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Generalized Lindhard power cross-section

G. FALCONEand F. PIPERNO

Dipartimento di Fisica, Università della Calabria - 87036 Rende (CS), Italy Unità INFM - Cosenza, Italy

(ricevuto il 4 Novembre 1996; approvato il 7 Luglio 1997)

Summary. — In the slowing-down of ions through the matter the Lindhard power cross-section is an elastic cross-section that has been very useful for finding analytical results of several transport equations when nuclear and electronic collisions can be treated as if they were unconnected events. Here, we present a generalization of the Lindhard power cross-section when the inelasticity is strong enough to influence the collision dynamics.

PACS 79.20 – Impact phenomena (including electron spectra and sputtering).

1. – Introduction

During the last two decades computer simulation studies and, in general, numerical solutions of transport equations have reduced the interest in searching for possible new analytical solutions of the same equations. Although, in several cases, only numerical approaches are possible, nevertheless the research of analytical approaches presents some advantages. We shall use the Lindhard work [1-3] to illustrate the advantages of analytical results and, probably more importantly, we illustrate the complementary aspect of analytical and numerical approaches. First of all, analytical results of linear Boltzmann or Boltzmann-like balance equations, according to the Lindhard scheme, have been used with a reasonable success in radiation damage, ion implantation and related phenomena [4-7]. Second, the Lindhard analytical solutions [1-7] predicted physical dependences on specific physical parameters, that experimental results [8, 9] and computer simulations studies confirmed [10]. Finally, the analytical approaches become themselves object of investigation by computer simulations, aimed at establishing the range of validity of the simplifying assumptions underlying the analytical theories.

Therefore, in the spirit of the above discussion, we shall present a new cross-section which represents a generalization of the Lindhard power cross-section, when the inelasticity is strong enough to influence the collision dynamics.

2. – The elastic case

Among the several conceptual contributions of the Lindhard school there is the introduction of the differential power cross-section [3, 6]:

dsel(E , T) 4CE2mT21 2 mdT ,

(1)

which enables to obtain analytical results for balance equations. In eq. (1) E is the energy of the incident particle, before the collision, T is the recoil energy of the stuck particle after the collision (the stuck particle is at rest before the collision), C is a well-defined parameter [3, 6]

(

see also eq. (19)

)

and m the power of the interaction potential. Equation (1) is an elastic cross-section, which presents a singularity at small energy transfer.

In the slowing-down of ions in solids, elastic and inelastic processes can take place. As was pointed out by Sigmund [11] an important feature of the Lindhard work has been “the clear recognition and identification of a wide range of particle energies

where electronic and nuclear stopping were competing processes”. Moreover [11]

“electronic processes enter into the theory via continuous stopping term, while the full

statistics of nuclear energy loss and scattering enters the general description”.

Equation (1) is the basic ingredient of the nuclear-energy loss and scattering.

It is evident that, when the inelasticity is strong enough to influence the collision dynamics, the previous scheme is not applicable [3].

The purpose of the present paper is to generalize the Lindhard power cross-section, eq. (1), when strong inelastic effects can influence the collision dynamics.

The following summary of the idea underlying eq. (1) can clarify what needs to be done to extend the Lindhard power cross-section in the case of strong inelastic effects.

Since the expression of the scattering angle a, in the center-of-mass system, cannot be integrated in a closed form, for a screened Coulomb potential, the small-angle approximation is used [3]: a 42 1 mv2 ¯ ¯b



2Q Q V

g

k

b2 1 z2

h

dz , (2)

where m is the reduced mass, v the velocity of the incident particle, b the impact parameter and z the direction of the incident particle. Moreover, V(r) is the interaction potential between the two colliding particles. If Z1and Z2are the atomic numbers of the

colliding particles, e is the electron charge and a is the screening radius, a screened Coulomb potential can have the following form:

V(r) 4 Z1Z2e 2 r ks s

g

a r

h

s 21 , (3)

where ks is a constant [3] and s 41/m. Equation (2) can now be integrated and we

obtain a 4 ksgs eL

g

a b

h

s , (4)

where eL4 a Z1Z2e2 M1 M11 M2 E (5)

is the Lindhard dimensionless energy variable [3] and

gs4



0

p/2

dh coss_{h 4} G( 1 O2) G

(

(s 11)O2

)

2 G(sO211) (6)

with G(z), the Euler gamma function [12]. In eq. (5), M1and M2are the masses of the

colliding particles.

