Pairwise additive semi ab initio potential for the elastic scattering of He atoms from the LiF(001) crystal surface

Pairwise additive semi ab initio potential for the elastic scattering of Heatoms from the LiF(001) crystal surfaceVittorio Celli, Dieter Eichenauer, Achim Kaufhold, and J. Peter Toennies Citation: J. Chem. Phys. 83, 2504 (1985); doi: 10.1063/1.449297 View online: http://dx.doi.org/10.1063/1.449297 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v83/i5 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Pairwise additive semi ab initio potential for the elastic scattering of He atoms from the LiF(001) crystal surface

Vittorio Celli ,a) Dieter Eichenauer, Achim Kaufhold, and J. Peter Toennies Sondeiforschungsbereich 126, Max-Planck-Institut/iir Stromungs/orschung, Bottingerstra{3e 4-8, D-3400 Gottingen, Federal Republic o/Germany

(Received 27 February 1985; accepted 21 May 1985)

The interaction potential for the elastic diffractive scattering of low-energy He atoms from the highly corrugated LiF(OOI) crystal surface is derived from semi ab initio pair potentials in the framework of the recently developed Tang-Toennies potential model [J. Chem. Phys. 80, 3726 (1984)]. In addition to the sum of all He atom-crystal ion two-body potentials the induced dipole potential caused by the electric field of the ion lattice is taken into account, leaving only one free parameter, the C6 dispersion constant ofthe He-F- interaction. By simple adjustment of this parameter, it is possible to fit all ofthe experimental bound states of the atom-surface potential well, to within experimental error. Diffraction probabilities calculated by the close coupling method with this potential are shown to be in good agreement with the available experimental results. Two different empirical potential models based on the Morse potential are also investigated, but do not provide as good a description of the bound states and diffraction intensities.

I. INTRODUCTION

The system He-LiF(OO 1) has played a seminal role in the current understanding ofthe interaction of inert low energy atoms with crystal surfaces. Diffraction,1 bound state resonances (selective adsorption),2 and the quantum rainbow effece were all revealed by experiments on this system. The hard corrugated surface (HCS) model of atom-surface scattering was introduced to explain the diffraction data in this system, with considerable success.4 Detailed explanations and predictions of the line shapes of elastic resonances were obtained on the basis of a simple model: a hard corrugated surface with an attractive well (HCSW).5-8 With the advent of time-of-flight measurements, well-resolved surface phonon spectra9- 11 and inelastic resonant processes, 12 corresponding to one-phonon assisted desorption and onephonon assisted adsorption, were first obtained for HeLiF(OOI), and could also be understood, even quantitatively, using a simple model of a vibrating hard corrugated surface with a static attractive potential well. 13

The success of the HCS model and its extensions, partly accidental as it may be, has apparently discouraged the widespread use of more realistic potentials that have been proposed to describe the interaction of He with the LiF(OO 1) surface,I4-22 and of noble gas atoms with ionic crystal surfaces in general. 23-28 The advantage of the HCSW model is that the calculations are greatly simplified. Moreover this model leads to simple procedures for the determination of the corrugation profile from the diffracted intensities. 29-3 1 Realistic soft potentials are, however, needed to explain the observed energy dependence of the diffracted intensities. Recently developed approximations for soft potentials,32 going beyond the distorted wave Born approximation and resonant scattering theory,33 may yet prove useful. At present, however, the accurate theory of diffraction from a soft po-

a) Permanent address: Department of Physics, University of Virginia, Charlottesville, Virginia 22901

tential with a sizable corrugation requires a close coupling calculation using a considerable amount of computer time. 15,20,3~7 Because it is too expensive to fully optimize a soft potential, the HCSW model with an adjustable corrugation has in the past given better fits to the experimental diffraction intensities from ionic crystals.

The earliest soft potential model, which was developed to explain bound state resonances in He-LiF(OO 1) scattering, goes back to Lennard-Jones and Devonshire48 who assumed that the potential is simply given by

V(R,z) = Voo(z) + 2VIO(zl(cos 2:X + cos 2;), (1.1)

where a is the closest distance of two like ions at the surface, R is the projected position of the atom onto the surface plane, and z is the distance of the atom from the surface [r = (R,z)]. In this model the uncorrugated term V 00 is approximated by a Morse potential and the corrugation term VIO by a simple exponential. This convenient form has recently been replaced by another soft potential model introduced by Armand and Manson49 which incorporates the hard wall corrugation:

V(R,z) =Do[e- 2a [z-zo-Q(R)1 - 2e- a ,(Z-Zo)] , (1.2)

with the corrugation function Q (R) given by

Q(R) = L 2.hGe'GoR, (1.3) G#o2

where the G's are the surface reciprocal lattice vectors: Gmn = 21T/a.(m,n). This potential has been shown to provide a very good fit of He atom-metal surface diffraction intensities.49

Even these soft potential models, successful as they may be for fitting diffraction data, have several important drawbacks. For one they do not easily lead to empirical rules for guidance in the prediction of potentials for unknown systems. Also because of the approximate modelling of the long-range attractive potential they do not provide a very

2504 J. Chem. Phys. 83 (5), 1 September 1985 0021-9606/85/172504-18$02.10 © 1985 American Institute of Physics


Celli et al. : Potential for He-LiF(001) 2505

accurate description of the bound states, which are important for understanding desorption and adsorption processes. More important is that they are not easily adapted to the calculation of inelastic phonon scattering data. Such calculations are feasible if an effective two-body potential is known. 50 Some progress has recently been made in developing approximate two-body potentials for atom-noble metal surface scattering. In these models the electron density at the surface is estimated by simply taking a superposition of atomic charge densities. 5 I-53 The repulsive potential is then assumed to be proportional to the electron density at the position of the atom.54,55 This simple prescription has been shown to provide a useful description of diffractionS 1,56 as well as inelastic scattering data. 57 The attractive part of the potential is either neglected or approximated by its lateral average, which is given by a modified Lifshitz formula.58

The best developed a priori method for dealing with noble gas-metal surface potentials, however, is based on the theory of the inhomogeneous electron gas. It was effectively initiated by Zaremba and Kohn,58 and as recently applied by Harris and co-workers, 59 it provides a good description of He diffraction and bound states.60 Because these potentials treat the crystal as a large rigid molecule, they are not better suited for phonon inelastic scattering calculations than the semiempirical potential (1.2). It should be possible, however, to establish a relation to effective two-body potentials. In this connection, it is interesting that Lahee, Allison, and Willis61

have investigated the corrugation of the attractive potential by summing effective van der Waals two-body potentials.

The assumption of pairwise additivity should be a relatively good approximation for describing the He-LiP(OOl) potential because of the closed shell nature of the ions in the crystal and the absence of a free electron gas. This approach has, therefore, been used many times in the past. Tsuchidal5

and Davies and Ullermayer18 used a summation of LennardJones 12-6 potentials with parameters derived from the known He-He and Ne-Ne interactions, in which one of the He and Ne atoms is used to approximate the Li + and pions, respectively. Similar approaches have been tried by Cabrera and Goodman 16 with a superposition of repulsive and attractive Yukawa pair potentials as well as by Goodmann 19 with a Morse two-body potential and Chow and Thompson20 with a Yukawa-6 pair potential. The most recent attempt in this direction has been made by Miglio, Quasso, and Benedek62 who summed anisotropic repulsive Born-Mayer and attractive dispersion potentials. All these calculations have been restricted to rather simple empirical models for the atom-ion potentials and no serious attempt was made to compare the results with molecular beam scattering data.

The aim of this paper is to find a realistic set of two-body potentials which reproduce both the elastic63~6 as well as the inelastic9

-12 data for He-LiP(OOl). We first construct a po

tential based on the superposition of He-Li + and He-Ppotentials, which are determined from ab initio calculations leaving only one adjustable parameter. Specifically, we use the form of the two-body potential recently proposed by Tang and Toennies,67 which includes both the overlap repulsion and the van der Waals attraction due to dispersion forces. The novel feature of this potential is that the long-

range dispersion coefficients, which are valid asymptotically for interatomic distances p-+ 00, are modified at short atomion distances p by universal damping functions containing no new adjustable parameters. The effect of the static induced dipole forces is added separately.

This article is organized in the following way. The method used to generate the atom-surface potential from pair potentials is discussed in Sec. II. The expressions derived there are perfectly general and can be used for the calculation of the interaction of an atom with any crystal. The formulae are then applied to the calculation of the He-LiP(OOl) potential in Sec. III. All the pair potential parameters are obtained from ab initio calculations with the exception of the C6 dispersion constant for the He-P- potential which could be determined uniquely by a fit of the experimentally known vibrational levels of the attractive uncorrugated potential. This semi ab initio potential as well as potential models based on Eqs. (1.1) and (1.2) were then used to calculate diffraction intensities and also angular distributions showing special resonance effects (Secs. IV and V). In the model potentials the parameters were fit to the experimental data whereas the parameters of the semi ab initio potential were not further adjusted. Nevertheless the semi ab initio potential provides a slightly superior fit of the diffraction data than the model potentials. Prom a comparison of the different descriptions it has been possible to estimate those features of the potential which still are not quite correct. With the aid of another simplified soft potential model we have investigated the transition from the soft potential to the hard wall limit where some unexpected problems with the hard wall are encountered. The paper closes with a discussion of the many additional mostly higher order terms which have been neglected in the present model (Sec. VI). Because of the success of the semi ab initio potential in explaining both the diffraction and phonon inelastic data it appears as if these correction terms largely cancel each other.

