Z. Physik B 36, 245-250 (1980) Zeitschrift for Physik B © by Springer-Verlag 1980

Inelastic Atom-Surface Scattering: A Comparison of Classical and Quantum Treatments*

R. Sedlmeir and W. Brenig

Physik-Department, Technische Universit/it Mtinchen, Garching, West-Germany

Received October 4, 1979

The validity of the classical approximation is investigated for the calculation of the energy transfer Ae and sticking coefficient of particles scattered by harmonic solids. If in zeroth order the static approximation for the force between particle and solid is valid, the energy transfer can be calculated classically, if the WKB approximation for the matrix element of the force holds. This is the case except at low impact energies. The characteristic energy below which quantum effects become important depends strongly on the long range behavior of the force. For the sticking coefficient one has to distinguish quantum effects of particle and solid. Quantum effects for the particle can again be neglected if the WKB approximation for the matrix elements is valid. The solid, however, can be treated classically only, if the number of excited phonons is large compared to unity. This requires energy transfers At large compared to the Debye energy of the solid.

1. Introduction

After the pioneering work of Lennard-Jones and Strachan [1] on the exchange of energy between gas atoms and solid surfaces a large number of investi- gations of gas surface scattering appeared. The theoretical literature up to 1971 is covered by a review article [2] of R.M. Logan. Of the more recent articles let us mention a few [3-5]. The situation of the theory to our knowledge is still pretty well characterized by a quotation of Logan's article: "Unfortunately there is no single unified theory of gas-surface collisions which deals in a manageable way with all the features of interest. In fact, there is a great abundance of theories in current use, ranging from very simple to complicated, some using classical mechanics and some using quantum mechanics. This vast array of theories is a source of some confusion to people working in the field, not to mention outsiders. Some of these theories are approximately valid, or at least useful, in certain regimes; some of them are not."

* Extract from doctoral thesis of R.S. submitted to Fakultgt fiJr Physik, Techn. Univ. Mttnchen, 1979. Work supported in part by DFG, Sonderforschungsbereich 128

Although there exist a number of comparisons of classical and quantum treatments there are usually additional approximations introduced with, for in- stance, the quantum treatment which obscure the results. We therefore thought it would be interesting to have a comparison of classical and quantum calcu- lations in which everything else is kept fixed. We are going to do the comparison for a number of different potentials and it will turn out, that the results depend strongly on the potentials. The results will be com- pared for two quantities of experimental interest: The average energy transfer in a single collision (closely related to the energy accomodation coefficient) and the sticking coefficient.

2. The Average Energy Transfer

In calculating the average energy transfer from the gas to the solid one has to distinguish situations with and without trapping. The complete treatment of the trapping case is most conveniently done using kinetic equations, while the case without trapping is more or less a mechanical problem. In classical mechanics this

0340-224X/80/0036/0245/$01.20

246 R. Sedlmeir and W. Brenig: Inelastic Atom-Surface Scattering

case corresponds to a single round trip in the in- teraction potential V between gas atom and solid, which we assume to be given by a sum of two body interactions

v = V ( x - r.) (1)

where x is the position of the gas particle and r, of the n-th substrate atom. The energy transfer Ae to the substrate is given by the work done to the solid

Ae= o~ Zfim(t)ri~(t) dt (2t - - c X )

where fi,,(t)=c~V(x(t)-rm(t))/c~xl is the i-th com- ponent of the force on the m-th substrate atom. If the solid is treated in harmonic approximation, there exists a linear relation between r,,(t) and the forces f,(t):

rim(t) = FOre (t) -[- 2 ~ )~im, in( t -- t') fj,(t') dt' (3)

with the dynamical susceptibility )~im, j~(t) of the lat- tice. Here o r~(t) describes the unperturbed (thermal) motion of the lattice. The expressions can be simplified considerably by assuming a head on collision with a single substrate atom, i.e. assuming vertical incidence of the gas par- ticle on a single atom and sufficiently short ranged potentials so that only one force component, say fl0 survives. One then can omit all indices and write (2) after inserting (3) as

