Sudden decoupling approximations for atom-surface scattering

Chemical Physics 31(1978) l-9 0 North-Holland Publishing Company

SUDDEN DECOUPLING APPROXIMATIONS FOR ATOM-SURFACE SCATTERING R.B. GERBER, A.T. YINNON and J.N. MURRELL* Physical Chemistry Department. The Hebrew Universityof Jerusalem, Jerusalem, Israel Received 5 January 1978

Two versions of the sudden approximation are introduced to decouple and solve the equations that describe atomsurface scattering with many open diffraction channels. Both approximations require a high incident beam wave number comparedwith the magnitude of the reciprocal space vector of the lattice. In this framework, simple explicit expressions are obtained for the observabIe diffraction intensities, making calculations feasible eve! for systems with hundreds of open diffraction channels. Further considerable simplifications ensue when the approximations arc specialized to the case of a Lennard-Jones-Devonshire potential, or to that of a weakly corrugated surface. The approximations were applied to the systems He/LiF(OOl); Ne/LiF(OOl) and Ne/W(llO) and the results are compared with other calculations or with experiment. The sudden approximationis found to be of good accuracyin these cases.

1. Introduction The intensive experimental activity in recent years in molecular beam studies of atom--solid surface scattering has stimulated considerable lnterest in the theory of such processes fl] _One approach to the calculation of the diffraction intensities is the numerical solution of coupled-channel equations where the basis consists of diffraction channel wavefunctions [l-3]. Whilst this method is exact if a sufficient number of channels is included, the computational effort prohibits, at present, the use of more than about 50 channels. Only one or two converged close-coupling calculations have been reported to date, on the very favourable case of He scattering from LiF(OO1) at very low energies [2]. Another approach is to use the semiclassical methods that have been recently applied to atom-molecule scattering [4]. ‘The extension to atom-surface scattering was first made by Doll [5] and by Masel et al. [6,7]. A test-calculation on He/Lii(OOl) suggests that the method is quite accurate [8] but although the computational effort is less than that of the close-coupling method it is nevertheless considerable. There is therefore need for a computationally simple me-hod that l

Permanent address: School of Molecular Sciences, University of Sussex, Brighton, Sussex, England.

can be widely applied to the interpretation of acrual experiments, even at the cost of reduced accuracy. In this spirit Goodman et al. [9,10] proposed an approximation related to the distorted-wave method which is valid for weak coupling (low corrugation surfaces) and Garibaldi et af. [ll] have described an eikonal approximation for the special case of an infmitely hard surface whose form is defmed by a shape function. With such a model they also produced most of the qualitative features of the scattering for the He/LiF system. Another study of the hard surface model was made by Garcia 1121. In this paper we propose a new approximation that requires little computational effort and which can be applied also to cases where the interaction is strong enough to couple a large number of diffraction channels. The approximation employs realistic atom-surface potentials and is expected to give good accuracy for the conditions realized in most present day experiments. The method uses a “sudden” approximation to decouple the multichannel equations, together with a WKB phaseshift for each decoupledstate. It is described in section 2 for two different but closely relatedversions. As we shall show, the expressions obtained can be even further simplified in two special cases: (i) a LennardJones-Devonshire interaction potential and (ii) a surface having low corrugation (e.g. most metals).

2

R. B. Gerber et aLlSudden decoupling approximations for atom-surface

In section 3 the two “exact” sudden versions and the simplified ones are applied to the systems He/Lii(OOl), Ne/LiF(OOl) and Ne/W(llO) and the results compared to other calculations or to experiment. In this paper we confine our attention to a phononfree model. Phonons clearly play a significant role in such processes in that they broaden the diffraction peaks and attenuate their intensities. However, before attempting to include them in the method (and the sudden equations appear to have a suitable structure for this), it is important to assess first the accuracy of the sudden liiit in a simple phononless framework.

