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γ factor in dechanneling theory
Volume 58A, number 7
PHYSICS LETTERS
y
18 October 1976
IN DECHANNELING THEORY
M. KITAGAWA Department of Electronics, North Shore College, Atsugi 243, Japan Received 6 July 1976 The relation between the Ellegaard—Lassen model and the diffusion model usually used in dechanneling theory is interpreted. A y factor, which is necessary to derive the diffusion coefficient under the channeling condition, is discussed.
Since the Lindhard theory [11, the theoretical treathas been developed by many
ment of dechanneling
[2—41mainly according to the diffusion model. Recently, the theoretical works concentrate in deriving the E1-dependent diffusion coefficient in higher degree of approximation [5,6], and solving the diffusion equation numerically [2,6]. However, in spite of many authors’ effort, the agreement between calculated results and experimental data is poor in general. This discrepancy between theory and experiment is discussed by Gemmell [7]. On the other hand, while many authors’ theoretical works cannot explain experimental data quantitatively, Ellegaard and Lassen [8] obtained the analytical result which gave a good fit to their experimental data according to their simple model. Taking into account the beam divergence and the surface effect, Fujimoto et al. [9] extended this model. The Ellegaard—Lassen model has been considered as a different approach from a usual diffusion model. It is based on the following two assumptions, (i) The distribution crystal has angular the Gaussian form. of the beam inside the
o f(0 z) dO ,
=
02
2D
8z exp
(— ~)
do.
(I)
authors
(ii) ç~2= (z/L) ~ where ~2 and z are the mean square angle of the Gaussian distribution and the depth from the surface respectively. L is the depth required for ~2 to grow as large as ~ under the channeling condition, and iL/~is the critical angle. The Ellegaard—Lassen model can be interpreted within the diffusion model as follows. Under the condition that the diffusion coefficient D0 in the angular space is constant and the initial distribution of the beam is taken as a delta function, the solution of the diffusion equation in the angular space is written
From eq. (1) and two assumptions of the Ellegaard--Iassen model, we obtain ~,2
D0
~
~,2 =
y
~—
,
(2)
0
where L = L0/’)’, and L0 is the depth required for ~22 to grow as large as 4i~under the condition of the random incidence. Alternatively, transforming the diffusion equation into the E1-space [2], we obtain the diffusion coefficient DEI(EI) in the E1-space from eq. (2), E DEI(E!)
~
(3) 0
where E~ E~I~ is the
critical energy and E is the energy of a projectile. As far as the nuclear part of the y factor (~yn)is concered, Ellegaard and Lassen take it as2),exp (_a2/u~), which are mdcand Fujimoto et al. as (0.1 u~)/(L na pendent of E 1, where a is the Thomas—Fermi screening radius, u~is the mean square thermal vibration amplitude perpendicular to the string, and Ln ln 1.29 e, c being the reduced energy in the LSS theory [10]. It is an interesting point that the y factor also appears as a quantity to be dependent on E1 in the usual dechanneling theory [1—6]. Therefore, it can be interpreted that the y factor appearing in the Ellegaard—Lassen model has a property to be the mean value of the E1dependent y factor (y (E~))over E1. This follows that the diffusion coefficient D1(E1) is approximated as the linear function of E1. 487
Volume 58A, number 7
PHYSICS LETTERS
18 October 1976
En fig. 1 we show the present result together with ‘y used by Ellegaard and Lassen, and that used by 08
Ellegaard and Lassen
Fujimoto et al. As far as the experiment performed by Ellegaard and Lassen for Bi, the best fit values de-
~_
rived from their experiment are 0.15 at 90K and 0.50 at 300 K, which are somewhat smaller at 90 K and larger at 300 K than the present result. On the other
06 90K
3
90K
hand, for Ge, the present result and the values used by Ellegaard and Lassen, and by Fujimoto et al. are too
0.4 Pres~nt
large in comparison with experimental values [9].
result
The averaging y0(E1) over E1 has a somewhat
0.2
Fujimoto et a 05
parameter-like property. However, taking it into the
consideration that the numerical results of the usual diffusion approach are in poor agreement with expe-
10
riments generally, and the relation between the Ellegaard— Lassen model and the diffusion model usually used in dechanneling theory can be interpreted through the
2
C a Fig. 1. 2a2) Thefor nuclear 5-MeV part proton of thein~y Bi. factor as a function of
ytorfactor, by averaging we notey(E that the determination of the y fac1) for various materials accurately
u~j(C Next, we calculate the averaged value of ‘y~(E1) over E1 for the many-phonon excitation [5], and compare it with that used in the Ellegaard—Lassen model
quantitatively. For the many-phonon excitation, y~(E1)is written down 2a~ I C ‘y~(E 1 (3) 1)= exp ~ —~-—(exp(c1) U l)~
is the useful approach for analysing the experimental data.