(Ae)=<~f( t ) i . ° ( t )dt+~f( t )2( t - t ' ) f ( t ' )d tdt '> (4)

where the brackets <) denote averaging over the thermal motion of the substrate. The determination of f(t) in general, of course, still amounts to a complete solution of the coupled equa- tions of motion. In the limit of small mass ratio # =mg/m s of gas and substrate particles, however, a perturbation procedure is possible, in which to lowest order only the motion of gas atoms in a static poten- tial is considered. In our comparison of classical and quantum results we shall use the same static potential and not introduce any differences by averaging the original potential over thermal and/or zero point motions of the substrate. For a static potential the first term of the r.h.s, of (4) averages out to zero, the second term can be sim- plified by Fourier transformation to

<A ~> = ~ ] f(co)l 2 co Z"(c9) dc9/2 ~ (5)

where Z"(co) is the absorptive part of the dynamical susceptibility.

An expression similar to (5) is obtained in a quantum mechanical first order distorted wave Born approx- imation. We write it as

< Ae> = ~ If~,vlZ C% ~ Z"(c~ ~,~) Acod2 ~z (6) v

Here co~, ~ = co~- co~ = ( e - e,)/h is the transition fre- quency from the initial scattering state le) to the final bound state iv>. In general there will be contributions from inelastic scattering events to final scattering states which have to be added to the r.h.s, of (6). Ac% =(e~-e~_l)/h is the transition frequency between neighboring bound states and

f~,v=<e[flv> (7)

the single particle part of the transition matrix ele- ment. The notation in (6) has been chosen so as to make the analogy between (5) and (6) as close as possible. In particular, the normalization of the states Iv), le> in the matrix element (7) is chosen such as to lead to a semiclassical (W.K.B.) approximation

f~,~ = f(co~, ~) (8)

where f(co) is exactly the Fouriertransform of the classical force occuring in (5). The first order distorted wave Born approximation which so far has been used to. derive (6) allows only for the emission or absorption of a single phonon. The validity of (6) is more general, however. It turns out that the condition for the validity of (6) is that the force f(t) can be approximated by the static calcu- lation described above. This requires an energy trans- fer Ae to the snbstrate sufficiently small compared to the energy of the scattering particle in the attractive well which is of the order o r e + V 0 (V 0 the well depth). We shall come back to this point in a forthcoming publication [6] and contend ourselves at present with the statement, that (6) can be valid, even though (Ae) is considerably larger than the maximum energy h o D of the phonon spectrum of the substrate [7].

3. The Sticking Coefficient

The situation is quite different for the sticking coef- ficient. It depends strongly on the number of phonons participating in the scattering event. Suppose, for instance, the probability of exciting two and more phonons is small, then the energy per phonon trans- ferred to the solid is hco~,v and the factor of this quantity in the r.h.s, of (6) is the probability of a transition from hco~ to hco~. The corresponding en- ergy distribution after the transition thus is given by

p(e)=~lf~,~lz)('(c%~)Ae)v6(e-ho~v)/2~h+... (9) v

R. Sedlmeir and W. Brenig: Inelastic Atom-Surface Scattering 247

Here the dots after the plus sign on the r.h.s, of (9) indicate contributions from final continuum states occuring in inelastic scattering events which are not contained in the sum over bound states v. In semiclassical approximation the matrix elements in (9) are replaced by the r.h.s, of (8). One may also go beyond this and neglect the quantization of vibra- tional states• Replacing the sum in (9) by the cor- responding integral and expressing the final state energy ~ in terms of the initial state energy hco~ and the phonon energy ha) as e=hco~-hco one then ob- tains

P(hco~-hco) = I f (co)l 2 )('(CO)/2 n h (10)

