Resonant inelastic scattering of atoms from surfaces

surface science ELSEVIER

Surface Science339 (1995) 205-220

Resonant inelastic scattering of atoms from surfaces S. M i r e t - A r t 6 s * Instituto de Matem~ticas y Fisica Fundamental, CSIC, Serrano 123, 28006 Madrid, Spain

Received 15 February 1995; accepted for publication 19 May 1995

Abstract In this paper, we present a new formulation of inelastic scattering of atoms from surfaces within the close-coupling formalism using a classical description of the surface motion. In this formulation, the diffraction channels exhibit not only moving thresholds but are also shifted (or dressed) by amounts of energy equal to the excitations of the active mode, taking into account all the creation and annihilation processes accesible by energy conservation. These inelastic diffraction channels can be arranged in blocks (Floquet blocks) differing by one quantum. When a continuum model for the surface and a soft potential including an attractive well is considered for the interaction, good agreement with the thermal attenuation of the specular and diffracted intensities and resonance line shapes is obtained compared to the experimental ones. Within the single-phonon approximation, multiphonon contributions of the same excited mode are also analyzed for non-resonant, resonant and critical scatterings. Application to the weak corrugated metal surfaces Cu(ll0) and Cu(ll3) is presented and discussed. Keywords: Atom-solid interactions, scattering, diffraction; Copper

1. Introduction Many efforts are being addressed to the elucidation of gas phase-surface interaction. Thanks to the large variety and richness of experimental data, together with an adequate theoretical framework, knowledge of this interaction is improving continuously. In this theoretical framework, and in order to describe elastic a n d / o r inelastic scattering, thermal averages over surface temperature are of fundamental importance and are at the origin of the D e b y e Waller (DW) factor. This factor arises when the atomic displacements of the lattice are taken into

* Corresponding author. E-mail: [email protected].

account explicitely. For atom-surface scattering, these displacements have been interpreted by means of a proper many-body statistical average. For the elastic case, a thermal average of these displacements was implemented leading to modifications of the potential parameters according to the theory developed by Cabrera et al. [1]. An ulterior extension to inelastic scattering was given by Manson and Celli [2] where the reflection coefficient was expressed in terms of a D W factor by assuming a distorted wave Born approximation to the T-matrix. This is equivalent to neglect of multiple scattering. The theoretical methods developed in the seventies were applied mainly to very simplified models of the gas-surface interaction. Thus, for example, Beeby [3] solved formally the scattering of an incident atom by a surface considered as a moving infinitely repul-

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S. Miret-Art~s / Surface Science 339 (1995) 205-220

sive step inside a stationary, attractive well, slowly varying in space. An eikonal approximation was applied by Garibaldi et al. [4] for solving the scattering of atoms by a hard corrugated wall where the corrugation or shape function was assumed time-dependent through the thermal motion of the wall. A further study along this line was carried out by Armand and Manson [5]. On the contrary, Wolken [6] chose to use a phonon description of the solid within a non-perturbative treatment, the close-coupling (CC) method. Due to the large number of accesible states of the solid, and in order to get tractable the problem numerically, the phonon quantum number was replaced by a continuous variable, an isotropic Debye model for the solid was considered and, before performing the calculations, averages over phonon directions and polarizations were done. With all of this, but neglecting diffraction and within the single-phonon approximation, only qualitative agreement with the experimental results was obtained. In a way, this work showed the inadequacy of this formalism to treat this scattering; the number of relevant coupled channels was so important (in order to obtain numerical convergency) that the very high time consuming of this kind of calculations prevented them. In fact, very few theoretical studies within this formalism were published after, except for the works by Chow and Thompson [7] which also introduced an optical potential for describing the inelastic scattering according to the first suggestion given by Hamauzu [8]. But, in any case, surface temperature effects were always introduced by means of a DW factor. Due to a lack of a well established DW theory for this scattering, Levi and Suhl [9] discussed within what limits this theory remains applicable and proposed new limits to go further. Phenomenological DW factors were also calculated in the literature introducing the Beeby correction [3] and what is sometimes called Armand effect [10]. Doing this, effective surface atom mean square displacements were reported for different surfaces. Nowadays, experimental work is still based on the assumption that the thermal attenuation of diffraction intensities is given by this factor [11]. Moreover, from the experimental data, particularly for He atoms scattered from metal surfaces [12], strong deviations from the linear behavior were observed by plotting in a logarithmic scale the diffraction intensities ver-

sus the surface temperature, being much more pronounced at high temperatures. This fact pointed out that it was inconvenient to attempt to represent this attenuation by using a DW factor. Armand and Manson [13] presented a fully quantum-mechanical theory of the inelastic scattering of low-energy atoms from surfaces based on the perturbation series and an expansion in the number of phonons exchanged. Further extensions to this work were also carried out by the same authors [14] including the multiphonon contribution and applied to flat metal surfaces working quite well. More recently, Manson and Celli [15,16] have presented very general treatments of inelastic scattering for the case of defects and adsorbates on a surface and flat surfaces filling the gap between the zero- and one-phonon contributions and the multiphonon one. Also, Stiles et al. [17] have developed a CC formulation in the one-phonon approximation for surfaces with weak corrugation. Finally, time-dependent treatments for inelastic scattering [18,19], where no explicit DW factor is needed, have been also reported in the literature. Inelastic scattering of thermal helium atoms from crystal surfaces has also proved to be a powerful method for the characterization of low energy vibration excitations of the surface and, in general, of their dispersion curves [20]. In the angular distribution patterns, additional to specular and diffraction peaks and selective adsorption resonances (SARs), many sharp maxima and minima have been explained by different mechanisms such as the kinematic focusing and Van Hove singularities. The corresponding kinematical conditions have been derived using diagrammatic schemes such as Celli diagrams and those representing scan curves a n d Rayleigh wave dispersion curves. These experimental results have turned out to be a good test of lattice dynamical calculations. Another very important aspect of this inelastic scattering is the analysis of resonance features. Experiments using quasi-monoenergetic beams and well characterized surfaces showed a very important variety of results about manifestation of SARs [21,22]. The elastic theory of resonant scattering was shown to explain the occurrence of minima, maxima and mixed extrema [23-25] by establishing some general rules for predicting them. However, it is well known that these rules fail many times. In particular, they

