the excited state can decay to both the ground state Zeeman levels with transition probabilities given by the spontaneous

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

109

FIG. 20. Schematic representation of optical pumping with circularly polarized D radiation from an alkali vapor. Pumping cycle results in inverted ground state populations.

emission theory. If the process is repeated several times atoms will accumulate in the nz = ++level of the ground state. Kastler has shown that the ratio of the populations in the 3 zS,,2 Zeeman levels can attain the value 4 in a pumping time of the order of a tenth of a millisecond [ I 191. Consideration of the nuclear spin complicates the mathematics but presents the intriguing possibility of polarizing the nucleus. The effect of cr + D, light on the F = 2 level of the ground state is to increase the population of the m, = 2 level. Since the electron and nucleus are a coupled unit the average value of m, must also increase. A sample of polarized atoms could thus serve as a source of oriented nuclei for the determination of P-decay symmetries, as a source of polarized electrons and as a means of enhancing nmr signals. Orientation of an ensemble of atoms was first detected by purely optical means [120, 1211. It is also possible to detect the orientation in the ground state by disorienting the atom via radio-frequency transitions between the Zeeman levels and observing the change in the scattered light [122]. A particularly simple setup for this type of experiment is shown in Fig. 21. CIRCULAR

POLARIZSR

AHPLIF

PI SCOPE

SODIUM ARC

ABSORPTION

BULB

~HOTKELL

FIG.21. Setup for observing orientation of an alkali ground state.

Resonance radiation from a sodium lamp is incident on a bulb containing sodium which is placed between a pair of Helmholtz coils (not shown) and surrounded by an rf loop. The transmitted light is viewed with a photocell. The bulb containing oriented atoms is transparent to the o+ pumping light.

110

B. Budick

+

Rf resonance transfers atoms to the lower m levels which can absorb the IJ light, thus rendering the bulb opaque. Resonance is detected as a decrease in the transmitted light. This technique was applied to an alkali metal vapor by Bell and Bloom [123] following a suggestion by Dehmelt. it is also possible to disorient the atoms by inducing AF = I, Am, = 1 transitions between the hfs levels [124]. A useful second-order effect permits observation of the field independent mF= 0 + m, = 0 transition and leads to precision values for the hfs splitting [124]. it is not necessary to polarize or filter the incident radiation provided a difference in the light intensity exciting atoms out of the two ground state hfs levels can be achieved [125]. This can be accomplished in the following way. Sodium light is incident on an absorption cell along its length. The component pumping atoms from the F = 2 level in the ground state is more strongly absorbed than the component exciting the F = 1 level by the ratio of the statistical weights. At the rear of the cell the light exciting atoms from the F = 1 level is more intense and a correspondingly greater population exists in the F = 2 level. Radio-frequency resonance equalizes the populations and can be detected as a change in the transmitted light. This experiment using self-absorption in the exciting light to achieve a population inequality in the ground state is reminiscent of the experiment of Bucka [68] in which selfabsorption was used to detect rf resonance in excited states.

1. Bufer Gases A number of collision processes serve to disorient the atoms and hence to decrease the amplitude of the radio-frequency signals. Two oriented sodium atoms are coupled by the dipole-dipole interaction between their electron spins. This leads to a definite transition probability to other Zeeman levels, which is usually negligible at the low vapor pressures employed. An oriented sodium atom can interact with paramagnetic foreign gas atoms or with the motional magnetic field resulting from its motion through the electric fields of molecules or rare gas atoms. Finally, collisions with the walls of the container occur with a frequency of the order of lo4 per second. These may have a serious disorienting effect on the atoms and may lead to broad resonance lines as well. It was discovered [I261 that the addition of a foreign buffer gas ennanced the orientation by preventing wall collisions. in addition, oriented zS1,z states were found to be extremely insensitive to depolarization by collisions with the buffer gas due to their spherical symmetry. Enhancement factors of 10 and 15 were obtained. The addition of a buffer gas was originally proposed as a means of reducing the Doppler width of microwave spectral lines [127]. The effect of collisions

111

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

is to change the line shape from Gaussian to Lorentzian and to reduce the linewidth to the value [128] Av

L

= 2.8 - x

;1

normal Doppler width,

(28)

where L is the mean free path of the oriented atom in the gas and A is the wavelength of the microwave radiation. Linewidths of 20 cps have been obtained for the 4.4-cm line in the Rb ground state using neon and helium pressures of several Torr. The absorption lines were a factor of 300 narrower than the normal Doppler width [128]. A buffer gas will in general produce a shift in the resonance frequency as well as an enhancement and a narrowing. Such shifts have been studied in the zero-field hyperfine splitting of Cs for a variety of buffer gases [129,130]. The light gases H, , He, N, , and Ne produce positive shifts while the heavier rare gases Ar, Kr, and Xe produce negative shifts. The pressure shift for a mixture of gases 6 is approximately given by 6 = I i d i p i, where d i is the shift of a pure gas and p iis its partial pressure ( x i p i = 1). A mixture can be chosen whose total shift is very small [129]. Table IX presents information on the hfs intervals in alkali ground states that has been obtained by optical pumping orientation techniques. TABLE IX Hfs INTERVALSAND NUCLEAR MOMENTS FROM O R l E N T A n o N EXPERIMENTS Atom

State

NaZ3 Rbs7

3 'SIlz 5 2Sl/Z 6 2S1/z

Cd"' Cd113 Hgzol Hg199

5'S0 5 'So

csl=

6'So 6'S0

Av

v(Cd)/v(H)

1771.6262(1) 6834.682608(7) 9192.6320(5) 0.21 1782(2) 0.211543(2)

p(Hg)lp(W

Ref.

[I241 ~ 9 1 [130,131] 11351 [I351 0.197421(4) [134] 0.178273(4) [134]

2. Nuclear Orientation Early attempts to orient nuclei in the manner suggested above were unsuccessful [132]. The failure was attributed to the imprisonment and consequent depolarization of the resonance radiation. Once nuclear orientation was achieved it became clear that the early work suffered from insufficient pumping intensity [133]. Thus far work has been confined to orienting nuclei in the ground states of the group IIb elements. Linewidths as narrow as one cycle have been achieved [134] for Hg'99 ( I = 3). HgZo1( I = 3) on

112

B. Budick

the other hand produces a linewidth of 8 cps. It is speculated that the principal mechanism responsible for the linewidth in HgZo1 are collisions in which the electron cloud of the mercury atom acquires an electric field gradient which interacts with the nuclear quadrupole moment. Results of recent nuclear orientation experiments are given in Table IX. Note that the Cd values are for nmr frequencies in Cd vapor compared to proton frequencies in a mineral oil sample of the same dimensions [135]. Ratios of the nuclear moments of the mercury isotopes to the proton moment are without diamagnetic corrections. B. SPIN-EXCHANGE EXPERIMENTS O N LIGHTSYSTEMS In the discussion of collision mechanisms given above one very important type of disorienting collision was deliberately left out. This mechanism called spin exchange has been treated by Wittke and Dicke [128] for the case of atomic hydrogen and has led to a new form of high-precision radio-frequency spectroscopy. Spin exchange functions as a disorientation mechanism since two hydrogen atoms, for example, passing within a distance of less than about 3.8 x cm have a probability of 3 of interchanging spin coordinates. Dehmelt was the first to suggest that spin exchange between electrons and oriented sodium atoms could be used to polarize the electrons [136]. Electron spin resonance serves to disorient the electrons which in turn reduces the atomic polarization. Dehmelt used the apparatus shown schematically in Fig. 21 to measure the free electron spin g factor, gs. The transmitted circularly polarized sodium light was monitored as a function of the radio frequency. A recent extension of this technique using Rb as a polarizing agent has yielded results for the gyromagnetic ratios of hydrogen, tritium, free electrons, and Rbss with a precision of one part in lo6 or better [137]. The efficiency of the spin exchange process depends on the cross section for electron exchange. Knowledge of spin exchange cross sections is also important in determining the effect of spin exchange collisions on the resonance linewidth. Typical cross sections are in the range from 2 x to 5 x cm'. A list of references in which spin exchange cross sections have been measured or estimated is given by Balling et al. [138, footnote 121. Spin exchange polarization in the alkali-alkali systems Na-K and Rb-Na was subsequently detected [139,140]. However, the real utility of this new form of optical pumping was exploited in experiments to measure the ground state hfs of hydrogen, deuterium, tritium, nitrogen, and phosphorus. The sensitivity of the method (approximately 10" atoms suffice) minimizes the dissociation problem of the gases. Straightforward optical orientation is ruled out by the short wavelength of the resonance lines of these atoms. Spin exchange has also been used to orient ions [141].

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

I13

The experiments on the gases are sufficiently similar not to warrant separate detailed description. A bulb is prepared containing the gas to be studied in molecular form, a small quantity of sodium, rubidium, or cesium and a buffer gas, typically argon or helium, at pressures up to a few cm of Hg. Tungsten or aluminum electrodes can be discharged with millisecond pulses within the bulb to dissociate the gas molecules. The alkali atoms within the bulb are oriented by circularly polarized D, resonance radiation. It has been shown that pumping with this single component of the doublet is much more efficient in achieving orientation than using unfiltered light [142]. Atoms of the gas under investigation are polarized by spin exchange collisions with the oriented alkali atom. The same mechanism will serve to depolarize the alkali atoms when a radio-frequency field succeeds in depolarizing the gas atoms. This is detected by observing a decrease in the transmitted light as the alkali atoms suddenly begin to absorb light again. In the experiments of Novick and coworkers on nitrogen, a flow system was used in which the nitrogen was dissociated prior to entering the resonance bulb and molecular nitrogen was used as the buffer gas [143]. In experiments on (He3)+ the ions were oriented by spin exchange collisions with oriented Cs atoms, but radio-frequency resonance was detected by monitoring the rate of ion neutralization in a manner described by Fortson et al. [141]. The hope of completely understanding one-electron atoms and of revealing quantum electrodynamic and proton structure effects are the incentives for studying the hydrogen isotopes [144,145]. The atoms of nitrogen and phosphorus have p 3 4S3,2ground states and should possess no hfs. However, the three unpaired p electrons can produce a magnetic field at the nucleus via core polarization. A pseudoquadrupole interaction resulting from secondorder effects and the effect of intermediate coupling have been treated by one TABLE X RESULTS OF SPIN-EXCHANGE EXPERIMENTS ON LIGHTATOMS AND IONS Atom

State

A

(Mc/sec) ~~

Av (Mclsec)

B (CPd

~~

~

H D

1420.405749l(60) 327.384349(5) 15 16.7014768(60) 8665.649905(50)

T

He3 + N14 N' N14 NL5 P3'

10.450925(20)

- 7(20)

- 14.645441(20)

~

10.450930(8) 14.645457(5) 55.055691(8)

+1(5)

Ref.

114

B. Budick

set of authors [146] while another has determined the hfs anomaly by measuring the nitrogen nuclear moments in a separate experiment [147,148]. A compilation of hfs constants and ground state splittings that have been obtained with the spin exchange technique is presented in Table X.

ACKNOWLEDGMENTS The author wishes to acknowledge the generosity of Dr. G . zu Putlitz who permitted the use of the tables from his review article and communicated many prepublication results. Professor G. W. Series allowed the reproduction of Fig. 13 in his review. I especially thank my wife for her patience during the preparation of this article and for her understanding of the requirements of research. While the article was being written the author was the holder of a National Science Foundation Postdoctoral Fellowship. Miss Susan Jacobson typed the manuscript. REFERENCES 1. A. Mitchell and M. Zemansky, “ Resonance Radiation and Excited Atoms.” Cambridge Univ. Press, London and New York, 1953. 2. S. Tolansky, “High Resolution Spectroscopy.” Methuen, London, 1947. 3. F. Bitter, Phys. Rev. 76, 833 (1949). 4. J. Brossel and A. Kastler, Compt. Rend. 229, 1213 (1949). 5. P. A. Franken, Phys. Rev. 121, 508 (1961). 6. G. W. Series, Repts. Progr. Phys. 22, 280 (1959). 7. A. Kastler, J . Opt. SOC.Am. 47, 460 (1957). 8. G. zu Putlitz, Ergeb. Exukt. Nnturw. 37, 105 (1964). 9. J. Brossel and F. Bitter, Phys. Rev. 86, 308 (1952). 10. R. Kohler, Phys. Rev. 121, 1104 (1961). 11. B. Budick, R. Novick, and A. Lurio, Appl. Opt. 4,229 (1965). 12. F. W. Byron, Jr., M. N. McDermott, and R. Novick, Phys. Rev. 132, 1181 (1963). 13. B. Budick, BUN. Am. Phys. SOC.10, 1214 (1965). 14. J. A. Giordmaine and T. C. Wang, In “Quantum Electronics” (C. H. Townes, ed.), p. 67. Columbia Univ. Press, New York, 1960. 15. A. Lurio and R. Novick, Phys. Rev. 134, A608 (1964). 16. C. A. Piketty-Rives, F. Grossetete, and J. Brossel, Compt. Rend. 258, I189 (1964). 17. J. Meunier, A. Omont, and J. Brossel, Compt. Rend. 261, 5033 (1965). 18. F. W. Byron, Jr., M. N. McDermott, and R. Novick, Phys. Rev. 134, A615 (1964). 19. F. W. Byron, Jr. and H. M. Foley, Phys. Rev. 134, A625 (1963). 20. M. N. McDermott and R. Novick, Phys. Rev. 131, 707 (1963). 21. F. D. Colegrove, P. A. Franken, R. R. Lewis, and R. H. Sands, Phys. Rev. Letters 3, 512 (1959). 22. A. Lurio, Phys. Rev. 42,46 (1966). 23. E. U. Condon and G. H. Shortley, “Theory of Atomic Spectra.” Cambridge Univ. Press, London and New York, 1951. 24. M. E. Rose and R. C . Carovillano, Phys. Rev. 122, 1185 (1961). 25. W. Hanle, Z . Physik 30, 93 (1924). 26. A. C. G. Mitchell and E. J. Murphy, Phys. Rev. 46, 53 (1934).

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

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(1964). 47. B. Budick and L. A. Levin, BUN. Am. Phys. SOC.11, 455,456 (1966). 48. H. Bucka, J. Ney, and G. Heppke, Z . Angew. Phys. 20, 354 (1966). 49. M. A Guiochon, J. E. Blamont, and J. Brossel, Compt. Rend. 243, 1859 (1956). 50. F. Boutron, J. P. Barrat, and J. Brossel, Compt. Rend. 245, 2251 (1957). 51. J. P. Barrat, Compt. Rend. 244, 2785 (1957). 52. J. P. Barrat, J . Phys. Radium 20, 541, 633 (1959). 53. J. P. Barrat and J. Brossel, Compt. Rend. 246, 2744 (1953). 54. A. Ornont, Compt. Rend. 252, 861 (1961). 55. E. Otten, Nalurwiss. 7, 157 (1964). 56. A. Omont, Compr. Rend. 260, 3331 (1965). 57. W. Happer and E. B. Saloman, Phys. Rev. Letters 15,441 (1965). 58. B. Budick and A. Levin, Private communication (1966).

59. B. Budick, H. Bucka, R. J. Goshen, A. Landman, and R. Novick,Phys. Rev. 147, l(1966). 60. E. S. Dayhoff, S. Triebwasser, and W. E. Lamb, Jr., Phys. Rev. 89, 106 (1953).

61. W. Cleland, J. M. Bailey, M. Eckhouse, V. M. Hughes, R. M. Mobley, R. Prepost, and J. E. Rothberg, Phys. Rev. Letters 13, 202 (1964). 62. W. E. Lamb, Jr., Phys. Rev. 105, 559 (1957). 63. R. C. Isler, S. Marcus, and R. Novick, Bull. Am. Phys. SOC.10, 1096 (1965). 64. N. F. Ramsey, ‘‘ Molecular Beams,” p. 84. Oxford Univ. Press (Clarendon), London and New York, 1954. 65. H. Kopfermann, “Nuclear Moments.” Academic Press, New York, 1958. 66. R. Sternheimer, Phys. Rev. 105, 158 (1957). 67. F. Bloch and A. Siegert, Phys. Rev. 57, 522 (1940). 68. H. Bucka, 2. Physik 151, 328 (1958). 69. G. J. Ritter, Can. J . Phys. 42, 833 (1964). 70. P. L. Sagalyn, Phys. Rev. 94,885 (1954). 71. J. N. Dodd and R. W. N. Kinnear, Proc. Phys. SOC.(London) 75, 51 (1960).

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G. J. Ritter and G. W. Series, Proc. Phys. Soc. (London) A68,450 (1955). W. M. Fox and G. W. Series, Proc. Phys. SOC.(London) 77, 1141 (1961). H. Bucka, H. Kopfermann, and J. Ney, 2.Physik 159,49 (1960); 167, 375 (1962). H. A. Schuessler, 2.Physik 182,289 (1965). H. Bucka, H. Kopfermann, M. Rasiwala, and H. Schussler, Z. Physik 176,45 (1963). H. Krueger and U. Meyer-Berkhout, Nuturwissenschuften (Nuturwiss.) 42,94 (1955). H. Bucka, H. Kopfermann, and A. Minor, Z. Physik 161, 123 (1961). H. Bucka, G. zu Putlitz, and R. Rabold, Z . Physik. To be published. G. zu Putlitz and A. Schenck, Z. Physik 183,428 (1965). K. Althoff and H. Krueger, Nuturwissenschuften (Nuturwiss.) 41,368 (1954). H. Bucka, H. Kopfermann, and E. W. Otten, Ann. Physik 4 , 39 (1959); Nuturwissenschaften (Nuturwiss.) 45,620 (1958). 83. H. Bucka and G. v. Oppen, Ann. Physik 10, 119 (1962). 84. B. Budick and L. A. Levin, Bull. Am. P h p . SOC.11, 168 (1966). 85. B. Budick and L. A. Levin, Private communication (1966). 86. B. Budick and L. A. Levin, Bull. Am. Phys. SOC.11, 328 (1966). 87. L. A. Levin, Private communication (1966). 88. G. Breit and L. A. Wills, Phys. Rev. 44, 470 (1933). 89. C. Schwartz, Phys. Rev. 105, 173 (1957). 90. A. Lurio, M. Mandel, and R. Novick, Phys. Rev. 126, 1958 (1962). 91. H. Bucka, H. Kopfermann, and G. zu Putlitz, Z . Physik 168, 542 (1962). 92. G. zu Putlitz, 2.Physik 175, 543 (1963). 93. H. Bucka, H. Kopfermann, and G. zu Putlitz,Z. Physik 165,72 (1961). 94. G. zu Putlitz, Ann. Physik 11, 248 (1963). 95. A. Lurio, Phys. Rev. 136, A376 (1964). 96. P. L. Sagalyn, A. C. Melissinos, and F. Bitter, Phys. Rev. 109, 375 (1958). 97. K. Boeckmann, H. Krueger, and E. Recknagel, Ann. Physik 20, 250 (1957). 98. P. Thaddeus and M. N. McDermott, Phys. Rev. 132, 1186 (1963). 99. R. F. Lacey, Ph.D Thesis, M.I.T., 1959. Unpublished. Values can be found in [loo]. 100. P. Thaddeus and R. Novick, Phys. Rev. 126, 1774 (1962). 101. F. W. Byron, Jr., M. N. McDermott, R. Novick, B. W. Perry, and E. B. Saloman, Phys. Rev. 136, B1654 (1964). 102. M. N. McDermott, R. Novick, B. W. Perry, and E. Saloman, Phys. Rev. 134, B25 (1964). 103. W. T. Walter and H. H. Stroke, Bull. Am. Phys. SOC.9,452 (1964). 104. 0. Redi, Quart. Progr. Rept. No. 74, p. 43. Res. Lab. Electron., M.I.T. (July 1964); W. J. Tomlinson 111 and H. H. Stroke, Nucl. Phys. 60, 614 (1964). 105. W. W . Smith, Phys. Rev. 137, A330 (1965). 106. A. C. Melissinos, Phys. Rev. 115, 126 (1959). 107. H. R. Hirsch and C. V. Stager, J. Opt. SOC.Am. 50,1052 (1960). 108. H. R. Hirsch, J. Opt. SOC.Am. 51, 1192 (1961). 109. C. V. Stager, Phys. Rev. 132, 275 (1963). 110. C. Brot, J. Phys. Radium 22,412 (1961). 111. G. S. Boyle, J. N. Dodd, and W. L. McLean, Proc. Phys. SOC.(London) B70,796 (1957). 112. G. Koster, Phys. Rev. 86, 148 (1952). 113. C. Schwartz, Phys. Rev. 97, 380 (1955). 114. T. Eck, Private communication (1963). 115. A. Abragam, J. Horowitz, and M. H. L. Pryce, Proc. Roy. SOC.A230, 169 (1955). 116. R. E. Watson and A. J. Freeman, Phys. Rev. 123,2027 (1961). 117. R. Winkler, 2.Physik 184,433 (1965). 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82.

OPTICAL PUMPING METHODS IN ATOMIC SPECTROSCOPY

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118. B. Budick, R. J. Goshen, S. Jacobs, and S. Marcus, Phys. Rev. 144, 103 (1966). 118a B. Bleaney, ConJ Magnetic Hyperjine Structure At. and Molecules, Paris, 1966. To be published. 118b P. G . H. Sandars and J. Beck, Proc. Roy. SOC.A289,97 (1965). 118c G. zu Putlitz, Private communication (1966). 119. A. Kastler, J. Phys. Radium 11, 255 (1950). 120. J. Brossel, A. Kastler, and J. Winter, J. Phys. Radium 13, 668 (1952). 121. W. B. Hawkins and R. H. Dicke, Phys. Rev. 91, 1008 (1953); W. B. Hawkins, ibid. 98, 478 (1955). 122. J. Brossel, B. Cagnac, and A. Kastler, J. Phys. Radium 15, 6 (1954). 123. W. E. Bell and A. L. Bloom, Phys. Rev. 107, 1559 (1957). 124. M. Arditi and T. R. Carver, Phys. Rev. 109, 1012 (1958). 125. W. E. Bell and A. L. Bloom, Phys. Rev. 109, 219 (1958). 126. J. Brossel, J. Margerie, and A. Kastler, Compt. Rend. 241, 865 (1955). 127. R. H. Dicke, Phys. Rev. 89, 472 (1953). 128. J. P. Wittke and R. H. Dicke, Phys. Rev. 103,620 (1956). 129. P. C. Bender, E. C. Beaty, and A. R. Chi, Phys. Rev. Letters 1,311 (1958). 130. M. Arditi and T. R. Carver, Phys. Rev. 112,449 (1958). 131. E. C. Beaty, P. L. Bender, and A. R. Chi, Phys. Rev. 112,450 (1958). 132. F. Bitter, R. F. Lacey, and B. Richter, Rev. Mod. Phys. 25, 174 (1953). 133. B. Cagnac, J. Brossel, and A. Kastler, Compt. Rend. 246, 1827 (1958). 134. B. Cagnac and J. Brossel, Compt. Rend. 249, 77 (1959). 135. M. N. McDermott, R. L. Chaney, and P. W. Spence, Bull. Am. Phys. SOC.11, 354 (1966). 136. H. G. Dehmelt, Phys. Rev. 109, 381 (1958). 137. L. C. Balling and F. M. Pipkin, Phys. Rev. 139, A19 (1965). 138. L. C. Balling, R. J. Hanson, and F. M. Pipkin, Phys. Rev. 133, A607 (1964). 139. P. Franken, R.Sands, and J. Hobart, Phys. Rev. Letters 1, 52 (1958). 140. R. Novick and H. E. Peters, Phys. Rev. Letters 1, 54 (1958). 141. E. N. Fortson, F. G. Major, and H. G. Dehmelt, Phys. Rev. Letters 16,221 (1966). 142. W. Franzen and A. G. Emslie, Phys. Rev. 108, 1453 (1957). 143. W. W. Holloway, Jr. and E. Luescher, Nuovo Cimento 18, 1296 (1960). 144. L. W. Anderson, F. M. Pipkin, and J. C. Baird, Jr., Phys. Rev. 120, 1279 (1960). 145. F. M. Pipkin and R. H. Lambert, Phys. Rev. 127,787 (1962). 146. W. W. Holloway, Jr., E. Luescher, and R. Novick, Phys. Rev. 126, 2109 (1962). 147. L. W. Anderson, F. M. Pipkin, and J. C. Baird, Jr., Phys. Rev. 116,87 (1959). 148. R. H. Lambert and F. M. Pipkin, Phys. Rev. 129, 1233 (1963). 149. R. H. Lambert and F. M. Pipkin, Phys. Rev. 128, 198 (1962). 150. J. Ney, 2. Physik 196, 53 (1966). 151. H. Ackermann, Conf. Magnetic Hyperjine Structure At. and Molecules, Paris, 1966. To be published. 152. G. Heinzelmann, G. zu Putlitz, and A. Schenck, Phys. Letters 21, 162 (1966). 153. H. Bucka, G. zu Putlitz, and R. Rabold, Z. Physik. To be published. 154. G. zu Putlitz and K. U. Venkatavamu, Z. Physik. To be published. 155. J. Korvalski and G. zu Putlitz, Z. Physik. To be published. 156. Ma Ing Fuinn, G. zu Putlitz, and G. Schuette, 2.Physik. To be published. 157. H. Bucka and H. H. Nagel, Ann. Physik 8,329 (1961). 158. H. Bucka and H. A. Schuessler, Ann. Physik 7, 225 (1961). 159. G. zu Putlitz, Private communication (1966). 160. 0. Redi and H. H. Stroke, Bull. Am. Phys. SOC.10,456 (1965).

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ENERGY TRANSFER IN ORGANIC MOLECULAR CR YSTALS: A SURVEY OFEXPERIMENTS H . C. WOLF Physikalisches Institut der Universitat Stuttgart, West Germany

........................................ ................................................. Crystal Structure.. ...... .......................... Purification ............ ..........................

119 121 A. 121 B. 121 111. The Optical Spectra .............................................. 122 IV. Experiments on Energy Migration in Mixed Crystals . . . . . . . . . . . . . . . . . 123 A. Basic Observations (Sensitized Fluorescence). ..................... 123 B. Principles of Measurement ..................................... 124 C. Experimental Difficulties and Problems .......................... 126 D. Results of Measurements for the Transfer Constant k . . . . . . . . . . . . . . 127 E. Analysis of k Measurements by Exciton Diffusion.. ............... 132 F. Additional Information from Decay Time Measurements. .......... 134 G. Delayed Fluorescence . . . . . . . . . . . . . . . H. Diffusion Coefficients. ............... ....................... 137 I. Electron Spin-Resonance Measurements ......................... 139 J. Davydov Splitting.. ........ .............................. 140 V. Summary . . . . . . . . . . . . . . . . . . . ................... 140 References . . . . . . . . . . . . . . . . ................... 141

I. Introduction In organic molecular crystals intermolecular forces are generally much weaker than the forces inside the molecules. Therefore, many properties of the molecules are altered only slightly if one incorporates a molecule into a crystal. For many problems of molecular physics the crystal is nothing more than a framework that fixes the oriented molecules in space without changing the molecules too much. In this way the crystal as oriented gas is an object of molecular physics. One makes use of the crystalline state in all kinds of investigations where one needs molecules in known orientations. One example is the measurement 119

120

H . C. Wolf

of the symmetry of the different excited molecular levels using polarized absorption and fluorescence spectra of single crystals. But from such measurements it became clear that the oriented gas model is too simple. In order to understand the electronic spectra of organic crystals one has to take into account the interaction between the molecules. If one is dealing with investigations of electronic excited states one has to deal especially with the interaction of an excited molecule with its originally unexcited surroundings. The result of this process is the breakdown of the oriented gas model. Energy cannot be localized in one single molecule during the whole lifetime of the excited state. This delocalization of excitation energy is the main aim of optical investigations on organic crystals. The most characteristic effects are splitting of lines (Davydov splitting) and sensitizedfluorescence. Therefore, the investigation of luminescence spectra is the most common method of studying transfer processes. In solid-state physics excited electronic states in organic crystals are the best example for that particular type of excitons where electron and hole are very close together, in contrast to the large radius Wannier excitons for instance in Cu,O. Many properties, processes, and interactions of excitons can be studied extremely well with molecular crystals because exciton states can be identified easily with excited molecular states. Furthermore, organic crystals are on the borderline between solid-state physics and biophysics. It seems highly probable that energy transfer processes in photosynthesis are very close to processes that are studied in organic crystals (Franck and Teller, 1938; Franck and Livingstone, 1949). These are some of the reasons for the constant interest in energy transfer processes in organic crystals for more than 30 years. There are many excellent review papers of different aspects of the problem. Optical properties are reviewed by McClure (1959), Wolf (1959), Hochstrasser (1962), Davydov ( I 964), and Knox (1 963). A very detailed review of energy transfer work up to 1964 is given by Windsor (1965). The energy transfer processes are treated theoretically by, for example, Forster (1960). The reader is referred to these papers for a discussion of the extensive theoretical and experimental background of the problem. It is the aim of the present article to discuss some current problems of luminescence and energy transfer in organic molecular crystals. It seems worthwhile to collect those experimental results which are reliable and well established in order to make it easier to compare theory and experiment. The discussion is more or less confined to a few substances, in particular, benzene, naphthalene, anthracene, and related compounds. These crystals have been investigated far more thoroughly than any other molecular crystals. Only for these crystals does it seem to be possible to know what the properties of

ENERGY TRANSFER IN ORGANIC MOLECULAR CRYSTALS

121

the pure crystal are and to separate influences from defects. One has to keep in mind that in these crystals the optical transitions which give rise to luminescence are quite weak. The oscillator strength never exceeds 0.1. Therefore, we are dealing only with medium and weak transitions. The question which we try to answer is: How far and how fast can excitation energy migrate in crystals ? Those experiments which help to answer this question are included in the present article.

11. The Crystals A. CRYSTAL STRUCTURE

The structures of the crystals that are treated in the following sections are well known (Robertson, 1953). Naphthalene and anthracene, as an example, are monoclinic with two molecules in the unit cell. Growth and orientation of the crystals are possible without difficulties, at least in principle. Several of the most important methods are collected by Wolf (1959).

B. PURIFICATION One of the most severe experimental problems is purification. This is a problem common to all branches of solid-state physics. It has been shown within the last few years that organic molecular crystals contain many organic impurities which are very specific for the substance, which often can be removed only with great difficulty, and which completely change the fluorescence spectrum of the host material by their sensitized fluorescence. There are many examples in the literature of the fluorescence of organic crystals in which impurities have not been recognized as such and have caused misinterpretations as a result. A few examples are as follows: All fluorescence spectra of naphthalene published before 1961 were mostly spectra of impurities like j?-methyl naphthalene, thionaphthene, and others (Propstl and Wolf, 1963a ; Shpak and Sheremet, 1966). So-called self-trapped exciton states in anthracene crystals are impurity-trapped excitons (Sidman, 1956; Benz and Wolf, 1964a). The quenching of energy transfer in anthracene crystals at low temperatures is caused by a competition between energy transfer to the guest molecules and to shallow traps which give rise to emission from impurities or disturbed exciton states at low temperatures. This quenching effect can be removed by further purification (Benz and Wolf, 1964a,b; Lacey and Lyons, 1964). A second group of impurities are inorganic atoms and ions (Sloan, 1966). So far, these impurities have not been as important in energy transfer and

H . C. Wolf

122

luminescence problems. But this may change. The most important method of purification is zone refining. The problems and difficulties vary from substance to substance. The effect of the procedure can be characterized by the fact that, in general, one is able to reduce the concentration of organic impurities below lo-’ mol/mol. In order to do this one needs more than 100 zones wandering through the substance. Methods to accurately measure concentrations lower than are unknown. A review on purification problems is given by Sloan (1963).

III. The Optical Spectra The energy level scheme (Fig. 1) gives a survey of those spectroscopic properties of organic molecular crystals that are most important for energy transfer. [For a more detailed review the reader is referred to the literature,

I

/

SFIG.1. Energy levels and transitions for the singlet S and triplet T state of organic molecules and crystals, schematic.

for example, Craig and Walmsley (1963).] This scheme is typical for crystals like benzene, naphthalene, and anthracene. The ground state is always singlet (So).Absorption is allowed to the vibronic states of the different singlet excited states S1,S2, .. . . These states are split

ENERGY TRANSFER IN ORGANIC MOLECULAR CRYSTALS

123

(Davydov splitting) and broadened into bands in the crystal (not shown in Fig. I). The lifetime of the vibrationless lowest level of S, is responsible for the fluorescence decay time which is of the order of lo-' + lo-' sec. The exciton band belonging to this state is by far the most important for energy migration since all the higher singlet states (vibronic states of S , , S, ,and S,) are transformed very rapidly (faster than lo-'' sec), and without radiation into the lowest state of S , (internal conversion). Fluorescence originates at low temperatures only in the vibrationless S , exciton band. Singlet-triplet absorption (So+ T , )is forbidden. The absorption coefficient is of the order of LO-' cm-' (Kepler et al., 1963). But the lowest triplet state T , which is always lower than S , can be populated by a radiationless process via S , with an efficiency of the order of lo-, + lo-' (intercombination, intersystem crossing). In the triplet state again all the electronic or vibrational energy in excess of the lowest purely electronic TI state is given up very rapidly without radiation. The radiative lifetime of T , is of the order of a few seconds. Processes in the solid phase, which are not yet completely understood, make the actual lifetime of T I in the crystal several orders of magnitude shorter. In fact, T I + So emission in the pure crystal has not been observed so far. The exciton band belonging to the vibrationless state of TI is most important for energy migration in the triplet state. Very characteristic for organic crystals is a defect fluorescence (X fluorescence), which is identical to the fluorescence of the pure material but redshifted (see Fig. 4 and Section IV,D).

IV. Experiments on Energy Migration in Mixed Crystals A. BASICOBSERVATIONS (SENSITIZED FLUORESCENCE)

The sensitized fluorescence in mixed crystals is the oldest method of studying energy migration processes. Reviews are given by Franck and Livingstone (1949) and Windsor (1965). The oldest and, even today, the best example is the anthracene-tetracene system (Winterstein et al., 1934). One measures the fluorescence of an anthracene crystal containing small traces of tetracene. The result is that the fluorescence intensity of tetracene is relatively much stronger than the anthracene emission (see Figs. 2 and 3). The blue-violet fluorescence of anthracene is quenched by the presence of tetracene. The green-yellow tetracene fluorescence appears instead. Qualitatively, this observation shows that the cross section for excitation of tetracene molecules has been increased by the incorporation of the tetracene molecules into the anthracene crystal. The tetracene molecules act as traps for the electronic excitation energy which is traveling in the anthracene

H. C. Wolf

124 h

80 X

,<. ,

,

,

,

, \, ,

,

,

,

,

-

,

FIG.2. Fluorescence spectra of anthracene-tetracenemixed crystals at two temperatures fordifferentanthraceneconcentrations(from Benz and Wolf, 1964a). Top: cT = 0.8 x 1O-j. Bottom: cT = 5.3 x Center: cT = 4.6 x

crystal. For quantitative results on the system see Bowen et al. (1959) and Benz and Wolf (1964a) (see Fig. 3). B. PRINCIPLES OF MEASUREMENT In general, one uses a host crystal H containing molecules of a guest G. The lowest electronic level of excitation of G must be lower than that of H. In this case G is able to act as a trap for the excitation energy (Fig. 4). One measures the intensity l o r the relative quantum yield Q of the host and guest emission (IH,l, ,Q H ,Q,) after excitation in the host absorption (Fig. 5). If the absorption coefficients of H and G are known one can calculate which fraction of the energy has been transferred from H to G. The energy transfer is analyzed by evaluating the concentration dependence of one of the following quantities: (a) Qc/QH (for example, Benz and Wolf, 1964a; Propstl and Wolf, 1963b). (b) QHCo,/QH(cG);where QfI,o,and QH(cc)mean host quantum yield with and without guest molecules in the crystal (for example, Zima et al., 1966).

125

ENERGY TRANSFER IN ORGANIC MOLECULAR CRYSTALS

103

10

cYq

\

L

0

.?

10

c?

5 S

r

0

10-

I

I

I

10-

10-6

10-~

I

10-3

10-2

lo-'

Concentration c , ( m o / / r n o l )

FIG.3. Fluorescence quantum ratio Q T /Q l+ as a function of the tetracene concentration CT in an anthracene mixed crystal at two temperatures. At the higher temperature, the emission from shallow traps ( X emission) is negligible (i.e., Q A + , = Q A )(from Benz and Wolf, 1964a).

Lowest Excilon Band

I

Ground 5 t a t e Shallow 7rop

Deep Trap

X

G

FIG. 4. Energy level scheme for sensitized fluorescence, involving guest emission and emission from shallow traps ( X emission).

H. C. Wolf

126

1

H+G

1

G + H - Emission

FIG.5. Scheme of sensitized fluorescence measurement.

(c) zH =f(c,), zG = f ( c c ) ; where zH and zG are the decay times of host and guest fluorescence, respectively, and cc is the guest concentration (for example, Schmillen, 1965; Bonch-Bruevich et al., 1961).

In this way one gets, as will be shown in the following, as primary experimental result k, the transfer constant. One needs special assumptions on the physical nature of the process of energy migration in order to calculate from k as secondary resuIts other constants which characterize the mechanism of energy transfer such as: D 2

tH P

L v

= F/d3

Diffusion coefficient. Number of hops per lifetime. Hopping time. Average radial displacement of an exciton. Diffusion length. Velocity of an exciton.

Experiments giving one or the other of these constants as primary results are described in Section IV,G to IV,J.

c. EXPERIMENTAL DIFFICULTIES AND PROBLEMS 1. Measurement of Concentration One measures the concentration cG either directly using the absorption spectrum of the crystal or after dissolving the crystal in a suitable solvent. In this way it becomes difficult to measure concentrations below mol/mol because the absorption is too weak. Therefore, one uses the sensitized fluorescence. One extrapolates the curve ZG/ZH =f (cG) to low concentrations. Obviously, this method is unsatisfactory. In order to understand the mechanism of energy transport one needs an independent method for measurement of concentration.

ENERGY TRANSFER IN ORGANIC MOLECULAR CRYSTALS

127

Unfortunately, one often finds in the literature only the impurity concentration in the melt (nominal concentration). Since the distribution coefficients are almost completely unknown these data do not contain much useful information. Furthermore, it is important to know if one is actually dealing with solid solutions without microcrystalline phases of different concentration inside the crystal. This can be checked by carefully measuring the structure of the low temperature fluorescence spectra. Generally, the solubility of guest molecules in organic crystals is very low. Examples for the limit of solubility are anthracene in phenanthrene, lo-’, and tetracene in anthracene, 5 x lo-’ (Benz and Wolf 1964a,b). 2. Measurement of Quantum Yield

Very few absolute measurements of quantum yields are available. Therefore, one measures mostly only relative quantum yields, using pure anthracene with a room temperature quantum yield of 0.95 k 0.05 as a standard (Kellogg, 1966). For the measurement of the transfer constant one does not need the absolute values of quantum yield. But one has to make sure whether or not the total quantum yield is independent of temperature.

3. Spectral Resolution In order to measure and separate the host and guest quantum yields one has to measure the total fluorescence spectrum with good resolution. Only in this way is it possible to separate unambiguously the individual contributions from host, guest, and traps to the total emission. [See, for example, the measurements on naphthalene by Propstl and Wolf (1963a) and on phenanthrene by Benz and Wolf (1964b).]

D. RESULTSOF MEASUREMENTS FOR THE TRANSFER CONSTANT k I . Concentration Dependence of Sensitized Fluorescence Examples of experimental results are given in the Figs. 2, 3, and 6. In a relatively broad region of concentration one can describe the measured concentration dependence in good approximation by a single transfer constant k . This transfer constant is defined by the following expressions :

H . C. Wolf

128

Y

100

90 80

70 60 50 40

30

20 10 -5

-4

-3

-2

log c

FIG. 6. Phase angle and decay time as measured using a phase fluorometer in the mixed crystal 2,3-dimethyl naphthalene-anthracene system (from Schmillen, 1965). T~ = 69.7 x sec; k = 1.05 x lo5 sec. Upper curve T ~ middle , T ~ lower , curve 7Gexcited directly.

[more exactly, cGpwith p = 0.8 or 0.9 f 0.1, after Benz and Wolf (1964a,b)], or

-'w0) - 1 + kc,

(Zima er a ~ .1966) ,

(2)

(Schmillen, 1965)

(3)

'H(CG)

or =

1

+ kc,

''H(fG)

where zH is the decay time of host fluorescence, Table I gives a summary of measured values of such transfer constants. It must be emphasized that only very few of the measurements are really reliable. The most important uncertainty is the real concentration of the solute. More careful measurements are required, including accurate measurements of the concentration. It is the purpose of the following discussion to explain the measured transfer constants theoretically.

TABLE I MEASUREMENTS OF TRANSFER CONSTANT k Substance Host Anthracene

Naphthalene

Guest

CG

Tetracene

10-7-10-4

Tetracene

10-5-10-2

Method of measurement

Result

k = (6 & 3) x lo4, p = 0.8 i 0.2

I x 10s

Reference Benz and Wolf (1964a) Korsunskii and Faidysch (1966)

Phenazine Anthanthrene Acridine Anthrachinone

7.5 x 104 2.5 x 104 1.5 x 103 7.5 x 103

Zima and Faidysch (1966

Anthracene /%Methylnaphthalene Acridine

2.5 x 104 3.5 x lo2

Zima er al. (1966)

Anthranilic acid

2,3-Dimethylnaphthalene Naphthalene Fluorene Phenanthrene

Concentration

2.4 x 104 10-6-10-3

4.35 x 104

Bonch-Bruevich et at. (1964)

1 x 105

1,CDiphenylbutadiene 1,&Diphenylhexatriene-l,3,5 Anthracene

3.22 x 104 1 x 105

Perylene Perylene Anthracene

10-6-10-2

Tetracene

10-6-10-4

8.7 x 104 8.5 x 1 0 3 k = (3 f 2) x lo4, p = 0.85 f 0.1 k = (3 & 2) x lo4, p = 0.9 f0.1

Schmillen (1965) Schmillen and Kohlrnannsperger (I 963) Benz and Wolf (196413)

130

H. C. Worf

2. Temperature Dependence

There have been several investigations on the temperature dependence of the energy transfer constant k. The question is: Is energy transfer more efficient at low temperatures or is it quenched? In a few cases one has observed a decrease of k with decreasing temperature, in other words, a freezing-in of energy transfer (Lyons and White, 1958; Avakian and Wolf, 1961; Gschwendtner and Wolf, 1961). One can show that in all these cases this decrease of energy transfer is caused by shallow traps. These are chemical or crystallographical defects in the crystal. At temperatures high enough so that their depth AE below the exciton band is lower than the thermal energy, these traps are not effective. At sufficiently low temperatures they compete with the guest molecules in getting excitation energy from the host and therefore reduce the host-guest transfer constant. The best example is thionaphthene in naphthalene (Propstl and Wolf, 1963a,b). Here the depth of the traps, which are created in naphthalene by the presence of thionaphthene, is AE = 29 f 1 cm-'. These traps are responsible for the decrease in energy transfer to other guest molecules below 10°K as observed by Gschwendtner and Wolf (1961). Evidence that such traps are responsible for the decrease of energy transfer to guests at low temperatures comes from removing this effect by further purification of the host crystal. In purest organic crystals there is no indication of a thermally activated energy transport process (Figs. 2 and 3). On the contrary, in extremely pure naphthalene crystals one observes an increase in the transfer constant with decreasing temperature (Hammer and Wolf, 1966) (see Fig. 7). In other words, the efficiency of energy transfer is higher at lower temperatures. Between 100" and 6"K, k is nearly proportional to l / n . This is expected if phonon scattering is the process limiting the exciton free path (Agranovich and Konobeev, 1963). To summarize the results on temperature dependence, it can be concluded that energy transfer cannot be described satisfactorily by a thermally activated hopping process. The temperature dependence as shown in Fig. 7 makes evident the wavelike motion of excitons in the crystal. This is discussed further in Section IV,E.

3. Influence of Traps There are different kinds of shallow traps which can be responsible for an apparent temperature dependence of energy transfer to guest molecules, as mentioned in the previous section. The most characteristic and the most important traps are disturbed exciton states [called " X-traps " by Propstl and Wolf (1963a)l (see Fig. 4).

ENERGY TRANSFER IN ORGANIC MOLECULAR CRYSTALS

131

One observes these disturbed exciton states as causing a fluorescence spectrum of the crystal which has exactly the same vibronic structure but is red-shifted by an amount AE against the intrinsic (undisturbed) crystal fluorescence. The value of AE is characteristic for the nature of the particular X center. Such X centers can be created by a chemical impurity [for example, thionaphthene in naphthalene, AE = 29 cm-' (Propstl and Wolf, 1963a,b)] or by crystallographic defects in the crystal [for example, in plastically deformed

0

50

100

150

2 00

25 0

300

Temperature [ O K 1

FIG.7. Quantum ratio Q A / Q N in the mixed crystal naphthalene-anthracene system as a function of temperature (from Hammer and Wolf, 1966). Concentration, cd = 5.3 x

naphthalene crystals with AE = 165 cm-' (Schnaithmann and Wolf, 1965)l. The value of AE can be determined both by the temperature dependence of intensities and by the quantum energy of the fluorescence lines. In such a ternary system (host, X center, guest), the quantum ratio characteristic for energy transfer is given by

where y is the rate constant for the phonon-assisted detrapping of the exciton and AE is the trap depth.

132

H . C. Worf

From the quenching ratio of host-guest energy transfer,

+ 1, one can determine the concentration of X centers (Propstl and Wolf, 1963b, Benz and Wolf, 1964a,b). Equation (5) makes possible a determination of trap concentrations as low as or lo-' by the temperature dependence of host-guest energy transfer.

E. ANALYSIS OF k MEASUREMENTS BY EXCITON DIFFUSION Now we have to go from the primary result k to other constants describing the mechanism, in other words, to the secondary results. We assume a twocomponent system H + G and a constant quantum yield of 1. In order to understand the experimental results on energy transfer reported so far one needs only two assumptions : (i) Host-guest energy transfer is competing with host emission. (ii) The transfer probability is proportional to cG = N,JNH, where N G , NH are the number of guest and host molecules per volume, respectively. Furthermore, in order to make the problem as simple as possible, we neglect exciton escape on the surface. For the sake of simplicity we take as equal the probability for exciton capture by a host and a guest molecule, respectively, nearest to an excited molecule. This assumption can be removed easily by the introduction of probability factors (Zima et al., 1966; Benz and Wolf, 1964a). An empirical analysis can be given using the following model (see Fig. 8).

Excilon Band Trap

Ground S t a l e

FIG.8. Model for describing energy transfer as measured by sensitized fluorescence.

ENERGY TRANSFER IN ORGANIC MOLECULAR CRYSTALS

133

Four processes which are competing with each other are taken into account. The characteristic times for these processes are : (i) Host fluorescence (time constant T ~ ) . (ii) Transfer of energy to the next guest site (time constant for one intermolecular transfer step: tH, hopping time in the hopping model). (iii) Guest fluorescence (time constant T ~ ) . (iv) Phonon-assisted detrapping of the guest, energy goes back into the band (rate constant y). To be taken into account only if AE 5 kT [see Eqs. (4) and (5). By using this model one readily finds for AE 9 kT:

This can be transformed, by substituting

IH(c= ~ )I C

1 IH(0) =

->

kc,

IG

+ IH(cc)

5

into

The corresponding relations for the decay time measurements are derived in the following way, The equation for the time dependence of the number N of molecules in the excited state is given by

dN/dt = -kl N - kZc,N

+

10,

(8)

where Zo is the excitation. For decay time measurements, 1, = 0. With Ilk, = T ~ ( and ~ ) l/(kl + kzcc) = 7 H ( C Gone ) , finds

This is a purely empirical analysis. The question is now: What is the physical nature of k? In many cases a satisfactory explanation i s given by the hopping model (see, for example, Propstl and Wolf, 1963b). Here one assumes statistical diffusion of “ small” excitons between the molecules in the lattice. This model is justified only if the diffusion length is large against the region of coherence of the exciton. One has to introduce the quantities r H , z, and a (the distance for one hopping process, in the most simple picture equal to the intermolecular distance in the lattice) to find the following equation: H k2 = t=tH

kl

=k.

(10)

134

H . C. Wo(f

Equation (10) gives the first interpretation for the physical nature of k . Furthermore, one can now calculate

F = ( a ’ ~ ) ’= / ~(3D7)’/’ D = a2/3tH,

(1 1) (12)

where F and D are the same as previously defined. A second and more elaborate attempt to apply diffusion theory to exciton migration takes care of the wavelike motion of excitons. If one forms wave packets, one can define a diffusion coefficient D and a trapping radius R of the guest molecule. In this way, Agranovich and Konobeev (1963) find

which means that

k = 471DRTHNH.

(14)

In other words, the rate constant for exciton capture by an impurity is given by 4nDR. This is the second physical interpretation of k. Since the value of R,the trapping radius of a guest molecule in the host lattice, is unknown, one can determine only the product DR.It seems fairly accurate to assume that R is of the order of lo-’ cm. Using this value, the k values of Table I are equivalent to diffusion coefficients for singlet excitons of the order of cm2 sec-’. In the band model, the diffusion coefficient is given by D = qh (where A is the free path, u the instantaneous velocity from +mv2 = +kT, and m the effective mass of the exciton). The observed temperature dependence of k (Hammer and Wolf, 1966) cannot be explained without making further assumptions in the most simple hopping model. In the diffusion approach of Agranovich and Konobeev (1963) the temperature dependence of D is due to exciton-phonon scattering as the predominant exciton scattering mechanism. This section may be summarized by remarking that sensitized fluorescence measurements give a value for the transfer constant. Additional assumptions are needed to calculate constants describing specific processes. Therefore, it is very useful to have additional and independent methods to measure such constants directly. The values of I,, D, and L calculated from k values are given in Table 11.

F. ADDITIONAL INFORMATION FROM DECAY TIMEMEASUREMENTS Measurements of decay times of guest and host emission in mixed crystals are equivalent to intensity measurements in deriving k [Eq. (9)] (Schmillen,

ENERGY TRANSFER IN ORGANIC MOLECULAR CRYSTALS

135

1965; Schmillen and Kohlmannsperger, 1963). The main experimental difficulty is that most of the z-measurements are made using phase fluorometers. In this way one measures directly only a phase angle. In order to find the decay time one has to assume an exponential decay. Therefore, the limit of error in decay time measurements using phase fluorometers is uncertain. TABLE I1 CHARACTERISTIC VALUES FOR EXCITON DIFFUSION, CALCULATED FROM MEASURED k VALUES"

Measured values Substance

k(=z)

Calculated values

T~

tli

(sec)

(sec)

Anthracene/ 6 x lo4 2 x lo-* 3 x Tetracene Naphthalene/ 2.5 x lo4 1 x lo-' 4 x Anthracene

a

D (cm2 sec-I)

L (cm)

5 x

1 x

5 x

5x

See Table I and Eqs. (l0)-(12).

The basic observations are as follows: (i) zH decreases with increasing guest concentration. (ii) zG is larger if the guest is excited over the host rather than directly. (iii) zG decreases with increasing guest concentration. The k values which have been determined by using this method are given in Table I. Bonch-Bruevich et al. (1961) suggest the following more detailed analysis :

(zc) ( H excitation) = z(migration H + G) + z(loca1ization) + zG . (15) They find from their measurements a value for tloca1. This is the time interval between trapping of the exciton by the guest center and its final localization. The measured values for tlocal are between 0.2 and 1 x lo-* sec. Unfortunately, this very interesting analysis has, so far, not been confirmed by further measurements using other systems and methods. Finally, it must be mentioned that decay time results using uv excitation are often different from those after excitation by other sources (a-particles, electrons, prays). Apparently radiation damage processes are important in these cases (Schmillen, 1962).

I36

H . C. Wo,f

G. DELAYED FLUORESCENCE A close analysis of the time dependence of emission decay has shown that

in many organic crystals there is a delayed fluorescence in addition to the normal fluorescence (Sponer et al., 1958; Blake and McClure, 1958). This delayed fluorescence is spectroscopically identical with the prompt fluorescence, but its decay time is much longer. In naphthalene, one measures for the delayed fluorescence a lifetime of milliseconds which has to be compared with the value z H x 100 nsec for the prompt fluorescence. The intensity of the delayed fluorescence is in many cases proportional to Zo2, this means to the square of the exciting intensity. The explanation by Sternlicht et al. (1963) assumes that the triplet state of the crystal is responsible for this delayed emission. Two triplet excitons combine according to the equation T" + T" + S" + So. The resulting singlet emission reflects the lifetime of the triplet state in the crystal. According to Kepler et al. (1963) this delayed fluorescence can be used to measure the velocity of triplet excitons in the crystal, the diffusion length, and the diffusion coefficient as secondary result. The primary experimental result is y, the bimolecular exciton interaction rate constant. Kepler et al. (1963) excited the blue anthracene fluorescence by the red light of a ruby laser. The resulting fluorescence has a delayed component. During the laser excitation the emission is at least partly due to two-quantum absorption. The delayed component results from the direct population of the triplet state (at about 14,720 cm-') by the red light, which is appreciable due to the high density of excitation in the laser beam. Exciton-exciton interaction causes the delayed fluorescence. This experiment is analyzed using the formula for the concentration n of triplet excitons: dn - = clIo - pn - yn2 dt where cl is the creation of triplet excitons by light, p the reciprocal lifetime of triplet excitons, and y the bimolecular interaction constant for the triplet exciton interaction. From an analysis of the measurements, Kepler et al. (1963) find a = 1 0 - ~an-', p = lo2 sec-', y = cm3 sec-l. In a kinetic model for exciton diffusion, y = OV (17) where 0 is the cross section for the triplet interaction process and 21 the velocity of excitons. With c = 9 A2 one finds u x lo4 cm/sec. The diffusion length 7 is given by

F = (UZA)'/~ x 1.5 x

cm,

(18)

137

ENERGY TRANSFER IN ORGANIC MOLECULAR CRYSTALS

where 7 is the lifetime of triplet excitons and ;ithe length for one displacement step (this means one molecular distance). Since this pioneering work of Kepler et al. (1963) and Avakian er al. (1963), y values for several other systems have been determined. Table 111 contains measured values of y. More measurements on triplet excitons are discussed in the next section. TABLE 111 VALUES OF THE TRWLET-TRIPLET ANNIHILATION COEFFICIENT FOR ANTHRACENE Substance

y (cm3 sec-

I)

4 x lo-''

Anthracene

5.5 x lo-"

5 x lo-"

Reference Avakian ei al. (1963) Kepler et af. (1963) Hall et al. (1963) Singh ef al. (1965)

H. DIFFUSION COEFFICIENTS Several methods of determining diffusion coefficient D of excitons as the primary result have been reported. Secondary results are in these cases values of t, or k or u. The method for measurement of D for singlet excitons used by Simpson (1956) and Callus and Wolf (1966) is illustrated in Fig. 9, where

u

I

I

I I

Principle

2

-

I

5 0 :

1 1

I I 0

4 00

800

2000

LoW

Boo0

6oW 0

Thickness of the pure phenanthrene film (A)-

+

FIG.9. Relative fluorescence efficiency of the phenanthrene-phenanthrene anthracene two-layer system as a function of thickness of the pure phenanthrene layer (from Gallus and Wolf, 1966). 0 :Experimental points. -: Theoretical curve with L = 78 A. ---: Theoretical curve without diffusion.

H . C. Worf

138

P is a layer of highly purified phenanthrene, and P + A is a layer of phenanthrene containing anthracene in a concentration lower than lo-'. The exciting light is absorbed mainly in the layer P, which has a quantum efficiency appreciably lower than 1 (Benz and Wolf, 1964b). Diffusion of excitons from P to P + A causes an increase in the total fluorescence quantum yield, since the quantum yield of the P + A layer is approximately 1. By measuring the total quantum yield as a function of the thickness of the layer P one gets evidence of exciton diffusion and a value for the diffusion length. This value is uncertain by a factor of the order two for the concentration of excitons on the front surface (Gallus and Wolf, 1966). Other methods for the measurement of D make use of photoconductivity and its dependence on the distance between electrodes and the illuminated part of the specimen. Such experiments are possible both for singlet and triplet excitons. Table IV contains results from Bassler (1966) on triplet excitons in liquid naphthalene that give an astonishingly h g h value for D. TABLE IV

MEASUREMENTS OF DIFFUSION COEFFICIENT D AND DIFFUSION LENGTH L OF EXCITONS

D Substance Anthracene microcrystals Anthracene crystals Naphthalene crystals Phenanthrene microcrystals Anthracene crystals Anthracene crystals Anthracene crystals Anthracene crystals Napthalene melt

State'

(cm2 sec-')

1.5 x 10-3

Lb (cm)

4.6 x 1.3

x 10-5

I x 10-4 8.5 x

2 x 10-4

2 x 10-4 0.4-2 x 1 f0.2 x lo-*

1.5

a b means the a b plane of the crystal.

Reference Simpson (1956) Khan-Magometova and Radzievskii (1964) Hammer and Wolf(1966)

8 & 3 x 1 0 - 7 Gallus and Wolf (1966) (1 0.5) x 10-3 Avakian and Merrifield (1964) 2-3 x 10-3 Ern et al. (1966) 10-3

Kepler and Switendick (1965) 7-15 x 10-3 Williams et al. (1966) 1

Bassler (1966)

* Nore: The formulas used in the literature for evaluating L differ for different authors by numerical factors 1/42or 1/45.

ENERGY TRANSFER IN ORGANIC MOLECULAR CRYSTALS

139

Diffusion coefficients for triplet excitons are measured using delayed fluorescence by the above-mentioned method or by other more elaborate experiments (Avakian and Merrifield, 1964; Kepler and Switendick, 1965; Ern et al., 1966; Williams et al., 1966). If the concentration of excitons created in the crystal is not homogeneously distributed or if diffusion to the surface becomes important, the number n of triplet excitons in the volume is given by the formula (assuming surface quenching) :

dnldt = al, - pn - yn2 + D d2nldx2.

(19)

For details of the experiments see the references given in Table IV. The diffusion coefficient D is related to 7 or the average displacement in one direction L by

F = ( ~ D T ) ’ ’ ~ and

L

= (Dt)’’’.

(20)

Table IV gives measurements of the diffusion coefficient and the length of singlet and triplet excitons.

I. ELECTRON SPIN-RESONANCE MEASUREMENTS Triplet states are paramagnetic. Electron spin resonance (ESR) measurements, especially by Hutchison and Mangum (1961), have given the ultimate evidence. In addition, they have given a lot of extremely useful information on fine structure and hyperfine structure. By using ESR measurements, it has been shown that energy transfer exists (Brandon et al., 1962; Hirota and Hutchison, 1965; Hirota, 1965). Brandon et al. (1962) made the following experiment. They investigated the ESR spectrum of naphthalene and phenanthrene molecules dissolved together in the same matrix of biphenyl. The energy of the naphthalene triplet state is a little lower than that of phenanthrene. One is able to observe the naphthalene triplet ESR if light has been absorbed by phenanthrene. The analysis of the fine structure tensor gives evidence that naphthalene and phenanthrene molecules were not nearest neighbors. This experiment shows that there is an appreciable energy transfer between molecules in the triplet state. So far, it has not been possible to measure transfer constants by its influence on energies of ESR transitions. By using ESR, one is able to measure exchange frequencies from the line shape and line width. As a primary result, one gets p, the matrix element for the pairwise energy exchange between molecules in the triplet state, or the hopping time t , = p-’. For naphthalene molecules in the lowest triplet state, one measures in this way fi = 5 cm-’ which means t H = 1.7 x lo-’’ sec (Schwoerer and Wolf, 1967). This result is consistent with the measurements of the diffusion

H . C . Wolf

140

coefficients given in Table IV. By using the relation D = a 2 / 3 t H ,one calculates D z 5 x cm2 sec-' for triplet excitons in naphthalene crystals. If one uses ESR just as a method to measure concentrations of triplet molecules, ESR spectroscopy can be used like optical spectroscopy. Using a kinetic analysis for the intensities and decay times observed, one can determine interaction rate constants and in this way energy transfer processes. This has been done by Hirota and Hutchison (1965) and Hirota (1965).

J. DAVYDOV SPLITTING The Davydov splitting AE of electronic excited states in crystals is a measure of the energy exchange interaction between an excited molecule and its unexcited neighbor in the crystal (Davydov, 1964). The primary result is t , . Using the uncertainty relation, one finds the hopping time 1, as t , w h/AE

(21)

where E = 100 cm-' corresponds to tH = 3 x values are given for the Davydov splitting.

sec. In Table V measured

TABLE V DAVYDOV SPLITTINGS A E BY SPECTROSCOPIC MEASUREMENTS Substance Naphthalene

State Transition A E (cm-') S,

04 1-0

7-1

Anthracene

SI

0-0 0-0

1-0 2-0 0-0 (extrapol.)

Phenanthrene Naphthalene

S,

0-0

TI

-

1501 5 30 5 5 10 f 2 220=t40 150 & 15

I

Reference Wolf (1959) Hanson and Robinson (1965) Wolf (1958)

40k 10 380

59 14 cm-'

=5

Lacey and Lyons (1964) Craig and Gordon (1965) Schwoerer and Wolf (1967)

(by E W

V. Summary From quite different types of experiments there seems to be plenty of evidence of the ability of excitons in organic crystals to migrate and to transfer energy. The diffusion length for singlet excitons is of the order of -+ lo-' cm, and for triplet excitons it is at least cm.

