Influence of phonon inelasticity upon atom-solid scattering intensities

251

Surface Science 203 (1988) 257-275 North-Holland, Amsterdam

INFLUENCE OF PHONON INELASTICITY SCATTERING INTENSITIES

UPON ATOM-SOLID

Gert Due BILLING Chemistry Laborarory III, H.C. Orsted Institute, Unwersity of Copenhagen. 2100 0 Copenhagen, Denmark Received

18 February

1988; accepted

for publication

25 April 1988

We have extended a semiclassical theory for atom/molecule-surface scattering such that the influence of phonon inelasticity upon diffraction peaks and scattering intensities between these peaks may be calculated. As an example the scattering of 4He from NaCl is considered.

1. Introduction Examination of intensity patterns of atoms scattered from solid surfaces gives information upon lattice geometry and the number of bound states supported by the atom-surface interaction potential. At small collision energies the elastic diffractive scattering and selective adsorption processes are the most important phenomena. At higher energies phonon inelastic processes start to influence the intensity pattern of the reflected atoms. The analyses of time-of-flight spectra is considerably more complicated if many inelastic channels are open. So unless one-phonon excitation is dominating some theoretical investigations appear to be necessary before such spectra can be interpreted. The number of transitions which are allowed increases with incident energy and surface temperature. Rapidly the number of accessible channels becomes so large that approximate methods must be introduced. We have previously formulated a quantum classical model in which the phonons are quantized whereas the motion of the incident particle is treated classically. However, in order to introduce a quantum description of the diffraction channels we quantize the motion along the X, y directions treating only the z-motion (perpendicular to the surface) classically. This approximation is expected to be valid if the kinetic energy in the z-direction is much larger than the well depth (cm), i.e. E cos28, 2 (5-10) x z,, where Bi is the incident angle. If this condition is fulfilled we are excluding situations where adsorption may occur. For smaller energies or grazing collisions it is necessary also to quantize the z-motion. The above approximation has previously been used by De Pristo and coworkers for rigid surfaces [l] and for vibrating surfaces using the 003%6028/88/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

258

G.D. Bdling / Influence ofphonon inelasticrty upon atom-solid scattering

GLE-description of the phonon motion [2]. In the same spirit is also the semiclassical perturbation (SCP) treatment by Miller and coworkers [3]. In the appendix we show how the present model in certain simplifying cases gives diffraction intensities which are identical to those derived from the SCP approach. It is therefore not surprising that about the same good agreement with exact quantum calculations [4] is obtained with both these models [1,3].

2. Theory If the motion along the z-axis, i.e. the axis perpendicular treated classically we may expand the total wavefunction problem as x(t,

x, Y) = Cxc(t>

G

exp[i(R+

G>

to the surface, is for the diffraction

‘~1,

(1)

where G = (n, m) denote the diffraction channels, R = (x, y) X= (K,, K.,.) and G = (2mn/a,, 27rm/a,) is a vector in the reciprocal lattice space. From the time-dependent Schrodinger equation iftax/at = [H,, + V(x, y, z)]x we then obtain a set of coupled equations in the expansion coefficients (2) where we have introduced Eo=

(h2/2m)(K+

GG = xG exp[(i/q)&t],

G)’

and that the potential

(3)

due to the x, y periodicity

can be represented

as:

&/0(x, y, z) = c V” exp(iG*R).

(4)

G

We note that by introducing

the average

wave vectors

K,=K,+:(G,,+G;)

(54

and j?,.=K,+:(G,,,+G,,,,)

(5b)

we can write

(6)

h-‘(EG-E,~)=(n-n’)ox+(m-m’)~~,, where w, = 2AaE,Jma, and similar for ox. Expression (6) is important for the analytical (2) given in appendix A.

solution

of the coupled

eqs.

