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Diffusion coefficient for dechanneling theory
Volume 93A, number 3
PHYSICS LETTERS
3 January 1983
DIFFUSION COEFFICIENT FOR DECHANNELING THEORY
H. NITTA and Y.H. OHTSUKI
Department of Physics, Waseda University, Ohkubo 3, Shin]uku, Tokyo 160, Japan Received 1 July 1982
A simple expressionof the diffusion coefficient due to the electronic excitation for energetic ions in planar dechanneling as a function of the distance from the channel wall is proposed.
1. Introduction. Since Lindhard proposed a somewhat phenomenological theory for dechanneling [ 1], many workers have studied the dechanneling phenomena by the use of Lindhard's diffusion coefficient, which is proportional to the electron density for electronic excitation. On the other hand, Waho and Ohtsuki [2] derived the diffusion coefficient on the basis of a quantum-mechanical treatment of the inelastic scattering process of electronic excitation in the crystal, and which agrees well with experiment quantitatively [3]. However, their expression of the diffusion coefficient as a function of the distance from the channel wall is rather complicated to be employed for the calculation. Ohtsuki et al. [4] proposed a linear-combination-type expression of the electron density, from the computer calculation of the complicated expression of Waho and Ohtsuki. However, this formula is not general. Here we derive another simple expression of the diffusion coefficient by taking a certain approximation for Waho and Ohtsuki's expression. 2. Derivation o f the diffusion coefficient. In planar channeling conditions, we may consider only the diffusion coefficient of the single-electron excitation part [5] which is given by [2] Dy (y) = (27rZ2e4/u2)N
L e = ln(qmax/qmin ) and = 1 +g2 2 3 I(gy) 4--~2(/32 - qmax) + ~ In I ¢
+lln 4R
q 2max_ g2 +~j2
,
2 2 [ qmax+gyR-/32 R+I 2 /32 R--2-r ' ' 2
]qmax - g;R
gy
l(gy)n(gy) exp(igyy) ), (1)
where 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
(2)
where 2 2 1/2 R = [1 + 4(qmin/g~) ] , 2 2 /32 = [(q2max _g2)2 +4qming~l
1/2
,
qmin and qmax are the cut-off wave number, qmin = I,/flVo ,
qmax = 2mvo/h ,
v0 and Z 1 being the velocity and the atomic number of the incident ion respectively, N is the density of the atoms in the channel plane, and n(y) means the electron density and n (gy) its Fourier component where gy is the one-dimensional reciprocal lattice vector. Although the position dependency of eq. (1) is complicated, it should be noted that the numerical calculation of eq. (1) for many cases indicates that Dy (y) is a linear function of the electron density n (y), i.e. [4]
Dy (y) ~--(2nZ 2e4/v2)N [L~n (y) + C] , X((L e - })n(y)+~
_____ 2q2ax
(3)
where L e is the factor obtained by modifying Le, and C is a constant which does not depend on the position y. We now derive eq. (3) in an analytical manner. 135
Volume 93A, number 3
PHYSICS LETTERS
3. Simplified formula. In high energy cases, the relation q2max >> q2in always holds. In addition, as gy increases, n (gy) approaches zero rapidly. Thus we can assume q2ax >>g2. Therefore, eq. (2) takes the form igyl l(gY)~2(g2
In (g2 + 4 q 2 i n ) l / 2 + g y
+ 4qLin)l/2
(g2 + 4q2in)l/2 _
I_I.~Si 2C
:I
n (gy)
I*
-l'h]
,
-2
2
(5) 1/2
2
I(gy) = ~ (qmin/qmax) ~ 0 f o r g y = 0. Thus we arrive at the following conclusion for the constants in eq. (3): L e=L e+I*,
C =-I*.
(7)
In eq. (6) only one uncertain value ~y is included. The value gv may be expressed by gy = CgyO, where c is a parameter which depends both on the crystal and on the incident ion. The parameter e takes the value of order ~1 in ordinary cases. In fig. 1 we show our results for the "half-thickness"
136
/ "
L,?.>.D
;
1'o (MeV)
Fig. 1. Half-thickness tll 2 for H+ in Ge(110) and Si(110). Top two curves represent our results and lower two curves are the results based on the Lindhard theory.