ENERGY TRANSFER IN ORGANIC MOLECULAR CRYSTALS

141

REFERENCES Agranovich, V. M., and Konobeev, Yu. V. (1963). Soviet Phys.-Solid State (English Transl.) 5, 999. Avakian, P., and Merrifield, R. E. (1964). Phys. Rev. Letters 13, 541. Avakian, P., and Wolf, H. C. (1961). Z. Physik 165,439. Avakian, P., Abramson, E., Kepler, R. G., and Craig, J. C. (1963). J. Chem. Phys. 39, 1127. Bassler, H. (1966). Z . Naturforsch. Zla, 447. Benz, K. W., and Wolf, H. C. (1964a). Z . Nuturforsch. 19a, 177. Benz, K. W., and Wolf, H. C. (1964b). Z . Naturforsch. 19a, 181. Blake, N. W., and McClure, D. S. (1958). J . Chem. Phys. 29, 722. Bonch-Bruevich, A. M., Kovalev, B. P., Belaev, L. M., and Belikova, G. S. (1961). Opt. Spectry. (USSR) (English Transl.) 11, 335. Bowen, E. J., Mikiewicz, E., and Smith, F. W. (1949). Proc. Phys. Soc. (London) A62, 26. Brandon, R. W., Gerkin, R. E., and Hutchison, C. A. (1962). J. Chem. Phys. 37, 447. Craig, D. P., and Gordon, R. D. (1965). Proc. Roy. SOC.A288, 69. Craig, D. P., and Walrnsley, S. H. (1963). In “The Physics and Chemistry of the Organic Solid State” (D. Fox, M. M. Labes, and A. Weissberger, eds.), Vol. 1 . Wiley (Interscience), New York. Davydov, A. S. (1964). Soviet Phys.-Usp. (English Transl.) 7 , 145. Ern, V., Avakian, P., and Merrifield, R. E. (1966). Phys. Rev. 148, 862. Forster, Th. (1960). “Comparative Effects of Radiation.” Wiley, New York. Franck, J., and Livingston, R. (1949). Rev. Mod. Phys. 21, 505. Franck, J., and Teller, E . (1938). J. Chem. Phys. 6 , 861. Gallus, G., and Wolf, H. C. (1966). Phys. Sturus M i d i 16, 277. Gschwendtner, K., and Wolf, H. C. (1961). Nuturwissenschuften 48,42. Hall, J. L., Jennings, D. A., and McClintock, R. M. (1963). Phys. Rev. Letters 11, 364. Hammer, A., and Wolf, H. C. (1966).Symp. Org. Scintillator, Chicugo,1966,to be published in Mol. Cryst. (1967). Hanson, D. M., and Robinson, G. W. (1965). J. Chem. Phys. 43,4174. Hirota, N. (1965). J. Chem. Phys. 43, 3354. Hirota, N., and Hutchison, C. A. (1965). J. Chem. Phys. 42,2869. Hochstrasser, R. M. (1962). Rev. Mod. Phys. 34, 531. Hutchison, C. A., and Mangum, B. W. (1961). J . Chem. Phys. 34,908. Kellogg, R. E. (1966). J . Chem. Phys. 44,41 I . Kepler, R. G., and Switendick, A. C. (1965). Phys. Rev. Letters 15, 56. Kepler, R. G., Caris, J. C., Avakian, P., and Abramson, E. (1963). Phys. Rev. Letters 10, 400. Khan-Magometova, Sh. D., and Radzievskii, G. B. (1964). Opt. Spectry. (USSR) (English Transl.) 16, 457. Knox, R. S . (1963). “Theory of Excitons.” Academic Press, New York. Korsunskii, V. M., and Faidysch, A. N. (1966). Opt. Spectry. (USSR) (English Transl.) Suppl. I, 62. Lacey, A. R., and Lyons, L. E. (1964). J. Chem. SOC.5393. Lyons, L. E., and White, J. W. (1958). J . Chem. Phys. 29, 447. McClure, D. S. ( I 959). Solid State Phys. 8, 1. Propstl, A,, and Wolf, H. C. (1963a). Z . Nuturforsch. 18a, 724. Propstl, A., and Wolf, H. C. (1963b). Z . Naturforsch. 18a, 822.

142

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c. wolf

Robertson, J. M. (1953). “ Organic Crystals and Molecules.” Cornell Univ. Press, Ithaca, New York. Schmillen, A. (1962). “ Luminescence of Organic and Inorganic Materials,” p. 30. Wiley, New York. Schmillen, A. (1965). J. Phys. Chem. 69, 751. Schmillen, A., and Kohlmannsperger, J. (1963). Z. Nuturforsch. 18a, 627. Schnaithmann, R., and Wolf, H. C. (1965). 2. Nuturforsch. 20a, 76. Schwoerer, M., and Wolf, H. C. (1967). Mol. Cryst. (in press). Shpak, M. T., and Sheremet, M. (1966). Opt. Specfry. (USSR)(English Trunsl.) Suppl. 1, 58. Sidman, J. W. (1956). Phys. Reo. 102,96. Simpson, 0. (1956). Proc. Roy. SOC.A238,402. Singh, S., Jones, W. J., Siebrand, E., Stoicheff, B. P., and Schneider, W. G. (1965). J. Chem. Phys. 42, 330. Sloan, G. J. (1963). In “The Physics and Chemistry of the Organic Solid State” (D. Fox, M. M. Labes, and A. Weissberger, eds.), Vol. 1. Wiley (Interscience), New York. Sloan, G. J. (1966). Mol. Crystals 1, 161. Sponer, H., Kanda, Y., and Blackwell, L. A. (1958). J . Chem. Phys. 29, 721. Sternlicht, H., Nieman, G. C., and Robinson, G. W. (1963). J. Chem. Phys. 38, 1326. Williams, D. F., Adolph, J., and Schneider, W. G. (1966). J . Chem. Phys. 45, 575. Windsor, M. W. (1965). In “The Physics and Chemistry of the. Organic Solid State” (D. Fox, M. M. Labes, and A. Weissberger, eds.), Vol. 2. Wiley (Interscience), New York. Winterstein, A., Schoen, K., and Vetter, H. (1934). Nuturwissenschufen 22, 237. Wolf, H. C. (1958). Z. Nuturforsch. 13a, 414. Wolf, H. C. (1959). Solid Stufe Phys. 9, 1. Zima, V. L., and Faidysch, A. N. (1966). Opt. Spectry. (USSR)(English Trunsl.) 20, 566. Zima, V. L., Kravchenko, A. D., and Faidysch, A. N. (1966). Opt. Spectry. (USSR)(English Trunsl.) 20,240.

A T O M E AND MOLECULAR SCA TTERING FROM SOLID SURFACES* ROBERT E. STICKNE Y Department of Mechanical Engineering and Research Laboratory of Electronics Massachusetts Insritute of Technofogy Cambridge, Massachusetts

I. Introduction .....................................................

143 144 144 B. Energy and Momentum Accommodation Coefficients .............. 147 111. Experimental Apparatus for Gas-Solid Scattering Studies. ............. 154 A. Chamber Design .............................................. 154 B. Beam Intensity.. . . . . . . . . . . . . . . . . . . . . . . . . 155 C. Lock-In Detection of M . . . . . . . . . . . . . . . . . . 158 D. Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 IV. Experimental Data on the Scattering of Atomic and Molecular Beams from Solid Surfaces ................................................... 161 A. Diffraction Experiments .. . . . . . . . . . . . . . . 162 B. Scattering of Vapor Ato Crystals.. .......... 163 C . Scattering of Rare Gas Atoms from Solid Surfaces ................ 167 D. Scattering of Molecules from Solid Surfaces ...................... 178 E. Summary of Experimental Data. ................................ 186 V. Classical Theories of Atomic and Molecular Scattering from Solid 187 Surfaces ........................................................ A. The Hard-Cube Theory. ....................................... 187 B. Theories Based on Lattice Models 198 VI. Concluding Remarks ............................................. Appendix. Classical Mechanics of Hard-Sphere Collisions . . . . . . . . . . . . 200 References .........................................

.................................. A. General Description of the Problem .............................

11. Preliminary Considerations .

I. Introduction Atomic and molecular processes occurring at the gas-solid interface are of first-order importance in a variety of scientific and technical fields. For example, discussions of such processes are generally included in basic texts

* This work was supported by the Joint Services Electronics Program under Contract DA36-039-AMC-O3200(E). In part, it is an outgrowth of a report (AEDC-TR-66-13) written by the author for the Aerospace Environmental Facility, ARO, Inc., Arnold Engineering Development Center, Tennessee. 143

144

Robert E. Stickney

on adsorption and condensation, catalysis and oxidation, sputtering and radiation damage, rarefied gas dynamics, electron and ion emission, friction and wear, vacuum technology, and space simulation. The science of gas-solid interactions is, however, far less advanced than that pertaining to the bulk gas and solid phases. This is to be expected, of course, because knowledge of interface phenomena generally requires a higher level of understanding of the fundamental processes in the individual phases. In this review we shall concentrate on one particular gas-solid process: the scattering of atoms and molecules of thermal energies from solid surfaces. Special attention is given to investigations performed during the period 19601966 because, through the use of modulated molecular beam techniques, many significant results have recently been obtained. Since the earlier work has been reviewed by Massey and Burhop (1952), by Ramsey (1956), and by Hurlbut (1962), only a brief summary is included for completeness and comparison. In the discussion of the theory of gas-solid scattering, we have chosen to concentrate on the simple " hard-cube " model (Logan and Stickney, 1966). This choice was based partly on the author's familiarity with the model and partly on the fact that, at present, theories based on more complex models do not yield analytical solutions that are convenient for comparison with the existing experimental data. Also, detailed reviews of the theory of gas-solid collisions have been presented recently by Trilling (1967b), Hurlbut (1967), and Beder (this volume). More than a dozen research groups are now developing new facilities for investigating the scattering of thermal and hyperthermal atomic and molecular beams from solid surfaces. [The new facilities and techniques have been reviewed by Knuth (1964).] The hyperthermal range is now receiving special attention because of its relation to the problem of aerodynamic drag in highaltitude flight, and the first experimental data obtained in this range indicate that the scattering patterns differ significantly from those observed at thermal energies (Alcalay and Knuth, 1967). In addition to providing us with information on the mechanics of gas-solid collisions, the new experimental techniques and facilities may provide better means for investigating (i) the intermolecular potential at a solid surface, (ii) surface catalysis, and (iii) gas-solid chemical reactions (e.g., oxidation).

11. Preliminary Considerations A. GENERAL DESCRIPTION OF THE

PROBLEM'

As illustrated in Fig. 1 for the case of Ar on Ag, the nature of the collision of an atom or ion with a solid surface varies markedly with Ei,the energy of incidence. The results of experimental investigations of the scattering of fast For a more detailed presentation of the material included in this section, see Stickney et al. (1967).

SCATTERING FROM SOLID SURFACES

145

m

L

t

DEEP PENETRATION WITH M A N Y COLLISIONS

RUTHERFORD

4-

+

WEAK

- SCREENING

-- lo5

i

THERMAL

1

-- lo3

APPRECIABLE DISPLACEMENT A N D SPUTTERING OF LATTICE ATOMS

Y

0

>

f

I_

-+ '-

HYPERTHERMAL

I

LATTICE ATOMS BEHAVE AS FREE PARTICLES

-L

HARD- SPHERE

(7)

EXCITATION OF LATTICE ELECTRONS

-- 108

t 2 0

I

1

(UNEXPLORED REGION)

1 I

EXCITATION OF LATTICE VIBRATIONS

REGIME

LATTICE TEMPERATURE CONDENSATION

1 DOMINANT ENERGY LOSS MECHANISM

SIGNIFICANT EFFECTS AND VARIABLES

FIG. 1. Interaction regimes for an Ar beam of energy Ei colliding with an Ag target.

-

(i.e., energies greater than 1000 eV) ions from metal surfaces indicate that a fraction of the collisions may be considered as an elastic' two-body interaction of the primary ion with a single atom in the solid. [See, for example, the recent reviews by Snoek and Kistemaker (1965) and by Kaminsky (1965).] It appears that the solid atom behaves as a free particle when the collision time is small relative to the period of lattice vibrations. At these high energies, it is likely that some ions will experience inelastic collisions (e.g., electronic excitation and ionization) and some will experience several collisions with lattice atoms before either emerging from the surface or coming to rest within the bulk of the solid. The energetic impacts also induce sputtering, electron emission, and lattice damage. These effects have been reviewed recently by Kaminsky ( I 965). Because of the difficulties of conducting scattering experiments with lowenergy ion beams, there are not yet sufficient results to determine how low the incident energy may be before the two-body collision model must be replaced by one that accounts for the e!Tect of the lattice on the interaction. Smith (1967) has recently reported that the two-body model is still valid at incident energies of 500 eV for H e f , Ne', and Ar' on Mo and Ni surfaces. Although several experiments have been performed at energies below 100 eV (Veksler, 1962a,b, 1963, 1965, 1966; Gurney, 1928; Read, 1928; Sawyer, 1930; By elastic we mean that the translational energy of the system, ion plus solid atom, is conserved (e.g., no electronic excitation). This does not necessarily mean, however, that the ion is scattered specularly with no energy transfer to the solid atom.

146

Robert E. Stickney

Longacre, 1934), the results are not in good agreement and the majority are unreliable because, as discussed by Kaminsky (19651, it is suspected that the surfaces were contaminated. Of these experiments, the most recent are the energy distribution measurements by Veksler (1962-1966) for 10-250 eV Cs' and Rb' scattered from Mo. The results led Veksler to assume that more than a single solid atom is involved in the elastic collision of a low-energy ion with the surface. Under this assumption, he estimates that at an incident energy of 20 eV the ion interacts elastically with a group of solid atoms having an effective mass approximately three times that of a single atom. The problem may be complicated, however, by the fact that the probability of multiple collisions is high in Veksler's experiments because p, the ratio of the mass of the ion to the mass of the solid atom, is approximately unity (e.g., p equals 0.9 and 1.4 for Rb' and Cs' on Mo, respectively). We shall return to this point in Section V,A,3. The experimental data for thermal-energy atomic beams are also insufficient to enable us to determine accurately the role of the lattice in the scattering process. In the limiting case, with Eisufficiently low that the atom is adsorbed (or trapped) on the solid, it is obvious that the "collision" time exceeds the vibrational period of the lattice and, therefore, it is incorrect to consider the process as an interaction of the atom with a single, free, solid atom. However, for the conditions of small energy transfer which are considered in this review, the collision time is of the same order of magnitude as the vibrational period (see Section II,B) so we are uncertain about the validity of the two-body approximation. On the one hand, we might intuitively feel that the lattice effects would still be dominant in this regime; on the other hand, we see that a simple two-body model provides a surprisingly good qualitative description of the experimental data (Section V). Hence, as will be discussed in more detail in Section V, even the form of the basic model for gas-solid collisions in the thermal energy range has not yet been firmly established. The goal of the theoretician is to determine the probability that a molt-cule having velocity ui will be scattered from the solid surface with velocity u,. If the internal state of the molecule is not altered by the encounter (e.g., no vibrational or rotational excitation), this differential cross section provides a complete description of the interaction. For example, the velocity distribution of the scattered molecules may be determined from the incident distribution and the differential cross section. At present, however, theoretical and experimental methods are not sufficiently advanced to provide an accurate determination of the velocity distribution for the conditions considered here. Since experimental measurements generally are proportional to a specific integral (or moment) of the distribution function, the results are often expressed in terms of the " macroscopic accommodation coefficients '' discussed in Section II,B.

SCATTERING FROM SOLID SURFACES

147

B. ENERGYAND MOMENTUM ACCOMMODATION COEFFICIENTS In the field of rarefied gas dynamics it is customary to describe the energy and momentum transfer between a gas and a solid in terms of three macroscopic parameters : the energy accommodation coefficient, the coefficient of tangential momentum transfer, and the coefficient of normal momentum transfer. (See, for example, Schaaf and Chambre, 1961.) These parameters are not completely unrelated because all three are functions of the velocity distribution of the scattered molecules. 1. Energy Accommodation CoeBcient

The energy accommodation coefficient (AC) may be considered from two viewpoints: near equilibrium and nonequilibrium. The majority of the experimental AC values have been obtained at near equilibrium conditions in thermal conductivity cells (Thomas, 1967; Wachman, 1962; Kaminsky, 1965). A typical thermal conductivity cell consists of a cylindrical tube which has a thin wire stretched along the axis and is filled with a gas at low pressures. A small temperature difference is established between the wire and the gas, and the resulting heat transfer rate is carefully measured. In this case, the energy accommodation coefficient is defined as the ratio of Q, the actual heat transfer rate, to Q, , the rate that would have occurred if the gas molecules had attained a state of thermal equilibrium with the solid before leaving the surface AC

=

QIQo.

(2.1)

The value of Q is measured directly, and Q , is calculated from kinetic theory and the measured pressure and temperatures. From Eq. (2.1) we can see that AC is essentially a measure of the efficiency of the energy transfer processes associated with gas-solid interactions. Since detailed summaries of the existing experimental data on energy accommodation coefficients may be found elsewhere (Thomas, 1967; Wachman, 1962; Kaminsky, 1965), it is not worth while to devote much space to this subject here. Results reported by Thomas (1967) for the rare gases on tungsten are more reliable than those reported for other gases because special care was taken to insure that the surface was clean. As illustrated in Fig. 2, the energy accommodation coefficient depends markedly on the gaseous species and on temperature. Note the general tendency of AC to increase with the molecular weight of the species. In the case of helium, the extremely low value of AC rr 0.02 indicates that there is very little energy transfer associated with this interaction. A qualitative interpretation of the experimental data shown in Fig. 2 has

Robert E. Stickney

148

0.9 Xe

?

:0.10 -

y.

‘0

0 ‘

I-

z

-0.6 Kr

w

I-

-0

-0.4 Ar

W

3 -

II. LL W

z

0

F 0.06a

-0.1

n

50

n

0

Ne 0

*a

z W

-0.2 2

z

a

L!

o-o/

0

-

V

2

-

>

a

cl

a W z

He

W

u a 0’ 0

I

50

I

100

I

150

I

200

TEMPERATURE, O

I

250

I

300

1 350

K

FIG.2. Temperature dependence of the energy accommodation coefficient of rare gases on tungsten (Thomas, 1958). Note: These experimental data were obtained with the temperature of the tungsten filament set 15” above the gas temperature.

-

been proposed by Stickney (1966). (Quantitative theories are considered in Section II,B,2.) This interpretation is based on the concept of adiabatic interactions which is commonly employed in the theory of gas-phase molecular collisions (Massey and Burhop, 1952; Herzfeld, 1955). An adiabatic interaction is one occurring so slowly that the system passes through a series of quasistatic equilibrium states, the result being a reversible process with no net energy transfer between the two interacting bodies. If the interaction is not sufficiently slow (i.e., if it is nonadiabatic), the inertia of the system causes the motion of the bodies to be “out-of-phase,” and this departure from a quasistatic process results in a transfer of energy. Hence, the important parameter is the ratio of the interaction time to the characteristic vibrational period of the lattice, which may be expressed approximately as

where I, the characteristic dimension of the repulsive portion of the intermolecular potential, is of the order of lo-’ cm, mi is the most probable speed of a gas of molecular weight M8 and temperature T 8 ,and v, the characteristic

SCATTERING FROM SOLID SURFACES

149

frequency of the lattice, is of the order of I O l 3 cps for a metal such as tungsten. Since the interaction becomes more adiabatic as T, decreases, it follows that the AC should decrease with T, as observed for helium and neon in the righthand portion of Fig. 2. This simple model breaks down, however, when the interaction time is sufficiently long (i.e., T, is sufficiently low) that a significant fraction of the energy may propagate from the surface into the bulk of the solid. In this case, the AC increases with decreasing T,, and, in the limit of AC+ 1, the energy transfer is sufficiently large that gas atoms are trapped (adsorbed) in the gas-solid potential well for a time long enough to result in thermal equilibrium. Since both the depth of the potential well and the magnitude of the interaction time [Eq. (2.2)] increase with molecular weight for rare gases, a heavy gas will attain the limit AC + I at a higher temperature than that for a light gas (Fig. 2). It should be emphasized that the present considerations are restricted to interactions that affect only the translational (kinetic) energy of the gas molecules. Very little is known concerning interactions that alter the rotational, vibrational, or electronic state of gas molecules, and it may be advisable to define a separate accommodation coefficient for each of these processes (Wachman, 1962; Kaminsky, 1965). The present considerations are further restricted to cases in which both the composition and the mass flux of the scattered molecules are equal to the corresponding values for the incident molecules (i.e., no chemical reaction, dissociation, adsorption, or condensation). Without these restrictions, the relationship between AC and the change of translational energy of the gas molecules is less clear. With the above-mentioned restrictions in mind, we may now derive a second expression for AC. Let ri represent the “arrival rate” (i.e., the number of incident molecules striking the surface per unit area and unit time), and Ei and E, represent the mean translational energies of the incident and scattered molecules, respectively. The energies Ei and E, are measured relative to the surface and it follows that their difference, Ei - E,, represents the mean energy transfer associated with the gas-solid interaction. Since we have restricted our consideration to the case in which li is the same for both incident and reflected fluxes, the rate of energy transfer, (7, is simply equal to n(Ei - I&), and Eq. (2.1) may now be expressed as AC =

ri(Ei - E,)

li(Ei - E,) ’

where E, is the value that E, would have if the molecules were “completely accommodated” (i.e., if the molecules leaving the surface were in a state of thermal equilibrium with the surface). We may cancel n in Eq. (2.3) and obtain AC=-

Ei - E, Ei - E,‘

(2.4)

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Robert E. Stickney

The energy accommodation coefficient is often defined by means of an expression similar to Eq. (2.4). (See, for example, Schaaf and Chambre, 1961; Wachman, 1962.) Note that AC is a measure of the degree to which gas molecules attain thermal equilibrium with the solid, and that its value is unity when E, = Es (i.e., complete accommodation) and is zero when E, = Ei (i.e., specular reflection, or zero energy transfer). Historically, the definition of AC represented by Eq. (2.4) was preceded by an expression based on temperature (Knudsen, 1934) which will now be derived. For the special case of an equilibrium gas at temperature T, the mean translational energy E of particles striking a fixed surface is given by

E = 2kT, (2.5) where k is Boltzmann’s constant. Therefore, Eq. (2.4) may be expressed as

where T, and T, are the temperatures of the impinging gas and the solid, respectively, and T, is the “ effective temperature ” of the scattered molecules. The representation of E, by a temperature T,lacks theoretical justification because the scattered molecules may have a non-Maxwellian distribution function, except in the limit T, = T,. Similarly, the substitution of 2kTg for Ei is valid only if the incident gas is Maxwellian, as is generally the case in thermal conductivity cells and thermal molecular beams. Because of these inherent limitations of Eq. (2.61, the more general expression given by Eq. (2.4) is to be preferred. An erroneous statement, which is often found in discussions of energy accommodation, is that all possible values of AC are contained in the interval 0 to 1. This point has been considered by Stickney (1962a) and by Goodman (1965). From the second law of thermodynamics we know that for a heat interaction between two systems that are in unequal equilibrium states initially, the net energy transfer is from the system of higher temperature to the system of lower temperature. From this it follows that AC cannot be negative if both of the interacting systems, the gas and the solid, are in equilibrium states when isolated from each other initially, i.e., before they are allowed to interact. This statement is not necessarily true, however, in the more general case where one of the systems is in a nonequilibrium state initially since the second law applies onZy to the interactions of systems that were initially in equilibrium states. Therefore, we see that thermodynamics does not restrict the value of AC to the interval 0 to 1 in nonequilibrium cases such as the interaction of a non-Maxwellian stream of molecules with a solid ~ u r f a c e . ~

’

If we assume that the hard-cube model provides a valid means of interpreting data, then existing experimental results indicate that, under certain conditions, the AC may be negative for molecular beams scattered from solid surfaces (Section V,A,2).

SCATTERING FROM SOLID SURFACES

151

A principal difference between the thermal conductivity cell and molecularbeam techniques is that the incidence angle of the impinging particles is random in the former case and directed in the latter. Since AC may depend on the angle of incidence, it follows that the value measured for a single angle of incidence is not necessarily equal to that for random incidence. Hurlbut (1962) has suggested that measurements associated with a specific incidence angle be referred to as “ partial accommodation coefficients.” 2. Energy Accommodation Theories

The development of an accurate theory of energy accommodation represents a formidable problem because the theories of lattice dynamics and atomic collisions, both of which are incomplete, must be employed. Only a brief survey of existing theoretical treatments of the problem is given below because detailed surveys have recently been presented by Trilling (1967b), Hurlbut (1967), and Beder (this volume). The simplest model for energy accommodation is suggested by the following form of Eq. (2.4): E, = (1 - AC)Ei

+ (AC)E, .

(2.7)

Following Maxwell (1890), it is assumed that the scattered molecules are divided into two classes, one consisting of the fraction (AC) of the molecules that are completely accommodated, and the other of the remaining fraction (1 - AC) that are specularly reflected. Since there seems to be no theoretical justification for this simple model, its popularity has waned in recent years. Also, the scattered data discussed in Section IV do not substantiate this model. A classical treatment assuming that both the gas molecules and the surface atoms behave as hard spheres was first presented by Baule (1914). Using this hard-sphere model, we may obtain (Appendix)

where p is the ratio of the mass of the gas molecule to the mass of the solid atom. [The expression derived by Baule (1914) differs from Eq. (2.8) by a factor of 2 because he assumed a Maxwellian velocity distribution for the gas and solid atoms.] In some cases this expression agrees qualitatively with the general trends of experimental data (Thomas, 1958). Quantitative agreement is not expected because the model is far too simple (see Section V). Goodman (1965) has employed a similar model recently in an interesting qualitative study of the dependence of AC on temperature and on the form of the velocity distribution function.

152

Robert E. Stickney

Although it is expected that a general theory of energy accommodation must be based on a quantum mechanical approach, classical mechanics has been employed in the majority of recent studies. Reasons for favoring the classical approach have been stated by Zwanzig (1960): (i) The mathematical complexities associated with quantum mechanical treatments tend to obscure the essential features of the problem; (ii) existing quantum mechanical solutions cannot explain the large energy transfers that have been observed experimentally. The validity of the classical approach has been a subject of continuing discussion (e.g., see Zwanzig, 1960; Gilbey, 1967; Howsmon, 1966). Classical theories of energy accommodation based on one-dimensional lattice models have been considered by Cabrera (1959), Zwanzig (1960), Goodman (1962), Leonas (1963), and McCarroll and Ehrlich (1963). To obtain a more realistic model of the solid, Goodman (1963, 1966) proceeded to a three-dimensional lattice; similar models have been employed by Trilling (1964, 1967a) and by Chambers and Kinzer (1966). In Goodman’s model the solid atoms are assumed to be interconnected by linear springs, and the spring constants are calculated from the Debye temperature of the solid. The intermolecular potential for a gas molecule and surface atom is described by the Morse potential function. Unfortunately, a computer solution is required for each new set of initial conditions and system properties. By a trial-and-error procedure Goodman has determined the values of the Morse parameters, a and D, which cause his theoretical predictions of AC to agree with experimental results. Although excellent quantitative agreement with the data shown in Fig. 2 is obtained for reasonable values of a and D, this result may be misleading because the theory is based on several questionable assumptions. (See Stickney, 1966; Logan and Stickney, 1966; Ehrlich, 1966; Hurlbut, 1967.) For example, since it is assumed that the solid atoms are at rest initially (i,e., the temperature of the classical solid is essentially zero degrees absolute), there is no mechanism for energy transfer,from solid to gas. In all AC experiments, however, the net energy transfer is from solid to gas because the temperature of the solid exceeds that of the gas. Therefore, it is not clear that it is reasonable to compare Goodman’s theoretical results with existing experimental data. The analog computer has proven to be a convenient means for studying the properties of gas-solid collision models. The surface is usually represented by a single solid atom connected to a spring, and a variety of gas-solid intermolecular potentials have been employed. Parametric studies of this model have been conducted by Rogers (1966) and Berkman (1965). Berkman has also used a simple spring-mass-damper system to represent Goodman’s three-dimensional model. As demonstrated by Hurlbut (1967), it is possible to design an analog computer system that can satisfactorily model a solid consisting of several atoms coupled together by nonlinear potential functions.

SCATTERING FROM SOLID SURFACES

153

Because of the inconvenience of having to use a computer to calculate AC, Goodman and Wachman (1966) and Oman (1 966) have used semi-empirical methods to determine closed-form equations that correlate their theoretical data. Trilling (1967a) also has reduced the computation requirements of his earlier work by adopting a simplified lattice model.

3. Momentum Coeficients The momentum transfer between a stream of gas molecules and a solid surface may cause the surface to experience forces both in the normal and tangential directions. In order to describe these forces, two parameters have been introduced: B , the coefficient of tangential momentum transfer, and B’, the coefficient of normal momentum transfer (Schaaf and Chambre, 1961):

(2.10) Here, z i and T~ are the mean tangential components of momentum for the incident and scattered molecules, respectively ; Piand P, are similarly defined for the normal components of momentum. The term P, represents the normal momentum component corresponding to complete accommodation. [Since T, is equal to 0 in the case of complete accommodation, it does not appear in Eq. (2.9).] Both B and B’ depend on the directions of the scattered molecules, as well as on the speeds. For instance, B is equal to unity as long as the scattering directions are diffuse, regardless of the degree of energy accommodation. An undesirable feature of B’ is that its value is not restricted to the interval 0-1 ;this point has been discussed by Stickney (1962a,b) and Goodman (1965). It is expected that the value of B for a specific gas-solid combination does not depend on the tangential component of the incident momentum alone; most likely, the magnitude of B is affected by changes in the speed and angle of incidence, when z i is maintained constant. An analogous statement could be made with respect to B’. As stated in Section II,B,I, the energy and momentum coefficients are not independent parameters because all three depend on the distribution functions of the incident and scattered molecules. Direct experimental measurements of momentum transfer have been accomplished by means of the rotating cylinder (e.g., Hurlbut, 1960), and force-balance techniques (e.g., Stickney, 1962a,b; Abuaf and Marsden, 1967). The results of these experiments will not be considered here because in all cases the surfaces were contaminated to an unknown degree. Indirect measurements

154

Robert E. Stickney

are provided by the results of the scattering studies discussed in Section IV. We shall defer consideration of the existing theories of momentum transfer to Section V.

111. Experimental Apparatus for Gas-Solid Scattering Studies Although many reviews of molecular beam techniques have been published, none includes a detailed description of the design features of the modulated molecular-beam apparatus for investigating gas-solid scattering. This may be explained by the fact that the use of modulated beam techniques in gassolid studies was insignificant before 1960. Since the discussion of experimental scattering data presented in Section IV assumes a familiarity with modulated beam techniques, the principal features of the existing apparatus are summarized in the present section.

A. CHAMBER DESIGN During the past five years (1961-1966), the most significant experimental investigations of the scattering of thermal beams from solid surfaces have been conducted at General Atomic, Oak Ridge, and United Aircraft. All three laboratories utilize the modulated molecular beam technique which is described in Section III,C. Also, mercury diffusion pumps are employed in order to eliminate the hydrocarbon contaminants that are often produced by oil-diffusion pumps. We have arbitrarily chosen the United Aircraft apparatus (Hinchen and Foley, 1966) shown in Fig. 3 as a typical example of a modulated beam system. By omitting the collimation chamber that is included in most systems, a more compact apparatus is obtained and one diffusion pump is eliminated. Since this simplification allows more gas to flow into the target chamber, a larger diffusion pump is required in order to maintain the chamber at sufficiently low pressure. Pressures of the order of 1 x lo-' Torr are obtained in the United Aircraft apparatus for normal beam intensities. A low chamber pressure is desirable because this reduces the background noise (Section II1,C) and the degree of contamination of the target (Section 111,D). A unique feature of the United Aircraft system is the use of metallic gaskets in the flanges of the target chamber. This eliminates the outgassing problem associated with the conventional elastomer O-rings, and it allows the stainless steel chamber to be baked during the evacuation process. As a result, pressures in the range of lo-'' Torr may be achieved when the beam is turned off. The molecular beam oven shown in Fig. 3 is an alumina tube, with a 0.4-mm diameter opening at the end. A simple resistance heater made from molybdenum wire is wrapped about the tube so that the oven temperature may be

SCATTERING FROM SOLID SURFACES

155

varied from 300" to 1700°K. [General Atomic and Oak Ridge both employ a tungsten oven so that higher temperatures (- 3000°K) may be obtained; see, for example, Fite and Brackmann (1958).] Gases of research grade are supplied to the oven through a metering valve. A typical value of the oven pressure is 0.1 Torr. The molecules effusing from the oven are collimated into a beam by the two apertures shown in Fig. 3. The diameter of both apertures is 1.0 mm, and the beam diameter is -2.0 mm at the target. The beam is modulated at 3600-cps frequency by a slotted disc (chopper) driven by a synchronous motor (Globe, SC53A107-2).4 TARGET CHAMBER

GAS RESERVOIR

I

7 I \\

LTARGET

PUMP

FIG.3. Schematic diagram of a modulated molecular beam apparatus for investigating the scattering of gas molecules from solid surfaces. The design shown here is that of the United Aircraft apparatus (Hinchen and Foley, 1966).

B. BEAMINTENSITY For purposes of illustration, we shall consider the order of magnitude of the detector signals for scattering measurements performed in the United Aircraft apparatus. If the source oven temperature and pressure are 300°K and 0.1 Torr, then the rate at which argon atoms flow through the 0.4-mm diameter aperture at the end of the oven is5 (Kennard, 1938) $A,nE

N

4 x 10l6 sec-l,

(3.1)

Hinchen and Foley have found that this motor operates satisfactorily in vacuum if Barden Bartemp bearings are used. These bearings do not require an oil lubricant because the Teflon races are impregnated with molybdenum disulfide. Equation (3. I ) is valid only if (a) the thickness of the aperture is small compared with its diameter and (b) A, the mean-free path of the gas in the oven, is large compared with the diameter of the oven aperture. (See, for example, Liepmann, 1961.)

156

Robert E. Stickney

where A . is the area of the oven aperture, and n and U are the density and mean speed of the argon gas, respectively. The flux within the solid angle dw at angle LY from the beam center line is given by cos LY dw 71

which, for LY pressed6 as

=0

($A,nij)

and a molecular beam of circular cross section, may be ex-

where d, is the diameter of the beam cross section at distance r from the oven aperture. For d, = 0.2 cm and r = 8 cm, the beam flux is

N, = 1.56 x 10-4($A,nG)

6 x 1OI2 sec-'.

(3.4) We shall assume that (a) the acceptance angle of the detector is sufficiently large that the area subtended on the target surface exceeds that subtended by the incident beam (Fig. 3), (b) the target area subtended by the incident beam may, from the viewpoint of the detector, be considered as a point source, and (c) the directions of the scattered molecules are diffuse (i.e., random). These assumptions permit us to apply Eqs. (3.2) and (3.4) to determine the flux of molecules scattered into the detector at angle 8, : N

In this case, dw = ndd2/rd2,where dd is the diameter of the detector aperture and rd is the distance from the target to this aperture. Hence, Nd =

dd2cos 8, 4r,2 N ,

(3.6)

Assuming, for example, that dd = 0.2 cm, r,, = 4 cm, and 8, = 60°,we obtain Nd = 3.12 x 10-4iV, N 2 x lo9 sec-'.

(3.7) If the detector is a stagnation-type ionization gauge, the detector signal will be directly proportional to N d .On the other hand, if a through-flow ionization gauge is employed, the detector signal will be directly proportional to the density of the scattered molecules at the detector, Nd nd = )ndd2ii,'

(3.8)

where Nd&dd2 is the intensity (i.e., molecules per cm2 per sec), and 5,is the We assume thatdw = &7ids2/r2;this is valid only if r % d, 3 do,where do is the diameter of the oven aperture.

SCATTERING FROM SOLID SURFACES

157

mean speed of the scattered molecules. In both cases the constant of proportionality is related to the gauge sensitivity or ionization probability. An advantage of the stagnation-type gauge is that its signal is generally larger than that of the through-flow gauge. The response time of the stagnationtype is, however, too slow if, as we shall see, one wishes to operate at the high modulation frequencies that are necessary for precise measurements of phase shift. [For a more detailed discussion of stagnation and through-flow detectors, see Smith (1969.1 Using Eqs. (3.7) and (3.8) with dd = 0.2 cm and U, N 5 x lo4 cm/sec, we find that the density of a scattered beam in a through-flow detector is of the order of 1 x lo6 molecules/cm3. (The effective density will be somewhat less than 1 x lo6 because the scattered beam generally does not fill the entire volume of the ionization region.) Since this is equivalent to a pressure of -3 x lo-" Torr, the detector output signal corresponding to the scattered beam will be an ion current of the order of 3 x amp if the detector efficiency equals that of the conventional ionization gauge. In most systems the density of the background gas greatly exceeds the density of the scattered beam. For example, the density corresponding to a typical background pressure of 1 x lo-' Torr is more than three orders of magnitude larger than the scattered beam density estimated above. Since the detector signal is proportional to the total density' in the ionization gauge, it is obvious that small fluctuations of the background density will obscure changes in the density of the scattered beam. This problem may be alleviated in several ways : (a) decrease the background pressure by increasing the pumping speed (e.g., use a condensable beam gas); (b) increase the beam flux by employing larger oven apertures, multiple apertures, higher oven pressures, or supersonic beam techniques (Anderson et af., 1965); (c) increase the beam density in the detector (at the expense of angular resolution) by using larger collimating apertures and shorter distances between the oven and the target, as well as between the detector and the target; (d) measure the difference in the readings of two similar gauges, only one of which directly intercepts the scattered beam (Hurlbut, 1953); (e) employ detectors that are capable of discriminating between the beam gas and the background gas (e.g., the surface ionization gauge and the mass spectrometer); (f) modulate the beam at a fixed frequency and use signal-processing techniques that reject all signals except those having the proper frequency and phase. Since the modulation technique has yielded the most significant results in recent investigations of gassolid scattering, we shall restrict our considerations to it in the following section.

'

It would be more accurate to state that the detector signal is proportional to the surnmation of the densities of each species multiplied by their respective ionization probabilities. In most cases, however, the beam gas is the main component of the background gas.

Robert E. Stickney

158

C. LOCK-INDETECTION OF MODULATED BEAMS The most elementary design of a detection system for modulated molecular beams is shown in Fig. 4. The signal from the detector consists of (a) the ac component associated with the modulated beam, (b) the dc component of the background gas, and (c) noise. Since the dc component is blocked by the condenser (Fig. 4), the rms value of the ac component plus the noise may be determined by using an ac voltmeter to measure the potential drop across the resistor. In practice, however, the simple voltmeter is generally replaced by the " lock-in " amplifier described below because this provides an effective means for separating the signal from the noise. We should also note that the

91 oc NOISE

r DETECTOR

(IONIZATION GAUGE1

~~,

CONDENSER

dc BACKGROUND

I

i

z

R

i

V VOLTMETER

,

,

I

FIG.4. Schematic diagram of a simple electricalcircuit for detecting modulated molecular beams.

condenser is generally replaced by a preamplifier which, in addition to blocking the dc component, changes the impedance of the voltage signal to a sufficiently low value that it can be taken outside the vacuum system without excessive losses caused by distributed capacity. [This point is discussed by Brackmann and Fite (1961); a preamplifier design has been described by Fite and Brackmann (19581.1 A lock-in amplifier consists of a narrow-band amplifier, a phase-sensitive detector, and an RC integrating circuit.* The narrow-band amplifier is tuned to the modulation frequency so that the fundamental component of the ac signal is amplified, while all noise components are rejected except those

* Since modulation techniques are employed in a variety of researchfields, lock-in amplifiers are now manufacturedby several companies, e.g., Princeton Applied Research Corporation and the Electronics, Missiles and Communications, Inc.

SCATTERING FROM SOLID SURFACES

159

having frequencies very near the modulation value. This signal is then rectified by a switching circuitg that is driven at the modulation frequency by a reference signal from the chopper. (In most systems the reference signal is obtained by placing a photocell and a light bulb on opposite sides of the chopper.) By including an electronic phase-shifting circuit in the reference-signal circuit, the phase of the switching operation may be adjusted to maximize the dc output signal. The final stage is a simple RC circuit serving to integrate the dc signal over many cycles. The integration time, which may be adjusted by varying the time constant of the RC circuit, is most influential in reducing the effective bandwidth of the total detection system. In practice, however, the integration time cannot be increased indefinitely because of the problems of drift and the inconvenience of waiting for long periods between measurements. The dc output of the integrator may be displayed on a meter and on a strip-chart recorder. Since a detailed analysis of lock-in detection of modulated molecular beams has been presented recently by Yamamoto and Stickney ( I 967), it will suffice to summarize the principal features:

(1) The dc output signal of a lock-in amplifier is proportional to A , , the average value of the fundamental Fourier component of the ac input signal. In the present case, the input signal is directly proportional to S,,(t), the instantaneous density (or flux, momentum, or energy, the choice depending on the detector design) of the scattered beam at the position of the detector gauge. (2) The magnitude of A , depends on the modulation frequency, the choppertarget and target-detector distances, the temperature of the incident molecular beam (which is assumed to be Maxwellian), the velocity distribution of the scattered beam (which may be non-Maxwellian), and 4,the phase of the switching operation relative to the reference signal. (Notice that 4 is no longer a free variable if, as is generally the case, we define A , to be the maximum value obtained by adjusting 4.)Since A,, depends on the speed distribution of the scattered beam, which itself depends on the direction of the scattered beam, it follows that scattering distributions based on measurements of A , are not uniquely related to the density distribution. (3) Measurements of phase shifts do not provide a unique determination of the mean speed or temperature of the scattered beam unless a specific collision model is assumed. Hence, the interpretations of phase-shift measurements put forth by Hinchen and Foley (1965) and by Moore et al. (1966) may be misleading because they are based implicitly on collision models that are unproved. The designs of several switching and mixing circuits for phase-sensitive detection are described by Andrew (1955).

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Robert E. Stickney

D. TARGETS For fundamental investigations of gas-solid interactions, it is desirable that the chemical composition and crystallographic structure of the target be known so that we may formulate an accurate theoretical model. Hence, the obvious choice for experimental studies is a pure element in the form of a single crystal. It is also obvious that we want to be able to obtain and maintain clean surface conditions during the experimental measurements, and we would like to be able to vary the target temperature to determine its effect on the gas-solid interaction. With these factors in mind, we shall consider the various types of targets used in scattering experiments. In many of the first scattering investigations, alkali halide crystals were employed as targets because it was extremely difficult to obtain metals in single-crystal form. The principal disadvantages of these crystals are: (1) The problem of formulating a theroretical model of the surface is complicated by the fact that alkali halides consist of two elements rather than one. (2) The target temperature cannot be varied over a wide range, because of the low melting temperatures of the alkali halides. Experiments conducted recently by Whetten (1964) and McRae and Caldwell (1964) indicate that certain alkali halide crystals may be cleaned by heating alone. Unless cryopumping methods are employed to obtain extremely high pumping speeds, the residual gas pressure in the target chamber of a molecular beam apparatus, in most practical cases, is sufficiently high that a clean surface held at room temperature will be contaminated by adsorbed residual gases within a matter of seconds. There are, however, two methods for preventing surface contamination without going to the expense of cryopumping : (I) If the target material is deposited continuously upon a substrate, it is possible to form a fresh surface at a rate that exceeds the rate of contamination. (2) By selecting a target material that does not adsorb the residual gases strongly, the degree of contamination may be decreased by maintaining the target at an elevated temperature. [For a detailed discussion of the techniques for obtaining clean surfaces, see Roberts (1963).] The continuous deposition technique has been employed by Smith and Saltsburg (1964) to obtain highly oriented films of gold or silver. The target substrate is a mica disc, 2.5 cm in diameter, mounted on a copper heating stage. A crucible containing either gold or silver is located near the target, and the deposition rate is controlled by the temperature of the crucible. If the substrate is held at -580°K during deposition, the resulting epitaxial film is highly oriented with the (1 11) plane parallel to the substrate surface. The single disadvantage of this technique is that the target temperature may not be varied over a wide range because the smoothness and orientation of the films vary with the substrate temperature.

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As an example of the second method, platinum targets have been employed in several investigations, because of the belief that the common residual gases are readily desorbed from this metal when it is held at moderate temperatures. Datz er al. (1963) used a polycrystalline ribbon mounted under spring tension to reduce buckling and misalignment at high target temperatures. The temperature is easily controlled by passing a heating current through the ribbon. Hinchen and Foley (1965, 1966) have chosen to use a polycrystalline disc, 1.3 cm in diameter, which is heated by radiation from a tungsten coil mounted immediately behind the disc. Both groups report that, as a result of grain growth produced by prolonged heating in vacuum, the platinum targets consist of large crystals having an average dimension of -1 mm. Although the specimens investigated by Datz et al. (1963) were strained to the degree that the crystal orientations could not be determined by X-ray diffraction, Hinchen and Shepherd (1967) have recently reported that the crystals in their targets are relatively free of strain and are oriented with a ( I 10) plane parallel to the surface. Since a wide variety of metals is now available commercially in singlecrystal form, the experimenter is no longer restricted to using polycrystalline targets. The results of low-energy electron diffraction studies indicate that single-crystal ribbons having near-ideal surface structures may be obtained if suitable processing and vacuum techniques are employed. (See, for example, MacRae, 1963; Germer and May, 1966.) Although this type of target appears to be ideal for gas-solid scattering studies, it has not yet been employed by workers in this area.

-

IV. Experimental Data on the Scattering of Atomic and Molecular Beams from Solid Surfaces With few exceptions, the history of molecular beam studies of scattering from solid surfaces may be divided into four periods: Period I (ca. 1930). Quantum mechanical diffraction of He and H, by alkali halide crystals. This topic is considered only briefly in Section IV,A because it is covered in detail in other reviews (Ramsey, 1956; Massey and Burhop, 1952). Period I1 (ca. /932). Scattering of vapor atoms, such as Hg and K, from alkali halide crystals. A brief review of these investigations is presented in Section lV,B. Period IfI (ca. 1955). Scattering of gas atoms and molecules from a variety of polycrystalline and amorphous surfaces. Most of these studies were conducted by Hurlbut and his co-workers for surfaces that were unintentionally contaminated with oil vapors and residual gases. For this reason, and because

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ofthe existence of the review by Hurlbut (1962), little attention is given to these studies in Sections IV,C and IV,D. Period I V (at present). Use of modulated molecular beam techniques to investigate a wide variety of gas-solid interactions. Results for the scattering of atoms and molecules from solid surfaces are summarized in Sections IV, C and IV,D.

A. DIFFRACTION EXPERIMENTS The first significant investigations of the scattering of atomic and molecular beams from solid surfaces were performed around 1930 for the purpose of confirming the diffraction effects predicted by the newly developed quantum theory. According to this theory, atoms and molecules may be treated as wave packets characterized by the de Broglie equation,

1 = h/mu,

(4.1)

where h is Planck’s constant, and A, rn, and u are the de Broglie wavelength, mass, and speed of the atomic or molecular species. Therefore, diffraction should occur when an atomic or molecular beam impinges upon a ruled grating (in practice, a crystal surface) having a spacing comparable to A. The diffraction of He and H2 by alkali halide crystals was observed experimentaIly by Estermann and Stern (1930), Estermann et al. (1932), and others. (For a detailed review, see Ramsey, 1956, or Massey and Burhop, 1952.) Although the zero-order diffraction peak that is located at the specular angle was dominant, it was possible to establish the existence of higher order peaks. The angular positions of these peaks agree well with the values predicted by diffraction theory. It is interesting to note that diffraction occurred in these cases even though the crystal surfaces were probably contaminated to some degree by adsorbed residual gases. More recently, Crews (1962) investigated the scattering of He and Ar from the (001) plane of a LiF crystal. Although the scattering pattern for He was highly specular, as we would expect for diffraction, the pattern for Ar was neither specular nor diffuse,” as shown in Fig. 5. We would not expect Ar to be diffracted as strongly as He because its de Broglie wavelength, which is 1/3 that of He at the same temperature, is considerably smaller than the

-

l o We shall use the word “ nondiffuse” to denote scattering processes that deviate from the limiting case of diffuse scattering. A diffuse scattering pattern is one in which the amplitude varies as cos 8, (see Fig. 7), just as in the case of Lambert’s law for the reflection of light from a rough or “black” surface. For the equilibrium case of a target exposed to a gas at rest with T, = T,, the scattering must be diffuse, according to the principle of microscopic reversibility. Note, however, that diffuse scattering may occur even when the energy accommodation is incomplete (e.g., scattering from a rough surface).

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lattice spacing of the crystal. Crews also concluded from an examination of his experimental results that the Ar pattern does not consist of a simple superposition of specular and diffuse components. A primary purpose of this review is to consider the gas-solid interaction processes resulting in nondiffuse" scattering patterns similar to that shown in Fig. 5 for Ar on LiF. More specifically, we shall concentrate on the case in which the mass or speed of the incident atomic or molecular species is sufficiently large that the de Broglie wavelength is small compared with the lattice spacing.

SCATTERING ANGLE,

8, ( DEG)

FIG.5. Experimental patterns for He and Ar scattered from the (001) plane of LiF. [Data taken from Figs. 3 and 6 of Crews (1962).] A,/A,. is the ratio of the detector signal for the scattered beam to that for the direct beam. The Ar data have been multiplied by 50.

B. SCATTERING OF VAPORATOMS FROM ALKALI HALIDE CRYSTALS 1. Measurements of Scattering Patterns

A logical extension of the experimental studies of the diffraction of He and H, from alkali halide crystals was to use the same apparatus to investigate the scattering of heavier atoms having shorter de Broglie wavelengths. The results of these experiments are summarized here, although their validity is questionable because the test surfaces were probably contaminated to some degree by adsorbed gases. [Water vapor may be the most detrimental adsorbate, according to Zabel (1932) and Hancox (1932).] The scattering of Hg atoms from alkali halide crystals is interesting because the mass of Hg is large compared with the mass of the crystal atoms, and the average momentum of Hg atoms impinging with thermal energy is sufficiently

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large to cause their de Broglie wavelength to be very much smaller than the lattice spacing. From these points, and the fact that at thermal temperatures Hg condenses readily on a large number of substances, it would be expected that the energy transfer in this gas-solid interaction should be large and the scattering pattern diffuse. It is surprising, therefore, that Hancox (1932), Josephy (1933), and Zahl and Ellett (1931) have observed nondiffuse patterns for Hg scattered from various alkali halide crystals. Experimental results reported by Zahl and Ellett (1931) are shown in Figs. 6 and 7. Although nondiffuse patterns were obtained for Hg on NaCl (Fig. 6), the patterns for Hg on KI were quite diffuse (Fig. 7). A possible explanation of the last result is that KI may adsorb residual gases more tenaciously than NaCl, thereby causing gross

\ 40'

\\

\\

,

,

Hg on NaCl

I 1

30"

8,.= 36O 0

323°K 443'K 40'

50a

50" 60'

70'

60"

70'

80"

80'

90'

900

FIG.6. Experimental patterns for Hg scattered from NaCl (Zahl and Ellett, 1931). These data illustrate the dependence of the patterns on T, and T, , the temperatures of the target and the gas, respectively. Notice that the pattern moves away from the target normal as T, increases or T, decreases; the pattern for T, = 323°K and T, = 773°K is slightly supraspecular (Le., its maximum occurs at an angle that is slightly larger than the specular angle).

The experimental results for the scattering of Hg from alkali halide crystals may be summarized as follows: (a) A nondiffuse scattering pattern occurs when a significant fraction of the adsorbed impurities has been removed by thermal desorption. (b) The angular position of this lobe deviates from the direction of specular reflection, and the deviation generally increases with increasing angle of incidence." l1

Throughout this discussion the angle of incidence is measured from the surface normal.

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FIG.7. Experimental pattern for Hg scattered from KI (Zahl and Ellett, 1931). The data agree closely with the cosine pattern (dotted circle) corresponding to diffuse scattering.

(c) Although the lobe is usually located between the surface normal and the specular direction, it may lie below the specular direction when the angle of incidence is sufficiently small (Fig. 6). (d) The angular position of the lobe increases with increasing gas temperature and decreasing surface temperature (Fig. 6 ) . (e) As the surface temperature is reduced, the width of the lobe decreases, although its intensity falls off because a greater fraction of the atoms is scattered diffusely; it is believed that the first effect results from reduced thermal motion of the surface atoms, whereas the second is attributed to the adsorption of impurities (Hancox, 1932). (f) There are no significant changes in the scattering pattern when the crystal is rotated about its normal; this indicates that, contrary to the case of diffraction, there is no strong dependence on the spacing of the rows of surface atoms. The different attempts to explain these features of the scattering pattern by means of simple theoretical models will be discussed in Section V. It suffices here to emphasize the fact that these results clearly show that in this specific gas-solid interaction the momentum transfer is a function of the angle of incidence and the temperatures of the gas and the solid. Although the measurement technique12 employed in these studies does not provide a direct The scattering patterns were measured with a stagnation-type ionization gauge which was rotated about the crystal. This technique provides direct measurements of the intensity of the scattered atoms versus angular position; it does not provide a direct measurement of either the momentum or energy of the scattered atoms.

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Robert E. Stickney

means for determining the degree of energy accommodation, it is expected to be incomplete because momentum accommodation is incomplete. Nondiffuse scattering from various alkali halide crystals has also been observed for Cd, Zn, Sb, T1, and Pb vapors. (See Hancox, 1932; Zahl and Ellett, 1931; Kellogg, 1932.) We may conclude, therefore, that the effect is not peculiar to Hg. 2. Measurements of Velocity Distributions

The experimental results described above stimulated several attempts to determine the degree of energy accommodation of atoms scattered from crystal surfaces. Although it is obvious that measurements of the velocity distribution of scattered atoms would provide the most detailed information on the scattering process (Section 11), the difficulty of performing these measurements has discouraged all but a few investigators. The most significant results obtained are those of McFee and Marcus (1960) which are discussed below. (An increasing amount of velocity distribution data is expected in the near future because many of the molecular beam groups are now developing the necessary apparatus.) McFee and Marcus (1960) chose to employ a potassium beam in their investigation so that a surface ionization gauge could be used as the detector. [The advantages of this type of detector are described in most reviews of molecular beam studies, one of the most recent being that by Pauly and Toennies (1965).] Because of the size of the mechanical velocity selector, the angle of the incident was held constant at 45" and the detector was fixed along the specular direction. The velocity distributions measured for K scattered from Au, Cu, W, and MgO were Maxwellian with temperatures equal to that of the target (within the precision of measuring the latter temperature). This result of complete energy accommodation is to be expected because potassium adsorbs readily on many materials. The complete accommodation of K scattered from MgO was also reported by Ellett and Cohen (1937). Incomplete accommodation was observed, however, for K scattered from a LiF crystal. This result was unexpected because earlier experimental data obtained by Taylor (1930) indicated that alkali metal atoms scatter diffusely from alkali halide crystals. McFee and Marcus found that the velocity distribution of Kscattered from LiF was not Maxwellian and depended on the temperature of the incident beam. The deviation from Maxwellian was greatest in the low-speed portion of the distribution. On the basis of data obtained for target temperatures of 600-900°K and beam temperatures of 550-75OoK, McFee and Marcus estimated the energy accommodation coefficient to be 0.7 f 0.1 for K on LiF.

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167

C. SCATTERING OF RAREGASATOMS FROM SOLIDSURFACES I . Scattering from Metallic Single Crystals It appears that the first investigation of scattering from metallic single crystals is the recent work of Smith and Saltsburg (1964,1966). They approached the problem of obtaining clean target surfaces of known crystallographic orientation by employing epitaxially grown Au films that may be deposited continuously during the experiment. This technique, which is described briefly in Section III,D, has also been used in studies of scattering from Ag single crystals (Saltsburg and Smith, 1966; Saltsburg et al., 1967). The modulated molecular beam technique (Section II1,C) has been employed in all of these investigations. We shall consider the investigations of scattering from Ag first because the experimental results are more extensive for Ag than for Au. The scattering patterns shown in Fig. 8 were all obtained for an incident angle of Oi= 50", a beam temperature of Tg= 300"K, and a target temperature of T, = 560°K. We see that the scattered beam signal A attains a maximum value A = A' at an angle 0, = 0,' that is not necessarily equal to the specular angle 8, = Oi. Throughout this discussion we shall use

A0 = Bi - 0,'

(4.2)

as a measure of the deviation of a given scattering pattern from specular. We also see in Fig. 8 that the patterns exhibit different degrees of dispersion (or broadening); a convenient measure of dispersion is the angular width of the pattern at half-maximum. Although the He pattern (Fig. 8) appears to be highly specular, the patterns for the other gases are neither specular nor diffuse. The data indicate that both the deviation from specular (Ad) and the dispersion of these nondiffuse patterns increase with the molecular weight of the beam gas. We shall soon see, however, that these tendencies are not true for all scattering conditions. The dependence of the scattering pattern on T,, the temperature of the incident beam, is illustrated in Fig. 9 for Xe on Ag. It is apparent that, in this case, both A' and 0,' increase with increasing T, . In fact, 0,' increases to the extent that the sign of A0 changes from positive to negative as Tgis varied from 300" to 2500°K. (This tendency is also observed for Hg on NaCl in Fig. 6.) Following the suggestion of Saltsburg and Smith (1966), we shall use the term subspecular to denote a scattering pattern having its maximum located between the target normal and the specular angle (i.e., 0,' < Oi,hence A0 > 1); the term supraspecular will denote a pattern having its maximum located between the target tangent and the specular angle (i.e., 0,' > O i ,hence A0 < 1).

Robert E. Stickney

168

T,= 560° K 0.04 (He) 0.19 (Nel 0.37( A r ) 1.22 (Xe)

9

I

0

I

I

I

I

I

I

I

I

10

20

30

40

50

60

70

, \

80

90

SCATTERING ANGLE, 8, [DEG)

FIG.8. Experimental patterns for rare gases scattered from the ( 1 1 1) plane of Ag (Saltsburg and Smith, 1966).

The angular position of the maximum of the scattering pattern, gr', also depends on B j , the angle of incidence, as shown in Fig. 10. We see here that Smith (1966) show that experimental data for Ar on Ag are similar, qualitatively, to the data shown in Fig. 10. The dependence of the amplitude or magnitude of the maximum of the scattering pattern, A ' , on and TBis illustrated in Fig. 11 for Ar on Ag. Similar results were obtained for Xe on Ag. The data for Tg= 1090°K in

SCATTERING FROM SOLID SURFACES

169

Fig. 11 lead us to the conclusion that A' is not a monotonic function of Bi. At this point it is important to note that the detector employed by Saltsburg and Smith in their investigation of scattering from Ag was a stagnation-type ionization gauge (Section 111,B). Hence, the magnitude of the signal A in Figs. 8, 9, and 11 is proportional to the intensity (or flux) of the scattered atoms, and we denote this by adding the subscript f to A. The detector employed by Smith and Saltsburg (1964, 1966) in their investigation of scattering from Au, on the other hand, was a mass spectrometer with a throughflow ionization gauge (Section 111,s);in this case, the magnitude of A is proportional to the number density of scattered atoms, and we shall denote this by adding the subscript d to A . It follows, therefore, that the scattering patterns measured in these two investigations cannot be compared on a quantitative basis. As mentioned briefly in Section III,C, Yamamoto and Stickney (1967) have shown that the constant of proportionality relating A, to the number density (or A , to the flux) is not truly constant because it depends on the form of the velocity distribution function, and this effect becomes more

.

I

Xe on Ag ( 1 1 1 )

I

I

I

I

I

I0

20

30

40

I

50

I

I

I

60

70

80

' 3

SCATTERING ANGLE, Br (DEG)

FIG.9. Experimental patterns for Xe scattered from the (1 1 1 ) plane of Ag as a function of the gas temperature T, (Saltsburg and Smith, 1966).

Robert E. Stickney

170

t3

w n Q

Q

w n

ANGLE OF INCIDENCE,

ei ( DEG)

FIG. 10. Dependence of AO, the deviation of the angular position of the maximum of the scattering pattern from the specular angle, on gas temperature and angle of incidence for Xe scattered from the (111) plane of Ag (Saltsburg and Smith, 1966).

serious at high modulation frequencies. A rather low modulation frequency of 100 cps was used in both the Au and Ag investigations described here. Although the experimental results for scattering from Au are in many ways similar to the above-mentioned results for Ag, there are several significant differences. The patterns for Au targets shown in Fig. 12 were measured under approximately the same conditions as those for the patterns for Ag targets (Fig. 8), and we see a definite similarity between the two sets of data. When the temperature of the incident beam is increased from 300" to 2550°K with all other conditions remaining the same, however, the He pattern remains essentially unchanged, whereas the patterns for Ne and Ar become specular, as shown in Fig. 13. In the case of Xe, the intensity maximum also shifts to the specular direction, and the presence of a large diffuse component led Smith and Saltsburg (1966) to hypothesize that the pattern consisted of a superposition of diffuse and specular components. The Xe pattern is almost identical to those for the other rare gases if the diffuse component is subtracted from the data. An interesting difference between the Ag and Au data for high

SCATTERING FROM SOLID SURFACES

I

a

I

1

I

I

171

I

I

-

i= 300°K

8.5

I

O

-

[L

6.0-

W

G

-

0

5.5 -

2

2-

5.0

-

X

cl

=

LL

0 W

45-

-

2

4.0

2a

3.5

3.0

‘

0

I

I

10

I

I

I

I

Ar on Aa (I1

I

I

20 30 40 50 60 70 ANGLE OF INCIDENCE, 8i ( DEG)

I

I

80

90

FIG.11. Dependence of the amplitude of the maximum of the scattering pattern on gas temperature and angle of incidence for Ar scattered from the (1 11) plane of Ag (Saltsburg and Smith, 1966).

beam temperatures is that the patterns for Ag targets tend to be supraspecular, whereas the patterns for Au targets seem to approach a common limit of specular reflection.I3 Smith and Saltsburg also investigated the effects of surface contamination on the scattering patterns for Au targets. Additions of H,O and CO to the background gas resulted in increased dispersion of the patterns of all beam l 3 The term “specular reflection” is used here to denote the fact that 6,‘ inferring that the energy accommodation is necessarily zero.