G.D. Billing / Influence

ofphonon inelasticity upon atom-solid

259

scattering

For the z-motion we have the following classical equations of motion i=

PJm,

(7)

i: = - j$_,,(z,

t),

(8)

where &(z, t) = (x(t) 1V(x, y, z) 1x(t)) and the brackets denote integration over the quantum coordinates x and y. Using that $La.‘dx

exp[(2ri/a,)(n

+ h - n’)x]

= 8,,+h-n,;0,

we obtain V&r=: V,(z)

+

C

+T,,V,S_,$,

exp[(i/ti)(&-E,)r].

(10)

GG;;GGr Within the present time-dependent formalism it has previously been shown that the appropriate effective potential if phonon inelasticity is included is to first order in the phonon displacement given by: K,, = v,(x,

Y, z) + c @)(x3 K

Y? z)nlk(& T,),

(11)

where T/k(‘) denote the first derivative of the atom-surface with respect to phonon mode k. Thus we have [5]:

vP(X, Y, z) = -

C7;y;k(Y _

Cm;1/2(r;q)d1~

ar =l

i

interaction potential

p),

02)

Y

where y = x, y, z and r,, the distance from the atom to the ith solid atom and i’&;k defines the transformation from Cartesian to normal mode coordinates

(Qk), i.e. mf”(v, - fq) = CT,;,Q,.

03)

k

The function qk depends upon time and the surface temperature given by: 9k(t, T,) = - jdt’(h,)-‘$(AE;

and is

+A&) k

XIIcc.k(t’)

cos@,(t’)+I,,,(t’)

sinO,(t’)],

(14)

where

o,(t) I,,, =

= w,t + :wk-ljdt’V,‘;‘I,q, /

dt’ VL’) cos O,( t’)

(15) @a)

260

of phonon inelasticity upon atom-solid

G. D. Billing / Influence

scattering

and I,,, =

/

dt’ vi’) sin @,(t’).

(16b)

The integrals (16a) and (16b) defines the excitation function pk

(17) and thereby the excitation surface is given by [5,6]: AEint= c(AEz

+AE;)

of phonon mode k. The energy transfer +AE,z,

to the

08)

k

where we have divided the energy transfer in a term A El for phonon excitation and AE, for deexcitation. The term AE,: is the energy transfer due to phonon-W processes induced by the quadratic forcing V$ terms of the intermolecular potential. The phonon W contribution is usually small and may be estimated by a first-order solution of the problem [6]. The surface temperature enters into the W term in eq. (18) but only into the first term (the VT term) if a semiclassical velocity symmetrization is introduced [5]. If this symmetrization is omitted we have AE,+ + AE,

= Fiokpk,

(19)

i.e. independent of the surface temperature. Introducing the syrnmetrization we obtain energy transfer to the phonon which decrease with T, and as demonstrated in refs. [5,6] the surface may even transfer energy to the incoming particles. When calculating the forcing integrals (16) one could use the 3D classical trajectories or eventually the quantum expectation value defined as in eq. (10) for V,. Once the functions qk(t, T,) are defined we can expand the effective potential (11) in a Fourier serie I&=

c Q(z, G

r; T,) exp(iG*R),

(20)

where V,(z,

t; T,) = V:“(z) + &k(t,

T,)%:‘(z).

(21)

k

The expansion coefficient V, now enters into the coupled equations for the diffraction channels (2) instead of the coefficients I$. In this manner we can predict not only the influence of the surface temperature but also the effect of phonon excitation upon the diffraction intensities.

G.D. Billing / Influence of phonon inelasticity upon atom-solid scattering

261

3. Kinematics For atom-surface scattering from the following conservation Energy: A2k2 2rnf

we may calculate relations:

the final

scattering

ii2k? + AEint

=ti

angle

(22)

>

where AEint=Awk(n~-nk)

(23)

is the energy transferred to phonon mode k and where the initial and final m is the atom mass and ki, k, the phonon states are nt and nk respectively, initial and final wavevectors. Conservation of surface momentum: Kr = Ki + Gn, + Q,p,,: where

(24)

Ki and K, are the surface

Ki = (ki

sin 8,

COS 9i,

ki

projected

initial

and final wavevectors,

sin Bi sin pi)

i.e. (25)

and likewise for K,. The last term is the surface projected phonon momentum associated with the transition n: to nk_ In order to evaluate the last term in eq. (24) it is convenient to introduce the operator Qa,fak [7] which has the property that it when operating on the wavefunction ( nk) creates the parallel phonon momentum Q,,, i.e.

