A quasi-hard wall model for the scattering of atomic beams from solid surfaces

ChemicalPhysics65 (1982) i77-184 North-HollandPublishingCompany

A QI&WMi.ARD WALL _MODEL FQR THE SCATTE~G OF ~ATOMIC BEAMS J!TtCXMSOLID SURFACES Tsofar hE%NTV Department of Physics, Technion-Israel Instime

of Technology, Haifa, Israel

Received 1 June 1981

An. effective mirror model for the scatteringof atomic beamsfrom stiff, slightly corrngated solid surfaces is examined. A simple formula for the corrngation fnnction in terms of only two parameters, characterizing some general features of a gas-solid pafnvise interaction, is derived. Applications to diffrktion of He and Ne from LiF indicate that deviations from simple harmonic corrugation functions may be more importanr than previously assumed.

The interpretation of gas-surface phenomena, a task which is known to be a very difficdt one at a fundamental level, can be simplified substantially and yet remains rather satisfactory in many cases, if one appiies an impulsive approximation for the gas-sofid dynamics [l-r]. Within the framework of this approximation, the simplest models used for the gas-surface interaction are the so-called hard wall models. In such models the scattering of gas atoms from the surface is described in terms of reflection from a rippling, corrugated mirror [S, 61, which commonly appears as 2 purely phenomenological object. The interpretation of experimental data in terms of these models provides veti useful structural information about the ontermost surface layer 17-91, especially for reconstructed surfaces [lo] or for

adsorbed overlayers [ll], where the standard LEED technique is known to have difficulties. By using such models it is implicitly assumed that the repulsive core of the gas-surface interaction potential is so steep that the details of the gas-surface impact event are irrelevant in determining the scattering amplitudes; the only relevant quantity being the surface morpholo,v (or surface corrugation). The price paid for the great simplicity thus achieved is that the corrugation function remains a purely phenomenological quantity, which can only be determined by experiment. This simple approach may have difficulties for surfaces which show considerable deviations from a simple harmonic corrugation, where the number of adjustable parameters required is larger than one. It is totally inadequate for inelastid scattering, where thi: dependence of the corrugation function on the dynamical configurations of the surface atoms should be specified [2]. In this paper we show how the situation can be improved without paying too much in terms of complexity. It is shown that by using a reasonable definition for the corrugation function and by employing two models for p&r poientiaIs of quite different analytical structures, one can derive in the steep potential limit a simple “universal” formula-for the corrugation function (i.e. independent of the specific Eorm of the model -potentiaI used), which is expressed in terms of only two parameters characterizing some generai features of the gas-surface interaction. 0301-0104/82/0000-0000/$02.75

@ 1982 North-Hoiland

178

T. Mflnic / Scarrering of atomic beams from solid surfaces

any adjustable By applying interpretations components in

The resulting formula allows one to interpret experimental

data of various systems without using parameter. this method to diffraction of He and Ne beams from Ii?? [S] it is found that previous [7j of these data probably underestimated the importance of higher Fourier the corrugation functions appropriate for these systems.

2. Derivation

of the corrigation

function

Following a discussion appearing in ref. [l] it seems reasonable c(R) as a solution z = c(R) of an equation of the form V,,(W,

to define a corrugation

function

z) = constant,

(1)

where V,, is some function which resembles the essential behaviour of the bare gas-surface interaction potential and the constant on the right is something like the kinetic energy of the incident gas atom (see also refs. [2,12! 131). In eq. (1) z is the distance of the gas atom to the nominal surface plane and R = (x, y) is the projection of its position vector on this plane. By assuming small corrugation, i.e. writing 5(R) = 4-0+&Y%

(2)

with ~&(R)]G<~, it is possible to get an approximate so we simply integrate the equation

obtained from (l), with respect to R, after replacing first order one has

formula for c(R)

c(R)

without solving eq. (1). To do

with its first approximation-
Thus to

(4) where co is an average distance between a gas atom and a surface ion at the moment of impact. In what follows we shah work ottt two different model potentials for V,,+ The first potential is a Lennard-Jones Iike pairwise sum of the form V&L

y, z) = Vor’o”1(r, Y.Z),

(5)

where I(& y,z)=

; C:x-nIa)*+(Y-n?_~)Z+z=]-r z nl==-ca ,,+=-a3

(6)

v>O is an integer, r. is a typical length scale and V. a typical ener,v

scale. Eqs. (5) and (6) represent an interaction potential between a gas atom and the outermost layer. We assume that the surface ions are packed in a square array for simplicity. A reciprocal lattice expansion of I(R, z) = 1(x, y, z) takes the form I(&

z) = (2rr/a)2[(-?)V-*/2Y-1(V-1)!ZY-1]

where 6 = (27r/a)(m1, &), order ZJ 1143.

ml, m2 = 0, *l,

2 ? rn,=-co ,$L.

surface

eiC’RG”-lKV-t(Gz),

+2, . . . . G = ]Gl and KV is a modified Bessel function of

(7)

T.