=

6 , without

Robert E. Stickney

172

0

20

40

60

80 90

SCATTERED ANGLE, e, (DEG)

FIG.12. Experimental patterns for rare gases scattered from the ( 1 1 1 ) plane of Au (Smith and Saltsburg, 1966).

gases; additions of N, , 0 2 ,and C2H2had little effect. Contamination also caused the intensity maxima to be displaced toward the surface normal. Similar tests were performed with Ag targets, and the results indicate that contamination effects are much less significant for Ag than for Au. In fact, Saltsburg and Smith (1966) found that it was not essential to deposit Ag continuously during an experimental run. There is no direct proof, however, that the degree of contamination of the Au and Ag surfaces was negligible during the scattering experiments, even when the deposition was continuous. Hence, we cannot conclude with complete certainty that Saltsburg and Smith’s data represent scattering from clean surfaces. 2. Scattering from Polycrystalline Metals

a. Nickel. Before the scattering investigations described in the preceding section, Smith and Fite (1963) used essentially the same apparatus to study scattering from polycrystalline Ni targets. This work, which was subsequently

SCATTERING FROM SOLID SURFACES

173

extended by Smith (1964), provides an interesting illustration of the sensitivity of the scattering pattern to the composition, and resulting structure, of the target surface. Smith (1964) investigated the scattering of room-temperature beams of He, Ne, Ar, Kr, and D, from high-purity Ni. Initial measurements showed that the scattering was quite diffuse for the entire temperature range, -300" to 1500°K. Nondiffuse patterns were obtained, however, after a substantial amount (-0.3 % by weight) of carbon was introduced into the target by heating it in C,H, . These nondiffuse patterns were observed only when the target temperature was between 425°K and 1175°K; the scattering patterns were essentially diffuse for temperatures either above or below this range. The transition from diffuse to nondiffuse scattering was reversible and reproducible. Although a definite explanation of the temperature dependence of these data has not been found, some insight into the problem may be gained

Au (111)

8, 50' K Ts = 600' K Tg N 2550'

c

0.02 (He)

0.67 ( X e )

)! He,Ne,Ar

0

20

40

60

SCATTERED ANGLE,

8090

4 (DEG)

FIG. 13. Experimental patterns for rare gases scattered from the (111) plane of Au (Smith and Saltsburg, 1966). The symbols are defined in Fig. 12.

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Robert E. Stickney

from the results of low-energy electron diffraction studies of the effects of C and O2 on the structure of single-crystal Ni surfaces (MacRae, 1963). It appears that the C-0-Ni surface structure may depend strongly on temperature but in a highly reversible manner. The detector employed by Smith (1964) enabled him to determine, approximately, the dependence of energy accommodation on the angular position of the scattered particles. [See Smith (1965) for a more detailed description of this detector.] As would be expected, these results show that the degree of energy accommodation is lower in the direction of the scattering maximum than in the direction of the surface normal. Results of a supplementary experiment indicated that the probability of a molecule being scattered diffusely increases as its initial (i.e., incident) speed decreases. It is important to note, however, that Smith's measurements show that a diffuse scattering pattern is not necessarily indicative of a high degree of energy accommodation. In fact, the average14 value of the energy accommodation coefficient for Ar scattered nondiffusely from carbonized Ni was slightly higher than that measured under similar conditions for nearly diffuse scattering from high-purity Ni. The Ni target employed by Smith was polycrystalline with an average grain size of -0.1 mm after heating in vacuum. It was not expected that the surface was completely free of adsorbed contaminants such as oxygen, but supplementary tests proved that the scattering patterns are not sensitive to substantial increases in the pressures of various residual gases. Aside from the anomalous characteristics described above, the main features of the experimental results for scattering from carbonized Ni targets are similar to those associated with scattering from Ag and Au targets. That is, for subspecular scattering, the angular positions of the maxima of the scattering patterns deviate more from the specular direction as either the target temperature or the molecular weight of the beam gas is increased. In a more recent investigation of the scattering of Ar from high-purity polycrystalline Ni targets, Hinchen and Shepherd (1 967) observe nondiffuse patterns without carbonizing the target. [A more detailed account of this investigation has been presented by Hinchen and Foley (1965).] Since this does not agree with the results obtained at General Atomic (Smith and Fite, 1963; Smith, 1964), we see that the experimental apparatus and techniques of this field have not yet reached the level where the scattering data are independent of the laboratory. In this particular case the lack of agreement may stem from the fact that the pressure level of contaminants in the General Atomic apparatus exceeds that in the United Aircraft apparatus by at least an order of magnitude. l4 More specifically, the AC measured for fixed conditions (8, = a", T, Y 300"K,and T, 2 725°K) was averaged over 8, from -0 to 80".

SCATTERING FROM SOLID SURFACES

175

b. Platinum. Two research groups, one at Oak Ridge and the other at United Aircraft, have observed nondiffuse scattering for polycrystalline Pt targets. (The principal features of these targets are described in Section 111, D.) Since a greater variety of gases has been studied at United Aircraft (Hinchen and Foley, 1965, 1966; Hinchen and Shepherd, 1967) than at Oak Ridge (Datz er al., 1963; Moore et al., 1966), we shall concentrate on the United Aircraft experiments. Both groups employed the modulated beam techniques described in Section 111, C. Although there is evidence that the Pt targets were free of gross contamination, we cannot conclude that the effects of surface contamination on the scattering pattern were completely negligible. Using a density-sensitive detector, Hinchen and his co-workers at United Aircraft have conducted a detailed investigation of the scattering of He, Ne, Ar, and Kr from Pt. The resulting patterns, several of which are presented in Section V, are similar to those for scattering from Ni (Section IV,C,2) and from Ag and Au targets (Section IV,C,l). Hence, we shall consider here only the unique features of these experimental data. As mentioned in Section lII,C, the unusually high modulation frequency (3600 cps) of the United Aircraft apparatus is advantageous for measuring the dependence of the phase of the scattered atom signal on the principal variables, O i ,Tg, and T, . Since the phase is a function of the mean speed’ or “effective temperature” of the scattered atoms, it follows that these data may be used to estimate the magnitude of AC, the energy-accommodation coefficient, for various test conditions. The results shown in Fig. 14 illustrate

FIG. 14. Energy accommodation coefficients of rare gases scattered from polycrystalline Pt (Hinchen and Foley, 1966). l S The appropriate mean speed to be used varies with the type of detector (Yamamoto and Stickney, 1967).

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Robert E. Stickney

the dependence of the AC of He, Ne, Ar, and Kr on T,, the temperature of the Pt target, for the conditions O i = 67.5" and T, = 298°K. In each case, the detector was positioned at the angle 8,' corresponding to the maximum in the scattering pattern. Notice that the AC increases with the molecular weight of the beam gas and appears to be relatively independent of T,. Both of these characteristics are in general agreement with the characteristics of data obtained by the thermal-conductivity cell method (Section II,B,l). Although the absolute magnitude of the AC data in Fig. 14 may be inaccurate [see Section III,C and Yamamoto and Stickney (1967)1, these results illustrate, qualitatively, the characteristics of the scattering process. In a subsequent investigation of the dependence of A8 [Eq. (4.2)] on €Ii, Tg, and T , , Hinchen and Shepherd (1967) observed that Pt, like Ag (Section IV,C,l), may produce supraspecular patterns for Ar. We shall postpone the discussion of these results to Section V. These experiments also include a comparison of the scattering patterns for a polycrystalline Pt target with corresponding patterns for a single-crystal (1 11) Pt target. At present, the comparison is limited to the case of Ar shown in Fig. 15. The outstanding feature of these data is that the dispersions of the patterns for both targets are approximately equal, thereby furnishing additional evidence to support the assumption that a polycrystalline Pt target may be considered as a single crystal because of its large, smooth grains (Section 111,D). Hinchen and Shepherd conclude that the dispersion in these scattering patterns is produced primarily by thermal effects, rather than by surface roughness. This conclusion is strongly supported by the fact that, for the nondiffuse case, the majority of the scattered atoms are detected in the vicinity of the plane defined by the incident beam and the target normal. That is, the three-dimensional scattering patterns are not axially symmetric about the direction of the maximum. Datz et al. (1963) restricted their attention to two gases, He and D, , and found that the nondiffuse patterns for both gases degraded to diffuse patterns when the target temperature was reduced below -600°K. Since it was suspected that this degradation resulted from contamination of the target surface by residual gases, they investigated the effects of various gases on the scattering patterns. An 0, pressure of - 4 x Torr produced diffuse patterns for He and D, over the entire range of target temperature ( - 300°K to 1500°K). [Similar results were obtained by Hinchen and Foley (1965) for the scattering of rare gases from Pt; the nondiffuse patterns were restored by heating the target in vacuum.] More recently, Moore et af. (1966) reported results indicating that carbon impurities may exist on Pt surfaces. (The effect of carbon impurities on scattering from Ni targets was discussed in the preceding section.) The following treatment was employed to reduce the carbon impurities: the Pt target was cycled for several minutes between 900°K and 1600°K in a

SCATTERING FROM SOLID SURFACES

177

relatively high pressure of O2(mm Hg) and then maintained at 1300°K at mm Hg throughout the scattering measurements. The resulting He scattering patterns are anomalous, in that two maxima appeared, both of which are supraspecular (i.e., both are located between the specular angle and the surface tangent). Moore et al. (1966) stated that these results were unexplainable at the present time and that they plan to continue the investigation after replacing the polycrystalline specimen with a single crystal. It is important to mention that Hinchen and Foley (1965) were unable to reproduce these peculiar patterns in their apparatus.

4 0'

5 0'

7 00

8 00 9 OQ

FIG.15. Comparison of Ar scattering patterns for polycrystalline and single-crystal Pt (Hinchen and Shepherd, 1967).

c. Tungsten and Tantalum. Although Pt targets have been employed in the majority of the scattering experiments conducted at United Aircraft, a limited amount of data was obtained for W and Ta targets in order to demonstrate that nondiffuse scattering may be expected for any surface that is sufficiently smooth and free from gross contamination. Hinchen and Foley (1965) report that nondiffuse patterns are obtained for Ar on W and on Ta, even though the targets had not been heated to the high temperatures required to remove oxides from the surfaces. In general, however, these patterns were more diffuse than corresponding patterns for Pt targets.

Robert E. Stickney

178

3. Scattering from Liquids and Nonmetals Experimental data also exist for the scattering of rare gases from liquids and nonmetals. Since these data are, in general, less detailed than those for metallic targets, only a brief tabulation of the various investigations is presented here. (See Table I.) TABLE I SUMMARY OF EXPERIMENTAL INVESTIGATIONS OF SCATTERING FROM LIQUIDS AND NONMETALS

Target

Gas

Comments

Reference

Mica

He

Scattering pattern changed from diffuse Smith and to nondiffuse when T , was increased Saltsburg from -300°K to -500°K (1 964).

Alkadag (colloidal graphite in alcohol)

Ar

After degassing the target by heating in Smith (1964) vacuum, nondiffuse patterns were observed at all values of T , (- 300°K to

Teflon

Ar (also N2) Small deviations from a diffuse pattern Hurlbut (1959) are observed, especially for grazing incidence (i.e., large 8,)

Glass

Ar (also Nz Small deviations from a diffuse pattern Hurlbut (1959) and air) are observed, especially for grazing incidence

Indium (liquid and solid)

Ar (also N2)Diffuse patterns

1300°K)

Gallium (liquid) N2 Synthetic coatings

Nondiffuse patterns

Ar (also Nz Small deviations from a diffuse pattern and air) were observed

Hurlbut and Beck (1959) Hurlbut and Beck (1959) Jawtusch (1963)

D. SCATTERING OF MOLECULES FROM SOLID SURFACES Gas-solid interactions involving molecules, rather than rare-gas atoms, are considerably more complex in general, because of the reactivity and the internal degrees of freedom of molecular species. The reactivity increases the probability for adsorption which may lead to dissociation of the molecules (e.g., Smith and Fite, 1963), marked changes in the surface structure

SCATTERING FROM SOLID SURFACES

179

(e.g., MacRae, 1963; Taylor, 1964), and chemical reactions either with the solid itself (e.g., Smith and Fite, 1963) or with other species adsorbed on the surface (e.g., Bond, 1962). As mentioned in Section II,B,l, the internal degrees of freedom of molecules provide for additional modes of energy transfer.

I . Hydrogen and Deuterium The experimental investigation of hydrogen condensation by Brackmann and Fite (1961) represents one of the first demonstrations of the usefulness of modulated beam techniques to studies of gas-solid interactions. Beams of H and H, at a temperature of c 80°K were obtained from a Pyrex tube cooled by liquid nitrogen; the dissociation of H, to H was accomplished in the Pyrex tube by means of a radiofrequency electrodeless discharge. The copper target was attached to a liquid-helium reservoir so that temperatures ranging from -4°K to 300°K could be obtained. A mass spectrometer was used to detect the scattered beam, thereby enabling one to distinguish the H signal from the H2 signal. In these experiments the target temperature was sufficiently low to cause various components of the residual gas to condense on the target surface. Hence, the incident beam molecules interacted with the surface of the condensate rather than with the copper substrate. Measurements by Brackmann and Fite indicate that, although both H and H, scatter diffusely from condensate surfaces, complete energy accommodation did not occur for either species. As the target was cooled, the mass spectrometer signal, corresponding to H, ,decreased suddenly at a target temperature of 15"K, while the signal corresponding to H showed a similar decrease at -4°K. The exact values of these condensation temperatures depended strongly on the composition of the condensate surfaces. For example, the condensation temperature of H2 increased significantly when H 2 0 was permitted to condense simultaneously on the target. (It was observed that H,O had a similar effect on the condensation of He and of 0, .) It appeared that the composition of the condensate also influences the probability for an H atom to be scattered from the surface without recombination, this probability being higher for an H, condensate than for a condensate of air. Upon completion of the study of the interaction of H2 with a cold solid, the General Atomic group turned to the opposite extreme of the interaction of H, with a hot (-2500°K) solid. Although the fundamental problem of thermal dissociation of H, at a hot tungsten surface had been considered previously by many researchers using several different techniques, Smith and Fite (1962) were the first to apply modulated beam techniques. The apparatus was essentially the same as that used for the condensation study, except that the copper target was replaced by a sheet of tungsten, 0.0025 x 2.5 x 3.8 cm,

-

Robert E. Stickney

180

mounted between water-cooled electrodes that provided a means of passing an ac heating current directly through the target. The scattering patterns shown in Fig. 16 were obtained by Smith and Fite for an incident beam of H, with TB= 300"K, Bi = 60", and T, = 2500°K. 30"

20°

10"

0

loo

20°

3 0'

40"

50'

70"

80°

goo

FIG.16. H and H2scattering patterns resulting from the interaction of an H, beam with a polycrystalline W target (Smith and Fite, 1962). The magnitudes of the H and H2signals are not to be compared because the H, scale has been reduced for convenience in this figure.

Notice that the pattern for H is diffuse (cosine), whereas the pattern for H, is nondiffuse. This may be explained by the fact that a fraction of the incident molecules are " trapped" (Section II,B,l) on the surface by dissociative adsorption and attain thermal equilibriumI6 with the solid before desorbing as atoms; the remaining fraction of those that are not trapped is scattered as molecules with incomplete energy accommodation (AC N 0.07). On the basis of their experimental results, Smith and Fite report that, for tungsten temperatures above 2500"K,the probability of dissociation approaches a constant value of 30 %. A comparative study of the scattering of H, , D, , and He from the (1 11) plane of Ag was performed recently by Saltsburg et al. (1967). By decreasing both the angular divergence of the incident beam and the acceptance angle of the detector, the resolution of the General Atomic molecular beam apparatus was improved in this study so that the features of the scattering

-

l6 It may well be that a state of complete thermal equilibrium is not achieved when the mean adsorption lifetime is extremely short (i.e., comparable to the period of lattice vibrations).

SCATTERING FROM SOLID SURFACES

181

patterns could be observed in greater detail. The detector was a stagnationtype ionization gauge (Section 111,B). From the results of this investigation, Saltsburg el al. state that the scattering patterns for H, , D, ,and He seem to be composed of three components: (i) a truly specular component; (ii) a dispersed specular component, i.e., the maximum of the component is located at the specular angle, but the dispersion exceeds that expected for specular reflection from a perfectly smooth surface. [This type of component was also observed by Crews (1962) for the case of He scattered from LiF] ; (iii) a nonspecular component having the amplitude and angular position of its maximum strongly dependent on T,, the beam temperature. (It appears that this component is similar to the nondiffuse component described in Sections IV,B and IV,C for the scattering of heavy atoms from solid surfaces.) The effects of these components on the H, pattern may be seen in Fig. 17. The patterns for He and D, are similar to those for H, except that the truly specular component is dominant for He whereas it is nearly undetectable for D, . Saltsburg et al. estimate that the fraction of scattered molecules contained in the truly specular component is less than 5 % for He and H, ,and less than 0.5 % for D, . Hence, it is not surprising that the

-

6

m

t

z

3

>

a

5

2m a a

4

0

20

40 SCATTERING ANGLE,

60

8,

(DEG)

FIG.17. Experimental pattern for H, scattered from the (111) plane of Ag as a function of the gas temperature T, (Saltsburg et al., 1967).

182

Robert E. Stickney

specular component is often undetectable in experimental data for the scattering of He or H, from polycrystalline or contaminated targets (e.g., Smith and Fite, 1963; Datz et al., 1963; Hinchen and Foley, 1966). In fact, even in the present case of D, scattered from (111) Ag, the nonspecular component overshadows the specular component to the extent that the maximum of the pattern does not always occur at the specular angle. We may wonder why the specular component for D, is much smaller than that for either H, or He. The answer proposed by Saltsburg et al. (1967) is based on the assumption that the gas-solid interaction may produce changes in the rotational state of Dz . This assumption is supported by the fact that the energy associated with the rotational transition of ortho-D, is one-half that of para-H, ,and is approximately equal to the energy ofthe most energetic phonon of Ag (according to the Debye theory). Additional support is gained from the energy accommodation data of Schafer and Schuller (1963) which indicate that, for interactions with Pt surfaces, rotational transitions are much more probable for D, than for H 2 . Saltsburg et al. (1967) also report that, for the case of H2,D, , and He scattered from (111) Ag, the dispersion of the out-of-plane” scattering pattern is less than that of the in-plane pattern. (As mentioned in Section IV,C,2b, a similar result was obtained for scattering from polycrystalline Pt targets.) Since no changes in the in-plane or the out-of-plane patterns were observed when the target was rotated about its normal, it appears that the nonspecular components are not caused by high-order diffraction effects. The out-of-plane patterns measured in the transverse plane passing through the in-plane maxima were found to be quite insensitive to changes in T,, the beam temperature. Precise measurements indicated, however, that the dispersion of these out-of-plane patterns increased slightly with T, for He and H2 ,whereas the opposite trend was observed for D2. The scattering of H, from Ni and D, from Pt have been investigated, respectively, by Smith and Fite (1963) and by Datz et al. (1963). In both experiments the targets were polycrystalline and the patterns were nonspecular. Since these results are similar to those for heavier atoms (Sections IV,B and IV,C), we shall not consider them here. 2. Atmospheric Gases

Although the scattering studies performed at United Aircraft have generally beenlimited to the rare gases, Hinchen and Foley (1965)recently conducted an exploratory investigation of N, and 02.The data arenoteworthy becausethey l7 The “out-of-plane” scattering pattern is measured by moving the detector in a plane that (a) is perpendicularto the plane defined by the incident beam and the target normal, and (b) passes through the point of intersection of the beam and the normal.

SCATTERING FROM SOLID SURFACES

183

show that the major constituents of the earth's atmosphere scatter nondifusely from several of the metals that were investigated. This result is especially interesting because both N, and 0, molecules (i) adsorb readily on many metals, (ii) have internal degrees of freedom which may be excited by the gas-solid interaction, and (iii) since they are not spherically symmetric, ma.y have an infinite number of orientations relative to the solid surface. One or more of these factors should tend to make the N, and 0, scattering patterns more diffuse than those for the rare gases. Patterns reported by Hinchen and Foley (1965) for N2 scattered from polycrystalline Pt are shown in Fig. 18. Notice that the dispersions of these patterns

40'

4 0"

50"

50'

60°

60'

70"

70°

80D

800

goo

900

FIG.18. Experimental patterns for N, scattered from polycrystalline Pt as a function of the target temperature T,(Hinchen and Foley, 1965).

are approximately the same as those for Ar on Pt (see Fig. 15). Another similarity between the N, data and the rare gas data is the fact that Or', the position of maximum, decreases with increasing T, . Hinchen and Foley also report a similarity in the dependence of 8,' on Oi.There is, however, a significant difference: Ad', the magnitude of the maximum of scattering pattern, depends strongly on T, in the case of N, , whereas it depends only weakly on T,for the rare gases. As illustrated in Fig. 18, the magnitude of Ad' increases with T, in the range 521-1075°K. One possible explanation for this increase is that the adsorption of N, on the target decreases with increasing T,,

184

Robert E. Stickney

thereby changing both the nature of the surface and the probability for trapping an incident N, molecule. We see, however, that A,' decreases drastically if T, is increased to 1306°K. Hinchen and Foley state that this transition is irreversible; that is, the nondiffuse patterns that were observed in the range 521" IT, I1075°K before the transition could not be reproduced even after heating the target to 1500°K in vacuum. It was concluded from microscopic examinations of the target that a general attack of the surface had occurred. Hinchen and Foley suggest that this attack may be the formation of platinum nitride, although they were unable to find published references to establish the existence of this compound. Another possibility is that thermal etching resulted in a roughened surface (Hinchen and Malloy, 1967). The data for O2 scattered from polycrystalline Pt also indicate the occurrence of a chemical reaction of the beam gas with the target material. For an incident beam temperature of T, N 1190"K, the scattering patterns were nondiffuse when T, 2 958°K. At lower values of T,, the patterns changed with time from nondiffuse to diffuse. For example, this transition occurred in 13 hr when T, = 626°K. The transition occurred more quickly, and at higher values of T,, when the beam temperature was reduced to 298°K. Although a quantitative explanation was not attempted, Hinchen and Foley suggest that the diffuse patterns are associated with the formation of an oxide at the surface of the Pt target. The rate of formation of oxide is a function of the temperature-dependent rates of the principal processes : adsorption, reaction, and decomposition. Since the nondiffuse patterns may be reproduced after heating the target to 1300°K for several hours in vacuum, it appears that this procedure is effective in removing the oxides from the surface. By using a radio-frequency discharge to dissociate the O2 gas in the molecular beam source, Hinchen and Foley were able to extend their study to include the scattering of 0 from Pt. A mass spectrometer was employed as the detector in this experiment so that both the 0 and the 0,signals could be measured. With the target at 298"K, surface recombination was negligible, although the scattering pattern for 0 was diffuse. At higher target temperatures (e.g., 1100"K),however, it appeared that the surface acted as an extremely effective catalyst for the recombination of atoms to molecules. Hinchen and Foley also investigated the scattering of O2 and N, from polycrystalline Ta, W, and Ni. In all casks, the patterns were diffuse when the target temperature was not sufficiently high to prevent gross contamination of the surface. By elevating the target temperature, however, the N, patterns became nondiffuse for all three target materials. The O2 patterns were observed to be much more diffuse than those for N, as we would expect, since 0,is known to react with all three materials. In these exploratory experiments the target temperature was not varied over a wide enough range to provide for a comprehensive picture of the N, and 0,scattering processes.

-

N

N

SCATTERING FROM SOLID SURPACES

185

3. Ammonia and Methane

Saltsburg and Smith (1966) have reported a limited amount of data on the scattering of NH, and CH, from the (1 11) plane of Ag. These gases were selected because their molecular masses are approximately equal to that of Ne, while their heats of adsorption (AH) on Ag differ significantly from that of Ne. (AHNHn x 8 AHNe;AH,,, FZ 4 AHNe.) Hence, a comparison of the scattering patterns of these three gases (Fig. 19) may serve as an indication of the effect of the depth of the potential well associated with the gas-solid interaction. (The heat of adsorption is assumed to be an approximate measure of the potential well depth.) As expected, the dispersions of the patterns shown in Fig. 19 increase with the magnitude of AH.

g---I

I

CH,,

NH3,8 Ne on A g ( 1 1 1 )

6;=50° p

Tg=3OO0K

Ts = 560"K 0.15,0.16, @I 0.19

SCATTERING ANGLE,

er (DEGI

FIG. 19. Experimental patterns for CH,, N H 3 , and Ne scattered from the ( 1 1 1 ) plane of Ag (Saltsburg and Smith, 1966).

186

E.

SUMMARY OF

Robert E. Stickney

EXPERIMENTAL DATA

The experimental results described in the preceding sections indicate that there are many similarities in the scattering patterns of different gas-solid systems. It is remarkable that these similarities exist for such a wide variety of gases (e.g., the rare gases, the atmospheric gases, certain vapors, H2 and D, , etc.) and solids (e.g., ionic and metallic, monocrystalline and polycrystalline, with different atomic masses and surface conditions). The main characteristics that are common to the majority of the scattering patterns are summarized below. Characteristic ( I ) : a(A8)/aTgI 0. An equivalent statement is: aO,'/aTg 2 0. Characteristic (2) :a(AO)/aT,2 0. An equivalent statement is: aO,'/aT, I 0. (This characteristic is invaIid if the composition or structure of the target surface varies with T, .) Characteristic (3): dO,'/aOi 2 0. (This characteristic cannot be expressed conveniently in terms of A0 because, as illustrated in Fig. 10, a(A8)/aOimay be either positive or negative.) Characteristic (4): d(A8)/dmg> 0 when A8 > 0; a(A8)/amg< 0 when A0 < 0. An equivalent statement is: a8,'/amg < 0 when 8,' < 8,; ae,l/am, > 0 when 8,' > Bi . (It is assumed that AH, the heat of adsorption, is held constant in these partial derivatives.I8) Characteristic ( 5 ) :The dispersion of the scattering patterns increases with AH and/or with mg.18 Characteristic (6): Supraspecular patterns (i.e., 8,' > 8,) are most likely to occur when both of the following conditions are met: (a) Tg> T, and (b) Bi -4 90". Characteristic (7): Nondiffuse scattering is most likely to occur from target surfaces that are smooth and free of gross contamination. Characteristic (8) : The dispersion of nondiffuse scattering is greater for the in-plane pattern than for the out-of-plane pattern. (That is, the probability is low that molecules will be scattered out of the plane defined by the incident beam and the target normal.) We should not assume these characteristics to be universally true. For example, we would not expect them to be valid when quantum effects are l 8 Although it appears that the nature of the scattering pattern depends on both m, and AH (see, for example, Saltsburg and Smith, 1966), it is difficult to determine the effects of each, individually, because A H usually changes whenever we change m, . Hence, characteristics (4) and (5) are not as definite or as general as (11, (2), and (3).

SCATTERING FROM SOLID SURFACES

187

significant (e.g., at low temperatures) or when the energy of the incident beam is hyperthermal. It should be emphasized that these characteristics are based primarily on the existing experimental data for the rare gases in the thermal energy range.

V. Classical Theories of Atomic and Molecular Scattering from Solid Surfaces We shall review here only those theories of gas-solid scattering that are based on classical mechanics ;those based on quantum mechanical considerations are reviewed by E. Beder in this volume. Furthermore, we shall concentrate on the simple hard-cube model, partly because of the author’s familiarity with this model and partly because, at present, theories based on more complex models do not yield analytical solutions that are convenient for comparison with existing experimental data. Also, this subject has been reviewed recently by Trilling (1967b) and Hurlbut (1967).

A. THEHARD-CUBE THEORY As discussed by Logan and Stickney (1966), the hard-cube model was devised as an attempt to formulate a simple theory that would describe, qualitatively, the general characteristics (Section IV,E) of the existing experimental data. Hence, a primary objective was to formulate a theory to account for a majority of the obvious experimental variables (Oi, T B ,T,, m,, and mJ without becoming so complex that approximate closed-form solutions could not be obtained. Because of this emphasis on simplicity, it is not expected that the theoretical results should agree quantitatively with experimental data. By combining and simplifying several features of the gas-solid models of Baule (I 914) and Goodman (1 965), the hard-cube model provides for a theoretical expression of the physical mechanism suggested, in qualitative terms, by Datz et al. (1963), by Kellogg (1932), and by others.

1. Assumptions and Analysis

Since the basic assumptions of the hard-cube model will be examined in detail in Section V,A,3, it suffices to list them briefly and then proceed with the development of the analysis. Assumption ( I ) . The gas-solid intermolecular potential is such that the repulsive force is impulsive and the attractive force is zero (i.e., both the gas atoms and the solid atoms may be considered to be rigid elastic bodies).

Robert E. Stickney

188

Assumption ( 2 ) . The gas-solid intermolecular potential is uniform in the plane of the surface (i.e., the surface is perfectly “smooth”). As a result, collisions do not change the tangential component of the gas-atom velocity. Assumptions (1) and (2) may be represented by the single assumption that the gas atoms behave as hard (i.e., perfectly rigid and elastic) spheres, and the solid atoms behave as hard cubes oriented with one face parallel to the surface. This “ hard-cube ” model is illustrated in Fig. 20.

/

\

/

‘ n

/

f

I I FIG.20. Schematic representation of the “ hard-cube’’ model.

Assumption (3). The solid is represented by a collection (i.e., a gas) of hard cubes confined within the volume of the solid by a square-well potential. Each gas atom experiences a single collision with one of these hard cubes which is considered to be a free particle (i.e., uncoupled from the other solid atoms) during the interaction.” l9 This statement of assumption (3) differs slightly from the original statement given by Logan and Stickney (1 966) because, as we shall see in Section V,A,3, the model is not expected to be valid when a gas atom experiences more than one collision with a solid atom.

SCATTERING FROM SOLID SURFACES

189

Assumption ( 4 ) . The solid atoms (Le., hard cubes) have a Maxwellian velocity distribution. The analysis may be described by referring to Fig. 20. [The analysis presented here follows the simplified formulation of Logan et al. (1967).] A gas atom of mass m, , speed ui,and angle of incidence Bi collides with a solid atom of mass m, and speed ui and is scattered with speed u, at angle 8,. Since the tangential component uti is unchanged according to assumption (2), only the normal components of u and Y are affected by the collision. An expression for u,,, , the normal component of the scattered atom, may be derived simply from classical mechanics (see Appendix) :

where p is the mass ratio, mg/ms.A second relation for u,, follows from the geometry shown in Fig. 20: u,,, = u,,(cot e,lcot

ei).

(5.2)

Equations (5.1) and (5.2) may be combined to obtain an expression for u n i , the speed that a solid atom must possess if it is to scatter a gas atom of speed ui and incidence angle Bi into angle 0,:

(5.3)

uni= B l u i , where l + P . B , = -sin Bi cot 8, - -pcosei. 2 2

(5.4)

Since the derivative of vni with respect to 8, will be needed in the analysis, we present it here in the form

-au,,, = - u iaB, =Bu where

ae, ae,

2 i?

(5.5)

Because of the nature of the hard-cube model, the only component of vi that is of importance in this analysis is the normal component uni . Hence, without loss of generality, we may consider the motion of the solid atoms to be onedimensional along the direction of the normal. It follows that V R ,the relative speed of the gas and solid atoms, is given by

+ u , ~= (COSO i + B,)ui,

V R = uni

where Eq. (5.3) has been substituted for u n i .

(5.7)

Robert E. Stickney

190

Consider the rate at which gas atoms having speeds within du, of u icollide with solid atoms having speeds within dv,, of vni. This differential collision rate is proportional to V, ,the relative velocity, multiplied by the probabilities (i.e., distribution functions) that the atoms have speeds ui and v,,: d2R

-

VRF(u,)G(vni)dui dvni.

(5.8)

The appropriate forms of the Maxwellian distribution functions are (5.9)

and G(v,,) dvni= (?)‘I2 2nkTs

exp(&-) - m,v:, dv,,,.

(5.10)

Equation (5.3) may be used to change the variables in Eq. (5.8) from ui and vni to uiand 8,:

(5.1 1) We now obtain an expression for the scattering pattern by integrating Eq. (5.11) over ui:

(5.12) This expression has been normalized by dividing through by the intensity of the atomic beam which, in the present case, is numerically equal to the normal component of the mean speed,

(5.13) Substituting the appropriate equations in Eq. (5.12), we obtain, upon integration, 1 dR m,T, ‘ I 2 m,T, - 512 _ _ -- p , ( 1 B, sec Oi) [ I -B12] . (5.14) iinido, m,T, m,Ts Here we have an expression for the scattered flux as a function of 0, for given values of 8, and m,T,/m,T,. [As may be seen in Eqs. (5.4) and (5.6), both B, and B2 are functions of 8, and O i .] Logan et ul. (1967) have also derived a similar expression for the density of the scattered molecules. Since they report that the qualitative features of the density and flux patterns are identical under most conditions, we may use either one in our comparison of theory with experiment in the following section. In the same paper, Logan et uI. derive expressions for the velocity distribution and the mean speed of the scattered atoms.

+

[-]

+

191

SCATTERING FROM SOLID SURFACES

The analysis presented here is valid only when the mass ratio p is small. The discussion of this limitation of the hard-cube model is deferred to Section V,A,3 where we shall examine the basic assumptions of the model. 2. Comparison of Theory with Experiment

In Figs. 2 1-23, scattering patterns measured experimentally by Hinchen and Foley ( I 966) are presented beside patterns predicted by the hard-cube theory for corresponding conditions (Logan and Stickney, 1966). These figures illustrate characteristics (2), (3), and (4) of the experimental data (see Section IV,E). It is apparent that, in the cases selected here for comparison, the theoretical results agree qualitatively with the experimental patterns. By means of an approximate analysis, Logan and Stickney (1966) have shown that the characteristics of the hard-cube model are in general agreement with characteristics (1) through (4) of Section IV,E. 0"

10"

20"

30"

0"

10"

20"

30" 40"

5 0" 6 0" 70"

8O0 9 0" (a)

(b)

FIG.21. Comparison of the dependence of experimental and theoretical scattering patterns on T,, the target temperature. (a) Experimental results for Ar on Pt (Hinchen and Foley, 1966). (b) Corresponding theoretical results based on the hard-cube model (Logan and Stickney, 1966).

As may be seen in Eq. (5.14), the hard-cube theory predicts that the dimensionless parameter (m,T,/m,T,) should serve as a general parameter for correlating experimental data obtained for different temperature ratios and mass ratios. (This parameter represents the ratio of the mean kinetic energies of the gas and the solid atoms.) Hence, a. more stringent test of the validity of the hard-cube theory is to determine if this parameter is successful in correlating a wide variety of experimental data. In a very limited sense, such tests

Robert E. Stickney

192 0"

lo"

20"

30"

0"

10"

20"

30" 4 0"

50" 60"

70" 80"

90" (b)

(0)

FIG.22. Comparison of the dependence of experimenta and theoretical scattering the angle of incidence. (a) Experimental results for Ar on Pt (Hinchen and patterns on Foley, 1966). (b) Corresponding theoretical results based on the hard-cube model (Logan and Stickney, 1966).

el,

0"

10"

20"

30'

@

10"

20"

30° 40'

50" 60" 70' 80" 9oo (0)

(b)

FIG.23. Comparison of the dependence of experimental and theoretical scattering patterns on the mass of the incident gas atoms. (a) Experimental results for He, Ne, and Ar on Pt (Hinchen and Foley, 1966). (b) Corresponding theoretical results based on the hard-cube model (Logan and Stickney, 1966).

have been performed by Logan et al. (1967) and by Hinchen and Shepherd (1967), and the results are shown in Figs. 24 and 25. The qualitative agreement of theory with experiment is quite satisfactory.

193

SCATTERING FROM SOLID SURFACES

1 [ 1 1 1 1 1 1 1 1

' O t

3

FIG.24. Comparison of hard-cube theory with experimental data. Dependence of fy, the angular position of the maximum of the scattering pattern, on m,r,/m,T, and 0, for Ar scattered from Ag (1 11) (Logan et a/., 1967).

On the basis of the hard-cube model, we may postulate the following physical interpretation of the general characteristics of existing experimental data. It is obvious from Fig. 20 that subspecular scattering results for collisions that increase the magnitude of u,, above the incidence value uni, because Or decreases with increasing (u,,, - u,J. From Eq. (5.1), 2 (5.15) Aun = u nr - u . = (uni - uni), n' l+p and we see that Aun increases with increasing uni and decreasing uni.It follows, therefore, that 0, will decrease when (a) the target temperature is elevated, thereby increasing the probability that uni will be large, (b) the gas temperature is reduced, thereby increasing the probability that uni will be small, and (c) the incidence angle is increased, thereby reducing uniby the factor cos Bi. These characteristics of the hard-cube model are in agreement with characteristics (l), (2), and (3) of the experimental data (Section IV,E).

Robert E. Stickney

194

Ar on Pi Tg =vorioble

TI = 7 1 3 O K tL

=0.2

30

W

W -I

z a a

I

0 LL K

0

FIG.25. Comparison of hard-cube theory with experimental data. Dependence of AO, the deviation of the angular position of the maximum of the scattering pattern from the specular angle, on m,T&n,T, and 8) for Ar scattered from polycrystalline Pt (Hinchen and Shepherd, 1967).

Characteristic (4) follows, in part, from the fact that, for a Maxwellian velocity distribution, the probability that uniwill be large increases with decreasing mass of the gas atom. An analogous statement could be made in connection with the effect of the mass of the solid atoms on the scattering pattern. In a similar manner, a physical interpretation of the characteristics of supraspecular scattering may be obtained. According to the hard-cube theory, supraspecular patterns occur when the net energy transfer from the gas to the solid is positive, and subspecular patterns occur when this net energy transfer is negative. An interesting property of the hard-cube model is that,

SCATTERING FROM SOLID SURFACES

195

under certain conditions, net energy may be transferred from the solid to the gas, even though the temperature of the gas exceeds that of the solid. (This property is illustrated in Fig. 24 for the case Oi = 45" and m,T,/m,T, c 0.2.) Although the energy accommodation coefficient [Eq. (2.4)] is negative in this case, the second law of thermodynamics is not violated because it does not apply to interactions involving a nonequilibrium system such as a directed molecular beam. (This point is discussed in greater detail in Section II,B,l.)

3. Examination of Basic Assumptions As we have shown, the hard-cube model leads to theoretical results that agree surprisingly well with the general, qualitative features of existing experimental data. We suggest, therefore, that this elementary model may provide a valid base for the development of a more exact scattering theory. Assuming this viewpoint, we shall now examine the basic assumptions of the hard-cube model. Assumption ( I ) . The analysis of the hard-cube model is greatly simplified by the assumption that the gas and solid atoms behave as rigid, elastic particles. [As mentioned in Sections II,B,2 and V,B, the inclusion of a more realistic gas-solid intermolecular potential (e.g., the Morse potential or the LennardJones potential) in a scattering theory results in mathematical expressions which, in general, can be solved conveniently only by numerical methods requiring extensive use of an automatic computer.] Because of this assumption, we would expect the hard-cube model to be most valid for gas-solid systems having small heats of adsorption. (See Sections IV,D,3 and IV,E.) Logan et al. (1967) have investigated the qualitative effects of forces of attraction by adding a simple square-well potential to the hard-cube model. The results of this preliminary investigation indicate that an increase in the depth of the potential well causes the scattering pattern to shift toward the normal and to exhibit greater dispersion. Assumption ( 2 ) . Through the assumption that the gas-solid intermolecular potential is planar, the hard-cube model avoids the complexities encountered when the three-dimensional nature of the surface potential is included in a scattering model. This assumption appears to be quite valid for thermalenergy beams because existing experimental data show that the out-of-plane scattering pattern is narrow in comparison with the in-plane pattern (Sections IV,C,2,b and IV,D,l). In addition, Goodman (1967) has shown that a hard-sphere model leads to scattering patterns that are much broader than the experimental patterns, unless the diameter of the collision cross section of the solid atoms is assumed to be considerably larger than the lattice spacing. Since the collision cross section decreases with increasing energy of the incident

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gas atoms, we would expect that for hyperthermal energies the planar potential must be replaced by a three-dimensional potential (Stickney et al., 1967). Logan et al. (1967) have shown that the hard-cube model may be easily modified to include, in an approximate manner, the effect of a three-dimensional surface potential. By assuming that the deviation from a planar surface may be described by a Gaussian distribution, they obtained expressions for both the in-plane and out-of-plane scattering patterns as a function of the mean angular deviation. The results indicate that measurements of the width of the out-of-plane pattern would provide significant information on the topography of the surface potential because the width does not depend strongly on the gas or solid temperatures. Assumption (3). Unlike most of the other classical theories of gas-solid collisions (Section V,B), the hard-cube model is based on the simplifying assumption that a solid atom is essentially uncoupled (i.e., free) from the other lattice atoms during a collision with a gas atom. Although this assumption is valid when the energy of the incident gas atoms is sufficiently large to cause the interaction time to be small relative to the characteristic period of lattice vibrations, there is no direct evidence that it applies in the thermalenergy range. [For further discussion of this point, see Section II,A and Stickney et al. (1967).] On the other hand, the qualitative agreement of the theoretical results with experimental data may be considered as an indirect indication of the validity of this assumption. It seems obvious, however, that a logical refinement of the hard-cube model would be to include the effect of lattice coupling on the motion of a solid atom. The analysis described in Section V,A,l does not account for the fact that a certain fraction of the incident gas atoms will require more than one collision with the surface to reverse the direction of their motion. In effect, the analysis neglects these atoms by excluding them from the scattering pattern for 0 I 6, S 90”. As shown by Stickney et al. (1967), the analysis is satisfactory when p is less than -0.3 because only a small fraction of thegas atomsrequire more than one collision to reverse their direction. This fraction increases with p, however, and the present analysis is not applicable when p 4 0.3. It seems reasonable to expect that gas atoms experiencing more than a single collision with the solid will tend to be scattered more diffusely than those that experience only one collision. Existing experimental data show, however, that the scattering patterns for p 4 0.3 (e.g., Hg on LiF and Xe on Ag) are only slightly more diffuse than those for p I 0.3, and the general characteristics are similar for all values of p. This may be considered as an indication that the “effective mass” of the solid atoms increases to the extent that single collisions suffice to reverse the directions of the atoms. An increase in the effective mass could result from the fact that each gas atom interacts simultaneously with several solid atoms because (a) the collision time is

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sufficiently long relative to the period of lattice vibrations that the coupling between solid atoms is influential and/or (b) the long-range forces of the intermolecular potential extend over several solid atoms. It should be possible to modify the present model to include these effects. Assumption ( 4 ) . Although a Maxwellian velocity distribution was assumed here for both the gas and the solid atoms, it would be a simple matter to introduce any distribution function into the analysis of the hard-cube model. It appears, however, that the scattering pattern is rather insensitive to small changes in the form of the distribution function (Logan, 1967). Stickney et al. (1967) have studied the special case of the collision of a monoenergetic beam with a solid based on the hard-cube model. This investigation was motivated by the fact that several research groups are now planning to employ molecular beams having a very narrow distribution of velocities. In the same paper, the hard-cube model is used in an analysis of the lock-in detector signals resulting from scattering of modulated molecular beams from solid surfaces.

B. THEORIES BASEDON LATTICE MODELS The existing classical theories of energy accommodation were reviewed very briefly in Section II,B,2. Most of these theories are based on models that account for the fact that the motion of each solid atom is coupled to the other atoms in the lattice. This refinement, together with the inclusion of a gas-solid intermolecular potential, results in mathematical expressions that, in general, require numerical integration. To simplify the problem, it is common to consider only the case of normal incidence (i.e., Bi = 0') and to assume that the solid atoms are initially at rest (i.e., the temperature of the solid is 0°K). Both of these restrictions must be removed if one wishes to obtain theoretical results that may be compared with the existing scattering data described in Section IV, but this generally leads to expressions requiring excessive use of an automatic computer. As a result, there have been few attempts to compute scattering patterns. Trilling (1964) has, however, presented an analytical method for treating variable angle of incidence, but he retained the assumption that the temperature of the solid is 0°K. Using a high-speed digital computer, Oman et al. (1966) have simulated the collision of a gas atom with a solid surface. The solid is represented by a regular array of lattice atoms, and it is assumed that the intermolecular forces applied to an incident gas atom may be described by a Lennard-Jones 6-12 potential summed over all of the solid atoms. By using classical mechanics, the three-dimensional trajectory of a gas atom is determined by numerical computation. Results are obtained for the dependence of energy and momentum transfer on the energy and angle of the incident atom, the point of impact,

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the mass ratio, the depth of the potential well, and the crystallographic structure of the solid surface. By comparing the results for an uncoupled lattice model with corresponding results for a coupled lattice, Oman et al. conclude that the solid atoms may be treated as independent oscillators if the energy of the incident atoms is in the hyperthermal range. They assume the temperature of the solid to be 0°K because they are concerned only with collisions in the hyperthermal range. In subsequent work, however, the effects of adding thermal motion to the lattice model are briefly considered (Oman, 1966). A similar study has been performed recently by Goodman (1967). He has, however, simplified the problem by assuming a hard-sphere intermolecular potential for the gas-solid interaction. It is also assumed that the solid atoms behave as independent hard spheres that are initially at rest. The results illustrate the dependence of the scattering pattern, as well as the momentum and energy transfer, on the energy and angle of the incident atoms, the mass ratio, the diameters of the spheres, and the crystallographic structure of the solid surface. Since this zero-temperature model is not realistic in the thermalenergy range, Goodman does not compare his results with existing experimental data. The hard-sphere model has also been employed by Erofeev (1964).” A closed-form expression for the scattering pattern is obtained for the case of a zero-temperature lattice with p 4 1. In a subsequent paper, Erofeev (1966) considers the vibrational and rotational excitation of diatomic molecules by gas-solid collisions. [This problem has also been considered by Oman (1967) and Feuer (1963).] A detailed discussion of the various lattice theories of gas-solid collisions is not presented because the subject has been reviewed recently by Trilling (1967b), Hurlbut (1967), and Beder (this volume).

VI. Concluding Remarks As illustrated in Sections I11 and IV, the modulated-beam technique provides us with a powerful tool for investigating the collisions of gas molecules with solid surfaces. This technique, together with the nozzle beams and velocity analyzers that are now being developed in several laboratories, is expected to result in a marked increase in both the quantity and quality of experimental investigations in the general area of gas-solid interactions. We should be aware of the fact that modulated-beam detectors and velocity analyzers do not always present a complete picture of the state of the scattered gas because their sensitivity increases with the coherency of the signal. For example, consider a hypothetical case in which 10% of the molecules are 2o The author is grateful to R. A. Oman for providing translations of A. I. Erofeev’s papers.

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199

scattered specularly and 90 % are scattered diffusely. The scattering pattern determined by modulated-beam techniques may overemphasize the specular component (e.g., Fig. 17), thereby making it difficult to use the data to determine accurately the gross characteristics of the interaction, such as the energy and momentum transfer rates. Hence, for practical purposes, we must continue to supplement the measurement of the angular and speed distributions with data obtained by gross measurements of heat transfer and of normal and tangential forces. We should also realize that theoretical models, such as the hard-cube model, often consider only the most coherent component of the scattering process, thereby neglecting the more diffuse components which may, however, have a greater effect on energy and momentum transfer. To establish the reliability and correlation of experimental data obtained by various techniques in different laboratories, it would be convenient if each group, at one point in their program, would investigate a common gassolid system. The system Ar on Ag (1 11) is particularly attractive, because of the existing data and the fact that the effects of surface contamination appear to be small for Ag targets (Section IV,C,l). Ar on W (1 10) would also be a convenient reference system because W may be cleaned by heating in vacuum and single-crystal specimens are commercially available. The following questions may deserve special consideration. Why is supraspecular scattering observed from Pt targets but not from Au, even though the experimental conditions and molecular weights are approximately equal ? How does the amplitude of the maximum of the scattering pattern vary with Bi, T, ,T, ,p, and AH? (See Fig. I 1.) How does the presence of different impurities on the target surface affect gas-solid collisions? Does the continuous deposition technique used by Saltsburg and Smith provide surfaces that are truly clean? Are Hinchen et al. able to obtain clean Pt surfaces by heating alone? What is the nature of the scattering of hyperthermal beams from solid surfaces ? When do rotational, vibrational, and electronic excitation processes become important? How does the trapping or adsorption probability vary with the experimental parameters ? Similar questions exist for the probabilities of chemical reactions between various adsorbed gases (catalysis) and between the gas and the solid itself.

ACKNOWLEDGMENTS This review is an outgrowth of a report written for the Aerospace Environmental Facility, ARO, Inc., Arnold Engineering Development Center. (See AEDC-TR-66-13, entitled “A Discussion of Energy and Momentum Transfer in Gas-Surface Interactions.”) The initial work resulted from the encouragement and support of E. K. Latvala and K. E. Tempelmeyer of ARO, Inc.

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200

The author has benefited much from frequent communication with the research groups at General Atomic, United Aircraft, and Oak Ridge. J. J. Hinchen, E. Shepherd Malloy, H. Saltsburg, and J. N. Smith, Jr., were especially helpful through their willingness to discuss experimental results and techniques in great detail. A number of persons at M.I.T. have contributed, either directly or indirectly, to the writ'ing of this review. The exceptional ability and interest of two graduate students, R. M. Logan and S. Yamamoto, have helped in many ways, as have discussions with two faculty members, J. C. Keck and F. 0. Goodman. It was a pleasure to have F. 0. Goodman as a visitor a t M.I.T. throughout the summer and fall of 1965. Mrs. Rose S. Hurvitz deserves special recognition for preparing the manuscript and for organizing the references. One could not hope for a more skillful and conscientious assistant.

Appendix. Classical Mechanics of Hard-Sphere Collisions Consider the collision of two hard spheres (i.e., spheres that are perfectly smooth, rigid, and elastic). Since the collision affects only the velocity components in the direction of the line through the centers of the spheres, we shall restrict our consideration to these components. To be consistent with the convention used in Section V, A, we shall adopt a rather unusual sign convention for u, and u,, the component speeds of spheres of mass me and m,, respectively. The initial speeds, uniand vni ,are positive when directed toward the point of impact, and the final speeds, u,,, and v,,, ,are positive when directed away from the point of impact. With this convention, the momentum and energy equations are

+ msvnr, +

rnguni- msuni= -mgunr

+m,u;

+ +n,v$

= $m&r

$m&.

(Al) (A21

Equation (Al) may be expressed as

+ unr) = ms(uni + unr)

(A31

- UnrNUni + unr) = -ms(vni - U n r N V n i + U n r ) .

(A41

mg(uni

and Eq. (A2) as mg(uni

Dividing Eq. (A4) by (A3) gives

u,i - u,, =

-u,i

+ u,, .

(A51

By using Eq. (A5) to eliminate v,,, from Eq. (Al), we obtain, after rearrangement, 1-P u,, = -u,i

1+P

2 +1+P

Uni

SCATTERING FROM SOLID SURFACES

where p

= m,/m,.

20 1

Similarly, we can show that

Substitution of Eq. (A7) in (A2) leads to the following expression for the energy transfer from m, to m,:

If m, is initially at rest (uni = 0), this reduces to +m,(u,2, - u,”,) =

2 ~

649)

As discussed in Section II,B, the energy accommodation coefficient may be defined as Ei - E, AC=Ei - E,

For the hard-sphere model described above, Ei = +rngufifor head-on collisions. If we assume the temperature of the solid to be zero, then E, = 0 and Eq. (A9) may be substituted for the numerator of Eq. (A10) to give

Hence, according to this oversimplified hard-sphere model of head-on collisions of a gas with a zero-temperature solid, the AC depends only on the mass ratio p.

REFERENCES Abuaf, N., and Marsden, D. G. H. (1967). In “ Rarefied Gas Dynamics,” Proc. 5th Intern. Symp., Oxford, 1966 (C. L. Brundin, ed.), Vol. I, p. 199. Academic Press, New York. Alcalay, J. A., and Knuth, E. L. (1967). In “Rarefied Gas Dynamics,” Proc. 5th Intern. Symp., Oxford, 1966 (C. L. Brundin, ed.), Vol. I, p. 253. Academic Press, New York. Anderson, J. B., Andres, R. P., and Fenn, J. B. (1965). Advan. At. Mol. Phys. 1,345. Andrew, E. R. (1955). “ Nuclear Magnetic Resonance.” Cambridge Univ. Press, London and New York. Baule, B. (1914). Ann. Physik. 44,145. Berkman, E. F. (1965). S.M. Thesis, Mechanical Engineering, Massachusetts Institute of Technology. Bond, G. C. (1962). “Catalysis by Metals.” Academic Press, New York. Brackmann, R. T., and Fite, W. L. (1961). J. Chem. Phys. 34, 1572.

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Cabrera, N. (1959). Discussions Faraday SOC.28, 16. Chambers, C. M., and Kinzer, E. T. (1966). Surface Sci. 4, 33. Crews, J. C. (1962). J. Chem. Phys. 37, 2004. Datz, S., Moore, G. E., and Taylor, E. H. (1963). In “Rarefied Gas Dynamics,” Proc. 3rd Intern. Symp., Paris, 1962 (J. A. Laurmann, ed.), Vol. I, p. 347. Academic Press, New York. Ehrlich, G. (1966). Ann. Reu. Phys. Chem. 17, 295. Ellett, A., and Cohen, V. W. (1937). Phys. Rev. 52, 509. Erofeev, A. I. (1964). Inzh. Zh. 4, 36 (English Transl.: Grumman Research Dept. Transl. TR-38). Erofeev, A. I. (1966). Zh. Prikl. Mekhan. i Tekhn. Fiz. No. 3,42 (English Transl.: Grumman Research Dept. Transl. TR-39). Estermann, I., Frisch, R., and Stern, 0. (1932). 2.Physik 73, 348. Estermann, I., and Stern, 0. (1930). 2.Physik 61, 95. Feuer, P. (1963). J. Chem. Phys. 39, 1311. Fite, W. L., and Brackmann, R. T. (1958). Phys. Rev. 112, 1141. Germer, L. H., and May, J. W. (1966). Surface Sci.4,452. Gilbey, D. M. (1967). In “Rarefied Gas Dynamics,” Proc. 5th Intern. Symp., Oxford, 1966 (C. L. Brundin, ed.), Vol. I, p. 101. Academic Press, New York. Goodman, F. 0. (1962). J. Phys. Chem. Solids 23, 1269. Goodman, F. 0. (1963). J. Phys. Chem. Solids 24, 1451. Goodman, F. 0. (1965). J. Phys. Chem. Solids 26,85. Goodman, F. 0. (1966). In “Rarefied Gas Dynamics,” Proc. 4th Intern. Symp., Toronto, 1964 (J. H. de Leeuw, ed.), Vol. 11, p. 366. Academic Press, New York. Goodman, F. 0. (1967). In “Rarefied Gas Dynamics,” Proc. 5th Intern. Symp., Oxford, 1966 (C. L. Brundin, ed.), Vol. I, p. 35. Academic Press, New York. Goodman, F. O., and Wachman, H. Y. (1966). M.I.T.Fluid Dynamics Research Laboratory Rept. No. 66-1. Gurney, R. W. (1928). Phys. Rev. 32,467. Hancox, R. R. (1932). Phys. Rev. 42, 864. Herzfeld, K. F. (1955). In “Thermodynamics and Physics of Matter” (F. D. Rossini, ed.), p. 646. Princeton Univ. Press, Princeton, New Jersey. Hinchen, J. J., and Foley, W. M. (1965). Rept. D910245-7, United Aircraft Research Laboratories, East Hartford, Conn. Hinchen, J. J., and Foley, W. M. (1966). In “Rarefied Gas Dynamics,” Proc. 4th Intern. Symp., Toronto, 1964 (J. H. de Leeuw, ed.), Vol. 11, p. 505. Academic Press, New York. Hinchen, J. J., and Malloy, E. S. (1967). Private communication. Hinchen, J. J., and Shepherd, E. F. (1967). In “Rarefied Gas Dynamics,” Proc. 5th Intern. Symp., Oxford, 1967 (C. L. Brundin, ed.), Vol. I, p. 239. Academic Press, New York. Howsmon, A. J. (1966). I n “Rarefied Gas Dynamics,” Proc. 4th Intern. Symp., Toronto, 1964 (J. H. de Leeuw, ed.), Vol. 11, p. 417. Academic Press, New York. Hurlbut, F. C. (1953). U. C. Eng. Proj. Rept. HE-150-118, University of California. Hurlbut, F. C. (1959). In “Recent Research in Molecular Beams” (I. Estermann, ed.), p. 145. Academic Press, New York. Hurlbut, F. C. (1960). Phys. Fluids 3, 541. Hurlbut, F. C. (1962). U. C. Eng. Proj. Rept. HE-150-208, University of California. Hurlbut, F. C. (1967). In “Rarefied Gas Dynamics,” Proc. 5th Intern. Symp., Oxford, 1966 (C. L. Brundin, ed.), Vol. I, p. 1. Academic Press, New York. Hurlbut, F. C., and Beck, D. E. (1959). U. C. Eng. Proj. Rept. HE-150-166, University of California.

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QUANTUM MECHANICS IN GAS CR YSTAL-SURFACE VAN DER WAALS SCA TTERING E . CHANOCH BEDER Department of Fluid Physics. Aerosciences Laboratory. TR W Systems Redondo Beach. California I. I1

Introduction ..................................................... 206 The Interaction Potential .......................................... 214 2.1. General Remarks ............................................ 214 218 2.2. Molecular. Covalent. and Ionic Solids .......................... 227 2.3. Metals ...................................................... 230 2.4. Approximate Interaction Potentials ............................. 2.5. Breakdown of the Electronically Adiabatic Point-Mass Interaction Potential .................................................... 233 111 The Harniltonian of the Point-Mass System.......................... 234 The One-Dimensional Hamiltonian ................................. 234 IV. General Scattering Theory ..................... ................. 238 4.1. Scattered Beam Distributions .................................. 238 241 4.2. Elastic vs Inelastic Scattering .................................. V . Elastic Scattering ................................................. 243 5.1. Some General Features in the Perfect Crystal Approximation ....... 243 5.2. Selective Adsorption .......................................... 247 5.3. The LJD First-Order Theory .................................. 252 5.4. The 2D Line-Grating Square-Well Model ....................... 256 5.5. Elastic Scattering by a Thermally Excited Lattice ................. 256 5.6. Quantum vs Classical Elastic Scattering ......................... 260 VI . Inelastic Scattering ............................................... 261 6.1. General Remarks ............................................ 261 6.2. The Lennard-Jones-Devonshire Linear-Coupling Single-Phonon ExchangeTheory ............................................ 263 6.3. Gadzuk Higher Order Coupling Single-Phonon Exchange Theory ... 267 6.4. The Allen and Feuer Two-Phonon Exchange Theory ............. 270 6.5. The Forced Harmonic Oscillator Theory for Multiphonon Exchange 271 and for QM-Classical Comparisons ............................. 6.6. The Effect of the Surface Frequency Spectrum on QM 1D Energy Exchange ................................................... 275 6.7. The Rigid Rotator Gas Molecule .............................. 277 List of Symbols .................................................. 285 References ...................................................... 287 205

206

E. Chanoch Beder

I. Introduction The study of neutral, nonpolar gas atom-solid surface scattering, where free-free gas atom’ transitions predominate, is developing into a reasonably coherent science for some aspects of chemically inert or chemically weak gas-solid interactions.’ In this case gas-solid binding energies range from about 0.01 to about 0.3 eV. Gas atoms exhibiting these properties include the noble gases, some saturated-valency diatomic gases (e.g., N2), and Hg. The solids of interest here are perfect ionic and metallic crystals, and we shall use the term “van der Waals ” to include gas-solid interactions with these crystals where charge-induced gas-atom dipole moments occur (cf. Herzberg, 1950, p. 377 ff.). By contrast, for strong chemically reactive gas-solid interactions (binding energies up to several electron volts), or where catalysis is important, the study of scattering is in a much less advanced state and will not be considered here; these cases are treated in the article by Wise and Wood in this volume. We shall discuss QM theories and effects with brief references to experiment where essential. Related reviews are those included in the works of BonchBruevich (1954), Massey and Burhop (1952), Rogers (1962), Gilbey (1962), Allen and Feuer (1 964), and Leonas (1 964). For recent experimental work and classical theories, a review by Stickney appears in this volume. Ideally, the model of interest consists of a noninteracting monoenergetic beam of gas atoms impinging on a perfect crystal surface, the latter in thermal equilibrium. We exclude sputtering, and secondary electron emission from metals. Beam energies should, therefore, not exceed several electron volts. We also exclude the cryogenic domain where dynamically active magnetic moments may be of crucial importance and the higher solid temperatures where anharmonicity becomes important. For the important effects of polycrystallinity and adsorbed layers, cf. the article Stickney in this volume. Typical de Broglie wavelengths and crystal lattice spacings are shown in Fig. 1. Some solid melting temperatures are listed in Table I. On the experimental side, one must produce the monoenergetic beams and the well-defined crystal surfaces, and then measure the (elastic or inelastic) differential scattering cross sections. At present, it is relatively easy to produce a well-defined nearly uniform neutral beam in the 0.01-2,3 eVrange (cf. Knuth, 1966). On the other hand the preparation of a well-defined crystal surface is very difficult, and one finds the difficulty much greater for metals than for the We shall use “gas atom” henceforth to include “gas molecule” where appropriate. For brevity we shall also use g-s for gas-solid, nD for n-dimensional, QM for quantummechanical, and AC for accommodation coefficient. Reliable threshold incident energies for such transitions (critical trapping energies) are not as yet available in theory or experiment, cf. McCarroll and Ehrlich (1964) and Goodman (1966).

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

207

cleavage planes of alkali halides. In the latter the amplitude of the periodic positive and negative ion fields is relatively large; approximately 3 of the heat of absorption [cf. Lennard-Jones (1932) for Ar(KC1; NeINaF] and surface contamination is not very serious. Estermann, Frisch, and Stern (Estermann and Stern, 1930; Estermann, Frisch, and Stern, 1931;Frisch and Stern, 1933)3 were the first to produce well-defined diffraction patterns with He and H, on alkali halides with thermal and monoenergetic beams, cf. Fig. 13, and similar thermal beam experiments have recently been performed by Crews (1962, 1967). EFS measured the velocity in the scattered beams with a second crystal. These experiments confirmed, for neutral particles, the de Broglie waveparticle duality hypothesis. The experiments gave the locations of diffraction spots, showed the independence of wavelength for off-specular diffracted intensities, and gave the incidence angles for selective adsorption. Reliable intensity distributions (the ratio of intensities between different orders of diffraction) were not obtained. With increased crystal temperature EFS measured a broadening of specular diffraction peaks (He1LiF). This effect may be attributed to translational inelastic collisions caused by lattice vibrations, a well-known effect in thermal neutron scattering. The periodic amplitude of the inert gas-metal interaction is generated by the positive ion cores in the "continuum" background of negative charge. This periodic effect is less than about 6 % of the total g-s interaction energy (cf. Section 2.3), and this may account in part for the difficulty experienced in observing diffraction (cf. also Stickney, this volume, Section IV,C,l). It appears that in recent work Saltsburg and Smith (1966, 1967) have produced the necessary conditions for the production of reasonably welldefined Au and Ag crystal surfaces by epitaxial growth in high vacuum. Beams of He, H,, and D, scattered by these metal crystals do show some unmistakable properties of a diffraction pattern (cf. Fig. 20 of this article and Figs. 12 and 13 of the article by Stickney in this volume). The broadening of the specular peaks with decreased beam temperature would indicate an increase of translational inelastic collisions. Furthermore, the broadened H, and D, beams relative to He suggest rotational mode activation of translational inelastic collisions, with greater broadening for the small rotational quanta of D, . Scattered particle flux has been measured, but the simultaneous measurement of particle number flux and velocity in the scattered beams is only in an initial stage of development by Saltsburg and Smith as well as by others (cf. Moran et al., 1967). AH existing theories of gas-solid scattering assume the point-mass or electronically adiabatic model, which is more plausible for nonmetals than for metals. In this model the gas atom and the lattice nuclei appear as point masses, and the energy of the system is the corresponding sum of kinetic and potential Hereafter, frequently referred to by the abbreviations ES, EFS, and FS.

