Inelastic scattering of neutral particles by a solid surface

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Inelastic scattering of neutral particles by a solid surface

G. Armand

To cite this version:

G. Armand. Inelastic scattering of neutral particles by a solid surface. Journal de Physique, 1989, 50

(12), pp.1493-1520. �10.1051/jphys:0198900500120149300�. �jpa-00211011�

Inelastic scattering ^{of neutral} particles by ^a solid surface

G. Armand

Service de Physique des Atomes et des Surfaces, Centre d’Etudes Nucléaires de Saclay,

91191 Gif-sur-Yvette Cedex, ^France

(Reçu ^{le 12} janvier 1989, accepté ^{le 13} février 1989)

Résumé.

La diffusion inélastique ^de particules couplées ^à

champ ^de phonon par

potentiel d’interaction est analysée

^{détails. Le} développement

^{série de} perturbation ^de l’équation de matrice T permet ^de décomposer ^la probabilité ^de transition entre état initial et état final

composants élémentaires. Ces termes décrivent les processus d’échange ^de phonons ^réels

et virtuels. Ils sont

étroite relation

les éléments de matrice du processus élastique :

connaissant leurs expressions analytiques

^les diagrammes ^les représentant, ^l’on peut ^déduire les expressions

diagrammes ^des composants inélastiques correspondant. Les relations d’unitarité valables à chaque ^{ordre de} perturbation ^sont établies. La probabilité de transition

d’échange ^d’un phonon ^{réel est} donnée par

de termes élémentaires décrivant

l’échange ^d’un phonon ^{réel et} l’échange ^{de 0, 1,}

phonons virtuels. Ces termes sont

proportionnels respectivement ^à T, T2,

Tn + 1 ... à

température ^{T du} cristal suffisamment élevée. La probabilité d’échange ^{de deux} phonons ^réels peut être écrite de la même façon, ^les

termes étant alors proportionnels ^à T2, T3, ..., ^Tn+2. L’état final peut ^{être soit}

état du continu

ou un

état lié du potentiel. ^Dans

^dernier

^la probabilité ^de transition donne le coefficient de capture. ^Les probabilités d’échange ^d’un phonon ^réel comprenant ^{tous les termes} proportionnels

à T et T2 et celles d’échange ^{de deux} phonons ^réels comprenant ^{tous les termes}

T2 ont été calculées

représentant ^le potentiel d’interaction par

potentiel unidimensionnel.

La probabilite de transition d’un phonon ^réel présente

^maximum lorsque ^la température ^du

cristal varie. Sa position dépend ^{de la nature de la} particule ^{incidente et} des conditions d’incidence. Le coefficient de capture ^croît

^la température du cristal. Le domaine de validité de l’approximation dite DWBA est déterminée.

Abstract. 2014 The inelastic scattering ^of particles coupled ^to

phonon ^field by

interaction

potential ^is fully analysed. ^The expansion ^{of the T matrix} equation ^allows

^{to decompose the}

transition probability ^between

initial and

final state into elementary components. ^These components exhibit real and virtual phonon processes. They

closely ^{linked to} the elastic matrix elements : from their analytical expression

diagrammatic representation

^deduce

the corresponding expression

diagram of the inelastic transition probability components.

Unitarity ^{relations to} each order of expansion

demonstrated. The probability ^{for the} single

real phonon exchange ^is composed ^of

^of components describing ^{with the} exchange ^{of the}

real phonon ^the exchange ^of 0, 1, ^{..., n}

^virtual phonons. ^{These terms}

proportional ^to T,

T2,

Tn

1 ... respectively ^at sufficiently high crystal temperature ^{T. The} probability ^{for two}

real phonon exchange

^be decomposed ^{in the}

^{way in terms} proportional ^to T2, T3, ..., Tn+2

^{The final state}

^{be either}

^continuum

^{bound state.} In this last

the transition probability, gives ^the sticking coefficient. The transition probabilities ^{for the}

^real

Classification

Physics ^Abstracts

03.80

61.14D - 68.30

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500120149300

phonon exchange including ^{all the terms} proportional ^{to T and} T2 and for the two real phonons

process including ^{all the T2 terms} have been calculated using

dimensional potential ^model.

The single phonon transition probability ^exhibit

^maximum

^the crystal temperature varies. Its

position depends

^{the nature} of the scattered particles ^and

the incident conditions. The

sticking coefficient increases with the crystal temperature. The domain of validity of the DWBA is determined.

1. Introduction.

Among the different methods which allow calculation of the inelastic scattering ^{of an incident} particle coupled ^{to a} phonon ^field by ^an interaction potential, ^{the T} matrix formalism is

certainly ^{one of the most fruitful.}

To the author’s knowledge the first work in the field using this formalism is that of Manson

ans Celli [1]. The authors neglect the processes off of the energy shell and calculate the ^one-

phonon ^cross section. Since this work, calculations have in general been limited to the

simplest ^term given by considering the first order in the expansion of the T matrix equation,

which is called distorted wave Born approximation. The papers published are numerous and the list given ^{here is} certainly ^not exhaustive [2-8].

The state of art is most advanced as far as elastic scattering is concerned. From the work of Levi and Suhl [9], who demonstrate that the elastic intensity ^is given by ^the square ^{of the}

thermally averaged elastic matrix element, ^{an exact} theory of this process has been developed

which includes one [10], ^two [11] ^and multiphonon virtual processes [12].

Using ^the technique elaborated for this study ^we present in this paper ^a detailed analysis ^of

the inelastic processes where the particle exchanges energy via the transfer of real phonons.