Associated with the scattering angle, eq. (4), there is the following analytical differential cross-section: ds (a) 4pa2

g

ksgs eL

h

2 /s d

k

g

1 a

h

2 /s

l

. (7)

Finally, this cross-section can be written in terms of the energy transfer and extended to large angles by using the relation between the energy transfer and the scattering angle in the Center-of-Mass System (CMS):

T 4TMesin2 a 2 , (8) where TMe4 4 M1M2 (M11 M2)2 E (9)

is the maximum energy that can be transferred in a binary elastic collision. In conclusion, eq. (1) is a consequence of eqs. (4) and (8).

Since it is not difficult to modify eqs. (4)-(7), to include the inelastic energy loss, the basic problem is to obtain an expression equivalent to eq. (8), for the inelastic processes.

In the following section we shall investigate the binary inelastic processes in order to solve this problem.

3. – The inelastic case

In a binary inelastic collision between an incident particle of mass M1and energy E

and a second particle of mass M2, at rest before the collision, on the basis of the usual

principles of conservation, the equation for the recoil energy T can be written as follows:

T2_{2 T}Mecos2a2T 12TQT 1TQ24 0 ,

where TQis the recoil energy, at the limiting angle

(

see below eq. (12)

)

: TQ4 M1 M11 M2 Q . (11)

In eq. (11) Q is the energy lost in the elementary inelastic process; moreover a2is the

recoil angle with respect to the incident direction, in the Laboratory System (LS). As can be easily seen [13], in contrast with the elastic case, where 0 Ga2G p/2, we find the

following limiting angle:

a2M4 arccos

o

Eth E (12) where Ethf M11 M2 M2 Q (13)

is the threshold, in LS, for the inelastic process. This angle is a measure of the degree of inelasticity of the process. Among the several possibilities of representing this angle, the best choice is to set

e 4sin a2M

(14)

because this quantity has a precise physical meaning (see the appendix) [13]: e is the

restitution coefficient, at the microscopic scale, of the binary inelastic collision. In

terms of this new parameter we obtain for the recoil energy the following expression: T 4 TM e 4

k

cos a26

k

e 2 2 sin2a2

l

2 . (15)

Two recoil energy values, which represent an absolute maximum, TM and an

absolute minimum, Tmin, can also be derived:

TM4 TMe 4 ( 1 1e) 2 , Tmin4 TMe 4 ( 1 2e) 2 . (16)

These limiting values, together with the limiting value of the recoil angle, define the region of accessible values for the energy-angular variables of the recoil particles. An application of these boundary conditions has been discussed elsewhere [13], in connection with the sputtering phenomenon.

Moreover, the recoil energy, in terms of the scattering angles in CMS, can be written as

T 4Tmin1 eTMesin2

a

2 . (17)

This relation is the generalization of eq. (8) for the inelastic processes. With respect to the elastic case, the new relation includes two new parameters: the restitution coefficient, e, and the minimum recoil energy, Tmin.

4. – Result and conclusions

Since the extension of eq. (4) to the inelastic case is rapidly derived, we can obtain the generalization of the Lindhard power cross-section, for inelastic collisions, by using eq. (17) and the procedure used for the elastic case:

ds (E , T ) 4e23 m_CE2m_{(T 2T} min)21 2 mdT , (18) where C 4 p 2 lsa 2

g

M1 M2

h

m

g

2 Z1Z2e2 a

h

2 m (19) with ls4 2 s

g

ksgs 2

h

2 /s (20)

is the same quantity used in the elastic case.