S. Miret-Art£s / Surface Science 339 (1995) 205-220

were derived for weak coupling conditions and hard model potentials. Recently, we have shown [26] starting from a CC formalism that, due to the moving threshold multichannel nature of this scattering, any attempt in that direction will lead to rules of limited applicability since resonance signatures are very dependent on the incident parameters of the incoming atoms, the surface structure and lattice vibrations. Moreover, relatively few experimental studies exist on the surface-temperature dependence on resonances [27-29]. Thus, inelastic effects were invoked in order to change signatures following the Hutchison [30] method and that proposed by Mantovani et al. [31], applied to a hard wall potential and a soft potential, respectively. They are based on introducing inelastic effects by scaling the scattering amplitudes by means of a DW factor. According to these authors, the procedure employed was justified as a consequence of the good results obtained when comparing with experimental results. This procedure has not always been completely successful [32]. Very recently, measurements [33] of resonance linewidths for the Cu(115) vicinal surface as a function of temperature have been reported. This work has been accompanied by some theoretical calculations of these widths when one-phonon contributions are considered. However, no theoretical line shapes have been reproduced for different temperatures. The authors concluded that the width and signature of such resonances at low temperatures were strongly affected by disorder on the surface and much less so by inelastic effects. Phonon-assisted resonances with surface bound states were also shown to have dramatic effects on the atomic time of flight spectra [34-36]. In this case, an enhancement of resonance features was observed. New enhancement mechanisms could be envisaged by taking into account the kinematical focusing effect together with the resonance condition [37]. In this paper, a new general formulation of inelastic scattering, resonant or not, of atoms from surfaces within a CC formalism using a classical description of the surface motion will be presented and discussed. We think that this approach opens new perpectives of understanding this complex problem since very few attention has been concentrated on the corresponding theoretical developing. In this new formulation, the problem is rendered in a more

207

tractable way than that presented by Wolken. It is valid for general realistic potentials and no explicit introduction of a DW factor is needed. This new analysis involves a treatment of the elastic and inelastic processes on an equal footing. Moreover, this theory could account for multiphonon contributions as a result of several single-phonon transitions and when only one mode of the lattice is active. Generalization to more active modes could be easily extended although numerical implementation of this would be more difficult. This theory is based on the same backgrounds as that developed for multiphoton processes. In particular, it assumes that the total Hamiltonian is periodic both in space and in time leading to a scheme of coupled channels where now the diffraction channels are shifted in energy by an amount corresponding to the creation or annihilation of one or more phonons. In the literature of molecular fragmentation, it is said that the channels are dressed by the photon field. Here, these dressed channels would represent the inelastic diffraction channels. The number of these channels dressed by a given number of phonons form a block, called a Floquet block [38,39]. Thus, if only single-phonon scattering is considered, three Floquet blocks must be included in the calculations; the block dressed by minus one phonon describing the creation process, that dressed by zero phonons for the elastic contribution and that dressed by plus one phonon describing the annihilation process. The number of diffraction channels within a given block is formed at least for those used for obtaining numerical convergency in an elastic calculation. Furthermore, two couplings of different nature are now present: intrablock and interblock couplings. As we will show later, these inelastic channels are going to be responsible for the attenuation of the specular and diffracted intensities through the interblock coupling. No assumptions about the thermal attenuation are required a priori. Only, for a given surface temperature, we average over the frequencies of the active mode by means of the density of modes, the maximum frequency being given by the Debye frequency. For describing a resonant scattering, the same procedure is used and we compare our theoretical line shapes (position, width and signature) with the experimental ones observing changes in the signature and the correct behavior of the widths as a function of the surface

S. Miret-Art~s / Surface Science 339 (1995) 205-220

208

temperature. The He-Cu(113) system will illustrate very well this purpose since detailed experimental and theoretical information is available. Moreover, a resonant scattering under special conditions of incidence, the so-called critical resonant scattering, proposed by us elsewhere [26], has also been studied with the aim of stimulating experimental work since this analysis has a predictive value. In particular, this scattering was studied at elastic conditions for the Cu(ll0). The resonance widths at 21 meV were too small to be experimentally detected. With the critical incident conditions proposed, they could be easily detected. Here we are going to extend these calculations to the inelastic case. At critical conditions, the line shapes do not follow the typical behavior (Lorentzian- or Fano-type function). They exhibit different shapes predicting at the same time a very important enhancement of their corresponding widths (the critical kinematic effect, CKE).

where Q specifies a parallel phonon wave vector with frequency w~(Q), v designates the different modes for each Q value and B(Qv, T) the amplitude of this motion, including the phonon polarization vector and the dependence on the surface temperature. For most practical purposes, only displacements of atoms on one layer (or at most two) significantly contribute to V. As this displacement is generally small compared to the lattice constant, the interaction potential could be developed in a Taylor series as

V(r, t) =V(r) +u(R, t) • VV(r).

(3)

With respect to the static part of the interaction, V(r), due to the periodicity of the lattice surface, this function can be expanded in a Fourier series V(r) =

EVa(z)

exp (iG

.R),

(4)

G

2.

Theory

Consider the inelastic scattering of a gas atom from a moving corrugated periodic surface. If the incoming atom is a structureless particle of mass /z, the corresponding time-dependent Schr6dinger equation can be written as

ih

8tlt( r' t) ( h2 ) 0------~- - ~ Vr2 + V(r, t) ~(r, t),

where the 3D-vectors are denoted by lower case letters and capital letters will be used for vectors which are to be parallel to the surface ( r = (R, z)). Following the standard notation, the z-direction is chosen as the outward normal to the surface. In the model assumed here for the surface, the gas-surface interaction V turns out to be dependent on time through the instantaneous position of the surface atoms, R + u(R, t), u(R, t) being the deviation from equilibrium position. From the layer description of lattice dynamics, it is well known that this u-displacement can be written as cos Q - R

cos

~(r,

t) = e x p ( - i E / h )

E

t,ba+Q.,o,.(z)

Q,u,nQ~,G

x e x p [ i ( K + G + Q) • R]

(:)

u(R, t) = ~'~B(Qv, T)

G being the 2D-reciprocal lattice vector. The total wave function qr(r, t) has to take into account the double periodicity given by the total Hamiltonian, with respect to r and t. Thus,

~%(Q)t,

Q,v

(2)

×exp[inQ~W~(Q)t],

(5)

where nQ~ stands for the number of phonons of the mode ( Q u ) and tpa+Q.ne(z) for the diffracted wave in the direction G + Q. In Eq. (5), E is the total energy of the particle, its incident wave vector is denoted by k with parallel and perpendicular components to the surface given by K ( = k sin 0(cos qS, sin qb)) and kz(= k cos 0). The set of these three variables (E, 0, 4') accounts for the scattering geometry. Now, we assume that only one mode is active in this dynamics, say (Qu). If in Eqs. (5) and (2) the different modes are replaced by (Q'v') and (Q"u"), respectively, we can obtain the following CC-equations for the diffracted waves after substitution of Eqs. (2)-(5) into Eq. (1), premultiplication by

S. Miret-Art& / Surface Science 339 (1995) 205-220

exp(-inQ~o~(Q)t) and e x p [ - i ( K + G + Q) • R] and integration over t and the unit cell,

2----~-~" dz + e~+Q'nQ"(E' O, oh; w,(Q)) - Vo(z )

+

x

G'#:G

F(G-G'-T-Q', z)

G'

•[F(G-G'-Q,

z)+F(G-G'+Q,

• (~G,+Q,ne.+1(Z ) + ~,+Q..o_,(Z))

z)]

(6)

and 02

2---~------g dz + EG+Q'nQv+I(E' O, ¢~; ~%(Q))

-- Vo( Z) )

I( Z)

E G ' 4 ~G

1

+ ~ E B ( Q u , T) • [F(G - G ' - Q, z) G'

+ F ( G - G'+ Q, z)] • (~b~,+Q,,Q~+z(Z ) + tpC,+Q,,Q~(Z)).