Qmn,m ’n’ = A-l

scattering

Jexp(-iGm,

* R) aR)

UC

X exp(iG,~,~ - R)dR ,

(6)

whereA is the unit cell area, and the two-dimensional integral is over the unit cell. 2.1. l7re matrix diagonalization

sudden (MDS)

approximation The coupled equations (4) can be written in matrix

2. Sudden approximations

form Consider the scattering of an atom of mass 1-1with incident energy E from a static surface. The system is described by the Schriidinger equation: [-@2/202

+ V(x, _Y.2) - E]\k(x. Y, z) = 0 ,

(1)

where z is the coordinate of the atom perpendicular to the surface and x, y are coordinates parallel to the translational vectors of the surface. V is the atomsurface potential, which we assume to have the form Vx* Y. z) = V&I + VI(t) QG Y) .

(2)

No restrictions are yet made to the functions V,,(z),

V,(z) and Q(x, y). In the close-coupling approach the wavefunction is expanded in the form [l] :

qx,y,4 = m.n=O,t-l,z2... c @n&) Xexp[i(K+C,,).R],

[I d2/dz2 - g2 - I Uo(z) - Q (ii(z)]@ = 0 ,

where the components of the vector CD(z)are the wavefunctions amn(z), I is the identity matrix and g2 is a diagonal matrix, the nonzero elements of which are given by g2,, =k2 -(K+Gmn)2

where K is the component parallel to the x. y plane of the incident wave vector k, R is the vector (x, y, 6) andGmn are the two-dimensional reciprocal space vectors (i.e. Gz) = 27rm/ax, GF’ = ZrJa,, where ax, ay are the lattice constants). inserting(3) into (l), using the orthogonality properties of the wavefunctions exp[iG,. *RI and expression (2) for V we find

with

(4)

(8)

(9)

*

The above approximation and the subsequent procedure that follows are similar to those used in introducing the sudden approximation for rotationally inelastic, molecular gas phase collisions [l] . Let B be the unitary matrix that diagonalizes the (z-independent) matrix Q: QD = BQB-1,

where PI @mn,m’n’ =Qk,mn [d21~*+k2-(K+Gmn)*-Llg(Z)-CI1(Z)Qmn,mnl~mn

= ul(z) m’n’ C Qmn,m’n’ @mrpfCZ) 3

_

We assume that Ii&, the incident momentum in the direction perpendicular to the surface is large compared with IG,, [ for all the channels that significantly contribute to scattering. This condition is satisfied for most present-day experimental systems, in particular for those using high-energy supersonic beams. For such impulsive collisions we can introduce the sudden approximation 1131: 2 9 =&

(3)

(7)

(10)

6m,m’

‘n,na .

(11)

In view of approximation (9) we have: Bg2B-l

=:g&, I .

(12)

R.B. Gerber et al_ISudden decoupling approximations

foratom-surfacescattering

3

We now apply the transformation matrix B to eq. (7)_ Using (12), (10) and (9) we fmd:

(Y), we make the appro.ximation

[Id2/dz2 fg$

where i* is defmed as the diagonalpart of the lefthand side of (19). This leads to a result similar to (18), the sole difference being that& appears instead of &, in the integrals (17).

I - U,(z)1 - UI(z)QDJb=O,

(13)

where 6=68.

(14)

The system of equations (13) is uncoupled since the matrix QD is diagonal. For the component krnn of the vector 6 the equation is: [d*/dz* +&

- U,(z) - ~,(4Q;,,,,l~,,

i 0WI

In the decoupled system (13) there are no transitions between the new decoupled channels. The scattering matrix SD that arises from the system of equations (13) is therefore a diagonal matrix which represents elasticscattetingin each of the decoupled channels. Its elements SE,,, mn therefore satisfy: %Lmn

= wG%,,)6, ’

ml h,,,. ,

(16)

where qnn is the elastic scattering phase shift determined from the single channel equation (15). It can conveniently be calculated in the WKB approximation

qmn = Iim

j

( [g2, - UO(z) -

2-m C =rFln

1

QEn UI(z)] 1’2

-gOOIdz + gOOzmn3

(17)

where .zmn is the classical turning point associated with the rnn decoupled channel. We now have to express the physical scattering matdx S, that corresponds to the coupled system (7), in terms of SD. As shown in the familiar case of rotationally inelastic scattering in the gas phase [12], the required connection is through the inverse unitary transformation (IO): s = 6-I sDB J