II. CONSTRUCTION OF THE ATOM·SURFACE POTENTIAL FROM PAIR POTENTIALS

Following Tsuchida lS we write the potential energy of an atom at r near the surface of an ionic crystal, as

VIr) = VSP(r) + Vid(r). (2.1)

VSP(r) is the sum of atom-ion pair potentials

VSP(r) = Lv(S)(r - RJ - Sis)' (2.2) Jis

where RJ is a two-dimensional vector (parallel to the surface) specifying the center of the J th surface unit cell, and Sis indicates the position of the s th ion in the / th layer of ions. (Por example, s = 1 denotes the Li + , s = 2 the P- ions in a LiP crystal, I = 1 denotes the first surface layer, I = 2 the second, etc.) The coordinates are chosen so that the first layer of ions is centered at z = 0, and the crystal occupies the half-space z < O. The pair potential v(S)(p) contains the long-range attractive van der Waals interaction as well as the short-range exchange repulsion. The dependence of ds

) on the layer index I is assumed to be weak and is effectively neglected in our calculations. We shall discuss later (Sec. III B) how the long-

J. Chern. Phys., Vol. 83, No.5, 1 September 1985


2506 Celli et al. : Potential for He-LiF(001)

range van der Waals attraction can be decomposed, approximately, into a sum of two-body potential terms.

The induced dipole interaction V id (r) describes the effect of the Coulomb field E(r) of all the lattice ions on the atom, through the static atomic polarizability a. It is given by

(2.3)

For completeness, we provide in the next section, an explicit evaluation of the lattice sums in Eqs. (2.2) and (2.3), although these can be found in the literature.68.16.18.24--26

A. Summation of pair potentials

For given atom-ion pair potentials viS)(p) (see Sec. III B) we can calculate VSP(r) of Eq. (2.2) by introducing the twodimensional Fourier representation of the pairwise potentials vis)

dS)(r') = f d 2K Vis) (z')e,K.R·

(21Tf K , (2.4)

v~(z') = f d 2R 'viS)(r')e-'K'R', (2.5)

where following the usual convention, a three-dimensional vector like k = (kx,ky,kz) is decomposed into a two-dimensional vector K = (kx,ky) parallel to the surface and az-component perpendicular to the surface k = (K,kz ). Similarly, we write r = (R,z) and Sis = (Sis ,zls)' From Eqs. (2.2) and (2.4) we obtain

VSP(r) = L f d2~ e,K'iR - S,,)

Is (211')

'Le -'K'RJV~(Z - Zls)' J

(2.6)

Because of the lattice periodicity, the sum over the surface unit cells J in Eq. (2.6) can be simplified to

L e - ,K·RJ = (211')2 L 8(K - G), (2.7) J Auc G

where G are the surface reciprocal lattice vectors andAuc is the area of the surface unit cell.

Equation (2.6) can be rearranged to obtain the two-dimensional Fourier expansion of the periodic function VSP(r):

VSP(r) = LVg(z)eiGoR, (2.8) G

where vg(z) is expressed in terms of the Fourier coefficients of the pair potentials (2.5):

vg(z) = -I-Le - iGoS"V~(z - Zls)' (2.9) Auc Is

For a spherically symmetric pair potential vis)(r') = dS)(lr'I), Eq. (2.5) reduces to a one-dimensional integral containing the zero order Bessel function Jo:

v~(z - Zls) = 211' f" dR ' R 'Jo(GR ')

·ds) [~R ,2 + (z - zls)2] . (2.10)

It is sometimes convenient to change the integration variable in (2.10) fromR' top = [R,2 + (z - zlsf] 1/2. In particular,

the formula for the surface averaged sum of pair potentials (the G = 0 Fourier component) becomes then

vg;(z) = ~L dppv(S)(p). 2 1'" Auc Is Iz - zl,1

(2.11)

For given pair potentials viS)( p) and lattice geometry (entering the calculation via Sis and Zls) all the Fourier coefficients of the summed pair potentials can be directly calculated by numerical quadrature of the integrals in Eqs. (2.10) and (2.11).

B. The induced dipole interaction

The effect of the electric field of the ions on the atom cannot be obtained by simply adding up all electrostatic ionatom potential energies. We can, however, compute the electrostatic potential ip(r) due to the ions as a pairwise sum of point charge Coulomb potentials. (We neglect corrections due to the static higher multipole moments of the ions.) To avoid the singularities caused by the Coulomb potential in the intermediate steps of the calculation, we use a pairwise sum ofYukawa potentials

e-fJlr-RJ-s,,1

ip (r) = 2)s (2.12) Jis Ir-RJ -sisl

(with qs the electric charge ofthesth ion) and let/3---+O at the end. We follow the same steps as in the calculation of VSP(r) where now we have in place ofEq. (2.4),

e- fJr' f d 2K 21Te-~lz'l .' qs-- = qs -- e,KR,

r' (21Tf ~/32 + K2 (2.13)

and we obtain the potential

() 211' " "" iGoR - iGoS" ip r = -£..qs£.. £..e e Auc s I G

e-~lz-zl,1

(2.14) ~/32 + G 2

Note that in atomic scattering, the region of interest is outside the crystal. It is given by z > Zls for all/,s. The electric field is then

E(r) = - grad ip (r)

= 21TLqsLL( _iG,~/32+G2) (2.15) Auc s I G

-~iZ-ZI') . e - iGoS1, e e'G-R.

~/32 + G 2

To obtain the dipole interaction [Eq. (2.3)] we compute IE(rW and take the limit /3---+0 in the resulting expression, which gives

with

Vid(r) = - i.aIE(rW 2

= - ~~a ~qs,qs' I~ G~"g(G',G") uc

. exp j(G"SI's' - G"'SI"s")

·exp[ - G '(z - Zl's') - G "(z - ZI"S")] . exp i(G" - G')·R (2.16)

J. Chem. Phys., Vol. 83, No.5, 1 September 1985


Celli et al. : Potential for He-UF(001) 2507

{

I, if G' = 0 or Gil = 0,

g{G',G") = 1 + G'·GII

otherwise. IG'IIG"I

(2.17)

This formula can now be evaluated for the particular geometry of the surface, as shown in the next section for LiF{ool). There a simpler approximate expression for Vid(r) is also derived.

III. CALCULATIONS OF THE He-LIF(001) POTENTIAL

The expressions for VSP(r) and V id (r) from the preceding section are valid for an arbitrary crystal structure and are not restricted to a special choice of the atom-ion pair potentials vIS)( pl. In this section we evaluate the potential formulae for the geometry ofa (001) surface of an ionicfcc crystal [NaCIstructure as in LiF(ool)] and specify the pair potentials used in the present work. When this is done thez-dependent Fourier components VG (z) of the atom-surface interaction potential

VIr) = LVG(z)eiGoR (3.1) G

are calculated numerically so that the coupled channels method can be applied to solve the scattering problem.

A. Geometry of the (001) surface of an ionic fcc crystal

Figure 1 shows the definition of the coordinate system. The x axis is in the (110) bulk direction, also called the (10) surface direction. The assignment of the indices I and s is the same as in Sec. II. The hatched area marks the surface unit cell. Its size is Auc = a2

, where a is the surface lattice constant. For LiF(OO 1), a = 2.84 A. If we denote the positive ion

z

1=1.3.5 •... bJ

• Li + (s =11 o F- (s=2)

y

---------~ , ,

a 0 : ,

--'''--0-::---+-_ X

1= 2.4.6 •... c)

FIG. 1. Geometry of the LIF(OOl) surface, (a) coordinate system and assignments of ion index s and layer index t, (b) cut along layer with / odd, (c) cut along layer with I even. The surface unit cell is marked as a hatched square of area a2

, where a is the surface lattice constant.

with s = 1 and the negative ion with s = 2, the charges qs to be used in Eq. (2.16) are ql = + e, q2 = - e.

If we assume that the surface is not relaxed, which is a good approximation for LiF(001),69,7o the ions form equidistant layers parallel to the surface. The positions of these layers are denoted by

Z/s = - (/- l)al/i. (3.2) For the evaluation of the potential formulae (2.10) and (2.16) we further need explicit expressions for the surface reciprocal lattice vectors,

G = Gmn = 2tr (m,nj, m,n integer, (3.3) a

and the position of the sth ion in the I th layer (see Fig. 1)

{

(O,O) if I + s odd,

Sl = (1 1) if I (3.4) S "2 a'2 a +s even.

