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Integral kinetic equation in dechanneling problem
Volume 142, number 1
PHYSICS LETTERS A
27 November 1989
INTEGRAL KINETIC EQUATION IN DECHANNELING PROBLEM V. RYABOV Institute of Physics and PowerEngineering, Obninsk 249020, USSR Received 3 May 1989; accepted forpublication 23 June 1989 Communicated by V.M. Agranovich
A version of dechanneling theory, based on using an integral kinetic equation in both the phase and transverse energy space, is described. It is derived from the binary collision model and it takes into account consistently the thermal multiple and single scattering of axial and planar channeled particles. The connection between the method developed and that of Oshiyama and of Gartner is discussed.
The diffusion approximation usually used in the problem of dechanneling has two disadvantages restricting its application in applied investigations. The first one, which concerns the logarithmic accuracy of the diffusion approach, results from a logarithmic discrepancy of the diffusion coefficients at small collision parameters, or, and that is quite the same, with large impulses transferred in the binary collision [1— 31. Putting aside the question concerning the choice of the minimal impact parameter let us indicate only the fact that actually this phenomenological parameter is not comprised in the initial integral formulation of the multiple scattering problem. The seeond disadvantage of the diffusion approach, the impossibility oftaking into account single scattering, had the effect of unsuccessful attempts to use it in defect crystals [4]. The known variants of the integral kinetic approach in the transverse energy representation which are based on the statistic equilibrium hypothesis [4,5] do not have such disadvantages. However, in terms of the most consistent Gartner method the kinetic equation integral core in axial channeling was determined from a rather extensive numerical cal-
this approach fails to describe the multiple scattering effects for a small distance from the particle to a row (plane). Below, proceeding from the binary collision model we obtain an integral kinetic equation of the Boltzmann type using the particle transverse coordinates and impulses in axial and planar channeling. The transformation to the transverse energy variable using the statistic equilibrium space distribution allows one to determine the origin of the integral approaches in question and to simplify substantially the Gartner method, currently successfully used in applied investigations. The arguments of the derivation of the original equation varied in many papers [7,8,2,3]. Let us consider a small part of the trajectory of length E~tL in the vicinity of the displaced chain atom with penod d, the particle being at a distance r from the chain axis. Strong correlation between the subsequent atomic collisions of the channeled particle result in the requirement of constant r within the length ~.L. Fig. la presents the projection of this particle trajectory part in the transverse plane with fixed configuration of random thermal displacements u1. In
culation of transverse energy changes in the binary collision. This difficulty seemed to have been overcome in the Oshiyama—Mannami method [5] where, in accordance with the Lindhard treatment [61, the transverse energy change is taken proportional to the thermal displacement ofthe lattice atom. As a result,
each i-binary collision the particle acquires a transverse impulse (s1) =pz~O(s~)in the direction of the impact parameter s1. The deflection angle d ~ ~O= ~
—
—
S
41
Volume 142, number 1
PHYSICS LETTERS A
a)
~
S
~\
S~// S~\
~
27 November 1989
w(0’,0,r)=~ö(0’—0—60(s,r)).
(3)
~° (,~)
s~\
I
angular fluctuations 60 on the particle trajectory which The kinetic follow equation from the in the equation phase of space motion including [1] can the be put in the following way:
8) ~
~~XI
18U8F 8r28r80
8z
~
__Jd0’ w(r,0’,O)[F(O’,r)—F(O,r)]. Fig. 1. Projection
of particle trajectory
(a) axial; (b) planar channeling.
is expressed in terms of the continuum string potential U(r). The total impulse transfer in the sequence of N= LxL/d collisions is equal to Ap1 = ~ Ap~(s1). On dividing the mean impulse transfer A~±, averaged over thermal displacements, by AL= vAt, we obtain the equation for the particle motion in the field of a continuum potential averaged over transverse thermal atomic displacements,
~‘
2/2u2)U(s) 0(r) = s=r—u —
~
In this equation use was made of suitable dimensionless variables: z=vt~P, 0=pjpW and U(r)/ U( 0), in which the kinetic equation looks identical both for MeV ions and for light ultrarelativistic partides (W=f20(0)/pv is the critical angle). It is easily noticed that a power expansion in 60 of the collisional term (4) accurate to O(60~)results in the well-known diffusion equation with angular spread [1—3,81 pv Dap(r)= ~
4~_ 80 ~t
Jdu exp( —u (1)
.
