Photodesorption via stimulated Raman emission of coherent surface phonons

Photodesorption via stimulated Raman emission of coherent surface phonons


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Physica 142B (1986) l-10 North-Holland. Amsterdam




Ramat Aviv,

69978, Tel-Aviv,


S.H. LIN Department

of Chemistry,


Received 28 January 1986 Revised manuscript received

State University,


AZ 85287 USA

12 May 1986

A new mechanism of a desorption induced by infrared laser is presented. When intensity of the laser and concentration of adsorbates are large enough, conditions of stimulated emission of Raman (Stokes) surface phonons are satisfied. This leads to an exponential rise in time of numbers of phonons with frequencies close to Stokes frequency. These volatile phonons are able to rupture the adsorbate-surface bond.

1. Introduction Wavelength-dependent laser-induced desorption of molecules by resonant excitation of internal modes has been investigated both experimentally and theoretically by many researchers [l261. Experimental and theoretical studies of the laser-induced desorption (and the whole area of laser-induced surface processes) have been reviewed by Chuang [4]. Theoretical investigation of the Rochester group was reviewed by George and his coworkers [13]. Main results of the experiments are: (1) the desorption is stimulated at moderate intensities -1 MW/cm*. (2) The desorption yield is resonant at the vibrational frequencies of the adsorbate. (3) The frequency dependence of the desorption yield is narrower or the same as the absorption shape. (4) Dependence of the desorption yield on a laser intensity I is proportional to the power of the intensity I” (e.g. it is reported n = 2.8 [l-3] and II = 2.2 [4]). (5) The desorption has a collective character: it essentially depends on surface coverage, e.g. for C,H,N-KC1 the desorption yield increases nonlinearly with number of adsorbed monolayers S and is saturated for S b 30 [4]. Another manifestation of collective character of laser-induced

desorption was a study of laser excited C,H,N and C,D,N from KC1 and Ag surfaces [4,5, 6, 8, 241. It was observed that the photodesorption yields did not have significant molecular selectivity in the photodesorption of C,H,N and C,D,N coadsorbates when either type of isotopic molecules was initially excited. Dependence of the desorption yield on surface coverage and nonspecificity of the desorption imply that in these experiments laser-induced desorption has cooperative, collective character. It should be explained not as a property of one adsorbate but as that of a collective of the adsorbates. Several mechanisms may lead to a collective desorption. One of them is “a resonant (or indirect) heating mechanism” and was suggested by Chuang [4,8], Chuang and Hussla [7] and by Gortel et al. [14, 151. Energy levels of an adsorbed molecule may be presented an Ei where u are internal quantum numbers and n are quantum numbers of the desorption mode. Upon absorption of the laser photons, the molecule will be excited to a state Ei from which it can emit a phonon of energy &J such that Ez = Ei + fiw. From an excited state Ef of a desorption mode it can cascade down to n = 0, emitting more phonons and thus heating up the solid.

0378-4363 /86/$03.50 0 Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)


B. Fain and S.H.

Lin I Photodesorption

This heating might cause a thermal desorption. Conclusions of theoretical investigations of this mechanism by Gortel et al. [15] and by us in a previous paper [25] show that this mechanism is not a very efficient one. Two other collective mechanisms were investigated in our previous works [22, 23, 25, 261. In of the one of them [22,26] an internal vibration adsorbate is excited as a result of joined action of the laser and of intermolecular energy transfer. Another mechanism of collective desorption is connected with an instability of the desorption mode. When the desorption which is an anharmanic oscillator is excited to state n (with energy E,) it may become unstable and lead to generation of coherent phonons of frequency w = (Ei - Ei,)lTi. The instability of the desorption mode may mean its rupture, i.e. the desorption. The aim of the present work is to analyze a new possible mechanism of collective photodesorption. This mechanism takes into account the possibility of stimulated Raman emission of coherent surface phonons, provided a number of the adsorbates is large enough. Once coherent phonons are excited and their number is exponentially increasing during the laser pulse, this may cause a desorption. This desorption has the same relation to thermal desorption as an effect causing laser light to the usual white-noise, thermal source of the light. Coherent phonons are an analog of the laser source.