Qah

Id

(26)

= Qn, Id.

The quantity

of interest

is the change

in parallel

phonon

momentum

Q no,n,= Qn, - Qnu,= Q(nk - n”k)> where Q is sofar unspecified. mode k is now

= Cp,z(T)CC+niQ(nk 4

The average

change

in momentum

- nok)y

coming

from

(27)

nk

where P,o, is the probability for phonon state nf at surface temperature T, and P,,; ~“k the probability for the transition ni to nk. The summations in eq. (6) may easily be evaluated [5]. The result is

CQ> =

QPk.

The average parallel momentum may also be calculated the momentum of atom cy in direction y is: Pa, ==?H’,/‘c Tay,kPk, k

(28) using eq. (13). Thus

(2%

G.D. Billing / Influence

262

ofphonon inelasticity upon atom-solid

scattering

where

We now introduce (Uk) = ii

the expectation

value the operators

uz

and ak, i.e.

a: + X&j I

(30)

and

where the functions obtain

Qik relate to phonon

W

processes.

If we neglect

these we

(32) where the summation

runs over surface

atoms (a),

CY_:= - ib,/dr’

V$i) exp( +i@,(t’)),

and b, = (A/2ak)

‘I2 . The x and y components

Q=

(Q,,

Q,) =~-‘P;‘((P!~‘),

(33) of Q now obtains

by

(P;“‘)).

(34)

The phonon momentum may take the values + or - the quantity given by (34) and the quantity is furthermore shifted with the lattice vector *(2a/a,, 27r/a,) so as to lie in the first Brillouin zone. Inserting in eq. (24) we get k, sin 8r

cos

$q =

ki sin Bi cos

hi

+

(2rr/a,)n

+ Q,( nk - n:)

(35)

+ Q,( nk - HZ).

(36)

and k, sin 8, sin ~$r= ki sin Bi sin $i + (2a/a,)m

Eqs. (22), (35) and (36) determine the final angles (e,, +r) as a function of the diffraction channel (n, m), phonon mode and channel. The intensity or probability for this event is given by: (37)

p&Jp~:,-.~I~,,12~ where P,,;(T,) is the thermal distribution T, is sufficiently large we have Pny = (1 -z&P, where zk = exp( - ha,JkBTs).

of quanta

in mode

k. Assuming

that

(38)

G.D. Billing / Influence of phonon inelasticity upon atom-solid scattering

The phonon excitation processes) given by:

probability

is (in the absence

of phonon

263

W

(39) and the diffraction channel (n, coupled equations (2).

4. The He-NaCl

m)

distribution

is obtained

by solving the

system

Since we in the present calculations want to include the phonon inelasticity it is necessary to have information upon the variation of the potential with the atom-displacement. If the potential is represented by a sum of atom-atom pair potentials this information is in principle given. Although this dumbbell model not yet has been confirmed for atom-solid interaction by ab initio calculations it is at present the only easy way to get the necessary information. For a system as He + NaCl it has recently been shown [8] that the pair wise additive assumption is fairly good. Thus we assume that the atom surface interaction potential may be expressed as:

where t”‘,(p)=A

cbp-

(~~~P~~~OP~"+~~~P~~SOP~~).

(41)

The summation in (40) runs over atoms in the two first layers and the switching functions fzn are [8]

fzn= 1 -

eebp F

(l~p)~/k!