However we show that this exchange ^can be achieved with the simultaneous exchange ^of

virtual phonons.

Considering ^the possibility of simultaneous real and virtual phonon processes ^one feels that the elastic and inelastic processes ^are closely ^{linked to} each other as it is the case on a physical

intuition. Effectively ^{we show} that, knowing the elastic matrix element, ^{one can deduce}

inelastic transition probabilities ^{and one can} demonstrate some useful

unitarity

^relations.

The formalism being ^well established we write the transition probability component proportional ^{to T and T2} (T ^{is the} crystal temperature) ^for scattering by ^a simple potential

model. Then the reflection coefficient to a continuum state and the sticking coefficient are

calculated for typical ^cases.

2. Général theory.

Let us consider a particle ^scattered by ^a surface where the interaction potential ^is thermally displaced ^{due to} the thermal agitation ^{of the} crystal ^atoms. This thermal agitation is taken into account through ^the displacement operator u ^{and the} potential ^can be written as

V (R, ^z, u ) where R and

are the particle coordinate parallel and normal to the surface, respectively. The static potential ^obtained by performing ^the thermal average of V over the

crystal phonon field and denoted by V, ^is, generally speaking, periodic. ^{Hence the} particle

is scattered elastically ⁱⁿ diffraction channels or inelastically ^{in numerous} open channels

corresponding ^{to the} exchange ^{of real} phonons.

The T matrix equation which describes the scattering process completely ^can be written as :

with

where Ef ^and Eic ^{are the initial} energy of the particle ^{and the} crystal, respectively, He ^{is the} crystal Hamiltonian and

In Ho ^{m is the mass of the incident} particle ^and U(z) ^{the distorted} potential ^{chosen for}

instance equal ^{to the zero} order Fourier component of «V ) .

In order to get ^a solution of this equation ^{it is} convenient to decompose ^the quantity V (R, ^z, u ) - U(z ) ^{into two} parts :

with

Then following ^a procedure presented in many textbooks [13] and used in a previous paper ^to

study the elastic scattering by ^{a surface} exhibiting adsorbed defects [14Î, ^one gets :

with

and :

h is the matrix which describes the elastic scattering by ^the periodic ^{surface. The h matrix}

elements have non zero value on the energy shell, only for the open diffraction channels

corresponding ^to the initial conditions.

t is the matrix which describes the inelastic scattering ^{as a} perturbation of the elastic diffraction _process or the scattering by ^the potential v (R, ^z, u ) ⁺ U(z ).

The transition rate fw; between incident (i) ^{and final} (/) particle ^{states is} given by :

where Ey ^{and E,P are} the final and initial particle energy, respectively, ^and fT, is the T-matrix element calculated between the final and initial particle ^states.

Replacing the T-matrix elements by ^their expression (3) the thermal average factor in (8)

can be written as :

The term «fLi(,k» ^{does not} contain correlation functions of the u operator ^{for two} different k values and the integration ^{over À with the} exponential ^factor yields ^a 8 function.

One gets :

The first and second terms in this expression correspond obviously ^to elastic and inelastic

scattering respectively.

In order to simplify ^the subsequent analysis ^we consider the simpler ^{case of a} flat surface or

quasi-flat ^{surface as the dense faces of CFC} crystal for which the diffracted peak intensities

are very small. In this ^case the diffraction process is unefficient and ^{one can} takes :

The reflection coefficient or the transition probability i.e. the transition rate divided by ^the

incident flux hx ²

pi is given by :

incident flux ( 2 7rm ) is given by :

where 2 X is the range parameter of the short range repulsive part ^{of the} potential U(z) and p;

kf/x with kiz the normal component of the incident particle wavevector.

3. Cohérent and incohérent scattering

sticking coefficient.

Let us consider the first term in expression (11). It has been demonstrated previously [10, 11]

that the «fti> ^{values are} solutions of the integral equation :

with C

EP - Ho ^and v (R, z, u ( a » ^{is now the} potential v in the interaction picture. ^The

solution of this equation by ^{an iteration} procedure shows that the matrix element t is equal ^to

the sum of elementary contributions, each of them involving only ^virtual phonon processes.

Each elementary contribution is a matrix element tt/m, n) ^{which can be} represented univocally by ^a diagram and labelled by ^{two numbers} (m, n ) : ^m is the order in perturbation expansion and n the number of virtual phonons. ^{Such a} matrix element or a diagram (m, n ) contain the product of n Bose-Einstein factors and consequently ^are proportional ^to

Tn at a sufficiently high crystal temperature ^T. Furthermore one has shown [12] ^{that the}

minimum number nm, of phonons involved in a m-order term in the interaction is

m/2 ^and (m ⁺ 1 )/2 ^{for m even or odd} respectively. ^{Then we define :}

For illustration and further reference figure ¹ gives ^{all the} diagrams corresponding ^{to one and}

two virtual phonon processes.

The detailed form of the different matrix elements depends obviously ^{of the chosen} potential V (R, ^z, u ) ^which represents ^the particle crystal interaction but the general analysis

in terms of diagrams ^remains independent ^{of this} particular ^choice.

As the «fti > matrix element contains only elementary processes of phonon ^virtual

exchange ^{it can have nonzero} value for final states on the energy shell, equal ^to open

Fig. ^{1. - Elastic} diagram ^for

^{and two virtual} phonon exchange.

diffracted channels. Then it is easy ^to show that the coherent reflection coefficient in a

diffracted beam G is given by :

in which now the matrix elements are dimensionless, ^{that is to} say ^are those given by equations (6) ^or (7) multiplied by the factor 2 m/h2 x2.