(4)
in the transverse plane:
(AOaA0flA0aA0P),
a,fl=x,y. (5) In planar channeling the particle incidents on some low-index atomic chain lying in the given atomic plane YZ at an angle exceeding the critical one, 0~,>~1’. Therefore we confine ourselves to the consideration
Here u2 is the root-mean-square amplitude of ther-
of only the trajectory path of length AL=d
mal displacements. When neglecting the correlations between the thermal displacements of atoms each occasional change pöO(s, r) =p[A0(s) —A0(r) 1 appears independent. Therefore actually the above-mentioned discussion concerns the r=const requirement only between two adjacent collisions. With the trajectory slope 0 =p /p at the point r this requirement will have the following form:
1/~Pbeing on one atomic chain. In fig. lb this is illustrated by a straight line segment. The condition of strong correlation between collisions in planar channeling reduces to the requirement of the constant distance between the particle trajectory and the atomic plane x within a length AL. From fig. lb it is evident that a mean nonvanishing change of the transverse impulse within the path AL is possible along the X axis. At the same time, in analogy with (1) we get the equation of planar motion [7]
Od<
.
(2)
The condition of collision independence permits determine at once theprobability ofthe change of particle direction within the trajectory part of length AL=d: one to
42
d0~ ~
1 8V(x) ~
where the mean value
(6)
Volume 142, number 1
~(x)=~-$dyU(r)
PHYSICS LETTERS A
(7)
coincides with the continuous atomic plane potential averaged over the transverse thermal displacements u~.Here the dimensionless variables 0~=p~/ pW0, z=L~t’~and ?(x)/I?(0) are also introduced (~P~=~/2?(0)/pv, L is the depth). As for the angular spread it is inherent in both directions of the transverse plane. Performing calculations identical to that of the axial case we get the kinetic equation ÔF 8F 1 0? OF + 0—— Oz Or 2 Ox OO~
—
=$do’ w~(0’,0,x)[F(0’,x)—F(0,x)],
(8)
with local scattering probability w~(0’,0,x)=~_$dyô(0’_0_60(r,s)). The second order series expansion of the~collision term results in a diffusion-type equation with diffusion coefficients 2), a=x,y; Da(X)DpoJdY (A0~—~ 2/ (9) D~0=Nd~vp where Nis the atomic density ofthe plane and dis ~‘~‘
27 November 1989
C dr W(~’,~)=~-’--a$dcoJ~s, x ö( 602_2 ö0(s, r) ~(r) cos ~,) ~‘
0(r)
—
—
=~J~— 0(r)
(11)
.
For large r the change of angle is equal to 60=
— -~-
ôr
,~ ôr
Hence (10) tends to the known expression derived by Oshiyama and Mannami [5]. The range of application of this equation is restricted only by the condition (2) fulfilled both for MeV ions and for ultrarelativistic particles. To realise the connection between the developed approach and the Gartner method let us turn to fig. 2 which depicts the geometry of the scattering ofions by a lattice displaced atom. In contrast to ref. [4] where an identical procedure included the continuous potential U(r) the averaged potential 0(r) will be used here as the first approach. It seems to be more consistent in terms of the trajectory equation (1). According to fig. 2 the particle coordinates in the transverse half-way planes between r adjacent atoms are given by 1 =r+~[~+A0(s)]d; r2=r—~d. (12) Expanding the transverse energy change ~~_f=(ê~+Ao)2±U(r 2)__g2_.U(r1)
the distance between the planes: Nd~=l/dd1. Thus, within the atomic chain row model [61 the stochastic angular spread compared that following from is thesuppressed amorphous planewith model (Da(X) u2, x—~oc) [7]. The discreteness contribution is unwarrantedly large in the last case (see the discussion in ref. [10]). The transformation of eq. (4) to the kinetic equation in the transverse energy variable ~= 0~+ U may be made by using the statistical equilibrium distribution function F=2irp(, z)n~(r).Integrating the obtained equation over the available region of area S, (r
,
(13)
in a power series of A0 and omitting terms of order 3) provides O(A0 ~ ~ 60+260 ê~(r) —~=
‘~‘
Oz
J
[W(E,~’)p(~’)—W(~’,E)p(~)],
with transition probability
(10)
7 ______________
/
2
/
~/
__________________
Fig. 2. Binary collision geometry. The position of the atom is marked by a circle.
