Scattering of helium atoms by surfaces at low energies and temperatures

Surface Science 171 (1986) 515-526 North-Holland, Amsterdam

515

S C A T r E R I N G O F H E L I U M A T O M S BY S U R F A C E S A T L O W E N E R G I E S A N D T E M P E R A T U R E S

Frank O. G O O D M A N

Department of Applied Mathematics and Department of Physics, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI

Nicolas G A R C I A

Departamento de Fisica Fundamental, Universidad Autonoma de Madrid, Canto Blanco, 28049-Madrid, Spain

and

Jesus R E Y E S and Jesus G A R C I A - F E R N A N D E Z

Departamento de Fisica, Unwersidad Autonoma de Puebla, Apartado Postal J-48, Puebla. Puebla, Mexico

Received 30 July 1985; accepted for publication 14 January 1986

It is well known that the existing calculations for helium scattering by solid surfaces, made on the first-order distorted-wave Born approximation, within the one-phonon approximation, dis- agree qualitatively with experiment at low energies and temperatures. Generally, the calculations give sticking probabilities which are too low and which have the wrong trend with incident energy: further, the calculations are incorrect in the limit of zero incident energy. We investigate whether those calculations may be modified in order to correct them and to help them agree better with experiment. The modifications we make are, essentially, the introduction of reflection coefficients and the consideration of surface excitations (modes) different from those considered before.

1. Introduction

We address ourselves to the p rob l em of the scat ter ing of hel ium a toms (He) by surfaces, in the regime of low incident energy E and low surface t empera - ture T, and we pay par t i cu la r a t ten t ion to the s t icking p robab i l i t y S under these condi t ions , a l though we cons ider also the energy a c c o m m o d a t i o n coeffi- cient a. When E is discussed, the energy associa ted with the " p e r p e n d i c u l a r m o m e n t u m " ( componen t of m o m e n t u m pe rpend icu la r to the surface, in the z-di rec t ion) of the inc ident a tom is general ly impl ied.

We have a l ready cons idered [1] the case of scat ter ing of He by l iquid He, and that case is not covered in the present paper . Here, we restr ict ourselves to

0039-6028 /86 /$03 .50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing, Division~

5 1 6 F.O. Goodman et aL / Scattering of He bY surfaces at low E and T

cases in which the single-atom effective potential V(z) for scattering yields complete reflection of atoms by the surface in the absence of inelastic processes, that is, cases in which V/E is large in the bulk. For He-l iquid He, V/E is negative in the bulk (where V--- -0 .62 meV), and atoms which reach the surface are lost to inelastic processes at the surface or subsequently in the bulk [11.

Essentially, all existing experimental data relevant to the present paper are for the scattering of He by solid surfaces, the indication [2-41 being that both S(E) and a ( E ) are decreasing functions for small E, although there are no data at the very low energies at which we have data [5,6] for liquid He surfaces. For incident atoms more massive than He (for example, Ar, Kr and Xe), experiment indicates that both S and a tend to unity as E --* 0 (at least in the presently-accessible range of E). We believe that similar behaviour would be found for He at sufficiently low values of E, in the presently inaccessible range; the data of Sinvani et al. are exceptional [4], the inequality ~ < S < 1 being reported for E = 1 or 2 meV.

The calculations of inelastic atom-surface scattering, made for He-surface scattering, with the one-phonon approximation in the first-order distorted-wave Born approximation (FODWBA), presented and reviewed in refs. [7-15], do not agree with experiment at low energies (E < = 4 meV, say), giving values for both S and a which are too small, and which have the wrong trends with E. We ask ourselves whether this model may be modified in order to reasonably interpret the experimental data referred to above. That the previ- ous calculations on this model are not valid in the limit of small E has been pointed out, for example, by Boheim et al. [16] The WKB approximation (WKBA) presented in section 3 of ref. [15] is not valid in the limit of small E because a condition for its validity breaks down; this condition is [17]

mhV'(z) C - << 1, (1.1)