From Eqs. (3.3) and (3.4) we get

(3.5)

so that Eqs. (2.8)-(2.10) for the sum of pair potentials become

VSP(r) = 2: L L L( - 1)(m + n)(/+s-l)eiGmn'R a mn / s

-l""dR 'R 'ds) [ ~R,z + (z _Z/s)2]

.JoCtr:' ~m2 + n2). (3.6)

In a similar way the induced dipole potential from Eq. (2.16) can be evaluated, using Eq. (3.5), the relation

qs.qs- = ( - 1)s' +s' e2 (3.7)

(e is the elementary charge) and the fact that according to Eq. (3.2) Zis does not depend on s (which means neglect of surface relaxation). The sums over s' and s" in Eq. (2.16) give

2 2 L L (_I),,+s'+(m'+n')(/'+s')-(m'+n')(/'+s')

s'=ls#=1

= {4'( - 1 Ii' + /- if m' + n' and m" + n II both are odd,

0, otherwise. (3.8)

The sums over the layer indices I' and I" form geometrical series as a consequence of Eq. (3.8) and the equidistance of the layers expressed in Eq. (3.2):

L(-lfexp __ ~m'2+n'2 Z "" [2tr [ l' I a

(/'-1)~]]

1 + exp [ tr~2(m'2 + n'Z) ] (3.9)

Because ofEq. (3.8) only terms for which neither G' nor G" is zero contribute to the potential, so that only the second equality in Eq. (2.17) is relevant and we can set

g(G',G") =g(m',n',m",n")

m'm" +n'n" = 1 + -;:;:::::;;::===::;;;::;=;;:;;:::::::::;;;;

~(m'2 + n,2)(m"Z + n"2)

Putting Eqs. (3.7H3.1O) into Eq. (2.16) we obtain

(3.10)




Vid(r) =

exp[ - 2; (~m'2 + n,2 + ~m"2 + n"2)z] .

. exp z(Gm"n" - Gm.n, )oR . (3.11) [1 + exp[ -1T~2(m'2 + n,2)]] [1 + exp[ - 1T~2(m"2 + n"2)]]

V id (r) depends on z via exponentials. Thus in many cases it is a very good approximation to keep only the terms in the sums over m', n', m", n" for which the exponential containing the z dependence in Eq. (3.11) has its maximal value. Under the restriction thatbothm' + n' andm" + n" are odd integers the leading terms are given by the 16 possible combinations of

(m',n') = (± 1,0),(0, ± 1) and (m",n") = (± 1,0),(0, ± 1).

(3.12)

Rearrangement of these 16 terms leads to the lowest order expression26

V id ( ) 641rae2 - 4; z r;:::: - e

a4(1 + e-1I'{2)2

. ( 1 + cos 2:X cos 27) . (3.13)

This approximate formula can be used for the calculation of the induced dipole potential if the distance z of the atom from the first surface layer is large compared to al(41T). For LiF(OOI) we have a = 2.84AandEq. (3.13) is estimated to be valid for z>0.2 A.

B. The atom-Ion pair potential model

The atom-ion pair potentials v(s)( p) defined in Eq. (2.2) include two different parts: (a) the short range repulsive interaction arising from the overlap of the electron clouds of the atom and the ion, and (b) the long range attractive dispersion interaction. According to Eq. (2.1) the forces resulting from the electric field of the ions are treated separately via the induced dipole interaction potential given by Eq. (2.3). All other pairwise forces are described by the new potential formula of Tang and Toennies,67 which has been extensively tested for gas phase atom-atom interactions and recently for ion-metal surface interactions71 :

(3.14)

Here the repulsive part is represented in the Born-Mayer form with the strength and steepness parameters A and b. The terms of the dispersion series, which are proportional to p - 2n(n = 3,4,5, ... ) for large values of the internuclear distance p, must be damped for smaller values of p because of the effect of the electron clouds overlap on the dispersion forces. This is described by the damping functions f2n (p) which depend only upon the steepness parameter b of the repulsive potential contribution and are given by67

2n (b )k hn(P) = 1 - L ~-bp. (3.15)

k=O k!

This form yields the realistic boundary conditions

hn(p)~1 for p~oo (3.16)

and

hn(P)-o + o (p2n+ I), for p-o. (3.17)

The first condition (3.16) states that the damping is only active for small values of p, while the second condition (3.17) implies that the dispersion term f2n (p). C2n (p)1 p2n itself is zero atp = O.

In the present application we have to keep in mind that the major second order effects in the SCF potential have already been accounted for by the term V id (r). Thus for the repulsive term we use instead of the usual second order SCF calculations the results of a first order SCF calculation indicated by an index (1). Moreover, we have found it sufficient to keep only the leading (2n = 6) term of the dispersion series in Eq. (3.14):

(s)( ) -A (s) -b!~If' 1 ~ (I) -half' 6 [

6 (b (s)p)k ] C (s)

v p - (I)e - - £.. e -6 . k=O k! P

(3.18)

The values of all potential parameters are listed in Table I. The Born-Mayer parameters A g" and b l~" for He--Li + (s = 1) and He--F-(s = 2) interactions are derived from first order SCF calculations which were kindly performed for us by Ahlrichs and B6hm.72 The dispersion constant C~) for He--Li+ is taken from the literature.73 Unfortunately C~) (He--F-) is not available presumably because of the difficulties connected with an ab initio calculation of the diffuse outer F- electron orbitals. As also semiempirical formulae74

yield rather widely differing results, we had to determine this dispersion constant empirically. We did this by fitting the calculated bound state energies of the He--LiF(OOI) atomsurface potential to the experimental values from Derry et aUs For different values of C~) used in the pair potential V(2)( p) the surface-averaged interaction potential V oo(z) has been calculated as described in the previous section. Then

TABLE I. Potential parameters for the two-body potentials used in the semi ab initio He-LiP(OOI) potential model.

Atom-ion

He-Li+(s = I) He-P-(s=2)

650.9" 5.092" 4408" 4.435"

a Prom a first-order SCP calculation by Ahlrichs and Bohm (Ref. 72). bprom Dalgarno and Davison (Ref. 73). C Determined from a best fit of the bound state energies. d An estimate based on several different semiempirical combining rules

(Ref. 74) yielded 3600 ± 1200 meV ).6.



Celli et sl. : Potential for He-LiF(001 ) 2509

..... > GI E

..... -4

-6

-8L----4~O~O-O------4~50-0-------5-0LO-O~

FIG. 2. Variation of the bound state energies Eo of the surface-averaged semi ab initio He-LiF(OOl) potential Voo with the C~) value of the He-F- twobody potential The four experimental bound state energy levels (Ref. 75) are represented by horizontal lines and error bars. labelled by the corresponding vibrational quantum numbers v = 0, ... ,3. The vertical arrow indicates that for C~) = 4425 me V A 6 the calculated and measured energy levels agree within the experimental error bars for all four bound states.

the bound state energy levels belonging to this potential well have been calculated by a numerical eigenvalue program.76

The result is shown in Fig. 2. Fortunately a good fit with all four bound state energies within the small experimental errors has been found for a unique value of C~i. The best fit values are compared in Table II with the experimentally determined results and the bound state energy levels obtained with the Morse corrugated potential of Sec. III E.

It is clear that even for a nearly perfect ionic crystal such as LiF the van der Waals attraction is strictly speaking not pairwise additive, because the fluctuating dipolar fields that mediate pairwise dispersion forces are screened by the other ions in the crystal. In other words, the bare He dipole field must be replaced by the local field acting on each ion. For a dilute medium, the local field caused by a point charge outside the crystal is equal to the bare field times 2/(£ + 1),77 where E is the frequency-dependent dielectric constant of the medium at the frequency of the fluctuation. For a dense medium, there are further corrections that in the bulk are given by the Clausius-Mossotti formulae78

; these local field cor-

TABLE II. Bound state energy levels with vibrational quantum numbers v.

v

o 1 2 3

Experiment (Ref. 75)

(5.90 ± 0.06) meV (2.46 ± 0.05) meV (0.78 ± 0.(4) meV (0.21 ± 0.02) meV

Present theory, semi ab initio potential

5.92meV 2.44meV 0.84meV 0.22meV

rections are hard to compute for the surface layers, due also to possible surface relaxation. The precise relation between the sum of pairwise van der Waals forces and the Lifshitz formula for the atom-surface dispersion force has been discussed elsewhere79 and is far from settled. Here we take a practical approach and lump all local field effects, plus the poorly known polarizability of F-, into one adjustable parameter. With this empirical modification the effective van der Waals forces can apparently be described adequately by a sum of effective two-body potentials, as also postulated by Lahee, Allison, and Willis.61

C. Features of the semi ab initio He-LiF(001) potential

With the parameters for the atom-ion pair potential given in Table I, the potential terms VSP(r) from Eq. (3.6) and Vid(r) from the approximation Eq. (3.13), valid for z> 1 A, have been calculated numerically. For the summation over the layer index I in Eq. (3.6), convergence was reached by taking the first ten layers into account for m = n = 0, whereas for the other Gmn components it is sufficient to retain only the first two layers. The polarizability a of the helium atom needed in Eq. (3.13) is a = 0.206 A3.80

The results for the lowest order Fourier components VG ofthe atom-surface interaction potential V(r) [as defined in Eq. (3.1)] are shown in Fig. 3. The magnitudes of the Va's are rapidly decreasing with increasing values of IGmn I = (m2 + n2)1/21GIOI which guarantees good convergence of

the Fourier series (3.1). Because ofthe choice ofthe coordi-nate system according to Fig. 1 all the higher Fourier components shown in Fig. 3 are positive, which means that the potential maxima situated on top of the F- ions, corresponding to R = (0,0) in Eq. (3.1) are enhanced compared to simple sine-curve shaped maxima. Note that because of the symmetry ofthe crystal surface a large number of the VG's are equal to each other, namely,

Vmn = Viiin Vmn = Viii;; = Vnm (3.19)

Figure 4 shows the contributions of the different atomion potentials and the induced dipole potential to the surface averaged He-LiF(OO I) potential V ()()(z). It is apparent that the main contribution comes from the summation of the HeF- pair interactions, whereas the He-Li + pair interactions and the induced dipole potential play only minor roles. Note that because of the long range dispersion attraction of the Fion also the ions of the second surface layer yield a considerable contribution to the total potential.