208

E. Chanoch Beder

energies of these masses only (electron coordinates are implicit in the potential energy). The quantum mechanical effects that should be included in the theory are: (1) The gas-solid interaction potential energy. (2) Diffraction when particle wavelength compares with projected lattice spacing. (3) Energy transitions of one or more quanta which break down the pointmass model and require dynamically active electrons for description. Among these are electronic excitations in nonmetals, secondary electron emission in metals, and strong ( 51 eV) gas-solid chemical reactions which occur with transfer or sharing of charge between gas atom and solid, cf. Fig. 2. (4) Lattice phonon statistics (bosom) for energy mode populations and for the lattice nuclei vibration mean-square-amplitude effect (Debye-Waller factor) below the Debye temperature. (5) For g-s energy exchange, a QM treatment has been shown to be important in some specific cases depending upon the g-s interaction, impact energy, mass ratio, M J M , , solid temperature, and lattice mode spectrum [see Gilbey (1967a,b), Gilbey and Goodman (1965), and Shuler and Zwanzig ( 1960)].

The status of Q M theory is as follows. Numerous gas-solid interaction potentials have been derived by treating the detailed structure of gas atom and crystal [cf. de Boer (1956), Young and Crowell (1962), and Part I1 of the present article]. These potentials include important parameters of the noninteracting gas atom and solid, e.g., atom polarizabilities, electronic resonance frequencies, lattice geometry, ionic charge in crystals, and electronic states in metals. In all cases these potentials vanish far from the surface and repel strongly near the surface, allowing little or no penetration. A negative potential minimum, at a distance from the surface comparable to lattice spacing, allows for adsorbed states. Lattice periodicity has been reasonably well included for alkali halides but not for metals. The lattices are assumed rigid and these potentials may consequently apply for elastic scattering. For a thermally excited lattice where gas-solid energy exchange may occur, the interaction potential is in the following very primitive form of development: the ID motion of the gas atom is coupled directly to the motion of one lattice atom only. Finally, the theory of thermal effects on the interaction as developed for thermal neutron scattering (e.g., the Debye-Waller factor effect) has only been initiated. The complexity of the problem is such that the more refined potentials referred to have not been used ; rather, for mathematical reasons, approximations have been adopted, mainly by Lennard-Jones and Strachan (1935) and by Lennard-Jones and Devonshire (1936a,c), which retain some gross aspects

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

209

of the i n t e r a ~ t i o nAmong .~ these are the gas-solid bound-state energies, the lattice geometry, the range of interaction, the amplitude of the periodic potential, the lattice temperature, and the lattice frequency spectrum (as a function of distance from the surface). These approximations may form one of the weaker links of existing g-s scattering theory. For elastic scattering we have firstly the excellent work of LJD (1936c,e) using Morse potentials. In this work we find formulas for intensities in the specular and the four first-order diffracted beams, which show the independence of intensities with wavelength in a confirmation of the ES work, and an explanation of the selective adsorption phenomenon confirming that of EFS. Secondly, Beder (1964) gives exact intensities for a square-well 2D line-grating model, approximating the LJD one. Finally, Howsmon (1966,1967) has some preliminary results for an interaction in which particle penetration is assumed to be important, as in thermal neutron scattering, and where the Debye-Waller factor is included in the interaction. No 3D (or 2D) QM inelastic scattering theory has been developed. Nevertheless, some important essentially 1D work has been done in thermal accommodation theory. In this single-collision theory an extremely simplified model has been developed for the inelastic collision of the gas atom with the thermally excited lattice. This model is shown in Fig. 15. The gas atom moves onedimensionally normal to the surface and is coupled in a “ binary ” interaction to the normal component of the motion of one lattice surface atom rather than to the lattice as a whole. As first proposed by Jackson and Mott (1932), the gas-solid interaction is purely repulsive and the lattice an Einstein crystal. LJD (1936a-f) and LJS (1935, 1936a,b) greatly improved upon this model by introducing the Morse potential and the harmonically excited Debye lattice.’ Using first-order perturbation theory, the transition probabilities for free-free scattering are derived. This work, therefore, provides an excellent point of departure for the far more complex 3D work, which is yet to be done. In the QM thermal accommodation theory to which these results have been applied, the incident gas atoms as well as the scattered ones are equilibrated at temperatures that approach the lattice temperature. Fairly reliable experimental AC’s are now available for comparison, and good quantitative agreement between theory and experiment is no longer uncommon for either QM or classical theories. The Boltzmann averaging, to which the scattered beam is subjected in thermal AC theory, renders the final integrated result relatively insensitive to the details of the collision model or to the method of dynamical analysk6 Hereafter frequently referred to by the abbreviations LJ, LJD, and LJS. The constants of this ‘‘ binary ” interaction are those of a 3D rigid lattice g-s potential (cf. Sections I1 and VI). Goodman (1963), Trilling (1964, 1967) and Stickney et af. (1967) have had remarkable success with classical treatments of very simple, widely differing models.

210

E. Chanoch Beder

It follows that some effects that are important in scattering may be hardly detectable in accommodation coefficients. Conversely, we should expect to find effects that are important in thermal AC theory to be important in scattering theory as well. The reliability of experimental AC's now available

10-2

1

lo-'

x

10

cx,

FIG. 1. Free-particle energy versus de Broglie wavelength. A, = h/(2M,Ev)1/2,A,,, = hc/Ephoron. Range of lattice constants and nearest-neighbor distances in solids indicated by thin rectangle.

W D (1936f) assumed a gas-solid perturbation energy that is linear in the lattice target atom displacement. This leads, in the harmonic approximation, to single-phonon-only energy exchanges in addition to elastic collisions. In accord with first-order perturbation theory, one must, therefore, expect the probability of higher order coupling or multiphonon exchanges to diminish

211

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

rapidly compared with one-phonon exchanges. This has been shown for the two-phonon case by Allen and Feuer (1967) [and by LJS (1936b) for transitions to bound states with higher order coupling]. Since most Debye temperatures of interest here are less than about 500"K, the lattice phonon energies will be less than about 0.04 eV, with the highest density of states (in the spectrum) in the upper half of the phonon energy range. It follows that for beams TABLE I SELECT PROPERTLES OF SOMBCRYSTALLINE SOLIDS

Substance

Structure

Lattice constant

Nearest neighbor distance

Melting temp.

(A)

(A)

(OK)

C' (graphite) Cd4 Aulg7 M

O

~

NiS9 Pd106 ptig5 Ag1OB W184 V5'

Li7FI9 Li7CPS K39C13s K39F19 NaZ3C13' Na23F'9

fcc fcc bcc fcc fcc

3.61 4.07 3.14 3.52 3.88

2.55 2.88 2.72 2.49 2.74

fcc fcc bcc bcc fcc with basis fcc with basis fcc with basis fcc with basis fa with basis fcc with basis

3.92 4.08 3.16 3.03 4.02

2.77 2.88 2.73 2.63

~

Dissociation energy (kcal/mol)

Debye temp. ("K)

Sublimation 298 1356 1336

171.3

420

1726 Sublim 1825 2045 1234 3653 21 90 1121

93.2 88.7

5.13

883

6.28

1043

5.34

1129

5.63

1073

4.62

1285

75.91 83.46

343 164 450 450 300

127.5 63.42 199 114

240 226 400 360 732

Sublim. 49.3

235

47.2

321

with incident energies of the Debye cutoff value or less (where trapping is likely to be significant) one would expect the one-phonon exchange theory to characterize the gas-solid interaction. Furthermore, following the LJD (1936f) one-phonon exchange selection rule which couples the gas atom with one lattice normal coordinate, one might then expect QM theory to be important since the discreteness of energy levels in a normal coordinate spectrum is then important. Neither theoretical nor experimental thermal AC's have been

212

E. Chanoch Beder

able to establish such a preference. Significant improvements of the LJD theory have been achieved by Gadzuk (1967) through the inclusion of a thermal averaging pseudo Debye-Waller factor effect (for one-phonon exchange), and byGoodman(l965a)and Gilbey(1967b) through theintroductionofthe surface frequency spectrum of the lattice. In some preliminary QM theory, diatomic gas molecules approximated by rigid rotators have been incorporated into the point-mass model (Jackson and Howarth, 1935; Feuer, 1963). This latter is the onIy work to date on gas particles with structure in QNi theory. For classical theory cf. Oman (1967).

-

I

I

I

I

I

,

I

1

Atomic Resonance CUI

I

Metallic work functions

-

Hg2 Dissociation, diatomicr

i Electronic

@-• M e t a l l i c cohesion

j

fine structure:

12

0 2 N2 H 2

-

vibrational quanta

@-Diatomic

HelC

ArlNaCl XelC XelW

-Exp.

L~F

Pb Au

Kr

heats of phyrirorption

C (diamond) Debye temperature phonons

Xe

Binary v w der Wools bond

Diatomic rotational resonance I

I

I

IO-~

Io

3

I

t

10-1

I

I

1

I

I

10

1

Energy ( e V )

FIG.2. Characteristic energies of gas particles and crystal lattices affecting gas-solid van der Waals scattering. Atomic resonance from Hg to Ne shown. Diatomic ionization for 02,H2,and N2shown; 1 eV = kT("K)/11,610= 1.6020 (lo-") erg = 23,063 cal/mole = 8068.3 cm-I (photon wave numbers).

Multiphonon exchange may occur for incident energies well above 0.04 eV. The experimental evidence for multiphonon exchange has not as yet been obtained with sufficiently well-controlled conditions to be conclusive for perfect crystal scattering [cf. experimental work of Devienne (1954) and the surveys of Estermann (1966), Wexler (1958), and Knuth (1967)l. Nevertheless,

213

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

many-quantum transitions are commonly observed in high-energy impacts, e.g., the binary ionization impacts of gas atoms, and one would likewise expect to find multiphonon exchange in high-energy (~0.5-3eV) gas-solid impacts with well-controlled crystal surfaces. On the theoretical side, QM calculations for the 1D head-on impacts between gas atom and harmonic oscillator with nonlinear coupling show probabilities for multiquantum exchange of the same order as for those of one-quantum exchange (cf. Shuler and Zwanzig, 1960; Gilbey, 1967a). Although one would expect the multiphonon exchange to be a classical phenomenon, this has not been conclusively

lo2-

Earth escape speed

-earth orbital spee

I

I I

I

I I

1

SPEED v q , CM/SEC

FIG.3. Translational kinetic energy versus speed of some atmospheric particles; +kT, at 300°K indicated by arrow.

214

E. Chanoch Beder

demonstrated as yet (cf. Gilbey, 1967a). A multiphonon QM theory may require a number of considerations beyond those encountered in the LJD work and the subsequent small corrections. These might be second-order perturbations with nonlinear g-s coupling, lattice anharmonicity, and multiple collisions. The 2D and 3D gas-solid van der Waals scattering theory is certain to benefit substantially from that for thermal neutrons, now in a rather sophisticated state of development (cf. Kothari and Singwi, 1959; Maradudin et al., 1963; and Kittel, 1963). To summarize, the experimental techniques (especially ultra-high vacuum) are now being perfected to an extent where experiments with well-defined incident and scattered beams and well-defined crystal surfaces can be carried out. Theoretical QM point-mass models for the gas-solid interaction are fairly well developed for elastic scattering by ionic crystals and in a preliminary state for metals and for inelastic scattering. We may expect these efforts to produce increasingly detailed information on the gas-solid interaction and in time to play a role similar to that of X-ray, electron, and thermal neutron diffraction. We note in passing that considerable work in this field has been done recently by rarefied gas dynamicists, motivated by the conditions shown in Fig. 3.

II. The Interaction Potential 2.1. GENERAL REMARKS Gas-solid van der Waals interactions, in the direction normal to the surface, are found to be similar for nonmetallic and metallic solids. In this component the potential vanishes at infinity, has a negative minimum near the surface, and becomes strongly repulsive beyond the surface and into the crystal. For gas atom energies below 3 eV, penetration into the surface seems to be quite unimportant, certainly beyond the first few layers. In the component parallel to the surface where the periodicity of the lattice comes into play, important distinctions are observed between nonmetallic (ionic and nonionic) and metallic solids. For example, one finds a strong periodic effect in ionic lattices compared with a relatively weak one in metals. The theories of these interactions have also important differences. To molecular, covalent, and ionic solids the Born-Oppenheimer adiabatic approximation may be applied. In practice the interaction potential is derived using the additivity assumption, where a summation or integration is carried J out over all binary interactions between gas atom and (assumed noninteracting) lattice atoms. For molecular and covalent solids, only the “neutral” van der N

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

215

Waals forces appear in the binary interactions. To these forces we must add, in the case of ionic solids, the induced dipole interactions, and electrostatic interactions for nonpolar gas molecules possessing a permanent quadrupole moment. In Fig. 4(a) we show some typical binary van der Waals interaction parameters ; for comparison chemical interaction parameters are shown in Fig. 4(b) and (c). Of the simple analytical forms for binary potentials commonly encountered, that of Buckingham (2.2.2) is closest to basic theory. However, many approximate forms are found more convenient in scattering analysis. Some examples are discussed below and are shown in Fig. 5. Margenau has shown that additivity applies to dispersion forces [neglecting third-order effects, see Young and Crowell (1962, p. lo)]. On the other hand, additivity breaks down for van der Waals repulsions between three molecules at close range (Hill, 1960, Appendix IV). We should expect a similar difficulty in gas-solid scattering. Additivity should apply effectively where the lattice atoms retain a spherical ground-state configuration essentially undisturbed by the lattice bond. This condition is satisfied best for molecular or van der Waals bonded noble gas solid^.^ It is not quite as good for a molecular solid such as N, ,or for covalently bonded solids. In alkali halide crystals, the ions have spherical noble gas, diamagnetic configurations; the additivity assumption should apply here as well, but one should expect some loss of lattice ion sphericity near the surface. The amplitude of the periodic field in this case has been rather extensively investigated. Additivity is likely to suffer most when the gas atom is closest to the lattice, i.e., within about a lattice spacing. In gas-metal interactions on the other hand, the conduction electrons provide the major interaction effect. The positive ion cores account for a rather smalI perturbation. The relatively smaller periodic term is yet to be successfully calculated. Binary potentials are treated very comprehensively in the excellent work of Hirschfelder, Curtiss, and Bird (1964), (hereafter, HCB), and in that of Pauly and Toennies (1965). Additive gas-solid nonmetal interaction potentials are reviewed in the recent works of Young and Crowell (1962), (hereafter, YC), and Kaminsky (1965). de Boer (1963, 1956) is another excellent source. We shall discuss some of this material very briefly in this chapter. The gas-solid interaction potentials mentioned above assumed a rigid lattice. They may, therefore, be used for elastic scattering (no exchange of energy between gas atom and solid) and for the calculation of heats of adsorption, etc. For inelastic scattering we must include lattice nuclear coordinate motions in the interaction. Finally, there is a temperature effect in both elastic and

’Because of the weak van der Waals lattice bond in noble gas and molecular solids, their behavior as scatterers must differ appreciably from that of the strongly bonded ionic and metallic solids, and will not be treated here.

+I40 . 1120.

H+ H

+loo.

+80.

0

-

Kr + Kr Xe t Xe 02

+

2.869 3.287

3.12 3.822 4.04 4.60 4.151

3.88

02

4.285

hv

-

I :; & 10.22

range repvlrive potential

8, ( O K ) Ro (4 --

R

H e + He

-&Short

3.405 3.60 4.10 3.698

N+0

H + Br H+ I

CI

+

I+I

CI

3200 310

85.4 2.86 2.07 2.77 2.42 15.2 12.1 9.0 0.346 0.116

0.054

43.3 -

0.740 1.095 1.204 1.128 1.150 1.275 1.414 1.604 1.989 2.284 2.667 ,742

-

5.29

3.60 2.75 2.48 1 .Y

(a 1

FIG.4. Binary potentials, van der Waals and chemical bonds compared. (a) Weakly interacting and van der Waals potentials: Lennard-Jones (1-12) parameters, Eq. (2.2.5), compiled by Hill (1960, Appendix IV) from empirical data in HCB (1964). (b) Strongly interacting chemical bond potential parameters (ground state). Compiled by Hill (1960) (cf. also Henberg, 1950). (c) Theoretical H H potential contributions. From Pauly and Toennies (1965).

+

9

V (R)+ D

6 -

7-

V (R) +

D

(Cm-l)

6-

I

5-

40.000i

A

a

R = 20.4

w34-

30,000

$ 1; 4:

3210-1

'

0.8 0.9

' 1

1.1

1.2

1.3

1.4

0.5

1.0

R/it

1.5

R

2.0

2.5

1 .o

""..".."' 2.0

3.0

4.0

6) (b)

Frc.5. Binary potentials, approximations. (a) Repulsive potential of Jackson and Mott [Ce--nR-Dl compared with the Lennard-Jones (6-12) potential. After Herzfeld and Litovitz (1959, see for further comparisons). (b) Morse potential (broken curve), Eq. (2.4.1), compared with ground-state H H empirical potential (u = 0, 1 , . . , of vibrational levels shown). After Herzberg (1950, Fig. 50). (c) Parabolic (harmonic) and cubic (first anharmonic) approximations to the ground state of H C1. After Herzberg (1950, Fig. 46).

+

.

+

E. Chanoch Beder

218

inelastic scattering which has been introduced into the interaction potential by averaging processes over lattice vibrations. The Debye-Waller factor is of this nature. We shall treat these considerations briefly here. Finally, we shall conclude with the approximate interaction models which have actually been used (cf. Introduction). 2.2. MOLECULAR, COVALENT, AND IONIC SOLIDS To the gas-nonmetallic solid system, we apply the electronically adiabatic approximation. This theory was first given by Born and Oppenheimer for molecules. An excellent review is given by Schiff (1955, p. 298). Using the fact that the electrons move much more rapidly than the nuclei (a result of the small electron-nucleus mass ratio), the energy of the electrons, kinetic and potential U(R), may be computed quantum mechanically for any fixed separation of the nuclei. The potential energy of the system V(R)is then obtained by adding the Coulomb energy of the nuclei: V(R)= U(R) + Z,Z,e2/R2 (diatomic case for atomic numbers 2, and Z 2 ) . This potential may then be applied to the dynamics of the nuclei provided electronic excitations do not occur. This interaction neglects electronic fine structure, cf. Fig. 2. In molecules spin-orbit coupling splits electronic energy levels when the spin (Z) and orbital angular momentum (A) do not vanish (Herzberg, 1950, p. 214). The splitting increases with the number of electrons. For BeH first excited state 213,the splitting is only 2 em-' (approximately 0.000025 eV), whereas for HgH it is 3684 cm-' (approximately 0.5 eV). Some similar effect should be anticipated in gas-solid interactions and may be neglected only when small compared with the energy transitions included in the scattering. Hyperfine structure splitting (electron spin-nuclear spin coupling) is about two orders of magnitude smaller than fine-structure splitting and is of much less importance here. Finally, as Born and Oppenheimer have shown, the adiabatic approximation is justified so long as not too high vibrational and rotational modes are excited (Schiff, 1955, p. 301). A straightforward application of the above approach to the gas-solid system becomes a many-body problem. Solutions have been achieved for three- and four-body systems, e.g., H, H, and H, ,H, (cf. Figs. 7 and 8) but not for more complicated ones. For gas-nonmetallic solid systems it is common to apply the additivity assumption discussed in Section 2.1 : N

Y ( r ) = c Vj(lr-Rjol); or

i

Y(r, R,, . . .) =

j = 1,2, ...

(2.2.Ia)

N

c Vi(lr- Ril). .i

(2.2.lb)

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

219

V j is the binary interaction potential for the gas atom and the isolated atom of the solid. In (2.2.la) the lattice atoms are all at their equilibrium sites and we have the ideal lattice model. In (2.2.lb) the lattice is thermally excited. a. Binary Interactions The theory of two-particle interactions divides the domain of nuclear separation into two regions. For each region appropriate approximations make a solution possible. At long range R 2 2R,, where R, is the equilibrium separation,8 the interaction is attractive and is characterized by negligible overlapping of the electronic wave functions of the two atoms. Antisymmetrizing effects of electrons are not explicit. In this range the particles interact as oriented point masses which may be charged. These point masses will have an electric dipole moment, permanent if a polar molecule and induced if otherwise. For these long-range interactions there is a large body of accurate theory. The parameters of the interaction energy function are properties of the isolated particles : charge, electric and magnetic dipole moments, quadrupole moments, polarizability, characteristic frequencies, and optical transition probabilities (HCB, 1964, p. 24). A complete list of the theoretical long-range interaction potentials is given by HCB (1964, p. 25, paragraph 1.3b). The contributions to these long-range interactions when the gas atoms are neutral and nonpolar, may be the induced or dispersion ones. Let the neutral atom have polarizability LY. Then it will interact with a point charge e according to -e2c(/2R4, with a point electric dipole p according to -p2u/R6 (averaged over orientations of the dipole), and with another neutral atom according to -C/R6 where three proposed forms of C are given in Table 111. The latter interaction is the dispersion one and depends upon polarizability and resonance frequency v, or number of electrons in the outer shell R, or magnetic susceptibility x, in each interacting atom. Finally, a gas molecule may have a permanent point electric quadrupole moment Q which will interact with a charge according to eQ(3 cos2 0 - 1)/4R3. The short-range interactions (R 5 2Ro) are characterized by the overlapping of electron wave functions and the consequent need to antisymmetrize these functions for a stable bond to result, in almost all cases. The theory is vastly more complicated than that of the long range. There is no general approach, and each system is treated separately, cf. Herzberg (1950, Chap. VI). We shall discuss the binary interaction between two neutral inert atoms which gives rise to the dispersion forces that are present as well for any two interacting particles, be they atoms, molecules, or ions with at least one electron. To these interactions the induction forces must be added when charge

* HCB (1964, p. 917) use “two collision diameters.”

E. Chanoch Beder

220

or electric dipole or quadrupole moments are present. In view of the uncertainties as to the form this binary interaction should take, approximations are usually adopted. The most compatible with QM theory seems to be the Buckingham form

V(R)= Bexp(-aR)

- C6-exp/R6.

(2.2.2)

The first term is the close-range repulsion. The second is the dispersion term of attraction resulting from ground-state induced dipole-dipole interaction. Higher order multipole interaction terms have been neglected. LennardJones proposed an alternative two-parameter repulsive term, which has no theoretical basis but provides a good fit with many experimental data:

V(R)= -- -. R"' R6 Bl.J

c6-m

(2.2.3)

The most commonly used value for m is 12. More convenient forms for (2.2.2) and (2.2.3) are

(2.2.5) where R, and -D or V(R,) are the equilibrium (dV/dR = 0) separation and energy, respectively. A list of about 25 cases for which QM calculations have been carried out is given by HCB (1964, pp. 1052-1054),9 and most of these are chemically bonded. This meager list, within which agreement with experiment ranges from poor to good, is evidence of the limitations of the theory. We show for example, in Fig. 6, Ne + Ne and Ar Ar. For helium (HCB, 1964, Fig. 14.2-1) theoretical potentials differ from one another by as much as a factor of 2. HCB regard the experimental curves as the most reliable. Mason and Vanderslice (cf. Young and Crowell, 1962, p. 18) report a number of conflicting theoretical values for B and a of noble gases and regard only those of He as reliable. For another comparison, Tables 11 and 111are given (from Young and Crowell, 1962, p. 21). Table I1 is empirical. In Table 111, C6-expand c6-1, are derived from Table I1 to show how various theoretical values compare with experiment. It is evident that discrepancies may exceed

+

' H + H , H e f H e , N e + N e , Ar+Ar, H + H Z , H Z + H 2 , H + H * , H z + H * , H e + H f , He++He', H + H e , N + N , Be+Be', Li+Li, B e f H , B e ' f H , L i f H , Li' H, Li Li', K H, Na Na, K K. The prime indicates an excited state, f denote ions.

+

+

+

+

+

30

3-

h

P al

u)

-

2-

20

t

1-

7

I

0

v n

n

P 10 al

c-

-

In

W

> -1-

I

0 v 7

n

-2 -

Lxc

v

> -

-3

- 10

-4 -

-5 -61

2

I

3

4

I

I

5

6

(a 1

I

7

-20

+

+

FIG.6. Binary potentials for noble gas atoms, theory and experiment compared: (a) Ne Ne and (b) Ar Ar. Theoretical curves have encircled numbers. Other curves are experimental. After HCB (1964, Figs. 14.2-2 and 14.2-3).

222

E. Chanoch Beder

+

a factor of two. In Fig. 7 the theoretical H H, curves are shown, but no comparison with experiment is reported. In Fig. 8, the theoretical curves for H, H, are shown to agree rather well with experiment. For close-range repulsion the binary interactions between an atom and a molecule or between

+

TABLE I1 EMPIRICAL BINARYPOTENTIAL PARAMETERS' Buckingham (6-exp) Potential [Eq. (2.2.4)]

Neon Argon Krypton Xenon Methane

LJ (6-12)Potential [Eq. (2.2.91

a

DlkC'K)

Ro(&

DlkC'K)

Ro(&

14.5 14.0 12.3 13.0 14.0

38.0 123.2 158.3 231.2 152.8

3.147 3.866 4.056 4.450 4.206

36.3 119.3 159.0 228.0 148.0

3.16 3.87 4.04 4.46 4.22

"Determined by Mason and Rice (1954) from experimental data including that of Comer (1948).Adapted from Table 2.1 of Young and Crowell (1962). TABLE 111 COMPARISON OF EMPIRICAL AND THEORETICAL BINARYPARAMETERS'

Neon Argon Krypton Xenon Methane

0.392 1.63 2.47 4.01 2.58

21.6 15.8 14.0 12.1 13.1

8 8 18 18 8

11.6 31.6 48.1 71.5 29.9

4.0 8.7 50.0 73 102 205 233 422 105 145

11 125 289 698 188

8.7 99.6 190 451 204

10 111 191 496 231

' C 6 - e x p and C6-12 derived from empirical data of Table I1 and compared with theoretical values of London, Slater-Kirkwood, and Kirkwood-Muller. From Table 2.2 of Young and Crowell (1962)l. * Polarizabilities after Beattie and Stockmayer (1951). Ionization potentials after Herzberg (1944)for the inert gases and appearance potential after McDowell and Warren (1951)for methane. Number of outer electrons assumed in Slater-Kirkwood formula. Diamagnetic susceptibilitiesafter Selwood (1956).

'

C L = Qa1a2ZIW(I1

+

IZ),

CKM = 6mc2alad[(adxl) + (a&4l.

c,, = (3eh/4rrml/2>a,a,/[(al/31)1'2 + (~rz/32~2)''~1,

35

I

I

30-

25

-

20-

I5

' 2

-

.

s

1

;10 -

h

&

'

v

5-

--

0-

-5

-10

-

I

2

aI

I

3

4

R

6)

FIG.7. Theoretical, threebody weakly interacting H + Hspotential after Margenau.Adapted from HCB (1964, Fig. 14.3-3).

-%i

I 3.18

I 3.70

I

I

4.23

4.76

Center of mass separation

R (A)

FIG.8. Theoretical, four-body weakly interacting H, + H, potential after de Boer, compared with expenment. Adapted from HCB (1964, Fig. 14.4-1). (---)cpI and (-) vII; I and I1 employ different assumptions for the valence energy.

224

E. Chanoch Beder

two molecules require a three or more center theory, which has not been produced except for the above-mentioned cases of hydrogen. b. Molecular and Covalent Crystals

Although molecular and covalent crystals are not of primary interest here, they provide the simplest case of additivity, since the gas atom-lattice atom binary interactions are the neutral inert ones only. We shall treat the case of a monatomic gas molecule.” When additivity is applied to the above binary equations by assuming a continuous semiinfinite solid, the following results were obtained by London and Hill (cf. YC, 1962, pp. 12,23,24) for the Buckingham and LJ potentials, respectively: 2nn B nnC6 -exp V(z) = -(za + 2) exp( -az) 6z3 a3

(2.2.6)

and V(z) =

nnC6-, 2nnB,, z 3 - m - ~. 6z3 ( m - 3)(m - 2)

(2.2.7)

Equation (2.2.7) may be more conveniently written, for m = 12, (2.2.8)

where n is the number density of nuclei in the solid. Here z is the distance of gas atom from the surface defined in principle by the plane of equilibrium sites of the surface atoms. In practice this reference datum is a bit vague and may be a source of appreciable error (especially in metals), cf. de Boer (1956, p. 24). Furthermore, the dispersion forces are two to three times too small near equilibrium, an error of the integration replacing the summation (YC, 1962, p. 12). Some important numerical results obtained with the above additivity potentials are given in Table IV for graphite and for alumina. These data, as well as the data in other tables of this article, are extracted from the very comprehensive work of Young and Crowell (1962), which must be consulted for important details. Table IV(a) is self-explanatory. The variation of the potential in Table IV(b) over the lattice cell is obtained by using the summation l o This treatment may, in principle, be extended in a straightforward way to the diatomic gas molecule by treating it as two coupled atoms, capable of rotation or vibration or both. In practice this becomes very difficult, however, and has only been done quantum mechanically in a 1D rigid rotator case, Section 6.7.

TABLE IV. NONMETALLIC NONIONIC SOLIDS-ADDITIVITY POTENTIAL AND ZERO-COVERAGE THERMODYNAMIC POTENTIAL PARAMETERS ~~

(a) Heats of adsorption from various LJ potentials, Eq. (2.2.Q by various authors. [From Table 2.3 of Young and Crowell (1962).] Adsorbent Carbon black

Graphitized carbon black

Alumina (AlK)a)

Gas

-Eo (3-CO) (cal/mole)

-Eo (3-9) (cal/mole)

600"

537" 459b 1,280b 4,O6Ob 3318'

Helium Helium Neon Argon Argon

1,360" 4,340"

(cal/niole)

-Eo (410) (cal/mole)

- E o (3-12)

Argon Neon Argon Krypton Xenon Hydrogen Deuterium Methane Deuteromethane

2,410' 864" 2,440' 3,280e 4,200' 1,298f 1,306" 3,297'

2,174' 758" 2,200' 2,900' 3,800' 1,!4!" 1,151* 2,947f

760' 2,200' 2,9 1Oe 3,820e 1,145" 1,155" 2,955"

729' 2,130' 2,790' 3,690' 1,097" 1,109' 2,846"

3,227"

2,881"

2,889"

2,846f

Argon Krypton

2,800" 3,460"

3,034'

* DeMarcus et a/. (1955).

Steele and Halsey (1954).

' Constabaris and Halsey (1957).

Sams et al. (1960).

Freeman (1958). "Constabaris et al. (1961).

(b) Argon on graphite. Variation of heats of adsorption and equilibrium distance over surface cell. [From Tables 2.4 and 2.6 of Young and Crowell (1962).] Site type" A B C

3.55 3.60 3.60

- U,b

- E(Zdb

zo(hb

3.18 3.25 3.25

1804 1775 1771

2,280 2,160 2,180

1,750 1,710 1,710

2,470 2,360 2,380

a A: above the center of a lattice hexagon. B: above the midpoint between two carbon atoms. C : above a carbon atom. Left-hand entries after Crowell and Young (1953). Right-hand entries after Pace (1957). In cal/mole.

(c) Various gases on graphite. Comparison between experimental heats of adsorption and U,,after Crowell and Davis. [From Table 2.5 of Young and Crowell (1962).] Experimental heats (cal/mole) Calculated Uo(cal/mole) Helium Neon Argon Krypton Xenon a Greyson and Aston (1957). Beebe (1958).

357 746 1,800 2,720 3,370

Beebe and Young (1954).

340" 830" 2,700* 3,900'

Amberg, Spencer and

226

E. Chanoch Beder

in the additivity equation (2.2.la) for the nearest lattice atoms and an integration for those several spacings away. We note the factor of about 0.8 between the results of Young and Crowell, and Pace. Also the variation of the potential over the cell is small, 2-4%. Table IV(c) shows good agreement with experiment for light gas atoms, and only fair for heavier ones.

c. Ionic Crystals For ionic crystals, to the neutral inert interactions, Eq. (2.2.6) or (2.2.7), we must add the binary interactions between the lattice ions and the induced dipole moment in the gas atom. This sum has caused difficulty because of its slow convergence. It was first treated by Lennard-Jones and Dent (1928) and LJ (1932). In a later treatment, Lenel(l933) included the spatial distribution of charge in the gas atom. The work of Lennard-Jones and Dent is shown in Fig. 9 and that of Lenel in Table V. Other work is shown in Tables VI and VII. The comprehensive work of Young and Crowell should be consulted for details of these calculations. The variation of the interaction energy (compared with the average value) over the lattice surface cell is approximately 3 for ArlKCl and NelNaF, Fig. 9. Tables VI-VII show variations ranging from

1

t

\I

i

O-*

FIG.9. The additivity potential for a noble gas atom outside the (100) plane of an ionic crystal. (- -) Ar and KCI (d = 6.28A and (-) Ne and NaF (d = 4.68 A). (Includesinduced dipole and Lennard-Jones(6-12) van der Waals contributions).A, above midpoint of lattice cell; C, above lattice point. After LJ and Dent (1928) and LJ (1932).

-

227

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

TABLE V

IONIC SOLIDS-ADDITIVITY

POTENTIAL HEATS OF ADSORPTION (AFTER LENEL)"

Total Experimental Dispersion Induced Quadrupole calculated System heats energy ED energy E, interaction energy (Adsorbate-adsorbent) (cal/mole) (cal/mole) (cal/mole) (cal/mole) (cal/mole) Ar-KCl Ar-K I Ar-LiF Ar-CsC1

2,080 2,520 1,770 3,560

Kr-KCl COZ-KCI C02-KI

2,620 6,350 7,450

1,500 1,420 1,230 2,400b 1,700' 1,930 2,900 3,300

370 680 540 1,100b 800' 5 50 740 1,140

(2,700)' 3,250

1,870 2,100 1,770 3,5w 2,5Ooc 2,480 (6,340)d 7,690

Experimental and theoretical values compared. Dispersion energy : induced dipole energy: permanent quadrupole energy w 1 : 1/3 : 1 . [From Table 2.9 of Young and Crowell (1 962).] Calculated for (100) face of cesium chloride. Calculated for (1 10) face of cesium chloride. * Quadrupole interaction determined as difference between experimental heat of adsorption and the sum of the dispersion and induction interactions. This difference was then used to calculate the quadrupole moment of carbon dioxide, which was used in turn to find the electrostatic quadrupole interaction of carbon dioxide on potassium iodide.

& to f of the average energy. In Table VI(c) a comparison is made between theory and experiment, with the former values ranging from 0.85 to 0.92 of the latter. Where gas molecules have a permanent electric quadrupole moment, the binary interactions between the lattice ions and this quadrupole moment must be added to the dispersion and induced dipole interactions above. The magnitude of this contribution may be large. For N,IKCl it may be 3 of the total adsorption energy and may vary by a factor of 3 over the surface cell, Table VII(b). Additional examples are shown in Table VII(c). 2.3. METALS For metallic solids, the principal interaction is between the gas atom and the sea of electrons in the solid. The ion cores contribute about 5 % to the interaction (YC, 1962, p. 17). There are a number of theoriesfor the dispersion

E. Chanoch Beder TABLE VI ARGONON IONICSOLIDS-ADDITIVITY POTENTIAL AND ZERO-COVERAGE THERMODYNAMIC POTENTIAL PARAMETERS (a) Argon on the (100) face of octahedral KCI. After Orr, Lennard-Jones, and Dent. [From Table 2.10 of Young and Crowell (1962).] E(~o)mar/E(~o)min w 1.28. -E(zo) X 10’’ (cal/mole)

Site Above center of lattice cell Above midpoint of lattice edge Above K ion Above C1- ion +

EI/ED at zo

-0 0 (cal/mole)

3.215

1630

0

1,593

3.44 3.48 3.74

1345 1461 1270

0 0.04 0.02

1,314 1,423 1,233

(b) Argon on the (111) face of octahedral KCI. [From Tables 2.11 and 2.13 of Young and Crowell (1962).] First five columns calculated by Young (1951). Site Surface type” layer A

-

+

-

B C

+ +

zo

-E(zo) x 10”

(A)

(cal/mole)

2.80 2.61 2.78 2.67 3.81 3.77

2102 1411 2246 1365 1084 706

El/& at zo

- Uo

-AH0

(%) (caI/mole) (caI/moIe) 2 4 1 1 1 2

2,060 1,450 2,210 1,340 1,050 680

-AHo (Hayakawa) (cal/moIe)

2,220 1,610 2,370 1,500 1,210 840

2,140 1,420 2,100 1,350 1,250 810

A: at center of lattice surface triangle of ions, over second layer ion of same charge.

-, + denote sign of change. B: same as A with opposite change in second layer. C: Directly over surface ion.

(c) Argon on KCI, comparison between experiment and theory. [From Table 2.14 of Young and Crowell (1962).]

(1 11) face (100) face Ho(l1 l)-Ho( 100)

- AHo (expt)

-AHo (calc)

(cal/mole)

(cal/mole)

Young

Hayakawa

Young

Hayakawa

2,600 2,100 500

2,460 2,080 380

2,300 1,800 500

2,140 1,900 240

229

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

TABLE VII NONPOLAR MOLECULES ON IONICCRYSTALS-ADDITIVITY POTENTIAL AND ZERO-COVERAGE THERMODYNAMIC POTENTIAL PARAMETERS (a) Variation of potential over surface cell, after Hayakawa (see Table VII(b) for site definitions). [From Table 2.12 of Young and Crowell (1962).] Site A System (Adsorbate-adsorbent)

- AH0

(cal/mole)

Ar-NaC1 N-NaC1 CO -NaCI Ar-KCl N-KCI COZ-KCl Ar-KBr N-KBr C0,-KBr (The site giving the

Site B - AH0 (cal/rnole)

Site C - AH0

(cal/mole)

Site D -AHo (cal/mole)

-AHo (expt) (cal/mole)

1,875 1,624 1,978 1,510 2,190 1,901 1,532 2,398 1,376 2,670 3,572 5,419 3,181 2,567 6,070 1,615 1,898 1,637 1,473 2,080 2,117 1,737 2,010 1,483 3,010 4,148 3,871 4,970 2,806 6,400 2,325 1,867 1,854 1,614 2,440 2,482 1,954 1,555 3,280 2,348 4,927 4,131 5,298 2,923 6,780 greatest interaction is shown in italics in each case.)

(b) N, I KCI variation of quadrupole contribution over surface cell after Drain. [From Table 2.16 of Young and Crowell (1962).] Adsorption Total energy for EQ for Orientation interaction argon energy nitrogen correction (cal/rnole) (cal/mole) (cal/rnole) (cal/mole)

Position of nitrogen molecule ~

~

(A) (B) (C) (D)

Above center of lattice cell Above midpoint of lattice edge Above a K + ion Above a C1- ion

1,590 1,310 1,420 1,230

820 870 1,150 400

0 -300 - 500 0

2,410 1,880 2,070 1,630

(c) Variation of quadrupole contribution over surface cell after Hayakawa. [From Table 2.17 of Young and Crowell (1962).]

EQ zo EQ over center of over center of over System lattice cell (A) lattice cell (A) alkali ion (C) (Absorbate-adsorbent) (calimole) (A) (calimole) N-NaCl CO2-NaC1 N-KCl COZ-KCl N-KBr C0,-KBr

550 1,328 815 2,432 1,130 2,915

3.38 3.28 3.25 3.09 3.05 2.99

1,738 4,184 1,522 4,390 1,664 3,924

20

over alkali ion (C)

(A) 3.12 3.03 3.31 3.18 3.29 3.27

E. Chanoch Beder

230

forces, proposed by Lennard-Jones, Bardeen, and by Marganau and Pollard (see YC, 1962, p. 13), all of which vary inversely with z3. We present here only that of Margenau and Pollard: 1 (2.3.1) EMp = - A,(v~) 8z3 2nm vo 2rs

[(fi)

where A,(vo) is the polarizability of the metal per unit volume at frequency vo, fo the oscillator strength of gas atom, Co x 2.5, r, the radius of a sphere containing one conduction electron, and ci the polarizability of the gas atom, Although repulsive forces have been worked out by Pollard (see YC, 1962. p. 24), it is at present conventional to use the same repulsion form as for nonmetals. In Table VIII, YC give the interactions of helium and hydrogen with metals of different values of r, (Ni : r, = 1.38 A). The HzlNi case compares favorably with experiment. TABLE VIII HELIUM AND HYDROGEN ON METALLIC MODELSOLIDSTHEORETICAL HEATS OF ADSORPTION AFTER POLLARD“

1 .o 1.5 2.0 2.5 3.0

1.9 2.3 3.0 3.7 4.3

765 325 140 68 37

2.3 2.7 3.3 3.8 4.6

865 730 375 200 110

a r, is the radius of a sphere containing one conduction electron. [From Table 2.8 of Young and Crowell (1962).]

A detailed study of the periodic amplitude of the potential field for metals has not been carried out (cf. YC, 1962, pp. 16 and 17). At best we have the work for argon on graphite, the latter being “metallic” in one set of crystal planes and “ molecular in those perpendicular. The results, Table IV(b), indicate a periodic amplitude of about 2-4 % of the average potential energy. ”

2.4. APPROXIMATE INTERACTION POTENTIALS

It is noteworthy that in QM gas-solid scattering theory none of the above theories have been used directly. For mathematical reasons simpler forms have been assumed, which retain some principal features of the more accurate

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

23 1

ones above. Normal to the surface the Morse potential is widely employed for metals as well as for nonmetals. This potential was first proposed by Morse (1929) for diatomic molecules and used very successfully for obtaining vibrational energy levels : V(R)= D[exp{-2~(R - R,)} - 2 exp{-Ic(R - R,)}].

(2.4.1)

Herzberg (1950, p. 99) has compared the Morse potential with the exact one for the chemically bonded case of H + H, cf. Fig. 5(b). The attractive part vanishes more slowly than the dispersion-force potential ( N R - ~ )cf. , Schiff (1955, p. 304). The author has not seen a similar comparison with "exact" cases of weak van der Waals binary interactions.

a. Elastic Scattering LJD (1936c,e) replaced R by z in (2.4.1) and assumed that the resulting expression, with additivity parameters as in Fig. 9, represented the normal component of the gas-solid interaction with the crystal as a whole:

V(z)= D[exp{ - 2 4 ~- z,)} - 2 exp{ - ~ ( -z z,)}].

(2.4.2a)

For collisions where EBB D, Jackson and Mott (1932) and Jackson and Howarth (1935) used the simplified purely repulsive form, which contains one parameter less than (2.4.2a):

V(z)= C exp[ -a(z - z,)].

(2.4.2b)

This binary form is compared with the LJ potential by Herzfeld and Litovitz (1959), cf. Fig. 5(a). For elastic scattering by alkali halides, LJD (1936c,e) added to (2.4.2a) the following first-order approximation to a periodic potential field : U,(x, y , zj = 2Be-"

X

cos 27~( d

3

+ cos 2z -

(2.4.3)

where the value of the constant p is obtained from the work of LJ (1932) mentioned above, or preferably from the many recent data reported by YC (1964), cf. Section 2.2. (No periodic term has been proposed for metals.) The LJD (1936c,e) interaction potential for elastic scattering from alkali halides is, therefore,

w ,y , z ) = V(z)+ U d x , y , z),

(2.4.4)

where V(z) and U l ( x , y , z ) are taken from (2.4.2a) and (2.4.3), respectively. Evidently with the use of these approximate interaction forms, the scattering can be correlated with fewer physical parameters of the interaction than in the more exact cases discussed above. We might expect to obtain values

E. Chanoch Beder

232

for heats of absorption and energies of excited bound states, lattice spacing d, range of interaction K - ' , equilibrium position z,, , and finally the magnitude of the periodic effect, p. A simplified approximation to the LJ interaction (2.4.4) that retains the main features in a crude way is the square-well version proposed by Beder (1964), cf. Section 5.4:

X

=

co,

z < 0.

(2.4.5)

Finally, a Debye-Waller correction factor to elastic scattering to account for the average effect of lattice temperature has been proposed by Howsmon (1967). The gas-solid potential used, however, is quite different from that of LJD (2.4.4), in that it vanishes outside the lattice and provides for considerable penetration into the periodic part of the lattice, cf. Section 5.5. Howsmon has reported preliminary results and expects to have some numerical values shortly.

b. Inelastic Scattering The dynamical incorporation of lattice coordinates is accomplished in a straightforward way with (2.2.lb), where an additive potential is justified, cf. Section 2.2. With the Morse potential, the binary interaction is Vj(lr - Rjl)

=

D[exp{-2~ Ir - Rjl} - 2 exp{ --I< Ir - Rjl}]

=

D[exp(-2~ Ir - uj - RjoJ)- 2 e x p { - ~ [r - u j - RjoI)], (2.4.6a)

and with the purely repulsive potential we have

Yj(lr- RjJ)= Cexp{-a Ir - RjJ)= Cexp(-a Jr- uj - RjoJ}. (2.4.6b) As discussed at length in Sections 2.1-2.3, the additivity assumption applies to molecular, covalent, and ionic solids. In metals, the conduction electrons dominate the gas-solid interaction. Nevertheless, virtually the only inelastic gas-solid interaction potential which has been used is the ID model proposed by LJS, Section 6.2, where the gas atom interacts explicitly with one lattice atom only and is, therefore, essentially a binary interaction' [an assumption The interaction parameters are the rigid-lattice additivity ones, however, as in Eq. (2.4.2a).

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

233

which may be justified for high energy impacts (cf. the article by Stickney in this volume)]. This form has been used indiscriminantly for ionic crystals and for metals. The additivity assumption with a thermally excited lattice, Eq. (2.2.1b), has not been used. Furthermore, no periodic effect has been proposed for the LJS model. One innovation is that of Gadzuk (1967), who applies a pseudo-Debye- Waller factor correction to account for lattice temperature, Section 6.3. 2.5. BREAKDOWN OF THE ELECTRONICALLY ADIABATIC POINT-MASS INTERACTION POTENTIAL In the gas-solid scattering under consideration, only nuclear coordinates are dynamically active. Electrons contribute implicitly to the potential energy. (In Section 2.2 we have discussed the error resulting from the neglect of finestructure splitting in the electronically adiabatic derivation of interaction potentials, as well as the restriction to low rotational and vibrational excitations.) Any phenomenon in which electrons become significantly dynamically active is beyond the present scope. This excludes chemisorption interactions, in which electrons are transferred and/or shared if the resulting bond energy appreciably exceeds 0.1 eV, the magnitude of physisorption energies. It excludes energetic impacts that cause appreciable secondary electron emission from metals. The thresholds for secondary electron emission compare with work functions, i.e., several electron volts, cf. Kaminsky (1965) and Fig. 2 here. In nonmetals we must exclude electronic excitations (exitons). Fox and Schnepp (1955) showed the spectral lines of the molecular solid, benzene, to be slightly split compared with those of the gas spectrum. We may deduce that collisional electronic excitations in molecular solids are not appreciably unlike those in a gas. For neutral beams, the evidence (cf. Massey and Burhop, 1952; Hasted, 1964) shows thresholds far in excess of the resonance energies and may, therefore, be disregarded here. The exiton threshold in alkali halides, on the other hand, may be as low as several electron volts (Kittel, 1956, Chap. 18). The lattice model obviously breaks down with sputtering. Lattice dissociation energies should provide a lower bound for sputtering threshold energies, cf. Fig. 2. For alkali halides we find, for example. 7.83 eV for NaCl and 9.35 eV for NaF (Pauling, 1960, pp. 506-511). In metals, sputtering thresholds with ion beams are found to be as low as 10 eV, approaching the lattice binding energies (cf. Kaminsky, 1965). It appears that, for the incident atom energies under consideration, sputtering could be important only for molecular solids. Finally, at very low temperatures, nuclear symmetry may have a predominant effect [ortho and para hydrogen, for example (cf. Hill, 1960, p. 466)] as

E. Chanoch Beder

234

well as paramagnetism [Ref. 11 of Kittel (1956, pp. 136-138)]. We shall not consider gas-solid scattering at these temperatures.

111. The Hamiltonian of the Point-Mass System We have discussed at length in Parts I and IT the justifications and limitations of the point-mass model. We shall now consider the dynamics of such a gas-solid system. The point-mass system is the model of almost all gas-solid scattering theory to date, and of this work the larger part is classical. We shall discuss a number of quantum theories and quantum effects inherent in this model, which have been considered. Many of the important concepts are manifested in the 1D model shown in Fig. 10. We shall use this model wherever it lends itself to a

M

-

0

FIG.10. Schematic of gas atom incident on one-dimensional lattice. Equal spacing, identical lattice atoms, and longitudinal motion assumed. Small circles denote equilibrium sites.

clear discussion of the 3D aspects which may be found in accompanying references. The analytical complexity of the 3D model is such as to frequently obscure the physics. Nevertheless, we must regard the 1D model with extreme caution as it may be qualitatively as well as quantitatively incorrect. THE ONE-DIMENSIONAL HAMILTONIAN For mathematical simplicity we assume all atoms of the solid to be identical and the lattice to have characteristic bulk spacing d (Bravais lattice). (2D and 3D models we shall assume to be square and cubic, respectively.) Lattices with bases have not been treated. Many lattice vibrational frequency spectra (including surface modes in some cases) have been determined using both analysis and experiment (cf. Brillouin, 1953; Maradudin et al., 1963). We shall not discuss the details of these determinations and shall use the results to the extent that the analysis permits. In practice only acoustic phonon

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

235

spectrums have been used.12 We assume the QM system of point masses to have a classical analog. Therefore, the Hamiltonian is the sum of the pointmass translation and potential energies, the latter a function of instantaneous mass positions. Using position coordinates R j in some inertial frame we have N p.2 P2 H = C L + 4 Y ( r , R l , ..., R N ) + - * (3.1) 1 2Mc 2Mt7 Thermally excited solids assume a stable thermal equilibrium configuration, and the lattice nuclei execute small oscillations from " equilibrium sites." In a strict sense this configuration is defined by the minima of the free energy and the thermodynamic potential (Landau and Lifshitz, 1958, p. 48). However, in the harmonic approximation we may use the classical result, namely, that the minimun of the potential energy defines the equilibrium configuration (Ziman, 1964, p. 63):

a q a R i = 0. (3.2) The ( N + 1) equations of (3.2) should furnish the equilibrium coordinates {ro , R , , ,. .. , RNO}. Thermal expansion, being very small for almost all solids below the melting temperature, will not affect the equilibrium configuration insofar as it relates to gas-solid scattering. The lattice spacings are very little affected by proximity of nuclei to the surface. Germer and MacRae (1963) showed the first and second layers to have a 5 % greater separation than the bulk. A good approximation to the equilibrium configuration of the lattice is one of equal spacing. The equilibrium coordinates of the lattice become Bravais lattice vectors: R , + , - R,, = I,,, = md, (3.3) where I, is a lattice vector and d is the lattice spacing. Lattice atoms move about their equilibrium sites with amplitudes that are usually small compared with lattice spacing (except at temperatures approaching melting). It is therefore convenient to introduce displacement coordinates : u =r ~j

- ro,

= Rj

(3.4)

- Rjo.

The potential energy 2'4 may now be written %(r, R , , . . . , RN)x Uc(Rl,.. . ,RN)

+

N

j= 1

(3.5)

Vj(r - R j )

N

. , , uN, d ) + C Vj(r - ~j

z Uc(ul,.

j=1

= l2

uc+ Y .

Consistent of course with the Bravais lattice assumption.

- Rjo)

(3.6) (3.7)

E. Chanock Beder

236

In (3.5) we assume additivity in two ways: Firstly, that the potential energy may be separated into that of the isolated crystal plus that of the gas-solid interaction. Secondly, that the gas-solid interaction is the sum of binary interactions. In (3.6) we introduce displacement coordinates and assume equal spacing in the lattice. We may now write (3.1) in the form:

H

= H,

+ 9'" + Pz/2M,,

(3.8)

where H, defines the Hamiltonian of the lattice: (3.9) The lattice potential energy U,may be expanded about equilibrium as follows (Pauling and Wilson, 1935, p. 292):

We rewrite (3.10) as follows (3.11)

where (3.12)

In (3.11) the constant term, having no physical significance, has been ignored, and the linear term must vanish since, approximately, no forces act on the particles while at their equilibrium sites in the equal spacing approximation. The quadratic terms account for harmonic oscillations, the cubic and higher order for the anharmonic ones. These latter terms account for a number of important effects including heat conduction (Ziman, 1964, pp. 63 and 204). In the absence of anharmonicity as we shall discuss below, the lattice energy becomes a noninteracting phonon gas, interacting only with the boundaries of the (isolated) solid. The harmonic approximation has been used effectively for bulk or macroscopic properties where surface effects are at most secondary. However, the average thermal excitation energy, approximately 3kT, is about 0.13 eV for 500"K,and therefore considerably less than some of the gas atom

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

237

impact energies under consideration. Whether anharmonic lattice effects in gas-solid scattering are important or not has not been investigated. In all QM theories the harmonic lattice approximation has been used. The lattice Hamiltonian in the harmonic approximation is (3.13) The importance of the harmonic approximation is, of course, in the transformation to normal coordinates {(, ,p,} whereupon H, becomes the sum of N uncoupled harmonic oscillators. When we assume that the lattice of N atoms is part of an infinite periodic one with a cyclic behavior in Ndintervals (the Born-von Karman condition), the transformation becomes (cf. Pines, 1963): (3.14) (3.15) and the resulting Hamiltonian is (3.16) In this model the dependence of the physical space coordinates (u, ,P,)upon the normal coordinates differs only in the phase factor. The lattice is known to be less stiff near the surface, however. By inclusion of a free surface in the lattice model, the coefficients in (3.14) and (3.15) become dependent upon location relative to the surface. There is a very rapid exponential decay of this dependence away from the surface such that, beyond about the fifth layer, the lattice vibrations have virtually bulk properties. The surface effect on the frequency spectrum is manifested in the localized surface modes which are active in the first few layers. These effects have been treated rather extensively In the recent work of Germer and MacRae (1963), MacRae (1964), Maradudin and Melngailis (1964), Maradudin et a/.(1963), Wallis (1964), Wallis and Gazis (1964), Goodman (1965a), and Gilbey (1967b). A comparison of surface and bulk frequency spectrums is shown in Fig. 11 by Gilbey (1967b). Furthermore, scattering is affected by the mean-square-displacement amplitudes, and the effect of the frequency spectrum on these amplitudes is shown by Goodman (1956a), cf. Fig. 18. In gas-solid van der Waals scattering the surface layers of the lattice are the main scatterers, and therefore the above surface effects must be considered. Goodman (1965a) and Gilbey(1967b) have treated this problem, cf. Section 6.6. We may now write the Hamiltonian of

E. Chanoch Beder

238

0

w/w

D

FIG.11. Theoretical normal mode spectra reproduced from Gilbey (1967b, Fig. 5). (1) Debye, nondispersive,(2) Rubin (1960) bulk atom (nearest-neighbor, central, and noncentral forces), (3) Rubin surface atom. Curves shown are summations over longitudinal and transverse modes.

the gas-solid system in the harmonic lattice and additivity potential approximations :

P2

N

C {tp,Z + fo,25;>+ 1Vj(r - uj - Itj,) + -.

(ID) (3.17) *M, Numerous gas-solid scattering solutions for this model have been proposed, but in every case additional approximations are made. These solutions are discussed in the following chapters. H=

4

j=1

IV. General Scattering Theory 4.1. SCATTERED BEAMDISTRIBUTIONS The steady-state scattering of a monoenergetic collimated beam of gas atoms by a solid surface is properly described by the energy eigenstates of the closed gas-solid system. Ideally, therefore, we would solve the following time-independent Schrodinger equation : HY =EY,

(4.1.1)

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

239

where E is the energy eigenvahe of the gas-solid system. The problem is simplified by taking a semiinfinite lattice for all considerations other than the lattice total energy and its frequency spectrum. This assumption is justified by the short range of the gas-solid interaction, which is at most several lattice spacings. Furthermore, the perfect lattice assumption will produce coherent scattering only. The energy eigenfunction will always be restricted by the following: (1) It is everywhere bounded. (2) It is everywhere continuous. (3) It has continuous first derivatives. (Exception: at an infinite potential discontinuity, conservation of probability current density is sufficient.) (4) It will vanish wherever V +a. ( 5 ) It will include an incident (unit-number density), constant-energy plane wave exp(ik, r) where ki,< 0, and kXi,k,,, are assumed positive. An approximate expansion of Y in gas-atom momentum eigenstates for the spatial domain far from the surface gives the desired momentum distribution in the scattered beam13: Y r exp(ik,*r)

P

+ CA,,,,exp(ik,.r) + k,,,

rl

J

A,(k,) exp(ik;r)

d3kr, (4.1.2)

where k,, > 0, and q represents parameters and states of the scatterer (crystal). IAk,I2 is the number density in momentum state of k,, and IAk,12k,,/k,i the corresponding scattered relative number flux [cf. Eq. (5.1.10)] when k, assumes discrete values. The summation over q is symbolic and may include an integration. In the case of elastic scattering from a perfect (rigid) crystal, only the sum in (4.1.2) is present, giving sharp diffracted beams. If thermal oscillations of the lattice are active, then both elastic and inelastic scattering occur, corresponding to the discrete sum and continuous integration in (4.1.2). Alternatively, the result may be expressed by means of a differential scattering cross section. An exact solution for (4.1.1) has been obtained for only one very simple case of elastic scattering, see Section 5.4. In all other solutions the first-order perturbation theory is used. (No second-order theory has been developed.) Accordingly, the Hamiltonian is separated into two parts : H = H0 + H'.

(4.1.3)

H o is the unperturbed Hamiltonian of the gas-solid system. H' is the perturbation, which is small compared with H o and which has been formulated l 3 To satisfy the uncertainty limitations on simultaneous measurements of position and momentum. Only in the limit of z -+ 00 is the expansion precise.

E. Chanoch Beder

240

to represent all or part of the gas-solid interaction energy. The choice of H' depends largely upon the specific scattering problem of interest. Obviously, the smaller H',the better the approximation. The energy eigenstates of H o are obtained from

H o Y o = E0Yo.

(4.1.4)

In all theory to date H o may be separated into two parts, one containing gas atom observables only (H,) and another containing crystal observables only (H,). We may therefore write HO = HB

+ H,,

Y o= Y g Y c ,

(4.1.5)

E o = E, -k E,. The nondegenerate energy eigenfunctions of H, belong to the eigenvalues ( E g ,k x i ,kyi),or to ( k X i k, y i ,kZi)by conservation of energy: 2M,E,/h2

= k2i

+ k:i + k:;.