2. Description

of the model

Before starting the analysis of the possibility of stimulated Raman emission of phonons in the crystal [27], we will remind the reader of the basic notions related to the problem. First, let us recall the meaning of the usual Raman emission by impurities (adsorbates in our case) in the crystal. Let E, and E, be two levels of the molecule (E, < E,). We assume that the molecules are subjected to the electromagnetic field of frequency w > 02, = (E, - E,)lTi. Then, as is known, at the frequency wk = R - w2, (Stokes component) the so-called Raman emis-

via stimulated Raman emission of phonons

sion may emerge. If the molecules were initially at level E, and the number of the photons at the Stokes frequency wk = 0 - w2r was nk, then the intensity of the Raman emission should be proportional to (nk + 1). The term in the intensity, the stimulated proportional to nk, describes Raman emission, and that proportional to unity, describes usual (spontaneous) Raman scattering. Now, the molecules embedded in a crystal interact not only with photons but also with vibrational excitations of crystal - phonons. It means that the adsorbate subjected to the electromagnetic field at the frequency 0 > wZ1 may also emit the phonons at the Raman (Stokes) frequency wq = of this emission fi - w21 (fig. 1). The intensity should be proportional to (ny + l), where nq is the number of phonons with the frequency wq = w2, and wave vector 4. nIt is quite obvious that a monolayer or a number of monolayeers of adsorbates can hardly excite phonons in the bulk of the crystal. On the other hand, the Rayleigh surface phonons (which are mainly concentrated near the surface of the crystal) could be excited by electromagnetic field with frequency R > 02,. Our aim is to find a threshold of excitation of stimulated emission of phonons with energies h(0 - ozl). This threshold is determined by the concentration of adsorbates excited to level 1 in the crystal domain where surface phonon modes are essentially nonvanishing (about surface phonons, see Appendix A). To analyze the problem analytically we will choose a certain model. This model will be characterized by effective Hamiltonian [27] representing transitions with emission (absorption)


Fig. 1. Raman R-Cl&,.


of phonons

of frequency

w,, = w, =

B. Fain and S. H. Lin I Photodesorption

of phonons

at Stokes


R, =

0, = n - wzl .


The energy that may be emitted or absorbed at the Stokes frequency depends on the state of the molecule. If the molecule is in the low energy state E,, it may emit the quant of the energy fiw,; if the molecule is in the state with higher energy E,, it may absorb the quant of the energy ho,. The effective Hamiltonian of the molecule (apart from is interaction with phonons) will have two eigenvalues, the difference of which should be equal to ho,. Such a Hamiltonian may be represented in the form HMj = -rr,jhos


[r,, r21 = ir,;

[r2, r31 = ir,;


Now we are able to write an effective Hamiltonian for molecules occupying levels 1 and 2 and subjected to laser field of frequency 0 [27]. + c



- C [D$‘l~L?)r,~aq i4


where the second sum presents energy of lattice vibrations of the crystal in harmonic approximand annihilation ation; a:, a4 are creation operators of phonons in the qth vibrational mode of the crystal. The third sum in Hamiltonian (2.4) is the interaction energy between the molecules and phonons in the presence of the laser field of frequency R. The effective matrix elements D”’qmn( 0 ) ma Y be found from theory of stimulated Raman scattering [27, 29, 301 Db’,l,(fl)

= D;;;(-0) =__ ;x k


A,,,V”‘(l2) kn. n-wkn++ 2

v!i:(fi>A,,n +




w mk-a-+ 21

where v;;

= v;;(n)

eifit + v;;(-n)



and T2 is a transverse relaxation time of transitions nk and mk and V(‘) is the energy of interaction between the laser field and jth molecule. Quantities A, are coefficients in the interaction energy between the molecule and phonons:

[r,, r,l = ir, ,

r: = ri = r: = + .

H = - R,ho,

Cr,j , i


where r,j is the effective spin operator of jth molecule. The effective spin [29] operator has the same properties as the usual operator of spin l/2. The only difference is that it is not necessarily connected with real spin. As for usual spin operator, the eigenvalue of operator r3 are equal to -l/2, l/2. The spin in eq. (2.2) is chosen in such a way that r3 = -l/2 corresponds to lower energy E, (and r3 = l/2 to energy E,). In this case, (when r3 = -l/2 and energy equals E,) the molecule may emit the Raman quanta tie,. The other components yl, r2 of the effective spin operator are isomorphous to the usual spin operator and obey the relations


via stimulated Raman emission of phonons

+ D/I’,<-O)rTa,‘]








+ Ay’ai]



Since we are interested in phonons of frequency the term (2.8) is not included into w =o H~mil&ian (2.4). The term (2.8) leads to transitions between levels E, and E2 (and between others as well) involving phonons with frequencies

. wq = (E, - E,)lh



(2.4) Here

These transitions mal equilibrium.

lead to the relaxation to therThey will be taken into account

B. Fain and S. H. Lin


I Photodesorption

by introducing corresponding relaxation times T; and T, for the components R’ and r3 respectively (see fig. 1).