(7)

The 1,/4 factor comes from the exponential representation of the two cosine functions in Eq. (2). Integration over t leads to the appearance of ~QQ'6v'v ~n'~z~z~n~ for the diagonal elements, that is, when the integrals do not contain the term u(r, t). V V(r); on the contrary, when this term is involved in the corresponding integrals the result is given by 6Q.Q, 6Q,Q 6u,, v, 6v, v 6 n,Q~n~ ± ± 1. . The z-component kinetic energies of the diffraction channels are gwen by the function EG+Q,nQv(E , O, ~);

wv(Q))

= ..+Q(E, O, c~) - nQ~hw~(Q) = E - (h2/21.z) l(21,zE/h2) 1/2 ×sin 0(cos 4', sin 4~) + G + Q I 2 -

These channels are represented by effective potentials formed by Vo(z) (the laterally averaged potential or bare potential) plus an asymptotic energy depending on (E, 0, qS) (moving thresholds) as in the elastic case [26] plus an amount of energy due to the presence of phonons shifting in energy these effective potentials (dressing picture). Finally, the contribution of the gradient of the interaction potential is expressed through the vector function F(G -

G'-T-Q', z),

1

+ ~ E B ( Q v , T)

=

209

nQ~hO~(Q).

(8)

= [i(G-G'-TQ')V~_~,~Q,(Z), V~_G, TQ,( Z)],

(9)

where V~_~,~Q,(Z) represents the first derivative with respect to the z-variable. The two signs Taccompanying Q come from the integration over R including the argument of the cosine function, Q • R, which is present in the integrand. The structure of Eqs. (6) and (7) is very similar to the elastic case [26]. These equations describe the most general inelastic scattering of atoms from surfaces within the single-phonon approximation (Eq. (3)). Attenuation of diffracted intensities and resonant scattering assisted or not by phonons could be treated from these equations. Interesting aspects of this scattering can be drawn out from a close inspection of these equations. Thus, two different couplings are distinguished. The first one is the same as that for the elastic case, V~,_a(z), and valid only for the diffracted waves with equal value in the number of phonons (nQ~hwv(Q)). In other words, all the Gchannels dressed by the same number of phonons can be considered to form a block with inner couplings Va,_a(z) (with G 4= G') and we could call it intrablock coupling (those dressed by zero phonons form the elastic block). For different values in the number of phonons, we have the second coupling which will be responsible for attenuation because it involves the coupling among the inelastic channels and, therefore, the corresponding diffracted intensities will decrease. This attenuation will be more important as the phonon frequency decreases because the Floquet blocks will be closer in energy, even mixed, increasing the effective coupling with the elastic block. Notice that only those blocks dif-

210

S. Miret-A rt~s / Surface Science 339 (1995) 205-220

fering by + 1 in the number of phonons nQ~ are coupled. We could call it interblock coupling. In the literature of molecular fragmentation, each block dressed by a given number of photons is called a Floquet block [38,39]. The same designation could be followed here. Hence, for example, for a singlephonon exchange, only three Floquet blocks must play a relevant role: two corresponding to the creation and annihilation of the phonon and the third one to the zero-phonon block. Morever, as this interblock coupling is governed by ~ n Q~,n Q~,+ ~, whenever be the value of nQ~, the different orders of this single-phonon approximation are also considered in this formalism by adding more Floquet blocks until numerical convergency. Furthermore, as each Floquet block has its own structure (formed by a given number of G-channels dressed by the same number of phonons), the coupling between two G-channels coming from different Floquet blocks is through the scalar function B(Qu, T) • F(G - G' -T- Q, z). Finally, the following notation will be used here for the inelastic diffraction channels: G ±, Qv, where +__nQ~ indicates the number of phonons of the active mode dressing the diffraction channel G; also, in a short notation, (k, l).:l: , .~ v, where k and l are integer num. . bers representmg the G-vector. The elastic diffraction channels will be represented by a subscript 0. Two kinds of resonant scattering can be envisaged in this dynamics. If p stands for the resonant diffraction channel and its corresponding z-component kinetic energy is given by the Ep+Q,~eFfunction (expression similar to Eq. (8)), the resonant kinematic condition is then expressed by 0,

=

< 0,

(10)

where e~°~ is one of the eigenvalues of the bare potential Vo(z) (at zero order) and is symbolized by the label ~. Depending on the value of nQ~ (0, 1, 2, etc.), the dynamics taking place is zero, assisted by one or several phonons of the same frequency. For nQ~ = 0, the same notation for a SAR as in elastic scattering will be employed, ( m~m~ ,~ -), where (m 1, m 2) here represents the p-vector. At each scattering geometry (E, 0, ~b), manifestation of the same SAR will be different for the same surface temperature, T, since the arrangement of all diffraction channels will not be the same. Near a

resonance position, diffraction intensities display a very sharp peak. In general, the S-matrix can be expressed as (if a generalization from the elastic case [26] to the inelastic one is used) S({¢a+Q,,e(E,

0,~b; ~%(Q))},T)

=Sbg({,C+Q,ne~(E,

(

0, qS; oJ~(Q))},T). iA

I - Eo+e,"~(E' 0, 4'; ~oAQ))- ep~ + i ~ / 2

) ' (11)

Sug being the background S-matrix, ! the unit matrix and A a complex matrix of dimensions the number of open channels. The S and Sbg matrices are functions of all kinetic energies of the inelastic diffraction channels, 0, q~; oJ,(Q))}. The dependence on T is through the amplitude of the atomic displacements. The internal SAR position and width are eo,~ and Fm~, respectively. To the total internal width contribute two different kinds of open channels and, accordingly, we can differentiate two contributions: elastic and inelastic. For the former case, we have to consider those open diffraction channels coming from the Floquet block dressed by zero phonons and, for the second case, those open ones coming from the other Floquet blocks. Clearly, the resonant channel belongs to the zero-phonon block when we are considering elastic SAR and to one of the non-zero-phonon blocks when the resonance is assisted by a given number of phonons. Thus, we can write

{eG+Q,nav(E,

~

= Fp~ + Fore' ,

(12)

with = E

(13)

G'

Foi"~'= ot

E

Fm~.c'ne, . v

(14)

G',nQv

where the elastic and inelastic contributions to the total internal width have been expressed by means of the corresponding partial internal widths. The sums over G' are carried out for the number of open diffraction channels inside each Floquet block, respectively. In order to represent the S matrix we could also explicitely use the E, 0 and ~b external variables,

S. Miret-A rtOs/ Surface Science 339 (1995) 205-220 their associated widths being FE,~, F0, and Fo,, which are called external widths (their dependences on p have been dropped to simplify the notation). Then, we can transform into these external variables by developing up to the linear term the Taylor series of the E -function with respect to one of them. Thus, we have for the E variable

r)

0,

211

Debye model is assumed and the Debye frequency is labeled by w D, then we have that

(I6+Q( E, O, qb; nQv, T)) = fo '°0 doJ~(Q)p(w~(Q)) × l Sc+Q,o(E, O, qb; w~(Q), nQ,, T)[ 2, (17)

iA 1 •

I-

) (15)

E_ ~ + iFEJ 2 '

where ffS stands for the external position of the resonance in this variable. Similar equations can be written for the 0 and 4) external variables. In these variables, SARs can also display the CKE [26], that is, a coalescence of two line shapes corresponding to the same resonance. This occurs at the critical points of the Eo+,~, V ~, Qv-function with respect to one of those variables (first derivative equal to zero). Hence, when the CKE is present, Eq. (15) has to be modified according to

s((,o+o

0.