(18)

and the intensity of the (mn) diffraction peak is given by&o-mn = I$,m,ool*The numerical effort involved in the MDS approximation involves therefore only the diagonalization of the matrix 0, from which QD and B are obtained, and a calculation of the WKB integrals (17) for each of the decoupIed channels. A slightly modified MDS is obtained if, instead of

Bg2 B- 1-c. -g -2 ,

(19)

2.2. The MDS for the Lennard-Jones-Devonshire potentialand analyticdiagonajizattion of the marrix Q The Lennard-Jones-Devonshire periodic function Q(X,y) = COS(2?7X/Q)

+

potential uses the

cos(27ryyla) -

(20)

The following discussion is not limited to any specific forms of V*(z) and V,(z) that appear in (2) and moreover can be extended in some aspects to the case where the lattice spacings are not equal in the x and y directions. The elements of the matrix Q are, from (6) Qm’n’,mn = %m’m 6n’,,J, + 6,t,*1

6n.<‘).

Czl)

This is a Hiickel-type matrix in the sense that if the channels are represented by points in a two dimensional Cartesian frame, then there are non-zero matrix elements only between nearest neighbours and all such matrix elements have the same value. Analytical expressions can be found for the elements of QD and B for many of these nets (we note in passing that they are altemant systems) and, in particular for any rectangular net. This case we now consider in detail. The well-known Hiickel solutions of ap-atom chain with interaction matrix element fl are [14]

where tP are the vectors of the standard basis, xr the eigenvectors of the Hiickel matrix. The eigenvaIues are given by E,=2pcos[~/(ptl)],

r=l,2

)... p.

(23)

For a rectangular arrayp X Q it is easy to show that the eigenvectors can be identified by two integers which determine the periodicity along the length and breadth of the array. The eigenvectors vrs are related to the

RB. Gerberet al./Suddendecouplingapproximations for atom-surfacescattering

4

standard basis vectors .$” by: Xrs= 2(p + l)-qq

x

ax, y). Thus, using bracket notation, we have in the coordinate representation:

+ 1)-l/2

5 5 sin[r/.m/(p + l)] sin[swl(q + l)]

p=l

v=l

(24) gMv ,

and the eigenvahres are: E,, =2fl{cos[nrj(p

= Q(x, y) 6(x’ - x) qy

r=l,... p;s=l,... q.

W)

For an array of diffraction channels whose corners are (e. f), (e * R, 13, (e, f + 4). (e f p. f + 4) the elements of 6 and ClB can be found from (24) and (25) by a simple change of label (Itu) + (or- e + 1, 0 - f+ 1) and takingp=i Bii = 2(p + 1)-l/2(/~ i-1)-‘/2

The eigenfunctions of ax, y) are xR(R’)=6(R’ -R) x)601’- y), whereR = (x, y). The matrix 6 that diagonahzes Q is the matrix of the coefficients of transformation from the basis in (3): .&,(R) =Ae112 exp(iG,, lR) to the eigenstates 6(R’ -R). Thus

(R’IBlmn) = BR,,,,, = A-II2 exp(iG,,

l)l(q + l)]

111vii ,

(27)

where i m (r, s) andj = ((u,/3)_The ordering of the eigenvectors in these arrays is arbitrary. As is typical in any Hiickel problem there are usually degeneracies in the elements of QD and hence the elements of B are not unique. This causes no difficulty m the sudden approximation (9) because the diagonal elements of Bgz B-l are assumed to be independent of B and oirly QD enters the uncoupled equations (13). However, if we use approximation (19) the diagonal elements of the matrix $j*depend on B. fn this version of the MDS, and for a degenerate QD, the different possible choices of B may give rise to different results. We expect, however, such differences to be reIatively small.