-
+gYI
4q---2min)l/2 - ~ 2 - + ~ 2 - - - ~ ' (6) + (g; 4qmi n) --gy[ h being the mean electron density n(gy = 0). #y is the mean value of the reciprocal lattice vector which was effective in the summation in eq. (1). In the above, we used 2
~IC_r
ENERGY
(g2
l
izl I
o
in I gY + 4qmin)
gY
~'ZI'IJ
exp(igyy))
where =
RESULTS/l( /
I-
(27rZ2e4102)N
× [(L e + I * - l ) n ( y )
o
gy(14)
Dy (y) ~ (2nZ2e4/u2)N
=
L=
T
gy¢O
(110) - Ge(110) . . . . .
PRESENT /o / /
.
Further, as can be seen from the direct numerical calculations of eq. (2), I(gy) changes more slowly than n(gy) in the region o f q 2 a x >>g2, then we obtain
X((Le-})n(y)+I*
3 January 1983
tl/2
[3] calculated by eq. (5), comparing it with the Lindhard dechanneling theory and with the experimental results by Feldman et al. [6]. The agreement between our theory and experiment is fairly good. One of the authors thanks Dr. M. Kitagawa for his useful discussions.
References [1] J. Lindhard, Mat. Fys. Medd. Dan. Vid. Selsk. 34 (1965) n. 14. [2] T. Waho and Y.H. Ohtsuki, Radiat. Eft. 21 (1974) 217. [3] T. Waho, Phys. Rev. B14 (1976) 4830. [4} Y.H. Ohtsuki, T. ()mura, H. Tanaka and M. Kitagawa, Nucl. Instrum. Methods 149 (1978) 361. [5] T. Waho and Y.H. Ohtsuki, Radiat. Eft. 27 (1976) 151. [6] L.C. Feldman and B.R. Appleton, Phys. Rev. B8 (1973) 935.
PHYSICS LETTERS
3 January 1983
DIFFUSION COEFFICIENT FOR DECHANNELING THEORY
H. NITTA and Y.H. OHTSUKI
Department of Physics, Waseda University, Ohkubo 3, Shin]uku, Tokyo 160, Japan Received 1 July 1982
A simple expressionof the diffusion coefficient due to the electronic excitation for energetic ions in planar dechanneling as a function of the distance from the channel wall is proposed.
1. Introduction. Since Lindhard proposed a somewhat phenomenological theory for dechanneling [ 1], many workers have studied the dechanneling phenomena by the use of Lindhard's diffusion coefficient, which is proportional to the electron density for electronic excitation. On the other hand, Waho and Ohtsuki [2] derived the diffusion coefficient on the basis of a quantum-mechanical treatment of the inelastic scattering process of electronic excitation in the crystal, and which agrees well with experiment quantitatively [3]. However, their expression of the diffusion coefficient as a function of the distance from the channel wall is rather complicated to be employed for the calculation. Ohtsuki et al. [4] proposed a linear-combination-type expression of the electron density, from the computer calculation of the complicated expression of Waho and Ohtsuki. However, this formula is not general. Here we derive another simple expression of the diffusion coefficient by taking a certain approximation for Waho and Ohtsuki's expression. 2. Derivation o f the diffusion coefficient. In planar channeling conditions, we may consider only the diffusion coefficient of the single-electron excitation part [5] which is given by [2] Dy (y) = (27rZ2e4/u2)N
L e = ln(qmax/qmin ) and = 1 +g2 2 3 I(gy) 4--~2(/32 - qmax) + ~ In I ¢
+lln 4R
q 2max_ g2 +~j2
,
2 2 [ qmax+gyR-/32 R+I 2 /32 R--2-r ' ' 2
]qmax - g;R
gy
l(gy)n(gy) exp(igyy) ), (1)
where 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