(42)

k=O

The potential used by Hutson and Fowler [8] contained additional small contributions from three body interaction and induction terms which have been omitted here. The Fourier transform (4) is calculated at N,N, points in the interval [ - +a,; + :a,[ and [ - ia,; + :a,[ where a, = uY = 5.64 A are the lattice constants for the NaCl crystal. Due to the rather modest corrugation it turned out to be sufficient to use N, = NY= 9 which yielded an average relative rms Fourier representation of the surface of about 0.5%. The 81 points were then used in the multidimensional Fourier transform NAG library routine CO6 FJF to obtain the coefficients V,“,(z) or vn,,,(z) in eqs. (4) or (11) respectively. The V;(z) term has a well depth of 4.75 meV at z = 3.6 A if we use the potential parameters given in table 1 (see fig. 1).

264 Table 1 He-NaCl

G.D. Billing / Influence ofphonon

potential

He-ClHe-Na+

parameters

inelasticity

used in the present

670.4 2500

upon atom-solid

scattering

work (see ref. [8])

3.204 5.082

9.14 0.82

56.41 2.065

l< = 100 kJ/mol.

For the Na+Cll potential:

interaction

in the crystal

we use a simple

Uj,=Ae-aR-q2/R,

Madelung

(43)

where the parameters obtains from ref. [9]. From this expression we calculate the force constants for nearest and next By diagonalizing the second derivative matrix we nearest + - interaction. obtain the normal mode spectrum for the crystal. In these calculations

m2 15

Fig. 1. Dominating

Fourier

expansion

coefficients as a function for He-NaCl.

of the distance

t from the surface

G.D. Billing / Influence ofphonon

inelasticity

upon atom-solid

265

scattering

-

I L -

.O

0.2

-

Fig. 2. The fraction of phonon frequencies in Aw = 0.05 T-’

0.5

i

0.6

0.:7

intervals as a function of 0.

“crystal” sizes of 72 or 108 atoms were used. The atoms are placed in 2 or 3 layers with 36 atoms in each. The incoming He atom interacts with the two top layers of atoms giving the potential expansion terms shown in fig. 1. Fig. 2 shows the phonon density n(w), i.e. the fraction of modes in a frequency range w f Aw where Aw = 0.0025 7-r (1~ = lo- l4 s). The frequencies are clearly separated in a small frequency acoustical and a high frequency optical part. The incident angles were chosen to be Bi = C#B; = 45” and the kinetic energy E,, = 0.1 g (12= 100 kJ/mol). Under these conditions many diffraction and phonon channels are energetically allowed. We included all

G.D. Billing / Influence

266

ofphonon

inelasticity upon atom-solid scattering

Table 2 Probability P,,,, for various diffraction channels (nm) as a function of surface temperature and number of phonon modes included in the atom-surface potential (initially we have Pnm = S,,,6m,) m

n

Pnn, Elastic ‘)

-2 -1

-2 -1 0 1 2

0 1 2

a) No phonon

0.026 0.014 0.017 0.019 0.0023 interaction

No of phonon

modes 210

No of phonon

modes 318

T,=lOOK

T, = 300 K

T,=lOOK

T, = 300 K

0.032 0.016 0.023 0.014 0.0041

0.038 0.015 0.027 0.011 0.0059

0.035 0.012 0.033 0.012 0.0049

0.046 0.0073 0.046 0.0084 0.0082

included.