(2%

=S’(R) 6(R’-R) .

00)

S(R) is determined from the asymptotic behaviour of

the solutions of eq. (15) that assumes the following form in the R-representation:

td2b2+&$ -~o(z)--l(z)e(x.~)l~x,,(z)

=O, (31)

where &,(z) depends parametrically on x. y. Eq. (3 1) shows that the sudden approximation treats the coordinates x, y in an adiabatic approximation so that S(R) can be written: S(R) = exp [2iq (R)] .

(32)

The elastic-scattering phase shift q(R) can be evaluated in the WKB approximation:

I

2.3. The coordinate-representation sudden (CRS)

--goow+goozR

approximation

This approximation was developed by Secrest [15] for rotationally inelastic atom-molecule collisions. To derive it in the present context, consider the extreme sudden Iimit in which all (infmitely many) channels are included, and the entire basis in (3) is retained in the calculation. If Q is not truncated, it is clear that it becomes diagonal in the coordinate representation, since it is made of matrix elements of the function

* R) .

As is evident from section 2.1 within the framework of the sudden approximation Q and S are diagonal in the same basis. Thus S is diagonalized also by transforming to the coordinate representation: S” =(R’ISlR)

C$ = ccos [rrrl(P + 01 f cosrs~/(s +

(28)

= 6(x’-

+ 1) + cos[sn/(q + l)]);

Xsin [r(or- e + l)/(p + l)] sir@@ -f+

- y) .

9

(33)

where zR is the classical turning point pertaining to (3 1). Finally, substituting (29), (30) into (IS), and using the coordinate representation we fmd for the scattering S-matrix:

sm’n’,mn =

JJh’n’lB-llR’XR’lSIRHRIBlmnMR’dR

R’ R

(34) = i fexp [i(G,, -Gmsnr)R] exp[Ziq(R)]dR . R

RB. Gerberet al/Sudden decoupling approximations for atom-surface seaher@

We see that within the CRS the S-matrix element for scattering into the (mul) diffraction channel is given by the (mn) Fourier component of an elastic S-matrix component evaluated as an adiabatic function ofx, Y. Expression (34) bears a formal similarity to the eikonal expression Smn co, obtained by Garibaldi et al. [11,12] for the case of scattering by an infinitely hard, corrugated surface. In their case the shape function of the surface plays the role analogous to q(R) in (34).

Table 1 Parameters relevant to the calculations a)_ All quantities are given in atomic unitp

System

energy of incoIning particle k_

2.4. I7te CRS in the limit of low comqation

l&ice

We consider now the special case of scattering from a surface of low corrugation, for which le(x,y)Ur(z)l Q 1U,,(z)]. Expanding the phase shift I&, y) to fust order in Q(x. Y)Ur(z)/&,(z), we fmd:

constant number of open channels potential P pammete& D

f

tl(x,Y)=+r+jxx,Y)nl

He/LiF(OOl)

Ne/W(llO)

Ne/LiF(OOl)

2.2842 X 1O-3

2.3245

3.3245

5.792

x

1O-3

13.066

13.066

5.37

5.18

5.37

59

365

385

0.582 0.1 0.00028

0.582 0.01 0.00063

0.529 0.06

*

x

1W3

0.00049

(35) a) AUcalculations were for a normal incident beam (k, = k,, = 0).

where

v”=

5

lim [ j

z-+-

Qo(z)l“2dr - goozo]9 (36)

Go -

20

.a0being the turning point for the integtand in (34), and (37)

We assume further that Q(x, Y) is separable into x and Y contributions (as in the case of the Lennard-JonesDevonshire potential). We can write: Mx. Y) = G,(x) + &(Y) *

(38)

The S-element for diffraction into the (mn) channel becomes:

ax

Smn,OO=eziqo

2s

exp[iG,(X)x] exp[2i&(x)$]dx

0 “Y

Xj

exe[iG$‘)yl exePik&Wslldy~ (39)