3. Self-excitation conditions for stimulated emission of coherent phonons Now, we will examine conditions of self-excitation of stimulated Raman emission of phonons. To find conditions of self-excitation of coherent phonons (or condition of instability of phonon system) we should derive equations of motion for mean values of quantities for corresponding operators. Equations of motion for the operators a a+ r,? follow from Hamiltonian (2.5) accordir$ tz ;he well-known rule



where A is an arbitrary operator having relation to the system represented by the Hamiltonian. From this rule, Hamiltonian (2.5), commutation rules (2.3,4) and those for a+ aq: [a:, aq.] = of 6 ‘9 we obtain the follow!ng equations mqZtion for operators r,: , a:, aq: + i C 2D~~\(fl)r3jaq ,

i,? = i(w,, - 0)r,!



via stimulated Raman emission of phonons

Now we assume that the molecules are distributed uniformly over the volume of the crystal. In our problem, having in mind an excitation of the Rayleigh surface phonons, it means that the molecules are uniformly distributed in the surface layers. In the case of uniformly distributed molecules one can linearize eqs. (3.2) replacing operators r3j by their equilibrium meaning which are the same for each position j: 0 r3j = r3 .


According to the definition of r3 [(2.2), (2.3)] its average value has the meaning (in our case) r3, = (1/2)(P,

P2 + P, = 1 .

- Pi);


P, and P2 are populations of states 1 and 2, respectively. Since we assume that P2 = 0 (before the instability develops) we have R3j = ri = -(1/2)P,

= -l/2.


Substituting (3.5), (3.7) into (3.2), averaging these equations over appropriate quantum ensemble and taking into account relaxation processes (see, e.g., ref. 30) we obtain

R, + i(w,, -


+ fRi

+ :2ria,

= 0,


‘,r = -i(o,, ti; - i(w, + ir,)a:

+ iDq~;



Here, 7, is rate of relaxation of ai and we have used the orthogonality property of functions D(j).


a, (3.3) ci, = -io,a,

+ i c Di{‘,(-fl)r,?





From Hamiltonian (2.9) it follows that Aq( j) must be proportional to the phonon eigenmode. The same relates to D, (0). It implies that we can expand operators ‘:’f over these quantities: r,? = c RADIO\; 4

ri = c R,Dk{L(-0) 4

. (3.4)

where D, = 2



i Here, we designate averaged values of operators Ri, a: by the same letters as operators themselves. Now we can investigate stability conditions of

B. Fain and S.H.


I Photodesorption

our system: the phonons coupling with the molecules subjected to electromagnetic field. This problem and eqs. (3.8) are isomorphous to the problem of self-excitation conditions of a laser [28,29]. Let us assume that there is a small perturbation which has the form


via stimulated Raman emission of phonons

ation of the molecules. When D, exceeds a certain critical value, the instability of phonons arises (cxY:, > 0) at frequencies close to 0, = 0 wzl_ Outside this region the phonon oscillations are damped (cx~

A system of phonons becomes unstable provided (3.12)


In this case, a small perturbation is exponentially increasing with time. Of course, this condition depends on q and frequency of phonon modes. Only those phonon modes are unstable for which condition (3.12) is fulfilled. Substituting (3.11) into (3.8) we obtain the following complex characteristic equation for a4 :

a;+aq[ f +i(w -

oq) + r,




i(w - w,) T2






On the other hand, at the frequency wq =




- ;(T;’


Y~Y,)~ -

+ yJ.





As we see a condition of instability (i.e. of existence of real boundaries (3.16) and positive ai has the form 2D,ri



a positive real part of tiq has its maximum value


For real and imaginary parts of complex quantity





> yqT;’



Thus, we come to the conclusion that a threshold

two equations follow from (3.13):


2riD, 2 h



cws -4





where w~=CJ-W~~. A quantity D, (see Appendix B) depends on the laser field and concentr-

Fig. 2. Dependence of ui, a real part of the exponential coefficient exp (a,t), on phonon frequency wq.

