- Email: [email protected]
Numerical calculation of parametric X-ray radiation by relativistic electrons channeled in a Si crystal
*H CL
Nuclear Instruments
and Methods in
Physics ResearchB 115(19%) 3%~400 FYlOKil
B
Beam Interactiaa38 with Yateriale 8 Atoms
EISEVIER
Numerical calculation of parametric X-ray radiation by relativistic electrons channeled in a Si crystal
*,
Y. Matsuda a, T. Ikeda a, H. Nitta b, H. Minowa ‘, Y.H. Ohtsuki
a
a Department of Physics, Waseda University Ohkubo 3-4-1, Shinjuku-ku. Tokyo, 169, Japan b Department of Physics. Tokyo Gakugei University Nukuikita-machi, Koganei, Tokyo, 184, Japan ’ Department of Physics, Teikyou Heisei University Ohoyatsu. Uruido, Ichihara, Chiba, 290-Ol( Japan Abstract Recently, a new type of radiation called “parametric X-ray radiation,” is of special interest and several experimental studies have been reported. By using the kinematical theory, we calculate differential scattering cross section of parametric X-ray radiation (PXR) and diffracted channeling radiation (CR) by (111) planes in silicon for 15-50 MeV electron beams channeled along (110) planes.
1. Introduction The phenomenon of parametric X-ray radiation (PXR) is that the X-ray radiation is observed near the Bragg angle from relativistic electron beams in a single crystal. In 1972, this phenomenon was predicted by Ter-Mikaelian [l]. The process of PXR is considered to be coherent polarization radiation from atoms polarized by the field of the incident electron. However, the origin of PXR is not yet very well understood. PXR is monochromatic and has a narrow angle distribution theoretically. In 1985, PXR was observed in Tomsk, for the first time. On the other hand, there are coherent radiations in a crystal due to the periodic modulation of the trajectories of incident electron, i.e., channeling radiation (CR). When the energy of the incident electron is under 50 MeV, the energy of CR is under 100 keV. Thus, CR is diffracted in a crystal. Therefore, in this paper, we calculate the angular distribution of PXR under the channeling condition (we call PXRC) when the Lorentz factor of the incident electrons: y 5 100. In this case, as the quantum effects appear remarkably about the electron motion, we use a semi-quantum approximation. To solve the wave equation of a channeling electron, we use a continuous approximation for the longitudinal motion parallel to the planes and use a many wave approximation for the transverse motion. We show that the angular distribution of PXRC of electron beams has an interesting shape, and a very strong peak compared to PXR
author. Tel.: + 8 1 3 5286 3440; fax: + 8 1 3 * Corresponding 3200 2567; e-mail: [email protected]. 0168-583X/%/$15.00 Copyright PII SO168-583X(96)00286-8
is obtained at the Bragg angle at a certain energy of the incident electron. We compare the intensity of PXRC in the case of a positron with that in the case of an electron.
2. Theory Not considered about spinor, the interaction Hamiltonian is given by H’=
-
&(A .P)*
and the transition probability per unit time is given by
where p is the momentum of the incident electron, u is the polarization-direction, Jl(r) is the wave function for the incident electron written as
where L is the length of the crystal. The transverse motion of channeled particles can be described by a S&r&linger equation with the relativistic mass.
(-&VT 1
0 1996 Elsevier Science B.V. All rights reserved
+U(r,) cp,(r.)-4,cpn(r,).
(4)
Y. Matsuda et al. / Nucl. Instr. and Meth. in Phys. Res. B I15 (1996) 396-400
Since the planar potential is periodic, it may be expanded in a Fourier series and the eigenfunctions become Bloch waves: IJ(r,)=
CG, h,
exp(ih,
cp,(rl)
=exp(k,
.r,)>
.ri)CC?j h,
(5) exp(h,
.r,).
Summing up the polarization of emitted X-rays, the number of photons per unit time:
dx,,12w dfi dNk= 27rc’2(1 -lVp;)
‘(
(6)
&(X1+q2 +
c u, I _
cp;
c$y’ = 0.
h’,
x [k;
1
- E’;‘(k,)
InX
[z,l(w-ni()h-(w/c*)2v,l]
+[(kh.z_)h-(W/C*>2z,,)] I’)
The band index n distinguishes the solutions of Eq. (4). which are assumed to have the many-beam matrix form
[
397
- (w/c*)‘]
-2
(13)
where I,,=((cpr, IexP(ikh.r.)lcpi,),
(7)
h’,
Z,=(q+,
(14)
lexp(-ik,.r,)(-ih)V,
IcPi.)