0

an expression that requires little computational effort. 3. NumericaI resuks The methods described in the previous section have

been applied to the systems He/LiF(OOl), NelLiF(001) and W(110). In ah the cases we used a Lennard-JonesDevonshire potential where Q(x, y) is given by (20), and V,(z), V,(z) are assumed to have the forms: V,(z) = D exp [o(zu - z)] {exp [a(20 - 2) - 2-O}, V,(z) = -2fl D exp [2Ly(zi - z)] _ The potential parameters, the lattice constants, the energy E and the z component of the wave vector, fez, for which tire calculations have been made are listed in table 1. Listed also is the number of open channels in each case. For He/LiF(OOl), the potential parameters were taken from ref. [12]. For the other cases, the potential parameters are not known, and physically reasonabIe values were adopted. The general lack of exact calculations on atomsurface scattering makes it difficult to assess the accuracy of approximations in this field. The only exception, in this respect, is Wolken’s close-coupling calcuIation on He/LiF(001)2. This calculation was, however, carried out at a very low energy (to reduce the number of channels that must be included), and is therefore in a regime for which the sudden approximation ceases to be valid. The accuracy of the sudden approximation will therefore be tested in the following two ways, neither of which is entirely satisfactory: Firstly by comparison with experimental results. These,

6

R.B. Gerberet al./Suddendecoupling approximations for atom-surfacescattering

Table 2 Diffraction peak intensities for He/LiF(OOl) Peak

MDS

MDS with modified wavevectors

Classical trajectories

Experiment a)

HCS b,

00 10 20 30 40 11

0.906 0.004 0.018 0.014 0.010 0.024

0.003 0.009 0.016 0.011 0.006 0.026

0.02 0.015 0.021 0.017 0 0.016

0.003 0.004 0.011 0.003 0 0.001

0.013 0.006 0.025 0.008 0 0.004

;: 12 13 14 23 24

0.031 0.035 0.010 0.00 1 0.004 0.024 0.017

0.037 0.025 0.013 0.004 0.005 0.025 0.013

0.038 0.016 0.023 0.018 0 0.021 0

0.021 0.001 - c) -

0.068 0.004 - c)

(mn)

a) Ref. (18).

@Refs. [11.12].

-

‘) Not calculatedor not measured.

however, are affected by experimental error and by phonon participation in the process and, most irnpor-

sample sets which are muchsmaller than 6000 the results have a large statistical error. Wenote here that the

tant, the true atom-surface

MDS. for imtance. is more than hvo orders of magnitude faster than the classical trajectory method for the systems shrdied. (We used a relatively efficient in-

potential is nGt known.

Secondly by comparison with results of classical trajectories and of the hard-corrugated surface (HCS) model [ll]. However, it must be remembered that the HCS shape function was chosen to reproduce experiment as far as possible, whereas our potentials were taken from sources unrelated to the scattering experiment. 3.1. Classicaltraiectov cakulations Classical trajectory calculations were carried out on the systems listed in table 1, for comparison with the results of the sudden approximations. In each application, 6044 trajectories were computed, with initial conditions sampled by Conroy’s prescription [16] _ The nUmeriCd

solution of the equations of motion was

based on an integrator due to Gear 1171. The assignment of trajectories to diffraction peaks was by “box quantization”: A trajectory with fmal momentumcomponentsPX,P,,Isuchthat:2(m-$ir/a9PX<2(m+ &)a; 2(n - &lr/a G P < 2(n f &/a, was counted as a transition into the Qm, n) diffraction channel. The classical trajectory calculations are extremely time consuming (=$ hour on a CDC6460 for each system) because of the large number of trajectories required. For

tegrator for the classical trajectories, thus this conclusion should be generally valid). This nnderlines the potential usefulness of the sudden approximation for practical computations. 3.1.1. He/LiF(OOI) The calculations on this system were made for the same energy and k,, kv as in the experiment of Boato et al. [ 181. As the data in table 1 shows, the sudden condition k,/2n/a > 1 is satisfied in thii case (although to a smaller extent than for the other systems). Table 2 lists the results of the MDS and other methods for He/LiF(OOl). The second column in the table gives the diffraction peak intensities as calculated in the MDS approximation, according to eqs. (16)-(18). The third column lists the results of the version of MDS, that uses the modified wave vectors&,,,, of (19). The other cob~rnns of the table give the results of the classical trajectory calculations, the experimental diffraction peak intensities reported by Boato et al. [ 181, and the results obtained by the HCS model [ 111. We note that the two versions of the MDS give very similar results and this is a consistency check on the va-