Although f(co) and )~(CO) are classical quantities, the quantum nature of the solid manifests itself strongly by the factor 1lb. An entirely classical approximation results, if the energy Ag transferred to the solid is large compared to the maximum energy hCO D of the phonon spectrum, since then a large number of phonons have to be excited. Quantum fluctuations then may be neglected and the energy distribution P(e) can be approximated by

P(e) = (~(e - A e - h co~). (H)

In order to distinguish (10) from (11) let us denote (10) as "continuum approximation" and (11) as "classical approximation". The zero temperature sticking coefficient can be ob- tained from P(e) by integrating over all final energies below zero

0

s(g)= 5 P(d)dd (12) oo

At nonzero temperature it is not sufficient only to take the temperature dependence of P(e) into account [8]. One rather has to solve a kinetic equation in- volving "upward" processes in addition to the "downward" processes described by P(e). We restrict ourselves to zero temperature.

4. Numerical Results

The foregoing discussion has led to two ingredients of the theory: The susceptibility Z(CO) being a property of the harmonic solid, and the matrix element £,v being a property of the scattering particle in a static potential (V) . With these two ingredients one may distinguish four different approximations: 1. The complete quantum mechanical one phonon theory, 2. The semiclassical approximation (8) for the matrix elements, 3. The

continuum approximation (10), in which the scatter- ing particle is treated completely classically (quanti- zation of vibrational states being neglected), but the solid is treated quantum mechanically, and 4. The fully classical treatment (11). Since we are more in- terested in a comparison of classical and quantum results for (A~) rather than in absolute values we contend ourselves with the Debye approximation for the susceptibility leading to

)('(co)=3rtco/2msco~; Ico[ <COD (13)

and Z"(CO)=0 for Ico[>co o- One then is left with the calculation of the matrix element £,v which, of course, depends on the poten- tial ( V ) = V(x)• For V(x) we choose the following models

oo, x / l<O V(x)= - D , O < x / l < l (square well) (14.a)

O, x/l < 1

V(x)={~, l (x / l )2-1j , X<lx>l (14.b)

(truncated oscillator)

V(x) = D (e- 2 xl, _ 2 e xll) (14.c)

(Morse potential)

V (x) = D [ 4(x/l)- 2n _ 4(x/l)n]. (14.d)

The potential (14.d) can be treated only numerically. It is of interest in particular for n= 3, since it repre- sents the correct van der Waals type long range behaviour. For the numerical results we have chosen D =0.1 eV for all potentials. At room temperature one then has e~D. The length parameter l has been chosen as 1 = I A for (14.b and c). For (14.a and d) l has been chosen so as to yield the same eigenfrequency of vibration, the classical eigenfrequency of a square well being defined as the round trip frequency of a

particle with e=0, i.e. C O o = r C 2 l / ~ / l . The parameters can be considered representative for the situation of the precursor state of molecular hydrogen on tungsten. The Fourier transform f(co) of the force f ( t ) for the potentials (14.a.b.c) can be determined analytically. One finds

f (co) = 2]/~nD- [2 ~ e/D - 2 (]/1 + e/D - ] f ~ )

• cos(re CO~coo)] (15.a)

f(co) = 21 /~D ~ { sin [(zr - el/~)/(co/co°co/coo - 1 - 1)]

sin [(~ -1/~)(CO/COo + 1)] ; (15.b) -~ CO/COo + 1 J

248 R. Sedlmeir and W. Brenig: Inelastic Atom-Surface Scattering

Flco}

/ ..\. / . ." ~ . .

•

S-':"'" --2". ~__ 2 co 0

Fig. l a. Low frequency behaviour of f(e)) for impact energies e =0.01D. (Morse), - . . . . . . . . (truncated harmonic Oscillator), - - - (square well)

IFILo) 12 2mD

, , ,

1 2 3 u~ 0

Fig. lb. Comparison of If(co)l 2 for the potentials (14a, b, c). Upper curve: Square well, middle curve: truncated harmonic oscillator, lower curve: Morse potential

FIoJ)

2

1 n =2

I I 1 2 3 u) 4

coo Fig. It. f(o)) for the potentials (14.d) with n=2, 3,4 and the Morse potential

rico) = 2l/~D(2nco/COo) cosh [(n - 1 /~)co/(COol/~) ] sinh [nco/(COo l / ~ ) ]

(15c) respectively. A comparison of these functions and their squares is shown in Figs. la and 1 b.