(‘5)
A good description of many experimental results is given by the following Doyle-Turner expression for a single plane, which is obtained by fitting the electron-scattering factor determined by a Hartree-Fock calculation with three parameters to experimental results:
and R ;,= ( Ej
U(r,)=
where Fh is the structure factor, v, is the volume of unit cell, A is the wavelength of the X-ray. We obtain the angular distribution of photons near the Bragg angle from a channeled electron in the case that the final state is different from the initial state,
(8)
I --IT, I )/fL, OL is the fine structure constant, n = k/] k I, xh is the Fourier component of the electric susceptibility. For X-rays, we have
Xh= _i
--
(‘6)
IT
aN
a
afw, -zc*2
E,=E,=\l(p,~)*+(rnc~)*, (9)
tq+(e2
co?
28, -2(n,,/oB))e:
[e:+et-2(R,,/o,)+Y-?+
where N is the atomic density in the crystal, d, is the interplanar distance, ai and bj = (2rr)‘B, are the coe.fficients tabulated by Doyle and Turner, pt is the one-dimensional mean square vibrational amplitude, Ei and Er are the initial and the final energy of the incident electron. The vector-potential A(r) is explained by Bloch waves:
+(n,,/oe)2
X
E,-E,+iio=/~+h.
Ixo/]2
('7) PX is radiated without any relations to a transverse state and the angular distribution of photons is already explained [S] as: aN
a
oB
-=~-+h12
A(r) = c h
c k’
c [Ah(k’,
ae,ae,
a’)
d
Xexp(ikL
(COG 20,)@-8; . r,)a,,,,
+ c.c.1.
(‘0) (11)
kh(
h.AO(k))
k:
-
(w/c*
)2Ao(k)
,
(W/c*)Z
(18) x(2e,+8~+y-2+lxol)2' where 8, and 8, are measured from the direction of the Bragg angle, rlif = (‘p, I ri I cpi>. (i.e., the final state corresponds to the initial state.) As a result, near the Bragg condition, the angular distribution of photons is obtained by adding Eqs. (17) and (18).
(‘2)
is the phase velocity of light in the where c * = c/G crystal, l k. is the polarization-vector, V is the volume of the crystal, k is the wave-vector of a photon, k, = k + h corresponds to that of a diffracted wave.
3.Calculations
and discussion
If the value of the denominator is zero in Eq. (13) (i.e., the value of Eq. (13) is infinity), then “a real photon” is V.
CRYSTALASSISTED
PROCESSES
Y. Matsuda et al./Nucl.
398
Insrr. and Meth. in Phys. Res. B II.5 (1996) 396-400
not diffracted. Thus, from the intensity of CR, the intensity of diffracted X-rays is obtained. The angular dis~bution of CR is written as [7]: -dwif = dO
e=r= iiic):,[(l -(l
- /&,12) sin’ 6 cos’ +]
X[2nhc3(1-
-30 -2
I
y,
,
n=O .
0
j 2
a.u. Fig. 1. The (1 IO)planar potential evaluated from Eq. (7) with the contributions from neighboring planes included. The vibrational amplitude for room temperature 300 K, as obtained from the Lkbye model with To =L625 K, is p, = 0.126 a.u. (0.067 A>. Energy levels for bound states of y = 100 (51.2 MeV) electrons are indicated, and the intensity dist~butions across the plane are shown.
diffracted. Therefore, the value has an upper limit for this condition. According to two-wave approximation of dynamical diffraction theory for X-rays, the mean intensity of diffracted X-rays is equal to the intensity of X-rays that is
- lP,,lcos a)*
lp,,lcos
a)“]-‘.
(19)
Assuming that the z axis is directed along the longi~dinal motion and x axis is perpendicular to the channeling planes and where 6 is the angle between k and the z axis, + is the polar angle of the vector k in the x, y plane. When the electron is under channeling conditions, the angular dis~bution of photons near the Bragg angle is different from PXR. We calculate the angular distribution under the following conditions. 15-50 MeV electron beams are channeled along (110) planes and CR photons are diffracted by (111) planes in silicon. The Bragg angle 0,=7.9” (1.379X 10-l r-ad), dllO>1.92 I;i, d(lll)= 3.135 A, wu = 14.38 keV, x0 i=i4.625 X 10e6, x,, = 2.479 X 10p6. Fig. 1 shows the potential of an incident electron, corrected for the contribution from neighboring planes, for a (110) plane in silicon; the squares of the wave-function from Pq. (4) are also shown. In the case of y = 100, photons forming four concentric circles are obtained from diffracted CR in addition to PXR (Fig. 2). However, the
(rad.) O-O3 ” Fig. 2 The angular distribution of PXRC forms four concentric circles near the Bragg angle for y = 100. From inside to outside, each circle represents the intensity of diffracted radiation for each transition (4 --f 3). (3 + 21, (2 --) 1) and ( 1 -+ 0).