[ 2 M ( E - V(z)) I 3/2

where M is the atomic mass, V(z) is the atom-surface interaction potential, and 2~rh is the Planck constant. It is instructive to consider a general inverse-power attractive potential defined by

V ( z ) = { - D ( f l / z ) ' " z>~fl, (1.2a) - D , z ~< fi, (1 .2b)

where D is a well-depth, 13 is a length, and s (> 0) is the inverse-power, when C defined by (1.1) becomes

,$ '~X ( ~ - 2 ) / 2

C = 2 ( c x ' + 8) ~/2 ' (1.3)

F O. Goodman et al. / Scattering of He by surfaces at low E and T 517

where (dimensionless) c, 8, x are defined by

c = 2 M E f l 2 / h 2, (1.4)

8 = 2 (1 .5 )

x = z / f t . (1.6)

If s > 2, then, for sufficiently large values of z (that is of x), there exists a value of E (that is of ¢) below which C is large.

This breakdown of the W K B A may be worded in terms of the reflection coefficient, R, of the attractive potential, for a toms incident from z = oo [18]; the value of R is the fraction of a toms which is reflected, from the attractive potential, back to z = oo. (To make contact with ref. [18], the R used in the present paper is the same as the JR[ 2 therein; in fact, a toms are incident f rom z = - oo in that reference, but this gives the same value for [ R [ 2.) In the WKBA, R = 0, and so the W K B A is expected to break down when R is not negligible.

For the inverse-cube attractive potential, that is s = 3, it is only at very small incident energies that R is not negligible, and a physical " r eason" for this may be given as follows. It is known that R ~ 1 as E ~ 0 (see ref. [18] for a clear demonstra t ion of this point) and that R -~ 0 as E ~ oo (a classical limit); if we ask over what range of E does R change from being non-negligi- ble to being negligible, then we should be interested in a "na tura l scale" for E. For the potential (1.2), this natural scale is not E itself [8], but rather E ~s-2~/z'. For s = 3, this means that we should expect R to be non-negligible for reasonably small values of ( E l / D ) 1/6, but not necessarily for reasonably small values of E / D ; that is, R is a " reasonable" function of ( E l / D ) 1/6, but not of E / D . This situation is illustrated in fig. 1, where R is shown versus

10

8-

6"

4"

2

1OR

100 [E/DlllBsgn (E/D)

; i & & Fig. 1. R versus I E/DI 1/6 for the UVA potential (1.7) with a = 1.44 A, fl = 1.22 A, D = 16.8 meV and M = 1.008 u.

518 F O. Goodman et al. / Scattering of He by surfaces at low E and 7"

[ E / D [ w6 for the same special case as in considered in ref. [18], that is for the UVA potential, which is defined by [19]

)3 v(~-) = f -D(~+fl)3/(~+= =>~fl" (1.7a)

I D : ~ [3. (1.7b)

with a = 1 . 4 4 , ~ , f l = 1 . 2 2 A,, D = 16.8 meV, M = 1.008 u. In this particular case, for E > 0, we see that R is non-negligible ( R > --- 10%, say) for values of ( E / D ) I/6 less than about 10%, that is for values of E less than about 0.02 ~eV.

Thus we are led to conclude that the WKBA is expected to be valid under all presently accessible experimental conditions, because E is always much larger than 0.02 ~eV, which corresponds to a temperature of about 0.2 mK. However, to modify the calculations in question [7 15] in order that they give the correct behaviour in the limit of small E, we assume that those results may be correctly applied to the fraction, 1 - R, of atoms which is transmitted by the attractive part of the potential to the repulsive part. For example, if P is the sticking probability of an atom, calculated as before, then we assume that the "correct" sticking probability, S, is given by

S = ( 1 - R ) P . (1.8)

This assumption seems reasonable: if R is small, then the previous calcula- tions are valid (because the WKBA is valid), and S = P; if R is not small, then S is substantially less than P, as it should be; if R ~ 1, then S ~ 0, which is correct. We emphasize that the assumption (1.8) does not affect the results of the previous calculations [7-15] at experimentally accessible values of E, because R is then negligible; it is hoped that (1.8) will enable WKBA calculations made under conditions where R is not negligible to give results which are more physically realistic.