Let ^us consider now the second term in expression (11). ^The thermally averaged ^factor

contains the correlation functions «u (0) - u (À » ^and consequently ^the integration ^{over Àwill} yield ^a variation of the particle energy by exchange ^{of real} phonons. ^This variation, ^{as well as} that of its parallel momentum to the surface, ^{will be} equal, respectively, ^{to the} algebraic ^sum

of energy and ^momentum of the phonons created and annihilated during ^the scattering ^event.

It can happen that these two sums are equal ^{to zero and the} particle is scattered incoherently

in the specular direction. It can also happen ^{that the} particle ^conserves its energy without

conservation of its parallel momentum to the surface. This particular process supposes the

exchange ^{of at least two real} phonons ^{and can be called} elastic incoherent process. It

corresponds ^to the elastic cross section measured in an experiment ^where the scattered

particles ^{are resolved} in energy for ^a scattered angle ^{different from the} specular direction. In such an experiment ^{one also measures the cross} section in the vicinity of this elastic event.

One can expect that this quantity ^{does not exhibit} any singularity ^{at the elastic} position.

Therefore if a peak is observed for this position ^{it could be due to another} effect and can be ascribed to the scattering by surface defects.

Another important point should be outlined. The set of final states are the eigenstates ^of Ho defined with the potential U(z). Within the assumption ^{of the} quasi-flat ^surface they ^are approximately ^{the same as those of a} Hamiltonian defined with the potential « V (R, ^z, u » .

Hence if we take a final state such as exp (iKf ’ R) Ob ^{where Ob is a bound state of}

U(z ), the reflection coefficient Kf,,Ri ^{gives the} probability ^{that the incident} particle ^{makes a}

transition to a state bound to the surface. As discussed previously [15] ^this probability ^is equal

to the sticking coefficient for the bound state considered.

The coherent component ^{of the} scattering ^{has been} fully analysed ⁱⁿ previous studies and the specular intensity ^{as a} function of the crystal temperature ^{has been} calculated when the

particle is scattered by ^a one-dimensional potential [11, 12]. It remains to perform ^{the same} analysis and calculation for the inelastic component ^{of the} scattering including ^the sticking

coefficient. This will be done in the following ^sections.

4. Inelastic scattering.

The iteration of equation (7) gives :

where the number under each term labels the expansion ^or perturbation ^order.

Replacing t by ^this expression ⁱⁿ equation (11) the bracket under the integration ^{over the À}

variable becomes :

or in short hand form, which used the perturbation order notation :

as « 1 > = «v» ^{= 0} (Eq. (2)).

The general ^{term is} «m’ m’ » ⁺ «m’+ m> ^{for m’= m and} «m+ m> ^{with m’}

^{m. Hence}

the notation m denotes a matrix element taken between the particle ^states f ^{and i of an} operator ^which contains m potential operators v and m - 1 Green operators. ^{We have :}

where we assume the potential ^{v to be} real, ^{and 0 is an} eingenstate ^of Ho namely exp (i K - R) 4> p or b ^{where the} subscript p ^{or b refers to a continum or bound state} respectively.

m + can be written as

and the sum

4.1 Let us consider the first term in expansion (14). ^From expression (11) ^its contribution to the reflection coefficient is given by :

It is interesting ^to compare this expression ^{to the} second order expansion ^term of the elastic matrix element «;t; » . Equation (12) gives :

In each of these expressions ^the thermal average bracket contains ^two potential operators.

Therefore the thermal average operation ^will yield ^{the same} function of the correlated

displacement, namely f( «u (0 ) u (A )» ) ^and f ( « u (,k ) u (0 ) » ) ^for (16) ^and (17), respectively.

To go further ^one expands f with respect ^to the correlation function and one gets ^a general

term proportional ^to («u(0) u(A )» y ^or «u(,k)u(O»)" ^{which are} proportional ^to

exp - iA Mhwq ^{and exp} + iA M hwq respectively. ^{Then the} subsequent integration

over k gives :

- for the inelastic reflection coefficient (16) ^{2 zr ô} Ey - ^{EP -} hWq ^which corresponds

to the exchange ^{of n real} phonons.

-

for «it.(2» ^{(17) a dressed} propagator G+ located in between two potential operators

now independent ^{of u.} G’ ^{writes :}

and corresponds ^{in the} elastic process ^to the exchange of n virtual phonons. This matrix element is denoted «itf2, n» .

Due to the form of the correlation function there remain n integrations ^{over w such as}

with p a spectral density ^and « N ( w ) > the Bose-Einstein factor. Therefore these two terms are proportional ^{to T" at} sufficiently high crystal temperatures ^T.

The calculation is achieved by introducing ⁱⁿ //2, n) ^the projector of 03A6) ^{states. This} gives the matrix element for an elastic event involving n ^virtual phonons. In the inelastic reflection coefficient, equation (16) shows that the projection ^is performed only ^{on the final}

state f. Therefore if ^we know the dimensionless matrix elements itl’ n> ^{we can write the}

corresponding inelastic reflection coefficient by using ^the following ^{rules :}

-

replace the dressed propagator by ^{a 6} function which insures the conservation of energy with the phonons involved in the virtual process described by ^this propagator. ^{This is}

equivalent ^to taking ^the part ^{of the} propagator ^on the energy shell ;

-

replace ^the projection ^{on all states} by ^the projection ^{on the final state ;}

-

multiply by ² 7T2/Pi.