R.B. Gerberet aLlSuddendecouplingapproximations for atom-surfacescattering

lidity of the sudden assumption. Perhaps the most reliable comparison is between the MDS results and those of the classical trajectories, since both calculations used the same potential function. The agreement is good, as individual peak intensities in the two methods do not differ for most channels by more than a factor of two. The two peaks for which agreement is poorest are (00) and (40). For the (00) peak the MDS agrees much better with the experimental value than the classical calculation, while for the weak (40) the opposite is true. Taking into account the considerable experimental error, the neglect of phonon contributions in the calculations, and the uncertainty about the reliability of the potential fimction used, the qualitative agreement betweenthe MDS and the experimental results is as good as could be expected. The agreement with experiment could have been improved much further by fitting the parameters (as done in ref. [l l] for the HCS model). However, so far as phonons are not included such further improvement may be artificial. Finally, we note that the MDS conserves the symmetry inherent in the problem, i.e. it yields the same result for intensities of symmetry-equivalent channels. For the normal incidence systems studied Smn, oo = Sm-n.00 =LVz,oo- Therefore in table 2 we have listed only a single channel from each symmetry-equivalent set. 3.1.2. Ne/LiF(OOI) At the energy of the calculation, this system has 385 energetically open channels. The problem can still be tackled within the MDS if use is made of the analytic diagonalization given in section 2.2 for cases where Q(x. y) is of the form (20). However, it is more natural to apply the CRS version of the sudden for a problem with such a large number of channels_ Table 3 compares the results of the CRS, with those obtained from classical trajectories, with experiment [18], and with the HCS results of Garibaldi et al. [ll]. Since

Boato et al. [18] caution that tire experimental results may

involve very large errors in the present system, it is even more true here than in case (a), that the most meaningful comparison is with the classical trajectories method. The agreement found between the classical trajectories and the CRS is fairly good, except for diffraction peaks of high order ((m, rz) = (6,6), (6, S), (6,4)). For these channels the basic sudden assump-

I

Table 3 Diffraction peaks intensities for Ne/LiF(OOl) Peak &on)

CRS

Classical trajectories

Experimenta)

KCSb)

a0 10 20 30 40 50 60 11

O.&J5 0.002

0.003 0.003

0.00035 0.00005

0.003

0.005

o.ooo49

0.006 0.002 0.001 0.001 0.005

0.005 0.007 0.005 0.004 0.004

0.00017 0.00015 0.0006 0.0003 0.00003

0.007 0.001 0.007 0.004 0.002 0.013 0.011 0

:; 44 55 66 21 31 32 41 42 43 51

0.002 0.007 0.016 0.014 0.003 0.003 0.004 0.004 0.010 0.004 0.004 0.009

0.004 0.005 0.011 0.020 0 0.005

0.00042 0.00008 0.0001 0.0006 0.0003 - c)

0.006 0.001 0.002 0.025 0.009 _ c)

0.005 a_005 0.007 0.008 0.009 0.006

-

52 53 54

0.004 0.003 0.015

0.006 0.009 0.015

-

61 62 63 64

0.005

0.003 0.002 0.007

0.004

0.004 0.003 0.001

65

0.006

0

_ -

_ -

a) Ref. [18]. b, Ref. [ll,lZ]. c) Not calculated or not measured

tion k, 3 IG,,, 1 does indeed break down. However, for more than half the peaks the CRS and the classical results do not differ by more than 50% and for only 25% of the peaks do the results of the two methods differ by more than a factor of two. 3.1.3. Ne/W(I12) For this system there are 365 open channels at the energy used, but because of the low corrugation of the surface the coupling is weak and relatively few channels are actually populated in the scattering process. In the MDS calculation it is sufficient to include 49 channels (convergence of the results was tested with respect to increasing the number of channels in the computation). We also applied the CRS, in order to compare it directly with the MDS.