:Sbg({E-G+Q,nQv(E, O, ~); o)v(Q))}, T )

•

( ,

I-

i A2 (E-E)2+iF~/4

)

.

(16)

Analogously, similar expressions can be derived for the other two external variables. As was already reported elsewhere [26], a loss of the typical Lorentzian- or Fano-type behavior of resonance profiles is predicted for such cases. In Eqs. (15) and (16), A~ and A 2 a r e the A matrix multiplied by the first and second derivative of the Eo+Q.,Qv-function with respect to one of the external variables, respectively, and evaluated at the resonant values of the remaining ones. In this model, the attenuated diffraction intensities and resonance profiles are finally obtained by averaging out the square S-matrix elements through the phonon spectral density, p(w,,(Q)). Therefore, if a

which represents the ensemble average over the crystal states. Here, the square S-matrix elements give the probability of incoming in the specular channel and exiting in the channel G + Q at a given scattering geometry, surface temperature and phonon frequency, involving a transition with nQ~ phonons interchanged. This ensemble average does not lead to the presence of an explicit DW factor where drastic approximations have to be made. A similar expression, but in the transition matrix T formalism, is obtained for the diffraction intensities [13,14]. These intensities are expressed as an integral over the phonon spectrum of a function depending, among other things, on the phonon frequency. This function can be expressed analytically at different orders of approximation according to a Taylor expansion of an exponential of a displacement correlation function. Here, in our formalism, this is replaced by the S-matrix elements which contain all of this information but no analytical solution is furnished due to the proper character of the CC-equations. This analysis also permits us to know the importance of the contributions of the phonons of low and high frequency since the elements of the S-matrix are expressed as a function of this frequency. Hence, the attenuation will appear in a straightforward way due to both the couplings among the Floquet blocks (interblock coupling) and the contributions of low frequency phonons. The relative disposition in energy of these blocks for the low frequency phonons will be very close leading to strong effective couplings and, whence, important variations in the specular and diffracted intensities will be manifested. In a sense, the effect of attenuation described at first order by the DW factor is described in this formalism by the

S. Miret-Art6s / Surface Science 339 (1995) 205-220

212

coupling between the elastic block and the remaining dressing blocks. Regarding the scattering at resonance conditions, the following comments have to be made. The square modulus of the S-matrix elements, given by Eq. (11), furnishes the line shapes of the SARs for different open diffraction (elastic and inelastic) channels. The line shapes of the diffraction intensities are Lorenztian- or Fano-type functions (with the same internal resonance position and width) but with different signatures depending on the phonon frequency. Thus, for the elastic case, if the resonance position and width are assumed to be weakly dependent on the phonon frequency, the integration given by Eq. (17) will provide us, in general, with a new signature of the SAR. In other words, we have an incoherent superposition of different shapes coming from the presence of the phonon spectrum. This result is also derived in Refs. [13,14] in the T-matrix formalism. Moreover, knowledge of the external widths is through Eq. (15), or similar, but is expressed in angular variables. All of these points will be applied and discussed in more detail in the next section.

3. Results and discussion

As we have mentioned above, we are interested in reproducing the thermal attenuation of the specular and diffracted intensities as well as the line shapes of SARs for the H e - C u ( l l 0 ) and Cu(ll3) systems. For this goal, previous to any calculation, we must specialize the CC-equations (Eqs. (6) and (7)) to the case of elastic scattering with Q = 0. Therefore, the CC-equations to be solved are h

d2

2----~----£ d z + eo,,( E, O, qb; to) - Vo( z ) ~bo,,( z ) =

E

1 V~ o,(z)q,o,,,(z) + ~ E B

G' 4=G

G'

• F(G-

+

G', z)(~0c,,,,+,(z ) + ~0~,,,,_,( z ) ) , (18) O, 4,;

-

G'~ G

1

EB

+2 G'

. F ( G - G',

+ tp~, , ( z ) ) ,

z)(~a,,,+2(z ) (19)

where now w~(Q) and nQ~ have been replaced by w and n, respectively, in order to simplify the notation. At this level, it is interesting to comment the interaction potential used for these systems. Experimentally [40], it was found that in-plane diffraction dominates, these measured intensities being normalized to unity. The interaction potential was modelled by a corrugated Morse potential and the parameters were found by extrapolating the experimental results to zero temperature and performing elastic calculations. In all of these calculations, the ~b angle was fixed to its zero value. Thus, in this work, the same interaction potential has been chosen for the static part and a one-dimensional model for the surface will describe the corresponding corrugation. It has been widely argued [5,12,41] that a correct description of the atom-surface inelastic scattering implies the use of soft potentials including an attractive well, to go beyond the Born approximation, as well as an adequate treatment of thermal motions of surface atoms. Clearly the last point is critical in this kind of studies. A detailed analysis of the interaction potential in terms of instantaneous positions of the incident atom and surface atoms is not known at present and, therefore, one has to use some phenomenological approach or more or less crude approximations to solving this scattering. In this sense, effective mean square displacements of lattice atoms or the hard wall were introduced in the literature. Obviously, these effective values are very related to the model potential and coupling assumed. The main goal is to fit the experimental data and compare with previous values issued from some theoretical calculation based on a many-body statistics (if they exist). Thus, for example, only for some faces of copper (111, 100 and 110) some mean square displacements of a lattice atom have been calculated. The discussion about whether only one atom is involved in the dynamics or the mean correlated displacements of neighboring atoms have to be included is at the origin of the Armand effect.

S. Miret-Art£s/ Surface Science 339 (1995) 205-220

Concerning the systems considered here, we are going to take advantage of a previous work [12] about the use of such effective displacements. The u-displacements were replaced by a quantity uz,eff characterizing the local thermal displacement of the interaction potential. The authors concluded that the tangential term of this effective displacement is very small compared to the normal one. Therefore, in our model, the coupling B • F ( G - G ' , z ) is reduced to B . F ( G - G', z ) = B z V d _ o , ( z ),

T=70K 1I

"

.J'~

"

(21)

Here a is the unit cell length and the Qc parameter is a fitting parameter accounting for the width of the cut-off Gaussian factor. This factor is associated to the charge deformation due to the scattering of the atom from a soft potential. This amplitude is very much related to that given by Bortolani et al. in Ref. [42]. The expression used for <(Uz,eff)2> is given in the Appendix. A Debye model for the solid has been assumed, the spectral density being given by 3to 2 p(w)-

co 3

,

i

~ ...

I

;

_!

I

*"

,,

"

u .

',

10-1 Too

10 -= I

.~

10-'

.