This prescription ^{can be} applied ^{to the} diagrams ^which represent ^the il/2, n) ^matrix

elements. This is equivalent ^to cutting the dressed propagator ^{between two vertices as shown} in figure ^2.

Note that the inelastic diagram thus obtained represents ^a reflection coefficient. They ^are

denoted (1+ ¹ n ) with n the number of real phonons exchanged. ^At this order n can take on

all integer ^values.

The above analysis suggests that there exists a relation between the specular intensity

calculated at second order and the total inelastic process calculated ^at first order. This is demonstrated in appendix ^I. One finds :

where the sum of fR/l, 1) ^{is extended over all final states and can be separated into two terms,}

one for continuum and the other for bound states, ^{as indicated} by ^the subscript ^{c and b} respectively. As in the calculation of it/2> ^{we use just one projector} this matrix element is also equal ^{to the sum of} components ^{due to} continuum and bound states. One deduces easily

that :

Fig. ^2.

Procedure for obtaining inelastic reflection coefficient diagrams (on ^the right) from elastic

diagrams (on ^the left). The vertical dashed line is the cutting ^{line. The} ¡Ri ^{diagram labelled}

(1+ 1, 1 ) ^{is the DWBA term.} The last column gives ^the proportionality ^to the power of crystal

temperature ^T (for ^T sufficiently high ^{such that} «N (n ) > f-- ^T).

This ^means that from the calculation of the specular inensity ^to second order including ^or disregarding the influence of bound states one can obtain the total sticking coefficient

corresponding ^{to the} 1 + 1> inelastic processes.

4.2 The contribution to the inelastic reflection coefficient of the term 1 + 2> ^is given by expression (16) in which the thermally averaged ^{factor is} replaced by :

The corresponding matrix element of the coherent elastic process is « itf3) 1 » . ^{Its thermally}

averaged factor is :

For the «2’ 1 > ^{term the} expressions corresponding ^to (20) ^and (21) ^are respectively :

and

The latter is equal ^to (21) in which the time has been reversed.

Because these four expressions contain the same number of potential operators the results demonstrated in the preceding paragraph ^are valid. The same rules allowing ^{one to obtain the} fR; ^values fR fl, ^{2) +} fR f2, 1)

^{2 Re} (fRi1’ ^{2) and to} deduce the diagrammatic representation ^can

be applied. ^{The Green} operator ^{to be} suppressed ⁱⁿ expressions (21) ^and (23), ^{or the} cutting position ^{in a} diagram, ^is obviously that which corresponds ^{to the} position ^{of the} projection ^on

the final state in (20) ^and (22) respectively.

As indicated in paragraph ^{3 the} «iti(3)> matrix element contains components involving n

virtual phonon processes with n::- 2 (expression (13)). Therefore the matrix elements

« it1(3, n) > ^{and the} corresponding reflection coefficient R/I, 2, n) ^are proportional ^{to Tn and at}

least proportional ^{to T 2 at} sufficiently high ^{T value.} Consequently ^the only component ^of

fR; proportional ^{to T is} given by ^{the term denoted} (1+ 1, 1 ) (Fig. 2) which is the well known distorted ^wave Born approximation (DWBA).

Figure ³ gives ^{the different} diagrams derived from the coherent elastic (3, 2) processes. The

¡Ri components ^{exhibit one real and one virtual} phonon processes, ^or the exchange ^{of two}

real phonons ^{and are} proportional ^to T2. Diagrams involving ^three phonons ^{are obtained} by cutting ^{the seven} (3, 3) ^elastic diagrams [12] ^{and are} proportional ^{to T3 : there are 2, 3 and 2}

¡Ri components ^with 1, 2 and 3 real phonons exchange respectively.

A unitarity ^relation analogous ^{to relation} (18) ^{for the} 1+ 1> ^{term, can} be established

(appendix I). ^One gets :

However to get the matrix element « it1(3 » ^{we should introduce two} projectors of the final states f. Therefore it is not a linear combination of continuum and bound ^states and the

decomposition leading ^to expression (19) ^{cannot be done.}

4.3 The extension of the above procedure ^{to the} general ^term «m"+ m’> ⁺ «m’+ m"> ^is

straightforward ^{and the same rules} allowing ^{one to} deduce the inelastic reflection coefficient

Fig. 3. - Inelastic reflection coefficient diagrams obtained with the three elastic diagrams (3, 2) ^with

the procedure ^{used in} figure ^2. Cutting

propagator ^located

^the right of the middle vertex yields

diagram with time reversed.

from the elastic matrix element hold. Hence, knowing ^{the latter, one can} write the former. In this procedure ^the diagramatic representation is very helpful because it gives ^{in a} simpler way the number of terms in the inelastic reflection coefficient and the number of real and virtual

phonons involved in each inelastic component.

To go into detail, ^{let us} consider the elastic matrix element itfm» - It contains

m - 1 Green operators ^and therefore should be associated with the inelastic terms deduced by suppressing ^{one Green} operator, ^{that is to} say with all the ^terms such that

As pointed ^out in 4.2 when m " >. mi we should associate the «it i(m) > matrix element in which the time has been reversed. However, this fact is included in relation (15) ^{where the}

two terms m"+ m> ^and «m’+ m"> ^{are added and give} twice the real part of the former.

Hence it is sufficient to consider the terms for which m" m’. Then for m odd there are

(m - 1 )/2 ^{terms. For m even there are} (m - 2 )/2 ^{terms such that m" = m’ to which one}

should add the term mi = m" : one the whole there are m/2 ^terms.