8

R.B. Gerber et al. jsudden

decoupling approximation for atom-surface

Table 4

Diffraction peaksintensitiesfor the Ne/W(llO) system Peak (mn)

MDS

00

0.400 0.110 0.006 0.030 0.002

10 20 11 21

CRS

0.400 0.110 0.006 0.030 0.002

Weak coupling CRS

Classical

0.442 0.110 0.006 0.025 0.001

0.138 0.117 0 0.099 0

trajectories

Finally, as can be verified from-the parameters of table 1, the weak coupling condition of section 2.4 is satisfied. We could therefore apply the low-corrugation limit of the CRS, which is computationally much faster than the CRS. The results of all sudden methods and those of the classical calculations are listed in table 4. We note that (i) the MDS and CRS yield identical results. This Implies that the inclusion of Inftitely many channels which is implicitly done in the CRS does not lead to significant loss of flux from open channels. The fact that methods that differ so much from a computational point of view give the same results also suggests that the sudden model is indeed justified for this system. (ii) The low-corrugation results are in excellent agreement wifh the CRS calculations. This result may be typical for many metals, that form low-corrugation surfaces. (iii) Agreement with the classical trajectories results Is fair. Because of the small number of diffraction peaks that are significantly populated, the results of the classical method are relatively sensitive in the present case to the method adopted for “quantizing” the diffraction peaks. It is very possible therefore that the classical method is !ess accurate in the present case than the sudden schemes_

scatteting

fast, simple method it may be especially suitable for the interpretjltion of experiments. The connection between the atom-surface potential and the diffraction intensities is transparent and simple in the sudden approximations. It should be practically easy therefore to use these methods for the determination of atom-surface interactions by fitting of experimental data. However, the applicability of the sudden methods is severely hindered by the fact that the present versions do assume 2 phononless surface. At high energies this seems particularly unjustified. We believe, however, on the basis of studies in progress now, that phonons can be incorporated into the sudden approach with relative simplicity, and that computationally simple expressions can be derived for both the elastic and inelastic scattering intensities in that framework. We hope to report on this in a future publication.

Acknowledgement We should like to thank Dr. U. Minglegrin for help ful discussions on several aspects of this work. We also thank Miss S. Feliks for help with the programming. J.N.M. is a-visiting research professor at the Hebrew University under the sponsorship of the Royal Society and the Israel Academy of Sciences. A.T.Y. would like to thank the Vulkani Research Institute for a graduate student fellowship. This research was supported by a grant from the Commission on Basic Research of the Israel Academy of Sciences.

Note added in pmof Preliminary results from exact coupled-state&lcuthat we did recently, show that the sudden is accurate within = O-l%(!) in the favourable case of Ne/W. lations

4_ Discussion It has been shown that the sudden method, in several versions, is an efficient tool for the caIznIation of diffraction intensities in atom-surface scattering at relatively high energies. The applicability range of the method, for typical experimental conditions, seems to be wide, encompassing perhaps nearly alI the systems hitherto studied by experiment. The accuracy of the sudden schemes is reasonable, to the limited extent that this could be studied. As a computationally

References [l] G. Wolken, & Dynamics of Molecular Collisions, Pert A ed. W.H. Miller (Plenum Press, New York, 1976). [2] G. Wolken Jr., J. Chem. whys. 58 (1973) 3047. [3] H. Chow, Surface Sci. 62 (1977) 487; 66 (1977) 221.

[4] W.H.hIiller, Advan. Chem. Phys. 25 151I.D. Doll, Chem. Phys. 3 (1974) 257.

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