The author thanks Prof. M. Mannarni for discussing the theoretical aspects of dechanneling, and also thanks Prof. Y.H. Ohtsuki for valuable discussions.
,
—
—
1
References
/
where C 3 and c1 = E1/2E~. We define the mean value ~n as follows, 1
‘Yn — ~
[1] J. Lindhard, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 34
Ec ~
J 7n(Ei) dE1
(4)
.
c0
Therefore we obtain from eqs. (3) and (4),
[41 K. Morita, Radiat. Eff. 14 (1972) 195.
[51 M. Kitagawa and
Y.H. Ohtsuki, Phys. Rev. B8 (1973) 3117. Datz, B.R. andcollision C.D. Moak (Plenum, [61 eds. H.E. S.Schiott et al.,Appleton in: Atomic in solids, vol. 2,
=
1
—
=
~
2—1)
l/(e =
J
r
1
exp (—Bt) t(t
4 1\ e~Ei I\
+
e2
/ —
e2 — 1)
1) dt /
B
—
1
Ei I
\
—
—i—— B
e2 —1
(5)
J/
New York, 1975) p. 843.
[71 D.S. Gemmell, Rev. Mod. Phys. 46 (1974) 129.
[81 C. Ellegaard and N.O. Lassen, K. Dan. Vidensk. Slesk. Mat.-Fys. Medd. 35 (1967) N. 16. [91 F. Fujimoto, K. Komaki, H. Nakayama and M. Ishii, Radiat. Eff. 13 (1972) 43. [101 J. Lindhard, M. Scharff and HE. Schiott, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 33, N. 14 (1963).
‘
where B = C2a2/u~,and Ei(—x) is the exponentialintegral function which is defined as
et
(6)
488
No. 14 (1965). 121 E. Bonderup et al., Radiat. Eff. 12 (1972) 261. [31 V.V. Beloshitsky, M.A. Kumakhov and V.A. Mulalev, Radiat. Eff. 13 (1972) 9.
PHYSICS LETTERS
y
18 October 1976
IN DECHANNELING THEORY
M. KITAGAWA Department of Electronics, North Shore College, Atsugi 243, Japan Received 6 July 1976 The relation between the Ellegaard—Lassen model and the diffusion model usually used in dechanneling theory is interpreted. A y factor, which is necessary to derive the diffusion coefficient under the channeling condition, is discussed.
Since the Lindhard theory [11, the theoretical treathas been developed by many
ment of dechanneling
[2—41mainly according to the diffusion model. Recently, the theoretical works concentrate in deriving the E1-dependent diffusion coefficient in higher degree of approximation [5,6], and solving the diffusion equation numerically [2,6]. However, in spite of many authors’ effort, the agreement between calculated results and experimental data is poor in general. This discrepancy between theory and experiment is discussed by Gemmell [7]. On the other hand, while many authors’ theoretical works cannot explain experimental data quantitatively, Ellegaard and Lassen [8] obtained the analytical result which gave a good fit to their experimental data according to their simple model. Taking into account the beam divergence and the surface effect, Fujimoto et al. [9] extended this model. The Ellegaard—Lassen model has been considered as a different approach from a usual diffusion model. It is based on the following two assumptions, (i) The distribution crystal has angular the Gaussian form. of the beam inside the
o f(0 z) dO ,
=
02
2D
8z exp
(— ~)
do.
(I)
authors
(ii) ç~2= (z/L) ~ where ~2 and z are the mean square angle of the Gaussian distribution and the depth from the surface respectively. L is the depth required for ~2 to grow as large as ~ under the channeling condition, and iL/~is the critical angle. The Ellegaard—Lassen model can be interpreted within the diffusion model as follows. Under the condition that the diffusion coefficient D0 in the angular space is constant and the initial distribution of the beam is taken as a delta function, the solution of the diffusion equation in the angular space is written
From eq. (1) and two assumptions of the Ellegaard--Iassen model, we obtain ~,2
D0
~
~,2 =
y
~—
,
(2)
0
where L = L0/’)’, and L0 is the depth required for ~22 to grow as large as 4i~under the condition of the random incidence. Alternatively, transforming the diffusion equation into the E1-space [2], we obtain the diffusion coefficient DEI(EI) in the E1-space from eq. (2), E DEI(E!)