00o

,., ~'~ -~ ~

',

(20)

where the amplitude B z of this displacement contains explicitely the dependence on the surface temperature. This amplitude is assumed to be related to this effective mean square displacement as follows (see Appendix): 2f~ B ~ = aQc 732((Uz,eff)2)

213

(22)

which has been employed for the average prescribed by Eq. (17). A value of T D = 236 K has been taken for the surface Debye temperature in Ref. [12] and the same value has been used in this work. 3..1. Thermal attenuation o f diffracted intensities

In this section, we are going to show a detailed numerical analysis of the behavior of the specular and diffracted intensities as a function of the phonon energy and surface temperature T for the H e C u ( l l 0 ) scattering with incident energy of 21 meV and incidence angle of 67 °. These behaviors are expected to be followed, in a global way, for any inelastic scattering. Elastic and inelastic diffraction channels present

~o"

"~'-~-t kI I', I ', l ',, 00+, , 0

,,,, ~

,"~ 5

Phonon

,, - . . ', "

oo_, "~

", I"0+, , t0

,, ,~, 15

Energy

I0_, 20

(meV)

Fig. 1. For a surface temperature of 70 K, specular and diffracted intensities as a function of the phonon energy (in meV). The Debye frequency is h wo = 20.34 meV. Solid lines correspond to the elastic diffraction channels, large and small dashed lines correspond to the same diffraction channels dressed by - 1 and + 1 phonon, respectively. Values for the He-Cu(110) scattering with E i = 21 meV and 0i = 67°. The experimental unitarity (0.88) [40] has not been considered in this plot.

thresholds which are not fixed. These thresholds depend on the scattering geometry (E,0,~b) as well as the number of phonons (and, therefore, their energies) dressing them. Effects of these moving thresholds with the scattering geometry have been analyzed extensively elsewhere [26]. Here, we focus our attention on a new aspect: variations of the specular and diffracted intensities versus the phonon energy. If only one mode is expected to contribute to this dynamics, and the Debye frequency for the C u ( l l 0 ) surface is taken to be 20.34 meV (T u = 236 K), these intensities must be very dependent on the number of phonons dressing the diffraction channels as well as the value of the phonon frequency. These behaviors are plotted in Fig. 1. For T = 70 K, the numerical convergency is good only with three Floquet blocks, including three diffraction channels in each block: the three G-vectors ( + 1,0) and (0,0). We display the variations of these intensities only for those channels which are open for the different phonon energies: solid lines for the elastic diffraction channels (000 and 100); large dashed lines for the inelastic diffraction channels dressed by minus one

S. Miret-Artgs / Surface Science 339 (1995) 205-220

214

Table 1 Specular and diffracted intensities for the H e - C u ( l l 0 ) at two different surface temperatures, 70 and 470 K

I c ( T = 70 K)

I ~ ( T = 4 7 0 K)

10_ 1

1.72(-3)

1.53(-2)

00_ l 1O_ 1

2.36( - 3) 6.59( - 2)

1.31( - 2) 3.36( - 1 )

~0 o 000

2.43( - 2) 7.72( - 1)

1.52( - 2) 4.32( - 1)

100

-

_

10+l

00+ ~

1.58(-2) 4.06( - 6)

7.11(-2) 1.89( - 5)

lO+t

-

G

scattering,

The incident energy and incidence angle are 21 m e V and 67 °, respectively. The first three r o w s correspond to the creation process of one phonon, the following three r o w s to the elastic process and the last three ones to the annihilation process of one phonon. The experimental unitarity is 0.88. N u m b e r s in parentheses indicate negative p o w e r s of ten.

phonon (00_ 1, 10_ t and 10_ 1) and small dashed lines for plus one phonon (00+ l, and 10+ t). Whereas the behavior of the elastic channels is very smooth, strong variations are observed in the inelastic ones. Energies around 3, 5.5 and 12 meV give us the thresholds for which some inelastic channel becomes open or closed with the subsequent redistribution of the incident flux of atoms into the several Bragg directions. For example, at each phonon energy and for the specular direction, three contributions of different orders of magnitude are present. Obviously the contribution coming from the elastic specular channel, 000, dominates. The inelastic specular channels are scarcely favored due to the fact that the inter-

change of phonons with the surface with no change in the x-component of the k-vector is not very likely. On the contrary, the 10 1 channel is much more favored than the elastic process 100 and the rest of the open channels for this scattering and for all phonon energies. The same is true for the 10+~ channel at certain energies of the phonon (the lowest ones). Similar conclusions are drawn from a CCcalculation with T = 470 K, where the single-phonon contribution should be still valid. In this temperature range (from 70 to 470 K), Qc has been set to 0.562 ~ - 1 which is a reasonable value for the Cu(ll0) system. Typical values of this parameter have been estimated in the literature to be between 0.6 and 0.7 In Table 1, we present the diffracted intensities averaged over phonon frequencies (Eq. (17)) for 70 and 470 K, this average being performed for 15 points of a Gauss-Legendre integration. These intensity values have been renormalized to the experimental unitarity 0.88 [41] and obtained using three Floquet blocks containing five diffraction channels each. As expected, contributions from the inelastic processes begin to be important as the temperature of the surface increases. Above 500 K, more Floquet blocks should be included in the calculations. However, it has been reported [12] that this surface face presents a thermal roughening transition around 423 K. Therefore, from about 500 K, the effect of the temperature is not only to introduce anharmonicity in the atomic displacements but also to create structural defects. This should lead to important modifications of the potential parameters for the static part. Thus,

Table 2 Specular and diffracted intensities for a surface temperature of 670 K with incident energy of 21 m e V and incidence angle 67 ° for the H e - C u ( l l 0 ) scattering

G

-3ho~

-2hw

-lhw

Ohto

+lhw

+2hw

40

1.43( - 3)

30

1.73( -- 3)

1.58( - 3)

1.53( - 3)

8.52( - 3)

2.20( - 3)

4.22( - 3)

1.30( - 3)

2.39( - 3)

1.47( - 3)

1.09( - 2)

20

6.53( - 4)

1.10( - 3)

1.49( - 3)

1.41( - 3)

1.17( - 2)

6.78( - 3)

2.72( - 2)

10 00 10 20

3.14( 2.76( 1.20( 4.34(

1.91( 6.92( 7.61( 2.29(

6.94( - 3) 7.75( - 3) 2.62( - 1) -

7.61( - 3) 2.38( - 1) -

3.51( - 2)

1.66( - 4)

30

8.75(- 2)

-

-

- 3) - 2) - 2) -- 3)

-

-

3) 3) 3) 1)

The c o l u m n s c o r r e s p o n d to the creation of three, two and one phonon, the elastic process and the annhilation o f one and two phonons, respectively. O n l y contributions greater than 1 . 0 ( - 4 ) are reported. Again, n u m b e r s in parentheses indicate negative powers of ten.

S. Miret-A rt~s / Surface Science 339 (1995) 205-220

* "~ 10

O0

, . ,.,,,,~

l 10

10 -~

, . . . . . . .

,i

. . . . . . .

200 Surface

,,i

....

, , , , , i , , , , , , , , , I

400 600 800 Temperature ( K )

Fig. 2. Elastic specular (00) and diffracted (10) intensities as a function of the surface temperature for the He-Cu(ll0) system with E i = 21 meV and 0 i = 67°; ( + ) CC-calculations and ( × ) experimental data. At these conditions, the experimental unitarity is 0.88.