B. Fain and S.H.


Lin i Photodesorption

condition of excitation of coherent determined by the relation



2D,riT, (3.19)

=l. fi2Y, Let us designate



the condition

of instability


via stimulated Raman emission of phonons

data (such as coverage dependence of the desorption yield, nonspecifity of the desorption, like C,H,N vs. C,D,N when either type of isotopic coadsorbates was initially excited) show that laser-induced desorption has collective character. In addition to existing collective laserinduced desorption mechanisms, this paper suggests a new one. The molecule is subjected to a laser-field of frequency 0 (fig. 1). This field is the source of stimulated Raman emission of phonons at the frequency

the form WY =





n>l and eqs. (3.16), (($

w, =


&j)2 =


are rewritten

(l+ hT2) (n -

in the form

1) )



These phonons which come to increase exponentially in time may give rise to the desorption, i.e. to a rupture of the adsorbate-surface bond. Estimates of conditions of phonon instability we find from (3.21), (3.20) and (B.12) with r3 = -l/2 D,T,

1] .

X [w_

Thus, when condition with frequencies


is satisfied,






75”W - ~21)~+ T,‘lrJ,


As we see, this condition the coverage 0. For




4 1


> I ’ depends

(4’2) on

(4.3) (4.2)


the form





(1 + y,T,)(n


- 1)



are unstable and the phonon rise exponentially


ii, = n,(O) e2ai’.




A maximum value of (Y; is achieved at central frequency w, of instability region and equals to that given by eq. (3.23).


h2[(R - w~,)~ + T,‘]

E ‘--K&Z>l.


0 Y,


Indeed, there are unknowns in this estimate. One of them is a reorganization energy E,. In the theory of rate processes in condensed media this quantity is usually compared with Debye energy r’& in order to divide between so-called weak and strong coupling cases. We may tentatively put E, - hw,

4. Conclusions,


There is a need for considerating lective laser-induced desorptions.

various colExperimental

In this case factor E,I?‘iT,’ may be much larger than unity, the same relates to w4/‘ys. This provides a hope that even at moderate fields

B. Fain and S.H.

Lin I Photodesorption

]V,,l’ stint,*, condition (4.4) may be fulfilled. On the other hand, a left-hand side of (4.2) saturates at coverages 83--_


In this case an estimate for condition (4.2) takes place IGI’ h2T,2

t;’ - hD>

wq E, y, hT;*=

1 ‘d

05(1/T,) -

Comparing In

(4.9) and (4.10) we obtain that for 2 wD




In the case when n > 1 we can compare thermally induced desorption rate with that induced by instability in phonon system. As an example we will use a formula (15) of paper [34] for thermally induced (by surface phonons) desorption rate which we rewrite in the form (4.7) where f(op) is some function defined in [34] and we assumed here the Debye frequency distribution and tip is an equilibrium number of phonons. For the case of stimulated Raman emission of phonons using (3.22), (3.23) and )2> 1 we will get by the same token

(4.8) Just as an example let us consider -y,T2 - 1 and k,Tfiw,. Then for (4.7) we get



2 WD


which may be satisfied more easily, since it contains two large factors 04/y, and E,IfiT,’ and does not contain factor Karl which is small in case (4.3). To estimate condition (4.5) one should take into account an estimate for K = (1 la) (wq/q,). This means that saturating condition (4.5) takes the form 6JD f33---. wq


and for (4.8)




via stimulated Raman emission of phonons




a desorption rate induced by nonstable Raman phonons is larger than thermally induced desorption. It is expedient to mention that this estimate does not depend essentially on the details of estimation of the desorption rate, due to the Thus, logarithmic nature of the estimate. taking w,-lO*cm-‘; w,-l/T,-lOcm_’ and f(OD)


f(%> we




It means that a laser pulse of duration ta4T2


will lead to a desorption rate exceeding a thermal one. It is expedient to mention that the abovedeveloped theory and its estimates may be germane to the analysis of new experiments [35,36]. In these experiments infrared-laserinduced photodesorption of NH, and ND, physisorbed on NaCl, Cu and Ag crystals was investigated. In these cases one-photon desorption was observed. According to formula (4.2), a condition of exponential rise of non-equilibrium phonons is a one-photon process. This process essentially depends on the coverage 8, saturating at large values of 8. Once phonon instability is developed, it may cause desorption of both NH, and ND, (while initially only levels of NH, are excited). This process has a resonant character: only phonons with frequencies wq - T,’ are excited. For laser pulse duration or 6 x 10m9 s

B. Fain and S. H. Lin / Photodesorption


bulk-like modes

[3.5,36] and T2’ - 10 cm T,‘t-

via stimulated Raman emission of phonons

10”. (A.4)

This has to be compared

with condition




Here, S = AB is the area of the surface and C is a depth of the crystal: we consider the case


The authors wish to thank U.S.-Israel Binational Foundation for financial support of this work.