(4.1.6)

For this reason it is convenient to follow the widely used convention in perturbation scattering theory of labeling the unperturbed gas-atom energy eigenfunctions by their incident momenta at infinity : IEg 3

kxi 3 kyi) = Ikxi, kyi kzi) = Iki). 7

(4.1.7)

Furthermore, in no case treated has Hg included a periodic effect (the energy eigenfunction becomes a purely specular elastic reflection), therefore, kXi= k,, , kZi= - k,, , and the labeling with incident momenta at infinity is not essentially different from labeling with reflected momenta at infinity. We may, therefore, omit the subscripts i or r in Ik). By another convention unprimed and primed symbols are used to denote states before and after transition, respectively. Accordingly, we shall use the symbols Ik) and Ik') for the gasatom energy eigenfunctions. Elastic scattering theory for a perfect ionic crystal has been rather extensively treated, and intensity distributions in the scattered beams have been obtained. The most sophisticated of these is the LJD (1936c,e) first-order perturbation treatment for the first five diffracted waves, in which the Morse potential (2.4.4) is used. The selection rules and probabilities of selective adsorption are obtained in this theory. Selective evaporation receives only a thermodynamic treatment and is omitted from scattering. A simplified squarewell model of the LJD Morse potential was used by Beder (1964) with which an exact solution of the Schrodinger equation (4.1.1) is readily obtained as well as the intensities following (4.1.2). Howsmon (1966, 1967) has produced a solution including the thermal oscillation effect (Debye-Waller factor) using

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

24 1

an interaction potential quite different from the LJD one, cf. Section 5.5. This solution is yet to be evaluated numerically. Inelastic scattering theory per se has not been produced. The possible exception is an essentially one-dimensional treatment incorporated in thermal accommodation theory. This limited work will be discussed in Part VI. 4.2. ELASTIC vs INELASTIC SCATTERING

Before discussing specific QM solutions, it is pertinent to inquire into the conditions for which the scattering is elastic, for then the treatment is very greatly simplified. Estermann et al. (1931) demonstrated the elastic nature of the scattering of helium by LiF experimentally. Criteria for such scattering have only recently been considered (none of which, however, seem to offer a plausible explanation for the Estermann et al. results). Three characteristic times enter into the discussion: Zc

-

(4.2.1)

K-l/Vg

is the "collision time" where K-' is the "range" of the gas-solid interaction and vg is the speed of the incident gas atom; Z,

-

2d/v,

(4.2.2)

is the lattice sound-speed propapation time for a distance of two lattice spacings, the shortest lattice wavelength possible; zp

-

(4.2.3)

is the lattice oscillation period and is of the order of the inverse Debye cutoff frequency. Using the Debye spectrum, it is easy to show that Z, T ~ since ,

-

But

N / P - 0(1024~ r n - ~ ) and therefore, 7,

-

d - lo-* cm;

Z*.

a. Slow Adiabatic Collisions

When T~ B z P we have a slow collision. It is well established for the head-on collision between a gas atom and a diatomic molecule, that a collision of this type is adiabatic (or elastic) and no energy is exchanged between the translational kinetic and the internal vibrational mode of the molecule. As explained

E. Chanoch Beder

242

by Landau and Teller (1936), the vibrational structure, undergoing many oscillations, adjusts continually to the slower moving incident particle (cf. Clarke and McChesney, 1964, Section 7.14). Zwanzig (1960) reasoned that. the situation was quite different when an atom strikes a lattice. In this case the energy imparted to a surface atom is propagated away, at approximately the speed of sound, by means of the coupled lattice nuclei. It follows that a long duration collision is more likely to be inelastic. Goodman (1962a, p. 1271) contends that Zwanzig’s conclusion is valid for the 1D model but not so for the relatively rigid 3D one, in which case the slow collisions tend more toward the adiabatic behavior. This conclusion is supported by the lower AC’s in the 3D case, which compare favorably with experiment (Goodman, 1963). Gilbey (1962) has reached a similar conclusion for continuum models. Stickney (cf. this volume, Part 11) has distinguished between “ slow ” collisions which have adiabatic character and “ very slow ” collisions which are nonadiabatic by virtue of sound-wave energy propagation. This distinction is shown to agree qualitatively with recent reliable thermal AC data of Thomas (Stickney, this volume, Fig. 2). Stickney has thus indicated a way to reconcile the seemingly conflicting conclusions of Zwanzig, and Goodman and Gilbey above.

b. Fast Adiabatic Collisions For high energy impacts, the uncoupled (Einstcin) crystal model is believed to apply. The interaction is then sensitive primarily to the steep repulsive part of the interaction potential, and the hard-sphere impulsive collision model is a reasonable approximation. The collision time 2, is then small compared with z, , and the slow adiabatic criterion is not fulfilled. This model may accommodate large energy transfers; nevertheless, in a limiting condition, the collisions tend to be elastic as we shall now discuss. In the classical 1D treatment, the solid is replaced by an harmonic oscillator. For p = M J M , 4 1 and the oscillator initially at rest, average energy exchange is AE/E, = 4p/(l p)’ + 4p. For very small p, the collisions are approximately elastic. Similar conditions are found in the experiments of Smith and Saltsburg (1966) where He4/Au19’ gives 4p = 0.081. Consider the case with gas temperature 2550°K and crystal temperature 600% (Stickney, this volume, Fig. 13). The specularity of the scattering is indeed quite pronounced. In a more sophisticated treatment, impulsive collisions of this nature with nonlinear hard-sphere coupling have been treated by Shuler and Zwanzig (1960) for p = f and Gilbey (1967a) for p = 0.1 and 0.OI.Shuler and Zawnzig give detailed QM transition probabilities, but p is too high for present purposes. Gilbey has obtained the average energy exchange for both classical and quantum theories, with the oscillator in initially excited states. Comparing

+

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

243

ground state oscillators for Eg/hw= 10 we find an average of 3.2 and 0.37 quanta transferred for p = 0.1 and 0.01, respectively. This is the expected trend towards specularity with decreasing p. On the other hand, for a given p and Eg/hw,the result becomes more specular with higher initial excitation levels of the oscillator. The detailed transition probabilities of Gilbey's theory should be examined for the occurrence of elastic scattering.

V. Elastic Scattering 5.1. SOME GENERAL FEATURES IN THE PERFECT CRYSTAL APPROXIMATION The simplest gas-solid scattering model is that of the rigid lattice with all atoms at their equilibrium sites. The lattice atoms are then dynamically inactive and the gas atom moves in a fixed potential field. Gas atom coordinates only appear in the dynamic equation. We shall use r = r(x, y, z ) with coordinates as shown in Fig. 12. The basic theory was developed by LJD (1936c,d,e) for alkali halides. The crystal appears in this theory as a simple " square lattice." This representation is justified by (1) the assumption that gas atoms virtually do not penetrate the lattice and (2) the diffraction experiments of ES (1930) which showed that the (100) face of LiF acted as a grating of one set of ions only (cf. LJD, 1936d, p. 243). The Hamiltonian (3.17), expressed three-dimensionally, reduces to N

H=

V(lr - Rj,l)

P2 = V ( x , y ,z ) + -,P 2 +2M,

j= 1

(5.1.1)

2wJ

and the Schrodinger equation (4. I ) becomes

(vz + 2M,(E, - V ) Y = 0. h

I

(5.1.2)

In the semiinfinite lattice model the potential field is periodic in the (x, y ) planes parallel to the surface ~ ( xy,, z ) =

: 31

C B,,,,(z) exp 2ni.(rn - + n -

;

(5.1.3)

rn, n are integers and d is the lattice spacing in x, y. From the asymptotic form of (5.1.2) for large z, (5.1.4)

244

E. Chanoch Beder

I

b)

FIG.12. Schematic of molecular beam diffraction by a crystaI lattice. (a) Coordinate system in rigid, simple cubic Bravais model. (---) trace of incidence plane in surface. (b) Incidence plane parallel to ( x , z) plane in (a). This is one experimental arrangement employed by EFS.The specular and first-order diffraction angles are shown.

245

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

we get [cf. (4.1.2)] : 3'' = exp(ik, * r)

+ 1A k p exp(ikr - r) +

z-m

kr

r

J

A(k,) exp(ik, r) d 3 k r . (5.1.5)

By conservation of energy we have

+ k;y + k,?,, 2MgEg/A2= k,Z = k,Z, + k:y + k:, . 2M,Eglh2 = ki2 = k k

(5.1.6) (5.1.7)

The periodicity of the lattice requires I'PI to be periodic. The most general form satisfying this requirement is (cf. Beder, 1964): Y = exp(ik,. r)

+ m,n

(5.1.8) where m, n = 0, f 1, + 2 , . . . , and the conservation of energy becomes

The scattering might be described as coherent parallel to the surface. It is now evident that, for sufficiently large rn and/or n, k:, will have to be negative and give an attenuation exp(- lkzrl 2). This obviously limits the number of diffracted waves. The diffracted waves are discrete and their intensities are given by IA,,12,. ..of (5.1.8) applied to (5.1.5). Thus,

\ \

(5.1.10)

where 8,. is the particle number flux in the scattered beam k, relative to the incident particle number flux. A,, are obviously to be obtained from the full solution to (5.1.2). Diffraction Spots

The required form of Y , (5.1.8), gives the well-known diffraction spots: k,,

+ 2nm/d;

= kXi

kyr= kyi

+ 2nnld.

(5.1.11)

Using the wavelength relation

k = 2n/A,

(5.1.12)

E. Chanoch Beder

246

we may put (5.1.11) in Laue form where cos a, cos p, cos y are direction cosines : m cos u, - cos ai= -1; d

n cos p, - cosp, = - 1, d

(5.1.13)

where ai,p i are measured relative to -x, - y , cf. Fig. 12, and therefore 0 5 cos a j , cos pi I 1. Diffraction spots are designated by (m,n) = (O,O),

specular reflection,

= (0, f I),( & I, 0),

first-order diffracted beams,

From (5.1.9) and (5.1.12) we have, for nonattenuating diffracted waves,

’)’-

(COS

pi + II 4)’ 2 0. d

(5.1.14)

It is evident that for wavelengths exceeding 2d, m and/or n may assume the value 0 only.14 We thus have specular reflection in the limit of long wavelengths. Otherwise the finite number of diffraction spots is given by inequality (5.1.14). The existence and location of diffraction spots is common to all ideal lattice theories independent of the approximation used in the scattering solution and is required by the lattice periodicity. The intensities in diffracted beams will of course depend upon the approximation used. The diffraction spots, specular (0, 0) and first order (0, & I), were observed in the pioneering experiments of Estermann and Stern (1930), which confirmed the de Broglie wave-particle duality hypothesis for neutral atoms. This work is reviewed by Massey and Burhop (1952) with a somewhat different emphasis than the present. The geometry of the ES (1930), EFS (1931), and FS (1933) experiments is shown in Fig. 12. Beams of helium and H, were scattered off the cleavage pIanes of alkalai halides, LiF, NaF, and NaCl. Typical results for thermal incident beams are shown in Fig. 13. The specular and first-order diffracted beams are very distinct. A second crystal was used to diffract the scattered beams, and by means of this second diffraction the “ elastic ” character of the scattering was clearly established. The spread in the diffraction shown is attributable to the Maxwellian distribution in the incident l4 Evidently, specular values only may occur for even smaller values of h, depending on the incidence angles aI and fli (cf. ES, 1930, appendix). In the special case of cos fll = 0, n = 0, then cos fl, = 0 and there is no coherent scattering out of the plane of incidence. Inequality (5.1.14) becomes 1 - (cos at rnh/d)z 2 0. No ( + l , 0) first-order diffracted beam will occur if (1 - cos ul - h/d) < 0 and no (- 1, 0) if (1 cos a, - x/d)< 0. The latter inequality is plotted in Fig. 14(b).

+

+

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

247

beam with a correction for beam width. The intensity (particle number flux) in the first-order diffracted beams was shown to be proportional to the particle number flux in the incident beam and therefore independent of wavelength, cf. Fig. 13(a). We shall return to this result later. The locations of first-order diffracted beams were sharply reproduced in many experiments by EFS. In a few cases monochromatic incident beams were used, cf. Fig. 13(e) (EFS, 1931). The " monochromatic " diffraction spot positions were sharply reproduced, but intensities varied considerably in different experiments, presumably because of crystal surface conditions, Estermann (1966). The thermal beam data have been reproduced in the recent experiments of Crews (1962, 1967) with He1 LiF. It is easy to show from Fig. 12(b) that, for elastic scattering, (5.1.15) Combining the measured value of )I with the direction cosine relation (cos2a + cos2jl+ cos'y = 1) and the known values of cxi , pi, and 1 in the experiment, it is possible from the (0, f1) diffracted beam locations to deduce the lattice period dfrom (5.1.13). From ES (1930) we have, for LiF, d = 2.85 A. In subsequent experiments, Frisch and Stern (1933) revealed the phenomenon of selective adsorption. This consisted of marked attenuations, or dips, in some parts of the scattered beam anornolous to the elastic scattering theory of Eq. (5.1.14), cf. Figs. 13(a) and (b). The explanation of this phenomenon is found in the energy relation (5.1.9), once we recognize the possibility of negative values for kzr. The corresponding " trapped " states in z are a characteristic of the gas-solid potential minimum, normal to the surface. The theory follows. 5.2. SELECTIVE ADSORPTION To explain the attenuations mentioned above, LJD (1936c,e,g) applied firstorder perturbation theory to the elastic scattering problem. In this section we shall derive the transition matrix to get the selection rules for incidence angles which produce selective adsorption (cf. also Massey and Burhop, 1952, p. 604). The probability of selective adsorption is discussed in Section 5.3. The interaction potential (5.1.3) is rewritten retaining the following terms only : V(x,y , z ) = +B,,(z) =

+

+ B,,(z)e(2ffi)x/d+ Bol(z)e(2"i)x/d X

V(z) 2U,(Z) cos 2n -

(

d

+ cos 2n

(5.2.1) (5.2.2)

*

*

FIG.13. Diffraction experiments of EFS (cf. Section V). In examples shown the crystal surface is a cleavage plane of LiF oriented as in Fig. 12. All incident beams are thermal except (d) which is monochromatic. The ordinates of (a) through (e) are the scattered intensities in arbitrary units. All curves are measured data except for the dashed theoretical curve in (a). (a) The specular and first-order diffracted beams are indicated. The two selective adsorption dips are evident. (---) “theoretical ” particle number flux in first-order diffracted beam assumed proportional to incident number density of corresponding wavelength with correction for incident lateral spread, cf. Section 5.1 (FS, 1933, Fig. 1). (b) Effect of incidence on scattered intensities (FS, 1933, Fig. 6). (c) Increased crystal temperature attenuates the specularly reflected beam (ES,1930, Fig. 23). (d) Scattered H2for two beam temperatures (specular and first-order attenuation with crystal temperature shown). a(= 11.5” (ES, 1930, Fig, 16. (e) Two runs of an experiment with a monochromatic incident beam (EFS, 1931, Fig. 17). (f) Observed parabolic relation between components of momenta in incident beams corresponding to the two dips (selective adsorption) in the scattered beams, pzz - 2p, = 0, - 1.25 (FS, 1933, Teil II, Fig. 1).

i

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

a

249

E. Chanoch Beder

250

Here V(z)vanishes at infinity, has a potential minimum, and becomes strongly repulsive into the crystal. The Hamiltonian is separated as follows : D2

(5.2.3) X

(5.2.4)

For the transition matrix (k‘ IH’I k) we solve Eq. (5.1.4), where E o is evidently the gas atom energy E, : 2M

(5.2.5)

by separating variables in (5.2.5) we get (5.2.6)

(5+

k;)Y,,”

(5.2.7)

= 0,

(5.2.8) Yo = YxoYyoYzo or

Ik) = Ik,, k,, k,) = Ik,) Ik,,) lkz), (5.2.9a)

and 2M E h2

88= kk,2

+ k; + kz2.

(5.2.9b)

Since V(z) must vanish at large z, Yo may be a specularly reflected wave: = exp{i(k,x

+ k,y)If(k, ,4,

(5.2.10)

where f(kk, z ) +e - i k z z + eikzz z-+m

(k,z > 0)

(5.2.1 1)

is the asymptotic solution for Ik,) when kz2> 0. Alternatively, when V(z)< 0, kZ2may be negative and Ik,) becomes a “standing wave” in the potential well. In this case we write k,2 = - Ik,12 = -2M, lE,,l/h2,

(5.2.12)

where 1E.I are the positive discrete energies of z-bound states. The “boundstate ’’ wave function is %#““d

= exp{i(k,x

+ k,,Y)If(i IU 4,

(5.2.13)

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

25 1

where

f ( i lk,l, z ) -,0,

z --t

00

(kZ2c 0).

For a transition Y:ree-,Y&nd,the matrix becomes

(v:‘ IH’I):y

=(i

X

Ik,‘l lU1(z, Ik,)

= ( i l k , ’ ] IV,(z)l k,)

k,’ - k, - bd) ] d ( k ;

- k,)).

(5.2.14)

Equation (5.2.14) yields selection rules which relate the x and y components of momentum in the same manner as first-order diffraction [cf. Eq. (5.1.11)ff.l:

k,’ = k,

and

k,’

- k, =

+211/d

(5.2.15)

k,’ = k,

and

k,‘ - k, = +211/d.

(5.2.16)

or We now combine the above with the conservation of energy where a final bound state is possible:

k,’

+ ky2+ k:

= k:’

Consider case (5.1.16); then

k:=

+ -41k1, + d

+ ki2 - lkz’I2.

6)--yIE,J.

(5.2.17)

2

(5.2.18)

Thus the components (k, ,k,) of the incident beam have a parabolic relation for selective adsorption with a separate parabola for each value of lEnl. Evidently the values for IE,I depend upon the form for V(z). LJD used the Morse potential (2.4.2). In the FS (1933) experiments with HelLiF, two experimental curves were obtained, Fig. 13(f). By fitting the theoretical parabolas to these curves LJD obtained values for El, E2 and consequently the parameters K and D in the Morse potential:

HelLiF

Ell (cal/mol)

(crn- I)

D (cal/mol)

-57

1.1ox lo8

175

1.44x 108

261

K

- 129

HelNaF

-80 - 193

E. Chanoch Beder

252

5.3. THELJD FIRST-ORDER THEORY The selective adsorption theory above was a part of the elastic scattering theory developed by LJD (1936c,e) which supplies the intensities in the freefree scattered beams. We shall be able to make a partial comparison of these results with the experimental ones of ES (1930). The Hamiltonian is separated as in (5.2.3) and (5.2.4), and the specific LJD-Morse potential (2.4.4) is used. As we shall specify Eo, k,, k, of the incident wave (where Eo = Eg),the case under consideration is nondegenerate. It is evident from H o that the unperturbed eigenfunctions are independent of the periodicity of the lattice. Nonspecular diffraction is therefore affected by the small perturbation of Yo by H'. We have y! = y o + y(1) (5.3.1) and E = Eo a,, (5.3.2)

+

where Y(') and a, are the first-order corrections to Yo and Eo, respectively. These corrections satisfy a, = ( E o IH'I Eo), (5.3.3)

+

( E o - H o ) Y ( ' ) a,Yo = H'Y'O.

(5.3.4)

For the unperturbed eigenstate (k, , k, , k, continuous) LJD obtained the following solutions [as summarized by Gilbey (1967b); compare this source for concise discussion of solution] : Yo = exp{@,x

+ k,Y)}f(k,, 4,

(5.3.5)

where

f(kz,z) = N(kz)(2de-KU)-'/2 Wa,ir(2de-6U), u =z

N*N

=

- zo,

d = (2MgD)112/uh, p

= k,/lc,

(5.3.6)

r(+- i p - d)r(++ i p - d)/r(2ip)r( -244,

and

where u = 2de-". For large z, f has the asymptotic form (5.2.1I), and correspondingly the asymptotic form of Yo (5.3.5) is Yo x exp{i(k,x 2-m

+ k,y + ~ K +U q ) } + exp{i(k,x + k,y - plcu - q ) } ,

(5.3.7)

and the energy equation becomes

+

2MgE,/h2 = kX2 k,'

+ p2u2;

(5.3.8)

253

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

with (5.3.5) the first-order correction to the energy a, in (5.3.3) is easily seen to vanish:

Equation (5.3.4) then becomes

(EO- HO)y“”

= H’YO

(5.3.9)

or ( E o - H0)Y“”= Ul(z)f(kz,z)

+ exp[i(

k, -

2).+

+ exp[ ik,x + (k,

ikyy]

- ;)y])

+ exp[ik,x + i(ky + $ ) y ]

.

(5.3.10)

The solution to (5.3.10) is = exp[i[ (k,

+ 2 ) x + kyy]}g(k:,) + exp[i[(k,

-2 ) x

+ kyy]]g(k:J

(5.3.11) where the g’s have the asymptotic form 1 ex~(ik‘z)7(kz’ I ui(z)I kz)c r-tm 2kz

g(k,’)-

Energy is conserved through

+

+

(

2’ +

kX2 kY2 k,’ = k, & -

ky2+

or

4z

kii,2 = k: f .;tkx

-

($)

2

(5.3.12)

+

kX2 k,Z

+ k,’ = kX2+ (.-” k + - + k::,, or

k::,,

= k,

2

- 47r

+d k, -

E. Chanoch Beder

254

The first-order perturbation theory thus gives the four first-order diffracted beams which are included in the very general asymptotic form (5.1.8). The terms ki1,2correspond to (m,n) = (f1, 0) and k:3,4to (m, n) = (0, f 1). By inserting the asymptotic form of g(k,, z ) into Y’”, (5.3.1 l), the wave equation Y = Yo + Y“) assumes the approximate momentum eigenfunction expansion form (5.1.5) far from the crystal surface. The intensities in various beams are then given by (5.1.10) or in our present form of (5.3.11): (5.3.13) The complete formula is given in Eq. (17) of LJD (1936e). For a special incidence, k, = 0, Eqs. (5.3.12) reduce to 2

(5.3.14) for the (0, & 1) beam,15 or according to (5.1.15), (5.3.15)

The intensity formula then assumes the simple form

R = /12G($).

(5.3.16) We thus have the interesting result that the (0, f l ) diffracted intensity is independent of wavelength, in agreement with the ES (1930) result, Fig. 13(a) mentioned above. The function G($) was computed for HelLiF (LJD, 1936e, p. 267). The values of d, K, and D were obtained from the diffraction and selective adsorption theory and experiment discussed above. Using the theoretical value /I = 0.04 obtained by LJS (1932) for ArlKCI X NelNaF we may add the resulting values for 8, Eq. (5.3.16), to Table I (from LJD, 1936e). One observes that a does not vary much (about a factor of 3) over the range of $ or k, . These are the only elastic scattering 3D free-free transition intensities that have been calculated. LJD compared these results with those of Frisch and Stern (1933). They concluded that the theoretical results (relative independence of with k,) and the value of /3 = 0.04 compared reasonably well with the experiments cited. Estermann (1966), upon reexamination of the above experimental thermal beam data, characterized the scattered intensity distribution very roughly as follows : Specular (090) -20% First-order (0, fl), 10 -20 (+ 1, fl), etc. -20 Higher order Diffuse background 5 50. l5

There appears to be an error in the LJD (1936e,p. 266) paper, where px = 0 is used

instead of pLy = 0.

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

255

These results varied considerably from experiment to experiment, presumably because of variations in the surface properties of the crystals. For these reasons and because thermal beams were used, Estermann (1966) believes the comparison made by LJD (1936e) with the experimental results is not reliable and at most one can conclude that the order of magnitude is the same. Estermann also believes that, with present day techniques for producing a monochromatic beam and especially ultra-high vacuum, the diffuse background may be appreciably reduced. We therefore lack here a crucial experimental test of the use of the Morse potential (2.4.2a) for the g-s interaction, cf. Introduction. TABLE I X

INTENSITIESR IN (0, f 1) DIFFRACTED BEAMSHe ILiF”

9.5” 14.0 18.4 24.0 29.1 33.7 39.8 a

341 231 175 147 132 124 117

0.545 0.370 0.280 0.235 0.211 0.198 0.187

45.0” 51.3 57.0 63.4 68.2 78.7

113 109 107 106 105 91

0.181 0.174 0.171 0.170 0.168 0.146

2?r/d= 2.20 x lO-*cm-’; D =O.O076 eV; K = 1.10 x 1OBcm-’-

Selective adsorption affects free-free scattering through the inverse process of “selective evaporation.” In the former, incident atoms go into a z-bound state according to the selection rules of Section 5.2. These atoms are later elastically evaporated, also according to Section 5.2, where the transition is (k,, k,, , - i lk,l) -+ (kx’,kyl, k,’). The probability for selective evaporation is obtained in a first-order treatment parallel to that above, in which boundstate functions f ( - i l k z l , z ) appear. LJD (1936~)give the rate of selective evaporation per adsorbed molecule as (5.3.17)

with

(5.3.18) These contributions to elastic scattering must be included with those of (5.3.13), but this has not been done explicitly. Rather the effects of selective

256

E. Chanoch Beder

adsorption and evaporation have been included in the thermodynamic treatment of a gas in contact with a surface of similar temperature (LJD, 1936e). The important transition matrices of (5.3.17) have been evaluated in this work and are available for scattering theory. 5.4. THE2D LINE-GRATING SQUARE-WELL MODEL Using a square-well potential approximation to the LJD-Morse potential (2.4.5) with a 2D model, Beder (1964) obtained the exact scattering distribution corresponding to this model. Calculated numerical values for He1LiF are shown in Fig. 14(a). A QM formulation was proposed for Euler’s momentum theorem in the paper cited. This theorem was then used to compute the tangential force on the surface which attends the scattering of the incident beam, Fig. 14(c). A comparison of the QM and classical forces, Figs. 14(b) and (c), showed the expected divergence toward the longer wavelengths. As the QM forces become more specular, the classical become more diffuse. An extension of these calculations to three dimensions is straightforward [a similar calculation has been done by Morse (1 930) for electron diffraction1. Although the 2D model is not quantitatively reliable, it does show the trend of increased specular intensity with increasing beam energies in the 0.010.06 eV range. This trend was observed by Saltzburg et al. (1967) for He, D,, and H, scattered from the (111) plane of Ag crystals in the plane of incidence. 5.5. ELASTIC SCATTERING BY A THERMALLY EXCITED LATTICE The diffraction or coherent scattering of thermal neutrons and electrons has been intensively studied (cf. Maradudin et al., 1963; Ziman, 1964; Pines, 1963; Kothari and Singwi, 1959; Kittel, 1963). Coherent scattering is the result of interference from the regularly periodic features of the lattice. In the first-order perturbation theory it is assumed that : (1) The particle is diffracted from one free state Yo = exp(ik.r) to another. (2) The diffraction is affected by an extended residence of the particle in the interior of the lattice between free states. (3) Additivity is assumed for the particle-lattice interaction (2.21). The rigid lattice gives elastic or Bragg diffraction only, such that k’- k = g where g is a reciprocal lattice vector and k’ = k by conservation of energy. When the lattice is thermally excited, some inelastic scattering must occur. This continuum (Umklapp process) inelastic scattering occurs at the expense of the elastic Bragg spots, which are attenuated by the Debye-Waller factor. The phonons absorbed or emitted by the lattice satisfy K = k’ - k = g + q

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

257

and E,‘ - Eg = h a , (in the one-phonon exchange case). q is a lattice displacement propagation vector. The continuum scattering centers about the elastic Bragg spots and drops off rapidly away from these spots. The Debye-Waller factor exponent given by Ziman (1964, p. 60) for a Debye spectrum is

W = 4 C IK.k41 = & K Z ( u j Z ) 4

(5.5.1)

2 ~ Z w---23 hM,ke,z

~ c for high temperatures,

T, 4 O D ,

for T, + 0.

(5.5.2) (5.5.3)

Equation (5.5.3) is of particular interest because it gives the effect of the purely QM zero-point motion of the lattice on the scattering, discussed extensively by Goodman (1 965a), cf. Fig. 18. Howsmon (1966, 1967) has attempted an application of these thermal effects t o the Bragg spots in neutral van der Waals gas-solid scattering. He gives the following formulation of an interaction potential :

V(r)

=0

outside lattice

-

V(r) = [ X Y , exp(ig r)] e - w

inside lattice

(5.5.4)

where W is the Debye-Waller factor and

-

d

Y g= - Y(r) exp( - ig r) dr

(5.5.5)

where R is a lattice unit-cell volume. Assuming an incident plane-wave beam, the reflected beam is then obtained by first-order perturbation theory with the usual electron-diffraction or thermal-neutron separation for the Hamiltonian, H’ = P 2 / 2 M , = H,, H’

=

v.

(5.5.6)

The perturbation here is the total gas-solid interaction, compared with the periodic term used by LJD, Eq. (5.2.4). To justify the proposed model, Howsmon (1 966) cites experimental evidence for the appreciable penetration of particles (mainly ions) into the lattice before they are ultimately expelled. At thermal energies this would conflict with the remarkably successful selective adsorption theory and experiment where virtually no penetration is assumed, cf. LJD interaction potential (2.4.4). Howsmon is currently

E. Chanoch Beder

258

(/

m. kx, * 1 -I

0

I

hnm

E = .01 eV 9

E =.03eV

259

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

0 O 0 5O

cos

T

e,

/ /

/

1

20

1'5

10

04

0 5

03

O P

h/d

FIG.14. Elastic scattering by a two-dimensional rigid lattice with a square-well interaction. Helium on LiF (Beder, 1964). (a) Particle number densities in diffracted beams (for unit incident number density). Circle radius equals d/h. Radially inward arrow denotes incident wave. Radially outward arrows denote directions of reflected plane waves. Numbers on arrowsareindexrn in relativeparticlenumber density [cf. Eqs. (5.1.8)and(5.1.10)]. Numbers outside semicircle are relative particle number densities to three decimal places. Waves without numbers have densities less than 0.001; k is nondimensionalized here with d. (b) Distinguished domains of the hid-incidence angle plane for classical and QM scattering theories (domain of " strong diffraction " depends upon specific gas-solid system). Rough low energy (uncertainty) limits are taken as cos 8[ < h/d < cos 8,. The right-hand curve is cos 8, = X/d - 1. (c) Tangential surface force. Quantum mechanical and classical theories compared. (- -) interpolation. < : cos 0, , < : (1/2) cos 0,.

-

260

E. Chanocli Beder

calculating numerical values for scattering intensities, which should be of considerable interest for particle energies in the range above 1 eV. 5.6. QM vs CLASSICAL ELASTIC SCATTERING The QM elastic scattering has wave and diffraction properties, absent in the classical treatment. The domain of the QM theory is determined as follows: A classical theory requires the particle trajectory concept in terms of a simultaneous precise specification of the conjugate components of position and momentum: (I, p ) . The QM wave packet corresponding to the classical particle has a spread in position and momentum, AI, Ap, according to (5.6.1) where p is the average momentum of the packet and L is some reference length with respect to which the particle is localized. L is of course related to lattice spacing since a classical particle may be located within a lattice cell. To Eq. (5.6.1) we add the de Broglie wave relation and the uncertainty relation

p1 = h,

(5.6.2)

Ap A1 2 h.

(5.6.3)

Combining (5.6.1), (5.6.2), and (5.6.3) we get L % 1.

(5.6.4)

The above result is quite general.16 We now apply it to the simple model of a particle incident at pi = n/2 on a simple cubic Bravais lattice with spacing d as shown in Fig. 12. We take the conjugate position and momentum coordinates ( x , p,). If we should require particle localization within Ax I d/2, the criterion for classical elastic scattering becomes :

1<< fd cos cli (classical). (5.6.5) This result is confirmed in a comparison between classical and QM elastic scattering theories, cf. Fig. 14(b) and (c), where the two-dimensional model of Fig. 12 was used for a HelLiF system. Conversely, when 19 fd cos ai

(QM)

(5.6.6)

we should expect QM elastic scattering. This is consistent with the conclusions following inequality (5.1.14), especially that defining the QM domain of 1 S 2d. The rigid lattice surface is an ideal plane with a microscopic structure of the order of 1 10 A. Long wavelengths (A B 2 4 are insensitive N

Equivalently we could use the requirement that the wave packet be small compared with L and that h be small compared with the wave packet.

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

261

to this structure and the reflection is, therefore, specular (as in continuum electrodynamics). Deviations from the ideal continuous plane are also caused by thermal oscillations normal to the surface. To be insensitive to this additional “ microscopic structure,” Massey and Burhop require that the incident wavelength satisfy I % ( u 2 ) l 1 2 sin c l i . They show that this requirement is confirmed in ES (1930) where sharper specular peaks were obtained with lower crystal temperatures, Fig. 13(c). Estermann (1966), however, attributes this result to inelastic collisions at the higher crystal temperatures.

VI. Inelastic Scattering 6.1. GENERAL REMARKS We shall consider only first-order perturbations which represent the present state of inelastic collision theory. We first discuss the I D model with longitudinal motion as in Fig. 10 and then briefly indicate the additional considerations needed for 3D treatment. Let the unperturbed gas-solid energy eigenfunction be denoted by Y = YgYcor Ik, n) = Jk)In) where Ik) and In) are energy eigenfunctions of the gas atom and lattice, respectively, and are obtained from Eq. (4.1.4). From the harmonic approximation for lattice vibrations, we have

In> =

n kq),

rq = 0,

. .,

(6.1.1)

4

where Ir,) is the harmonic oscillator eigenfunction belonging to the eigenvalue (r4 + +)hw,of the qth mode of the lattice frequency spectrum. For later use we write well-known matrices of harmonic oscillations (Schiff, 1955, pp. 65 and 155): (6.1.2) (rq I s 4 ) = &q,sq (6.1.3)

(6.1.5)

(6.1.7)

E. Chanoch Beder

262 Consider the transition

Ik, n)

Ik’, rn).

-+

By conservation of energy

E,’

h2k2 - E, = h2k” -- - C s4hw4, 2Mg

2Mg

s4 = 0, k l , .

. ..

(6.1.9)

4

The transition probability per unit time is given by

The probability of the lattice being in the initial state In) is

14n) = I1w(rq),

(6.1.11a)

4

where

+

w(rq) = Z - ‘(kT,)exp{ - (r4 t.)Rw,/kT,}

(6.1.1lb)

and Z(kT,) is the partition function of the harmonic oscillator: Z(kT,) = exp{ -+hw,}[l - exp{ -hw,/kT,}]-’. With (6.1.1 I), (6.1.10) becomes, after summing over all initial states, 2.n 6W(k’, k) = - p(k’) h

1~ ( nl(k’, ) ml H’ Ik, n)I2 6(E,’ - E, - C s4hoq) 4

n

(6.1.12) in the simplest case of transitions affecting one lattice mode only, where s phonons are exchanged, w(n) + w(rq). The frequency spectrum of lattice vibrations may be assumed continuous with C(w) modes per unit range of frequency. Thus (6.1.12) takes the form 6W(k, k) 211 = - p(k’)C(w) 60 fi

1r w(r) l(k’, rnl H’ Ik, n)I2 6(E,’

- E, - sho). (6.1.13a)

For transitions affecting two modes, for example two-phonon transitions h(w, 02)or h(w, - w2), we get w(n) + w(rl)w(r2)and

+

w(rl)w(r2)l(k’, rnl H‘ Ik, n)lz 6[E,‘ - E, - h(wl f 02)]. (6.1.13b)

x ~1.12

263

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

Finally we introduce the density-of-states per unit range of E i , per unit length, p(k)

= M,/2nh2k'

(1D)

(6.1.14)

and divide (6.1.13a), for example, by the incident particle number flux, Z(kJ

= IYi12~i = 1 *hk/M,.

(6. I. 15)

The result is the differential scattering cross section, per unit range of lattice frequency, per unit length. Equivalently it is the probability per collision for the transition of interest. For one-phonon transitions only it becomes, with the help of (6.1.1 lb), do

-=-

do

2Mg2 C(O) C ~ ( rl(k', ) ml H Jk,n)lz 6(E,' - E, - hw). (6.1.16) h4k'k r

The 3D form of (6.1.16) is the differential scattering cross section per unit range of lattice frequency, per unit solid angle, per unit volume. It is obtained as above with the following modifications. The 3D density of states, per unit range of energy EB) per unit solid angle and per unit volume, is p(k')

= M,k'/(27~)~h~

(3D).

(6.1.17)

Next the Bragg relation for conservation of momentum in coherent inelastic scattering must be satisfied. In one-phonon processes it has the form

k' - k = 9

+ g,

(6.1.1 8)

where q must lie in the first Brillouin zone. Finally, the distinction must be made between the longitudinal and two transverse polarizations in the lattice vibration spectrum. We also write (for a bulk atom only) the 3D lattice displacement relation which plays a central role in the scattering theory uj = (NMc)-

C ehl)51(9) exp(iq * RjO),

(6.1.19)

f4.A

where A = 1, 2, 3 denotes the three polarizations of a lattice wave vector q and en(q) is a unit vector belonging to the wave q and the polarization A. The summation over q should satisfy the conservation of momentum, e.g., (6.1.18). The most sophisticated of inelastic scattering theories to date employ the LJD model shown in Fig. 15. We discuss these theories in the remainder of this chapter. 6.2. THELENNARD-JONES DEVONSHIRE LINEAR-COUPLING SINGLE-PHONON EXCHANGE THEORY Jackson and Mott (1932) formdated the one-collision approach which couples the motion of the gas atom with a single target atom of the solid, Fig. 15, and which has been used in subsequent theories. They used a purely

264

E. Chanoch Beder

FIG. 15. Schematic of QM inelastic single-collision model used by LJD. (--), 1D trajectory of gas atom. Distance between gas atom and single surface target atom assumed to be R= Iz- ulrl.

repulsive interaction and an Einstein crystal. LJD (1936a-f) and LJS (1936a,b) extended this work to Morse potentials (2.4.1) and Debye solids, a much more realistic model, but by virtue of its complexity they were restricted to a linear g-s coupling approximation.” The assumptions of the LJD theory are the following: (1) The gas atom moves normal to the surface only ( z direction). LJS (1936b) treated the full nonlinear case for bound-state excitations and evaporation and found the one-phonon exchange dominant in first-order perturbation theory.

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

265

(2) The gas atom motion is coupled directly to that of one target surface atom of the solid only (whose equilibrium site is taken as origin of coordinates, i.e., R,, = 0). (3) The gas-solid coupling of (2) includes the z-component of the target atom motion only, ulz. The distance between gas atom and lattice atom is assumed to be Ir - ulZl.The g-s interaction has a “binary” form with additivity potential parameters. (4) The gas-solid coupling is linear in ulz . ( 5 ) The target atom has the properties of an infinite-lattice or bulk atom. (6) The solid is a 3D simple cubic lattice, and is isotropic and nondispersive (Debye spectrum). Lattice waves have the long-wavelength polarization of an elastic continuum. The first restriction evidently limits the above treatment to a scattering theory in which the component of momentum of the gas atom parallel to the surface is unaffected. Although the lattice is treated three-dimensionally we shall frequently refer to the one-dimensional configuration for simplicity. The ID Hamiltonian (3.17) is written as follows:

(6.2.1) The second form of (6.2.1) includes assumptions (I), (2), and (3). We define the unperturbed H o and the perturbation H’ as follows: (6.2.2)

(6.2.3a)

H o represents the isolated crystal and the gas atom moving in the rigid lattice potential field. H then represents that part of the gas-solid interaction

266

E. Chanoch Beder

which the thermal motions add to the rigid lattice field. When ulr + 0 the perturbation vanishes, and in this way it has the property of being “ small.” The second form of (6.2.3a), an expansion about ulz = 0, is justified for sufficiently small ulz. The third form expresses ulz in normal coordinates where the superscript in Ti’) takes cognizance of the surface location of the lattice target atom. LJD used this form and retained the first term only with the added assumption that the lattice atom has bulk properties as in the fourth form.’” The LJD transition matrix factors into a lattice displacement matrix and a gas atom matrix:

(k‘, ml u,,V,’(z) I k n> = (4U l z In>
(6.2.4)

Using (6.1.2), (6.1.3), and (6.1.4) the lattice displacement matrix becomes

if one normal mode only suffers a one-phonon transition. It vanishes otherwise. Formally the very same result would be obtained from the onedimensional interaction of the gas atom with a harmonic oscillator with displacement coordinate &. For this “ two-particle ” system, the Hamiltonian becomes

The gas-solid interaction then becomes the sum of such one-dimensional gas atom-harmonic oscillator interactions, one for each normal mode. Gilbey and Goodman (1965) and Gilbey (1967b) have employed this concept very For convenience we list the derivatives of Eq. (6.2.3) for the potential of Jackson and Mott, (2.4.6b): Vdz - UIA = Cexp{-a(z - ~ 1 . ) ) ~ Vl’(z)= -aaCexp(-az) = -aVl(z), (6.2.3b) Vl”(z)= a2Cexp(-az), and for that of LJD and LJS,(2.4.6a): vi(Z-Ui,)= L)[eXp{-2K(Z-Uj,))- 2eXp{-K(Z-U1r)}], (6.2.3~) vi’(2)= 2KD[eXp(-2KZ) - eXP(- KZ)], Vi”(Z)= (2K)’D[eXp(-2KZ) - 4 eXp(-KZ)].

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

267

extensively in their accommodation coefficient theory (see below). Finally, LJD (1936f) evaluated the gas atom matrix for the Morse potential (2.4.2):

V1’(z)l k ) I 2

=

MgZh2wqZ sinh 2np sinh 2np’ -t KZh2 (cosh 2np’ - cosh 2 ~ p ) A,A,, ~

(6.2.7)

where and pz

=~M,E,/IC~~~.

To summarize, the linear-coupling transition matrix has the form

for the absorption or emission of one phonon; otherwise the matrix vanishes. With (6.1.11b) and (6.2.8), (6.1.16) becomes

da

-=--

dw

2Mt 1 ( L ) C ( w ) J(k’I Vl’(z) Ik)12[(efio/kTc - 1)-’ h4k’kNMc 2w

+ (45- 4)]

(1D) (6.2.9)

The thermal AC theory to which similar ID-3D results [following assumptions (1)-(6) above] have been applied are compared with the experiments of Thomas by Gilbey (1962), cf. Fig. 16. 6.3. GADZUK HIGHER ORDER COUPLING SINGLE-PHONON EXCHANGE THEORY Gadzuk (1967) has made a significant extension of the LJD single-phonon theory by writing the linear-coupling lattice matrix (6.2.5) in the following 3D way: a-(ml -ul In) = a-(rq l -ul lrq rt 1) = (r,l 1 - a u1 lrq k 1)

= (rqlexp(-a*u,)Ir,

k 1)

(6.3.1)

where a is a range factor in the interaction of the order of K in the Morse potential.” The first form is essentially (6.2.5). The third form is justified l9 The scalar range factor la1 or K is somewhat artificially reformulated as a vector here to play a role similar to that of K = k’ - k in thermal neutron scattering and is assumed restricted to K < I a I < 2 ~The . objective appears to be to facilitate thecalculation of < U Z > rlater, although it is not clear why this is necessary. Formally this implies a noncentral g-s interaction force contrary to the central forces assumed in the LJD theory.

.a3

I

1

I

I

I

1

I

He1 W

x

i

x

x X

0

0

1

.2

I

.4

I

I

.8

.6

I

1.0

I

1.2

0

I

1

I

I

I

I

I

1.4

Tc’@D

(a1

FIG. 16. The single-phonon pseudo-Debye-Waller factor correction of Gadzuk to the LJD inelastic scattering matrices. (a) Computed for two values of la1 : K and 2~ of Morse potential. (b) Applied to thermal accommodation coefficients. (- -) LJD theory computed by The Gilbey-LJD curve corrected with the Gadzuk factor using la1 = 2 ~ (x) . Unpublished experimental data of Gilbey (1962). (-) Thomas. Gadzuk (1967). The Gadzuk curve is adjusted to match the experimental data at 300°K.

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

269

by the smallness of the lattice displacement, i.e. a - u , 4 I. In this form the interaction has been extended from linear coupling to all high order coupling terms (of the purely repulsive potential form, see below). In the final form Gadzuk makes use of a result of Glauber (1955): " If there exists an operator of the form e', where U is the sum of creation and annihilation operators ( a q +a,)20 , for all normal modes of the field, then, when the field is in thermal equilibrium at a temperature T, that operator deriving from eu which induces n quantum transitions is given by:" (eU)(T.)= ( ~ " / n exp{+(U'),}. !)

(6.3.2)

In the present case U = -a * ul,n = 1, and ( U2), is the expectationvalue of U 2with respect to the thermally excited state. For n = 1, (6.3.2) becomes (exp(-a*u,))',"

z -aulz exp{+((a-~,)~),}.

(6.3.3)

The full 3D form a * u1 has been retained in the thermal averaging factor, but a u1 z aulT for the linear coupling factor, as in the LJD theory, Section 6.2. The odd higher order terms in the expansion of exp( - a ul) contribute single phonon exchanges as follows: A cubic term, for example, may be of the form a,.a,+a; with the result that ho,' is virtually emitted. The net effect is an exchange of hw, only. Finally we have the LJD transition matrix modified by a pseudo DebyeWaller amplification factor as follows:

-

(k', ml ui,Vi'(z) ik, n )

Til)(rql

tqirq

k 1) e x ~ [ f ( ( a u i J ~ >
The single-phonon linear coupling AC of LJD likewise gets multiplied by this factor. Gadzuk has derived ((a u ) ~ ) ,for a three-dimensional nondispersive bulk lattice spectrum with longitudinal mode transfer only2' :

-

In a numerical calculation for HelW, Gadzuk took a = 2~ = 2.6 A-' and 0, = 300"K, with the results shown in Fig. 16(a): The effect becomes as large as 70 % for T OD, but the crucial assumption a * u1 4 1 may begin to fail at T >, 0, as Gadzuk points out. Using this factor as a correction to the LJD (Morse potential) AC's, Gadzuk has achieved a closer agreement with experiment, Fig. 16(b). It should be pointed out that the pseudo Debye-Waller factor thus obtained results from anharmonicity in the gas atom-one target atom coupling, in

-

20

21

+

6, = ( h ~ , / 2 ) ~ ' ~ ( ua?,). ,, See Pines (1963, p. 18). The LJD matrices include longitudinal and transverse modes. p, = (hw,/2)"2i(u,+ - u-,);

E. Chanoch Beder

270

contrast with a similar effect in electron and neutron scattering, which results from gas atom-total lattice coupling [cf. additivity, Eq. (2.2.1b)]. It is of interest that for the Jackson and Mott purely repulsive potential, Eq. (6.2.3b), the lattice matrix in the third form of Eq. (6.3.1) is exact since 1 Vl(z - uI2) - Vl(z) = - - Vl’(z)(e‘U1z - 11. d The corresponding gas-atom matrices are known, cf. Allen and Feuer (1966), and could be used in the above theory. 6..4 THEALLENAND FEUERTWO-PHONON EXCHANGE THEORY

In an extension of the above work of LJD, Allen and Feuer (1967) included both Iinear and quadratic coupling in the interaction H’, cf. (6.2.3a). As the first-order perturbation theory is used, one should expect from this work a small correction to the linear coupling one-phonon results. In this way the transition matrix becomes :

(k’,ml { - u,,Vl‘(4

+ tu:lVl”(z)>Ik, n>

= (k’, ml-. ulzVl‘(z)Ik,n )

+ +(k’, ml u:,Vl”(z)

Ik, n).

(6.4.1)

The first term has been evaluated above, (6.2.8). The second or quadraticcoupling term factors into a lattice matrix and gas atom matrix:

W‘,mlu:,Vl”(Z) Ik, n> = f(ml 4,InXk’I Vl’’(Z)

Ik),

where

(ml u:,

c T!$ytqtq, 1 = -+I c exp[i(q + 4 ’ ) ~ 1 , 1 t q t q * NMC

In> = (ml

In>

In>.

(6.4.2)

For any initial state In), the lattice matrix will not vanish in two cases according to relations (6.1.2) through (6.1.8)”: (1) either one mode only makes a two-phonon transition:

22

Equation (6.1.7), although nonvanishing, belongs to elastic scattering.

27 1

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

(2) or two modes only make a one-phonon transition each:

The second selection rule, (6.4.3), shows that the normal lattice modes are coupled in the quadratic interaction, and it is therefore no longer possible to treat an assembly of isolated gas atoms interacting with single harmonic oscillators, as in the linear coupling case. To summarize, when the linear and quadratic coupling terms of the LJD collision theory are included, the inelastic transitions which may occur are (1) one-phonon of one mode only, (2) two-phonons of one mode only, (3) one-phonon in each of two modes only.

Allen and Feuer (1967) compared the one- a(1) and two-phonon a(2) exchange contributions to the thermal accommodation coefficient: a = a(1) a(2), where the gas temperature approaches that of the crystal. They used the purely repulsive potential of Jackson and Mott, Eq. (6.2.3b). They find the ratio 42) 3a2k2Tc2 Z2 cc(l) = 32n4M,hvD3T;

+

'

I2 and Zl are two complicated integrals. The numerical values obtained by Allen and Feuer for T, = 300°K are given in the following tabulation:

He(W He I Pt He I K

3.2 3.2 1.6

Mc

00 ("K)

184 195 39

330 230 100

I1

2.52~ 2.08 x 1.21 x

12

5.36 x 3.89 x 1.05 x

a(2)/a(l) 0.00458

0.0133 0.0897

From the latter case one may conclude that two-phonon exchanges in scattering may be significant.

6.5. THEFORCED HARMONIC OSCILLATOR THEORY FOR MULTIPHONON EXCHANGEAND FOR QM-CLASSICAL COMPARISONS The LJD linear-coupling theory decomposes the gas-solid interaction into 1D isolated interactions as in (6.2.6). We may discuss energy exchange in these interactions without temperature considerations. The gas atom

E. Chanoch Beder

212

+

trajectory z(t) may be obtained from the Hamiltonian H, = +P;/M, Vl(z). The derivative V,’[z(t)]may be written Vl’(t) or -F’(t) with the classical interpretation of force. In any case, the interaction may now be reinterpreted either classically or semi~lassically~~ as a forced harmonic oscillator with the Hamiltonian

H(0,)

P 2 2

= J-

+ )0&2

-

TpS,F(t).

(6.5.1)

The subscript q will be suppressed for convenience. For a Morse potential, Gilbey (1967b) gets (6.5.2) F(t) = ~ K D ( u-- U~- I ) , where

In the strictly classical case, the gas atom-harmonic oscillator energy transfer, averaged over initial phases, is independent of the initial energy of the oscillator, Gilbey (1967b) :

-

AEs = l F ( 0 ) 1 ~ / 2 M ~ ,

(6.5.4)

where (6.5.5) and for the Morse potential Gilbey (1967b) gives the formula

F(w) = (8M,EN)1/22wz cosech(2wz) cosh pwz, where wz = nM,w/2~(2M,E,)’/~,

p=

142 tan-’ 7c

(:)li2.

(6.5.6)

The results are shown in Fig. 17. In the semiclassical treatment, transition probabilities are determined using time-dependent perturbation theory for the system of (6.5.1) where the Hamiltonian is split as follows: (6.5.7) Solutions have been obtained by Gol’dman and Krivchenkov (l961), Cottrell and Ream (1955), and Gilbey and Goodman (1965). For an arbitrary initial 23

Semiclassical because a precise trajectory defines F(t).

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

273

2.8 I

Languid

2 f c T osc

FIG.17. Average energy per collision transferred to a harmonic oscillator by an incident gas atom (1D Morse potential interaction) calculated by Gilbey (1967b): (-)classicaland semiclassical and (---)Quantum ( K X = 0.1). The QM theory is the single-phonon exchange theory of LJD, 7,= ~ / K Z and I ~ T . . ~= T~ = l / v o s c .

state of the oscillator Gilbey and Goodman (1965) have derived the average energy transferred :

1s

00

(2MJ-I

-m

F(s) exp(iws) ds

I2

and find it identical with the classical result (6.5.4). The average number of phonons exchanged is of course

1. 2

F(s) exp(ios) ds

(6.5.8)

274

E. Chanoch Beder

The author is not aware of a calculation of (6.5.8) for a specific system and varying gas atom impact energies. In a nondimensional calculation, however, Gilbey has compared the classical result with the strictly QM, LJD linear coupling, one-phonon-only exchange theory. For the latter he gets the following average energy exchange :

The author interprets this to be the following: An oscillator in the excited state Ir), after impact with an atom of energy Eg , may remain unchanged or suffer a loss or gain of one phonon, hw. The probabilities for finding the oscillator in each of these three possible final states are known; therefore, the average energy transfer may be evaluated. Using the transition probabilities in the form given by Allen and Feuer (1964) and by Takayanagi (1962), the author has rederived results (6.5.9) in all essentials save possibly for a pure numerical factor. We reproduce Gilbey’s numerical comparison in Fig. 17. We have also given the abscissa in Fig. 17 the interpretation of collision time over lattice vibration period in order to examine the process for adiabatic collisions. Following the discussion of Section 4.2 for head-on atom-diatomic molecule collisions, the adiabatic trend with slow collisions is evident. Very fast collisions (2, -+ 0) on the other hand produce a constant value of average energy exchange dE/E, -+ j)Mg/Mc,independent of the parameter DIE,. From this limiting result one may infer that the calculation includes the assumption M J M , 1. In earlier work, Gilbey (1962) suggested that the linear-coupling, firstorder theory of LJD should allow multiphonon transitions, in light of the above forced-harmonic-oscillator theory. However, Allen and Feuer’s (1967) two-phonon work, Section 6.5, clearly shows that resolution into isolated gas-atom normal mode oscillator interactions breaks down for higher-order g-s coupling and thereby invalidates the multiphonon, forced-harmonicoscillator approach as a rigorous theory for a coupled lattice. For multiphonon exchange, where cIassical mechanics should apply, some rigorous work is now available, cf. final paragraph, this section. In conclusion we observe the following. The linear gas-solid coupling theory does not appear feasible for approximating multiphonon exchange : indeed one might there prefer the simpler and more rigorous classical theory of Zwanzig (1960) and Goodman (1962a,b, 1963, 1965b) where nonlinear gas-solid interactions may be treated in a straightforward way. On the experimental side, there is very little inelastic scattering data to facilitate the choice between single and multiphonon scattering. The above multiphonon exchange

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

275

theory would of course be valid if an Einstein crystal model were valid. Gilbey (1967a) has produced some calculations of this type. However, Feuer (1967) believes the Einstein crystal model is appropriate only for a light impurity target, and it is obviously in conflict with the known frequency spectra of crystal lattices, cf. Leonas (1964, p. 141). Stickney (this volume, Part 11) justifies the dilute gas or zero frequency Einstein crystal for hyperthermal fast collisions, but not for thermal collisions. Nevertheless, the “ hard-cube ” classical theory evolved for this model by Logan and Stickney (1966) has produced scattering distributions in good agreement with some thermal measurements. Thus the controversy over Einstein vs. coupled lattice remains unresolved. 6.6. THEEFFECTOF THE SURFACE FREQUENCY SPECTRUM ON QM 1D ENERGYEXCHANGE The surface layers of lattice atoms are the main scatterers in gas-solid van der Waals scattering, cf. discussion following Eq. (3.16). Therefore, the lattice frequency spectrum to be used should be characteristic of the surface atoms. The lattice frequency spectrum affects the scattering cross section in two ways : Firstly, it appears as a weighting factor C(o)in the differential scattering crosssection. cf. Eq. (6.1.13a), (6.1.13b), and (6.1.16) and discussion following. In the linear-couphg, gas atom-one target atom interaction theory of LJD Section 6.2, it appears only in this way. The Debye and Rubin (1960, 1961) spectra, Fig. 11 compared by Gilbey, have been mentioned following (3.16). Gilbey (1967b) applied these spectra to the LJD thermal AC theory without numerical evaluation. Replacing the Debye spectrum with the more realistic “ Rubin” spectrum should have an appreciable effect on the accommodation coefficient. On the other hand, this coefficient is not likely to be very sensitive to the difference between the surface and bulk Rubin spectra. The differential scattering cross section, on the other hand, should be quite sensitive to this difference. For the extensive work which has been done on surface frequency spectra, the references following Eq. (3.16) should be consulted. Secondly, the frequency spectrum C(o)may affect the matrix

I@’, ml H’P, n>12 through a thermal averaging over lattice vibrations in a manner similar to the “ Debye-Waller factor ” effect. In this case the effect of C(w) depends upon lattice temperature as well as upon the location of scatterers in the lattice.

E. Chanoch Beder

276

Gadzuk, Section 6.3, produced an effect of this type by extending the LJD gas atom-single target atom, linear coupling interaction H’ to include higher order terms. In this case, the matrix gets multiplied by the pseudo DebyeWaller factor, Eq. (6.3.3) and Fig. 16. In quite a different way, one would expect to find a factor of this type when the gas-solid interaction H’ is the result of additivity over the entire lattice, Eq. (2.2.1 b), as occurs in electron and thermal neutron scattering. Howsmon, Section 5.5, has initiated an investigation of this type. In either case one finds, in the exponent of this factor, the mean-square amplitude of lattice atom displacements over the frequency spectrum, cf.

10.

0.5

I 0

‘ _ I _ c

2

4

6

8

ION,

-

‘I I

0

2

6

4

(C)

8

I O N ,

(d)

FIG.18. Mean-square harmonic displacements <xz> of semi-infinite lattice atoms calculated by Goodman (1965a). The lattice iscubic with one noncentral and one central nearestnormal neighbor force constants in the ratio 0.3 (tungsten) for the above calculations. (-) to surface (-- -) tangential. Here x = au where a w K in Morse potential. b is the dimensionless parameter a2(k&,)/4Mcuo2,where yo is the natural frequency of the lattice. Nt = 0, 1, denotes successive layers starting from the free surface. (a) Surface atoms, (b) surface-tobulk ratio, (c) QM limit Tc/& -+ 0, and (d) classical limit T,/& 03. -+

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

277

Wallis (1964). Goodman (1965a), using a lattice model similar to that of Rubin (1960) for surface and bulk spectra, has computed these mean-square displacements, as shown in Fig. 18. The conclusions of Goodman's work are the following (cf. Goodman, 1965a, p. 411): Figure /8(a). The ( ~ ~ ' ( 0 ) )of surface atoms is essentially classical above T/8, z 0.5 and a zero-point QM problem below T/e, z 0.1. Classical theory begins to fail below about Tie, z 0.2. The difference between normal and tangential components is small. Figure /8(b). The normal component mean-square amplitude of surface atoms is 20% greater than bulk at TJ8, z 0 and increases sharply at TJ8, % 1 to 50 % greater. Tangential components behave similarly with a weaker surface effect. Figure /8(c). In the QM limit (7'JdD+O), the surface effect disappears beyond the third layer. Figure 18(d). In the classical limit (Tc/OD-+ 03) the surface effect disappears beyond the eighth layer.

From these results it is evident that the surface frequency spectrum should have an appreciable effect on the transition matrices in first-order scattering theory. The effect of the frequency spectrum in classical theory may be separated from the nature of the gas-solid interaction force as shown by Goodman (1965a). The important function derived by Goodman to study this effect is the " velocity response function." This work should be of considerable importance in multiphonon exchange theory. 6.7. THE RIGIDROTATOR GASMOLECULE In view of the paucity of work on QM gas-solid inelastic scattering for point-mass gas particles, it is not surprising that particles with structure have received almost no attention. The vibrational and/or rotational states of diatomic gas molecules may be incorporated into the point-mass model in the form of two interacting point masses. The magnitude of some diatomic rotational quanta (first level from ground) and vibrational quanta are shown in Figs. 2 and 4. Molecules in the vibrational ground state may be treated as rigid rotators with a spectrum of A2 ho(;) = -[2(j 8n21

+ 111 = 2 ( j + i)ke, = 2 ( j + I)B,~,

where j = 0, 1 , . . . , and 8, and Be are the forms given by Hill (1960) and Herzberg (1950), respectively. The quanta are inversely proportional to the

E. Chanoch Beder

278

moment of inertia I. The harmonic vibrational spectrum has the single frequency w, with a rotational mode fine structure, since the vibrational quanta are approximately two orders of magnitude greater than the rotational. Let the incident particles be in the vibrational ground state. Below some threshold value of impact energy for excitation of vibrational modes, the molecule will act as a rigid rotator. If this threshold value is above critical energies for trapping, the rigid rotator molecules may then be a valid model for scattering theory. These threshold values are not available for gas-solid collisions. They probably exceed the vibrational quanta. We note that gaskinetic vibrational thresholds may be several vibrational quanta. For example, thermally equilibrated N, molecules with an average translational kinetic energy of 0.715 eV require 4800 collisions for a vibrational excitation, Table X (cf. Herzfeld and Litovitz, 1959). For lack of data at present, rigid rotator scattering should not be applied where impact energies exceed about two or three vibrational quanta, or about 1 eV. TABLE X AVERAGE NUMBEROF COLLISIONS NECESSARY FOR TRANSLATIONVIBRATION ENERGY TRANSFER IN DIATOMIC GASES' Nz; 8, = 3336°K

TCK)

pkT(eV)

300 550 68 1 763 778 1020 1168 1273 3480 3910 4200 4630 5540

0.039

0.715

>0.4 x lo6 2.8 x 107 1.2 x 107 1.0 x 107 6.5 x lo6 4.5 x 106 2.3 x lo6 4.2 x lo5 28,000 20,000 12,000 8300 4800

Oxygen; 8, = 2228°K

TCK)

+kT(eV)

288 1173 1260 1640 1910 2370 2870

0.037

0.37

2 x 10' 250,000 44,000 19,000 12,000 5900 1800

From Tables 50-1 and 50-2 of Herzfeld and Litovitz (1959).

QM inelastic gas-solid scattering for a plane rigid rotator molecule has been treated by Jackson and Howarth (1935) and Feuer (1963). All motion is in a plane normal to the surface, as shown in Fig. 19(a). The center of mass of

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

279

the molecule and the Einstein lattice atom, assumed in this theory, execute 1D motion. The gas-solid interaction is assumed purely repulsive, Eq. (2.4.6b), and specifically for the diatomic model of Fig. 19 we get Vl(z,u,,,6) = Cexp{-a(z-u,, = 2C exp{ -a(z

-+fsine)}

+ Cexp{-a(z-u,,

- u,,)} cosh(+af sin 0).

++fsine)} (6.7.1)

For mathematical convenience it would appear, the interatomic distances rl, r , were approximated by L,, L,. This could restrict the analysis to those cases where z B 1. The specific matrices derived by these authors are restricted to one-phonon transitions. For the one-phonon case, however, it is possible in a straightforward way to derive matrices for a coupled lattice as in the LJD work, Section 6.2, to get a more realistic model than the Einstein lattice, at least for thermal energies. The Hamiltonian is

or

Before splitting H into H o and H' for perturbation theory, Jackson and Howarth did a classical trajectory analysis as shown in Fig. 19(b) and (c). E, is the initial rotational energy of the gas molecule and E, its translational energy. By this analysis, two modes of collision are distinguished-for high translational energies E, > E, , gas molecule oscillations are possible, but not for E, < E,. This classification should be reviewed for the more realistic Morse interaction. Where oscillations are possible, the appropriate perturbation energy is

H' = VI(Z, Ulz, 0) - Vl(Z, 0) = 2Ce-"z(e""'z -

1) cosh(+al sin 0)

(6.7.3)

and corresponding

For small oscillations, cosh($af sin a) w 1 - f(#af)202, (6.7.3) may be simplified, as well as for small vibrations, ulz . Where complete rotations only are

280

E. Chanoch Beder

Q

t: 0.075

-

0.050

-

0.025

-

TEMPERATURE

TEMPERATURE

OK

(d

OK

I

FIG.19. Single-phonon translational thermal accommodation coefficientsfor a plane rigid-rotator gas molecule scattered by an Einstein

crystal (Jackson and Howarth, 1935). (a) Schematic of collision model. For interaction potential R * L assumed. (- -) 1D trajectory of lattice atom and gas molecule center of mass. (b) E,,, < El,,,, and (c) E,, > E,,,,,: Classical domains of gas molecule trajectory for repulsive potential; z,, point of translational reversal and z,, minimum for complete rotations. In (b) 8 oscillations only between z, and zt . In (c) complete rotations only. (d) Translational thermal accommodation coefficients for Hz/W and OzlW, C = erg, a= 4 x lo8 cm-' assumed for both, and OD = 205°K (cf. Section 6.7).

x

2 2

E '2 n

E. Chanoch Beder

282

possible, the appropriate perturbation energy is

H' = vi(z, uizr 8) - vi(z) = 2Ce-aZ[eaU1z cosh(+aZ sin

6 ) - 11

(6.7.4)

and corresponding

H o= H,,,

+ 2 -+ H, + V,(z). 1 2

247

For small vibrations, eaulZ= 1 + au12,and (6.7.4) reduces to the LJD case with rotation:

H' = 2Ce-"[(l

+ au12)cosh(+al sin 0) - 11.