In Fig. 5 the atom-surface potential V (z) above the Li +

and F- ions is plotted for the potential models considered

Morse corrugated potential, Eqs. (3.23), (3.25), and (3.26)

5.97meV 2.53 meV 0.57 meV



2510 Celli et a/. : Potential for He-LiF(001)

15

V22 He - LiF (0011 10

V21 Vll

5 V20

~ .§ 0

c >E

-5

-10

-15 0 2 4 6 8

z [AI

FIG. 3. Fourier components Vmn of the semi ab initio He-LiF(OOl) potential as a function of the distance z from the center of the first surface layer. V 00 is the surface averaged part of the potential. The higher Fourier components VIO' Vll' V20, V2 1> and V22 are all positive, which means that the potential maxima situated on top of the F- ions are enhanced compared to simple sine-curve shaped maxima.

here together with the surface averaged potential V oo(z). At the position R = (0,0), corresponding to a F- ion in the first surface layer, the potential energy has its maximum as a function of R with z = constant, whereas it has its minimum at R = (a/2,a/2) corresponding to a Li+ ion in the first surface layer. From the horizontal distance of the two extremal semi ab initio potential curves in Fig. 5(a) one gets the peak to peak corrugation amplitude ofthe He-LiF(OOl) potential of about 0.5 A, depending only weakly on the perpendicular energy over the whole range of interest from - 5 to 60 me V. This value is significantly smaller than the value of 0.614 A suggested by Garcia4 on the basis of the contact hard spheres model and a HCS fit to experimental results. The relation between the corrugation of the turning point of a soft potential and the HCS corrugation is discussed in Sec. V.

A contour plot of equipotential lines for a diagonal cut through the surface unit cell is given in Fig. 6, where a comparison with the empirical potential models to be discussed in the next subsections is made. Finally in Table III the im-

15 I I

a) semi 10 ab initio

I b) Lennard -Jones I

Devonshire I I I

5 I I I

> ., 0 Voo E I

I

> I above U+--1

·5 I I I I I 88 I I

·10 \ I '~/ - -

I I I I I I I I I I \ \ \

He - Li F (001) \ , \ I \J

-15 0 2 3 4 5 6 7 0 2 3 5

z IAI z [AI

I I I I I I I I

15

He - LiF (001) 10

I I

5 /~

Vsp (He-Li+) \ 00 I

1st layer I

~ \ \

.§ 0 , , --------

0 Vsp (He-F-) ,/ >0 00 1"'-2nd layer /(

-5 -/ . Vsp (He-F-) / I

'" 00 I . 1st layer

-10 ! 'iid rOO ,

-15 i 0 2 4 6 8

z IA]

FIG. 4. Different contributions to Voo(z) for the potential of Fig. 3: F- ions in the first surface layer (-), Li + ions in the first surface layer (._.), F- ions in the second surface layer (- -I. and induced dipole interaction (_._).

portant parameters of the semi ab initio He-LiF(OO 1) potential are listed. For the C3 constant of the long-range atomsurface van der Waals interaction there is a significant difference between our value of 137 meV A3 and the result C3 = 93 meV A3 of a calculation based on the Lifshitz formula. S

! A discussion of this discrepancy and the limitations of our potential model in general is given in Sec. VI.

D. The Lennard-Jones-Devonshire potential

In order to examine the uniqueness of the pairwise additive potential and in an attempt to obtain a better description of the experimental diffraction intensities we have also tried to construct a model atom-rigid surface potential by fitting both the energetic locations of the surface resonances and the elastic diffraction data. Because of the expense of repeated close coupling calculations the bound states were fitted first and then only one or two free parameters were optimized.

We first tried to approximate the atom-surface potential by a few terms in its Fourier series expansion (3.1). This functional form is convenient because the Fourier coefficients

I FIG. 5. Comparison of the surface averaged po-c) Morse \ corrugated tential V 00(-)' the potential above the Li + ion (---)

I and that above the F- ion (- -) for three differ-

\ ent He-LiF(OOl) potential models: (a) semi ab ini· I \ tio potential from Sec. III C; (b) Lennard-Jones-I \ Devonshire potential from Eq. (3.22) with I I ,

D= 7.63 meV, a = l.lA-I,ZO= 1.0A,P=0.1, I \ I \ (c) Morse corrugated potential from Eqs. (3.23) I I \-A and (3.25) with Do = 8.03 meV, a = 1.35 A -I; I I , r

a , = 1.08 A -I, ZO = 1.0 A, h = 0.102 A. The z \

, / axis is perpendicular to the surface. In (b) and (c) \ I

\ .. ./1 its origin is arbitrary. whereas in (a) z = 0 denotes the center of the first surface layer.

0 2 3 5

z [AI




F- Li+ F- F-

5

a) semi ab initio -3

-L

4 -5

-6

-7 -8

N

c) Morse corrugated -3

-4

2 -6 -5 -6

OL-~~~-L-L __ -L __ ~~~~~ __ ~

o 2 4 6 8

Vx2+y2' [A1

FIG. 6, Contour plots of equipotential lines for the same potentials as in Fig, 5: (a) semi ab initio potential, (b) Lennard-Jones-Devonshire potential, and (c) Morse corrugated potential, The cut through the surface unit cell is along the (100) crystal direction along the line y = x of adjacent F- and Li + ions (see Fig, 1). The z axis is perpendicuJar to the surface, in (b) and (c) its origin is arbitrary, whereas in (a) z = 0 denotes the center of the first surface layer. The numbers at the contour lines are the corresponding potential energy values in meV. The labelled arrows at the top of the figure indicate the positions of the ion centers,

V G (z) appear directly in the close coupling equations. Following Lennard-Jones and Devonshire,48 Wolken,34 and Drolshagen et al., 42 we took the lateral average to be a Morse potential

Voo(Z) = De-a(Z-z"i[e-a{z-z"i - 2]

and the other Fourier coefficients to be exponentials

V. ( ) -/3 D - 2aG{z- zo) G Z - G e .

(3.20)

(3.21)

TABLE III. Potential parameters for the semi ab initio He-LiF(OOl I potential.

Surface averaged part V oo(z): position of the potential minimum

well depth

Asymptotic behavior for large values of z: V oo(z) - C~(z - zl'

Peak-ta-peak corrugation height at V=20meV at V=60meV

zmin =3,04A D=8.72meV

C, = 137 meV A'a z= 1.00 A

~z=O,518A ~z=0.527 A

a A calculation based on the Lifshitz formula yields C, = 93 meV A' (Ref, 8!).

Unfortunately, as discussed in Sec. V, this potential model did not prove to be useful. For comparisons with the other potential curves in Fig. 5 and the contour plots in Fig. 6 the original simple form suggested by Lennard-Jones and Devonshire48 was used:

V(r)=D[e-2aIZ-Zol[1 +2P(cos 2;X + cos 27)] - 2r a (Z-Zol] , (3.22)

with the values D = 7.63 meV, a 1.1 A-I from Devonshire,82 and Zo = 1.0 A, /3 = 0.1 as chosen by Wolken.34 The most recent fit by Drolshagen et al.42 gave quite similar values.

E. The Morse corrugated potential

A more convenient representation suggested by Armand and Manson49 is provided by a modified Morse potential with a rigidly shifted repulsive part:

VIr) = Do[e- 2a[z-zo-Q(Rll _ 2e- a ,(z- zolJ,

Q(R) = I..!..hGeiG-R. G#o2

(3.23)

(3.24)

Note that in contrast to the case of the original Morse potential, the range parameters of the repUlsive (a) and attractive (a 1) terms are allowed to be different. If the attractive part is neglected, Q (R) represents the surface corrugation. The Fourier expansion of Q (R) is often found to require only one or two coefficients; in particular, the simple sinusoidal form

( 21TX 21T)' ) Q(R)=h cos-a-+cos-

a- (3.25)

already provides an adequate representation for the LiF(OO 1) surface. With this functional form, however, the Fourier coefficients VG(z) must be numerically computed. In practice, only a finite number of terms are again kept in the expansion (3.1), but the functional form of VG (z) is now fixed, and there are fewer adjustable parameters.