A candidate for increasing the sticking probability at low energies was once the electron-hole pair mechanism, although this has been shown [20,21] to be negligible. The calculations of Boheim et al. [16] show an increase of the sticking probability with decreasing E, due to higher-order corrections to the inelastic scattering amplitude, but this modification is still inadequate to give agreement with experiment. Boheim et al. [22] considered diffraction of a He atom into its bound states at the surface, with subsequent inelastic scattering, and obtained reasonable agreement with experiment; in fact, they claimed [22] that the disagreement of theory with experiment was removed by their work. Using a formalism developed by Doyen and Grimley, [23], Doyen [24,25] claims to have shown that the sticking probability is unity when T = 0 and E = 0; because our result contradicts that claim, we must conclude that it is wrong.

Another shortcoming of the previous calculations is that they apply to scattering of atoms by only a special case of excitations at surfaces, that is to

F.O. Goodman et al. / Scattering of He by surfaces at low E and T 519

ordinary phonons with a three-dimensional (3D) distribution property (al- though 2D and 1D properties are discussed briefly in ref. [151). By the name "ordinary phonons" is meant excitations, the frequency-wavevector relation ~(q) of which is given by

o~ = c q , (1.9)

where c is a sound speed; by the term "3D distribution property" is meant the fact that the wavevectors q are distributed in 3D space, that is the distribution Q ( q ) of q is proportional to q2:

Q 0c q2. (1.10)

[It follows from (1.9) and (1.10) that the distribution I2(o~) of ~o is propor- tional to ~o2:

I2 ( ~o ) = Q ( q ) ~-fl~q oc ~2, (1.11)

which is the result for phonons in a 3D bulk-continuum model.] In other cases of excitations, results different from either or both of (1.9)

and (1.10) are relevant: important cases may be ripplons at a liquid surface and certain other types of excitations at a solid surface. In the present paper, we suggest how the previous one-phonon calculations should be modified if one or both of (1.9) and (1.10) do not hold. Particular attention is paid to the problem of the sticking probability S ( E ) at low E, although the accommod- ation coefficient a ( E ) is also discussed. We set T equal to zero throughout the discussion; although this is not necessary in quantum-mechanical calculations such as those presented here, it leads to considerable simplification of the working, and we believe that it will not lead to spurious results in view of the evidence [26,27] that accommodation of He at surfaces is essentially indepen- dent of T.

2. Analysis

2.1. Transit ion probabil i t ies

We restrict attention for the moment to the transition probability, Pi m, of an incident atom's being scattered into the bound state m associated with the atom-surface interaction potential. The previous result (for the model incor- porating a 3D continuum modal frequency distribution) is eq. (2.10) of ref. [15]:

Pi ' ' = 121r~th 2 ( 2 M ) - ' /2 (bO) -3 B2i 6ol(ho~), (2.1)

where ~0 is a phonon frequency, b is the Boltzmann constant, @ is a

520 1"~ O. Goodman et aL / Scattering of He b.v surfaces at low E and T

characteristic temperature defined from

b O = h w . . . . (2.2)

is the mass-ratio,

I~ = m / m ~ , (2.3)

where M S is the surface atomic mass, B,, is a matrix element defined by eq. (2.11) of ref. [15], and the function I is defined by