215

370 K, where important deviations from the linear behavior are observed both in the experimental data and our CC-calculations. The agreement is very good in the range 70-470 K. As mentioned before, at higher temperatures the agreement has to be considered more artificial. However, these results have been presented to show that increasing the coupling strength (or decreasing the Qc value), the CC-results are able to produce very strong attenuation, as displayed in the experiments. With this behavior, we think that the feasibility of this formulation is clearly justified. Remember that no explicit DW factor has been included in these calculations. It is expected that when the interblock coupling and the corrugation of the surface increase (and, therefore, the number of diffraction channels), the range of validity of this DW theory diminishes. At least, this is what is observed from an experimental and theoretical point of view. 3.2. Thermal attenuation at resonant conditions

in our CC-calculations, these facts should be taken into account. However, we have chosen to modify Qc (from 0.562 to 0.394 ,~-1), instead of the corrugation parameters, in order to show the behavior of the theoretical intensities as a function of T. Obviously, the physical meaning of such results has to be taken with some objection since this new value of Qc is somewhat arbitrary. Thus, in Table 2, diffraction intensities are shown for the different open channels when T = 670 K. The elastic intensities are very much attenuated in favor of the other inelastic processes since the theory is unitary. In particular, the 20_ 2 and 10 i channels display intensities of the same order as the specular one. The CC-calculations have been performed with nine diffraction channels (G = ( + 1,0),( ___2,0),( + 3,0) or ( + 4,0) and (0,0)) in each Floquet block (in all, 63 channels). Finally, in Fig. 2, the elastic specular (000) and diffracted (100) intensities are plotted as a function of T for a range of temperatures between 70 and 770 K issued from our CC-calculations ( + ) and those measured ( × ) by Lapujoulade et al. [12]. The unitarity in this case is [40] equal to 0.88. Up to 370 K, the linear behavior displayed in Fig. 2, where intensities are plotted on a logarithmic scale, shows that a phenomenological DW factor could be justified for this system. This is not the case for T greater than

In this section, we tackle the resonant scattering problem for different values of T. In particular, we are going to focus on the H e - C u ( l l 3 ) system because it is a case where experimental [29] and theoretical [29,31] data are available. Moreover, it is an example where the signatures predicted by the Wolfe and Weare rules [23] fail and some controversy can be found in the literature and, in our opinion, it is not clarified yet. Signatures of SARs are usually discussed in terms of a quantity b which is defined to be the ratio between the resonant and direct amplitudes, evaluated at the resonance position [24]. In the CC-formalism, this concept can be extended to any diffracted peak (not only to the specular one) and the line shapes can be expressed as [26] IsGoI: = l + i

bGO 2 I SGobgl:,

(23)

x+i where b~o determines the signature [24,26] and x represents a normalized deviation from the resonance position: 2 x = ~ - - ( ,o+Q,no" -- ¢o~)"

(24)

216

S. Miret-Art~s / Surface Science 339 (1995) 205-220

This equation is valid for any multichannel (elastic or inelastic) scattering, irrespective of the model potential used. Here, I sbg0]2 gives the background contribution to the line shape, very important for the case studied here. From the experimental information on Cu(113) and for the specular peak at E i = 21 meV, three minima appear at 0 i = 52 °, 53 ° and 60.5 °, corresponding to the resonances ( ~ ) with a = 2,1,0, respectively. According to the second rule of Wolfe and Weare, mixed extreme (maximum predominant) are to be expected. After different theoretical attempts for understanding these observations and where, even, the calculated widths were much smaller than the experimental ones, it was concluded that a purely elastic theory was unable to correctly predict the resonance features. On the contrary, for the H e - C u ( l l 5 ) system, the same theory predicted the correct signature (a maximum) observed experimentally. With respect to broadening origins, several sources of these discrepancies were mentioned. Firstly, inelastic effects. Following the method of Hutchison, and for the (~o) resonance, the correct signature together with a value of the internal width of F = 0.12 meV versus the experimental value of 0.25 meV was obtained [29]. Secondly, the presence of resonances coming from other diffraction channels ((0,1) and ( 1 , - 1)) at about the same incidence angle could perturb the diffraction pattern observed. And, finally, the incoherent scattering which could be due to some irregularities in the step lattice. In an ulterior work [31], the calculated line shape was narrower and deeper than the experimental one. Taking into account the 0.2 ° angular dispersion in the incident beam, the minimum calculated was still narrower than the experimental one. With these antecedents, it would be interesting to know what is the result issued from our CC-calculations using Eq. (17). At T = 70 K and for the (lO) resonance, we have plotted in Fig. 3 the line shapes as function of different phonon frequencies (dashed lines) and the final line shape (solid line) after performing integration over to in Eq. (17). As it has been mentioned above for the elastic case, the resonance position and width are not very sensitive to to but, on the contrary, the signature changes drastically with the phonon frequency. In fact, it is to be noticed that all kinds of resonance profiles are found in our

0.70

w =

20.2B1

meV

w =

0,0~6

meV

w =

4,192

meV

0.60 .,..,

~

/

0.50

.~

. . . .

~J

0.40 /

0.30

~-

-

d

~

%/t

i

t

0 0.20 "/

~ t

\.

w =10.169

meV

0.10

0.00

, . . . . . . . .

58

i , , , , , , , , , i , , , , ,

59

60

Incident

....

i , , , , , , , , , i , , ,

61

62

. . . . . .

i

63

Angle

Fig. 3. Line shapes for the specular intensity of the (1o) resonance and for the H e - C u ( l l 3 ) system at E i = 21 meV and different phonon frequencies, oJ (dashed lines). Solid line corresponds to the resonance profile after integration over the phonon spectrum according to Eq. (17).

calculations. For small frequencies, these profiles present greater distortions. This again confirms that low phonon frequency contributions are very important in the resonance problem. According to Eqs. (17) and (23), if x is in a first approximation independent on to, the corresponding integration modifies only the numerator and, therefore, only the signature is changed. This modification appears in a natural way without necessity of introducing any scaling factor in the scattering amplitudes. Finally, in Fig. 4 we plot the line shapes at different T (0, 70 and 170 K; solid lines) together with the experimental one (dashed line) at T = 70 K. The theoretical resonance position is 60.84 °, slightly different from its experimental value 60.5 ° and elastic value 60.34 °. This difference is easily understood because the bare potential (Morse potential) has been fitted to have that resonance position corresponding to the average value observed from the different experimental diffraction intensities. As a result of the interaction, the resonance position is shifted. So the displacement of the theoretical profiles is not a drawback of this theory because it is possible to give the correct position changing the Morse parameters only a little. Concerning the background contribution, we do not reproduce exactly the same one because we are omitting the diffraction channels (0,1) and ( - 1 , 1 )

S. Miret-Art~s / Surface Science 339 (1995) 2 0 5 - 2 2 0 1.0

,.•0.8 "~0.6

T :

,.,-~ 0.4

T = 70 K

0 K

~J T = 170 K f,~ 0,2

0.0 58

59

60

Incident

61

62

63

Angle

Fig. 4. Line shapes for the specular intensity of the (lo) resonance and for the H e - C u ( l l 3 ) system at E i = 21 meV and different surface temperatures. Solid lines, CC-calculations, and dashed line, experimental profile only at 70 K.

which are not considered here due to our one-dimensional model assumed for this surface. As has already been mentioned, at T = 0 K a maximum profile with an internal width of F = 0.02 meV and angular width of F0 = 0.05 ° is obtained. At T = 70 K, the maximum is transformed into a minimum but with an internal width F = 0.14 meV, compared to 0.12 meV issued from the Hutchison method and 0.25 meV, the experimental value. The deep of the minimum is more or less the same as the experimental one without taking into account the angular dispersion of the incident beam. This result has been obtained with Qc = 0.516 •-1. Furthermore, as has already been reported [33] and confirmed here, the thermal attenuation is substantially stronger at resonance than in off-resonance conditions. Increasing T up to 170 K, a very attenuated line shape is displayed in Fig. 4. For this scattering, the experimental unitarity is equal to 0.78.