The parameters Do, a, and a l in Eq. (3.23) were chosen to fit the three lowest bound states for He-LiF(OOI), as measured by Derry et al.'5 By changing a (and simultaneously adjusting a I to retain a good fit to the three lowest bound states) it was possible to reproduce the diffraction data of

J. Chern. Phys" Vol. 83, No, 5, 1 September 1985



Boato et al. 63 for a value for h of about 0.1 A as shown in Sec. V. The best overall fit was obtained for

Do = 8.03 meV, a = 1.35 A -I, 0_ 1 0

a l = 1.08 A , h = 0.102 A. (3.26)

With these parameters, the lateral average of the potential (3.23), i.e., V oo(z), gives three bound states with binding energies 5.97, 2.53, and 0.57 meV. As shown in Table II the two lowest bound states of our potential agree with the experimental data, within the stated experimental uncertainty. The third bound state is not far off, but the fourth is missing, as expected. Potential curves for this model are shown in Fig. 5c, a corresponding contour plot in Fig. 6(c).

IV. CALCULATIONS OF DIFFRACTION PROBABILITIES

From Sec. III we obtain the Fourier components V G (z) of the interaction potential V (r) of a He atom in front of a static LiF(OOl) surface. We solve the time-independent Schrodinger equation for the atom moving in this potential and calculate the probabilities for elastic scattering into the different possible diffraction channels. The accuracy of all three potential models is then judged by comparison with experimental measurements.

A. The close coupling method

In order to calculate the diffraction probabilities we numerically solve the time-independent Schrodinger equation for the atom with mass m interacting with the surface via the potential V(r):

[ - ;:: V2 + V(r) ]tf!(r) = E;¢(r). (4.1)

Since we deal with a scattering problem we must solve Eq. (4.1) for the positive eigenvalue Eo which is equal to the kinetic energy of the atom before the collision

(4.2)

where k; = (k;x,k;y,k;z) = (Kok;z) is the incident wave vector.

Because of the surface periodicity the wave function tf!(r) can be expanded in a Fourier series in a similar way as the potential V(r) (Bloch's theorem):

.,,( ) _ 1 ~ () ;(K,+G)'R ."r ---.£..lPG ze . ~Auc G

(4.3)

HereA uc is the area of the surface lattice unit cell as already introduced in Eq. (2.7). Substituting the wave function from Eq. (4.3) and the Fourier representation(3.1) of the potential into the Schrodinger equation (4.1), we obtain the usual close coupling equations20

:

(d

2 2) 2m dzZ +kGz lPG(z)-~~VG_G.(z)lPG'(z)=O (4.4)

with

k~z =k;-(K; +Gf, (4.5)

The scattering boundary conditions on lPG (z) are

( ) - ;ktzZ £ + A + ;kGzZ lPG Z -+- e UG,O Ge (4.6)

z~""

for k ~z > 0 (open channels) and

lPG(z) -+- AGe - KGZ (4.7) z~""

for k~z <0 with K Gz = (- k~z)l/2 (closed channels). Each open channel corresponds to a diffracted beam for which the scattering probability P G is related to the scattering amplitudeAG by

kGz I 12 PG =--AG . Ikjz I

(4.8)

The P G fulfill the unitarity condition

(4.9)

which means that the summation of intensities over all open channels is 1 in elastic scattering.

B. Numerical details

In the actual calculations the system (4.4) of secondorder differential equations is integrated numerically using a predictor-corrector Numerov computer program.S3 If not given analytically, the Fourier components VG_G.(z) in Eq. (4.4) are interpolated, using cubic splines from a tabulation of the potential components compiled as described in Sec. III. As we cannot take the infinite number of all (open and closed) channels into account, we must restrict the calculation to a finite basis set ofG vectors. The size of this basis set has been determined by checking the convergence of the results with increasing number of channels N included in the calculation. The number of channels needed for convergence depends on the energy E;, and is increasing with larger values of E;. It is important that not only open, but also some closed channels are included, because the often observed phenomenon of bound state resonances23 can significantly affect the diffraction probabilities due to the coupling of the open to the closed channels.

If the scattering is along a high-symmetry direction of the, crystal surface ( 100) or (110) direction) the number of channels to be taken in the calculation can be reduced by exploiting the symmetry of the wave function. 36 For example, the basis set of N = 41 channels where all Gmn with Iml + Inl<4 are included, is reduced to N = 23 for the (100) or N = 25 for the (110) direction. As the calculation time is roughly propqrtional to N 3, this reduction to the so called symmetry adapted basis set decreases the required computer time considerably. In the final calculations different values for N have been used as listed in Table IV. Thus N=69 includes all Gmn with Iml+lnl<6 except (m,n) = (± 5,0),(0, ± 5),( ± 5, ± 1),( ± 1, ± 5),( ± 6,0), and (0, ± 6). N = 41 corresponds to the condition I m I + I n I < 4, whereas in the case of N = 26-29 only those channels with Iml + Inl<4 have been kept for which (If/2m)k~z >-25 meV (which means closed channels with perpendicular energies ofless than - 25 meV are neglected). Whenever possible, the symmetry of the basis set of G vectors has been utilized in the actual calculations to reduce N to the symmetry adapted values given in Table IV. In the Fourier series expansion (3.1) of the semi ab initio atom-surface potential the



Celli £It a/. : Potential for He-LiF(001) 2513

TABLE IV. Kinematical conditions of the He--LiF(OOI) ditfraction experiments and numerical parameters for the corresponding calculations.

Experiment by Boato et al. (Ref. 63)

See figure S 9 Incident wave vector k j 10.95 A.-I

Incident energy E, 62.61 meV Exp. velocity spread ,ju/u -1O%FWHM Crystal temperature SOK Scattering along (100) (110) Incident azimuthal angle t/J, 45· 0-Incident polar angle OJ oo-SO·

Number of channels N 69 Symmetry adapted N 3S 39 Integration start point z. a 1.6SA. Integration end point z •• lOA.

Computer time per 164-173 s 175-185 s Integration run

"Perpendicular" Debye--Waller factor e - 2 Wi 0.25

Well depth parameter D 0 for Beeby correction

• Only for the semi ab initio potential of Sec. III C.

following 21 Gmn components have been taken into account: (m.n) = (0,0).( ± 1,0).(0. ± 1).( ± 1. ± 1),( ± 2.0).(0, ± 2), ( ± 2, ± 1). and ( ± 1. ± 2). It was found that the effect of the higher Fourier components on the diffraction probabilities can be safely neglected with regard to the numerical accuracy of the calculations, whereas it is important to include the ( ± 2, ± 1) and ( ± 1, ± 2) terms. Taking all the possible numerical errors into account the relative accuracy of the calculated diffraction probabilities is estimated to be better than 5%.

The integration is started at a value z., which is far enough inside the solid, so that we can safely set (jJG (zs) = 0 for all the N G vectors included in the calculation. As we can only solve initial value problems with the Numerov program we must propagate N linearly independent solutions with N independent sets of derivatives (jJ G (z) at Z = Zs out to Z Ze'

where the potential is essentially zero. There the linear combination of the N solutions is formed, which fulfills the boundary conditions (4.6) and (4.7) and yields the scattering amplitudes AG and via Eq. (4.8) the diffraction probabilities. 20

The Numerov integration program used in the present works3 is a two-point predictor-corrector routine with variable step size. About 200-400 steps are needed for one integration run from Z = Zs to Z = ze' In order to stabilize the solutions during the course of the integration (which is very important for the exponentially increasing closed-channel components of the wave function) the N solutions propagated by the integrator from Z = z. to Z = Ze are orthogonalized to each other every 20 steps. This guarantees that the linear independency of the solutions in maintained over the whole range of integration.

The calculations were performed on the UNIVAC 1100 computer of the "Gesellschaft fUr wissenschaftliche Datenverarbeitung" in Gottingen. The computer time required for

Frankl et al. (Ref. 64)

II 12 5.76 A.-I

17.325meV -2%FWHM 125 K

(110) 00-450 00

70· 32·-So-

26-29 41 25

1.72 A. lOA.

40--60 s 36-53 s

0.094

S.72meV

one integration run (i.e., calculation of all diffraction probabilities for one fixed set of values for E;, (J;, and rp;) is also reported in Table IV.