Maniv

/

of aton&

Scattering

beams fromdid

sqfaes

179

U&g the identity [I41 aCK,_1(Gz)/z’-*];a~

= (G/z’)KAGz),

(8)

we may write the ratio @V,JtR)/@VV,n/&)

in the form

WR, z)/M?= i XC exp (iG - R)G(Gz)‘-‘K,-1CG.z) N(K, z)& xG exp (id; . R)(Gz)“K,(Gz)

(9)

’

The limit of an infinitely hard wall is obtained for u-r, 00. Our aim here is to study the case of a steep but finite potential barrier. Thus we consider a large value of v (the precise meaning of this statement will be discussed below) and use the asymptotic expansion of Ky(Gz) for large order [14], We may further expand this asymptotic form in powers of (Gz/v)*, provided that (Gz/v)‘< 1, so that to second order we get

Note that the second order term in the exponent above is sma!i compared to the first order term if (Gz/v)‘
[2(v- l)/e]“-‘[~r/2(~ - l)]“’ (2v/e)“(7r/2v)“’

& exp (iG - R)G exp [-(Gz)‘/~(v - l)] L- exp (iG * R) exp [-(Gz)‘/~Y]

l/v)“-ai In z exp [iG * R - (Gz)*/~v] I G

. I

(11)

TJsing this expression in eq. (4) and then integrating both sides with respect to 62 we get (after approximating (1 -l/v)” ==e-‘) c(R) = (&/2v) In 1C,exp [iG - R - (G&)“/&]I

tconstant.

w

We define the effective range, c, of the gas-surface forces as: (13)

Following closely the procedure used above for calculating <(I?) we get for c:

mc 60)

=

CGe iG~R(GSo)v-lK~-l(GSo)

=ar@f, co)/aio = lo CG eiG‘R(G~o)'Kv(G~o)

(14)

so that 5(R) may be written in the form: 5(R)= c In g exp (iG . R -$G*~oc) fconstant. I I

(15)

The form of c(R) appearing in eq. (15) is free of any specific feat&e of the model potential used. To see whether this reflects some sort of universality, we shall work out in what follows a different type

180

T. Manti

of model potential,

/ Scar&ing

of atomicbums frvm solidswfaces

e.g. a pairwise sum of the form [2]

V&cr, y, z) = vo nl=_m if ~~~_mex~{-a[!x-n,R)2~(~-~~u)2+*2~,

(161 :

where again V, is a typical energy scale and CY-I’~has the units of leng&. The equation V&R, 2) = E:; where Ei Is the gas atom incident energy, OR be. sol%-ed directly for this model potential and the result is (17) where n = (nt, n2). XII the limit of a steep potentiai

aa’>

1 and fVo/Ei)>

1 so that

ln(Vo/Ei)~~InI~exp[-a(~-na)2]~I.

(18)

Thus to first order in the ratio of the right hand side to the left hand side of (18) we get C(R) =

&~-i$cx

In (Vc/EJ-“’

In 1; exp [-a (R - i~a)~]1,

(19)

where &, = [CT-’ In ( VO/Ei)j”“. Note that 50 above is a very weak function of Ei. Using (19) in (13) we find for c the value c a 1/2i&.

(20)

An application

of Poisson sum rule [2] to eq. (19) with eq. (20) finally gives

Ic

b(W) = c In 1 exp (iG - R -$G”&c)

I

tconstant,

which is exactly the formula (eq. (15)) obtained

3. Application

(21) from the first model potential.

to diffraciion by LiF

We consider diffraction of rare gas atoms from the (100) plane of LiF. We assume that the projectiles encounter only F- ions. This can be checked to be a reasonable assumption for the range of energies under consideration?. To make this assumption even more realistic; we shall restrict our consideration only to scattering along the (110) azimuth [8]. ThI e corrugation function is-given by eq. (15) where the values of c and &I may be inferred from available data on the interaction between rare gas atoms and closed shell ions &5]. As we shall see later in the cases uoder consideration here only the first few Iattice vectors in the

reciprocal lattice expansion of eq. (1.5) contribute significantly; to second order we thus get. 3(x, y)==c In Il+2e-‘[cos

(27rx/a)+cos (27iy/a)]+(4 e-” cos (2m/a) cos (2sry/a)l+constant,

(22)

? The distances of c!osest approach for the systems Sk-ii+ and Ne-Li’ at the relevantimpact energies can be &t+ated from ref. [IS! to be r, = 1.64 %, and q,= 1.85 _& respectively. These values are smaller than the respective values, focnd in the discussConof the present paper. for the distance of closest approach between a project% and the ou?ermost layer af the Fions.