diffraction channels from ( - 5, - 5) to (5, 5) i.e. 121 channels and all 3N - 6, i.e. 210 or 318 phonon modes of the K 72 or K 108 crystal. Since each phonon mode has 5-20 open energy levels we notice that a full quantum treatment would require a solution of a 200000 (or more) channel problem. In the present semiclassical formulation the problem is reduced to a set of coupled equations for the diffraction problem but in a phonon perturbed effective potential. Table 2 shows that the possibility of exchanging energy with the quantum heat bath has an effect upon the diffraction peak intensities. They may change both in absolute but also in relative magnitude by a factor of two or more. Fig. 3 shows that it is mainly the low frequency acoustical modes with wk < 0.33 7-l which are excited by the collision and that the low frequency surface mode with ok = 0.032 7-l is “isolated” but not more excited than higher frequency modes. The excitation strengths pk shown in fig. 3 are not available - at least not directly - experimentally. If that had been the case valuable information upon the intermolecular phonon forcing would have been available since pk through the integrals (16) are directly related to the first derivative of the interaction potential with respect to phonon mode k. What can be measured however is the intensity between the diffraction peaks. In figs. 4 and 5 we have shown the calculated intensity pattern between 30 o 4 Br 4 60 0 for K 108 at T, = 100 and 300 K respectively. The intensities were calculated using final conditions for which pf c [45 o f 0.051 and by using the expressions given in section 3. The structure (intensity) between the diffraction peaks is about a factor of lo4 smaller and has therefore been multiplied by lo3 before being plotted as a function of the scattering angle 8,. In many experiments the angle 8; + 8, is fixed and equal to the detector angle 0,. Here the intensity is shown as a function of 8,. Although this is in principle also possible to simulate it would be very expensive. Each initial 8; value takes with 121 diffraction channels and 318 phonon modes included in the calculation about 6 min CPU on an Amdahl 1100 vector machine. But a complete 8,

G.D. Billing / Influence ofphonon inelasiiciiy upon atom-solid scattering

.3

0’.4

O’.S

0’.6

261

I 0.:7

FREQUENCY/lO-14SEC Fig. 3. Excitation strengths pk as a function of phonon frequency for He colliding with K 108. The kinetic energy is E,, = 0.1 i, incident angles Bi = & = 45 o and T, = 300 K.

scan can be obtained from such a calculation - this is therefore what has been shown in figs. 4 and 5. Each of the diffraction peaks is surrounded by phonon excitation/ deexcitation structure which if we are close to a peak “belongs” to that diffraction channel. The structure to the left of a peak is usually associated with phonon deexcitation and to the right with phonon excitation processes. Further away e.g. for 8, I 41” on figs. 4 and 5 the high frequency phonon excitation for the (- 1, - 1) diffraction channels start to mix with the

268

G.D. Billing / Influence ofphonon

inelasticity

upon atom-solid

scattering

1 t

s 55

I-1-1

I

(0 01

(

11

Fig. 4. Intensities as a function of final scattering angle 0,. The surface temperature and the diffraction peaks are located at 0, = 36.5 O, 45.0 o and 55.0 O.

T, is 100 K

high frequency deexcitation from the (0, 0) peak. If the surface temperature is lowered the population of the high frequency modes decreases and the high frequency structure between the diffraction peaks diminished (compare fig. 4 with fig. 5). With the small magnitudes found for the excitation strengths pk (see fig. 3) the dominating transitions are one-quantum transitions. This may be used to extract phonon information from the intensity plots. Thus if we

G.D. Billing / Influence of phonon inelasticity upon atom-solid scattering

269

c

5b

is

I T&R

c-1-1 1

50

IO 01

55

Cl 1)

Fig. 5. Same as fig. 4 but with T, = 300 K.

have only single phonon processes the intensity for deexcitation forced oscillator model) given by [5]: I_ =

zY’I,(t;)

exp ( -p $g

is (within the

270

G.D. Billing / Influence ofphonon

inelasticity

upon atom-solid

scattering

and for excitation I,

=~;~‘~I,(tk+) exp ( -p :$)!

(44b)

where zk = exp( - tzw,/k,T,), tk’ = 2pt&/(l - zk) and I, a Bessel function. If for this qualitative analyses we neglect the difference between pl for excitation and pk for deexcitation we obtain the intensity ratio for deexcitation over excitation: I-/I+

-z,
(45)

41

41.5

42

42.5

TETA Fig. 6. High resolution

scan of fig. 5 between

40 o and 45 o

G.D. Billmg / Influence of phonon inelasticity upon atom-solid scattering Table 3 Dominating

40.45 42.17 42.23 42.38 42.98 44.95 45.60 47.33 47.82 48.20 48.40

“phonon”

intensities

between

40 o 2 19,I 50 ’ (7” = 300 K)