3.3. Thermal attenuation at critical conditions Recently [26], a means for making weak resonances visible has been reported for the elastic scattering of atoms from surfaces. This situation is particularly convenient for the H e - C u ( l l 0 ) system, where no trace of resonances has been experimentally observed at 21 and 63 meV. The special inci-

217

dence conditions for which they could be visible have been called by us critical conditions, and the corresponding effect over the resonance profiles the critical kinematic effect (CKE). In the external variables (E, O, 49), SARs can sometimes display a coalescence of two diffraction peaks corresponding to the same resonance; in other words, a remarkable loss of the typical (Lorentzian- or Fano)-behavior of intensity profiles with the subsequent enhancement of the resonance widths has been theoretically predicted. The goal in this section is to know the modifications of these profiles as a function of T and to stimulate experimental work for the observation of this singular manifestation• From Eq. (23), we can transform into external variables (E, 0, q5) by assuming a linear approximation in the Taylor series of the eo+Q,,Qv function on each variable around the resonance position. If the derivative of this function with respect to some external variable is zero, we have to develop up to the second order in the external variable and Eq. (23) has to be replaced by + 1 ~ ISG012= 1

•

2

GO

Xq+i Is~gl

2

,

(25)

where q designates one of the three external variables. In the energetic version of this formula, the line shape for the specular intensity has the following form (an alternative way of rewriting Eq. (16)):

IsoolZ=(l+{[(b~oo)2+ 2b~oo+(b~o)2]F4/16

+bi~(E-E) ['i/Z} -- 2

2

X { ( e - ffS)4 + [ ' {4 / 1 6

}-1)IS0obgl2, (26)

b~o and b~0 being the real and imaginary part of boo, respectively. This is exactly the type of profile reproduced in our CC-calculations for the (to) resonance of the He-Cu(110) system. In Fig. 5, at different T (0, 70, 170 and 670 K), the profiles theoretically predicted are displayed. This type of line shapes (dependence on the resonant variable with the fourth exponential) is very singular and has not been reported before, only for elastic scattering [26]. Due to this functional form, the manifestation of a given resonance is by means of two peaks, or only one

218

S. Miret-ArtOs / Surface Science 339 (1995) 205-220 0.90

0.80

T=OK

-F-I

~

I

0.70

=

70 K

T =

170

K

T

670

K

~.4~ 0 . 6 0

0,50 ¢,,,) 0,40

=

0.30

0.20

,,,,,,,,,~,,,,,,,,,F,,,, 8

Incident

.... 10

,i,,,,, 12

E n e r g y (meV)

Fig. 5. Line shapes of the (10°) resonance for the He-Cu(110)

system at critical conditions (0 i = 54.58° and E i = 9.37 meV) and different surface temperatures. depending on the values of the real and imaginary part of the boo parameter. For this resonance, the critical conditions are: 0 i = 54.58 ° and E i - - 9 . 3 7 meV. At T = 0 K, F E = 1.19 meV with bh0 = 0.03 and b~o = 0.054 (two maxima). A very important enhancement of resonant features was predicted with respect to the scattering at 21 meV (only one peak, maximum, with a width of F E = 0.03 meV). However, in order to compare with experimental results, we need to introduce the surface temperature explicitely. Thus, for T = 70 K and at critical conditions, the resonance position is practically the same but the new value of /'E is 2 meV now. It is also predicted that the signature of the resonance is changed (two minima). As expected, this external width increases with T. This is also what we obtain at T = 170 K, where a similar profile, but attenuated with respect to the previous case and with a width of 3.2 meV, is obtained. At very high temperatures (for example, T = 670 K) the attenuation is so important that any trace of the two peaks disappears and only a more or less smooth baseline is observed, like a background contribution around the value 0.5. Notice that, as mentioned before, at this temperature the number of defects gathered in domains increases due to the onset of the thermal corrugation [12,33] and, therefore, the interaction potential should be changed. However, reference to this temperature has been chosen mainly to know again the feasibility of the

method to calculate resonant scattering involving several single-phonon contributions. Finally, we would like to comment that the cases analyzed in this work illustrate that this new formulation of the inelastic scattering of atoms from surfaces, within the CC-formalism, works quite well. As far as we know, it is the first time that CC-results have been contrasted with experiment (and with good agreement) for different types of scattering (non-resonant, resonant and critical) and surface temperatures. Extension from these calculations to more corrugated surfaces such as Cu(ll5) and Cu(ll7) is now in progress since very detailed experimental information is also available. As has been recently pointed out [43], for these systems the study of the inelastic sticking is also particularly important and this interesting process could be easily adapted to this formulation. Finally, the single-phonon approximation used in this work could also be relaxed in order to study multiphonon processes.

Acknowledgements I would like to thank to Professors F. GarciaMoliner, V.R. Velasco and especially J.R. Manson for their very interesting and helpful discussions. This work has been supported partially by CYCIT under Grant No. PB92-0053 and by Comunidad de Madrid under Grant 064/92.

Appendix Following Ref. [12] the u-displacements were replaced by an effective quantity, uz,eff, characterizing the local thermal displacement of the interaction potential. Those authors reported that the tangential component of this effective displacement was very small compared to the perpendicular one. Now, if we further suppose that this effective displacement is periodic both in x (with period a, - a / 2 <~x <~ a / 2 ) and in t, presents only one frequency to corresponding to the active mode, and behaves as a Gaussian function within the unit cell, we can write Uz,eff( X, t ) = Uzo

e x p ( - x 2/ x Z?) cos

tot,

(A.1)

S. Miret-A rtOs / Surface Science 339 (1995) 2 0 5 - 2 2 0

where Uzo is the initial amplitude and x c controls the width of the Gaussian function, this amplitude being limited by the unit cell length. If the spatial factor in Eq. (A.1) is developed in a Fourier series, according to Eq. (2), the amplitude Bz, z associated to the 3' mode is then obtained by integration over the unit cell to give 1

B w = - fa/2 dxuzo e x p ( - x -~/ x ~ ') cos Qvx a ~

a/2

2f~ . ~-Uzo----x-exp(-Q~/Q~) ,

(A.2)

a~c

where Qc = 2/xc is known as the cut-off parameter which is an adjustable parameter. For elastic scattering, Qz, = 0 and B:, z has the following expression: 2x/~ Bz ~- u ~ 0 - -

(A.3)

aQc

The u. 0 value can be deduced from the effective mean square displacement, (U~,eff), as [14b] (Uz,eff(O ,

t)

• Uz,eff(0, 0 ) )

= 4 ( U 2 e, f f ) = 211 4 U z 02,

(A.4) where the 1 / 2 factor comes from the time average of the cosine function and the 1 / 4 factor from an average over polarizations. Thus, Eq. (A.3) becomes 2x/~ B z •

- -

aQc

~/32(U2z.eff) .