According to 1311, the Rayleigh modes satisfying normalization condition (A.4) take the form


Bulk-like and Rayleigh surface phonon modes


A lattice displacement at point j at time t may be presented by the known expansion 112 u(j,




where R, = (x, y),

Cw,’ 4


x [a,W,(i)

+ a,‘@)$

For infinite solid the eigenmode I~~(T,t) = +$

(i>l .






K = (Y - n)(~ - n + 2~77~)/2~77~>







uq equals


eiqr ,




n=j/l-C,lCJ2. where V is the volume of the solid, and eqV is a unit polarization vector. For a crystal bounded by a surface, there are two kinds of surface phonon modes uq : (a) bulk-like modes, containing a wavelike component propagating into the bulk of the crystal, and (b) the Rayleigh modes propagating over the surface and decaying in the direction z < 0 which is perpendicular to the surface of the crystal. One can show that bulk-like surface modes are of the same order of magnitude as (A.l), i.e., ]U)jULkl - ]uq] = +v




The situation is quite different when we consider the Rayleigh mode decaying inside the surface. They have the same normalization conditions as

C, and C, are constants related to the Lame elastic constants: A and p, and the mass density of the solid: c, =j/Q+2/J)lp




these are identified as the propagation velocities of the longitudinal and the transversal waves, respectively. Comparing (A.l) and (A.5), we can write an estimate for the Rayleigh mode U,(R,, 0) -- &



where the estimate for an effective volume of the Rayleigh mode takes the form

B. Fain and S.H.




Lin I Photodesorption

SK-l .


Thus, while infinite solid modes and bulk-like surface modes have a volume V= ABC = SC

which tends to infinity with C+ 00 and S finite; the Rayleigh modes have volume (A.13) which is finite for S finite and has a volume of a layer of depth K-~ near the surface area.

Here 8 is a number of monolayers (which may be non-integer) of adsorbates (a coverage); a is a thickness of a monolayer; r] = NIV is a density of adsorbates which is assumed to be constant up to a depth 0a. A quantity (A,,, - Aq,,12 may be found from Hamiltonian (2)-(S) which we represent in the form (2.9)


-c ~:Q!,&, 4i


and (3.5) the

lD$‘z(W)]’ ,


c[ AqlkVg+on)

= ;




- ok2 -i/T,





The leading term in this sum is with lowest denominator 0 - wzl + i/ T2:


_&;Q;q/-$(aq qi =

+ ‘,‘) 4

c At&(a,

+ a,‘).


Now, we see that the interaction energy has the form conventionally used in the theory of rate processes in condensed media. It has been shown [32,33] that for low frequency acoustic phonons quantity ( QL - Q$) equals (for bulk modes)




IA,,, - A,,, t2= fiw,E,;



x V,,(-W(A,,, - A,,,).


where E, is so-called reorganization

Thus the expression for Dq has the form

Dq=C fi’[(a i

W’ _ w21)2 + l/T;]

IA‘222 - Ad2





E, = ;

c o;

Having in mind that A, is proportional to the eigenmode we may approximate an expression IA,,, - Aq,,12 for the Rayleigh surface phonons in the form


IA,22- Aq,,12=


I d+Iq22 0







- a;,)‘.


(B.4) IY2l’

Dq =



1 = h(o _ wzl + i/T2)


Ak(‘,(a, + a,‘)


where D;;‘z(_a)



for system in the state 1, and correspondingly the state 2:

An estimate of quantity D,




Appendix B

D, = c


w',Q;,Q, =



According to definitions quantity Dq has the form


via stimulated Raman emission of phonons

s~_~jv , e-2rz



B. Fain and S.H.

Lin I Photodesorption

where K iS of order of magnitude YK (7K) in (A.6). Expression S(~K-~) in the denominator is of order of magnitude of the effective volume of the Rayleigh mode (A.13), and S(2~-‘)77 = NR is a number of adsorbates in the layer of thickneSS eqUd t0 K ‘. Substituting expression (B .11) into (B.5) we get the estimate for D, (B.l) in the form D,