C. The Debye-Waller factor

For a quantitative comparison of our calculations with experiment we must take account of the effect of the lattice vibrations in reducing the diffraction probabilities. This is done by a standard Debye-Waller correction

(4.10)

where P G is the diffraction probability as calculated by the close-coupling method and P G is the corrected value. In the conventional theory the Debye-Waller exponent depends upon the momentum transfer fiq of the atom to the crystal and the mean square displacement of the surface atoms < u;) in the direction of q63:

2WG = tf<u;). (4.11)

Including the Beeby correction84 we have -----.

q= (G'~k~z + ~D + ~k~ + ~D ), (4.12)

where D is the well depth of the atom-surface interaction potential. The Debye-Waller scaling actually used in the calculations is specified by two parameters: the well depth D and the "perpendicular" Debye-Waller factor

e- 2W1 exp[ -4(k:+ ~D )<u;>], (4.13)

which can be interpreted as the probability for an atom. which hits the surface perpendicularly ((J; = 0°) to be coherently reflected back into the same direction (k ~z = k ~ k n. For this Debye-Waller factor we use the



2514 Celli et at. : Potential for He--LiF(001)

value estimated by Boato et al.63 which is e - ZW" = 0.25, independent of q, for Ei = 62.61 meV He atoms diffracted from a LiF(OO 1) surface at 80 K. The same authors show that this is consistent with (a) the sum of all measured diffraction probabilities divided by e - zWG being equal to 1, (b) the ratio of experimental data and results from HCS calculations, and (c) an independent estimate from a Debye model in the low temperature range.

For the high incident energy of Ei = 62.61 meV the Beeby correction can be neglected (D is set to zero) and we get a simple formula for the angular dependence of the Debye-Waller factor:

-Zw. ( 20i+OG) e G = exp - 2W1 cos 2' (4.14)

Here OG is the polar angle for the diffracted beam. For the lower energy of Ei = 17.325 meV (see Table IV) we take D = 8.72meVfrom Table III, whereas in thiscasee - 2W1has been adjusted to fit the absolute intensities of Fig. 11 in the following section. In the case of the specular beam (G = O,OG = 0i) we get the expression

-2W ( COSZOi +DIEi ) e = exp - 2W1 •

1 +DIEi (4.15)

v. COMPARISON WITH EXPERIMENT

In order to test the validity of our potential models we calculated the experimental diffraction probabilities reported by Boato et al.63 and Frankl et al.64 The various experimental conditions and numerical parameters entering the calculation are listed in Table IV. The experimental velocity

25

"'g 20

x

.rl? lS

:0 c 10 .a e D-

c: .g ~ ;; -c

He - LiF (001)

Ei = 62.61 meV

9 j = 0°

T , , I , I

: I ; I I I I I

T

l experiment

I theory, semi ab initio potential

r theory

Morse corrugated potential

(30) (40)

FIG. 7. Histogram of measured and calculated diffraction probabilities for He-LiF (001), Ei = 62.61 meV, (Ji = 0°. The experimental data are taken from Table 2 of Ref. 63. The theoretical results are obtained from a close coupling calculation with the semi ab initio potential of Sec. III C and the Morse corrugated potential of Sec. III E, includinJ a Debye-Waller correction according to Eqs. (4.10) and (4.14) with e - 2 1 = 0.25.

c:

.a!=

>---.0 ~ .0 0 ... Co

c .2 -u ~ ... --"0

10-3

10-4

10-1

10-2

10-3

10-4

10-1

10-2

-4 10

a) experiment

(33)

He- LiF (001) <100>

Ej = 62.61 meV

b) theory, semi ab initio potential

c) theory, Morse corrugated (001 potential

(ll)

(22)

20° 60° 80°

incident angle OJ

FIG. 8. Diffraction probabilities as a function of the incident polar angle (Ji

for He-LiF(OOI) scattering with Ei = 62.61 meV along the < 1(0) direction ( tfJi = 45°). (a) The experimental data are taken from Fig. 10 of Ref. 63, (b) the theoretical data were calculated by the close coupling method using the semi ab initio potential of Sec. III C, and (c) the Morse corrugated potential of Sec. III C. A Debye-Waller correction according to Eq. (4.14) with e - 2W

, = 0.25 is included. The single data points, which were calculated in

OS intervals, are connected by straight lines. The sharp structures appearing in the theoretical curves are due to resonances which are not resolved in the experiment.




.&J

" .&J o

Q. 10-4

c: o :;; u ~ ---0

10.2

10-3

o·

10-1-

10-2 -

He-LiF(001) <110> Ej = 62.61 meV

.0. experiment

0(00)

0(10 )

20· 40· 60·

incident angle 9j

I

b)

80· a·

I

+-+P22 1O-3'--____ ....L1 ____ ----'1'--___ ---'

1 2 3 4

J. [A-I]

FIG. 10. Dift"raction probabilities as a function of the softness parameter a ofthe purely repulsive model potential of Eq. (5.1) calculated by the closecoupling method for Ei = 62.61 meV, B, = 50" in the (100) direction without Debye-Waller correction. The dashed lines indicate that the accuracy of the results decreases with increasing values of a.

c)

d)

e)

20. 40· 60·

incident angle 0j

80·

FIG. 9. Diffraction probabilities as a function of the incident polar angle Bi for He-LiF(OOI) scattering with Ei = 62.61 meV along the (110) direction

( ¢Ii = (0). (a) Results are given for the specular beam (b) the (10) (c) the flO) (d) the (20) and (20), and (e) the (30,(30), and (40) beams. The experimental data points (e,O,O) are taken from Table 4 of Ref. 63. The theoretical results were calculated by the close coupling method using the semi ab initio potential of Sec. m C (full lines) and the Morse corrugated potential of Sec. III E (dashed lines). The same Debye-Waller correction as in Fig. 8 was applied. The single data points, which were calculated in 0.5' intervals are connected by straight lines.

spread in the incident beam given in Table IV was not taken into account in the calculations.

The theoretical results including the Debye-Waller correction of Sec. IV C are presented in Figs. 7-9, 11, and 12 together with the experimental data. Figure 7 shows a comparison of experimental diffraction probabilities for E; = 62.61 meV and perpendicular incidence (0; = 0·) with the

results calculated using the semi ab initio potential and the Morse corrugated potential. The parameters of experiment and theory are the same as listed in Table IV for the results presented in Fig. 8. The agreement between the experimental and both calculated results in good, not only for the lower, but also for the higher order diffracted beams. Note that in the case ofthe semi ab initio potential the theoretical results are obtained without any further adjustment of parameters in the course of the scattering calculations. For this potential the largest deviation of the theory from experiment occurs for the (20) and (30) beams reaching about - 35%. There are deviations not only in the negative, but also in the positive direction, as for example in the (11) and (31) beams. For the Morse corrugated potential the situation is quite similar. These calculations demonstrate the strong sensitivity of the diffraction probabilities to changes in the potential param-




0.2 r----,---__ ,-----.--------,--,

He - LlF (0011. Ej:: 17.325 mIN

ei = 70°

0.1

0 IQ..0

>-oJ experiment -'iii 0

c \II - 0.2 c ... tJ :::I U \II Q. If)

0.1

b) theory semi ab initio potential

OL-____ ~ ______ ~ ______ ~ ____ ~ __ _J

00 100 200 30° 400

<110> azimuth «Pi <100>

FIG. 11. Specular intensity as a function of the azimuthal angle I/J; for HeLiP (001) scattering with E; = 17.325 meV, 8; = 70·. (a) The experimental data are taken from Fig. 1 of Ref. 64, (b) the theoretical data were calculated by the close-coupling method using the semi ab initio potential of Sec. III C. A Debye-Waller correction according to Eq. (4.15) with D = 8.72 me V and e- 2W

, = 0.094 is included. The single data points, which were calculated in OS intervals, are connected by straight lines.

eters. For example, the specular intensity was found to drop by almost one order of magnitude, if the rather small Fourier component VZ1(z) (see Fig. 3) was omitted from the calculation of Fig. 7.

The angular distributions of the diffraction peak intensities for the same initial energy as in Fig. 7 are presented in Fig. 8. The scattering is along the (100) direction of the LiF(OOl) surface. A similarly good agreement as for the case of OJ = (f in Fig. 7 is obtained between the experimental data [Fig. 8(a)] and the results calculated using the semi ab initio potential [Fig. 8(b)]. The shapes as well as the absolute values of the angular dependence of the diffraction probabilities are reproduced well by the theory. The calculated results contain a number of sharp structures due to bound state resonances. Because of the velocity spread of ..:ivlv-lO% in the experiment (see Table IV) which has not been taken into account in the calculations, these selective adsorption resonances are washed out in the measurements. Only two weak minima in the (00) and {I I) beams at OJ z60° are left in the experimental curves of Fig. 8(a). They correspond to the cou-

11) -.c ... d

>--'Vi c cv -c ... ,g ::J u cv 0-11)

... ,g

He -UF (001) <110>

E j = 17.325 meV

a) experiment

b) theory, semi ab initio potential

B 0.1 cv 0-11)

O~~~-----L----~--~----~ WO ~o ~o roo MO 000

incident angle 9j

FIG. 12. Specular intensity as a function of the incident polar angle 8, for He-LiF(OOI) scattering with E; = 17.325 meV along the {lID} direction ( I/J; = 0"). (a) The experimental data are taken from Fig. 3 of Ref. 64 and (b) the theoretical data were calculated and represented as in Fig. 11 including the same Debye-Waller correction.

pIing with the two lowest bound states Eo and E I of the HeLiF (001) potential well via an out-of-plane (10) or (01) reciprocallattice vector. The angular positions of these minima in the measured curves are as predicted by the theory.