(6.7.5)

In either case above, the perturbations have been properly formulated to vanish when internal motions of molecule and lattice vanish. The unperturbed states are the products Ik, mrot,n ) = Ik) Imrot)In), where Imrot) is the energy eigenfunction of the rigid rotator HP,, = +(h2/Z) d2/iW2. The transition matrix for ( k ,mrot,n ) + lk', mLot,n ) is ( k ' , miot, n J H' Ik, rnrot, n). Consider the case (6.7.5). When the lattice exchanges a phonon with the molecule, the transition matrix becomes 2aC(k'I e-" lk)(m;o,l cosh(4af sin 8) lmrot)(n'l ulz In)

(6.7.6)

by virtue of orthogonality of the oscillator eigenfunctions. Otherwise, there may be exchange of energy between translational and rotational modes of the molecule. To (n'1u12 In) we may apply the well-known one-phonon transition work of Section 6.2 and finally incorporate a realistic lattice frequency spectrum into the energy exchange theory as well as the pseudo DebyeWaller factor correction of Gadzuk, Section 6.3. The matrices are available in the original papers cited, and the reader is referred to this work for the translational mode AC theory to which it has been applied. Even multiples only of rotational quanta may be exchanged. The results obtained for Hz and O 2 on tungsten are shown in Fig. 19(d). a,(osc) is the AC of a gas satisfying Fig. 19(b) where rotational transitions have a negligible effect on a, and the result is approximately that of a monatomic gas. For a gas satisfying Fig. 19(c), a,(l) = a,(l/O) + a,(1/2) + .. + a,(l/n) denotes the contribution to translational thermal accommodation made by collisions with n rotational quantum transitions, The " 1 " indicates onephonon-only transitions in the lattice. The collisions with rotational transitions, i.e., a,( 1/2)shown in Fig. 20(d), have a smaller effect on translational accommodation compared with no rotational transition collisions, a,( l/O).

-

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

A

FIG.20. Monochromatic beams scattered by silver crystal surfaces (Saltsburg et al., 1966). In examples shown yi = 50" (cf. Fig. 12a), and y, is in the plane of incidence. T, = 560(&20) "K,and beam temperatures are shown on curves. (a) He4jAg'OB ( l l l ) , (b) H,' IAglo8 (11 l), and (c) DzZIAg1O* (111).

283

284

E . Chanoch Beder

Furthermore, the large rotational quanta of H, are virtually inactive compared with the smaller ones of 0 2 at , least in the temperature range shown. Since the ID average gas kinetic energy for 400°K is about 0.02 eV, we note that the rotational excitation threshold for H, with hw, = 0.015 eV has not been reached according to this AC calculation. The first rotational quantum of 0, is about 0.00025 eV, and, therefore, much lower than the average kinetic energy of the gas-solid vibrations. In this case, the theory predicts appreciable inelastic translational accommodation caused by rotational transitions. We may compare the above theoretical results with related high-resolution measurements of Saltsburg et at. (1967). They scattered monoenergetic beams of He, H, , and D, from the (1 11) plane of epitaxially grown Ag films, for which ho,= co,0.01, and 0.005 eV, respectively. In Fig. 20 the total (translational) particle number flux in the scattered beams is shown. The comparison can only be rough since the crystal temperature (560°K) is quite different from beam temperature. To compare the effect of rotational structure only, Dr. Saltsburg (1967) has suggested that H, and D, be compared since these have the same g-s interaction potential and that the mass difference is likely to be of secondary importance. At each beam temperature the D, specular broadening is the more pronounced. This tends to confirm the Jackson and Howarth analysis showing rotational mode activation of translational energy exchange. Feuer (1963) has generalized the matrix formulas to accommodate a general gas-solid interaction potential. She has also analyzed the important distinction between translational energy accommodation z, and rotational energy accommodation mi and has compared these coefficients for the Hz and 0, work of Jackson and Howarth.

ACKNOWLEDGMENTS The past five or ten years have seen a flurry of activity in the quantum theory of gas-solid interactions which had hardly been touched since the 1930's. Professor I. Estermann requested that I review this work to include important new material presented at the Oxford (July, 1966) and San Diego (December, 1966) Symposiums. The review has, therefore, the virtue of being current at the expense of some care with detail. Professor Estermann as well as Professors Joshua Zak (Technion, Haifa) and Otto Schnepp (University of Southern California) have made considerable time available to me for advice and criticism for which I am greatly indebted. For many ideas, I also wish to thank Professor Eldon Knuth (UCLA), Professor Hendricus G . Loos (U.C. Riverside), Dr. Richard P. Treat (Giannini Scientific Corporation), Professors A. A. Maradudin and R. F. Wallis (U.C. Irvine), Drs. Saul Altshuler, D. Schecter, and P. Csavinsky (TRW Systems), Dr. J. W. Gadzuk (MIT), Professor Paula Feuer (Purdue), Dr. H. Saltsburg and J. N. Smith, Jr. (General Atomic, San Diego), and Professors S. Ben-Avrahani, S. Rosendorf, A. Ron, and M. Revson (Technion, Haifa). For the main effort in typing I wish to thank Sandra Bigos, Ruth Esser, and Sharon Bryant.

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

285

LIST OF SYMBOLS Range factor in interaction potentials (2.2.2) and (2.4.2b), (cm-') C Speed of light in vacuum C(W) Density of states in frequency spectrurn, per unit range of frequency (sec) C Parameter in interaction potential (2.4.2b), (eV) d Cartesian coordinate spacing in cubic lattice (cm) D (minus) potential minimum (2.2.5), (2.4.1), (2.4.2a), (eV) e Charge of electron Heat of absorption from disED persion forces E Energy eigenvalue of gas-solid system (eV) Heat of absorption of gas EO particle on crystal surface with '* zero " coverage E(zo)= Eo eV 1 electron volt = 1.60199 (10-lz) erg, = 23.063 kcal/gm mol, = kT ergs for T = 11,610 K , = 8068.3cn-' (photon wave numbers) E, Energy eigenvalue of gas atom Hamiltonian H, (5.5.6), (5.2.31, (6.2.2), (eV) E, Energy eigenvalue of isolated crystal (3. I6), (eV) - 1 E,I Bound-state energies of gas atom in field of rigid lattice (5.2.12). (eV) Bound-state energy of gas atom on surface of ionic crystal from induced electric dipole moment (eV) Bound-state energy of gas molecule on surface of ionic crystal from permanent electric quadrupole moment (eV) Force Reciprocal lattice vector (cni- ') = C(w)/(3N)(2Mc), (sec gni-') n

= 6.6252 erg sec (Planck's constant) = h/2T Hamiltonian ofgas-solid system Hamiltonian of unperturbed gas-solid system Perturbation Hamiltonian Gas atom Hamiltonian, (5.2.3), (5.5.6), (6.2.2) Isolated crystal Hamiltonian (3.16) Differential heat of adsorption with zero coverage Gas molecule moment of inertia Ionization potential = 1.38042 (10-l6) erg OK-', Boltzmann's conslant Gas atom propagation eigenvector ( k y , k , , k,) before transition in perturbation theory, cf. Eq.(4.1.4)ff.,(cm-') Gas atom propagation eigenvector (kx',ky', kz')after transition in perturbation theory, cf. Eq. (4.1.4) ff., (cm-') = k' - k (cm-') Bravais lattice vector (cm) Mass of electron (gm) Mass of gas atom (gm) Mass of crystal atom (gm) Number density of atoms in crystal (cm-3) Total number of atoms in crystal Number of electrons in outer shell of atom Transition probability per unit time from state In> to Im> Momentum of qth normal coordinate (3.14) Momentum in physical space (gm cni sec-') Lattice wave propagation vector (cm - ') Electric quadrupole moment Gas atom position vector (cm)

286

E. Chanoch Beder jth lattice atom position vector (cm) jth lattice atom equilibrium site (cm) Particle number flux in scattered beam (5.1.10) per unit incident particle number flux Crystal temperature (OK) Potential energy of gas-solid system (eV) Internal energy of adsorption at a given site at absolute zero and zero coverage (eV) Potential energy of isolated crystal (eV) Displacement of atom from its equilibrium site (cm) Speed (cm sec-') Volume @m3) Potential energy of gas atomthermally excited crystal interaction (eV) Potential energy of gas atom in field of rigid lattice (eV) Potential energy of binary (twoparticle) interaction (eV) Statistical probability factor, (6.1 .I la), (6.1. I 1b) Transition probability per unit time (6.1.10); Debye-Waller factor (5.5.1) Gas atom Cartesian coordinates: (x, y ) parallel to crystal surface, z normal, with origin at surface atom equilibrium sites, Figs. 10 and 12 Polarizability of atom or molecule; gas-atom trajectory angle, Fig. 12(a); thermal accommodation coefficient Constant in LJD interaction potential (2.4.3); gas-atom trajectory angle, Fig. 12(a) Gas atom trajectory angle, Fig. 12(a) Thickness of potential square well (2.4.5) Dirac delta function Angle between line connecting centers of mass and dipole or quadrupole moment

Debye crystal temperature (OK) Incidence angle in 2D scattering (measured from surface) Range parameter in Morse potential (2.4.1), (cm-') de Broglie wavelength (cm) k Z / K ; electric dipole moment; MdM, Frequency of lattice vibrations; of photon in electronic transition (sec-') Normal coordinate of qth lattice mode Differential scattering cross section Characteristic time, cf. (4.2.1), (4.2.2), and (4.2.3), (sec) Energy eigenfunction, coordinate representation Scattering angle, Fig. 12(b) Frequency of lattice vibrations (sec-') Solid angle (sterad) SUBSCRIPTS AND SUPERSCRIPTS c Crystal D Debye g Gas atom or molecule i Incident beam 0 Classical equilibrium configuration r Reflected beam; molecular rotation u Molecular vibration ( ) O Unperturbed state, (6.2.2) ( )' Condition of gas atom after transition in perturbation theory; derivative with respect to lattice atom displacement uI (6.2.2); cf. H' MISCELLANEOUS AC Accommodation coefficient Ar 1 KCI Gas-solid system of argon gas and KCI crystal g-s Gas-solid nD ti-dimensional(1y) I k) Gas atom energy eigenfunction Eq. (4.1.7) IT. LJ Lennard-Jones LJD Lennard-Jones and Devonshire

GAS CRYSTAL-SURFACE VAN DER WAALS SCATTERING

LJS Lennard-Jones and Strachan ES Estermann and Stern EFS Estermann, Frisch, and Stern

287

FS Frisch and Stern YC Young and Crowell HCB Hirschfelder, Curtiss, and Bird

REFERENCES Amberg, C., Spencer, W. B., and Beebe, R. A. (1958). Can. J. Chem. 33, 305. American Institute of Physics Handbook (1963). McGraw-Hill, New York, 2nd Edition. Allen, R. T., and Feuer, P. (1964). J. Chem. Phys. 40, No. 10, 2810. Allen, R . T., and Feuer, P. (1967). In “ Rarefied Gas Dynamics,” Proc. Fifth Intern. Symp. Oxford, 1966 (C. L. Brundin, ed.), Vol. I, p. 109. Academic Press, New York. Beattie, J . A,, and Stockmayer, W. H . (1951). In “Treatise on Physical Chemistry” (H. S. Taylor and S. Glasstone, eds.), Vol. TI. Van Nostrand, Princeton, New Jersey. Beder, E. (1964). Surface Sci.1, 242. Beebe, R. A., and Young, D. M. (1 954). J. Phys. Chem. 58,93. Bonch-Bruevich, V. L. (1954). National Research Council of Canada, Ottawa, Tech. Transl. 509 from Usp. Fiz. Nuuk 40 (3), 3 6 9 4 6 (1950). Brillouin, L. (1953). “Wave Propagation in Periodic Structures,” 2nd ed. Dover, New York. Clarke, J. F., and McChesney, M. (1964). “The Dynamics of Real Gases.” Butterworths, London and Washington, D.C. Constabaris, G., and Halsey, Jr., G. D. (1957). J . Chem. Phys. 27, 1433. Constabaris, G . , Sams, Jr., J. R., and Halsey, Jr., G. D. (1961) J. Phys. Chem. 65,367. Corner, J. (1948). Trans. Faraday SOC.44, 914. Cottrell, T. L., and Ream, N. (1955) Trans. Faraday SOC.51, 159. Crews, J. C. (1962). J. Chem. Phys. 37, 2004. Crews, J. C. (1967). In “Fundamentals of Gas-Surface Interactions,” Proc. Symp., San Diego, I966 (H. Saltsburg, J. N. Smith, Jr., and M. Rogers, eds.), p. 480. Academic Press, New York. Crowell, A. D., and Young, D. M. (1953). Trans. Faraday SOC.49,1080. de Boer, J. H. (1956). Aduan. Catalysis VIII, 18. de Boer, J. H. (1953). “The Dynamical Character of Adsorption.” Oxford, Clarendon. DeMarcus, W. C., Hopper, E. H., and Allen, A. M. (1955). A.E.C. Bulletin K-1222. Devienne, F. M. (1954). Compt. Rend. 238,2397. Dirac, P. A. M. (1958). “The Principles of Quantum Mechanics,” 4th ed. Oxford Univ. Press, London and New York. Estermann, I. (1966). Private communication. Estermann, I., and Stern, 0. (1930). 2. Physik 61, 95. Estermann, I., Frisch, R., and Stern, 0. (1931). Z . Physik 73, 348. Feuer, P.(1963). J . Chem. Phys. 39, 1311-1316. Feuer, P.(1967). Private communication. Fox, D., and Schnepp, 0. (1955). J . Chem. Phys. 23,767. Freeman, M. P. (1958). J . Phys. Chem. 62,723. Frisch, R., and Stern, 0. (1933). Z . Physik 84, 430-443. Gadzuk, J. W. (1966). Private communication. Gadzuk, J. W. (1967). Phys. Rev. 153,759. Germer, L. H., and MacRae, A. U. (1963). Intern. Sci. Tech. August, 34. Gilbey, D. M. (1962). J . Phys. Chem. Solids 23, 1453-1461. Gilbey, D. M. (1967a). In “Rarefied Gas Dynamics, Proc. Fifth Intern. Symp., Oxford, 1966 (C. L. Brundin, ed.), Vol. I, p. 101. Academic Press, New York.

E. Chanoch Beder Gilbey, D. M. (1967b). In “Rarefied Gas Dynamics,” Proc. Fifth Intern. Symp., Oxford, 1966 (C. L. Brundin, ed.), Vol. I, p. 121. Academic Press, New York. Gilbey, D. M., and Goodman, F. 0. (1965). Am. J. Phys. 34, 143. Glauber, R. J. (1955). Phys. Rev. 98, 1692. Gol’dman, I. I., and Krivchenkov, V. D. (1961). “Problems in Quantum Mechanics.” Pergamon, London. Goodman, F. 0. (1962a). J. Phys. Chem. Solids 23, 1269-1290. Goodman, F. 0. (1962b). J. Phys. Chern. Solids 23, 1491-1502. Goodman, F. 0. (1963). J. Phys. Chem. Solids 24, 1451-1466. Goodman, F. 0. (1965a). Surface Sci. 3, 386-414. Goodman, F. 0. (1965b). J. Phys. Chern. Solids 26, 85-105. Goodman, F. 0. (1966). In “ Rarefied Gas Dynamics,” Proc. Fourth Intern. Symp., Toronto, 1964 (J. H. de Leeuw, ed.), Vol. 11, p. 366. Academic Press, New York. Greyson, J., and Aston, J. G. (1957). J. Phys. Chem. 61, 610, 613. Hasted, J. B. (1964). “ Physics of Atomic Collisions.” Butterworths, London and Washington, D.C. Herzberg, G. (1944). “Atomic Spectra and Atomic Structure.” Dover, New York. Herzberg, G. (1950). “Molecular Spectra and Molecular Structure I Spectra of Diatomic Molecules,” 2nd ed. Van Nostrand, Princeton, New Jersey. Herzfeld, K. F., and Litovitz, T. A. (1959). “Absorption and Dispersion of Ultrasonic Waves.” Academic Press, New York. Hill, T. L. (1960). “An Introduction to Statistical Thermodynamics.” Addison-Wesley, Reading, Massachusetts. Hirschfelder, J. O., Curtis, C. F., and Bird, R. B. (1964). “Molecular Theory of Gases and Liquids.” Wiley, New York. Howsmon, A. J. (1966). In “ Rarefied Gas Dynamics,” Proc. Fourth Intern. Symp., Toronto, 1964 (J. M. de Leeuw, ed.), Vol. 11, p. 417. Academic Press, New York. Howsmon, A. J. (1967). In “Rarefied Gas Dynamics,” Proc. Fifth Intern. Symp., Oxford, 1966 (C. L. Brundin, ed.), Vol. I, p. 67. Academic Press, New York. Jackson, J. M., and Howarth, A. (1935). Proc. Roy. SOC.(London) A152, 515. Jackson, J. M., and Mott, N. F. (1932). Proc. Roy. SOC.(London) A137, 703. Kaminsky, M. (1965). “Atomic and Ionic Impact Phenomena on Metal Surfaces,” Vol. 25: ‘‘Struktur und Eigenschaften der Materie in Einzeldarstellungen.” Springer, Berlin. Kittel, C. (1956). “Introduction to Solid State Physics,” 2nd ed. Wiley, New York. Kittel, C. (1963). “Quantum Theory of Solids.” Wiley, New York. Knuth, E. L. (1966, 1967). Private communications. Knuth, E. L., and Kuluva, N. M. (1966). AGARD Colloquium, Oslo, May 16-20, 1966. Kothari, L. S., and Singwi, K. S. (1959). Solid State Phys. 8, 109. Landau, L. D., and Lifshitz, E. M. (1958). “ Statistical Physics.” Addison-Wesley, Reading, Massachusetts. Landau, L. D., and Teller, E. (1936). Phys. Z. Sowjetunion 10, 34 Lenel, F. W. (1933). Z. Phys. Chem. B23, 379. Lennard-Jones, J. E. (1932). Trans. Faraday SOC.38, 333. Lennard-Jones, J. E., and Dent, B. M. (1928). Trans. Faraday SOC.24, 92. Lennard-Jones, J. E., and Devonshire, A. F. (1936a). Proc. Roy. SOC.A156,6. Lennard-Jones, J. E., and Devonshire, A. F. (1936b). Proc. Roy. SOC.A156, 29. Lennard-Jones, J. E., and Devonshire, A. F. (1936~).Proc. Roy. SOC.A156,37. Lennard-Jones, J. E., and Devonshire, A. F. (1936d). Proc. Roy. SOC.A158,242. Lennard-Jones, J. E., and Devonshire, A. F. (1936e). Proc. Roy. SOC.A158, 253. Lennard-Jones, J. E., and Devonshire, A. F. (1936f). Proc. Roy. SOC.A158,269. Lennard-Jones, J. E., and Devonshire, A. F. (19638). Nature, June 27, p. 1069.

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Lennard-Jones, J. E., and Strachan, C. (1935). Proc. Roy. SOC.A150, 442. Lennard-Jones, J. E., and Strachan, C. (1936a). Proc. Roy. Soc. A150, 456. Lennard-Jones, J. E., and Strachan, C. (1936b). Proc. Roy. Soc. A158, 591. Leonas, V. B. (1964). Soviet Phys.-Usp. (Engli.sh Trans/.) 71, 121-144. Logan, R. M., and Stickney, R. E. (1966). J . Chem. Phys. 44, 195. Loos, H. G. (1965). Private communication. MacRae, A. U. (1964). Surface Sci. 2, 522. Maradudin, A. A., and Melngailis, J. (1964). Phys. Reo. 133, 1188. Maradudin, A. A., Montroll, E. W., and Weiss, G . H. (1963). Solid State Phys., Suppl. No. 3. Mason, E. A,, and Rice, W. E. (1954). J. Chem. Phys. 22, 843. Massey, H. S . W., and Burhop, E. H. S. (1952). “Electronic and Ionic Impact Phenomena,” 1st ed. Clarendon, Oxford. McCarroll, B., and Ehrlich, G. (1963). J . Chen?.Phys. 38, 2, 523. McCarroll, B., and Ehrlich, G. (1964). General Electric Res. Lab. Reprint, 4918. McDowell, C. A., and Warren, J. W. (1951). Discussions Fahraday SOC.10, 53. Messiah, A. (1961). “Quantum Mechanics.” North Holland Publ., Amsterdam. Moran, J. P., Wachman, H. Y . , and Trilling, L. (1967). In “Fundamentals of Gas-Surface Interactions,” Proc. Symp., San Diego, 1966 (H. Saltsburg, J. N. Smith, Jr., and M. Rogers, eds.), p. 461. Academic Press, New York. Morse, P. M. (1929). Phys. Rev. 34, 57. Morse, P. M. (1930). Phys. Rev. 35, 1310. Oman, R. A. (1967). In “Rarefied Gas Dynamics,” Proc. Fifth Intern, Symp., Oxford, 1966 (C. L. Brundin, ed.), Vol. I. Academic Press, New York. Pace, E. L. (1957). J . Chetn. Phys. 27, 1341. Pauling, L. (1960). “The Nature of the Chemical Bond,” 3rd ed. Cornell Univ. Press, Ithaca, New York. Pauling, L., and Wilson, E. B. (1935). “Introduction to Quantum Mechanics.” McGrawHill, New York. Pauly, H . , and Toennies, J. P. (1965). Advan. A t . Mo1. Phys. 1, 195. Pines, D. (1963). *‘ Elementary Excitations in Solids.” Benjamin, Inc. Rogers, M. (1 962). “ Gas-Surface Phenomena.” Johns Hopkins Univ. AFOSR 2342. Rubin, R. J. (1960). J . Math. Phys. 1, 309. Rubin, R . J. (1961). J . Math. Phys. 2 , 373. Saltsburg, H. (1967). Private Communication. Saltsburg, H.,Smith, Jr., J. N.,and Palmer, R. L. (1967). In “Rarefied Gas Dynamics,” Proc. Fifth Intern. Symp., Oxford, 1966 (C. L. Brundin, ed.), Vol. 1, p. 223. Academic Press, New York. Sams, Jr., J. R., Constabaris, G., and Halsey, Jr., G. D. (1960). J . Phys. Chern. 64, 1689. Schiff, L. I. (1955). “Quantum Mechanics,” 2nd ed. McGraw-Hill, New York. Selwood, P. W. (1956). “ Magnetochemistry.” Wiley (Interscience), New York. Shuler, K. E., and Zwanzig, R. (1960). J . Chenz. Phys. 33, 1778. Smith. Jr., J. N . , and Saltsburg, H. (1966). In “Rarefied Gas Dynamics,” Proc. Fourth Symp.. Toronto, 1964 (J. M. de Leeuw, ed.), Vol. 11, Academic Press, New York. Smith, Jr., J. N., and Saltsburg, H. (1967). 1tz ‘‘ Fundamentals of Gas-Surface Interactions,” Proc. Symp., San Diego, 1966 (H. Saltsburg, J. N. Smith, Jr., and M. Rogers, eds.), p. 370. Academic Press, New York. Steele, W. A., and Halsey, Jr., G. D. (1954). J . Chem. Phys. 22, 979. Stickney, R. E., Logan, R . M., Yaniamoto, S., and Kcck, J. C. (1967). In “Fundamentals of Gas-Surface Interactions,” Proc. Synip., San Diego, 1Y66 (H. Saltsburg, J. N. Smith, Jr., and M. Rogers, eds.), p. 422. Academic Press, New York. Takayanagi, K. (1962). Sci. Rept. Saitarna Univ. Ser. A 4, No. 2, 51.

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Trilling, L. (1964). J . Mecan. 3, No. 2, 215. Trilling, L. (1 967). In “Fundamentals of Gas-Surface Interactions,” Proc. Symp., San Diego, 1966 (H. Saltsburg, J. N. Smith, Jr., and M. Rogers, eds.), p. 392. Academic Press, New York. Wallis, R. F. (1964). Surface Sci. 2, 146-155. Wallis, R. F., and Gazis, D. C. (1964). Surface Sci.3, 19-32. Wexler, S. (1958). Rev. Mod. Phys. 30,402. Young, D. M. (1951). Trans. Faraday SOC.47, 1228. Young, D. M., and Crowell, A. D. (1962). “Physical Adsorption of Gases.” Butterworths, London and Washington, D.C. Ziman, J. M. (1964). “Principles of the Theory of Solids.” Cambridge Univ. Press, London and New York. Zwanzig, R. W. (1960). J. Chem. Phys. 32, 11 73-1 177.

R EA CTI V E COLLISIONS BETWEEN GAS AND SURFACE ATOMS HENRY WISE A N D BERNARD J . WOOD Stanford Research Institute Menlo Park, Califarnia

Introduction. .................... Energetic, Mechanistic, and Kinematic erations . . . . . . . . . . . . . . . . 292 A. The Energetics of Atom-Surface Interaction ...................... 292 B. Mechanisms of Atom Reactions at a Surface . . . . . . . . C. Kinematics of Atom-Surface Interactions ........................ 300 111. Experimental Methods .............. 311 A. Catalytic Recombin . . . . . . . . . . . . . 311 B. Chemical Reactions toms . . . . . . . . 334 IV. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 344 List of Symbols.. ................................................ 347 References ...................................................... 348 I.

11.

I. Introduction The interaction of gases with solid surfaces includes such a large area of research that it is necessary to restrict this review to a specific segment of the general topic in order t o evaluate the work which has been completed and to point to the problems which deserve further study. Within the framework of the discussion, emphasis is placed on two specific concepts: (i) reactive collision at a gas/solid interface, and (ii) gaseous atomic species containing unpaired electrons. A reactive collision involves the formation and destruction of chemical bonds during the encounter of the gaseous atom with the solid. As a result, the binding forces effective in the interaction are of a magnitude greater than that encountered in physical adsorption (van der Waals forces). The gaseous atom comes sufficiently close to the solid so that covalent bond formation becomes feasible. In the extreme case of interpenetration of electron shells of the adatom and the solid, a heteropolar force may result with actual electron transfer and the formation of charged species on the surface of the solid. The gaseous reactant of special interest to this discussion is the atom with its unpaired electron, as produced by the dissociation of a diatomic molecule. Such atoms may undergo two basic types of reaction with the solid: (i) a 29 1

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heterogeneous catalytic reaction leading to the formation of a diatomic molecule, and (ii) a chemical reaction with the lattice atoms of the solid resulting in the formation of a product composed of atoms of the solid and of the gas. Specifically excluded from this review are studies of adsorption processes, reactions of charged particles, and problems related to diffusion of gases through solids.

11. Energetic, Mechanistic, and Kinematic Considerations A. THEENERGETICS OF ATOM-SURFACE INTERACTION As a starting point for the discussion, we may consider the interaction potential between a gaseous species and a metal as first proposed by LennardJones (1932). The potential energy diagrams shown in Fig. 1 demonstrate the s + 2x

s-xo+x ENERGY

0

s t x 2

I

DISTANCE

-

FIG.1. Energy diagram for atom-surface interaction.

energetics of the elementary surface process encountered during dissociative chemisorption of a diatomic molecule. In this diagram the binding forces involved in the collision of a molecule are seen to be of two types: (i) physical adsorption and (ii) chemisorption. The small minimum in the potential energy curve (A) at large distances from the metal surface is related to the van der Waals forces active during physisorption of the molecule and associated with dipole interactions. On the other hand, if the molecule in the gas

GAS AND SURFACE ATOM REACTIVE COLLISIONS

293

phase is dissociated into its constituent atoms and an atom is allowed to approach the solid, its potential energy follows the second curve (B) in the diagram. In this case, the atom comes much closer to the solid surface and is more strongly bound to it-chemisorbed-as evidenced by the depth of the potential well. The binding force associated with chemisorption is in the form of an exchange force in which a “ binding orbital” is formed by interaction of the electron shell of the adatom with that of the solid. In recent theoretical analyses, the formation of a chemisorbed state has been described in terms of the overlap of molecular orbitals which are directional due to the existence of the crystal field (Bond, 1966). It is to be noted that as an alternate path for chemisorption a molecule with sufficient energy to “climb” the repulsive potential of the physisorbed state may cross over to the potential curve associated with the chemisorbed state at the point P. The height of P above the ground state is, in fact, the activation energy for adsorption. A molecule such as H, will dissociate at P into two atoms. Because of the high density of electron levels in the metal, the quantum-mechanical limitations imposed upon radiationless transition, such as spin conservation and conservation of angular momentum, do not apply in this process. For chemical interaction between surface and gaseous atoms, a n approach of the gaseous atom corresponding to chemisorption is required. Elucidation of the elementary processes associated with reaction requires, therefore, some information on the bond strengths between atoms and surfaces. Such estimates of the covalent binding energies, based on Pauling’s (1960) formulation of bond energies have been made. The binding energy Q, is given by Qs=+Qc+~QD+~,

(1)

where Q , represents the bond energy of the metal atoms, Q , that of the gaseous molecule, and (T the electronegativity term. Eley (1950) chose Q , as a fraction of the heat of atomization of the metal‘ and computed the electronegativity term from the contact potential associated with the adsorbed gas. Subsequently, Stevenson (1955) proposed the use of work function data in the calculation of the (T term. In Table I we have compared the binding energies of hydrogen atoms on metals calculated on the basis of Eq. (1) with those obtained from experimental measurements of the heat of adsorption QA, which, as is evident from Fig. 1, is given by Qs = HQD -

QA).

(2)

The data indicate that the bond formed between the metal and hydrogen atom is relatively strong and hardly affected by the nature of the solid. The bona strength is comparable in magnitude to the bond energy of a metal hydride molecule. It is to be expected, therefore, that if the cohesion between Taken to be one-sixth of the atomization energy for the closest packed lattices.

294

Henry Wise and Bernard J. Wood TABLE I

BINDING ENERGIES OF HYDROGEN ON METALS QS

Metal

Heat of adsorption" Q A (kcal/mole) Exptlb Calc'

cu co Cr

Fe Ir Mo Ni Pd Pt Rh Ru Ta W a

-28 -24 -45 -33 -26 -40 -30 -26 -21 -26 -26 -45 -46

66 63 14 68 65 72 61 65 65 65 65 14 75

65 61 64 68 12 73 67 63 63 68 70 17 15

Values refer to low surface coverage (Stevenson,

1955). Eq. (2). Eq. (1).

adatom and metal atom exceeds that between the lattice atoms, irreversible adsorption may occur and during desorption a surface atom of the solid may be torn away. It can be seen from the potential energy diagram (Fig. 1) that the process of dissociative chemisorption from the gaseous molecular state to the surfacechemisorbed state may involve passage over an energy barrier EA, the activation energy for adsorption. Thus, although the thermodynamics of the adsorption process may favor the formation of the chemisorbed state, the rate of activated adsorption may be limited because of kinetic factors. Such considerations must also include the observed fact that the heat of adsorption is a function of the degree of surface coverage with adsorbate. As a result, the potential energy curve shown in Fig. 1 must be replaced by a family of curves (Fig. 2). These curves demonstrate how the heat of adsorption may diminish and the activation energy for adsorption may grow with increase in surface coverage (Eley, 1957). At the same time, it should be appreciated that adsorption of a gaseous atom on a solid surface may proceed without any activation energy along line B (Fig. l), by way of a process in which the diatomic gaseous molecule is first dissociated and then allowed to come in contact with the solid.

GAS AND SURFACE ATOM REACTIVE COLLISIONS

295

DIS T A N C E

FIG.2. Formal representation of the dependence of EA on coverage (after Bond, 1962).

Desorption of an adatom represents essentially the reverse of the adsorption process. Thus, either an atom or a diatomic molecule may be desorbed depending on the particular conditions. The activation energy for atom desorption to the gaseous atomic state is equal to the binding energy Q, given by Eq. (2). An alternative route is molecular desorption by way of interaction of two adatoms for which process the energy barrier is given by EA, = EA - QA (Fig. 1). Because of this energy requirement, the rate of associative desorption is relatively slow except at elevated temperatures where a mobile surface adsorbate acquires sufficient energy to overcome the high energy barrier.’ For most metals of interest, the molecular desorption process becomes limited by the low equilibrium density of adatoms prevalent at such high temperatures. Obviously, if the activation energy for desorption is less than that for adsorption, EA, < E, (Fig. l), we are dealing with endothermic chemisorption. Another process which may lead to the removal of surface adatoms is that of heterogeneous recombination. In essence, such a reaction may be treated as a catalytic step in which an atom from the gas phase collides with an atom adsorbed on the surface and a diatomic molecule is evolved into the gas phase. We have presented the energetics of this process in the potential energy diagram shown in Fig. 1 (curves C and D). Here a gaseous atom (X) approaches a solid covered with a surface layer of adatoms (s - X). It may form The energy barrier for surface diffusion is of the order of 6Qs (Ehrlich, 1959).

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Henry Wise and Bernard J. Wood

a second layer of adatoms (Xs) which may be stable under certain conditions (curve C), as discussed in more detail in Section 111. Alternatively, formation of the diatomic molecule X, may also proceed along the repulsive potential surface (curve D, Fig. 1) which leads to a crossover point with the potential surface represented by curve A (Fig. 1). The net reaction may be written as

s - x + x-is + x,.

B. MECHANISMS OF ATOMREACTIONS AT A SURFACE From a mechanistic viewpoint, two types of chemical interactions need be considered: (i) catalytic recombination of gaseous atoms in which the solid essentially plays the role of an energy sink and (ii) reactive collision between a gas atom and a solid surface leading to a product molecule composed of atoms of the gas and the substrate. These are represented schematically in Fig. 3. 1. Catalytic Recombination

In recent years, investigators have given differing weights to the importance of the various atoms, molecules, and surface states involved in heterogeneous recombination and to the elementary interactions which might occur. Table I1 illustrates the span of such considerations. By confining our present consideration, however, to surface kinetics (the kinematics of gas-surface interactions is discussed in a later section) in a temperature region where weakly adsorbed species are essentially absent we may limit our discussion to the mechanism of recombination associated with chemisorbed atoms. One, the Eley-Rideal mechanism (E-R) postulates the direct reaction of an atom chemisorbed on the surface (s - X), with an atom from the gas phase (X): s-x

ki

+ x +! s + x,. k-i

(3)

The other, the Langmuir-Hinshelwood mechanism (L-H), involves surface recombination of two adatoms: k2

2(s - X) P 2s

+ x,.

(4)

k-2

It should be noted that at high surface coverage with adatoms, a distinction can be made between these two mechanisms on the basis of the experimentally observed order of reaction with respect to the chosen species in the gas phase. For the E-R mechanism, the rate of molecule formation would be expected

297

GAS A N D SURFACE ATOM REACTIVE COLLISIONS

TABLE I1 ELEMENTARY RATEPROCESSES CONSIDERED IN HETEROGENEOUS ATOMRECOMBINATION

Process

deBoer Shuler and Tsu and van and Laidler Steenis Sat0 Ehrlich Boudart (1949) (1952) (1956b) (1959) (1960)

X Dissociative chemisorption Recombination of 2 adatoms ( t - H x mechanism) in first layer Recombination of gaseous atom and adatom (E-R mechanism) in first layer x Evaporation of adatom in first layer Recombination of 2 adatoms in second layer Recombination of gaseous atom and adatom in second layer Evaporation of adatorn in second layer Recombination of first- and second-layer adatoms Formation of first layer by gaseous atom adsorption Formation of second layer by gaseous atom adsorption Diffusion of gaseous atoms toward surface Diffusion of gaseous molecules away from surface Desorption of (physisorbed) molecules Migration of atoms in second layer

X

x

x

x

x

X

Hardy and Linnett (1967)

X

x

X

x

X X

X

X X X

X X

x

x

x

X

x

x

X X X

x

x X

to depend on the first power of the gas-atom density, whiIe zero-order dependency would result from the L-H mechanism under these conditions. On the other hand, at low surface coverage when the surface density of adatoms is proportional to the atom density in the gas phase, both mechanisms would yield second-order kinetics and a distinction between them would be difficult. An analysis of the kinetics of the encounter between an atom and a solid may be made (de Boer and van Steenis, 1952; Ehrlich, 1959; Hardy and Linnett, 1967) on the basis of the reactions represented by Eqs. (3) and (4) and on the adsorption step from the gaseous state, described by Eq. (5): k3

x +s *s

- x.

(5)

k-3

The rate of removal of adatoms from surface sites per unit area, 9, may be expressed in terms of the collision rate, per unit surface area, of gaseous atoms

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and molecules, Z, and Z, , respectively; the number of surface sites N occupied by adatoms 9N; the rate constants for the various transitions k,, k - l , k, , etc. ;and the mean velocities of the gaseous atoms and the molecules c1 and c2 :

W = 4kl -9NZI + 2k2e2N2+ k-39N C1

For practical purposes, several of the terms in Eq. (6) are of negligible magnitude. In the temperature and pressure regimes in which a large number of atom-surface interaction studies have been made, atom desorption [reverse of reaction ( 5 ) ] may be neglected and the approximation of complete monolayer (0 + 1) coverage applied. Thus, Eq. (6) becomes

W = (4kl/cl)9NZ1 + 2k2(eN)’.

(7)

From an examination of the potential energy surface (Fig. 1) one must conclude that reaction (4) would be thermochemically less favorable than reaction (3). Experimental evidence by many investigators (Smith, 1943; Linnett and Mardsen, 1956a; Wood and Wise, 1961a) has supported this conclusion by demonstrating first-order kinetics with respect to the gaseous atoms for recombination, i.e., 92 cc Z, for 0 = 1. For such cases, only the first term on the right-hand side of Eq. (7) is significant and the ratio W/ZL becomes a constant which is commonly referred to as the recombination coefficient y. It should be noted, however, that the same E-R mechanism will yield secondorder kinetics if the degree of surface coverage is proportional to the gaseous atom pressure. Under these conditions, it would be difficult to distinguish it from the L-H mechanism whose rate is given by the second term on the righthand side of Eq. (7). 2. Chemical Reaction with Surface Atoms

In contrast to the catalytic recombination process, where the over-all chemical change is independent of the nature of the surface, chemical reactions between gaseous atoms and surface lattice atoms can involve an array of chemical processes. Thus, it is impossible to discuss the mechanistic character of such reactions except with respect to specific reactants and products. However, a valuable distinction may be made in terms of gas-surface reactions resulting in volatile products and those in which the products are nonvolatile. a. Volatile Products. The case in which reaction products desorb rapidly from the site of formation on the surface is exemplified by the reaction of oxygen atoms with graphite and with some refractory metals. The mechanism

299

GAS AND SURFACE ATOM REACTIVE COLLISIONS

of such a reaction may involve elementary steps similar to those postulated for catalytic recombination. Thus, the surface-incident gaseous atom may react directly with an atom in the solid surface to produce a volatile product following rupture of the lattice bonds (cf. Fig. 3). Such reactions are kinetically GAS

SOLID

---Q

B -w A

-@-@

---@---a Le-@--@ FIG.3. Schematic presentation of intermediate states for reactive collision. 0 :lattice atom. 0: gas atom.

first order with respect to the gaseous reactant. However, in order to obtain unequivocal information on the reaction kinetics, special attention must be paid to the limitations imposed by kinematic considerations such as diffusional and convective mass transport of reactants and products in the gas and in the pores of the solid. A complicating factor in such a reacting system is the competition between catalytic recombination of gaseous atoms and their reaction with the surface atoms. Empirical evidence from many studies of atom-surface reactions verifies the kinetically predominant role of catalytic recombination over that of atom-solid chemical reaction. Indeed, it seems likely that in many systems, the absence of observed chemical reactivity between the gaseous atoms and the solid can be attributed to an overwhelming rate of catalytic recombination. It is apparent that any study of an atom-solid chemical reaction is incomplete without an assessment of the catalytic activity of the solid for recombination. b. Nonvolatile Products. When the product of reaction between the gaseous atom and the solid surface is nonvolatile under the imposed conditions of temperature and pressure, it will accumulate on the reactive surface. Usually, the kinetics of this type of chemical process become obscured by those of the transport processes by which reactants and products move through the product layer (Gulbransen and Wysong, 1947). Such molecular gas-solid reacting

300

Henry Wise and Bernard J . Wood

systems have been analyzed and discussed at some length by Hauffe (1965) and Kubaschewski and Hopkins (1963). It should be noted, in summary, that even though empirical rate equations can be deduced for many systems, the elucidation of a mechanism is difficult, for the product of the reaction is not ordinarily capable of discrete characterization. That is, the composition, density, and other properties are likely to be changing with respect to both time and with position relative to the surface. Recently, a more detailed analysis (Ong, 1964) based on the diffusion of ions and vacancies through a solid electrolyte matrix in the absence of a net electric current has been proposed to account for these variations in the product scale layer.

c. KINEMATICS OF ATOM-SURFACE INTERACTIONS From both the analytical and the experimental viewpoints, it has been found convenient to study the surface interaction of atoms in cylindrical or tubular systems in which gaseous atoms are transported by convective and/or diffusive flow to the surface from a source. Historically, the first detailed analysis of the kinematic problem was made by Paneth and Herzfeld (1931) in their study of the interaction of gaseous organic free radicals with metal surfaces (mirror removal technique). Since that time, a number of analytical models have been developed which will be examined in this section.

I . Surface Reaction without Convective Flow The steady-state analysis of the kinetics of interaction of gaseous atoms with a solid surface can be carried out most conveniently by utilizing a model of diffusive flow inside a cylinder with radial and longitudinal concentration gradients originating from a source of atoms located at one end (Fig. 4). r

4 r=R

I

I I

.

* X

X'L

FIG.4. Schematic presentation of cylindrical system used in analysis.

Analytic treatments of this model by a number of authors are summarized in Table 111. We shall discuss in detail the development of Wise and Ablow (1958) and Motz and Wise (1960) which is applicable to the majority of experimentally realizable systems.

GAS AND SURFACE ATOM REACTIVE COLLISIONS

30 1

TABLE I11 ANALYSES OF STEADY-STATE DIFFUSIVE FLOW OF ATOMS INTO CATALYTIC WALLS

Investigators

Atom concn. gradients Axial only

Smith (1943)

Greaves and Linnett (1959a) Axial only

Diffusion equation Linear

Nonlinear

Wise and Ablow (1958), Motz and Wise (1960)

Axial and radial Linear

Dickens et al. (1960)

Axial and radial Nonlinear

A

CYLINDER WITH

Practical limitations and applications Atom concentrations <20%; low wall activity; 4DlycR 9 1 9 AIR Atom concentrations >20 % ;low wall activity ; 4D/ycR& 1 &AIR Atom concentration <20 %; short tube with catalytic closure (probe) Atom concentration >20 %; short tube with catalytic closure (probe)

Atom concentration gradients in such a system are due to the catalytic properties of the cylinder wall3 (r = R) and of a closure at the end of the cylinder (x = L). In this system the atom density n at any point in the cylinder is governed by the balance between diffusion from the source and atom loss by heterogeneous r e a ~ t i o n . ~ Under steady-state conditions the number flux of particles (Di in the positive and negative directions relative to the Concentration gradient is given by

a+* = +ni(+ci+ w)

+ D iIVn,l, (8) where the diffusion coefficient Di = f l i c i , liis the mean free path of species i with respect to collisions with the unlike species, ci is the root-mean-square random velocity of species i, and w is the mean particle transport velocity. Accordingly, the net flux of particles in the direction of the concentration gradient is mi = mi+ - ( D i - = - D iVni+ niw. (9) If the species i represents atoms, a fraction y of which recombines at the surface, the next flux of atoms is (Dl =

a,+- (D1-= y(D1 = - D iV , n + nw,,

(10)

See List of Symbols for definitions. Atom removal by homogeneous recombination may be neglected when the ;ecombination of atoms by termolecular, gas-phase collision is reduced to a negligible contribution by limiting the total gas pressure to a low value, i.e., reducing the concentration of third bodies.

Henry Wise and Bernard J. Wood

302

and by proper substitution in Eq. (8) ncy/4 = - D , [ I - (y/2)]V,n

+ nw,[l - (y/2)1.

(1 1)

Equation ( 1 1 ) represents the boundary condition at a catalytic surface with a recombination coefficient y,’ and the subscript c denotes the component normal to the boundary. In the model chosen, the resultant mass flow is zero, and M1q

+ M,@, = 0,

( 1 2)

where M , is the atomic and M , the molecular mass. For a chemical reaction involving the recombination of two atoms with a diatomic molecule as a product @, 2@, = 0, or - D1 Vn + nw = 2(D2 V m - mw), (13)

+

where m is the concentration of molecules. Since the total gas pressure is constant throughout Vn + V m = 0 , and from Eq. (13) w = [ ( D , - 2D2)/(n 2m)l Vn.

+

By substitution into Eq. ( 1 1 ) one obtains for the boundary condition -Vcn = (ncy){2 - [n/(n+ m ) I l / W 1 2 [ 1 - (Y/2)117

where the interdiffusion coefficient Dp, is defined by D 1 2E (nD2

+ mDl)/(n+ m).

When the concentration of atoms is small so that n/(n f rn) -4 1 -Vcn

%

( ~ c Y ) / { ~ D ,, U(rl2)ll.

It is convenient to define a dimensionless parameter6

6 E 4D,2[1

- (y/2)]/yCR.

It should be noted that Eqs. (8) and (9) are derived on the assumption that the mean particle transport velocity is small compared with the random velocity. This assumption is justified when n -4 ( n + m). For high atom concentration more accurate transport equations might be required. The probability of reaction between an adsorbed atom and a gaseous atom arriving at the surface may be expressed in terms of a recombination coefficienty defined as the fraction of atoms striking the surface which react. The wall of the cylinder ( r = R ) and the closure located at x = L (Fig. 4) may have the same or different catalytic activities. In order to distinguish between the cylinder wall and the closure, the recombination coefficient y refers to the former, while y’ refers to the latter. Similar considerations apply to 6 and 6’.

303

GAS AND SURFACE ATOM REACTIVE COLLISIONS

In the absence of convective flow the diffusion equation in the presence of a catalytic boundary is of the form A,,

+ (V,,)’/(n + 2m) = 0.

(18) The properties of this nonlinear equation have been explored in a series of solutions obtained by numerical integration (Greaves and Linnett, 1959a-c). The results show that the linear form of the diffusion equation An

(19)

=0

is a satisfactory approximation for n/m < 0.2. The solutions to Eq. (19) with various boundary conditions for first-order surface reactions [Eqs. (20)-(35)] are presented in Table IV. The functional behavior of these solutions is demonstrated graphically in Fig. 5. Of special 8 7

6

t 5

- 4 I

-13

W

3 2 I

0

0

10

-20

30

40

L R

FIG.5. Catalytic effect of cylinder walls and probe on concentration profile of reactive species (Wise and Ablow, 1958).

interest is the simple approximate form Eq. (33) applicable to a catalytic cylinder of infinite length. It is identical to that derived by Smith (1943) for a system without radial diffusion. However, it should be noted that the solution as given by Eq. (33) is an adequate approximation only under the conditions of 6 % 1 and L/R % 1. i.e., the case of low catalytic wall activity and large distances from the atoms source.

w

0 P

TABLE IV SOLUTIONS TO DIFFERENTIAL EQUATION An = 0” (0’

Physical model Cylinder of finite length with catalytic wall and catalytic end plate (Wise and Ablow, 1958)

Cylinder of finite length with noncatalytic wall and catalytic end plate (Wise and Ablow, 1958)

Boundary conditions n,(O, x ) = 0 (20) (21) n(r, 0) = no n,(R, x) = -n(R, x)/SR (22)b nJr, L) = -n(r, L)/6’R (23Ib

n,(O, x) = 0 (20) n(r, 0) = no (21) nr(R, X)= 0 (24) n=(r,L) = -n(r, L)/6’R (25Ib

a

G

Solutionb ur=0=2c 1=

al(l

sinh al[(L - x ) / R ]+ 6’al cosh al[L - x)/R] + 6’alz)Jl(ai)[sinh al(L/R) + 6’al cosh ai(L/R)]

m

I&=‘=

2 c 1=,

6’

(1

+ 6ZaiZ)Jl(a,)[sinh( a J / R ) + 6‘al cosh al(L/R)I +

u = 1 - {(x/R)/[S’ (LIR)I1 u,=

(27) (28)

3 (29)

Infinite cylinder with catalytic Wall (Wise and Ablow, 1958)

+ 2611 e~p-[(2/6)“~x/R]

2a (0

‘= 6’/[6‘ + (L/RII

u a [26/(1

3 8

(33)

s. 4

Infinite cylinder composed of two sections with different catalytic wall activities (Ablow et al., (1965)

Cylinder of finite length with catalytic wall and catalytic end plate of fractional area (Ablow et al., 1965)

n,(O, x ) = 0 (20) n(r, 0) = no (21) n,(R, .x) = -n(R, x)/S,R, 0 < x < L, (22a)b &(R, X ) = - 4 R , x)/&R, L , < x (22b)* n(r, a)) = 0 (26) n,(O, x ) = 0 (20) n(r, 0) = no (21) nr(R, X ) = -nn,(R, x)/SR (Wb -n(r, L)/S’R,

n&, L) =

a

i

cosh xs

+ B, sinhx,,

+

cosh Lss (6s/6,)1/2sinh Lss’

L, < x

(34)

0 I(

=1

- RS2xJ(6’+ Rs2L)

O
(23a)b 0, R , < r < R (23b)b . .

See Eq. (19). For recombination of atoms to form a diatomic molecule, the surface boundary condition

for n/[2(n

0I x I L,

i% (35)

5 u

*

2 z

rtT

+ M ) ] < 1 and (y/2) < 1 (Motz and Wise, 1960).

w 0 ul

306

Henry Wise and Bernard J. Wood

In certain applications, e.g., when the catalytic surface is located at such large distances from the source that the atom density is very low, the mathematical analysis may be simplified by assuming the atom concentration to be zero. Although this assumption contradicts the statement of Eq. (17), it appears to be an adequate approximation in the work reported by Schulz and Leroy (1962) and by Morgan and Schiff (1963b). Ablow et al. (1965) considered two modifications of the catalyst geometry which are of interest from an experimental viewpoint. First, atom diffusion was examined in an infinite cylinder composed of finite sections of different catalytic activities as shown schematically in Fig. 6. Second, an analysis was

FIG.6. Schematic representation of cylinder made up of (a) two catalytic sections, and (b) variable-area catalytic closure (Ablow et al., 1965).

made of the influence of changes in the cross-sectional area of the catalytic closure on the concentration profile of reactive species in the cylinder (Fig. 6). The applicable boundary conditions and solutions to the differential equation for these cases are presented in Table IV. These analyses demonstrate the care required in designing experiments to obtain quantitative information concerning the catalytic properties of surfaces. For example, the pronounced effect of variable catalyst area of the closure (Table V) demonstrates the magnitude of the error which may be introduced in the analysis of the experimental data in terms of the surface recombination coefficient.

GAS A N D SURFACE ATOM REACTIVE COLLISIONS

307

2. Surface Reaction with Convective Flow Of special interest to the dynamics of a system undergoing heterogeneous reaction is the contribution of convective flow to the diffusion gradients established in a cylinder containing catalytic surfaces. In this case, the model is similar to that shown in Fig. 4 except that convective transport at velocity v in the axial direction in the cylinder is superimposed upon the diffusion of atoms. A steady-state balance of the number of reactive particles entering and leaving an arbitrary volume leads t o

D An - vn,

= 0.

(36)

Wise and Ablow (1961) reported a series of numerical solutions for this equation for low atom densities (less than 25 at. %) under the following boundary conditions (see Table IV):

(20)

n,(O, x ) = 0 n(r, 0) = no

(21)

n,( R , x) = 0 nx(r, L ) =

(24)

- t ~ ( r ,L)/6'R.

(25)

TABLE V EFFECTOF PROBE AREAON CATALYST ACTIVITY PARAMETER'

1 0.7 0.5 0.2 0.1 0

0.17 0.30 0.45

1.o 2.0

0.83

25.0 100.0

0.95 1 .oo

4.0 co

a From Ablow et al. (1965). Calculations are based on the following conditions: 6' = 1, 6 = to ; L / R = 5 (see text).

Equation (24) implies the absence of a radial concentration gradient; such a condition offers a satisfactory approximation in cylinders of large radius. The numerical results are presented in Fig. 7 in terms of a dimensionless velocity parameter G'RvlD, a type of reactive Peclet number. The data indicate

Henry Wise and Bernard J. Wood

308

the degree to which the atom concentration profile may be " stretched " by the use of convective flow in the presence of a catalyst of specified activity located at x = L. 100

10

L R 8'

I

0.1

0.0

0.2

0.6

0.4

0.0

I .o

U

FIG.7. Heterogeneous atom recombination in the presence of convective flow (Wise and Ablow, 1961).

GAS AND SURFACE ATOM REACTIVE COLLISIONS

309

3. Transition from Surface- to Difiision-Limited Process An examination of the limiting conditions under which either the kineniatics of diffusive and convective transport or the kinetics of chemical reaction predominates is of special importance to the quantitative study of surface catalysis, surface ablation, and the application of catalytic probes for atom concentration measurements. It is apparent that in the system under discussion the observed rate of heterogeneous reaction may be determined partly by the rate of mass transport and partly by the rate of surface reaction. Wise et al. (1963) analyzed this problem by calculating the mass flux to the various catalytic surfaces under a variety of physical and chemical conditions. The results are summarized in Table VI [Eqs. (37)-(39)]. The criterion for the approximation of one-dimensional diffusional flow in a cylinder may now be deduced on the basis of the equations presented. It is found to be k x 2 / ( R D )< 1 for the infinite cylinder with catalytic walls

(40)

k'L/D < 1 for the cylinder of finite lengths with noncatalytic walls and catalytic end plate

(41)

k L Z / R D and k ' L / D < 1 for the cylinder of finite length with catalytic walls and catalytic end plate.

(42)

A simple calculation demonstrates that at total gas pressures greater than 3 Torr in a cylinder with a radius of 1 cm, the radial gradient becomes negligible at a distance of less than 2 radii from the atom source. It is apparent that in a system undergoing surface reaction and diffusion the evaluation of catalytic properties must be restricted to a region where mass transport is not rate-limiting. The data presented allow quantitative analysis of this parameter. In general, it is to be expected that the gas-pressure range suitable for catalytic measurements will extend to higher values as the catalytic activity of the solid diminishes. At the same time the use of probes for atom-density measurements in gas streams in the diffusion-controlled regime will require detailed knowledge of (i) the efficiency of the solid for atom recombination, (ii) the catalytic properties of the cylinder walls, (iii) the location of the probe, and (iv) the transport properties of the gas stream. The analytical perturbation introduced by the interplay of convective and diffusional mass transport on heterogeneous chemical kinetics is not a new subject in catalytic research. Rigorous analyses have been carried out by various authors (Frank-Kamenetskii, 1955; Chambrt and Acrivos, 1956; Rosner, 1964).

w

c

0

TABLE VI TRANSITION FROM SURFACETO DIFFUSION-LIMITED PROCESS"

Physical model

Limiting flux F,=

Mass fluxb F,=

Surface controlled Infinite cylinder with catalytic walls

kno e~pI--(2/6)'/~(x/R)l=

Diffusion controlled

5 9

4

no

(37) (l/k) +(2~*/ki?D)"~

Cylinder of finite length with noncatalytic walls and catalytic end plate Cylinder of finite length with catalytic walls and catalytic end plate (I

no (]/k')[l + (kL2/RD)] (L/D)[l

+

From Wise et a/. (1963). Surface rate constant (in units of cm sec-I): k = D/(6R);k'

=

+ (kL2/3RD)]

D/(6'R).

k'n,

(39)

[l

+ (kLZ/RD)]

(DndL)

11

+ (kL2/3RD)1

3

E

GAS AND SURFACE ATOM REACTIVE COLLISIONS

311

In. Experimental Methods and Results The preceding discussion of the kinetics and kinematics of atom-surface interaction demonstrates clearly that any experimental approach must consider the influence of fluid dynamic and kinetic processes on the measurement. Consequently, we shall discuss experimental methods and results from this over-all viewpoint, first for the case of catalytic recombination and second for atom-surface chemical reactions. A recent review by Jennings (1961) discusses extensively the techniques of atom production and detection but does not consider the kinematic processes which might be involved. A. CATALYTIC RECOMBINATION

I . Brief Historical Review The early work by Wood (1923), describing the production of gaseous atoms in a low-pressure electrical discharge tube, marked the beginning of extensive experimental studies by many investigators of the reaction of these labile species. The first workers in this field (Wood, 1923; Bonhoeffer, 1924), assessed the catalytic recombination efficiency of various materials by coating them on the bulb of a thermometer located in the effluent stream of the discharge tube. The steady-state temperature rise indicated by the thermometer was taken as an indication of the relative catalytic activity of the surface. The temperature rise of a surface located at a discrete point in a stream containing atoms will be dependent, however, on the diffusion rate of the atoms through the gas and on the fraction of the exothermic heat of reaction accommodated by the solid, in addition to the rate of recombination on the surface. Hence, such measurements are not likely to yield reliable quantitative values of surface catalytic efficiency. Indeed, because the interplay between transport processes and chemical reaction was generally ignored by these earlier investigators (Wood, 1923; Bonhoeffer, 1924; Taylor and Lavin, 1930; Roginsky and Schechter, 1934, 1937; Suhrmann and Csesch, 1935; Poole, 1937a-c; Buben and Schechter, 1939), these results are of questionable validity. A rather unique technique, based on the theory of exothermic heterogeneous chemical reactions developed by Frank-Kamenetskii (1959, was employed by Voevodski and Lavrovskaya (1948; Lavrovskaya and Voevodski, 1951) to evaluate the rate of recombination of oxygen atoms on a number of oxide surfaces. In any process in which both heat transfer and diffusion are occurring simultaneously, the quantity of heat released at a surface is governed by the rate of chemical reaction and/or the rate of diffusion while the rate of heat removal is determined by the rate of heat conduction. In an evaluation of the

312

Henry Wise and Bernard J. Wood

stationary temperature for an exothermic reaction, the possibility of critical phenomena, that is, transitions from one thermal regime to another, has been demonstrated (Frank-Kamenetskii, 1955). In line with this theory of “thermal heterogeneous ignition ” the experimental measurements of Voevodski and Lavrovskaya demonstrated that the low steady-state temperature of a coated glass capillary exposed to an oxygen discharge would quite suddenly rise to a very high steady-state value as the concentration of atoms was slowly increased. On the basis of the theoretical criticality parameter, the recombination coefficient was fitted to an equation of the form y

= y,,

exp( - E/R*T).

The results obtained (identified by footnote b in Table XIV) are substantially lower than those found by the steady-state diffusion tube method of Smith (1943). This discrepancy cannot be explained, although it may be attributed to differences in the surface composition of the materials employed in the respective experiments. The work of Smith (1943) can be regarded as a turning point in atom recombination studies. Smith recognized the importance of the diffusive atom transport process, and devised a steady-state experimental technique which has been adopted with various modifications by a number of subsequent investigators. 2. Steady-State Diffusion Tube Experiments a. Basic Experimental Technique. The experiment of Smith (1943) employed an experimental analog of the infinitely long tube (see Section 11), along which a steady-state atom gradient was maintained by the balance between atom removal by recombination on the wall and atom diffusion through the gas from a source at one end. The shape of the gradient (although not necessarily its absolute magnitude) is determined by moving an atom detector along the axis of the tube. The experimental apparatus constructed of glass or fused quartz, generally has the form shown in Fig. 8, with a movable atom detector that effectively approximates a closure for the tube. In this apparatus, the source of atoms is a gaseous electric discharge. Earlier investigators employed an ac or dc electric arc, sustained between aluminum electrodes inside the discharge tube, but in more recent investigations, the use of electrodeless discharges at radio and microwave frequencies has come into favor. Studies of factors which affect the efficiency of production of atoms in such discharges have been the subject of a number of papers (Wrede, 1928; Poole, 1937c; Finch, 1949; Jennings and Linnett, 1958a; Coffin, 1959; Shaw, 1959; Greaves and Linnett, 1959a-c; Bak and Rastrup-Andersen, 1962; Fontijn et al., 1964b; Fehsenfeld et al., 1965; Rony, 1966).

GAS AND SURFACE ATOM REACTIVE COLLISIONS

313

,DISCHARGE TUBE

d

TO BRIDGE

ATOM DETECTOR

t H?

FIG.8 . Schematic diagram of apparatus for measurements of catalytic activities of surfaces for atom recombination.

Solutions to the diffusion equation (see Table 1V) for such a system yield the fractional atom concentrations to be expected at the closed ends of cylinders of various lengths, radii, and catalytic surface activities for given gas temperatures and pressures. The theory predicts that the steady-state concentration of surviving atoms at the atom detector will be dependent on four related but independently variable parameters, viz., the respective recombination efficiencies of the tube wall and the probe (y and y'), the multicomponent diffusion coefficient of the reactive atoms D,and the radius of the cylinder R. The relationship between these parameters provides the dimensionless quantities 6 for the wall and 6' for the detector which are the basic variables evaluated from measurements of relative atom concentration at the detector as a function of detector position, i.e., tube length. The quantities 6 and 6' are evaluated by fitting the experimental values of relative atom concentration n/no as a function of tube length L to theoretical curves obtained from Eq. (28) (Table IV). The great value of this technique is that the evaluation of 6 and 6' does not require knowledge of t h e absolute atom concentration for a first-order surface reaction. This fact, which has been demonstrated experimentally (Linnett and Marsden, 1956a; Wood and Wise, 1961a,b) enables accurate determination of surface catalytic activity to be made from relative changes in the spatial distribution of the concentration of atomic species.