Although the general trends of the results for the three potentials a, b, c are similar there are marked differences in detail, for instance in the low frequency region. We have also done a numerical calculation of f(co) for the potentials (14.d) with n = 2, 3, 4. The results are shown in Fig. l c. They exhibit rather good agreement with the Morse potential results. From now on we therefore restrict ourselves to the "van der Waals" case n = 3 in the discussion of the potentials (14.d). Let us now consider the quantum mechanical matrix element of the force as introduced in (7). We chose the "WKB-normalization" explained in the context of (7). The result for the Morse potential can be given in closed form [1, 8] as

If~, vl2 = 7r (pZ __p2)2

Ir(d + l/2 + ip)l 2 sinh (2 rcp) F ( 2 d - v) F(v + I) cosh (2 rip) + cosh (2 n d)

Here we have chosen the notation of [8], i.e. d

=1 2]~]/~D/h, p=l 2]/2~/h, p ~ = l ~ / h . Figure 2a shows a comparison of (16) (upper curve) and the corresponding classical value (15.c) for a transition to the ninth vibrational state in a Morse potential with d=10 (D=0.1 eV, / = I A ) for various impact energies. Again a numerical calculation of the matrix elements for the "van der Waals"-potential (14.d) with n=3 shows good agreement (Fig. 2b) with the Morse potential result except at very low impact energies, a case that will be discussed below. The average energy transfer for our model system can be calculated from (5). One finds

(A e) ~0.12hc%. (17)

The one phonon approximation therefore can be expected to be reasonable. We can now compare (see Fig. 3) the correct quan- tum mechanical calculation with three approxi- mations in which an increasing number of (different) quantum effects are neglected. Firstly, the single particle states are treated in the WKB approximation. (We chose the Morse potential for our comparison). The sticking coefficient then no longer exhibits the correct low-energy behavior [9], but approaches a constant (not unity) at low energy. Secondly, the adparticle motion is treated completely classically. The dominant effect of this approximation as compared to WKB is the disappearance of the discrete vibrational levels in the final state producing the steps in the dependence of the sticking coefficient

(16)

R. Sedlmeir and W. Brenig: Inelastic Atom-Surface Scattering 249

8 ̧

6

4

2

IFIco)I 2 2mD

o.s _E D

- IFI~)I2/2mD

"'... ' lb~ g g /. 3 ~, i

o ~ o

w/w o I

6 6N

D

Fig. 2. a Comparison of quantum (upper curve) and classical matrix element of the force f between a continuum state and the ninth

vibrational state for a Morse potential with d=l]/2mD/h= 10. b Matrix elements of the force between a continuum state (e/D =0.1) and the vibrational levels for the "van der Waals" potential (dots) as compared to classical approximation for the Morse potential (solid line)

as a function of the energy. These steps, however, are a consequence of the use of the Debye approximation for the phonon spectrum (with a sharp cutoff) and of the neglect of a nonzero width of the vibrational levels of the adparticle, and would be smeared out in more realistic calculations anyway. Finally, the substrate is treated classically as well. This leads to a drastic change in the sticking coef- ficient. It is now zero down to an energy ~=Ae and then rises discontinuously to unity whereas it starts to rise continuously to a value of the order of Ae/hc% in a one phonon calculation, if the substrate is treated quantum mechanically. The energy integrated stick- ing coefficients in both approximations are the same, since they are given by the average energy transfer Ae which is much less sensitive to quantum effects than the differential sticking coefficient itself. Let us come back to the low energy behavior. Ac-

cording to (16) the sticking coefficient vanishes ,--1/~ for small impact energies (compare Fig. 3 a). Figure 4 shows the difference between Figs. 3a and 3b in more detail. It demonstrates that for the Morse potential the quantum mechanical deviations from the WKB