1, 0 < w~< w . . . . (2.4a)

l ( h w ) = 0 otherwise, (2.4b)

where Wm~ is the maximum modal frequency; the phonon frequency ~0 is given from

hw = E - E,,, (2.5)

where E,,, is the bound-state energy. (It must be remembered that we are considering only motion perpendicular to the surface.) The previous result for the sticking probability P is

e = E e, m. t2.6t m

As explained in section 3.6 on p. 295 of ref. [15], the wl(hw) factor in (2.1) is, for a more general class of models, to be replaced according to

W3a~12( w ) /3w, (2.7) where ~2(w) is normalized to unity, that is,

L~°l l (w) dw = 1. (2.8)

For a 3D bulk continuum model, 12(w) is given by 3w21(hw )/w-~ .... and so the relation (2.7) is verified. However, in more general models the dispersion relation w(q) is not necessarily (1.9), but rather

w cr. qP, (2.9)

where the exponent p depends on the model (for example, p = 1 for ordinary phonon modes, and p = ~ for ripplon modes on liquids, gravit~¢ being neglected). Further, the distribution Q(q) is not necessarily (1.10), but rather

Qcc qn 1, (2.10)

where n is the "dimensionali ty" of Q(q) (for example, n = 3 for bulk continuum modes, and n = 2 for surface modes, such as Rayleigh modes on solids or rippions on liquids).

In these more general cases, ~2(w) must be recalculated from the equality in (1.11), and it follows from (2.8)-(2.10) that

= 1( (2.11)

F O. Goodman et al. / Scattering of He by surfaces at low E and T 521

where ~, is defined by

p = n / p . (2.12)

It follows from (2.7) and (2.11) that the ~ factor in (2.1) is to be replaced according to

p tO ~ - 2 ~o--+ ~ ,_~, (2.13)

COma x

and hence, using (2.2) and (2.5), that Pi" becomes

P("=4~vgh ( 2 M ) - l / 2 ( b O ) - " B2~ ( E - E , , ) "-2 I ( E - E , , ) . (2.14)

In the limit of small incident energy, E, we get

P,"(E--*O)=4~rvgh ( 2 M ) - ' / 2 ( b O ) - " B 2 IEml "-z I ( I E , , I ) , (2.15)

where B,,, given by eq. (3.18) of ref. [15], is the appropriate limit of B,,i. A similar treatment may be applied to the transition probability Pifd Ef of

an incident atom's being scattered into a continuum state f, the energy of which lies in the interval ( E f , Ef + dEf); the appropriate analogue of (2.1) is eq. (2.5) of ref. [15]:

p t = 12g (bO) -3 Cf 2 ~ol(h~o), (2.16)

where Cri is defined by eq. (2.6) of ref. [15], and ~o is given in this case from

hto = E - Et (2.17)

(remember that T = 0, and hence E > Er). The analogue of (2.14) is

Pi t= 4vla(bO)-" C 2 ( E - El) "-2 I( E - Et). (2.18)

2.2. Accommodation coefficients

If "y is an accommodation coefficient, calculated as before, then we assume that the corresponding "correct" accommodation coefficient, a, is related to by the analogue of (1.8):

a = (1 - R)y . (2.19)

We write

"/= "/b + 3'~, (2.20)

where 3% is the contribution to y due to transitions of atoms into bound states, and 3'~ is the contribution due to transitions into continuum states. It follows from eqs. (2.12) and (2.14) of ref. [15] that, in our case,

y b ( E ) = P ( E ) , (2.21)

y~(E) 4ug(bO) " E - ' [ °° 2 ( E - E l ) "-I = - C n I ( E - El) dEf. (2.22) Jo

522 F.O. Goodman et aL / Scattering of l ie hv surfaces at low l=" and 1

in the limit of small E, we may obtain

y c ( l i _ _ , O ) = 4 p . ( b O ) - , , C 2 E,, L

w h e r e (:', g i v e n b y eq. ( 2 . 1 7 ) o f re f . [15 ] , is t he a p p r o p r i a t e l i m i t o f ( ' r , .