T. Maniv /Scattering of atomic beams frdm did su&ces

181

(22’) Expanding

the logarithm to third order in eqE we find that

E(x, y) s2c

e?[cos

(Zm/a)+cos

+ gc L-“[cos3 (2m/a)

(2?ryja)]-2c

+ qos3 (2iy/a)]+

e?[cos’

(2m/a)

+cos’ i%&y/ajl

constant,

(23)

which may be rewritten in the form

(2rx/a)+cos

&(x, y) -$cos

+h2[cos (6~x/a)+cos

(2zy/aj]-hJcos

(47x/a) fcos (4ry/a)]

(6ry/a)]+constant,

(24)

where h = 2c e-:(1 + e-a”),

(25)

Note that the nice separable form of eq. (24) is-due to cancellations of alI cross terms in the expansion of the logarithm. This is a very fortunate situation since the scattering cross section in the eikona! approximation [6] for a detector located within the X-Z plane (x is the (110) direction) is thus propoitionai to jS(k,, k,)j2; a/2 S(k,, k,j =

I

exp [ikz&(x)+ikxx] dx,

126)

<(x) = h cos (2?rx/a) - hi cos (4srxja) + h2 cos (6zx/a),

(27)

-a/2

where

and

k, z ki(sin 8; -sin &),

kz z ki(COS Si t

CGS 6f),

(28)

0i and 19rare the angles between the incident and the scattered beams and the normal to the nominal surface plane. Using eqs. (27) in eq. (26) we find for the intensity 4 of the I-diffraction peak Pl=(COS

@[/COS

@i)lS*l’;

I = 0, *l,

*2, . ..)

(29)

where (30) n =$?z f3(--l)‘nz’+(-1)“1], 19, L

m+3m’+l

=&en,

sin-l (sin 8i - 2nl/akj),

7), -2; =l; and J,,,(x)

m=O, mf0,

i

is a Bessel function of integer order.

klekj(COS~8i+COS

8,),

(alj

182

i: Miniu / Scanning of atomic beams from solid surfaces

4. Discussion The formula for c(R) derived in this paper shows that surface roughness and surface stiffness are closely related quantities. In other words, it suggests that the corrugation amplitude Al [i.e. the peak to peak variation of c(R)] scales with the effective range-c of the gas-surface forces:. In particular, eq. (15) yields Al + 0 in the limit where c + 0. This limit should be considered carefully; strictly spe&ing c may not go to zero without leading to violation of the classical criterion, eq. (I), from which eq. (15) is derived. This happens for values of c which are too small compared to the de Broglie wavelength of the incident beam. Eq.(15) also suggests that an increase in surface stiffness (i.e. a decrease in c) tends to increase the deviations from a simple harmonic corrugation function. This is due to the fact that the number of reciprocal lattice vec’.ors contributing sizable amounts to the sum in eq. (15) increases with decreasing values of c. This may be physically understood by considering the extreme case of a hard sphere model, where the corrugation surface is defined by the motion of 2 rigid sphere rolling on a layer of rigid spheres, representing the outermost surface layer. As can be easily seen the corrugation surface thus generated is smooth around the tops of the hills but is kinky down in the valleys. The presence of increasing numbers of higher harmonics in the expansion of c(R) for increasing degree of stiffness reflects this tendency of stiff surfaces to include such sharp morphological structure. To study the extent of this deviation in actual systems, we shall consider here diffraction of He and Ne from LiF (100) along the azimuth (110). The energies of the incident beams are 55.8 meV for He and 60.8 meV for Ne as in Boato et al. [8] (i.e. ki= 10.38 A-’ and ki=24.2 _&-l_ respectively). To minimize the effect of the attractive part of the gas-surface interaction and the effect of mu!tiple scattering 2s well we consider normal incidence only (i.e. Qi= 0). We use available data of the potential energy for the systems He-F- and Ne-F- 1151 to estimate the degree of stiffness of the potentials for the above mentioned energies. We find that for a model potential similar to eqs. (5) and (6) our best fits to those data give Y = 6 for He-F- and v = 7 for Ne-F-. The corresponding values of the impact diameter rb are found to be 2.3 A for He-F- and 2.6 pi for Ne-F-. Using the above values of Y we then calculate Pr by means of eqs. (29), (30). Our best fits to the experimental data are plotted in figs. 1 and 2 aiong with the results due to the best harmonic corrugation function. The vaiues of 60 found by this procedure are CO= 2.25 A for He/Lii and lo = 2.45 A for Ne/LiF. The corresponding peak to peak variation of the corrugation function is twice the values (h -hi + hr = ) 0.131 A and 0.12 A respectively. Thus our best fits for 10 are 2.381 A and 2.57 8, respectively_ The agreement with the values of r. obtained from the potential data of He-Fand Ne-F- is good for the former and very good for the iatter. The better agreement obtained here with the Ne diffraction data is not surprising: (1) The use of a classical criterion such as eq. (1) is more credible for the Ne beam than for the He beam used; (2) The assumption of neglecting the interaction with the Li’ ions is more satisfactory for the Ne atoms than for the smaller He atoms; (3) The effect of multiple scattering, neglected in the present work, should be smaller for Ne than for He. Note that our value for h essentially agrees with that obtained by Garcia in his extensive best fit calculations applied to the He scattering data [7]. His scheme, which takes into account the effect of multiple scattering, yields, however, zero for cC2.0)and ,J(s.o, (our hi and h:! respectively) and 2 nonzero value for the first order cross coefficient 5 (i.i), which vanishes identically in our scheme due to exact cancellations in the expansion of the logarithm in eq.,(22). “; In ref. [l] the condition c < hi is invoked to assure the validity of an effective mirror description. It can be shown, however, see ref. [16], that a weaker condition on the potential would be sufficient for this description to be valid.