Diffraction channel

Phonon transition

Phonon frequency

0 0 0 0 0 0 0 0 0 0 0

n+n-1 n-n*1 n+n-1 n-n-1 n-n-1 n+n-1 n-n+1 n-nil n-n+1 n-n+1 n-n+1

0.295 0.313 0.167 0.158 0.119 0.292 0.292 0.119 0.182 0.158 0.167

0 0 0 0 0 0 0 0 0 0 0

271

Intensity

Dominating transition

0.0089 0.0071 0.0104 0.0098 0.0069 0.0051 0.0051 0.0090 0.0065 0.0120 0.0127

1 j 0(19%) l+ 2(17%) 2 ---tl(lS%) 2 j 1(18%) 3 + 2(20&) 1 j 0(37%) 0 + 1(33%) 2 + 3(14%) 2 * 1(20%) l+ 2(13%) l---* 2(14%)

(7-l)

A@ = 0.017 o

Thus the peak hight to the left of the diffraction peak divided by the one to the right should be smaller than unity if single phonon processes dominates and if pz - pk. This is clearly the case at T,= 100K (fig. 4). From fig. 4 we can read off the ratio between the large peaks adjacent to the (0, 0) line to be about 0.64 which from eq. (45) gives wk - 0.06 7~‘. Since the correct value is 0.08 7-l the analyses shows that we may use it to get an approximate determination of the frequency, i.e. the phonon mode contributing to a given peak. From the peak hight we can through eqs. (44) determine the excitation strength pk. Thus we have shown that pk is in principle available from the intensity scans as those shown in figs. 4 and 5. At higher collision energies multiphonon processes begin to contribute significantly and the analysis is more complicated although in principle doable within the present model where the phonon excitation probabilities are given by analytical expressions. The lines shown in fig. 5 may be analyzed with a larger resolution than A(3 = 0.1”. Fig. 6 shows the region 40 o I f?rI 45 o with AB = 0.0166 O. The peak at 13,= 44.43O is connected to phonon inelastic n -+ n - 1 processes in the low frequency wla = 0.0316 7-i surface mode, the one at 43.62 is due to n + n - 1 phonon processes in mode w,~ = 0.079 7-i etc. (see also table 3). Table 3 also shows that these large peaks belong to the specular channel (0, 0) and that the dominating phonon transition contributes with 14-37s of the intensity. At 41.82O the high frequency phonon excitation processes from the ( - 1, - 1) peak starts to contribute. Likewise for 8 > 45 o where the dominating peaks are due to n + n + 1 phonon processes of low frequency modes close to the specular and mixing with n -+ n - 1 transitions in high frequency modes belonging to the (1, 1) diffraction channel at larger angles. Information upon the mode contributing to a given peak is available from

272

G. D. Billing /

Influenceof phonon , ,

I

I

I

I

rnelasliciry upon atom-solid

/,I

I I I I I I I I I I I I I I I I I I

Creation

0.05 -

0.01 -

z z 2 + z

0.05 ~

0.01 I[ i

---I Annihilation

Of = 18.L’

rl

Of = 12.3’

1

~

IL

Fig. 7. Time of flight spectrum

scartenng

as a function

15

kf/A-’

of k, for some of the dominating

peaks in fig. 5.

experimental time-of-flight (TOF) measurements [lo]. We have therefore in fig. 7 shown a TOF analyses of some of the large peaks in fig. 5. The simple structure shows that single quantum processes excitation or deexcitation dominates. To the left of the specular line 0, > 45 o we have mainly annihilation and to the right mainly phonon creation as mentioned above. The calculation yields of course the exact information upon modes and transitions which contribute (see e.g. table 3) and fig. 7 has only been given in order to mimic the experimental situation.