(2.23)

3. Applications

3.1. General

For interaction potential models with realistic attractive parts, the highest bound states m have energies E,, close to zero. Therefore, according to (2.6), (2.14), (2.15) and (2.20)-(2.23), the behaviours of the sticking probabil i ty P and the accommodat ion coefficient y. at low energies E, depend crucially on the parameter p, defined by (2.12). Now, we know that, in the strict limit of E ~ 0, the limits of both the "correct" sticking probabili ty S and the "correc t" accommodat ion coefficient a are zero [because of (1.8) and (2.19) and the strict result R ~ 1]; however, at the lowest energies currently accessible experimentally, the behaviours of both S and a are similar to those of P and V (because R is small at these energies), and hence also depend crucially on u.

An at tempt to illustrate the situation is made in fig. 2. In the n-p diagram therein, the two lines n = p ( u = 1) and n = 2p ( u = 2) separate regions of different behaviours of P and y,, ( remember that Yb = P). Values of P and y~. are shown in the different regions and on the two lines; for the purposes of the diagram, we are taking the limits as both E --* 0 and F,, ~ 0, as we are mainly

P=O v = 2 ~'~=o /

n t / 0 / / / p =oo

', ~ / 4 , o . .- - 3 . . . . . . . . . . . . . . . . ~ ~ ~ . 0

: : / o o 2 . . . . . . . . . . . . . . . ~ . , ' - - - - -~ .. . . . . : - ~

i i .... p = o o , • c - Y c = C 0

0 1.0 1.5 2 .0 P

Fig. 2. The n • p diagram. Points associated with di f ferent types o f excitations (modes) are shown as heavy dots. The lines ~, = ] and u = 2 separate regions of di f ferent behaviours o f P and 7,. See text for further explanation.

F. O. Goodman et al. / Scattering of He by surfaces at low E and T 523

concerned with the contributions to P from the uppermost bound states. Of course, if one (both) of P = oe and Tc = oe is (are) shown, this means that the calculations are invalid, but that one (both) of P and 3'c is (are) expected to be large. Also shown in the diagram are various points associated with different types of excitations which may exist at surfaces: the three points lying on the line labelled ~r, for n = 1, 2 and 3, respectively, are relevant to ordinary nD phonons (1.9); the point labelled p applies to 2D (surface) ripplons on liquids; the point labelled B applies to "phonons" associated with bending modes of a plate, which we suggest, in section 3.3, may be relevant to solid surfaces.

3.2. Liquid surfaces

As we have mentioned in section 1, there are no relevant experimental data on He-l iquid surface scattering, because the existing He-l iquid data [5,6] tell us little or nothing about the scattering at the liquid surface (because atoms which reach the surface may be transmitted immediately into the bulk liquid) [1].

From the location of point p in fig. 2, it is evident that, for atom scattering by ripplons at a liquid surface at low energies, we expect a large (calculated- as-before) sticking probability P, which in turn from (1.8), leads us to expect a large value of S (except at the very low energies at which R ---, 1, implying that S ---, 0), that is,

S = 1 - R, (3.1)

the sticking probability depending on only the reflection coefficient. The relation (3.1) holds also for He-l iquid He scattering, but for perhaps a different physical reason: for the scattering considered here, atoms which reach the surface are adsorbed by a tom-r ipplon processes occurring at the surface; for He-l iquid He scattering, while a tom-r ipplon processes may occur, atoms which reach the surface will be eventually adsorbed in any case because they may so easily enter the bulk liquid. Thus, in the relevant energy ranges, experimental data on the scattering of He atoms by liquid surfaces other than He may, in fact, be similar to data on the scattering of He atoms by liquid He surfaces, although it may be difficult to obtain such surfaces under the required conditions (for example, at sufficiently low temperatures).