314

Henry Wise and Bernard J. Wood

The applicability of the model to the experimental technique has been investigated by verifying the predicted interpedendence of the variable parameters (Greaves and Linnett, 1959a; Wood and Wise, 1961a; Wood and King, 1961). It should be noted that all measurements aimed at evaluating catalytic activity in such systems must be made under conditions where atom transport is in the viscous-flow regime and is governed by the surface recombination kinetics. b. The Atom Detector. Perhaps the major single aspect in which the basic technique of Smith has been modified and explored by various workers has been in the selection and design of the atom detector. Detectors employed in such experiments have varied widely in form, but most of them can be cataloged in one of four categories: (i) thermal detectors (probes) which measure atom population in terms of the heat released by catalytic atom recombination on their surface, (ii) chemical titration methods in which the atoms react with a metered gaseous reagent that produces a visible end point, such as a luminous glow, (iii) Wrede gages (Wrede, 1928) which measure the pressure differential across an orifice through which the atoms diffuse into a highly catalytic chamber, i.e., a region of zero atom concentration, and (iv) electron spin resonance (ESR) spectroscopy which responds to the unpaired electrons possessed by the atomic species of interest. Detection devices in the first three categories are themselves sinks for atoms, and therefore influence the atom gradient. The effect of such a device requires that an investigator be aware of the conditions of surface activity, gas pressure, and vessel dimensions which define the limits of applicability of a particular theoretical model in order to analyze the experimental data appropriately (see Section 11). Most of the atom detectors which fall into the first category (thermal devices) transduce the heat of atom recombination into a temperature rise of a thermocouple or into an enthalpy change at constant temperature in a metal filament calorimeter. A particularly interesting application of the "catalytic thermometer" to quantitative atom detection in high-velocity gas streams is to be found in an analysis by Rosner (1 962). Some questions have been raised by Tsu and Boudart (1961) concerning the validity of results obtained in short tubes ( L / R small) with such detectors on the assumption that with changes in the position of the atom detector relative to the source a perturbation in the concentration of atoms at the source may occur. The influence of the potential variation in source intensity of atoms due to such a catalytic probe surface was investigated with hydrogen atoms in two sets of experiments reported by Wise and Rosser (1963). In one set, a twoprobe system was employed. The first probe (A), consisting of a small tungsten filament, was located in a fixed position very near the atom source ( x = 0),

GAS AND SURFACE ATOM REACTIVE COLLISIONS

315

while the second probe (B), made of nickel wire, was movable. In this way the distance between the two catalytic surfaces could be varied. During steadystate conditions the atom density as detected by probe A was examined while the position of probe B was varied. The results obtained at a total gas pressure of 50 x Torr, given in Table VII, indicate that the atom source is virtually unperturbed by changes in axial position of the movable detector. TABLE VII PERTURBATION OF ATOM SOURCE BY CATALYTIC MOVABLE PROBEB” Response of probe A (arbitrary units)

Discharge power (watts)

Distance between probes A and B (cm)

65

0.3 0.6 0.9 1.2

0.415 0.412 0.412 0.41 1

24

0.4 0.9 I .2

0.384 0.384 0.388

, From Wise and Rosser (1963).

In the second type of measurement, the hydrogen atom concentration profile was determined by means of ESR spectroscopy in a quartz cylinder containing a tungsten filament detector located at fixed positions. Since the catalytic activities of the tungsten probe and the quartz wall had been determined previously, a theoretical analysis of the relative atom density at a given distance from the source as a function of the position of the probe was feasible. The agreement obtained between the measured values of the atom concentration and those predicted by the theory suggest that the perturbation of the atom source by the probe is small under the conditions of the experiment. In the same publication, these authors discuss another issue of some importance to systems employing an atom detector in the form of a filament calorimeter maintained at a temperature T, different from that of the gas at the source Tg. Under these conditions, the steady-state differential equation for diffusive transport of atoms is given by V ( D Vn) = 0. If we assume that the Lewis number is unity so that

A = CpD = (const)T’/2,

(43)

316

Henry Wise and Bernard J. Wood

we may write for the one-dimensional, steady-state conductive heat transfer

with the boundary conditions at

x=O;

T=T,

x=L;

T=T,.

Similarly, for mass transport by diffusion, one obtains

=o

dx "(p..;)

(45)

with the boundary conditions at

x=O;

n=n,

x

-dn/dx

=L;

= n/S'R.

Integration of Eq. (44) yields T3I2- T;/'

= (T,""

- T;")(x/L).

On integration of Eq. (45) and substitution, one finds (47) where the terms in square brackets on the right-hand side of Eq. (47) represents a correction term due to the nonisothermal model. A comparison of the theoretical atom concentration gradient at the catalytic filament surface under isothermal and nonisothermal conditions for two probe locations (Table VIII) suggests that the perturbation of the atom gradient due to a temperature difference between the probe and the gas is small. Consequently, it seems appropriate to employ the bulk gas temperature in the evaluation of the diffusive flux and the collision frequency rather than the probe surface temperature. A singular type of recombination energy transducer is the chemiluminescence detector, in which the heat of recombination of atoms recombining on a solid lumophor generates light in proportion to the incident atom flux (Sancier et al., 1959). Such a device has been employed to measure the recombination coefficients of nitrogen atoms on doped and undoped calcium oxide (Sancier et al., 1962). Another unique energy transducer employed to detect recombining atoms is a device similar to a Knudsen pressure gage which measures the torque imparted to a fiber-suspended vane by the momentum of desorbing molecules.

GAS AND SURFACE ATOM REACTIVE COLLISIONS

317

TABLE VIII EFFECTOF CATALYTIC FILAMENT TEMPERATURE ON ATOMCONCENTRATION GRADIENT"'^ Relative atom concentration gradient at probe (l/n,,)(dn/dx) Probe distance = I radius

Probe distance = 10 radii

0.167 0.164 0.162 0.161 0.160

0.066 0.064 0.060 0.058 0.057

300' 400 600 800 1000

From Wise and Rosser (1963). These Calculations based on a surface activity corresponding to 6'= 5 (see Section 11). Isothermal conditions. a

Larkin and Thrush (1963) employed this technique in attempting to measure the excess energy of a desorbing molecule over that of the surface on which it was formed. A variety of gas titration methods have been devised for quantitative detection of atoms. Clyne and Thrush (1961a, 1962) studied the reaction of hydrogen atoms with nitric oxide and found that the intensity of a resulting light emission in the red region of the spectrum is proportional to the product of the NO concentration and the H concentration. The observed luminescence is attributed to the radiative transition of an electronically excited species HNO* produced by the rapid reaction H

+ NO+HNO* + hv.

(48) (49)

+ H -+ Hz + N O

(50)

HNO* + HNO

This sequence is followed by the reaction HNO

regenerating the nitric oxide. All of these reaction steps are quite rapid; hence, nitric oxide may be employed to measure the relative concentration of hydrogen atoms in a gas stream by simply introducing NO at a known flow rate and observing the intensity of the light emission with a suitable photoelectric detector. Nitrogen dioxide may also be employed to detect and measure the concentration of hydrogen atoms. In this case, nitric oxide is generated by the reaction between hydrogen atoms [Eq. (51)] and NO,, and rapidly proceeds

318

Henry Wise and Bernard J. Wood

through steps (48) and (49) to give the characteristic HNO* emission. But OH radicals are also generated in the primary reaction, and they react rapidly in a series of steps to generate hydrogen atoms: H +NO,+NO

+OH

+ OH+HzO + 0 0 + OH+OZ + H.

OH

(51) (52)

(53)

As a result of this mechanism, the stoichiometry of the over-all reaction varies with the quantity of hydrogen atoms reacted; initially the ratio N 0 2 / H has a value of unity, but subsequent reaction of the OH gives an over-all stoichiometry of NOJH = 1.5, which is appropriate to the sum of the reaction steps, viz.: 2H

+ 3N02

+3N0

+ H20 + 0,.

(54)

Clyne and Thrush (1 96 1b) and Phillips and Schiff (1962) discuss the elementary steps involved in this reaction and consider its use as a titrant for hydrogen atoms. Westenberg and de Haas (1965) examined the stoichiometry of the reaction in detail by means of ESR techniques, and conclude that H atom titration with NO, is reliable in principle but somewhat ambiguous in practice due to the variation in stoichiometry and to the very low intensity of the red light emission. In a similar fashion, oxygen atoms react with nitric oxide in a light-emitting reaction : 0 + N O +NOz* +NO*

+ h~

(55)

and the NO is regenerated by reaction with another oxygen atom: 0

+ NO2 +NO +

0 2 .

(56)

In this case the emitted light, a continuum extending from 4000 8, to the near infrared, exhibits a maximum in the yellow-green region, the " Gaydon band," and is easily detected with the naked eye. Since no other reactive atoms or radicals are formed in steps (55) and (56), this mechanism forms the basis of an absolute titration technique of oxygen atoms using NO,. As NO, is added to the stream of oxygen atoms, the light emission intensity increases until the flow of NO, is one-half that of the oxygen atoms. Further addition of NO, results in a diminution of emitted light intensity until, when the flow of added NO, just equals the flow of oxygen atoms, the glow is extinguished. The reactions have been studied in detail by Kaufman (1958) and are discussed in his review of the reactions of oxygen atoms (Kaufman, 1961). The applicability of the NO, titration technique to systems containing oxygen atoms was

GAS AND SURFACE ATOM REACTIVE COLLISIONS

319

recently verified in experiments employing ESR spectroscopy by Westenberg and de Haas (1964). The light-producing character of the reaction of nitric oxide with oxygen atoms, together with the extremely rapid reaction of nitric oxide with atomic nitrogen, form the basis for a titration technique for nitrogen atoms. The NO + N reaction produces N, and 0 virtually instantaneously, but when all nitrogen atoms have been consumed by the NO, excess NO reacts with oxygen atoms to produce the easily distinguished greenish-yellow oxygen afterglow (Kistiakowsky and Volpi, 1957). In addition, the intensity of the afterglow has been correlated with the rate of disappearance of nitrogen atoms (Berkowitz et a/., 1956). The kinetics of these reactions are discussed in Mannella’s review (1963) and have been examined by the ESR technique (Westenberg and de Haas, 1964; von Wepsenhoff and Patapoff, 1965). Fontijn et al. (1964a) determined the absolute rate of the 0 + NO reaction and suggest that their results may be used as a calibration standard for other chemiluminescent reactions. Although considerably less sensitive than thermal detectors and chemical titration techniques, Wrede gages as atom detectors offer certain advantages, such as their lack of response to excited species and the absence of added chemical agents as takes place during titration. The devices are employed in pairs to measure the difference in atom concentration at two points along the tube with a catalytic wall in which diffusive atom flow is occurring. There is some disagreement in the literature about the proper design of Wrede gages, and these issues are discussed in some detail by Greaves and Linnett (1959a) and by Kaufman (1961). In contrast to the first three techniques, ESR spectroscopy is a measuring tool which does not perturb the steady-state atom gradient. Electron spin resonance also possesses the virtue of characterizing specifically the measured species; there is no chance of confusing atoms with energetically excited molecules, as could be the case with some of the other methods. Unfortunately, the ESR technique imposes rather severe restrictions on the diffusion tube length and diameter and also requires great care in calibration of the instrument. The great usefulness of this technique for recombination studies has been demonstrated, however, by a number of investigators. Krongelb and Strandberg (1959) developed an ESR technique for monitoring the concentration of atomic oxygen in quartz tubes in which atoms were transported by diffusion alone and by diffusion with bulk flow. Similar experiments were reported by Hacker et af. (1961) and also by Marshall (1962a,b) who additionally investigated the recombination of hydrogen atoms and nitrogen atoms with the ESR detection technique. Others who employed ESR to detect atoms in flowing gas streams are Shaw (1959), Hildebrandt et al. (1959), Wood et al. (1963), and von Wepsenhoff and Patapoff (1965). A critical study,

Henry Wise and Bernard J . Wood

320

evaluation, and development of the ESR technique for detection of atoms in low-pressure gases have been described in a recent series of papers by Westenberg and de Haas, for oxygen and nitrogen atoms (1964) and for hydrogen atoms and OH radicals (1965). Westenberg (1965) carried out a similar investigation of the ESR response to halogen atoms. The technique and method of instrument calibration are also described and discussed by Ultee (1966), and by Evenson and Burch (1966a,b). c. Results. With the exception of hydrogen atom recombination on glass and quartz at temperature extremes (Wood and Wise, 1962), the kinetics of atom recombination has been observed universally to be first order with respect to the gaseous atom concentration over a wide range of temperature and pressure. Thus it is convenient and significant to discuss the results of recombination kinetics in terms of the recombination coefficients y for various surfaces. (i) Glass and fused quartz. Because of the widespread use of glass in constructing experimental apparatus, an understanding of its catalytic properties for atom recombination is especially desirable. A detailed study (Wood and Wise, 1962) of the recombination rate of hydrogen atoms on Pyrex glass and fused quartz has been carried out over a temperature range from 70" to 1125°K. It was found that at temperatures above 550°K and below 120°K a transition occurred from first- to second-order kinetics in the surface reaction. The results of the measurements' are summarized in Fig. 9. In a temperature range from approximately 150" to 550°K the recombination coefficient of glass increased monotonically by a factor of 200. The observed kinetics may be interpreted in terms of two adsorption states with different binding energies (de Boer and van Steenis, 1952; Hickmott, 1960). These results are in substantial agreement with the data obtained by Smith (1943) and by Tsu and Boudart (1960) over a smaller temperature interval but are about an order of magnitude higher than the results reported by Sat0 (1965a) and by Green et al. (1959). These authors interpret their observations by postulating loosely chemisorbed hydrogen atoms, such as might be effected by one-electron bonds between the sorbed hydrogen and a lattice oxygen. Numerous investigators added water vapor to the hydrogen on the hypothesis that water acts as a surface poison (Wood, 1923; Smith, 1943; Finch, 1949). Recent measurements (Coffin, 1959 ; Rony and Hanson, 1966) question the inhibiting action of water and suggest an increase in the production of hydrogen atoms caused by changes in the characteristics of the discharge or by subsequent reactions in the gas phase. The results of recent measurements

' Reexamination of thedata obtained at 77°K (Wood and Wise, 1962)indicates somewhat lower values for the second-order recombination-rate constant and the recombination As a result, the binding energy of the weakly rather than 6 x coefficient (7 bound adatom would be slightly smaller than the 1.5 kcal/gm atom reported value. N

GAS A N D SURFACE ATOM REACTIVE COLLISIONS

32 1

of the effect of added water vapor on the concentration profile of hydrogen atoms in a tube indicate that OH radicals formed in the discharge generate hydrogen atoms in the gas phase by the reaction OH + H, + H,O + H (Kaufman and Del Greco, 1961 ; Wise ef a/., 1964; Dixon-Lewis et a/., 1966). The kinetics of this reaction is well established. 10.' : -

I

1

I

A

I

1

-

TSU 8 BONDART (1960)

--

-

0

-

I

I

I

2

4

6 ( I/T)

I 8

I

1 0 1 2 1 4

x103, O K - '

FIG.9. Temperature dependence of rate of hydrogen atom recombination on Pyrex glass and fused quartz surfaces.

The kinetics of recombination of oxygen atoms on glass (Linnett and Marsden, 1956a) and on quartz (Greaves and Linnett, 1959a-c) has been studied in some detail. Also, numerous investigators have determined values of y o in a variety of experiments with other primary objectives such as the study of gas phase kinetics. These values and references are listed in Table IX. In general, it is found that the recombination coefficient for oxygen atoms is smaller by approximately a factor of ten than y,, under comparable conditions. The exceptionally low value of yo for Pyrex reported by Morgan er al. (1960), Morgan and Schiff (1963a) (Table IX) cannot be explained.

Henry Wise and Bernard J. Wood

322

TABLE IX RECOMBINATION COEFFICIENT OF GLASSFOR OXYGEN ATOMS AT ROOM TEMPERATURE Surface Pyrex glass Pyrex glass Pyrex glass Pyrex glass Pyrex glass Pyrex glass Pyrex glass Soda glass Vitreosil Quartz (fused)

Yo 1.3 x 1.1 x 3.2 x 7.7 x 2.1 x 5.0 x 1.7 x 3.4 x 1.6 x 4.0 x

10-4 10-4 10-5 10-4 10-4 10-5

10-4 10-4 10-5

Reference Linnett and Marsden (1956a) Herron and Schiff (1958) Krongelb and Strandberg (19591 Elias et a/.(1959) Wood and Wise (1962) Marshall (1962a) Morgan and Schiff (1963a) Greaves and Linnett (1958) Greaves and Linnett (1959~) Hacker et a[.(1961)

Certain analogies are apparent between the catalytic properties of glass for oxygen atom and hydrogen atom recombination. First of all, it is noted that water vapor has little, if any, effect (Linnett and Marsden, 1956a; Greaves and Linnett, 1959b) on the value of y o . Second, quartz and Pyrex glass appear to exhibit the same catalytic efficiencies for this process. Also the variation of y o with temperature (Greaves and Linnett, 1959c) has certain characteristics observed in the case of the heterogeneous recombination of hydrogen atoms. The value of y o on glass rises from about at 300°K to about at 850°K. The change in slope of the curve depicting the variation of log y o versus the reciprocal temperature suggests a complexity in the mechanism of the surface reaction which, as in the case of hydrogen, may be related to a change in the order of the reaction due to the presence of different adsorption states. It would be desirable to extend these measurements to higher and lower temperatures in order to examine in more detail the kinetics under such conditions. For nitrogen atoms recombining on glass, various investigators have reported values spanning almost two orders of magnitude (Table X). In contrast to the recombination kinetics of hydrogen and oxygen on glass, the nitrogen system has not been studied in great detail. The problem is made difficult by the existence of long-lived metastable molecular states. No reliable data are available on the kinetics of OH destruction by glass surfaces. In his early experiments, Smith (1943) observed a large temperature rise on a KC1-coated glass surface when exposed to the products of a gas discharge through H,-H20 mixtures. However, no temperature increase was noted as a result of the interaction between KC1 and hydrogen atoms. From such measurements, yOH for Pyrex glass was evaluated. It ranged from

GAS AND SURFACE ATOM REACTIVE COLLISIONS

323

TABLE X RECOMBINATION COEFFICIENTS OF VARIOUS SURFACES FOR NITROGEN ATOMS AT ROOM TEMPERATURE Surface Pyrex glass Pyrex glass Pyrex glass Pyrex glass Pyrex glass Pyrex glass Fused quartz

YN

3.0 x 1.6 x 1.7 x 3.0 x 1.5 x 5.0 x 8.0 x

10-5 10-5 10-4 10-4 lo-" 10-5

10-4

Reference Wentink et at. (1958) Herron et al. (1959) Young (1961) Marshall (1962a) Wood and Wise-(unpublished results) Sancier et a/. ( 1 962) Marshall (1962b)

at 400°K to about 6 x lo-' at 800°K. It is not known whether the glass surface catalyzes the reaction H + OH, or OH + OH, or both, under these experimental condition>. Sanders et al. (1 954) subsequently examined the microwave spectra intensity of OH radicals and the temperature rise of a KCI-coated surface exposed to the products of an electrodeless discharge through water and found no correlation. It may be surmised from these results that the KCI surface may be sensitive to other species generated in the discharge besides OH. Recently, a number of investigators have examined the catalytic activity of various surfaces for halogen atom recombination. Ogryzlo (1961) noted that no chlorine or bromine atoms could be detected downstream from a discharge in a clean glass apparatus. The addition of a coating of phosphoric or other acid on the tube wall caused the atoms to appear, and Ogryzlo concluded that glass has a recombination efficiency approaching unity for chlorine atoms. In later work, Bader and Ogryzlo (1964) evaluated the chlorine atom recombination efficiency for phosphoric acid to be ycl = 5 x lop5.Hutton (1964) measured yc, = 3.5 x l o w 5on sulfuric acid-coated glass. Other workers, who have been interested in transporting chlorine atoms in low-pressure gas streams, have employed phosphoric acid- (Rosner and Allendorf, 1965b) or boric oxide- (Linnett and Booth, 1963) coated-glass tubes to inhibit heterogeneous recombination. (ii) Metals. A series of measurements of the catalytic properties of metals for atom recombination have been reported. The general problem of such measurements is complicated not only by the kinematic considerations, but also by the fact that in addition to atom recombination, chemical reaction between the solid and the gas may occur. At this point, we shall consider only those surfaces which are chemically stable in the presence of gaseous atomic species. The phenomenon of chemical reaction will be discussed in a subsequent section.

yOH =

Henry Wise and Bernard J . Wood

324

A significant number of reported measurements of recombination coeficients for atoms on metals are of qualitative value only, mainly because the rate effects of atom transport in the gas phase were not taken into account. This criticism would appear to apply to the investigations of Katz et al. (1949), of Nakada et al. (1955b), and of Nakada (1959a,b), all of whom evaluated relative catalytic activities by observing the temperature rise of metal films or disks situated downstream from a discharge in a flow tube. A number of investigators (Sato el al. 1955; Nakada et al., 1955a; Frantsevich and Lavrenko, 1963; Cottingham and Grosh, 1963) carried out similar experiments using Wrede gages situated near the metal surface to estimate the incident flux of atoms, but did not consider the effect on this flux of atom gradients in the apparatus. Wood and Wise (1961a) measured the hydrogen atom recombination coefficient of a number of metals over a range of temperature (Fig. 10). These 0.4

1A 1 I

I

I

I

Ni

z

50

0.05

f m

s

0.03 I

W

a

0

2

I

(fx

3

4

103) ( O K - ’ ) -

FIG. 10. Surface activity for hydrogen atom recombination on metals at various temperatures (Wood and Wise, 1961a).

results show no correlation between the catalytic properties of the metals and the electron distribution in the valence band of the solid (Pauling “ d ” character) as has been suggested by Pickup and Trapnell(l956). As a matter of fact, some of the nontransition metals exhibit higher catalytic activity than the transition metals. However, it is to be remembered that in a system containing atomic hydrogen the formation of a chemisorbed layer can proceed readily without any activation energy even for nontransition metals. Such an adsorbate may cause changes in the electronic properties of the solid, such as filling of holes in the d band, analogous to those produced by the formation

GAS AND SURFACE ATOM REACTIVE COLLISIONS

325

of solid solution composed of transition and nontransition metals. Also, it would be expected that in the presence of atomic hydrogen the formation of a chemisorbed layer on nontransition metals, such as gold, would occur without an energy barrier for the process. As a result, the formation of a hydrogen molecule by the Eley-Rideal mechanism would be feasible under these circumstances. Some indication exists that the dissipation of the energy released in the heterogeneous reaction may play an important controlling role in this process. The normal mode-frequency spectrum of a lattice ha5 an upper limit at the Debye frequency. It may be expected that those solids which exhibit a high velocity of propagation of longitudinal and transverse waves will tend to have high recombination efficiencies. Indeed, a correlation is found between catalytic activity and the Debye characteristic temperature of the solid OD (Fig. 11). In the collision between a gas atom and a solid the transport of 0.9 0.8

h’ 07

+ a

z

m

03

P V 0

w 0.2

n

01

n 0

100

200

300

400

500

DEBYE T E M P E R A T U R E

eI,

600

700

FIG.1 1 . Variation of hydrogen atom recombination coefficient with Debye characteristic temperature (Wood and Wise, 1961a).

energy by phonons in the solid becomes less efficient above a critical value of the incident energy (Cabrera, 1959; Zwanzig, 1960). During a highly exothermic reaction on the surface of the solid, similar limitations may prevail. As a result, partial accommodation of the energy released during heterogeneous

Henry Wise and Bernard J. Wood

326

atom recombination may result in the formation of a molecule containing excess kinetic or internal energy. Larkin and Thrush (1963) attempted to measure such excess energy in recombined atoms of hydrogen, nitrogen and oxygen desorbing from surfaces of silver, copper, nickel and platinum. Their measurements of the torque imparted to fiber-suspended vanes of these metals exposed to the atoms led them to conclude that the molecules could leave the surface with excess kinetic energy no greater than twice the normal thermal velocity. Wood et al. (1963) carried out a series of experimental measurements to examine quantitatively the coefficient of energy accommodation of various metals during atom recombination. For this purpose the atom flux incident upon the solid was determined by measuring the atom density by the ESR technique. The energy released as a result of the recombination process was found simultaneously by evaluation of the difference in electric power required to maintain a catalytic metal filament at constant temperature (i.e., constant resistance) in the presence of the atomic species and in the presence of molecular hydrogen. From such experiments the product of the accommodation p and recombination y coefficients may be obtained. For those metals for which the values of y have previously been determined, the energy accommodation coefficients may readily be computed (Table XI). It is to be noted that only a fraction of the energy released is transmitted to the solid, except in the case of tungsten, where the value of fl is close to unity. Of special interest is the result TABLE XI ENERGY ACCOMMODATION IN HYDROGEN ATOM ON METALS RECOMBINATION

Catalytic surface

T

Recombination" Accommodationb coefficient coefficient

Y

("K)

B

Ni

423

0.20

0.82

w

443 480 773

0.07 0.07 0.07

I .o 1 .o 1 .o

Pt

378 588 813

0.04 0.079 0.10

0.73 0.34 0.26

Wood and Wise (1961a). Wood et al. (1963).

GAS AND SURFACE ATOM REACTIVE COLLISIONS

327

for platinum, a material which exhibits a pronounced variation of y with temperature. In this case, it is found that an increase in temperature of the filament and in its catalytic efficiency is accompanied by a corresponding decrease in its ability to accommodate the energy released in the surfacecatalyzed reaction. As a matter of fact, for Pt, the product appears to remain constant in the temperature range examined. In effect, the same fractional amount of energy is transmitted to the solid per reactive collision. This result suggests, indeed, an upper limit of the amount of energy that can be transferred to the solid as a result of an exothermic reaction on its surface. It is apparent that a solid may exhibit high efficiency yet at the same time low energy accommodation for atom recombination. Further evidence of the nonequilibrium nature of the catalytic process is to be found in the detection of vibronically excited nitrogen molecules formed by catalytic reaction of nitrogen atoms on a solid surface, as reported by Harteck et al. (1960). The role of the chemical properties of the catalyst on the catalytic activity of the surface remains basically unresolved. One approach to this problem has been the investigation of the hydrogen atom recombination coefficients for alloys of palladium and gold (Wood and Wise, 1961a; Dickens et af., 1965). By analogy to hydrogenation catalysis (Bond, 1962), one might expect to see changes associated with filling of the d band in alloys composed of transition and nontransistion metals. Wood and Wise observed a maximum in the hydrogen atom recombination coefficient of metal alloy filaments at temperatures from 400" to 800°K in the vicinity of 30 at. % gold with a marked dependence on temperature. Dickens and his associates, working with Pd-Au foils, detected a similar maximum at room temperature, but found a significant dependence of catalytic activity on the length of exposure of the sample to atomic hydrogen. The trends observed by both investigators are similar, although the absolute magnitudes of the measured recombination coefficients differ by more than an order of magnitude. This discrepancy may be due to the physical nature of the surfaces employed in these catalytic studies, filament versus foil, and to differences in the experimental techniques. In spite of these variations, it does seem safe to conclude that the degree of filling of the d band is not the key to the mechanism of catalytic activity. In the absence of published Debye temperatures for this alloy system an interpretation of these results in terms of the energy transfer has not been made. A difference in catalytic activity attributable to the physical forms of the catalyst may, in part, explain the disparity in values of yH obtained earlier by Wood and Wise (1958) using evaporated films on thermocouple probes compared to those determined with wire filament probes (Wood and Wise, 1961a). In the case of oxygen and nitrogen atom recombination on metallic surfaces the experimental data are relatively meager. The catalytic properties of the B

e

y

Henry Wise and Bernard J . Wood

328

TABLE XI1 OXYGEN ATOMRECOMBINATION COEFFICIENTS (yo) ON METALS, NONMETALS, AND METAL AT ROOM TEMPERATURE” HALIDES Surface

Yo

Metals 2.4 x 10-l 1.7 x l o - ‘ 3.6 x 2.8 x 5.2 x 10-3 2.6 x 10-3 Nonmetals Antimony 8.2 x 10-4 Arsenic 4.6 x 10-4 Selenium 1.7 x 10-4 Halides Lithium chloride 1.9 x 10-3 Potassium bromide 1.3 x 1 0 - 3 Potassium iodide 0.74 to 3.5 x 10-3 0.57 to 1.9 x Barium chloride Sodium chloride 9.4 x 10-4 Potassium fluoride 9.2 x 10-4 Potassium chloride 7.8 x 10-4 Rubidium chloride 0.45 to 2.4 x 0.34 to 1.6 x Cesium chloride Silver Copper Iron Nickel Gold Magnesium

a

From Greaves and Linnett (1958).

solid are frequently obscured by the chemical reaction these metallic surfaces undergo when exposed to the gaseous atoms. Greaves and Linnett (1958) report oxygen atom recombination coefficients for several metals (Table XII) but caution that the metal surfaces used in their experiments may have become covered with oxide layers possessing different properties than the bulk oxides. In the case of platinum catalyzing the recombination of oxygen atoms the results indicate that at T > 2200°K the metal oxide is unstable (Fryberg and Petrus, 1960). At these temperatures, Wood and Wise (1961b) measured a recombination coefficient of yo = 0.04. At lower temperatures where the oxide is stable the measured value was yo x 0.01. However, the oxide surface is not well defined. Hacker et al. (1961), using ESR to measure oxygen atom flux incident on a platinum wire, computed values of yo = 0.1 at 850°C by assuming perfect accommodation by the solid of the energy of recombination. In a contrary fashion, the catalytic activity of aluminum (or aluminum oxide) was found to change from 0.17 to 0.08 as the temperature

GAS AND SURFACE ATOM REACTIVE COLLISIONS

329

was raised from 360" to 530°K (Wood and Wise, 1961b). The reason for this is not clear, but the phenomenon may be attributed to a change in phase of the film of oxide from an amorphous to a crystalline layer. The catalytic properties of gold and nickel surfaces in the presence of nitrogen atoms have been reported (Wood and Wise, 1961b). In the case of a nickel filament exposed to an atmosphere containing atomic nitrogen, a stable nitride Ni,N was produced on the surface. Its recombination coefficient was found to be yN = 0.1 for a temperature range 350°K < T < 800°K. Prok (1961) observed that the catalytic activity of platinum exposed to a stream containing nitrogen atoms slowly increased to a steady value of yN = 2.3 x lo-*. The nature of the surface under the conditions of the experiment was not characterized. (iii) Elemental semiconductors. Sancier et al. (1 966) investigated the effect of bulk doping of germanium and silicon on the catalytic activity of these elemental semiconductors for hydrogen atom recombination. In the temperature range 360"-440"K, no variation was found in the recombination coefficient in going from highly p-type to highly n-type materials. The recombination coefficient of germanium at 360°K was reported to be 0.25 & 0.06, which increased by about 50 % at the upper end of the observed temperature range. The recombination of hydrogen atoms on evaporated graphite films in the temperature range 360"-500"K was studied by King and Wise (1963), and found to increase with temperature according to the expression y = 0.38 exp( - 2300/R*T). (iv) Compound semiconductors. In his early experiments, Smith (1943) examined the catalytic properties of various salts coated on glass toward hydrogen atoms (Table XIll). Most of the materials exhibited recombination coefficients greater than glass with the exception of KCI. More recently, TABLE XI11 RECOMBINATION COEFFICIENTS OF VARIOUS GLASSCOATINGS FOR HYDROGEN ATOMS AT 300°K" Coating

Recombination coefficient yll

A1203 Na3P04 K2C03 KCI KZSi03 Pyrex glass a

From Smith (1943). Interpolated value.

4.5 x 1 0 - 1 b 2 lo-' 210-l 2 x 10-5 7 x 10-2 4 x 10-3

330

Henry Wise and Bernard J. Wood

Linnett and Marsden (1956b) measured the oxygen atom recombination efficiencies of potassium and lithium chlorides and of the oxides of molybdenum and lead. The measurements were later repeated and extended to a large number of metal oxides by Greaves and Linnett (1958, 1959b) (Tables XI1 and XIV). In a number of these experiments the catalyst was presented as a coating on the walls of the cylinder. The oxygen atom density was determined TABLE XIV (yo)ON OXYGEN ATOMRECOMBINATION COEFFICIENTS OXIDE SURFACES AT ROOMTEMPERATURE'

Periodic table SOUP

2A

Yo

Surface

Calcium oxide

2.5 x b9 x 10-4 1.6 x 10-3

3A

Boric oxide Aluminum oxide Gallium oxide

6.3 x 1 0 - 5 2.1 x 10-3 1.3 x 10-4

4A

Silicon oxide (Vitreosil)

5A

Transition

Magnesium oxide

Germanium oxide Tin dioxide Lead dioxide

1.6 x 10-4 bi.7 x 10-5 1.3 x 10-4 1.0 x 10-3 6.3 x 10-4

Phosphoric acid Arsenic Antimony Bismuth

1.2 x 8.1 x 2.7 x 2.2 x

Vanadium pentoxide Chromic oxide Molybdic oxide Tungstic oxide Manganic oxide Ferric oxide Cobalt oxide Nickel oxide Cupric oxide Zinc oxide

From Greaves and Linnett (1959b).

10-4 10-2

4.8 x 10-4 2.5 x 10-4 b 1 . 5 x 10-4 1.0 x 10-3 b6 x 1.7 x 1.3 x 5.2 x 10-3 4.9 x 10-3 8.9 x 10-3 4.3 x 10-2 4.4 x 10-4 b2.3 x 10-5

~

(I

10-4 10-5

* Results of Voevodski and Lavrovskaya (1948).

GAS AND SURFACE ATOM REACTIVE COLLISIONS

33 1

by means of a thermocouple probe whose surface was made up of silver oxide. By carrying out these measurements at large distances from the atom source and by dealing with wall surfaces of relatively low catalytic activity the authors were able to simulate the conditions of a catalytic cylinder of infinite length (see Section 11). However, reexamination of some of the data presented in graphical form (Greaves and Linnett, 1958) indicates that in some instances both the thermocouple probe and the cylinder walls effectively influenced the atom concentration profile in the cylinder. By taking into account this contribution of the probe as a sink for atoms we calculate somewhat different values for some of the metal oxide surfaces examined by Greaves and Linnett (1958). For example, the value of yo so found in the case of magnesium oxide is as compared to 2.5 x lo-’, and for aluminum oxide it is 2.5 x 5x as compared to 2.1 x (Tables XI1 and XIV). The authors cited above point out that the surfaces of low activity are acidic or amphoteric in nature and that in any group in the periodic table the recombination coefficients tend to increase as the atomic weight increases. They conclude that the catalytic activity may be associated with an ability of the surface to bond the oxygen atoms temporarily in some way. Wise and Rosser (1963), however, view these results as analogous to the hydrogen atom recombination activity of metals, and suggest that a high Debye characteristic temperature favors high recombination efficiency. The data are indeed in qualitative agreement with this notion (Fig. 12), which supports an interpretation of catalytic recombination activity for oxides of the same type as that proposed for metals. Dickens and Sutcliffe (1964) extended the investigations of Greaves and Linnett (1958, 1959b) by varying the temperature of the oxide surfaces and also measuring the electrical conductivity of the polycrystalline catalytic materials. The room temperature values of the oxygen atom recombination coefficient reported by Dickens and Sutcliffe agree with those determined by their predecessors. In addition, they observed temperature coefficients of activity corresponding to activation energies in the range 1-1 1 kcal/mole and noted that the catalytic activity of the oxides fell into the following scale of magnitude: p-type oxides > n-type oxides > insulating oxides. The catalytic behavior was too complex to interpret analytically in terms of the solid-state properties of the catalytic surfaces. In conjunction with an investigation of atom-excited luminescence in solids, the recombination of nitrogen atoms on pure and manganese-doped calcium oxide was studied in detail by Sancier et al. (1962). The results, reported to be precise to &30%, indicate little effect of the dopant on the recombination efficiency of the solid: ycaO = 5 x ~Ca0~Mn~0.0046~~C= I ~3 0 ,x 0~~~ Prok (1961) exposed lithium chloride and lead monoxide to a stream containing nitrogen atoms and estimated values of the recombination coefficient to be: yLic, = 0.0072-0.014; ypbo = 0.045.

332

Henry Wise and Bernard J. Wood

DEBYE TEMPERATURE eo FIG. 12. Oxygen atom recombination efficiency (Greaves and Linnett, 1959b) as a function of Debye characteristic temperature [derived from specific heat data of Kelley (1 950)l.

(v) Other surfaces. Largely in search of an effective surface recombination inhibitor for glass, numerous investigators have examined the catalytic behavior of a variety of materials which lend themselves to easy surface coating. The all-time favorite, rnetaphosphoric acid, was employed at one time or another by virtually all investigators who used gas discharge tubes. More recently, methylchlorosilanes (Dri-Film) (Shaw, 1959) and Teflon (Berg and Kleppner, 1962) have been tried. Bergh (1965) made an experimental comparison of the relative effectiveness for inhibiting hydrogen atom recombination of a variety of such surface coatings on Pyrex, and noted the following order of decreasing catalytic activity: Dri-Film S Pyrex

-

quartz > phosphoric acid > Teflon.

3. Non-Steady-State Experiments

A few investigators have developed techniques to observe the time-dependent decay of atomic species which occurs in a vessel following a pulse discharge. Some of these have been employed to evaluate the catalytic activity of surfaces.

GAS AND SURFACE ATOM REACTIVE COLLISIONS

333

Preston (1940) described the use of atomic resonance spectrophotometry using a Lyman-a light source to monitor the decay of hydrogen atoms in a glass bulb following a radio-frequency pulse to dissociate the gas. The technique has great sensitivity, stability, and simplicity, but was not pursued because of the strong Lyman-cr absorption by water vapor. Recently, however, Myerson et af. (1965) have demonstrated the applicability of the technique to measurement of the hydrogen atom density following a shock wave. The authors point out that unambiguous quantitative measurements will require further refinement of the technique. A catalytic surface in the form of a thin film or foil deposited on a fastresponse resistance thermometer was employed by Myerson (1963, 1965) in a different pulse technique to investigate the recombination behavior of oxygen atoms during the first moments of their encounter with the catalyst. Fresh metal catalysts, previously exposed only to molecular oxygen, showed identical heat-input rates during the first 50 psec of exposure to a step function flow of atomic oxygen generated by a radio-frequency pulse of 300 psec duration in a continuous flow stream of the gas. Continued exposure of the catalyst to atomic oxygen pulses caused the heat input rate to increase (recombination efficiency increasing) but to different extents in different metals (Table XV). Myerson suggests that the mechanism of recombination involves the formation of a surface oxide:

+

2Ms)

+

Ag2O(s),

which is subsequently alternately oxidized and reduced as atomic oxygen adsorbs and recombines: AgzO(s)

+ O(g)-t2AgO(s)

2AgOfs)

+ O(g)

+

Ag,O(s)

+

02(&

TABLE X V

RELATIVE SURFACE RECOMBINATION EFFICIENCY OF INITIALLY FORMED OXIDES AS A FUNCTION OF AVERAGE BONDENERGY'

Metal

Ag AU

Pt Pd a

Av. energy of metal-oxygen bond in oxide (kcal/mole)

Observed relative surface recombination heat transfer rate

32.9 39.0 75.5 78.9

100

From Myerson (1965).

40 I .O 0.9

Henry Wise and Bernard J. Wood

334

In support of this explanation, Myerson cites a correlation between the metaloxygen bond energy in the oxide and the observed heat input rates during atom recombination (Table XV). Energy accommodation efficiency of the surface is assumed to be unity. A similar technique was employed by Hartunian er al. (1965) to examine the catalytic behavior of a number of oxidized metals with respect to oxygen atoms and to nitrogen atoms. These investigators determined the incident flux of atoms on their catalytic surfaces by means of the standard NO titration technique; hence, they were able to compute numerical values of y. The results (Table XVI) are in reasonable agreement with the results of steadyTABLE XVI

RECOMBINATION COEFFICIENTS y OF VARIOUS SURFACES AT 300°K" Gas 3 N 0 N 0 N 0 0 0 0 0

Y

Surface (oxidized) Ag

0.15 i 15%

cu

0.15 i0.05 0.07 & 0.04

A1

<

< 10-2

Pt Ni SiO Dow-Corning No. 510 silicone oil Pyrex

0.004

* 0.003

0.04 0.02 < 10-4

< < 10-4

From Hartunian et al. (1965).

state experiments (cf. Tables IX, X, and XII). In more recent work, Hartunian et al. (1966) describe a glow discharge shock tube in which a shock tube driver is combined with a glow discharge flow tube to produce higher atom fluxes, temperatures, and densities than are normally attainable. These authors suggest the applicability of such a technique to the study of atom-surface interactions but do not report any experimental results. B. CHEMICAL REACTIONS OF GASEOUS ATOMSWITH SURFACE ATOMS In recent years, a number of studies of the kinetics of gas-surface chemical reactions have been published. Only a relatively small fraction of these, however, have involved gaseous atomic reactants. Excluded specifically from this review is the rather vast literature on corrosion processes, in which the gaseous reactant contacts the solid surface in the molecular state, even though it may subsequently dissociate and react as an atom on the surface.

GAS AND SURFACE ATOM REACTIVE COLLISIONS

335

The high reactivity of atomic species, particularly hydrogen atoms, generated in a low-pressure electric discharge has been long recognized. The metal mirror transport experiments of Paneth (1920) were followed by a large number of qualitative experiments where investigators looked for signs of chemical reaction in the physical and chemical properties of the solid phase (Bonhoeffer, 1924; Schultze and Miiller, 1930; Pietsch and Josephy, 1931; Pietsch, 1933; Pearson et al., 1933; Pietsch and Lehl, 1934). The subject of hydride formation in such reactions between a solid surface and gaseous atomic hydrogen has been comprehensively reviewed by Siegel (1961).

I . Metals, Oxides, and Semiconductors a. Hydrogen Atoms. In general, those metals which possess thermodynamically stable, well-characterized hydrides have been shown to react chemically with atomic hydrogen at low temperatures (Siegel, 1961). This includes the alkali metals and the alkaline earths (groups 1A and 2A in the periodic table). Ferrell et al. (1934) identified the hydrides of sodium, lithium and potassium formed by exposure of mirrors of the metals to atomic hydrogen. Quite recently, Anderson and Ritchie (1966) studied the kinetics of the reaction between atomic hydrogen and evaporated films of sodium in the temperature range 78"-178"K by measuring the rate of pressure change in a closed vessel when hydrogen atoms were generated on a hot tungsten filament. Under the conditions of the experiment, the hydride, assumed to be NaH, forms a solid layer on the surface of the metal, and the kinetics are diffusion limited. Based on initial rates, an activation energy E < 1.5 kcal/ mole was calculated. Since the mean free paths of the gaseous species in this experiment were relatively large compared to the vessel dimensions, kinetically excited hydrogen molecules leaving the hot filament could strike the surface. The authors suggest that these species contribute to the hydride formation reaction. The more electronegative elements of group 3A are less reactive; Siegel (1960) observed the formation of AIH, which condensed at -78°C when aluminum was vaporized in the presence of atomic hydrogen, but it is uncertain whether the reaction occurred at a surface or in the gas phase. Metals of the transition group are known to occlude hydrogen, but the existence of discrete hydrides is questionable except in a few cases. Harris and Tickner (1948) detected no volatile products by exposing lead and bismuth mirrors to atomic hydrogen, but did obtain a volatile, heat-labile tellurium hydride, condensable at 77"K, by reacting the hydrogen atoms with a tellurium film. Group 4A elements, however-particularly carbon, germanium, and tinform known hydrides, and have been shown to react with atomic hydrogen at low temperatures.

336

Henry Wise and Bernard J. Wood

Sancier et al. (1966) studied the rate of reaction of germanium single crystals with hydrogen atoms as a function of the electronic carrier type and doping level in this semiconductor. Since germanium forms volatile hydrides [(GeH,), or GeH,] (Amberger, 1959), the reaction rate could be followed by recording the change in mass of the germanium crystal. The reaction was found to be first order with respect to atomic hydrogen and possessed a rate constant k = 1.4 & 0.2 cmjsec at 2 5 ° C independent of doping level. This was a factor of about lo4 slower than the observed rate of catalytic recombination of hydrogen atoms occurring simultaneously. Bennett and Tomkins (1962) examined the interaction of atomic hydrogen with germanium films at 78°K and observed chemisorption of a monolayer of hydrogen, presumably without chemical reaction. Raising the temperature of the films resulted simply in desorption of the hydrogen. This observation is in agreement with the observed high ratio of recombination rate to reaction rate (Sancier et al., 1966). An early study (Bagdasar’yan and Semenchenko, 1935) demonstrated the reactivity of a number of metal oxides, sulfides, and halides with atomic hydrogen. Based on simple, qualitative observations, these authors empirically correlated the atomic hydrogen reducibility of compounds with their heats of formation and the ratio of their ionic radii to ionic charge. Bagdasar’yan (1937) later reported rate measurements of the reactions between atomic hydrogen and sulfur, selenium, tellurium, and antimony, in all of which the kinetics were first order with respect to hydrogen atoms. In the case of H + S, the number of effective atomic collisions were reported to be one-fifth. For Se and Te, the ratio of reactive collisions was one-tenth, but for Sb it fell to one-ninetieth. The low reactive efficiency for antimony was attributed to a high atom recombination efficiency for this element. Vdovenko reacted hydrogen atoms with SnCl, (1939) and with SnCl, (1945) to produce a condensible, heat-labile product which was assumed to be SnH, . Based on a tin analysis of the product residue, 1 in 330 collisions was estimated to be reactive. From a practical standpoint, the possible use of atomic hydrogen to remove oxide films from metals has been of some interest. Glemser et al. (1952) flowed a gas stream containing atomic hydrogen through stirred beds of powdered metal oxides and, by means of chemical analyses, identified products containing reduced forms of the metals (Table XVII). Bergh (1965) measured the minimum temperature at which a variety of oxides underwent reduction in a stream of atomic hydrogen by comparing the color change of the oxide to that observed in molecular hydrogen at much higher temperatures (Table XVIII). No kinetic information can be obtained from his data, except the observation that the activation energy for the reaction with atoms is much lower than that for the molecular reaction.

GAS A N D SURFACE ATOM REACTIVE COLLISIONS

337

TABLE XVII PRODUCTS

FORMED BY WITH

Oxide

a

REACTION OF

HYDROGEN ATOMS

METALOXIDES‘

Observed products

Glernser el al. (1952). TABLE XVIII REDUCTION TEMPERATURE FOR DIFFERENT OXIDES IN MOLECULARAND ATOMICHYDROGEN^

Oxide

Moo3 GeO,

wo3 Sn02 Fez03 PbO

cuzo NiO CUO

Reduction temp. Color change in H2 (“C) in Hz (“C) 500-725 500-700 475-650 400-675 250-500 225-475 225-450 225-325 10@225

610 560

535 490 310 300 265 250 140

Color change in H (“C) 43 35 25 100

40 25 25 52 25

From Bergh ( I 965).

For quantitative detection of atomic hydrogen in the gas phase, Melville and Robb (1949) employed the oxides of Mo and W i n their early work on the kinetics of interaction of hydrogen atoms with olefin molecules. These authors observed at room temperature a reduction of yellow tungsten oxide, WO, , to blue W,O, , when exposing the oxide to atomic hydrogen. More recently, Khoobiar (1964) demonstrated that a similar reduction of WO, occurs at 50°C when molecular hydrogen is passed over a mechanical mixture of platinum deposited on y-alumina and tungsten oxide. Under these conditions it appears that the hydrogen atoms formed on the Pt surface are migrating over the alumina support before reaching the WO, .

338

Henry Wise and Bernard J. Wood

b. Oxygen Atoms. A study of the rate of oxidation of copper exposed to atomic oxygen in the efflux of a low-pressure electric discharge, was reported by Dravnieks (1950), using the time variation of electrical conductivity of a metal sample as a measure of the reaction rate. At 5OO0C,and a total presssure of 0.5 Torr, the initial oxidation rate of copper is 50 % higher in dissociated oxygen than in molecular oxygen. The initial product of the reaction was Cu20,which remained on the surface of the metal as a scale layer. Continued exposure to oxygen atoms resulted in the formation of a mixture of CuO and Cu,O in the scale, complicating the kinetics, which was evidently controlled by mass transport through the oxide layer. Dravnieks examined the oxidation behavior of other metals also (Ta, Zr, Ni, and Mo) but found no consistent enhancement in the observed rates of oxidation when atomic oxygen was employed. A more thorough study of the oxidation of molybdenum by atomic oxygen was reported quite recently by Rosner and Allendorf (1964). These investigators employed the basic technique of Dravnieks but added several significant refinements. Perhaps most important was the selection of a temperature range (1050°-15000K) in which molybdenum oxides are known to be volatile, so that the kinetics of interaction between 0 and Mo would not be obscured by mass transport processes in the condensed phase. (The absence of rate control by diffusional transport in the gas phase was demonstrated experimentally.) In addition, they controlled the macroscopic temperature of the reacting metal surface by means of optical pyrometry, and they reliably assessed the incident atom concentration by titration with NOz (Kaufman, 1958). Their results are presented in terms of oxidation probabilities, defined as the ratio of the mass flux of molybdenum atoms (regardless of their chemical state of aggregation) away from the surface, to the collision flux of 0 or 0, with the surface. If the reaction product remains identical, the use of such a dimensionless rate parameter allows a direct comparison of the kinetics of oxidation by atomic and molecular oxygen, respectively (Table XIX). Rosner and Allendorf interpret their data to indicate that gas phase atomic oxygen enhances the oxidation rate by means of a Rideal mechanism in which reaction occurs by direct attack of the incident oxygen atoms on the surface chemisorbed oxygen. A Russian investigator (Nazarova, 1958) studied the kinetics of oxidation of several metals (Cu, Al, Fe, Zu, and Mg) by using them as electrodes in an oxygen discharge apparatus. Rates were measured by recording the mass change of the sample with a microbalance, and also by observing the decrease in oxygen pressure in the apparatus. It is of interest that the product formed depended on whether the electrode was used as an anode or a cathode. The oxide film formed on an aluminum anode was amorphous, while that formed on an aluminum cathode was crystalline. In the case of copper, Cu,O formed

339

GAS A N D SURFACE ATOM REACTIVE COLLISIONS

KINETICS OF O X I D A n b N

TABLE XIX OF MOLYBDENUM BY ATOMIC A N D MOLECULAR OXYGEN' Reactants

Kinetic parameters Order with respect to reactant

Surface temp. (OK)

Reactant pressure (Torr)

1150 and 1400

Activation energy (kcal/mole)

1050-1 500

Oxidation probability

1200

0.002-0.05 1.5 0.01 0.03

Mo

+ 0, -

Complex 26 0.001

-

-

Mo

+0

First -6 0.1 -

From Rosner and Allendorf (1964).

on an anode, but CuO formed on a cathode. However, as with other investigations carried out by immersing reactant solids within the gas discharge [Miyamoto (1932a-d, 1933a-c, 1934a-c, 1935, 1937) studied the effect of a hydrogen discharge on a number of metal salts in this way], it is impossible to distinguish the effects of atom-surface encounters from gas phase reaction, electric field transport phenomena, and reaction of short-lived excited molecular species with the surface. In a series of investigation spanning several years, Fryburg studied the reactivity (1956), the products (1965a) and the kinetics (1965b) of the platinum atomic oxygen system. Working in a temperature range where the reaction product, PtO, , is volatile (900"-1 500°C), Fryburg followed the reaction rate by chemically analyzing quantitatively the condensed residue which deposited on a cool, quartz cylinder within which the platinum ribbon specimen was situated. Oxygen atom concentration was determined by NO, titration. The kinetics was found to be first order with respect to atomic oxygen, and the reaction exhibited zero activation energy over the observed temperature range. Oxygen atoms were found to be about 300 times more reactive than oxygen molecules at 900°C (EOIEO, = 5 x 10-6/2 x lo-') but Fryburg about equally reactive at 1500°C ( E O / E O ~= 5 x 10-6/9 x postulates a Rideal mechanism (comparable to that suggested by Rosner and Allendorf ( 1964) for the atomic oxygen-molybdenum reaction) :

, \':Pt-O(ads)+O(g)+[ ,

'\

,'.o

,Xt+O,(g)

I

\\

'Pt, \

,

/

](ads)<

' '0 PtOa(g) It should be noted that at T > 850°C the rate of the oxygen atom recombination reaction is greater by a factor of about lo4 than the platinum oxidation reaction (cf. Wood and Wise, 1961b). An independent study of the desorption

340

Henry Wise and Bernard J. Wood

of oxygen from a heated, oxidized platinum filament (Mitani and Harano, 1960) verified Fryburg’s characterization of the oxide as PtO, . c. Chlorine Atoms. Until very recently there seems to have been little experimental interest in reactions of halogen atoms, possibly due to their apparent high recombination efficiency on glass (Ogryzlo, 1961) which seems to severely limit the attainable steady-state concentration of the atoms. Ogryzlo (1961) noted that acid coatings on glass inhibited chlorine atom recombination and observed that the atomic chlorine rapidly attacked silver and cobalt wires. The only kinetic study of an atomic chlorine-metal reaction is a recent investigation of the rate of attack of molybdenum wire by Rosner and Allendorf (1965b). These workers used the same technique which they employed in their earlier study of the molybdenum-oxygen reaction (1964). They found that the reaction was first order with respect to chlorine atoms at 1050”K, at 400°K but that the reaction probability gC1 increased from a value of to 3 at 1000°K. (Corresponding reaction probabilities for C1, never exceeded & at PCl2= 3 x lo-’ Torr.) The mechanism is postulated to be of the Rideal type, in which a gas phase chlorine atom strikes an adsorbed chlorine atom to form an activated complex leading to desorption of MoCl,(g) or to the desorption of a chlorine molecule (recombination). At temperatures above about 1200”K,the reaction probability decreases rapidly with increasing surface temperature, presumably because of an enhanced rate of evaporation of C1 and/or C1, from the metal surface. Rosner and Allendorf also investigated the rate of chlorination of a nickel wire, but found that the introduction of atomic chlorine did not enhance the reaction rate. They explain this by postulating that, in contrast to molybdenum, nickel possesses a high sticking probability for C1, and a low evaporation rate of unreacted chlorine. d. Nitrogen Atoms. Several comprehensive reviews on “Active Nitrogen ” have appeared in recent years (Raghavendra Rao, 1956; Jennings and Linnett, 1958b; Mannella, 1963), but none cite any references to investigations of the heterogeneous chemical reactivity of atomic nitrogen with metals or salts. Wood and Wise (1961b) noted that a nickel wire exposed to atomic nitrogen developed a scale which produced an electron diffraction pattern comparable to that reported for the interstitial nitride Ni,N, but made no attempt at kinetic measurements. 2. Carbon a. Hydrogen Atoms. Avramenko (1946) showed that carbon black deposited on the wall of a hydrogen discharge tube rapidly disappeared and at the same time the spectrum of the CH radical appeared in the discharge glow. A year

GAS AND SURFACE ATOM REACTIVE COLLISIONS

34 1

later, Harris and Tickner (1947) reported the results of a similar experiment, in which they estimated that the gas stream contained 20% atomic hydrogen at a total pressure of 0.4 Torr. These investigators analyzed their product stream and found it to be principally methane with small amounts in addition of C 2 to C , hydrocarbons. Blackwood and McTaggart (1959a) did a kinetic study in which they monitored reaction rate by measuring the rate of production of methane. The solid reactant employed in their experiments was eucalyptus wood charcoal pretreated with hydrogen or with carbon dioxide. At room temperature they observed rates of the order of mole CH, min-l gm-', which were practically independent of pretreatment. King and Wise (1963) chose graphite in the form of thin evaporated films deposited on glass supports as the carbonaceous solid reactant in a recent kinetic investigation. They worked in the temperature range 365"-500"K, with total gas pressures from 0.02 to 0.1 Torr. The reaction rate was measured optically by recording the change in optical density of the graphite film as it was consumed by reaction with atomic hydrogen. Two distinct rates with activation energies of 9.2 and 7.1 kcal/mole, respectively, were reported. The products of the reaction were low-molecular-weight hydrocarbons, principally methane and ethane, but the chemical reaction competed rather poorly with the catalytic recombination reaction in the temperature range studied. King and Wise observed first-order kinetics with respect to atomic hydrogen at low atom concentrations, which change to half-order kinetics at higher concentrations. They postulated a mechanism involving some additional surface intermediates which are formed by a first-order reaction of hydrogen atoms and disappear by a second-order reaction. h. Oxygen Atoms. A number of investigators have recently reported rate studies of the reaction between carbon in various forms and atomic oxygen over a large temperature range. A principal objective of these investigations has been to better understand the mechanism of carbon oxidation by molecular oxygen. Blackwood and McTaggart (1959b) monitored the rate of formation of carbon monoxide and carbon dioxide obtained from samples of graphite, eucalyptus wood charcoal, and industrial diamonds exposed to mixtures of atomic and molecular oxygen at low pressures. Their experiments were carried on at room temperature, under which conditions graphite and the charcoal were readily oxidized, but not diamond. Indeed, the latter exhibited little reactivity even when raised to a temperature of 200°C. The ratio of CO to CO, in the product stream increased rapidly as the pressure was lowered. The authors suggest that CO is the primary reaction product, and that even in the oxidation of carbon by molecular oxygen, dissociation of the 0, must precede reactions with the carbon.

Henry Wise and Bernard J. Wood

342

Somewhat contrary results were reported by Streznewski and Turkevich (1959). These workers reacted atomic oxygen with flame-deposited soot in the temperature range 20"-lOO"C, monitoring the removal of soot by optical density measurements. They found the rate of carbon removal to be proportional to the estimated oxygen atom concentration, but they detected only C0,-no CO-in the product stream. However, under the conditions of this experiment, whether CO, was the primary reaction product, has been recently questioned by Vastola et al. (1963). A high temperature study of graphite oxidation by both atomic and molecular oxygen was made by Rosner and Allendorf (1965a). They determined the rate of carbon removal from a 20-mil diameter graphite rod by recording the increase of electrical resistance at a constant temperature in the range 1 100"-2000°K. The results demonstrated that the reaction kinetics were first order with respect to atomic oxygen. Below 1400°K the data fit an Arrhenius plot (Fig. 13) suggesting activation energies of 6.7 and 31 kcal/mole for the atomic oxygen and molecular oxygen reactions, respectively. By computing the incident atom or molecule flux on the basis of the surface temperature of the graphite rod (using the Hertz-Knudsen equation), these TEMPERATURE

2000

I600

K

- O

I400

1600

I200

I .o

i

>-

z!

i m m

L E C U L A R OXYGEN

a

0 LL

0.1

a

z

0 F a

APHITE OXIDATION \ / 0 2 / 0 MIXTURES, p = l Tory,

E X

0

0.01

0.5

0.7

0.6 IOJ/T

-(

I

OK

0.8

0.9

1-1

FIG.13. Arrhenius plot of oxidation probabilities for the attack of graphite by atomic and molecular oxygen. (Dashed curves represent values of E based on incident particle flux computed for gas at 300°K.) (Rosner and Allendorf, 1965b.)

G A S AND SURFACE ATOM REACTIVE COLLISIONS

343

authors suggest that oxygen atoms react with unit probability in the temperature range 1500"-1700"K (Fig. 13). They conclude that the major reaction product is CO and that oxidation is dominant over recombination under these conditions. It seems questionable, however, that the rod temperature is appropriate for evaluating the incident particle flux when the bulk gas temperature is probably near 300°K (cf. Wise and Rosser, 1963; Table VIII). A recomputation of the reaction probabilities E reported by Rosner and Allendorf based on an incident particle flux evaluated for a gas at 300°K suggests that their values of E should be reduced by about a factor of 2 (dashed curves in Fig. 13). One could conclude from this recomputed data that CO, might very well be a primary product of the reaction. Indeed, the reaction probability E M 0.4 in the 1500"-1700"K temperature range strongly indicates that CO, might be formed by a Rideal mechanism in which an incident oxygen atom encountering an adatom leads directly to desorption of a CO, molecule. This interpretation is further supported by the reports of a number of other investigators (Blackwood and McTaggart, 1959a,b; Streznewski and Turkevich, 1959; Marsh e i al., 1965b) that CO, is a primary product of the reaction between graphite and atomic oxygen. Marsh er a/. (1965b) examined the oxidation of three amorphous carbons and an artificial graphite by atomic oxygen in the temperature range 14"250°C, by measuring the rate of production of products CO and CO,. Rates of the order of mole min-' gm-' were observed, but they were found to be dependent on the length of time the solid was exposed to atomic oxygen. An activation energy of about 10 kcal/mole was measured, but above 200°C the value of this parameter begins to diminish, becoming almost zero at 350°C. At this temperature the kinetics are first order with respect to oxygen atoms. Based on the observation that the activation energy of the reaction between carbon and atomic oxygen is about one-fourth that reported for the molecular oxygen reaction, these authors conclude that the rate of desorption of a carbon-oxygen surface intermediate cannot be a rate-determining step. They suggest rather that the formation step of such a surface intermediate governs the reaction rate, and that once formed, such an intermediate can desorb to give CO, or it can become stabilized on the surface as a surface oxide. Marsh et al. (1965a) used an electron microscope to examine the surface changes wrought on carbons and graphites by atomic oxygen. While molecular oxygen attacks the carbon or graphite at separated, discrete points (probably defect structures on the graphites), atomic oxygen produces a general background of conical pits on the surface. c. Nitrogen Atoms. Zinman (1960, 1961) studied the interaction of nitrogen atoms (also mixtures of nitrogen and hydrogen atoms) with graphite rods at 800"C, by analyzing the volatile products formed. The principal product

344

Henry Wise and Bernard J. Wood

observed was HCN, which appeared even when H, was absent from the gas stream, presumably as a result of an organic contaminant in the graphite. In no event was cyanogen (or paracyanogen) detected, which, Zinman suggests, indicates that the reactive collision efficiency between atomic nitrogen and carbon is less than lop4 at 800°C. 3. Organic Materials

A few investigators have studied the chemical interaction of atomic hydrogen with organic molecules. Reactions between hydrogen atoms and a number of solid hydrocarbons at 77°K have been investigated by coating the organic materials on the inside wall of a glass bulb containing hydrogen at low pressure and a hot tungsten filament. Atoms formed on the filament diffuse to the wall where they react with the organic film. Chemical analysis of the residual film discloses the extent of the reaction and identifies the composition of the products. This technique has been developed by Klein and Scheer (1958, 1961) and employed by them to study the low temperature chemistry of the propylene-atomic hydrogen system (Klein et a/., 1960) and the reaction between hydrogen atoms and four-carbon olefins (Klein and Scheer, 1963). These investigators found that hydrogen atoms added to the terminal double bonds of the olefins to form reactive radicals which rearranged internally, recombined with other radicals, or disproportionated. Thus, for example, butadiene-l,2 reacts with atomic hydrogen to produce butene-2, three isomeric eight-carbon dimers, and butyne-2. Some of the elementary steps associated with these reactions have been further investigated by employing deuterium instead of hydrogen as the gaseous reactant (Scheer and Klein, 1959). A similar experimental technique was used by Taylor and Malkin (1964) to study the reactions between atomic hydrogen and both long-chain alkanes and some aromatic compounds. These authors reported that polymerization of the organic reactants occurred under their experimental conditions. A practical application of this reactive characteristic of atoms has been demonstrated by Hansen and Schonhorn (1966), who enhanced the hardness and chemical resistance of polymer films by inducing cross-linking in thin surface layers by exposure to gaseous atoms. Cole and Heller (1965) produced free radicals by exposing malonic acid, naphthalene, and other compounds to a stream of atomic hydrogen. In some cases, hydrogen was abstracted from the organic molecules while in other cases it was added.