(2)
where 2 2 1/2 R = [1 + 4(qmin/g~) ] , 2 2 /32 = [(q2max _g2)2 +4qming~l
1/2
,
qmin and qmax are the cut-off wave number, qmin = I,/flVo ,
qmax = 2mvo/h ,
v0 and Z 1 being the velocity and the atomic number of the incident ion respectively, N is the density of the atoms in the channel plane, and n(y) means the electron density and n (gy) its Fourier component where gy is the one-dimensional reciprocal lattice vector. Although the position dependency of eq. (1) is complicated, it should be noted that the numerical calculation of eq. (1) for many cases indicates that Dy (y) is a linear function of the electron density n (y), i.e. [4]
Dy (y) ~--(2nZ 2e4/v2)N [L~n (y) + C] , X((L e - })n(y)+~
_____ 2q2ax
(3)
where L e is the factor obtained by modifying Le, and C is a constant which does not depend on the position y. We now derive eq. (3) in an analytical manner. 135
Volume 93A, number 3
PHYSICS LETTERS
3. Simplified formula. In high energy cases, the relation q2max >> q2in always holds. In addition, as gy increases, n (gy) approaches zero rapidly. Thus we can assume q2ax >>g2. Therefore, eq. (2) takes the form igyl l(gY)~2(g2
In (g2 + 4 q 2 i n ) l / 2 + g y
+ 4qLin)l/2
(g2 + 4q2in)l/2 _
I_I.~Si 2C
:I
n (gy)
I*
-l'h]
,
-2
2
(5) 1/2
2
I(gy) = ~ (qmin/qmax) ~ 0 f o r g y = 0. Thus we arrive at the following conclusion for the constants in eq. (3): L e=L e+I*,
C =-I*.
(7)
In eq. (6) only one uncertain value ~y is included. The value gv may be expressed by gy = CgyO, where c is a parameter which depends both on the crystal and on the incident ion. The parameter e takes the value of order ~1 in ordinary cases. In fig. 1 we show our results for the "half-thickness"
136
/ "
L,?.>.D
;
1'o (MeV)
Fig. 1. Half-thickness tll 2 for H+ in Ge(110) and Si(110). Top two curves represent our results and lower two curves are the results based on the Lindhard theory.
-
+gYI
4q---2min)l/2 - ~ 2 - + ~ 2 - - - ~ ' (6) + (g; 4qmi n) --gy[ h being the mean electron density n(gy = 0). #y is the mean value of the reciprocal lattice vector which was effective in the summation in eq. (1). In the above, we used 2
~IC_r
ENERGY
(g2
l
izl I
o
in I gY + 4qmin)
gY
~'ZI'IJ
exp(igyy))
where =
RESULTS/l( /
I-
(27rZ2e4102)N
× [(L e + I * - l ) n ( y )
o
gy(14)
Dy (y) ~ (2nZ2e4/u2)N
=
L=
T
gy¢O
(110) - Ge(110) . . . . .
PRESENT /o / /
.
Further, as can be seen from the direct numerical calculations of eq. (2), I(gy) changes more slowly than n(gy) in the region o f q 2 a x >>g2, then we obtain
X((Le-})n(y)+I*
3 January 1983
tl/2
[3] calculated by eq. (5), comparing it with the Lindhard dechanneling theory and with the experimental results by Feldman et al. [6]. The agreement between our theory and experiment is fairly good. One of the authors thanks Dr. M. Kitagawa for his useful discussions.
References [1] J. Lindhard, Mat. Fys. Medd. Dan. Vid. Selsk. 34 (1965) n. 14. [2] T. Waho and Y.H. Ohtsuki, Radiat. Eft. 21 (1974) 217. [3] T. Waho, Phys. Rev. B14 (1976) 4830. [4} Y.H. Ohtsuki, T. ()mura, H. Tanaka and M. Kitagawa, Nucl. Instrum. Methods 149 (1978) 361. [5] T. Waho and Y.H. Ohtsuki, Radiat. Eft. 27 (1976) 151. [6] L.C. Feldman and B.R. Appleton, Phys. Rev. B8 (1973) 935.