Expression (13) ^shows that itfm» ^{is a sum} of matrix elements «itm,’» ^where

n

is the number of virtual phonons involved in the process. Since there ^are different _ways to

exchange the n virtual phonons this matrix element is in fact the sum of elementary ^matrix

elements : each of them depicts ^a given process and their number N (m, n) has been calculated

previously [12]. ^Therefore each «itm, "> matrix element corresponds ^{to a number of}

components ^{in the inelastic} reflection coefficient equal ^{to :}

and

Each of them gives ^an exchange of p ^real and q ^virtual phonons with p + q ^{= n ;} p may vary from 1 to n. All of them are always proportional ^{to Tn at} sufficiently high ^{T values.}

However, the numbers given by (25) ^are overestimated. Among ^{the elastic matrix elements}

or diagrams ^some of them exhibit an undressed propagator ^{like :}

Cutting this line is equivalent, according ^{to our} rules, ^to replacing ^{the Green} operator by

the function 2 7T5 (EF - Epf). This leads to an elastic process which does ^not remain in the inelastic reflection coefficient expression (Eq. (14)) ^as it cancels with the same term contained in :

This fact is illustrated in figure ^{4a with a} (6, 3) diagram. Furthermore, ^{when m is} even, another fact reduces the above calculated number. Then a propagator ^{is a « center of}

Fig. ^4. - (a) ^The cutting operation ^of

^undressed propagator ^{leads to elastic} intensity. (b) ^{Case where}

a

cutting operation ^{leads to}

reflection coefficient ; ^the propagator ^{cut is in}

symmetric position. (c)

Case where two cut elastic diagrams yield

reflection coefficient component.

symmetry

namely ^{the Green} operator located in the middle of the elastic t matrix element :

Cutting ^the corresponding line leads to an inelastic reflection coefficient of the form

«m + m » and this has been assumed to give ^{N (m, n) inelastic} components. ^In fact, if the elastic

diagram ^is symmetric ^with respect ^{to this} propagator ^the cutting operation ^{will lead to one}

reflection coefficient (Fig. 4b). ^{On the} contrary it is necessary ^{to cut} the two elastic diagrams symmetric ^to each other to get ^one reflection coefficient (Fig. 4c). ^Hence cutting N (m, n) diagrams in the middle gives ^{a number} of reflection coefficients less than

N (m, n).

Table 1 gives the number Ni of reflection coefficient components deduced for different

elastic (m, n ) matrix elements and the number of these components depicting ^the exchange ^of 1, 2 or 3 real phonons. The table contains all the terms proportional ^to T, T2 ^or T3. One can remark that there is only ^{one inelastic} component proportional ^{to T and 9} proportional ^{to T2} among which 5 depict ^{one real} phonon exchange. ^The exchange ^{of n real} phonons ^{starts with terms} proportional ^{to Tn.}

Table I. m order in t matrix expansion and n number o f ^virtual phonon processes in ^the elementary elastic matrix element (t (m, n»), nm is the ^{minimum number} o f virtual phonon ^at

order m. N (m, n) number o f elementary t (m, n) matrix element or diagrams. Ni ^{is the number} o f

inelastic reflection coefficient components ^or inelastic diagrams. ^{The last} four ^{columns on the} right give ^the proportionality to the power o f crystal temperature (for sufficiently high ^T value)

and the number o f inelastic components for 1, 2 and 3 real phonons processes.

Figure ⁵ gives the inelastic diagrams ^deduced from the elastic (4, 2) matrix element. With those shown in figures ^{2 and 3 one} has all the single ^real phonon exchanges proportional ^{to T}

and T 2. This will be calculated in the following paragraph. ^{It is} interesting ^to give ^the unitarity

relation analogous ^{to relations} (18) ^and (24) for this order. One has (appendix 1) :

The last sum comes from the presence of undressed propagators ^{in some elastic} diagrams ^as

mentioned above.

Fig. 5. - Inelastic reflection coefficient diagrams yielded by cutting ^{the three} (4, 2) ^elastic diagrams

with the procedure ^{used in} figure ^2. Cutting

propagator ^located

^the right of the middle propagator yields

diagram with time reversed.

5. Flat surface.

As in preceding papers [10-12] ^we consider the scattering ^{of an incident} particle by ^{a flat}

surface. The interaction potential is modelled by ^a simple ^soft potential :

and the distorted potential U(z) = «V> is of the Morse type. ^The displacement operator ^u

acts only ^{on the} repulsive part ^{of this} potential in the direction normal to the surface. Hence the potential ^{is one} dimensional and the particle ^can gain ^or lose energy without any change ^of

its momentum parallel ^to the surface.

A scattered particle ^{in a continuum state} remains in the incident plane ^{and the} corresponding calculated reflection coefficient will be greater ^than that measured in an

experiment. ^{For such} experiments ^{where the} exchange ^of parallel ^{momentum is} small, ^one

can expect that the calculation gives ^the respective strength ^{of the one and two} phonon

reflection coefficients and reproduces their evolution with the crystal temperature.

For particles ^scattered into bound states one defines the sticking ^or capture coefficient as

equal ^{to the sum} of transition probability into all bound states. This calculated quantity ^will certainly ^be dependent ^on the normal component of the incident particle energy which is for this potential ^a characteristic quantity. However, ^{one can also} expect ^{to obtain a} good

evolution of the sticking coefficient as a function of the crystal temperature ^{and also to} study

the influence of the different potential parameters ^{on this} quantity.