= h2[(L!IV1212~~,4 (1 - w21)2 + T;*]



References 111 J. Heidberg,

H. Stein, E. Riehl and A. Nestman, Z. Physik. Chem. N.F. 121 (1980) 145. PI J. Heidberg, H. Stein and E. Riehl, Phys. Rev. Lett. 49 (1982) 666. H. Stein and E. Riehl, in: Vibrations and [31 J. Heidberg, Surfaces, R. Caudano. J.M. Gilles and A.A. Lucas, eds. (Plenum Press, New York, 1982) p. 17 [41 T.J. Chuang, Surf. Sci. Rep. 3 (1983) 110.5. PI T.J. Chuang, J. Chem. Phys. 76 (1982) 3828. 161 T.J. Chuang and H. Seci, Phys. Rev. Lett. 49 (1982) 382. [71 T.J. Chuang and I. Hussla, in: Proc. of 17th Jerusalem Symp, in: Quant. Chemistry and Biochemistry on Dynamics of Molecule-Surface Interaction (1984). J. Electr. Spectr. Relat. Phenom. 29 PI T.J. Chuang, (1983) 125. 191 M. Mashni and P. Hess, Chem. Phys. Lett. 77 (1981) 541. 1101 B. Schafer, M. Buck and P. Hess, Infrared Phys. 25 (1985) 245. [Ill B. Schafer and P. Hess, Appl. Phys. B38 (1985) 1. [121 M. Buck, B. Schafer and P. Hess, Surf. Sci. (in press). J. Lin, A.C. Beri and W.C. Murphy, [I31 T.F. George, Progr. Surf. Sci. 16 (1984) 139.

via stimulated Raman emission of phonons [14] Z.W. Gortel, H.J. Kreuzer, P. Piercy Phys. Rev. B27 (1983) 5066.

and R. Teshima,

[15] Z.W. Gortel, H.J. Kreuzer, P. Piercy and R. Teshima, Phys. Rev. B28 (1983) 2119. [16] D. Lucas and G.E. Ewing, Chem. Phys. 58 (1981) 385. [17] H.J. Kreuzer and D.N. Lowy, Chem. Phys. Lett. 78 (1981) 50. [18] H.J. Kreuzer and Z.W. Gortel, Phys. Rev. B29 (1984) 6926. [19] B Fain and S.H. Lin, Surf. Sci. 147 (1984) 497. [20] G.S. Wu, B. Fain, A.R. Ziv and S.H. Lin, Surf. Sci. 147 (1984) 537. [21] B. Fain, A.R. Ziv, G.S. Wu and S.H. Lin, in: Adv. in Multiphoton Processes and Spectroscopy, Vol. 1, S.H. Lin. ed. (World Scientific, 1984). [22] B. Fain and S.H. Lin, Chem. Phys. Lett. 114 (1985) 497. [23] B. Fain, Chem. Phys. Lett. 118 (1985) 283. [24] H. Seci and T.J. Chuang, Solid State Comm. 44 (1982) 473. [25] B. Fain and S.H. Lin, Physica B (in press). [26] B. Fain and S.H. Lin, Physica 138B (1986) 63. [27] B. Fain, J. Appl. Phys. 54 (1983) 4832. [28] B. Fain and Ya.1. Khanin, Quantum Electronics (Pergamon Press, Oxford, 1969) Vol. I. B. Fain, Photons and Non-linear Media (in Russian) (Sov. Radio, Moscow, 1972). [29] B. Fain and Ya. Khanin, Zh. Eksp. Teor. Fiz. 41 (1961) 1498; Sov. Phys. JETP 14 (1962)1069. [30] W. Heitler, The Quantum Theory of Radiation (Clarendon, Oxford, 1954). [31] H. Ezawa, Ann. Phys. (NY) 67 (1971) 438. [32] B. Fain, Phys. Stat. Sol. (b) 63 (1974) 411. [33] B. Fain, Theory of Rate Processes in Condensed Media (Springer-Verlag, Berlin, 1980). [34] E. Goldys, Z.W. Gortel and H.J. Kreuzer, Surf. Sci. 116 (1982) 33. [35] I. Hussla, M. Seki, T. Chuang, Z.W. Gortel, H.J. Kreuzer and P. Piercy, Phys. Rev. B32 (1985) 3489. (361 T.J. Chuang, M. Seki and I. Hussla, Surf. Sci. 158 (1985) 525.