Whereas the semi ab initio potential of Sec. III C can be used to calculate diffraction probabilities without adjusting parameters, the empirical potentials ofSecs. III D and III E




contain free adjustable parameters. In the Lennard-JonesDevonshire model of Sec. III D the Morse potential does not give a good representation of the attractive potential at large distances and, as a consequence, does not reproduce the higher bound state energy levels.23 This is illustrated in Table II for the case of the Morse corrugated potential. However, these shortcomings are not important for reproducing the diffracted intensities at energies greater than the well depth over a wide range of angles as in the experiments of Boato et al.63 Moreover, because of the short range of the Morse potential it is possible to terminate the numerical integration at a smaller Z than in the case of a Z-3 attractive potential. In order to reduce the number of free parameters we tried the form of the Lennard-Jones-Devonshire model [Eq. (3.22)] for which Drolshagen et al.42 obtained the best fit parametersD = 8.03 meV, a = 1.1 A-I and{3 = 0.086 from a comparison with measurements of the velocity dependence of the specular and diffraction peaks. However, despite a number of iterations in which the parameters D, a, and {3 were varied we were not able to fit the diffraction data of Boato et al.63 in a manner comparable to that of the Morse corrugated or the semi ab initio potential.

Using the Morse corrugated potential of Sec. III E, with the parameters (3.26), it was possible to fit the diffraction data of Fig. 8(a). The results are shown in Fig. 8c. Only the (00), (11), and (I I) beams were used in the fit because of computer time limitations. These intensities can be calculated with an estimated error ofless than 5% by keeping 41 coupled channels (which reduce to 23 when symmetry is exploited), whereas it is necessary to keep 69 channels (see Table IV) to obtain comparable convergence in all cases for the other diffracted beams. Also the results of Fig. 8(c) obtained with the Morse corrugated potential show sharp resonance structures. Their angular positions are the same as in Fig. 8(b). The amplitudes of the resonance peaks, however, are much larger for the semi ab initio potential and there are resonant features, above all in the specular beam, which do not show up in the curves of Fig. 8(c) for the Morse corrugated potential. This behavior is probably a consequence of the stronger coupling of the different channels in the close coupling calculations for the semi ab initio potential. In both calculations the specular intensity exceeds the experimental data for incident angles of ()i > 50·. In the case of the Morse corrugated potential a larger value of a yields better results for the specular peak in this range, but the fit of the bound states is correspondingly worse. We note however that this deviation occurs for both the semi ab initio and the Morse corrugated potential and that also the higher order beams drop off faster in the experiment than in the theory when ()i approaches 90°. Thus another possible explanation is that the smaller experimental intensities at large incident angles can perhaps be attributed to a frequently observed loss of measured intensity for grazing incidence of the He beam.

Figure 9 shows the angular dependence of the diffraction probabilities for scattering along the (110) crystal direction. For this direction only a small number of experimental data points are available. They are marked by the symbols in Fig. 9. The calculations using the semi ab initio potential (full lines in Fig. 9) reproduce the experimental

data as well as in Fig. 8 for the < 110) direction, and slightly better than the results obtained with the Morse corrugated potential (dashed lines in Fig. 9). Whereas, for example, the semi ab initio potential yields good agreement between experiment and theory even for the higher order beams in Fig. ge, the Morse corrugated potential does not reproduce the measurements so well. As in Fig. 8 for the (100) direction, here also the semi ab initio potential leads to more pronounced resonance structures than the Morse corrugated potential.

Overall, the empirical Morse corrugated potential of Sec. III E predicts the diffraction probabilities about as accurately as the semi ab initio potential of Sec. III C. The main difference between the two potentials is in the behavior of the specular beam in the range around () i = 40° in Fig. 8. It is known from the hard wall model,85 and from semiclassical calculations,86 that a minimum in this range occurs because of the destructive interference between reflection from the tops of the surface corrugation, corresponding to the positions of the F- ions, reflection from the bottoms, corresponding to the Li + ions, and reflection from the saddle points, corresponding to the bridge positions. The expected minimum in the specular intensity for ()i around 40° is very pronounced (semiclassically it reaches down to zero) if the bridge position is exactly halfway between the top and the bottom, corresponding to a purely sinusoidal corrugation, which is a good approximation in the case ofHe-LiF (001).4 However, the simple semiclassical argument is incomplete, because, for instance, it fails to explain the marked difference of the behavior of the specular between the < 1(0) and (110) directions. The amount of destructive interference that gives a broad minimum in the specular is critically dependent on the phase of the reflected intensity from the various points of the surface unit cell and cannot be safely predicted by the equipotential contours alone, because it depends on the "softness" parameter a, as well as on the corrugation parameter h.

To study this softness-dependent behavior more precisely, we have performed a model calculation with the purely repulsive potential

V(r)=Doexp [ -2a[z-zo-h(COS 2:X + cos 27)]]

(5.1)

with Do = 8.03 meV, h = 0.15 A, and Zo = 1.0 A. The parameters Zo and Do are actually irrelevant and are used only to scale the potential to a convenient numerical range, while h has been chosen to agree with the corrugation determined by a HCS fit to the data of Boato et al.63 (The peak-to-valley corrugation is 4 h = 0.6 A.) The results for three diffraction probabilities in the < 100) direction, without Debye-Waller correction, are shown in Fig. 10 for an incident angle of 50°, as a function ofthe softness parameter a. For a..;3 A-I the results are fully converged with respect to the starting point Zs of the integration. For a > 3 A-I we had to set Zs < 0.0 A in order to get fully converged results. However this is not possible because for such small values of Zs the numerical procedure breaks down due to very large values of the potential near Zs' For this reason we used Zs = 0.0 A for a > 3 A-I



2518 Celli et al. : Potential for He-LiF(OOl)

so that these results are only qualitatively correct. For a > 4 A -1 the error becomes so large that it makes no sense to continue the curves. However, the minimum at a;::::3.25 A -1 and the steep increase of the specular intensity curve for a> 3.25 A -I can safely be deduced from the calculations. This unexpected minimum is present only in the specular intensity, while the (22) and (iT) intensities vary smoothly and presumably approach uniformly the HCS value for a_ 00. The anomalous behavior of the specular beam shows that the HCS model must be used with caution when one of the low-index beams has a destructive-interference minimum.

As a further test of the semi ab initio potential model of Sec. III C we calculated the variation of the specular intensity with the azimuth <Pi of the scattering plane (Fig. 11) and the angle of incidence 0i (Fig. 12) for the low incident energy of Ei 17.325 meV. For these conditions experimental data with a rich structure of resonances are available,64 which are plotted in Figs. II(a) and 12(a). There the velocity spread of the atoms in the initial beam is smaller (.:;1v/v~2%) than in the experiments reported in Figs. 8 and 9 (see Table IV) so that the bound state resonance minima and maxima are well resolved. The absolute values of the calculated specular intensities were fitted to the expeimental data in Fig. 11 by adjusting the perpendicular Debye-Waller factor e - 2 wL in

Eq. (4.15) yielding /wL = 0.094. The experimentally observed structure is well reproduced by the calculated curves in Figs. l1(b) and 12(b) showing only small discrepancies with regard to position and shape of the resonant features. A similarly good fit has been obtained by Garcia, Celli, and Goodman6 using the HCSW (hard corrugated surface with a well) model for the azimuthal distribution shown in Fig. 11. However, their results for the polar angle distribution do not reproduce the experimental data as well as the present calculations with the semi ab initio potential presented in Fig. 12.

Overall then we conclude that the new potential presented here provides a fit of all the available diffraction and resonance data as good as or better than previously suggested potentials, which have been usually designed and adjusted to fit some selected set of data. The present potential model has however the advantage that it is realistic in shape and has been obtained in a predictive fashion that can be used for other related systems.

VI. DISCUSSION AND CONCLUSIONS

Despite the apparent success for He-LiF(OOI) the prescription of adding pairwise potentials for the constituent ions, as if they were free ions, is only a first approximation. We list below several possible sources of error of our potential. It will be clear that some of the corrections, if taken alone, are large enough to undo the agreement with experiment, and that accidental cancellation probably occurs among opposing effects. The estimated corrections to the potential well depth due to the various effects are summarized in Table V.

(1) The first order SCF calculations of the Hartree-Fock repulsive energy for He-F- provide only an upper bound to the true first order Hartree-Fock energy. The corresponding

TABLE V. Estimates of various sources of errors for the semi ab initio potential.

Source of error

1. First order SCF calculation 2. Tang-Toennies two-body potential 3. Charge transfer in the crystal 4. Surface relaxation and rumpling 5. Static dipole moments of surface ions 6. Static quadrupole moments of

surface ions 7. Higher terms in the dispersion series 8. Thermal motion of the crystal ions 9. Three-body forces

10. Static polarization of the atom and the surface

Estimated correction to the well depth

+10% -20% +20%

<±I% < ± 1% -30%

+20% -3% -5%

< ±l%

error for He-Li+ is probably smaller, and the contribution of Li + to the potential is also less important. A reasonable guess is that the Born-Mayer term for F- in our potential should be lowered by as much as 10%.