The subsequent ^different expressions for the reflection coefficient can be calculated directly (expression (11)) ^or deduced from the elastic matrix element «iti(’» ^{[11] using the rules}

exposed ^{in the} preceding paragraph. They ^are expressed ^{with the} dimensionless quantities :

-

phonon energy R = A 2 hw

-

eingenvalues ^of Ho j p 2 _f2b = A 2 2 ep _{= A 1 eb 1} ^continuum _bound state . ^state

The exchange of energy for the incident particle will be denoted

with

Op ^and 4>b being ^the eingenstates ^of Ho, for continuum and bound states respectively. ^The potential matrix elements are defined as follows :

The mass of ^a crystal ^{atom and the} spectral density ^{of the u} operator ^{are denoted} by ^{M and} p (d2 ) respectively.

In each fRi expression the final state can be either a continuum (pf) ^{or a bound state}

(b ).

5.1 FIRST ORDER. - The two components corresponding ^to the first and second diagrams ^of figure ^{2 are :}

The

unitary relation » (18) which is valid for each order of phonon exchange ^is

The imaginary parts ^of the iti ^{matrix elements are} always negative [11] ^and yield ^a decrease of the specular intensity ^{when the} crystal temperature increases. To a depletion ^{of the} specular intensity corresponds ^{an increase} of the inelastic scattering. Effectively, ^the product

p ( n ) « N ( f2 ) > ^is always positive ^{and the} total inelastic reflection coefficient is also positive.

03A9

However one should notice that a calculation of the latter quantity ^{does not allow a}

calculation of the specular intensity ^{as one does not} know the real part ^{of the} elastic matrix

elements.

5.2 SECOND ORDER. - There are two components ^{for a one} phonon ^real exchange (Fig. 3).

Their reflection coefficients are given respectively by :

with

and

where G (2, 1 ) is the dressed propagator ^{which accounts} for the virtual phonon exchange (Q), ^{the real} phonon process occurring ^after (f2’

^0, Eq. (30a)) ^{or before} (f2’ = ,Qo, Eq. (30b)) the virtual one.

The two real phonon exchange gives ^a component which is written as :

with

and

The particle exchanges ^{the real} phonon nt ^{on the first vertex} (third ^{term in} Fig. 3), ^then

propagates ^{with an} energy pi + f2l 1 and exchanges the second real phonon ^on the second vertex.

These three terms are proportional ^to the square of the crystal temperature ^{and to the ratio}

m/M.

For all the particle-crystal pairs considered [11], ^{exact numerical} calculations indicate that

lm itf3, 2) ^{values are} positive leading ^{to an} increase of the specular intensity ^with respect ^to the value given by the matrix elements « itf2, 1 1) ⁺ it.(2, 1 2) >. ^{In agreement} with these results the

« unitarity relation » (24) indicates that the total contribution to the inelastic reflection coefficient given by these three terms is negative ^{so that} they ^{tend to decrease} the inelastic

scattering. ^{However, as we} should take the real part ^{of the} fA1 expressions the contribution of

5 function generated ⁱⁿ G(2,1) or G(2, 1) ^{does not hold. The} only contributions come from the

principal values of these Green functions which depict intermediate processes in which the

.

particle energy is ^not conserved. One knows that this contribution is not negligible ^{but is in} general ^small compared ^{to the 6} function contribution which depicts ^{the « on} energy shell processes

Consequently ^{these three} components of the inelastic scattering ^will certainly ^be

small and probably negligible in many ^cases. This statement is confirmed by the fact that the elastic matrix element « itf3, 1 2) ^{in both} its real and imaginary parts ^{has been found to be} small compared to «iti (2,1) > ^or even «itf2, 2»

5.3 THIRD ORDER. - There are three components ^{for the} exchange ^{of one real} phonon. ^The

reflection coefficient is given by :

with fA 1 f4)(4) fA 1 (’)and f A(6) i corresponding ^{to the} first, second and third diagrams ^of figure ⁵ respectively.

With :

where the g ^{functions are} defined in the same way ^as the four preceding ^ones, for instance

In expression (34) corresponding ^{to the first} diagram ^of figure 5, the g ^functions depict ^all

the virtual phonon processes which end ^{on a} continuum state (qgi) ^{or a bound state}

(bgi). After these processes the particle propagates freely ^and exchanges ^{one real} phonon ^at

the last vertex.

Expression (35) corresponding ^to the second diagram ^of figure 5 describes the reverse

process. The particle exchanges ^{one real} phonon ^{at the first vertex,} propagates freely ^{with an}

energy jpi ⁺ Q0 and then passes through ^all elastic virtual phonon processes leading ^{to the}

same final state of energy pz ⁺ f2o. ^{It is} important ^{to notice} that, ^{when a} phonon of energy

pz ⁺ n b (Ei + 1 eb 1 ) ^{is created,} the denominator of the second term in (35) ^becomes equal ^to

zero. This is ^a kind of resonance on the final state f

b of energy - l1b (- 1 eb ). ^{Then it}

appears only in the calculation of the sticking coefficient on the bound state considered. In

particular ^{there is no} singularity ^{for a final state} in the continuum.

with

which depicts effectively the virtual phonon process (n) and the real phonon exchange (no) ^on the intermediate vertex (see ^third diagram ^of Fig. 5). For purposes of numerical calculation we can write :

which for

In general these three components ^{of the} single ^real phonon exchange contain the product

of two factors of the form :

The product of the two-5 function ^terms will yield ^{a real} negative number and their contribution to the reflection coefficient is effectively contrary ^to the second order terms

which, ^{as outlined in} 5.2, ^contained just ^{one 6 function.} Consequently the third order contribution to the reflection coefficient will be stronger than the second order contribution.