T. Maniu / Scattcrimg of atomic beams from solid surfaces

-Fig. 1. He diffraction from LiF(100) along the azimuth (110): Solid lie-experiment (from Bozto et al. IS]); full circles -our model with i0 = 2.25 A, c = 0.188 A (thus h = 0.1489 8, h, = 0.023 & h,=0.0055 A); empty circles and crosses -simple harmonic corrugation functions with h = 0.154 A and h = 0.135 A, respectively. Incident beam conditions: Bi= 0 (normal incidence) and ki = 10.38 A-‘.

183

from LiF(100) along (110): Solid lineexperiment [S]; full circles -our model with &,= 2.45 A , c = 0.175 8, (thus h = 0.136 A, hl = 0.021.&, hZ = 0.0049 A); crosses -a stiple harmonic corrngatian function with h = 0.15 A. Incident conditions: Bi= 0 and ki = 24.2 A-‘. Fig. 2. Ne diction

It should be noted here that above all quantitative arguments it seems rather clear that the qualitative structure of a diEraction pattern obtained from any purely harmonic corrugation function over-emphasizes the rainbow structure at the spectrum edges [7]. The values of the dominant corrugation coefficient h, obtained here for both He/LiF and Ne/LiF, are very close to those obtained by the best fit calculations presented in ref. [Xl. The appearance of nonzero, though smal!, second and third order coefiicients, hl and h2, in the expression for c(R) derived here, transfers intensity from the rainbow peaks to the specular one and improves the agreement with the experiment. Our conclusion is therefore that the deviation from a simple harmonic corrugation function is considerable. it is, howcver, still small enough so that the reciprocal lattice expansion converges quite rapidly. The applicability of this method to scattering from non-ionic crystals such as metals is questionable since an important part of the gas+olid potential in such systems cannot be described by a pairwise sum. However, since this part is usually very smooth a modified version of the present scheme, which treats separately the ion cores and the conduction electrons, may be applicable. The generalizadon of the method represented in this paper to inelastic scattering will be published elsewhere.

Acknowledgement The author is indebted to Professor B. Gerber and to Dr. C. Herscovici for helpful discussions.

T. Mania / Scarrering of atomic beams f~m

184

solid surfaces

References [l] [2] [3] [4!

T. Maniv and Morrei H. Cohen, Phys. Rev. B 19 (1979) 4883. J.E. Adams and W.H. Miller, Surface Sci. 85 (1979) 77. R.B. Gerber, A.T. Yiion and J.N. MurrelI. Chem. Phys. 31 (1978) 1. J.H. W-, J. Chem. Phys. 61 (1974) 2900. M.V. Berry, J. Phys. A: Math. Gen. 8 (1975) 566. U. Garibaldi, A.C. Levi, R. Spadacini aod GE. Tommei, Surface Sci. 48 (1975) 649. N. Garcia. J. Chem. Phys. 67 (1977) 897. G. Boato, P. Cantini and I.,. LMattera, Surface Sci. 5 (1976) 141; G. Boato, P. Cantinf and R. Tatarek, Phys. Rev. Lcners 40 (1978) 587. M.J. Cardilio and G.E. Becker, Phys. Rev. Letters 40 (1978) 1148. ,M.J. Cardillo and G.E. Becker, Phys. Rev. B21 (1980) 1497. K.H. Rieder and T. Engel, Phls. Rev. Letters 43 (1979) 373; 45 (1980) 824. J.R. Klein and M.W. Cole, Surface Sci. 81 (1979) L319. N;. Esbjerg and J-K_ Norskov, Phqs. Rev. Lertrrs 45 (1980) 807. M. Abramowitz and LA. Stegcn. eds., Handbook of mathematic functions (Dover, New York, Y.S. Kim ar,d R.G. Gordon, i. Chem. Phys. 6G (1974) 4323. T. Maniv, unpublished work.

1972) pp. X6-378.