(2) The Tang-Toennies formula neglects several interaction terms, one of which, the second order exchange, or exchange-correlation energy, is found to make a positive 17% contribution to the total potential near the minimum for triplet H2 and He2•

67 To account for this we expect that our potentials should also be increased by as much as 20% near the minimum. The Tang-Toennies two-body potential also uses a Born-Mayer fit of the SCF results and an approximate expression for the damping of the dispersion coefficients. Extensive comparisons with gas-phase data and calculations suggest that the Tang-Toennies formula approximates these terms to better than 1%.67 The use of this model for the highly deformable F- ion is less certain, but we still expect that this is not a significant source of error.

(3) The electronic charge density of the LiF crystal is known from x-ray studies to be significantly different from a simple superposition ofLi+ and F- charge densities.87 This difference has been attributed to the formation of a partial covalent bond between the F- ions. About 10% of the extra charge on F - is transferred to the bonds. If a similar bonding occurs for the surface F- ions, the charge concentration is correspondingly reduced in front of the F nucleus. This deformation can be described by adding quadrupolar and octupolar components to the F- charge distribution. The quadrupolar effect is similar to the one described in point six below, but of opposite sign. Roughly, the effect could be to decrease the repulsive potential by as much as 20% in front of the F- ion.

(4) Surface relaxation and rumpling have been taken to be negligible on the basis of the most recent theoretical work,69,70 which in tum is supported by the reasonable agreement with the measured phonon spectraY However, widely different values of the surface ion displacements can be found in the literature.4

•69,88,89 We have calculated the He

surface potential for several positions of the top layer ofnudei, assuming undeformed ions, and found that only an outward shift of the Li + -ions, relative to F-, by 0.1 A or more would have a significant effect on the He-surface potential. In particular, such a shift alters the shape of the equi-




potentials near the valley bottoms. Recalling the discussion in See. V we could use the position of the Li + as an adjustable parameter to improve the agreement between theory and experiment in Fig. 8, especially for the specular beam near 40". However, in view of the other uncertainties in the potential we do not feel that a reliable value of the Li + shift can yet be derived in this way.

(5) The static dipole moment of the surface ions is predicted to be negligible by the same calculations that give negligible relaxation and rumpling. Until new evidence is available, this effect can be neglected.

(6) The static quadrupole moment of the surface ions has been estimated recently on the basis of a simple point ion model for the crystal field of the surrounding ions.62 The induced quadrupole moment is large for the F- ions, which are predicted to assume a prolate spheroidal shape. The crystal-field-induced charge density op - is given approximately by

op

Po

5 aq_ (aEz ) 1 2 -- - --(3 cos ()-1),

4 e az (r_) (6.1)

where Po is the unperturbed charge density of the free Fion, aq its quadrupolar polarizability, a q = 2.937 AS, e is the magnitUde of the electron's charge, (aEzlaz) is the crystal field gradient at the position of the F nucleus, (r~ ) is the mean square radius ofF-, and () is the polar angle relative to the surface normal. Taking (r~ ) = (1.199 A)2, Miglio et al.90 find

o -L.. = 0.42(3 cos2

() - 1) . (6.2) Po

Actually, such a large deformation cannot be treated as a small perturbation and Eq. (6.1) is no longer quantitatively valid, but it will serve for the purposes of this discussion. If we make the reasonable assumption that the fractional change in the repulsive He-F- potential is given by the same Eq. (6.1) we find a very large change indeed. (The calculation is performed by an easy extension of the formalism that is described in detail in Sees. II A, II B, and IlIA.) The potential well depth is reduced to about 6 me V, as also found by Miglio et al.62 To compensate for this increased repulsion, we should make substantial changes in the other potential parameters. Before taking such drastic action, we note that a calculation by Ben Ephraim and Folman91 shows that the surface crystal field of a realistic charge distribution in a NaCI crystal is smaller than the surface field due to point ions by a factor of about 3. When the quadrupolar deformation ofEq. (6.1) is reduced by the same factor, it is ofthe same magnitude, but of opposite sign as the charge deformation due to the formation of a partial bond between F- ions [point (3)]. Clearly, only a complete surface electronic band calculation can provide an indication of the relative magnitude of these two competing corrections to the surface charge distribution. Meanwhile, we take the optimistic view that they nearly cancel, or that the quadrupolar deformation itself increases our repulsive potential by no more than 30% in front of the F- .

(7) The effects of screening and of many-body interactions on the dispersion forces are partially accounted for by

our use of an effective dispersion constant C~21 (He-F-). Since we do not know the corresponding value of C6 for the free F- ion, it is difficult to estimate how much error is introduced by this approximation. It would have been nice if the adjusted value of C~) gave a value of C3 in agreement with the Lifshitz formula (see end of Sec. III C). We have, of course, tried such a value of C6, and attempted to adjust Cs and CIO in order to obtain agreement with experiment. However, it was not possible to achieve such a good fit of the experimental bound state energy levels as shown for Cs = 0 and C 10 0 in Fig. 2. Experience with atom-atom potentials shows that a fairly good potential can be obtained by C6

alone, provided that one uses a value of C6 larger than the correct value. Apparently we have the same situation here. It is also possible, however, that there are significant corrections to the Lifshitz formula at intermediate ranges, in other words the values of C4 , Cs etc. for the surface-atom interaction are not negligible in the region of the potential minimum. Jiang, Toigo, and Cole92 have calculated a value of Cs = 0.0447 a.u. = 50.5 meV As for He-LiF (001) deriv~ from the surface dipole-atom quadrupole dispersion interaction. With z = 1.00 A and Zmin = 3.04 A from Table III we obtain a contribution of - CS/(zmin - Z)5 1.43 meV to the potential at the position Zmin of the minimum of V 00' This means an increase of the well depth by the order of 20%.

(8) The summation of the atom-ion two-body potentials has been performed assuming the positions of the crystal ions to be fixed. An approximation of the correction to be applied to the sum of pair potentials due to the thermal motion of the crystal ions has been derived in Sec. 5 of Ref. 33. For the Fourier component VG(z) of the sum of atom-fixed ion potentials this correction yields the thermally averaged potential

VG(z) = VG(z')exp - z dz', (6.3) e G'(u~)12 J+ 00 [(Z')2] ~21T(U;) - 00 2(u;)

where (u; ) and (u;) are the mean square displacements of the ions parallel and perpendicular to the surface. If (u;) is small compared to the square of the width of the potential well, Eq. (6.3) can be approximated by

- G2

(U2)/2 [ 1 d

2 ] Vc(z) = e x VG(z) +"2 (u;) dr VG(z) . (6.4)

Taking calculated values for (u~) and (u;) at a surface temperature of 300 K from Table II of Ref. 93, Eq. (6.4) decreases both the potential well depth and the corrugation amplitude by about 3%.

(9) The assumption of pairwise additivity excludes three-body forces from our potential model. Hutson and Schwartz46 have investigated the sensitivity of He-Xe coated graphite resonant elastic scattering to three-body forces, which they have included via an Axilrod-Teller triple-dipole term. The contribution of this term was estimated by them to yield an increase of the repulsive potential by the order of 0.5 meV in the region of the minimum, corresponding to a decrease of the well depth by about 5%.

(10) As a consequence of nonlinear surface effects a static dipole moment can be induced in the atom as shown by Galatry and Girard.94 Interacting with the electric field of



2520 Celli sf al. : Potential for He-LiF(001)

the surface ions this dipole moment may cause a correction to the potential which, however, is estimated to be negligible in typical cases of atom-surface potentials.94 Similar theoretical approaches have been used by the same authors to derive a spontaneous polarization of the free surface95 and a surface polarization induced by the adatom.96 It is very difficult to estimate the small effect of these mechanisms on the atom-surface potential, since their contributions are only indirect.

In conclusion, the semi ab initio potential of Sec. III C provides a good description of the elastic scattering process for He-LiF(OOl), as shown in Sec. V. The same potential has also been used for calculations of resonant phonon inelastic scattering with equally good results.97

In view of this success it is of interest to consider the application of the same simple sum of pairwise potentials to the construction of noble gas-surface potentials for other insulators, semiconductors, and even for metals. In general, the surface should be divided into appropriate subunits, which may be atoms, ions, molecules, or electron orbitals, each interacting separately with the impinging atom. Obviously, the method can be extended to compute the interaction of molecules with surfaces. The basic idea is to treat the solid as a large molecule and to make good use of the vast amount of work done by quantum chemists on interatomic and intermolecular potentials.98 An alternative approach that is currently fashionable is to regard the surface as the termination of a perfect solid, which is then often treated as a weakly inhomogeneous electron gas. We view the two approaches as complementary, as is generally the case for tight binding schemes vs nearly free electron models.

ACKNOWLEDGMENTS

We thank R. Ahlrichs and H. J. B6hm for providing us with results of their calculations of the first order SCF HeLi+ and He-F- potentials. We are grateful to A. A. Maradudin for assistance in solving the sums in Sec. II and to W. Diercksen for discussions on the He-F- potential.

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Celli et a/. : Potential for He-LiF(001) 2521

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Documents

Pairwise additive semi ab initio potential for the elastic scattering of He atoms from the LiF(001) crystal surface