Furthermore one can expect ^{that the} product ^{of the two d functions will} yield ^a contribution greater in absolute value than those yielded by ^the product ^{of the} principal parts. ^{In such a}

case the fRi(13,1 + 31,1 > value will be negative ^and proportional ^{to T2. Added to the}

D (il, i) proportional ^{to T will} yield ^a parabola shaped ^{variation of the one} phonon ^reflection

coefficient as a function of the crystal temperature always ^at large ^T.

For the two real phonon exchange ^{there are two terms and one gets :}

A d and A e corresponding ^{to the 4rd and Srd} diagrams ^of figure ⁵ respectively.

and

This expression ^{contains a term} equal ^{to the} product ^{of two} matrix elements ¡e q qe ⁱ associated with the factor (Cp ^{+ f2 +} i E )- 1, and another one equal ^{to the} preceding ^but

with time reversed. One phonon ^is exchanged ^on the incident vertex and the second phonon

on the final vertex.

fAi ^{is equal to} fA d ^{but the} C(3,2) ( Q ) ^{functions are} replaced by :

The order in which the two phonons ^are exchanged is inverted in the time reversed part ^of this expression.

One can demonstrate that the imaginary part ^{of these two term vanishs.}

For the purpose of numerical calculation it is advantageous ^to perform ^{first the} integration

over p and/or q variables. Then the product of the two 6 functions yielded by ^the G (3,2) functions gives ^a positive contribution. One cannot say anything ^{about the} principal part values but one can expect that, ^{on the} whole, ^{these two terms} give ^a positive ^reflection

coefficient component. As all the terms of third order, they ^are proportional ^to T 2and (m/M)2.

Now let us look at the unitarity ^relation (26). ^{For this} potential and the order considered it appears ^{as :}

For many systems and incident conditions the calculation leads to positive ^{values of}

Im {it/4, 2). This confirms our analysis showing ^{that the} single ^real phonon ^{term is} negative

and means that the decrease of the single ^real phonon scattering yielded by ^{this term is} large :

it should compensate the increase of the ^two real phonon processes and of the elastic specular

intensity yielded by ^the positive value of Im {it/4, 2) .

6. Results.

The numerical calculations ^are performed ^{under the same} conditions as those for specular intensity calculations reported and discussed in previous papers [11, 12]. We recall here that the ^u operator ^is equal ^to the average of the normal displacements ^to the surface of the four atoms belonging the unit cell of the (100) ^{face of} Copper. ^{This face can} be considered as a flat surface. The calculation is purely harmonic ; ^{we do not} introduce any anharmonic effects as in

previous papers.

The ten inelastic components written above are calculated and therefore the results include the contribution of all the terms proportional ^{to T and T2. When the} components ^{of a} given

order have been calculated one can verify ^the

unitarity relation » by calculating indepen- dently ^the corresponding elastic matrix element. For order one and two (Sects. ^{5.1 and} 5.2)

the error on the

unitarity

is less than 10- 5. For third order we have pointed ^{out that the} expression (35) corresponding ^to the second diagram ^of figure 5, ^{exhibits a} singularity ^when

the final state is a bound state. For this final state the singular ^term is omitted in the calculation and the calculated value of bR fl3,l + 31, 1)i.e. ^the sticking coefficient for one phonon

events is not correct. The

unitarity relation » at third order cannot be satisfied. In order to recover the « unitarity » ^the

unitarity » difference is added to the calculated sticking

coefficient value (6 ) ^{which is called now corrected} sticking coefficient (C 6 ). ^Otherwise stated, ^{this last} quantity ^is given by ^the difference Ilnel

¹ (1 ^{+ 2 + 3} )c, ^where I inel ^is given by

the sum of the left hand side of unitarity ^{relations for order} 1, 2 and 3 and I (1 + 2 + 3)c is the total reflection coefficient for continuum states at the three orders. With this procedure, ^one gets ⁱⁿ

a simple way ^a reliable value of the sticking coefficient. With a three-dimensional potential

like that given by ^a pairwise potential ^{summation we can} show that the above mentioned

singularity disappears. ^The sticking coefficients is then exactly given by ^{the above} procedure.

In the figures presented ^{below the curves labelled} (1 )t, (1 ⁺ 2 )t ^and (1 ^{+ 2 + 3} )t ^represent

the values of the total inelastic reflection coefficient calculated with the left hand side of the

unitarity relation for order 1, (1 + 2 ) ^and (1 ^{+ 2 +} 3 ), respectively. ^{A label} (1 ^{+ 2 +} 3 )c ^{1 or} (1 ^{+ 2 + 3} )c corresponds ^{to the total} reflection coefficient to continuum states calculated from the fRi expressions ^{with one and one} plus ^{two real} phonons processes, respectively. ^The

calculated sticking coefficient and its corrected value are labelled

and C o- respectively.

Finally, ^{the curve} labelled 1 - Ioo gives the total inelastic reflection coefficient as

I00 ^{is the} calculated specular intensity ^{which includes} multiphonon processes [12]. ^{Table II} gives ^the potential parameter values and the bound state energy.

Table II. - Potential parameters ^{and bound state} energy for the systems ^He-Cu (100) [16] ^and

Ne-Cu (100) [17].

Figures ^{6 to} 8 exhibit the results for an incident Helium atom of kinetic energy 21 mev and incident angles ^of 73.5, 55.5 ^{and 31.8} degrees respectively. The C6 values are given by ^the

difference between the values (1 ^{+ 2 +} 3 )t ^and (1 ^{+ 2 + 3} )c. Figure ⁹ gives the result for a

Neon atom, the incident angle being ⁷⁵ degrees.