43
Volume 142, number 1
PHYSICS LETTERS A
In this expression we used
27 November 1989
and for (11) (Ze2\J W(~,~)=t
—
A0=— d OU ——.
pvOr
~pvj
2
~
2
2
,
3exp(—r~/2u ) dS~I—~I
Accurate to the negligible term AG 60 the expression obtained coincides with the argument of the ôfunction in (11). 7(r) Thusand using the averaged neglecting smallcontinterms uous (. (whichpotential were considered in ref. [4]) in the expression for the transverse energy change immediately results in the integral core of a very simple type. Besides, when deriving (11) no assumptions were made concerningthe dependence on transverse kinetic energy only, in contrast with ref. [5]. The application ofthe master equation (11) in the dechanneling problem within the frame of the finite difference procedure [4] makes it possible in the simplest way to take into account both contributions of close single and diffusion type collisions without introducing any phenomenological parameters. Its main point is the partition of the infinite transverse energy scale into a finite number of intervals. The particle transition into the adjacent interval due to the dominant diffusion mechanism is given by the diffusion coefficients. In the case of close single collisions and Coulomb behaviour of the cross section o~(0) it is possible to write down the expression for the integral core (3)
(16) It is essential that the closed integral formulation of the dechanneling the axial planar case appeared possibleproblem withoutinusing anyand quantum notion of scattering processes. This remark is quite suitable here in view of the fact that all obtained resuits (see eq. (10)) follow in the quasiclassicallimit from the quantum kinetic equation for the density matrix [1,10] and for eqs. (4), (5) and (8), (9) from Wigner’s representation for the quantum distribution function [11].
W(0’ 0)=
[9]V.Ryabov,Phys.Tverd.Te1a58(1988)1479. [10] J.U. Andersen and E. Bonderup, Nucl. Instrum. Methods B 33(1988)34. [111 H. Nitta and Y.H. Ohtsuki, Phys. Rev. 38 (1988) 4404.
•
.
1 exp(—r2/2u2)o~(I0’—0I) 2,tdu2
(15)
44
References [1] V. Ryabov, Soy. Phys. JETP 55 (1982) 684. [2]Y. Yamashita, Phys. Lett. A 104 (1984) 109. [3] E. Fushini and A. Uguzzoni, Radiat. Eff. 69 (1983) 113. [4] K. Gartner, K. Hehl and G. Schlotzhauser, Nucl. Instrum. Methods 216 (1983) 275. 15] T. Oshiyama and M. Mannami, J. Phys. Soc. Japan 51 (1982) 2255. [6] J. Lindhard, K. Dan. Vidensk. Selsk., Mat. Fys. Medd. 34 [71V. Ryabov, Soy. Phys. JETP 36 (1972) 577. [81V. Ryabov, Proc. VII USSR Conf. on Charged particle interaction with crystals (Moskow, MGU, 1976) p. 11.
PHYSICS LETTERS A
27 November 1989
INTEGRAL KINETIC EQUATION IN DECHANNELING PROBLEM V. RYABOV Institute of Physics and PowerEngineering, Obninsk 249020, USSR Received 3 May 1989; accepted forpublication 23 June 1989 Communicated by V.M. Agranovich
A version of dechanneling theory, based on using an integral kinetic equation in both the phase and transverse energy space, is described. It is derived from the binary collision model and it takes into account consistently the thermal multiple and single scattering of axial and planar channeled particles. The connection between the method developed and that of Oshiyama and of Gartner is discussed.
The diffusion approximation usually used in the problem of dechanneling has two disadvantages restricting its application in applied investigations. The first one, which concerns the logarithmic accuracy of the diffusion approach, results from a logarithmic discrepancy of the diffusion coefficients at small collision parameters, or, and that is quite the same, with large impulses transferred in the binary collision [1— 31. Putting aside the question concerning the choice of the minimal impact parameter let us indicate only the fact that actually this phenomenological parameter is not comprised in the initial integral formulation of the multiple scattering problem. The seeond disadvantage of the diffusion approach, the impossibility oftaking into account single scattering, had the effect of unsuccessful attempts to use it in defect crystals [4]. The known variants of the integral kinetic approach in the transverse energy representation which are based on the statistic equilibrium hypothesis [4,5] do not have such disadvantages. However, in terms of the most consistent Gartner method the kinetic equation integral core in axial channeling was determined from a rather extensive numerical cal-
this approach fails to describe the multiple scattering effects for a small distance from the particle to a row (plane). Below, proceeding from the binary collision model we obtain an integral kinetic equation of the Boltzmann type using the particle transverse coordinates and impulses in axial and planar channeling. The transformation to the transverse energy variable using the statistic equilibrium space distribution allows one to determine the origin of the integral approaches in question and to simplify substantially the Gartner method, currently successfully used in applied investigations. The arguments of the derivation of the original equation varied in many papers [7,8,2,3]. Let us consider a small part of the trajectory of length E~tL in the vicinity of the displaced chain atom with penod d, the particle being at a distance r from the chain axis. Strong correlation between the subsequent atomic collisions of the channeled particle result in the requirement of constant r within the length ~.L. Fig. la presents the projection of this particle trajectory part in the transverse plane with fixed configuration of random thermal displacements u1. In
culation of transverse energy changes in the binary collision. This difficulty seemed to have been overcome in the Oshiyama—Mannami method [5] where, in accordance with the Lindhard treatment [61, the transverse energy change is taken proportional to the thermal displacement ofthe lattice atom. As a result,
each i-binary collision the particle acquires a transverse impulse (s1) =pz~O(s~)in the direction of the impact parameter s1. The deflection angle d ~ ~O= ~
—
—
S
41
Volume 142, number 1
PHYSICS LETTERS A
a)
~
S
~\
S~// S~\
~
27 November 1989
w(0’,0,r)=~ö(0’—0—60(s,r)).