(A.5)

Finally, the following expression for (u~.eff) will be used [14b]: 6/xT

(U~,eff)

Mk,nT2 ,

(a.6)

where TD is the Debye temperature, k'B is related to Boltzmann's constant k B by k~ = 2txkB/h 2, and M is the crystal atom mass.

References [1] N. Cabrera, V. Celli, F.O. Goodman and R. Manson, Surf. Sci. 19 (1970) 67. [2] R. Manson and V. Celli, Surf. Sci. 24 (1971) 495. [3] J.L. Beeby, J. Phys. C: Solid State Phys. 4 (1971) L359; 5 (1972) 3438.

219

[4] U. Garibaldi, A.C. Levi, R. Spadacini and G.E. Tommei, Surf. Sci. 48 (1975) 649. [5] G. Armand and J.R. Manson, Surf. Sci. 80 (1979) 532. [6] G. Wolken, Jr., J. Chem. Phys. 60 (1974) 2210; Y. Lin and G. Wolken, Jr., J. Chem. Phys. 65 (1976) 2634. [7] H. Chow and E.D. Thompson, Surf. Sci. 54 (1976) 269; 59 (1976) 225; 82 (1979) 1. [8] Y. Hamauzu, J. Phys. Soc. Jpn. 42 (1977) 961. [9] A.C. Levi and H. Suhl, Surf. Sci. 88 (1979) 221. [10] G. Armand, J. Lapujoulade and Y. Lejay, Surf. Sci. 63 (1977) 143. [11] V. Bortolani, V. Celli, A. Franchini, J. Idiodi, G. Santoro, K. Lern, B. Poelsema and G. Comsa, Surf. Sci. 208 (1989) 1. [12] J. Lapujoulade, J. Perreau and A. Kara, Surf. Sci. 129 (1983) 59. [13] G. Armand and J.R. Manson, Phys. Rev. Lett. 11 (1984) 1112. [14] (a) G. Armand, J.R. Manson and C.S. Jayanthi, Phys. Rev. B 34 (1986) 6627; (b) J.R. Manson and G. Armand, Surf. Sci. 184 (1987) 511; 195 (1988) 513. [15] J.R. Manson and V. Celli, Phys. Rev. B 39 (1989) 3605. [16] J.R. Manson, Phys. Rev. B 43 (1991) 6924. [17] M.D. Stiles, J.W. Wilkins and M. Persson, Phys. Rev. B 34 (1986) 4490; M.D. Stiles and J.W. Wilkins, Phys. Rev. B 37 (1987) 7306. [18] B. Jackson, J. Chem. Phys. 88 (1988) 1383. [19] G.D. Billing, Surf. Sci. 203 (1988) 257. [20] P. Cantini and R. Tatarek, Phys. Rev. B 23 (1981) 3030; G. Brusdeylins, R.B. Doak and J.P. Toennies, Phys. Rev. Lett. 46 (1981) 437; R.B. Doak, U. Harten and J.P. Toennies, Phys. Rev. Lett. 51 (1983) 578; G. Benedek, G. Brusdeylins, R.B. Doak, J.G. Skofronick and J.P. Toennies, Phys. Rev. B 28 (1983) 2104; G. Benedek, G. Brusdeylins, J.P. Toennies and R.B. Doak, Phys. Rev. B 27 (1983) 2488. [21] G. Boato, P. Cantini and L. Mattera, Surf. Sci. 55 (1976) 141. [22] G. Derry, D. Wesner, S.V. Krishnaswamy and D.R. Frankl, Surf. Sci. 74 (1978) 245. [23] K.L. Wolfe and J.H. Weare, Phys. Rev. Lett. 41 (1978) 1663. [24] V. Celli, N. Garcia and J. Hutchison, Surf. Sci. 87 (1979) 112. [25] F. Goodman, Surf. Sci. 94 (1980) 507. [26] M. Hernandez, S. Miret-Art6s, P. Villarreal and G. DelgadoBarrio, Surf. Sci. 251/252 (1991) 369; 274 (1992) 21. [27] G. Brusdeylins, R.B. Doak and J.P. Toennies, J. Chem. Phys. 75 (1981) 1784; U. Harten, J.P. Toennies and Ch. WBII, J. Chem. Phys. 85 (1986) 2249. [28] D.A. Wesner and D.R. Frankl, Phys. Rev. B 24 (1981) 1978. [29] J. Perreau and J. Lapujoulade, Surf. Sci. 119 (1982) L292; 122 (1982) 341. [30] J. Hutchison, Phys. Rev. B 22 (1980) 5671. [31] J.G. Mantovani, J.R. Manson and G. Armand, Surf. Sci. 143 (1984) 536.

220

S. Miret-Art~s / Surface Science 339 (1995) 205-220

[32] P. Cantini, S. Terreni and C. Salvo, Surf. Sci. 109 (1981) IA91. [33] G. Armand, J. Lapujoulade and J.R. Manson, Phys. Rev. B 39 (1989) 10514. [34] P. Cantini, G.P. Felcher and R. Tatarek, Phys. Rev. Lett. 37 (1976) 606. [35] D. Evans, V. Celli, G. Benedek, J.P. Toennies and R.B. Doak, Phys. Rev. Lett. 50 (1983) 1854. [36] S.A. Safron, W.P. Brug, G.G. Bishop, G. Chern, M.E. Derrick, J. Duan, M.E. Deweese and J.G. Skofronick, J. Vac. Sci. Technol. A 9 (1991) 1657. [37] G. Benedek and S. Miret-Art6s, Surf. Sci., submitted. [38] S. Chu, J. Chem. Phys. 75 (1981) 2215.

[39] S. Miret-Art~s, O. Atabek and A.D. Bandrauk, Phys. Rev. A 45 (1992) 8056. [40] D. Gorse, B. Salanon, F. Fabre, A. Kara, J. Perreau, G. Armand and J. Lapujoulade, Surf. Sci. 147 (1984) 611. [41] (a) J. Lapujoulade, Y. Lejay and G. Armand, Surf. Sci. 95 (1980) 107; (b) J. Lapujoulade, Surf. Sci. 134 (1983) L529. [42] V. Bortolani, A. Franchini, F. Nizzoli, G. Santoro, G. Benedek and V. Celli, Surf. Sci. 128 (1983) 249; V. Bortolani, A. Franchini, N. Garcia, F. Nizzoli and G. Santoro, Phys. Rev. B 28 (1983) 7358. [43] S. Miret-Art~s, Surf. Sci. 294 (1993) 141,