Fig. ^6.

Scattering ^{of He} by ^Cu (100 ) - Ei

²¹ meV, 0i,

^73.5 degrees, ^normal particle energy

Ei, = 1 .69 meV. ^The

^labelled (1 )t, (1 + 2 )t, (1 + 2 + 3 )t, 1 - 1 00 gives the total inelastic transition probability calculated with unitarity relations for first, ^{first +} second, ^{first + second + third} order and to all orders, respectively. ^The

^labelled (1 ^{+ 2 +} 3)c gives the total inelastic transition

probability up ^to third order for the continuum states.

In each case (Figs. ^{6 to} 9) ^the contribution of the two-phonon diagram ^of figure ^{2 is} completely negligible ^{so that the} (1 )t ^curve corresponds ^to the values given by ^{the DWBA}

term proportional ^{to T. One sees that as the} crystal temperature increases, the results yielded by ^this approximation ^{are no} longer good ^{and are} greater ^{than the true value 1 -} Ioo. Adding

the terms of second order (Fig. 3) ^{does not} modify this conclusion as they yield ^a negligible

contribution compared ^{to the DWBA term.} The addition of the third order contribution modifies considerably ^the preceding values. The total inelastic scattering ^becomes equal ^{to or}

lower than the 1 - Ioo values. The domain of validity ^of (1 ^{+ 2 +} 3 ) calculation extends to

higher temperatures and the limit of validity T3 ^{could be} appreciated ^{as the} temperature where the difference between (1 ^{+ 2 + 3} )t ^and 1 - Ioo ^{curves becomes} appreciable (Tab. III).

In particular the C6 values calculated are reliable in this temperature range. Even if this temperature range is reduced ^as in the Neon case we can see that the Ccr value increases with the temperature and decreases as the normal component of incident energy increases. This last effect can be seen by comparing the results of figures 6, 7 and 8 : for the last case of

relatively high normal energy component the Ca value practically ^vanishes.

For values of T greater ^than T3 ^the contribution of reflection coefficient components proportional ^{to T3} which contain one, ^two and three real phonon processes becomes ^non

negligible. They ^will probably ^{act in such a} way that the difference between the

Fig. ^{7. - Same}

figure ^{6 :} 8i

^55.5 Ei.L = ^{6.74 meV.}

Table III. - Crystal temperature below ^{which an} inelastic calculation gives acceptable ^results.

TD ^is for ^{the DWBA term} (first order) ^and T3 ^is for ^the first plus ^{second and third order with}

terms proportional ^{to T and} T2 only.

(1 ^{+ 2 +} 3)t ^{and 1 -} 100 ^curves will be reduced. Also the maximum exhibited by ^the (1 ^{+ 2 +} 3 )t ^{curve can} be shifted to higher temperatures.

Figures ^{10 to} 13 show the variation of the reflection coefficient with the crystal temperature for particles ^{scattered at a} given ^final angle Of

and for the same particles and incident conditions as above. In each case the particle exchanges ^{a small amont} of energy (negative ^or positive) corresponding ^{in the case of} single phonon exchange ^{to small} phonon ^momentum.

The curves labelled 1 and 2 give ^the quoted quantity ^{for one and two real} phonon exchange, respectively. The dashed line is the result yielded by ^{the DWBA term.}

As has been predicted ^{in the} preceding paragraph ^the two-phonon reflection coefficient

increases as T2. Also in agreement with the above conclusions, ^{the DWBA} contribution gives

Fig. ^8. Fig. 9.

Fig. ^{8. - Same}

figure ^{6 :} 6i

^31.8 EiJ. = ^{15.43 meV.}

Fig. ^{9. - Same}

figure ^{6 but for neon.} Ei = ⁶³ meV, 6i

⁷⁵ degrees-particle normal energy 4.22 meV. The sticking coefficient is given by ^the

labelled Co,.

acceptable values in a small domain of temperature, ^the temperature ^{above which this} approximation ^{fails is of the order of} TD (see ^Tab. III). ^{The addition to the DWBA term of}

the five contributions proportional ^to T2, ^and particularly ^{the three} coming ^{from third} order, yields ^a parabola shaped ^{curve for the one-real} phonon exchange, ^as expected. ^{As shown}

above these results are valid below the T3 temperature ^{which is close to the maximum} exhibited by ^the one-phonon ^curves. Beyond ^this temperature ^{one can} expect ^{that the} contribution from terms proportional ^{to r3} yields ^an increase of the one-phonon ^reflection

coefficient value and consequently ^also yields ^{a more or less} great ^{shift of the maximum}

position. ^{For the} two-phonon processes, this contribution will certainly ^be negative yielding

the appearance of ^a maximum, ^{whereas the} three-phonon reflection coefficient will increase

as r3.

7. Discussion.

Generally speaking the reflection coefficient is equal ^{to the} probability ^{that the incident}

particle undergoes ^a transition to the given ^{final state.} During ^this transition some real

Fig. 10. - Transition probability for He scattered by ^Cu (100) : Ei

²¹ meV, Oi

^73.5 degrees ; Of

^80.2 (a) ^{and 65.9} (b) degrees. Ef -- ^{Ep is the particle exchange of energy (meV).} Dashed line DWBA - curves labelled 1 and 2 : one plus ^second and third order calculation with terms proportional

to T and T2 only,

^{and two real} phonon exchange respectively.

Fig. 11. - Same

figure ¹⁰ but 8i = 55 degrees ; 8 f

^{62.5 (a) and 48.6 (b) degrees.}