(3)
~° (,~)
s~\
I
angular fluctuations 60 on the particle trajectory which The kinetic follow equation from the in the equation phase of space motion including [1] can the be put in the following way:
8) ~
~~XI
18U8F 8r28r80
8z
~
__Jd0’ w(r,0’,O)[F(O’,r)—F(O,r)]. Fig. 1. Projection
of particle trajectory
(a) axial; (b) planar channeling.
is expressed in terms of the continuum string potential U(r). The total impulse transfer in the sequence of N= LxL/d collisions is equal to Ap1 = ~ Ap~(s1). On dividing the mean impulse transfer A~±, averaged over thermal displacements, by AL= vAt, we obtain the equation for the particle motion in the field of a continuum potential averaged over transverse thermal atomic displacements,
~‘
2/2u2)U(s) 0(r) = s=r—u —
~
In this equation use was made of suitable dimensionless variables: z=vt~P, 0=pjpW and U(r)/ U( 0), in which the kinetic equation looks identical both for MeV ions and for light ultrarelativistic partides (W=f20(0)/pv is the critical angle). It is easily noticed that a power expansion in 60 of the collisional term (4) accurate to O(60~)results in the well-known diffusion equation with angular spread [1—3,81 pv Dap(r)= ~
4~_ 80 ~t
Jdu exp( —u (1)
.
(4)
in the transverse plane:
(AOaA0flA0aA0P),
a,fl=x,y. (5) In planar channeling the particle incidents on some low-index atomic chain lying in the given atomic plane YZ at an angle exceeding the critical one, 0~,>~1’. Therefore we confine ourselves to the consideration
Here u2 is the root-mean-square amplitude of ther-
of only the trajectory path of length AL=d
mal displacements. When neglecting the correlations between the thermal displacements of atoms each occasional change pöO(s, r) =p[A0(s) —A0(r) 1 appears independent. Therefore actually the above-mentioned discussion concerns the r=const requirement only between two adjacent collisions. With the trajectory slope 0 =p /p at the point r this requirement will have the following form:
1/~Pbeing on one atomic chain. In fig. lb this is illustrated by a straight line segment. The condition of strong correlation between collisions in planar channeling reduces to the requirement of the constant distance between the particle trajectory and the atomic plane x within a length AL. From fig. lb it is evident that a mean nonvanishing change of the transverse impulse within the path AL is possible along the X axis. At the same time, in analogy with (1) we get the equation of planar motion [7]
Od<
.
(2)
The condition of collision independence permits determine at once theprobability ofthe change of particle direction within the trajectory part of length AL=d: one to
42
d0~ ~
1 8V(x) ~
where the mean value
(6)
Volume 142, number 1
~(x)=~-$dyU(r)
PHYSICS LETTERS A
(7)
coincides with the continuous atomic plane potential averaged over the transverse thermal displacements u~.Here the dimensionless variables 0~=p~/ pW0, z=L~t’~and ?(x)/I?(0) are also introduced (~P~=~/2?(0)/pv, L is the depth). As for the angular spread it is inherent in both directions of the transverse plane. Performing calculations identical to that of the axial case we get the kinetic equation ÔF 8F 1 0? OF + 0—— Oz Or 2 Ox OO~
—
=$do’ w~(0’,0,x)[F(0’,x)—F(0,x)],
(8)
with local scattering probability w~(0’,0,x)=~_$dyô(0’_0_60(r,s)). The second order series expansion of the~collision term results in a diffusion-type equation with diffusion coefficients 2), a=x,y; Da(X)DpoJdY (A0~—~ 2/ (9) D~0=Nd~vp where Nis the atomic density ofthe plane and dis ~‘~‘
27 November 1989
C dr W(~’,~)=~-’--a$dcoJ~s, x ö( 602_2 ö0(s, r) ~(r) cos ~,) ~‘
0(r)
—
—
=~J~— 0(r)
(11)
.