5. Discussion The approximations introduced in the present approach are: (a) a classical treatment of the motion along the z-axis: and (b) a harmonic approximation for the phonon dynamics. Furthermore the present calculations uses a relatively small number of surface atoms. However it is possible to extend this number with a factor of 3 to 5 without major difficulties. Also anharmonic terms (when known) may be included by using effective force constants [12]. Thus the only serious assump-

G.D. Billing / Influence of phonon inelasticity upon atom-solid scattering

213

tion is the classical treatment of the z-motion, which limits the approach to situations where the energy in the z-motion is somewhat larger than the binding energy as mentioned in the introduction. An approximate and easy way of improving upon this aspect would be to introduce a Gaussian wave packet in the z-coordinate. This would lead to “quantum trajectory” approach which recently has been formulated for gas phase scattering [13]. The advantages of the method are that it treats the phonon processes exact to infinite order, the introduction of a surface temperature dependent potential makes it unnecessary to introduce Debye-Waller factors and it is furthermore easy to in the calculation. include steps and other surface “anomalities” Previous theoretical work mainly by Benedek and coworkers [14] uses Greens function methods to treat the surface vibrations and introduces distorted wave Born or eikonal approximations for the inelastic scattering processes. To account for temperature effects Debye-Waller factors are introduced. These methods which treat the phonon inelastic dynamics more approximate than the present approach are without doubt less time consuming from a numerical point of view and they have therefore been used to simulate experimental TOF data [14]. In order to model a beam experimental one should in the simulation introduce a velocity spread, e.g. described by a Gaussian function with a half width defined by the experimental apparatus. This “convolution” has not been performed in the present exploratory calculations. Since the kinematics used when resolving the experimental data only gives the total phonon energy and momentum exchange one cannot from the intensity scans determine the number of phonons participating in the process. But if the intensity at a given angle is plotted as a function of time the TOF spectra obtains and they yield additional information. However even here it may be impossible to distinguish between single phonon excitation of a low frequency and multiphonon excitation of a high frequency mode. The present theory offers a way of obtaining this mformation even at energies where multiphonon processes are dominating. By comparing with experimental data one may then obtain information upon the “phonon inelastic” part of the interaction potential. Acknowledgement This research Council.

was

supported

by the

Danish

Natural

Science

Research

Appendix Considering for simplicity the situation where the diffraction potential has the leading terms V,, = V,(t) and V,,, = V+iO = V,(z) we get from eqs. (2): iA$,,,

= V,(z)+,,

+ Vi(z)(e’i”~‘+.f,.,+

e*iwlr$,,,r,).

(A.1)

G.D. Bdling / Influence of phonon melastrcity

274

Inserting

a product

itt& = i(V,&

function

$,,, = &,,,

upon atorn-solid

scattering

we get

+ V, e+‘w“+RTl)

(A.2)

and likewise for +,,,. The &diagonal matrix coupling the &-states may be diagonalized transformation matrix T the elements of which are given by [ll]:

by a

(A.3) where 8 = r/( N + l), j? = exp(iw,t) eigenvalues are h,=2cos[km/(N+l)],

k=l,...,

and N the dimension

of the matrix.

N.

The

(A.41

If V,(t) = Vl( - t), i.e. the potential is symmetric in time around the turning point for the z-motion at t = 0 we may neglect the antisymmetric V, sin( w,t) term in (A.l) and the transformation matrix becomes time-independent with p= 1. In this case the solution is

where P, = i”mJ, (-p)

with P,(-p=

- i”-J,+(

-p),

00) = C$ ,,,,,,1, k_ = n -n,,

cc dt Vl(z(t)) J --Ix,

(A.5) k, = n + n, and

cos(w,t).

(A.6)

Usually Jk+ +c Jk_ and we finally obtain the probability initial diffraction channels nom,, to nm as:

for excitation

from

dt V, cos w,r

where J,( ) is a Bessel function. Expression (A.7) is identical to that obtained by Hubbard and Miller [3] using the semiclassical perturbation method.

References [l] [2] [3] [4] [5]

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G. D. Billing / Influence of phonon inelasticity upon atom-solid scattering [6] [7] [8] [9] [lo] [ll] [12]

215

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