As was expected after eq. (16), the singularity in the power cross-section is now shifted at Tminc0.

To summarize the results, we can say that, when the inelasticity is strong enough to influence the collision dynamics, the region of the accessible values of energy and angular variables of the recoiling particle is different from the elastic case. This difference will appear in all physical quantities. We shall show explicitly this difference between elastic and inelastic quantities, in relation with eq. (1) and (18), by discussing some aspects of the Kessel model for excitation [14].

Inner-shell ionizations, which are responsible for the Auger processes in atom-atom collisions, have been considered by several authors [15, 16] as a consequence of binary inelastic collisions. Although there exist several models based on molecular-orbital concepts, the Kessel model, due to its simplicity, provides a useful parametrization of experimental data. In this model the electrons are promoted with essentially unit probability for distances of closest approach less than some critical value. In ref. [17], the Kessel model was analyzed to include the binary excitation processes within the Lindhard scheme. Our discussion will be limited to the problem of the asymptotic boundary conditions on the recoil energy. Since there is a critical distance for excitations, this distance, through the relation between the impact parameter and the energy transfer, is transformed into a critical energy transfer T(bc), where bc is a

critical impact parameter. Then, a differential excitation cross-section is defined as [17]

dsex4 dsel(E , T) for T DT(bc) ,

(21)

otherwise dsex4 0. In eq. (21), dsel is given by eq. (1). Moreover, since to apply the

previous cross-section to the transport equations the boundary conditions on the energy transfer are necessary too, the following condition was derived [17]:

T D 1 E

g

Eth b

h

2 , (22)

where b 42m/gs. The previous result is valid in the region far from the threshold for

the excitation and for M14 M2. But more important, as explicitly mentioned in the

paper [17], the previous results are based on the assumption that the inelasticity is weak enough to not influence the collision dynamics.

If this last assumption fails, one has to use our cross-section, eq. (18). Through a different relation between the energy transfer and the impact parameter, within the same approximation used for eq. (22), we find

T D 1 e3 1 E

g

Eth b

h

2 . (23)

In conclusion, when the inelasticity is strong enough to influence the collision dynamics, to account for the different general boundary conditions on the energy and angular variables

(

see eqs. (12)-(17)

)

, we have to modify the cross-sections and the most part of the physical quantities involved.

* * *

The authors are pleased to acknowledge the valuable discussions with D. AIELLO,

L. FORLANOand R. MAZZUCA.

AP P E N D I X

In this appendix we shall prove that sina2M is the restitution coefficient at the

microscopic scale.

For macroscopic bodies the restitution coefficient e is defined as the ratio of the relative velocities after and before the collision [18]:

e 4

N

v128 v12

N

4

N

v18 2 v28 v12 v2

N

, (A.1)

where with the prime we indicate the quantities after the collision.

From the definition of the restitution coefficient, we can write for a binary collision, at the microscopic scale, the following equation:

e2 1 2mv12 2 4 1 2mv128 2_, (A.2)

whereas the standard Q-equation is

1 2mv 2 124 1 2 mv128 2 1 Q . (A.3)

The quantity m 4M1M2/(M11 M2) is the reduced mass of the two colliding particles.

By using eqs. (A.2) and (A.3), we arrive at the relation between the restitution coefficient and the Q quantity:

e2

4 1 2 2 Q

mv122

. (A.4)

If we restrict our discussion to those collisions in which, in the Laboratory System (LS), one of the colliding particles is at rest before the collision, we set v24 0 and the

previous equation reduces to

e2

4 1 2 Eth

E ,

(A.5)

where E is the kinetic energy of the incident particle before the collision and Eth is

given by eq. (13). A comparison between eq. (A.5) and eq. (12) enables us to obtain eq. (14).

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