~
(3) 0
where E~ E~I~ is the
critical energy and E is the energy of a projectile. As far as the nuclear part of the y factor (~yn)is concered, Ellegaard and Lassen take it as2),exp (_a2/u~), which are mdcand Fujimoto et al. as (0.1 u~)/(L na pendent of E 1, where a is the Thomas—Fermi screening radius, u~is the mean square thermal vibration amplitude perpendicular to the string, and Ln ln 1.29 e, c being the reduced energy in the LSS theory [10]. It is an interesting point that the y factor also appears as a quantity to be dependent on E1 in the usual dechanneling theory [1—6]. Therefore, it can be interpreted that the y factor appearing in the Ellegaard—Lassen model has a property to be the mean value of the E1dependent y factor (y (E~))over E1. This follows that the diffusion coefficient D1(E1) is approximated as the linear function of E1. 487
Volume 58A, number 7
PHYSICS LETTERS
18 October 1976
En fig. 1 we show the present result together with ‘y used by Ellegaard and Lassen, and that used by 08
Ellegaard and Lassen
Fujimoto et al. As far as the experiment performed by Ellegaard and Lassen for Bi, the best fit values de-
~_
rived from their experiment are 0.15 at 90K and 0.50 at 300 K, which are somewhat smaller at 90 K and larger at 300 K than the present result. On the other
06 90K
3
90K
hand, for Ge, the present result and the values used by Ellegaard and Lassen, and by Fujimoto et al. are too
0.4 Pres~nt
large in comparison with experimental values [9].
result
The averaging y0(E1) over E1 has a somewhat
0.2
Fujimoto et a 05
parameter-like property. However, taking it into the
consideration that the numerical results of the usual diffusion approach are in poor agreement with expe-
10
riments generally, and the relation between the Ellegaard— Lassen model and the diffusion model usually used in dechanneling theory can be interpreted through the
2
C a Fig. 1. 2a2) Thefor nuclear 5-MeV part proton of thein~y Bi. factor as a function of
ytorfactor, by averaging we notey(E that the determination of the y fac1) for various materials accurately
u~j(C Next, we calculate the averaged value of ‘y~(E1) over E1 for the many-phonon excitation [5], and compare it with that used in the Ellegaard—Lassen model
quantitatively. For the many-phonon excitation, y~(E1)is written down 2a~ I C ‘y~(E 1 (3) 1)= exp ~ —~-—(exp(c1) U l)~
is the useful approach for analysing the experimental data.
The author thanks Prof. M. Mannarni for discussing the theoretical aspects of dechanneling, and also thanks Prof. Y.H. Ohtsuki for valuable discussions.
,
—
—
1
References
/
where C 3 and c1 = E1/2E~. We define the mean value ~n as follows, 1
‘Yn — ~
[1] J. Lindhard, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 34
Ec ~
J 7n(Ei) dE1
(4)
.
c0
Therefore we obtain from eqs. (3) and (4),
[41 K. Morita, Radiat. Eff. 14 (1972) 195.
[51 M. Kitagawa and
Y.H. Ohtsuki, Phys. Rev. B8 (1973) 3117. Datz, B.R. andcollision C.D. Moak (Plenum, [61 eds. H.E. S.Schiott et al.,Appleton in: Atomic in solids, vol. 2,
=
1
—
=
~
2—1)
l/(e =
J
r
1
exp (—Bt) t(t
4 1\ e~Ei I\
+
e2
/ —
e2 — 1)
1) dt /
B
—
1
Ei I
\
—
—i—— B
e2 —1
(5)
J/
New York, 1975) p. 843.
[71 D.S. Gemmell, Rev. Mod. Phys. 46 (1974) 129.
[81 C. Ellegaard and N.O. Lassen, K. Dan. Vidensk. Slesk. Mat.-Fys. Medd. 35 (1967) N. 16. [91 F. Fujimoto, K. Komaki, H. Nakayama and M. Ishii, Radiat. Eff. 13 (1972) 43. [101 J. Lindhard, M. Scharff and HE. Schiott, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 33, N. 14 (1963).
‘
where B = C2a2/u~,and Ei(—x) is the exponentialintegral function which is defined as
et
(6)
488
No. 14 (1965). 121 E. Bonderup et al., Radiat. Eff. 12 (1972) 261. [31 V.V. Beloshitsky, M.A. Kumakhov and V.A. Mulalev, Radiat. Eff. 13 (1972) 9.