IV. Concluding Remarks It is apparent that the free-bound transitions of the atoms undergoing inelastic, reactive collision with the solid represent a complex pattern of

GAS AND SURFACE ATOM REACTIVE COLLISIONS

345

surface processes in which chemical forces are effective. Free-free transition, in which an atom colliding with the surface need not undergo adsorption, probably plays a lesser role in the processes of interest to the discussion. Energy transfer to and from the solid is very likely an essential step in the reactive collisions considered in this review since chemical bonds are formed and broken. However, the energy of concern is the potential energy of the reacting system rather than translational energy, which is of interest in inelastic, nonreactive collisions with solid surface, and which has been reviewed in some detail in several recent publications (Wachman, 1962; Hurlbut, 1963; Schaaf, 1964; and two of the articles in this volume, viz. R. E. Stickney and E. C . Beder). Different intermediate states for reactive collisions between gas and lattice atoms are depicted schematically in Fig. 3. It is apparent that the product composition is indicative of the type of bond broken during the reaction. Rupture of bond A (Fig. 3) results in the formation of a gaseous diatomic molecule composed of identical atoms, while breakage of bond B leads to a product composed of a lattice atom and a gaseous atom. For case A the amount of recombination energy transferred to the solid need not be very large since the molecule X, may be stabilized by a nonequilibrium process in a highly energetic state. For case B the amount of energy released is much smaller, and it is expended in the rupture of a bond characteristic of thelattice. Similar considerations apply to case C in which is postulated the formation of a polyatomic gaseous compound with the inclusion of a lattice atom. The course and kinetics of reaction are therefore dictated by thermodynamic considerations as well as those of relaxation time for energy transfer to the solid relative to the lifetime of the surface intermediates [s - XI or [s - X - XI. As yet no precise quantitative data are available for these lifetimes in the case of reactive collisions. However, from the kinetic data on surface-catalyzed atom recombination an estimate may be made of the mean adsorption lifetime of an atom on a metal. The average residence time t may be defined in terms of the ratio of the surface density of adatoms to the rate of catalytic reaction. For a typical system such as hydrogen on tungsten, z z sec under experimental conditions of temperature and pressure commonly employed in recombination rate measurements. These times are smaller by several orders of magnitude than the adsorption lifetimes of charged particles such as Cs' on tungsten (Scheer and Fine, 1962). From statistical considerations Frenkel (1924) derived an expression relating the residence time t to the surface temperature T and heat of adsorption QA 5 = ( W K T )exp(lQ~l/R*T). At a 1000°K where the pre-exponential term is 5 x sec, a residence time sec requires a value of I Q AI = 43 kcal/mole. This value is of the same t=

346

Henry Wise and Bernard J. Wood

magnitude as that derived from experimental measurements of the heat of adsorption (Table I). The phenomenon of surface-catalyzed excitation of the product particle leaving the surface is of considerable interest both in itself and also as a tool for the study of the properties of such metastable particles. Several recent publications have been concerned with the subject of excited molecular species. In the case of oxygen atoms and nitrogen atoms, some published data are available which identify certain aspects of the degree of nonequilibrium in the energy distribution between the solid and the gaseous products of reaction. For oxygen atoms in contact with nickel, spectroscopic investigation revealed emission associated with the forbidden transition A 'Xu++ X 3C,- of the atmospheric system and A 3C,,+ + X 'C,- of the Herzberg system (Mannella and Harteck, 1961). Similarly, for nitrogen atoms recombining on a cobalt surface, the formation of N, in highly excited vibrational levels has been reported (Harteck et al., 1960). These authors suggest that the mechanism of this reaction involves the recombination of two nitrogen atoms on the surface of the solid to yield a vibronically excited nitrogen molecule: 2N(4S) -+ N2(A 'Xu')

[U

2 81.

These molecules are collisionally deexcited to the crossing point of the B 'llg state, and a radiationless transition occurs : N2(A 3C,')

-+

[u 2 81

N,(A

3&,f)

+

[v = crossover]

N,(B 311e) [v I 81

.

Subsequently, the first positive system is observed in emission as a result of N,(B

'nS)+ N,(A

[v 2 81

'Xu+) + hv.

[t' = low]

Mass spectrometric analysis of vibronically excited molecules (produced in a discharge) has been performed by Foner and Hudson (1966) in an experimental analysis of the appearance potential of excited nitrogen molecules. In another study (Starr, 1965; Starr and Shaw, 1966) the transfer of vibrational energy from N, in the ground electronic state to electronic excitation energy of alkali atoms has been demonstrated. In chemical reaction kinetics studies Bauer and co-workers (Lifshitz et al., 1965; Bauer and Ossa, 1966; Watt et al., 1966) have explored the role of vibrationally excited molecules in isotopic exchange rates. Finally, Hirschfelder and Eliason (1957) have made some theoretical calculations on the effect of electronic excitation of atoms and molecules on transport properties such as viscosity and diffusion coefficients.

GAS AND SURFACE ATOM REACTIVE COLLISIONS

347

It is apparent that many questions relevant to the chemical behavior of gas-atom-solid systems await further investigation and clarification. For example, an analytical treatment of the role of the solid phase in such systems cannot be realistically approached without considering (i) the effect of gas sorption and occlusion on the reactive properties of the surface, and (ii) the variation between bulk and surface properties of the solid. A number of studies of gas sorption by solids have been carried out (Cupp, 1953) but most of them with solids exposed to molecular gases so that the sorption process was governed by kinetic limitations. The determination of surface properties of materials has been greatly stimulated in recent years by the utilization of new tools for examining the surface such as contact potential measurement, flash desorption with high-speed partial-pressure gas analyzers, low energy electron diffraction (LEED), and field emission microscopy. Undoubtedly, the next few years will witness major advances in the elucidation of the fundamentals of reactive interactions between gas atoms and solids.

ACKNOWLEDGMENT Preparation of this review was made possible through the sponsorship of Project Squid, supported by the Office of Naval Research, U. S. Department of the Navy.

LISTOF 3,=

-[sinh L,,

[cash L,,

+ (S,/8,)1/2 cosh L,,] + (6,/6,)112sinh L,,]

C Specific heat c Root-mean-square velocity of reactive species D Diffusion coefficient E Activation energy for reaction E* Activation energy for chemisorption E.4 ' Activation energy for desorption F Mass flux h Planck constant J o , J 1 , J 2 Bessel functions of the first kind k Chemical reaction rate constant L Length of cylinder L,, ~s(2/6~)1/2 i = s, t . . .

SYMBOLS

M rn

N n QA

Qc

Qs

Qw @

R

r R* T

Particle mass Concentration of molecular gaseous species Surface site density Concentration of gaseous atoms Heat of adsorption Bond energy of atoms in solid crystal lattice Bond energy of gaseous molecule (heat of dissociation) Binding energy of chemisorbed atom Heat of physisorption Rate of chemical reaction Radius of cyiinder Radial distance Gas constant Temperature

Henry Wise and Bernard J. Wood

348

OD

T, Temperature of surface T, Temperature of gas u n/no u Bulk flow velocity w Mean particle transport velocity x Axial distance x l = ~ ( 2 6 , ) ” ~i = s, t . . . Z Collision frequency cq

Roots of equation Jo(ccl)= sat.Jt(4

/3 Reaction energy accommoday

y’

A V 6

6‘ E K

0

tion coefficient Recombination coefficient of cylinder wall ‘ Recombination coefficient of cylinder closure Laplacian operator (for cylindrical symmetry) = (a2/ar2) (I/r)(a/ar) (a2/ax2) Gradient operator = 4D[1 - (y/2)l/(ycR) = 40[1 - (y’/2)]/(y‘~R) Reaction probability Boltzmann constant Fraction of occupied sites

+

+

h v p u T

Debye characteristic temperature Mean free path Frequency Density Electronegativity term Residence time Number flux of particles

Subscripts H Parameter associated with reaction of hydrogen atoms i Chemical species L Value of parameter at x = L N Parameter associated with reaction of nitrogen atoms 0 Parameter associated with reaction of oxygen atoms o Value of parameter at x = 0 r Derivative with respect to r s, t . . . Geometric limit of surface of uniform catalytic activity (see Fig. 6) x Derivative with respect to x 1 Atoms 2 Molecules

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AUTHOR INDEX Numbers in parentheses are reference numbers and indicate that an author's work is referred to although his name is not cited in the text. Numbers in italic show the page on which the complete reference is listed. Ablow, C.M., 300, 301, 303, 304, 305, 306, 307, 308, 309, 310, 321,348,352 Abragarn, A., 105(115), 116 Abramson, E., 136, 137, I41 Abuaf, N., 153, 201 Ackermann, H., 85(151), 93(151), 117 Acrivos, A., 309, 349 Adolph, J., 138, 139, 142 Agranovich, V. M., 130, 134, 141 Alcalay, J. A., 144, 20I Allen, A. M., 225, 287 Allen, R. T., 206, 211, 270, 271, 274, 287 Allen, S . J. M., 37, 48 Allendorf, H. D., 323, 338, 339, 340, 342, 351 Althoff, K., 85(33), 93(33, 81), 115, I16 Amberg, C., 225, 287 Amberger, E., 336, 348 Anderson, D. K., 85(37), I15 Anderson, J. B., 157, 201 Anderson, J. R., 335, 348 Anderson, L. W., 113(144), 114(147), 117 Andres, R. P., 157, 201 Andrew, E. R., 159, 201 Ardifi, M., 110(124), 111(124,130), I17 Aston, J. G . , 225, 288 Avakian, P., 130, 136, 137, 138, 139, 141 Avramenko, L. E., 340,348 Bader, L. W., 323, 348 Bassler, H., 138, 141 Bagdasar'yan, Kh. S . , 336, 348 Bailey, J. M., 86(61), 115 Baird, J. C . , Jr., 113(144), 114(147), 117 Bak, B., 312, 348 Balling, L. C., 54, 72, 112(137), I17 Barrat, J. P., 84(50, 51, 53), 85(38, 39), 115 Barrell, P., 346, 352 Barth, C. A., 319, 350 Bates, D. R., 8, 14, 29, 41, 43, 46, 47, 48 Bauer, S . H., 346, 348, 350, 352 Baule, B., 151, 187, 201

Bearden, A. J., 37, 48 Beattie, J. A., 222, 287 Beaty, E. C., 111(129, 131), I17 Beck, D. E., 178, 202 Beck, J., 107(118b), 117 Beder, E., 209, 232, 240, 256, 259, 287 Beebe, R. A., 225,287 Beleva, L., M., 126, 129, 135, 141 Belikova, G . S . , 126, 129, 136, I41 Bell, K., 13, 30, 48 Bell, R. J., 37, 38, 39, 48 Bell, W. E., 110(125), 117 Bender, P. C., 111(129, 131), 117 Bennett, M. J., 336, 348 Benz, K. W., 121, 124, 125, 127, 128, 129, 132, 138, 141 Berg, H. C., 332,348 Bergh, A. A., 332, 336, 337, 348 Berkman, E. F., 152, 201 Berkowitz, J., 319, 348 Bethe, H. A., 28, 50 Bely, O., 29, 48 Bichsel, H., 39, 48 Biermann, L., 43, 48 Bird, R. B., 215, 219, 220, 221, 223, 288 Bitter, F., 73, 74, 76, 79, 100(96), 102(96), 111(132), 114, 116, I17 Blackwell, L. A., 136, 142 Blackwood, J. D., 341, 342, 348 Blake, N. W., 136,141 Blamont, J. E., 84(49), 115 Bleaney, B., 107(118a), 117 Bloch, F., 92, 115 Bloom1 A. L., 110(125), 117 Boeckrnann, K., 102(97), 116 Bogan, A., 197,203 Bonch-Bruevich, A. M., 126, 129, 135, 141 Bonch-Bruevich, V. L., 206, 287 Bond, G. C., 179, 201, 293, 295, 327,348 Bonhoeffer, K. F., 311. 335, 348 Booth, F. B., 319, 350 Booth, M. H., 323, 350

355

356

AUTHOR INDEX

Boudart, M., 297, 314, 320, 352 Boutron, F., 84(50), I15 Bowen, E. J., 124, 141 Boyd, A. H., 41,42, 43, 44,48 Boyle, G. S., 102(111), 116 Brackmann, R. T., 155, 158, 179, 201, 202 Bradt, P., 323, 350 Brandon, R. W., 139, 141 Branscomb, L. M., 2, 45, 46, 48, 50 Breene, R. G., 46,48 Breit, G., 97, 116 Brillouin, L., 234, 287 Brog, K. C., 85(28), 86(28), 87(28), 105 115 Broida, H. P., 312,349 Brossel, J., 73, 74, 76, 78(17), 79, 84(49. 50, 53), 109(120, 122), 110(126), lll(133, 134), 114, 115, 117 Brot, C., 102(110), I16 Bruckner, K., 46,49 Buben, N., 311, 348 Bucka, H., 85(42, 48, 83, 153, 157, 158), 86(59), 87(59), 88(59), 92,93(68,74,76, 78,79,82,83), 94(48), 96(85,86), 98(91, 93), 105(42, 59), 110, 115, 116, 117 Budick, B., 77(11, 13), 85(42, 47), 86(58, 59), 87(59), 88(59), 94(84), 98(1 l), 105(42), 107(118), 114, 115, 116, 117 Burch, D. S., 45, 50, 320, 349 Burgess, A., 2, 5, 29, 30, 42, 43, 48 Burhop, E. H. S., 144, 148, 161, 162, 203, 206,233, 246,247, 289 Burke, P. G.,2,12.14,16,17, 19,30,31, 32, 34, 35, 47, 48, 48, 49, 50 Busch, F. V., 62, 72 Byron, F. W., Jr., 77(12), 78(18, 19), 85(18, 34), 1L'3(12), 102(12, 34, 101), 114, 115, 116 Cabrera, N., 152, 202, 325,349 Cagnac, B., 109(122), lll(133, 134), 117 Cairns, R. B., 47, 50 Caldwell, C. W., 160, 203 Caris, J. C., 136, 141 Carovillano, R. C., 82, I14 Carter, V. L., 41, 43, 44,49 Carver, T. R., 110(124), lll(124, 130), 117 Chaika, M. P., 85(46), 115 Chambers, C. M., 152,202 Chambrk, P. L., 147,150,153,203,309,349 Chandrasekhar, S., 6,44, 45, 49

Chaney, R. L., 111(135), 112(135), 117 Chi, A. R., lll(129, 131), 117 Chuck, W., 64,72 Chupka, W. A., 319, 348 Church. D., 71, 72 Clark, G. L., 60, 72 Clarke, J. F., 242, 287 Cleland, W., 86(61), 115 Clyne, M. A. A., 317, 318, 349 Codling, K., 19, 31, 39, 40, 50 Coffin, F. D., 312, 320, 349 Cogan, J. L., 85(38), 115 Cohen, V. W., 166,202 Cole, T., 334,349 Colegrove, F. D., 80, 86(21), 87(21), 114 Condon, E. U., 2, 49, 82(23), 114 Constabaris, G., 225, 287, 289 Coolidge, A. S., 41, 49 Cooper, J., 31, 39, 50 Cooper, J. W., 4, 15, 25,46, 49 Corner, J., 222, 287 Cottingham, W. B., 324, 349 Cottrell, T. L., 272, 287 Craig, D. P., 122, 140, 141 Craig, J. C., 137, 141 Crews, J. C., 162, 163, 181, 202, 207, 247, 287 Crowell, A. D., 208,215,220,222,224, 225, 227,228, 229, 230, 231, 287, 290 Csesch, H., 31 1, 352 Cupp, C. R., 347, 349 Curtiss, C. F., 215, 219, 220, 221, 223, 288 Cuthbertson, C., 37,49 Cuthbertson, M., 37, 49 Dalgarno, A., 5 , 8 , 1 4 , 2 7 , 2 8 , 3 7 , 3 8 , 3 9 , 4 5 , 47, 48, 49 Damgaard, A., 29, 48 Datz, S., 159, 161, 175, 176, 177, 182, 187, 202, 203 Davydov, A. S., 120, 140, 141 Dayhoff, E. S., 86(60), 115 de Boer, J. H., 208, 215, 224, 287, 297, 320 349 de Haas, N., 318, 319, 320, 352 Dehmelt, H. G., 54, 5 5 , 56, 72, 112(136, 141), 113(141), 117 Del Greco, F. P., 321, 350 De Marcus, W. C., 225, 287 Dent, B. M., 226, 288

AUTHOR INDEX

Dershem, E., 37, 49 Devienne, F. M., 212, 287 Devonshire, A. F., 208, 209, 210, 231, 240, 243, 247, 252, 254, 255, 256, 264, 267, 288, 289 de Zafra, R. L., 85(36), 115 Dibeler, V. H., 323, 350 Dicke, R. H., 5 5 , 72, 109(121), 110(127), 111(128), 112, 117 Dickens, P. G., 301, 327, 331, 349 Dirac, P. A. M., 287 Ditchburn, R. W., 2,42,49 Dixon-Lewis, G., 321, 349 Dodd, J. N., 93(71), 102(111), 115, 116 Doughty, N. A., 30,49 Dousmanis, J. C., 323, 352 Dravnieks, A., 338,349 Drees, T., 71, 72 Eck,T. G., 83, 104(114), 115, 116 Eckhouse, M., 86(61), I15 Ederer, D. L., 31, 32, 34, 35, 37, 49, 50 Ehler, A. W., 47, 49 Ehrlich, G., 152, 202, 203, 206, 289, 295, 297, 349 Eisenbud, L., 22, 49 Eley, D. D., 293, 294, 349 Elias, L., 322, 349 Elias, L., 321, 351 Eliason, M. A., 346, 350 Ellett, A., 164, 165, 166, 202, 204 Emslie, A. G., 113(142), 117 Ern, V., 138, 139, 141 Erofeev, A. I., 198, 202 Esterrnann, I., 162, 202, 207, 212, 241, 243, 246,247, 248, 252, 254, 255,261, 287 Evenson, K. M., 312, 320, 349 Ewart, R. W., 45, 49 Faidysch, A. N., 124, 128, 129, 132, 141, 142 Fano, U., 19, 20, 25, 26, 28, 49 Fehsenfeld, F. C., 312, 349 Fenn, J. B., 157, 201 Ferrell, E., 335, 349 Feshbach, H., 17, 49 Feuer, P., 198, 202, 206, 211, 212, 270, 271, 274, 275, 218, 284 ,287 Finch, G. I., 312, 320, 349 Fine, J., 345, 352

357

Fischer, E., 54, 58, 61, 62, 72 Fite, W. L., 155, 158, 172, 174, 178, 179, 180, 182, 201, 202,204 Forster, Th., 120, 141 Foley, W. M., 154, 155, 159, 161, 174, 175, 176, 177, 182, 183, 191, 192,202 Foldy, L. L., 83(27), 115 Foley, H. M., 78(19), 114 Foner, S. N., 346, 349 Fontiyn, A., 312, 319, 349 Fortson, E. N., 55, 72, 112(141), 113, 117 Fox, D., 233, 287 Fox, L., 1 5 , 49 Fox, W. M., 93(73), 116 Francis, W. E., 68, 72 Franck, J., 120, 123, 141 Franken, P. A., 74, 80(21), 81, 86(21), 87(21), 112(139), 114, 117 Frank-Karnenetskii, D. A., 309, 311, 312, 349 Franklin, J. L., 323, 350 Frantsevich, I. N., 324,349 Franzen, W., 113(142), 117 Fraser, P. A,, 30, 49 Fredericks, W. J., 316, 323, 331, 351, 352 Freeman, A. J., 105(116), 126 Freeman, M. P., 225, 287 Frenkel, J., 345, 349 Frisch, R., 162,202,207,241,246,247,248, 251, 254,287 Fryburg, G. C., 328, 339,349 Gadzuk, J. W., 212, 233, 267, 268, 287 Gallagher, A., 85(43, 44),86(44), 115 Gallus, G., 137, 138, 141 Gaponov, A. V., 60,72 Gazis, D. C., 237, 290 Geltman, S., 14, 17, 18, 19, 30, 31, 44, 45, 49,50 Gerkin, R. E., 139, 141 Germer, L. H., 16), 202, 235,237, 287 Gilbey, D. M., 152, 202, 206, 212, 213, 214, 237, 242, 252, 266, 267, 268, 272, 273, 274, 275, 287, 288 Giordmdine, J. A,, 77, 114 Gioumousis, G., 5 5 , 72 Glauber, R. J., 269, 288 Glernser, O., 336, 337, 349 Goldenberg, H. M., 54, 58, 62, 67, 71, 72 Gol’dman, I. I., 272, 288

358

AUTHOR INDEX

Goodman, F. O., 150, 151, 152, 153, 187, 195, 198, 202, 206, 209, 212, 237, 238, 242,257, 266, 272, 274, 276, 277, 288 Goodwin, E. T., 15, 49 Gordon, R. D., 140, 141 Goshen, R. L., 85(36, 42), 86(59), 87(59), 88(59), 105(42, 59), 107(118), 115, 117 Gough, W., 85(45), 86(45), 115 Greaves, J. C., 301, 303, 312, 314, 319, 321, 322, 327, 330, 331, 332,349 Green, L. C., 36, 49 Green, M., 320, 349 Greyson, J., 225, 288 Grosh, R. J., 324, 349 Grossetete, F., 78(16), 114 Gschwendtner, K., 130, 141 Guiochon, M. A., 84(49). 115 Gulbransen, E. A., 299,349 Gurney, R. W., 145,202 Haeker, D. S., 319, 322, 328, 349 Hall, J. L., 137, 141 Halsey, G. D., Jr., 225, 287, 289 Hammer,A., 130, 131, 134, 138, 141 Hancox, R. R., 163, 164,165, 166,202 Hanle, W., 83, 114 Hansen, R. H., 344,349 Hanson, D. M., 140,141 Hanson, D. N., 320,351 Hanson, R. J., 112(138), 117 Happer, W., 85(40), 86(57), 115 Harano, Y., 340, 350 Hardy, W. A., %97,349 Hargreaves, J., 29, 49 Harris, G. M., 335, 341,350 Hart, J., 30, 49 Harteck, P., 327, 346, 350 Hartunian, R. A., 334, 350 Hasted, J. B., 233, 288 Hatchett, J. L., 316, 323, 331, 352 Hauffe, K., 300,350 Hauschild, U., 336, 337, 349 Havens, G. G., 39,49 Hawkins, W. B., 109(121), 117 Heinzelmann, G., 93(152), 117 Heitler, W., 2, 37, 49 Heller, H. C., 344, 349 Henry, R. J. W., 5 , 16, 46, 47, 49, 50 Heppke, G., 85(48), 94(48), 115 Herron, J. T., 322, 323, 350

Herzberg, G., 30,49,206,216,217,218,219, 222, 231, 277, 288 Herzfeld, K. F., 148,202,217,231,278,288, 300,351 Hewitt, E. W., 334,350 Hickmott, T. W., 320,350 Hildebrandt, A. F., 319,350 Hill, T. L., 215,216,233,277, 288 Hinchen, J. J., 154, 155, 159, 161, 174, 175, 176, 177, 182, 183, 184, 191, 192, 194, 202 Hirota, N., 139, 140, 141 Hirsch, H. R., 102(107, 108), 116 Hirschfelder, J. O., 215, 219, 220, 221, 223, 288, 346, 350 Hobart, J., 112(139). 117 Hochstrasser, R. M., 120, 141 Holloway, L. W., Jr., 113(143, 146), 114(146), 117 Hopkins, B. C., 300,350 Hopper, E. H., 225,287 Horowitz, J., 105(115), 116 Howarth, A., 212, 231, 278, 281, 288 Howsman, A. J., 152, 202, 209, 232, 240, 257,288 Hudson, R. D., 41,42,43,44, 49 Hudson, R. L., 346, 349 Huggett, G. R., 67, 71, 72 Hughes, V. M., 86(61), 115 Hurlbut, F. C., 144, 151, 152, 153, 157, 162, 178, 187, 198,202, 345,350 Hutchison, C. A., 139, 140, 141 Hutton, E., 323, 350 Isler, R. C., 87(63), 88(63), 105, 115 Jackson, J. M., 209,212,231,263,266,278, 28 1, 288 Jacobs, S., 107(118), 117 James, H. M., 41, 49 Jawtusch, W., 178,203 Jefferts, K. B., 54, 72 Jennings, D. A., 137, 141 Jennings, K. R., 311, 312, 320, 340, 349, 350 Jones, W. J., 137, 142 Joseph, P. J., 333, 351 Josephy, B., 164, 203 Josephy, E., 335, 351 Jutsum, P. J., 42, 49

AUTHOR INDEX

Kabir, P. K., 8, 49 Kaminsky, M., 145, 146, 147, 149, 203, 215, 233, 288 Kanda, Y., 136, 142 Kapitsa, P. L., 60, 72 Kastler, A., 73(7), 109(119, 120,122), 110 (126), 111(133), 114, 117 Katz, S., 324, 350 Kaufman, F., 318, 319, 321, 338,350 Keck, J. C., 144, 189, 190, 192, 193, 195, 196, 197, 203, 204, 209, 289 Kelley, K. K., 332, 350 Kellogg, J. M. B., 166, 187, 203 Kellogg, R. E., 127, 141 Kennard, E. H., 155, 203 Kepler, R. G., 136, 137, 138, 139, 141 Khan-Magometora, Sh. D., 138, 141 Khoobiar, S., 337, 350 King, A. B., 329, 341, 350, 352 Kingston, A. E., 13, 37, 38, 48, 49 Kinnear, R. W. N., 93(71), 115 Kinzer, E. T., 152, 202 Kistemaker, J., 145, 204 Kistiakowsky, G. B., 319, 324, 348, 350 Kittel, C., 214, 233, 234, 256, 288 Kleen, W.. 68, 72 Klein, M., 46, 49 Klein, R., 344, 350,352 Kleppner, D., 332, 348 Knox, R.S., 120, 141 Knudsen, M. H. C., 150,203 Knuth, E. L., 144, 201, 202, 206, 212, 288 Kohler, R., 76, 102(10), 114 Kohlmannsperger, J., 129, 135, 142 Konobeev, Yu. V., 130, 134, 141 Kopfermann, H., 90(65), 93(74,76,78,82), 98(91, 93). 115, 116 Korsunskii, V. M., 129, 141 Korvalski, J., 85(l55), 117 Koster, G., 103(112), 116 Kothari, L. S., 214, 256, 288 Kovalev, B. P., 126, 129, 135, 141 Kravchenko, A. D., 124, 128, 132, 142 Krivchenkov, V. D., 272, 288 Krongelb, S., 319, 322, 350 Krueger, H., 85(30), 93(30,77, 81), 102(97), 115, 116

Kubaschewski, O., 300, 350 Kuluva, N. M., 288

359

Kurzius, S. C . , 312,349 Kuyatt, C. E., 28, 31, 37, 50 Lacey, A. R., 121, 140, 141 Lacey, R. F., 102(99), 111(132), 116, 117 Laidler, K. J., 297, 352 Lamb, W. E., Jr., 86(60), 87, 115 Lambert, R. H., 113(145,148,149), 114(148) 117 Larnkin, J. O., 13, 15, 16, 51 Landau, L. D., 60, 62, 72, 242, 288 Landman, A., 85(35), 86(59), 87(59), 88(59), 98(35), 102(35), 105t59). 115 Langmuir, R. V., 54, 58, 62, 67, 71, 72 Larkin, F. S., 317, 326, 350 Lassettre, E. N., 28. 31, 32, 34, 35, 50 Lavin, G. I., 311, 352 Lavrenko, V. A., 324,349 Lavrovskaya, G. K., 311, 330, 350, 352 Lecluse, Y.,85(38), 115 Lehl, H., 335, 351 Lenel, F. W., 226, 288 Lennard-Jones, J. E., 207, 208, 209, 210, 211, 226, 231, 240, 243, 247, 252, 254, 255, 256, 264, 267, 288, 289, 292, 350 Leonas. V. B., 152, 203, 206, 275, 289 Leroy. D. J., 306, 352 Levin, L. A,, 85(47), 86(58), 94(84, 85), 96(85, 86, 87), 115, 116 Levinger, .J S., 37, 39, 50 Lewis, D., 346, 352 Lewis, J. T., 8, 49 Lewis, M. N., 36,49 Lewis, R. R., 80(21), 86(21), 87(21), 114 Li, C. H., 197, 203 Liepmann, H. W., 155,203 Lifshitz, A,, 346, 350 Lifshitz, C., 346, 350 Lifschitz, E. M., 60, 62, 72 Linnett, J. W., 297, 298, 301, 303, 312, 313, 314, 319, 320, 321, 322, 323, 327, 328, 330, 331, 332, 340, 349, 350 Lipsky, L., 5, 49 Litovitz, T. A., 217, 231, 278, 288 Livingston, M. S., 28, 50 Livingston, R., 120 123, 141 Logan, R. M., 144, 152, 187, 188, 189, 190, 191, 192, 193, 195, 196, 197, 203, 204, 209, 275, 289

360

AUTHOR INDEX

Longarce, A., 146, 203 Loos, H. G., 289 Lowry, J. F., 31, 32, 34, 35, 50 Lubeck, K., 43, 48 Luescher, E., 113(143, 146), 114(146), 117 Lurio, A,, 77(1 l), 78( 15),85( 15, 36,44,95), 86(44), 97(22, 90), 98(1 I , 9 9 , 102(15), 114, 115, 116 Lutz, G., 336, 337, 349 Lynn, N., 27,49 Lyons, L. E., 121, 130, 140,141 McCarroll, B., 152, 203, 206, 289 McCarroll, R., 12, 50 McChesney, M., 242,287 McClintock, R. M., 137, 141 McClure, D. S., 120, 136, 141 McDar:el, E. W., 2, 50 McDermott, M. N., 77( I2), 78( 18), 80, 85(18, 34), 98(20), lOO(12, 20), 102(12, 20,34,98,101,102), lIl(135). 112(135), 114,115, 116,117 McDonald, W. M., 64, 72 McDowell, C. A., 222, 289 McDowell, M. R. C., 46,50 McFee, J. H., 166,203 McGuire, E. J., 29, 50 McLean, W. L., 102(111), 116 McRae, E. G., 160, 203 MacRae,A. U., 161,174,179,203,235,237, 287, 289 McTaggart, F. K., 341, 343,348 McVicar, D. D., 12, 17, 19, 30, 31, 32, 34, 35, 47, 48, 49 Madden, R. P., 19, 31, 39,40,50 Ma Ing Fuinn, 85(156), 117 Major, F. G., 54, 55, 72 112(141), 113 (141), 117 Malkin, I., 344, 352 Malloy, E. S., 184, 202 Mandel, M., 97(90), 116 Mangum, B. W., 139, 141 Mannella, G. G., 319, 327, 340, 346, 350 Maradudin, A. A., 214, 234, 237, 256, 289 Marcus, P. M., 166, 203 Marcus, S., 85(42), 87(63), 88(63), 105(42), 105(63), 107(108), 115, 117 Margerie, J., 110(126), 117 Markova, G. V., 85(46), 115 Marr, G. V., 42, 45

Marriott, R., 16, 50 Marsden,D. G. H., 153, 201,298. 313, 321, 322, 330, 350 Marsh, H., 343, 350 Marshall, S. A., 319, 322, 328,349 Marshall, T. C., 319, 322, 323, 350 Martin, J. B., 46, 49 Mason, E. A,, 222, 289 Massey, H. S. W., 28,50, 144,148,161, 162, 203,206, 233, 246,247, 289 Matsen, F. A,, 38, 39, 50 May, J. W., 161, 202 Maywell, J. C., 151, 203 Melissinos, A. C., 100(96), 102(96. 106), 116 Melngailis, J., 237, 289 Melville, H. W., 337, 350 Menasian, S., 67, 71, 72 Merrifield, R. E., 138, 139, 141 Meunier, J., 78(17), 114 Meyer, C. B., 319, 349 Meyer-Berkhout, U., 85(32), 93(32,77), 115, 116 Mielczarek, S. R., 31, 39, 50 Mikiewicz, E., 124, 141 Miller, M. A., 60, 72 Miller, W. F., 28, 50 Mills, J. S., 319, 326, 353 Minor, A., 93(78), 116 Mitani, K., 340, 350 Mitchell, A. C. G., 73(1), 78, 83, 84, 85(26), 114 Miyamoto, S., 339, 350, 351 Mobley, R. M., 86(61), 115 Montroll, E. W., 214, 234, 237, 256, 289 Moore, C. E., 159, 161, 175, 176, 177, 182, 187,202, 203 Moores, D., 29, 48, 48,50 Morgan, J. E., 306, 321,351 Moran, J. P., 207, 289 Morrison, S. R., 329, 336, 352 Morse, P. M., 231, 256,289 Mott, N. F., 28, 50, 209, 231, 263, 266, 288 Motz, H., 300, 301, 305, 306, 307, 348 351 Miiller, E., 335, 352 Mulder, M. M., 36,49 Murphy, E. J., 83, 85(26), 114 Myerscough, V. P., 46,47, 50 Myerson, A. L., 333, 351

AUTHOR INDEX

Nagel, H. H., 85(157), 117 Nakada, K., 324, 351,352 Nazarova, R. I . , 338, 35I Ney, J., 85(48, 150). 93(74), 94(48), 96(150), 115, 116, I17 Nieman, G. C., 136, 142 Novick, R., 77(1 I , 12). 78(15), 80, 85(15, 34, 35). 86(59), 87(59, 63), 88(59, 63), 97(90), 98(11, 20, 3 9 , lOO(12, 20), 102(12, 15, 20, 34, 35, 101, 102), 105 (59, 63), 112(140), 113(146), 114(146), 114, 115, 116, 117 Ogryzlo, E. A., 322, 323, 340,349,351 O’Hair, T. E., 343, 350 O’Malley, T. F., 17, 18, 19, 31, 50 Oman, R. A., 153, 197, 198, 203, 212, 289 Omont, A. 78(17), 84(54), 86(56), 114, I15 Ong, J. N., Jr., 300, 351 t)pik, U., 2,49 Oppen, G. v., 85(83), 93(83), 116 Osberghaus, O., 54, 58, 61, 62, 72 Ossa, E., 346, 348 Otten, E. W., 84(55), 93(82), 115, 116 Pace, E. L., 225, 289 Pagel, B. E. J., 45, 48 Palczewska, W., 327, 349 Palmer, R. L., 167, 180, 181, 182, 203,207, 256, 283, 284, 289 Paneth, F., 300, 335, 351 Patapoff, M., 319, 352 Paul, W., 54, 58, 61, 62, 71, 72 Pauling, L., 233, 236, 289, 351 Pauly, H., 166, 203, 215, 216, 289 Peach, G .,29, 50 Pearson, T. G., 335, 349, 351 Pekeris, C. L., 45, 50 Penning, F. M., 56, 72 Percival, I. C . , 12, 50 Perry, B. W., 85(34), 102(34, 101, 102), 115, 116 Peters, H. E., 112(140), 117 Petrus, H. M., 329, 349 Phillips, L. F., 318, 351 Pickup, K. G., 324, 351 Piech, K. R., 37, 39, 50 Pierce, T. R., 56, 72

361

Pietsch, E., 335, 351 Piketty-Rives, C. A., 78(16), 114 Pines, D., 237, 256, 269, 289 Pipkins, F. M.. 54, 72, 112(137, 138), 113(144, 145, 148, 149), 114(147, 148), 117 Platzman, R. L., 28, 50 Poole, H. G., 31 I , 312, 351 Prepost, R., 86(61), 115 Preston, W. M., 433, 351 Propstl,A., 121,124,127,130,131,132,133, 141 Prok, G. M., 329, 331, 351 Pryce, M. H. L., 105(115), 116 Rabold, R., 85(153), 93(79), 116, 117 Radzieuskii, G. B., 138, 141 Raghavendra Rao, K. S., 340,351 Ramsey, N. F., 88(64), 115, 144, 161, 162, 203 Rapp, D., 68, 72 Rasiwala, M., 93(76), 116 Rastrup-Andersen, J., 312, 348 Read, G. E., 145, 203 Ream, N., 272, 267 Recknagel, E., 102(97), 116 Redi, O., 102(104, 160), 116, 117 Reed, R., 343, 350 Reeves, R. R., 327, 346.350 Rice, W. E., 222, 289 Richardson, C. B., 54, 72 Richter, B., 111(132), I17 Ritchie, I. M., 335, 348 Ritter, G. J., 85(31), 93(31,69, 72), 104(69), 115, 116 Robb, J. C., 337, 350 Roberts, R. W., 160, 203 Robertson, J. M., 121, I42 Robinson, G. W., 136, 140, 141 Robinson, P. L., 335, 349, 351 Rogers, M., 152, 203, 206,289 Roginsky, S., 311, 351 Rohrlich, F., 5 , 6, 50 Rony, P. R., 312, 320, 351 Rose, M. E., 82, 114 Rosenbluth, M. H., 64, 72 Rosner. D. E., 309, 312, 314,323,338, 339, 340, 342,349,351 Rosser, W. A., 314, 315, 317, 331, 343, 352 Rothberg, J. E., 86(61), 115

362

AUTHOR INDEX

Rothe, H., 68, 72 Rubin, R. J., 238, 275, 289 Safron, S., 334, 350 Sagalyn, P. L., 93(70), 100(96), 102(96), 115. 116

Saloman, E. B., 85(34, 40, 41), 86(57), 102(34, 101, 102), 115, 116 Salpeter, E. E., 8, 49 Saltsburg, H., 160, 167, 168, 169, 170, 171, 172, 173, 178, 180, 181, 182, 185, 186, 203, 204, 207, 242, 256, 283, 284, 289 Sam, J. R., Jr., 225, 287, 289 Samson, J. A. R., 2, 31, 32, 36, 37, 38, 39, 40, 41. 47, 50 Sancier, K. M., 316, 321, 323, 329, 331, 336, 351, 352 Sandars, P. G. H., 107(118b), 117 Sanders, T. M., 323, 352 Sands, R. H., 80(21), 86(21), 87(21), 112 (139), 114, 117 Saraph, H., 16,50 Sato, S., 297, 320, 324,351, 352 Sawyer, R. B., 145, 203 Schaaf, S. A., 147, 150, 153, 203, 345, 352 Schafer, K., 182,203 Schalow, A. L., 323, 352 Schechter, A., 31 1, 348, 351 Scheer, M. D., 344, 345, 350,352 Scheffler, K., 85(30), 93(30), 115 Schenck, A., 93(80,152), 116, 117 Scherr, C. W., 38, 39, 50 Schiff, H. I., 306, 316, 329, 321, 322,349, 350,351 Schiff, L. I., 218, 231, 261, 289 Schmillen, A., 126, 128, 129, 134, 135, 142 Schnaithmann, R., 131, 142 Schneider, W. G., 137, 138, 139, 142 Schnepp, O., 233,287 Schoen, K., 123, 142 Schofield, D., 301, 320,349 Schonhorn, H., 344,349 Schott, D. J., 309, 310,352 Schuessler, H. A., 85(75, 158), 93(75, 76), 115, 116, 117

Schuter, D., 29, 50 Schuller, D., 182, 203 Schultze, G., 335, 352 Schulz, W. R., 306,352

Schwartz, C., 27, 44,45, 50, 97(89), 103 (1 13), 107(89), 108(89), 116 Schwoerer, M., 139, 140, 142 Seaton, M. J., 5 , 12, 14, 16, 29, 30, 42, 43, 46,47, 48, 50 Selwood, P. W., 222, 289 Seman, M., 46, 50 Semenchenko, V. K., 336,348 Series, G. W., 74, 85(31, 4 3 , 86(45), 93(31, 72, 73). 114, 115, 116 Sewell, K. G., 4, 37, 38, 39, 40, 50 Shaw, T. M., 312, 319, 332, 346,352 Shein, M., 37, 49 Shepherd, E. F., 161, 174, 175, 176, 177, 192, 194,202 Sherement, M., 121, 142 Shida, S., 324,351, 352 Shortley, G. H., 2, 49, 82(23), 114 Shpak, M. T., 121, 142 Shuler, K. E., 213, 242, 289, 297, 352 Sidman, J. W., 121, 142 Siebrand, E., 137, I42 Siegel, B., 335, 352 Siegert, A., 92, 115 Silverman, J. N., 28, 32, 34, 36, 38, 39, 50 Silverman, S. M., 31, 50 Simpson, J. A., 28, 31, 37, 39, 50 Simpson, O., 137, 138,142 Singh, S., 137, 142 Singwi, K. S.. 214, 256, 288 Sloan, I. H., 13, 50 Sloans, G. J., 121, 122, 142 Smith, D. P., 145, 203 Smith, F. T., 22.46, 50 Smith, F. W.,124. 141 Smith, J. N., Jr., 157, 160, 167. 168, 169, 170, 171, 172, 173, 174, 178, 179, 180, 181, 182, 185, 186, 203, 204, 207, 242, 256, 283, 284, 289 Smith, K., 12, 16,47, 48, 49, 50 Smith S. J., 45, 50 Smith, W. V., 298, 301, 303, 312, 320, 322, 322, 329, 352 Smith, W. W., 85(43), 102(105), 115, 116 Snoek, C., 145, 204 Spence, P. W.. 111(135), 112(135), 117 Spencer, W. B., 225, 287 Spitzer, L., 64,72 Starr, W. L., 346, 352 Sponer, H., 136,142

AUTHOR INDEX

Stager, C. V., I02(107, 109), 116 Steele, W. A,, 225, 289 Steinberg, M., 319, 322, 328, 349 Steiner, R. F., 324, 350 Steinwedel, H., 58, 71, 72 Stern, O., 162, 202, 207, 241, 243,246, 247, 248, 251, 252, 254, 261, 287 Sternheimer, R., 91, 102(66), I15 Sternlicht, H., 136, 142 Stevenson, D. P., 55, 72, 293, 294,352 Stewart, A. L., 5, 8, 14, 19, 28, 30, 32, 34, 41, 47, 49, 50, 51 Stickney, R. E., 144, 148,150,152,153,159, 169, 175, 176, 187, 188, 189, 190, 191, 192, 193, 195, 196, 197, 203, 204, 209 275, 289 Stockmayer, W. H.. 222, 287 Stoddart, E. M., 335,351 Stoicheff, B. P., 137, 142 Strachan, C., 208, 209, 211, 254, 264, 289 Strandberg, M. W. P., 319, 322, 350 Streznewski, J., 342. 343, 352 Stroke, H. H., 102(103, 104, 160), 116, 117 Suguira, T., 324, 351 Suhrmann, R., 31 I , 352 Sullivan, J. O., 323, 352 Sutcliffe, M. B., 331, 349 Switendick, A. C., 138, 139. 141 Tait, J. H., 41. 51 Takayanagi, K., 274,289 Taylor, A. J., 14, 48 Taylor, E. H., 159, 161, 175, 176, 177, 182, 187, 202, 203 Taylor, H. S., 311, 352 Taylor, J. B., 166, 204 Taylor, K., 344, 352 Taylor, N. J., 179, 204 Teller, E., 120, 141, 242, 288 Temkin, A., 13, 15, 16, 51 Thaddeus, P., 102(98), 116 Thomas, L. B., 147, 148, 151, 204 Thompson, H. M. 333,351 Thompson, W. P., 334,350 Thrush, B. A,, 317, 318, 326,349, 350 Tickner, A. W., 335, 341, 350 Tietz, T., 45, 51 Toennies, J. P., 166, 203, 215, 216, 289 Tolansky, S., 73(2), 114 Tomboulian, D. H., 31,32,34,35,37,49,50

363

Tomlinson, W. J., 111, 102(104), 116 Tompkins, F. C., 336, 348 Townes, C. H., 323, 352 Trapnell, B. M. W., 324, 351 Triebwasser. S., 86(60), 115 Trilling, L., 144, 151, 152, 153, 187, 197, 198, 204, 207, 209,289, 290 Tsu, K., 297, 314, 320,352 Turkevich, J., 342, 343, 352 Ultee, C. J., 320, 352 Van Steenis, J., 297, 320, 349 Vastola, F. J., 342. 352 Vdovenko, V. M., 336,352 Veksler, V. I., 145, 146, 204 Venkatavamu, K. U., 85(154), 117 Vetter, H., 123, 142 Vinti, J. P., 27, 36, 51 Voevodski, V. V., 311, 330,350, 352 Volpi, C . G., 319, 350 von Wepsenhoff, H., 319,352 Wachman, H. Y., 147, 149, 150, 153, 202, 204, 207, 289, 345, 352 Walker, P. L., 342, 352 Waller, J. G., 344, 350 Wallis, R. F., 237, 277, 290 Walls, F., 71, 72 Walmsley, S. H., 122, 141 Walsh, J., 301, 349 Walter, W. T., 102(103), 116 Wang, T. C., 77, 114 Warren, J. W.. 222,289 Watson, R.E., 105(116), 116 Watt, W. S., 346, 352 Webb, T. G., 19, 30, 32, 34, 51 Weibel, E. S., 60, 72 Weiss, A. W., 41, 47, 51 Weiss, G. H., 214, 234, 237, 256, 289 Weissler, G. L., 47, 49 Wentink, T., 323, 352 Westenberg, A. A., 318, 319, 320, 321, 349, 352 Wexler, S.,212, 290 Whetten, N. R., 160,2U4 White, J. W., 130, 141 Wieder, H., 83(27), 105, 115 Wightman, J. P., 342,352

364

AUTHOR INDEX

Williams, D. F., 138, 139, 142 Wills, L. A., 97, 116 Wilson, E. B., 236, 289 Wilson, G. 321, 349 Windsor, M. W., 120, 123, 142 Winkler, R., 105(117), 116 Winter, J., 109(120), 117 Winterstein, A., 123, 142 Wise, H., 298, 300, 301, 303, 304, 305, 306, 307, 308, 309, 310, 313, 314, 315, 316, 317, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 331, 339, 340, 343, 348,350,351, 352, 353 Wisendanger, H. U. D., 329, 336,352 Wittke, J. P., 111(128), 112, 117 Wolf, H. C., 120, 121, 124, 125, 127, 128, 129, 130, 132, 131, 133, 134, 137, 138, 139, 140,141, 142 Woll, L. W., 36, 49 Wood, B. J., 298, 311, 313, 314, 319, 320, 322, 323, 324, 325, 326, 327, 328, 329, 339, 340,352,353 Wuerker, R. F., 54, 58, 62, 67, 71, 72 Wray, K. L., 323, 352

Wrede, E., 312,314,353 Wyeth, C. W., 36,49 Wynne-Jones, W. F. F., 343, 350 Wysong, W. S., 299, 349 Yamamoto, S., 144, 159, 169, 175, 176, 196, 197, 204, 209, 289 Young, D. M., 208, 215, 220, 222, 224, 225, 227, 228, 229, 230, 231, 287, 290 Young, R. A., 323,353 Zabel, R. M., 163,204 Zahl, H. A., 164, 165, 166, 204 Zemansky, M., 73(1), 78, 84, 114 Zima, V. L., 124, 128, 129, 132, 142 Ziman, J. M., 235,236,256,257,274,290 Zinman, W. G., 343, 353 Zu Putlitz, G., 74(8), 85(153, 154, 155, 156), 92(8, 79, 80), 93(152, 159) 98 (91, 92, 93, 94), 100, 108, 114, 116, 117 Zwanzig, R. W., 152, 204, 213, 242, 289, 290 325, 353

SUBJECT INDEX A Absorption continuous, 1 cross section, 41, 42 window, 24 Accommodation in gas-surface interaction, 147ff, 281 Accommodation coefficient, see also Energy accommodation for energy, 147ff hard sphere model, 151 for momentum, 153ff temperature dependence, 148, 15 1 theories, 15 Iff Additivity assumption in gas-surface interaction, 214, 218 Additivity potential, 226, 227 Adiabatic interactions, 148 Adsorption, selective, 247ff Ammonia, 185 Anharmonic oscillators, 236 Antilevel crossings, 83 Approximation close coupling, 10, 12 Hartree-Fock, 10, 12, 39 one-electron, 46 Argon, 39, 46, 173, 177, 178 Atom detectors, 3 I4ff atomic resonance spectrophotometry. 333 chemical titration, 3 17ff chemiluminescence, 3 16 ESR spectroscopy, 3 19 ionization gage, 156 particle momen turn ( Knudsen gage principle), 316 tungsten oxide, 337 Wrede gage, 319 Atom production in electric discharge, 312 Atomic beams, see Molecular beams Atomic fine structure, 86 Atomic hyperfine structure, general, 88

extraction of nuclear moments, 90 hsf studies of alkali metals, 91 of group Ilb elements, 91 hsf measurements of chromium, 106 of stable alkaline earth metals, 96 Atomic lifetimes, 83 Atomic and molecular scattering from solid surfaces, 143ff Autoionization, 2, 17, 22, 39, 40

B Beryllium, 39 Binary interactions, 2 19ff Binding energies of sorbed atoms, 293, 294 Bragg diffraction, 256 Bromine atoms, recombination on glass, 323 Buffer gas, 110 effect on Doppler width, 111 shifts due to, I 1 1

C Carbon. 45, 46 Catalytic recombination of atoms energetics, 294 experimental measurements historical review, 3 1 1 kinetics, 297 mechanisms, 296, 297 non-steady-state, 332ff steady-state, 3 12ff Charge exchange, 68 Chemical reactions of atoms with solids. see Gas-solid chemical reactions Chemisorption, 292 activation energy, 293, 294 dissociative, 294 effect of surface coverage, 294 endothermic, 295 heat of, 293 Chlorine atoms reaction with metals, 340

365

366

SUBJECT INDEX

recombination on glass, 323, 340 on inorganic acids, 323, 540 Classical elastic scattering of atoms, 143ff, 260 Close-coupling approximation, 10, 12 Coherence narrowing, 84 Collision parameter, 68 reactions, 53 time, 145, 146 Collisions adiabatic, 241, 242, 274 elastic, 145, 243 electron-electron, 58 gas phase, 148, 149 hard spheres, 200 heating, 67 inelastic, 145ff ion-ion, 64, 65, 67 multiphonon, 212, 274 single-phonon, 210ff, 263ff spin exchange, 54 two-phonon, 2 1 1, 274 Configuration interaction, 20, 106 Continuity principle, 29 rule, 36 Continuum differential equations, 15 resonance in, 16 wave function, 9 Convective transport of gaseous atoms, 300, 307 Core polarization, 103 dependence on principal quantum number, 104 by d electrons, 105 by p electrons, 103 Correlation coefficient, 26 Coulomb phase shift, 9 Coupling intermediate, of I-s configuration, 97 L-s, 4 linear, 270ff quadratic, 270ff Cross section, general, 25, 28, 29 absorption, 42, 43 for ejection of photoelectrons, 2 for photodetachment, 46 Cyclotron frequency, 56

D Davydov splitting, 123, 140 de Broglie waves of atoms, 246 Debye-Waller factor, 256, 268, 269, 275 Decay time of excited states, 123, 135 Defect quantum, 29, 42, 43 many channel, 48 Desorption, 295 Deuterium, 179ff Diffraction Bragg, 256 molecular beam, 207, 244ff, 254 Diffusion of excitons, 137ff Diffusion limited kinetics in atom-surface interaction, 309 Diffusive transport of gaseous atoms, 300 Dipole accelerator operator, 6 length, 32, 33 velocity, 32, 33, 35 velocity operator, 6 Disorientation of electrons. 54

E Elastic scattering of atoms, 231, see also Scattering of atomic and molecular beams, Energy accommodation in gas-surface interactions, Collisions from perfect crystal, 239 by thermally excited lattice, 256 theory, 240 time delay in, 22 Electric quadrupole moment, 227 Electron disorientation of, 54 electron-electron collisions, 58 one-electron approximation, 5 spin resonance, 139 Elementary rate processes in atom recombination, 297 Eley-Rideal mechanism, 296, 338, 339 Emission continuous, 2 radiationless, 17 Energy accommodation in atomic beam scattering, 166 by catalyst, 325, 346 in gas surface interaction

SUBJECT INDEX accommodation coefficient, 147ff, 175 dependence on angular position, 174 temperature dependence, 148, IS1 theories, classical, 15lff hard sphere model, 15 1 one-dimensional lattice model, 152 three-dimensional lattice model, I52 Energy excitation of molecules, 120 Energy exchange multiphonon, 212 single phonon, 210ff two-phonon, 2 1 I Energy migration in organic molecules, 123ff Energy transfer in molecules, 1 19ff, 126fi Excitation energy of molecules, 120 Exci tons diffusion of, 132, 137 distributed states, 130 hopping model, 133 impurity trapped, 121 interaction, 136 self trapped, 121 triplet, 136

F Fluorescence, 120 delayed, 136 of organic crystals, 12lff Fluorine, 46 Frequency in crystals, 237 cyclotron, 56 of surface atoms, 275, 276

G Gas-metal interaction, 207 Gas-solid chemical reactions, 174, 298, 334ff competition with recombination, 299 kinetic studies, 334 mechanisms, 298ff Gas-solid scattering models electronic adiabatic, 207, 2 18 point-mass, 207, 2 12 Gas-surface collisions, high energy impact, 213 Gas-surface coupling, 263, 264 Gas-surface interactions, 23 Iff, see alsc:

367

Energy accommodation in gas-surface interaction, Gas-surface scattering accommodation, 147ff, 28 I binary, 232 coupling, 263ff, 281 potentials, 152, 206, 214, 224, 231, 255 Gas-surface scattering, general, 143ff, 20sff adiabatic interactions, 148 collision time, 145, 146 comparison between theory and experiment, 191ff elastic, 145, 243 hard cube model, 144, 187ff inelastic, 145 intermolecular potential, 152, 187, 206, 214, 224, 231, 255 lattice models, 197ff multiple, 146 targets, 160ff theory, 187ff, 238ff

H Half-width of resonance curves, 22 Halogen atoms detection by ESR spectroscopy, 320 recombination on glass, 323 on inorganic acids, 323 Hamiltonian for gas-surface interaction. 234 Hard-cube model, 144, 187 Hard-sphere collisions, classical mechanics of, 200 Harmonic oscillations, 236 Hanle effect, 83 in odd A isotopes, 86 Hartree-Fock approximation, 10. 12, 38. 39, 41 Heat of adsorption, 227 Helium atoms, 30, 32-36, 162, 254, 255 Helium ions, 54 Heterogeneous recombination of atoms, see Catalytic recombination of atoms Hopping model for exciton diffusion. 133 Hydride formation, 335 Hydrogen atoms detection by atomic resonance spectrophotometry, 333 by ESR spectroscopy. 319

368

SUBJECT INDEX

by titration, 3 17 by tungsten oxide reaction, 337 effect of water vapor on concentration, 320 reaction with carbon, 340 with metals, oxides and semiconductors, 33% with organic materials, 344 recombination energy accommodation by metals, 326 recombination on glass and quartz, 320 on graphite, 329 on metals, 324 on palladium-gold alloys, 327 on salts, 329 on semiconductors, 329 on Teflon and other coatings, 332 Hydrogen ions, photoionization of, 44, 45 Hydrogen molecules, 179ff I

Inelastic scattering of atoms, 163ff, 232, 261ff Interaction, see also Gas-surface interactions configuration, 20 between excitons, 136 between molecules, 120 potential, 223, 233 Intermediate coupling in s-1 configuration, 97 Ion beams scattering, 145 velocity distribution, 146 Ion cloud, 63, 64 Ion creation, 68 Ion traps circular, 71 cubic, 71 design, 58, 62 Penning, 71 quadrupole, 58, 61ff Ionization cross section, 69 Ions helium, 54 stationary, 64 stored, 53ff

J Jackson and Mott potential, 217

K Kinematics of atom-surface interactions, 300ff Kinetics of atom-surface interactions, see Catalytic recombination of ions, Gas-solid chemical reactions Krypton, 38

L Lamp scanning techniques, 100 in barium, 97 in mercury, 100 Langmuir-Hinshelwood mechanism, 296 Lattice periodicity, 208 Lennard-Jones potential, 23 Iff Level crossing spectroscopy, general, 81 experiments in copper, 94 in gold, 96 in lithium, 88 in noble metals, 93 in silver, 95 Lifetime of excited states, 123 of sorbed atoms, 345 Light sources for optical pumping, 76 Lithium level crossing experiments, 41 photoionization, 88 Low energy electron diffraction, 174 L-S coupling, 4

M Magnetic resonance, 54 Matrix elements acceleration, 8 length, 8 velocity, 8 Mean lifetime of excited states, 22 Methane, 185 Molecular beam, see also Scattering of atomic and molecular beam detectors, 156ff, 314ff diffraction, 214, 248, 254 hyperthermal, 144 intensity, 155 modulation, 144, 154, 157ff oven design, 154 thermal, 146 Momentum accommodation, 153, 165

SUBJECT INDEX distribution in scattered molecular beams, 239 Morse potential, 152, 231, 255 Multiplet strength, 5 Multiphonon exchange, 213 N

Negative ions, photoionization of, 45 Nitrogen atoms detection by ESR spectroscopy, 3 19 by titration, 319 photoionization, 47 reaction with graphite, 343 with nickel, 340 recombination on glass, 322 on metals, 329 on oxides, 331 Nitrogen molecules, 178, 183 Nonequilibrium energy distribution in surface interactions, 346 Normalization, I4 Nuclear orientation, 11 1

0 One-dimensional gas model, 266 One-phonon scattering, 2106, 263ff Operator dipole acceleration, 6 dipole velocity, 6 projection, 17, 19 Optical double resonance, general, 81 experimental setup, 75 intensity of signal, 78 excitation cross section, 78 transition probability, 79 magnetic field requirements, 78 radioactive isotopes, 100 resonance condition, 75 rf requirements, 78 shape of signal, 79 vacuum requirements, 78 zero magnetic field experiments, 91 Orbitals correlated, 14 polarized, 13, 33 Orientation of atomic ground states with circularly polarized light, 109 nuclear orientation, 108 with unpolarized light, 110

369

Organic crystals energy levels, 122 fluorescence, 122ff purification, 121 Oscillator strength, 3, 35, 36 effective, 36 generalized, 28 sums, 37 Oscillations in crystals anharmonic, 236 frequency spectrum, 237 harmonic, 236 thermal, 239 Oxygen atoms detection by ESR spectroscopy, 3 19 by titration, 3 18 photoionization, 47 reaction with carbon, 341 with metals, oxides, and semiconductors, 338 recombination on glass and quartz, 321 on halides, 328 on metals, 328 on oxides, 3 11, 328, 330 Oxygen ions, 46 Oxygen molecules, 182, 184

P Perturbation theory, 29, 210, 239, 252ff Penetration of particles into surfaces, 257 Penning trap, 56 Phase-sensitive detector, 158, 159 Phase shift, 9, 22 Photodetachment, 2, 46 Photodissociation, 55 Photoelectrons, cross section for ejection of, 2 Photoionization, 23, 47 hydrogen, 44 lithium, 41 potassium, 43 sodium, 41 Photons, polarized, 54 Point-mass system, Hamiltonian for, 234 Polarization potential, 41, 46 Polarized orbitals, 13, 33 Potential gas-surface, 23 1 interaction, 223, 225, 228, 229 intermolecular, 187

370

SUBJECT INDEX

cells (optical), 77 cross section of atoms, 263 elastic, 209, 260 of helium on lithium fluoride, 259 inelastic, 209, 261 one-phonon, 266 Scattering of atomic and molecular beams change in rotational state, 182 chemical reactions, 184 diffraction of helium and hydrogen on lithium fluoride, 162, 295 diffuse, 162 of molecules, 178ff momentum transfer in, 165 Q nondiffuse, 164, 166, 173, 175 of rare gas atoms, 167 Quadrupole specular scattering, 162, 171 interaction, 227 subspecular scattering, 167, 174 trap, 58, 61ff summary of experimental results, 186 Quantum defects, 29, 42, 43, 48 supraspecular scattering, 167, 171, 176 velocity distribution in scattered mechanics in gas-surface interactions, 205ff beams, 146 yields in fluorescence, 127 Selection rules for adsorption of atoms, 247 R for L-S coupling, 4 Radiationless emission, 17 Selective evaporation of adsorbed moleReactive collisions, 53, 291 cules, 255 Recombination Self-collision time, 66 of atoms on surfaces, see Catalytic Sensitized fluorescence, 120, 123 recombination of atoms Series, spectral di-electronic, 2 perturbations in, 29 Reduction of oxides by atomic hydrogen, Rydberg, 24-26, 30, 263 337 Sign of nuclear magnetic moment, deterRideal mechanism, see Eley-Rideal mechmination of, 98 anism Sodium Resonance absorption cross section, 41, 42 curve, 20 photoionization, 41 electron spin, 139, 319 Spectral lines magnetic, 54 profile index, 24 optical, 16, 17, 23, 30, 31, 34 shapes of, 24 line profile index for, 34 strength of, 2, 5 , 41 width of, 34 width of, 25, 34 Rigid rotator model for gas molecules, Spin exchange 277 cross section, 112 Rydberg series, 24-26, 30, 263 free electron g-factor, 112 of ions, 112 S of molecular gases, 113 Scattering Square well potential 232, 256 of atoms from surfaces, 231ff, 243ff Stability parameter for trapped ions, 61

Jackson and Mott, 217 Lennard-Jones, 231ff Morse, 152, 231, 232 for noble gas atoms, 221, 222 repulsive, 270 square well, 232, 256 van der Waals, 214, 216 Potential field of metals, pericdic amplitude of, 230 Principle of continuity, 29 Probes (atom detectors), 314, 315 effect of temperature, 315, 317 Profile index of spectral lines, 24 Projection operator, 17, 19

SUBJECT INDEX Sternheimer correction, 107 Stored ions, 53ff Strorngren method, 14 Sum rules, 26, 45 Surface lattice spacing of crystals, 235

T Thermal conductivity cell, 147 Time delay in elastic scattering, 22 Transitions, many quantum, 213 Trapping energy for atoms, 206 Traps see also Ion traps for excited electronic states, 123ff shallow, 130

V van der Waals forces, 206 interaction, 214 Variation principle, 10 Velocity dipole, 32, 33, 35 distribution in scattered beams, 166 formula, 44 non-Maxwellian, 150 response function, 277

W Wave function, 9, 239ff Wave-particle duality, 246

U Uncertainty principle, 23

37 1

X 'A Center, 131

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