3.3. Solid surfaces

We ask whether the failure of the theory to predict larger values of P, and hence of S from (1.8) under conditions where R is small, may be due to neglect of certain types of mode of vibration which may occur at solid surfaces. If such a type of mode is represented by a point below the line v = 2 in fig. 2, then it would lead to large values of P and S similar to those attributed, in section 3.2, to ripplon modes at liquid surfaces.

524 F.O. Goodman et al. / Scattering of He by surfaces at low E and T

" I m ___

Fig. 3. What a typical experimental solid surface may resemble on a scale of tens or hundreds of ~ingstroms. We suggest that the shaded parts may have associated with them localized modes of the plate-bending type.

A typical "clean" experimental solid surface may look, on a scale of tens or hundreds of &ngstroms, something like that shown schematically in fig. 3. Certain parts of the surface, for example the shaded parts in fig. 3, may have associated with them modes of types not considered previously. For example, it seems to us possible that the shaded parts in fig. 3 may exhibit some behaviour similar to that of plates, and may accordingly be associated with modes of vibrational character similar to that of plate-bending modes [28]. These modes are represented by the point fl in fig. 2, which implies large values of P for atoms scattered by phonons associated with them. If correct, this suggestion would help to make the quantum theory of adsorption agree better with experiment at low energies. For example, results of calculations which may be relevant to the system H e - W are presented in fig. 4: S and P are shown versus E for the 3-9 H e - W potential [15]

V(z) = ~D[(~/z) ~ - 3(~/z)3], (3.2a)

with D = 6.8 meV and ~" = 2.22 ,~t. P is calculated from (2.6) and (2.14), with

15" ,~=1

"'"'""',,,, .'~100 P lO- _ /s(o)=o

5 10 15 E (mev) 2'0 0 Fig. 4. Results of calculations for He-W: S ( - - -) and P( ) versus E for the potential (3.2) with ~" = 2.22 ,~ and D = 6.8 meV. See text for further explanation.

F O. Goodman et al. / Scattering of He by surfaces at low E and T 525

u = 1; S is calculated using (1.8), with R calculated from the attractive part (z > ~') of the potential (3.2a), with

V ( z ) = - D , z < ~'. (3.2b)

The qualitative behaviour of S ( E ) is now in better agreement with experi- ment [15]: S ( E ) is now a decreasing function of E, and its magnitude is considerably larger than before. Of course, as is discussed in section 3.1, these statements must be modified at very small values of E, where S ~ 0, although there are no experimental data at these energies. One further relevant point is that, if the ideas implied by fig. 3 are correct, then the modal properties of such a solid surface are expected to be a "mixture" of properties associated with different types of modes, only a certain fraction of which would be of the plate-bending type. Hence, the expected values of S are in fact somewhat lower than those shown in fig. 4.

As was correctly pointed out by a referee of this paper, "none of the recent high-quality experiments on surface phonon spectroscopy has revealed [plate- bending] modes"; see refs. [29-33], for example. However, there are accepted surface localized modes which have also not been revealed by such experi- ments [34], and there have been observed modal frequencies which are considerably different from those expected previously from theory [29,31], some of those anomalies having since received a theoretical explanation [34].

4. Conclusions

We have investigated whether the older FODWBA one-phonon calculations for He-surface scattering may be modified, using ideas associated with reflection coefficients and with surface excitations (modes) different from ordinary 2D or 3D phonons, in order to correct them and to help them agree better with existing and potential experimental data on scattering at low energies and temperatures. We conclude that this is, in fact, the case, after having considered scattering by ripplons at liquid surfaces, and by modes of the plate-bending type at solid surfaces.

Acknowledgments

We acknowledge the benefit of useful discussions with Dr. I. Romero. The work was supported by CONACyT and DGICSA-SEP (Mexico), and by the Natural Sciences and Engineering Research Council of Canada, under Grant No. A6282.

526 F.O. Goodman et aL / Scattering of He I~v surfaces at h)w E and T

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