For large r the change of angle is equal to 60=
— -~-
ôr
,~ ôr
Hence (10) tends to the known expression derived by Oshiyama and Mannami [5]. The range of application of this equation is restricted only by the condition (2) fulfilled both for MeV ions and for ultrarelativistic particles. To realise the connection between the developed approach and the Gartner method let us turn to fig. 2 which depicts the geometry of the scattering ofions by a lattice displaced atom. In contrast to ref. [4] where an identical procedure included the continuous potential U(r) the averaged potential 0(r) will be used here as the first approach. It seems to be more consistent in terms of the trajectory equation (1). According to fig. 2 the particle coordinates in the transverse half-way planes between r adjacent atoms are given by 1 =r+~[~+A0(s)]d; r2=r—~d. (12) Expanding the transverse energy change ~~_f=(ê~+Ao)2±U(r 2)__g2_.U(r1)
the distance between the planes: Nd~=l/dd1. Thus, within the atomic chain row model [61 the stochastic angular spread compared that following from is thesuppressed amorphous planewith model (Da(X) u2, x—~oc) [7]. The discreteness contribution is unwarrantedly large in the last case (see the discussion in ref. [10]). The transformation of eq. (4) to the kinetic equation in the transverse energy variable ~= 0~+ U may be made by using the statistical equilibrium distribution function F=2irp(, z)n~(r).Integrating the obtained equation over the available region of area S, (r
,
(13)
in a power series of A0 and omitting terms of order 3) provides O(A0 ~ ~ 60+260 ê~(r) —~=
‘~‘
Oz
J
[W(E,~’)p(~’)—W(~’,E)p(~)],
with transition probability
(10)
7 ______________
/
2
/
~/
__________________
Fig. 2. Binary collision geometry. The position of the atom is marked by a circle.
43
Volume 142, number 1
PHYSICS LETTERS A
In this expression we used
27 November 1989
and for (11) (Ze2\J W(~,~)=t
—
A0=— d OU ——.
pvOr
~pvj
2
~
2
2
,
3exp(—r~/2u ) dS~I—~I
Accurate to the negligible term AG 60 the expression obtained coincides with the argument of the ôfunction in (11). 7(r) Thusand using the averaged neglecting smallcontinterms uous (. (whichpotential were considered in ref. [4]) in the expression for the transverse energy change immediately results in the integral core of a very simple type. Besides, when deriving (11) no assumptions were made concerningthe dependence on transverse kinetic energy only, in contrast with ref. [5]. The application ofthe master equation (11) in the dechanneling problem within the frame of the finite difference procedure [4] makes it possible in the simplest way to take into account both contributions of close single and diffusion type collisions without introducing any phenomenological parameters. Its main point is the partition of the infinite transverse energy scale into a finite number of intervals. The particle transition into the adjacent interval due to the dominant diffusion mechanism is given by the diffusion coefficients. In the case of close single collisions and Coulomb behaviour of the cross section o~(0) it is possible to write down the expression for the integral core (3)
(16) It is essential that the closed integral formulation of the dechanneling the axial planar case appeared possibleproblem withoutinusing anyand quantum notion of scattering processes. This remark is quite suitable here in view of the fact that all obtained resuits (see eq. (10)) follow in the quasiclassicallimit from the quantum kinetic equation for the density matrix [1,10] and for eqs. (4), (5) and (8), (9) from Wigner’s representation for the quantum distribution function [11].
W(0’ 0)=
[9]V.Ryabov,Phys.Tverd.Te1a58(1988)1479. [10] J.U. Andersen and E. Bonderup, Nucl. Instrum. Methods B 33(1988)34. [111 H. Nitta and Y.H. Ohtsuki, Phys. Rev. 38 (1988) 4404.
•
.
1 exp(—r2/2u2)o~(I0’—0I) 2,tdu2
(15)
44
References [1] V. Ryabov, Soy. Phys. JETP 55 (1982) 684. [2]Y. Yamashita, Phys. Lett. A 104 (1984) 109. [3] E. Fushini and A. Uguzzoni, Radiat. Eff. 69 (1983) 113. [4] K. Gartner, K. Hehl and G. Schlotzhauser, Nucl. Instrum. Methods 216 (1983) 275. 15] T. Oshiyama and M. Mannami, J. Phys. Soc. Japan 51 (1982) 2255. [6] J. Lindhard, K. Dan. Vidensk. Selsk., Mat. Fys. Medd. 34 [71V. Ryabov, Soy. Phys. JETP 36 (1972) 577. [81V. Ryabov, Proc. VII USSR Conf. on Charged particle interaction with crystals (Moskow, MGU, 1976) p. 11.