Fig. 3, The dependence of the sticking coefficient s for substrate temperature T,=0 on the energy e of the incident particle. Parame- ters of the underlying model are chosen so as to represent the situation for the precursor state of Ha/W (further details of the model in section 4), c%=Debye frequency of W. Bottom ("quan- tum"): Result of a fully quantum mechanical first order distorted wave Born approximation. Upwards: Approximations neglecting an increasing number of quantum effects, ("discrete"): WKB ap- proximation for initial and final state of adparticle, substrate quantum mechanical, ("contin."): Adparticle classical, substrate quantum mechanical, ("classical"): Adparticle and substrate classical

s (T s O)

classical

0.1 1

0'2 t ~ contin

1

0 . 2 ~ ~ J ~ ~1 ~ ~ -- ~ discrete

o.21r L

quantum [.

e/hw o 1

250 R. Sedlmeir and W. Brenig: Inelastic Atom-Surface Scattering

0A

$

0.3

0.2

0.1 f [ ~ i i t I I I I

E ~K ~B

Fig. 4. Comparison of the two lower curves of Fig. 3 ("quantum" and "discrete") in more detail

result occur at impac t energies be low 1K, a region which, so far, has not been invest igated exper imenta l - ly. Moreover , for more long ranged potentials , such as the " v a n der Waa l s " potent ia l we could not detect vanishing sticking coefficients down to impac t en- ergies of 1 0 - S K . This is in agreement with calcu- lations of G o o d m a n [10].

5. C o n c l u s i o n s

We have invest igated the validity of classical approxi- mat ions for the calculat ion of energy transfers and sticking coefficients of particles scat tered by har- monic solids. As long as the static app rox ima t ion for the force between scat tering particle and solid is valid, the average energy transfer can be calculated classically, if the W K B app rox ima t ion for the matr ix elements of the force holds. This is the case except at

low impact energies. The characteris t ic energy below which q u a n t u m effects become impor tan t depends s t rongly on the long range behavior of the force. Fo r the sticking coefficient one has in addi t ion to distinguish q u a n t u m effects of the particle and the solid. Q u a n t u m effects of the particle can be neglect- ed again, if the W K B approx ima t ion for the matr ix elements holds. The solid, however, can be treated classically only, if the n u m b e r of excited phonons is large c o m p a r e d to unity. This requires energy trans- fers Ae large c o m p a r e d to h~o D.

R e f e r e n c e s

1. Lennard-Jones, J.E., Strachan, C.: Proc. R. Soc. (London) Ser. A 150, 442 (1935) Strachan, C.: Proc. R. Soc. (London) Ser. A150, 456 (1935)

2. Logan, R.M.: Solid state surface science, Vol. 3, pp 2-97, New York: Marcel Decker 1973

3. Goodman, F.O.: Prog. in surf. science, Vol. 5, pp 261-375 (1974)

4. Goodman, F.O., Romero, I.: J. Chem. Phys. 69, 1086 (1978) 5. Lin, Y.W., Adelman, S.A.: J, Chem. Phys. 68, 9 (1978) 6. Brenig, W.: Z. Physik, in print 7. This is in agreement with Gilbey's conjecture. (Gilbey, D.M.: J.

Phys. Chem. Solids 23, 1453 (1962) 8. Goodman, F.O.: Surf. Sci. 24, 667 (1971) 9. Brenig, W.: (to be published)

10. Goodman, F.O.: J. Chem. Phys. 5, 5742 (1971)

R. Sedlmeir W. Brenig Physik-Department Technische Universit~it Mfinchen - Theoretische Physik - James-Franck-Stral3e D-8046 Garching bei Mfinchen Federal Republic of Germany