Y. Matsuda et al./Nucl.
Table I The intensities
Instr. and Meth. in Phys. Res. B I15
of PXRC and PXR arc shown when CR satisfies
the Bragg condition Ya
Transition b
Number of photons
PXR (max)
PXRC 60 50 41 29
4.58X 4.90x 4.05x 2.84X
4-3 3-2 2-l I-O
lO-4 10-4 lo-4 1O-4
9.02X 6.27x 4.66X 2.18X
lo-’ IO- * IO-’ lo-*
a Lorentz factor of incident electron. b Transition state (initial-fmal).
intensity is weak compared with PXR. As the value of the Lorentz factor y decreases, the radius of the photon circles decreases. In the case of y = 60, 50, 41 and 29, the
f 1996) 396-400
399
photons radiated in the direction of the longitudinal motion of the incident electron satisfy the Bragg condition. (The radiated photons for the each transition of (4 + 31, (3 --f 2). (2 --$ 1) and (1 -+ 0) satisfy this condition corresponding with each value of r.> In this case, the intensity of PXRC is some thousands times stronger than PXR under the same conditions (see Table 1 and Fig. 3). When the value of y is little less than these values, these peaks are much smaller than that of PXR. These peaks are explained as diffracted virtual photons, kinematically. Incidentally, we compare the angular distribution of the radiation in the case of positron incidence with that of electron incidence. As the potential for a positron is almost harmonica1 in the crystal, the difference of the energy level is almost the same. Thus, the angular distribution of PXRC makes one thick circle. The transition probability is more than that in the case of electron. According to our calcula-
b-ad.1 _o,/l
0.4
N
0.2
0
-i
?ad.)
Fig. 3. The angular distributions satisfies the Bragg condition.
of PXRC (a) and PXR (b) in the case of y = 29. In this case, the radiated photon for (1 -+ 0) transition
V. CRYSTAL
ASSISTED
PROCESSES
400
Y. Matsuda
et al./Nucl.
Instr.
and Meth. in Phys. Res. B II5
tions, the intensity of PXRC by positrons is one order higher than that by electrons. As a result, we show the angular distribution of PXRC, but a kinematical theory does not predict the intensity of diffracted CR. To estimate the intensity, a dynamical theory must be used. However, the kinematical formula can easily explain the places where the intensity of PXRC is most strong.
References
[II M.L. Ter-Mikaelian, High-energy Electromagnetic Processes in Condensed Media (Wiley-Interscience, New York, 1972) p. 332. M.A. Kumakhov, Phys. Lett. A 57 (1976) 17. L-21 131V.G. Baryshevsky and I.D. Feranchuk, J. Phys. (Paris) 44 (1983).
(1996) 396-400
V.G. Baryshevsky and I. Ya. Dubovskaya, J. Phys. C 16 (1983) 3663. (51 I.D. Feranchuk and A.V. Ivashin, J. Phys. (Paris) 46 (1985) 1981. [6] H. Nina, Phys. Lett. A 158 (1991) 270. [7] V.V. Beloshitsky and F.F. Komarov, Phys. Rep. 93 (1982) 117. [8] Y.H. Ohtsuki, Charged Beam Interaction with Solids (Taylor and Francis, 1983). [9] A.W. Sknz and H. ijberall, Coherent Radiation Sources (Springer-Verlag. 1985). [lo] C. Kittel, Introduction to Solid State Physics, 6th ed. (Wiley, 1986). [ 1l] M.A. Kumakhov and R. Wedell, Radiation of Relativistic Light Particles during Interaction with Single Crystals: Spectrum (Akad. Verl., 1991). [ 121 J.A. lhers and W.C. Hamilton, International Tables for X-my Crystallography, (Vol. 4) (The International Union of Crystallography, 1974). [4]
Nuclear Instruments
and Methods in
Physics ResearchB 115(19%) 3%~400 FYlOKil
B
Beam Interactiaa38 with Yateriale 8 Atoms
EISEVIER
Numerical calculation of parametric X-ray radiation by relativistic electrons channeled in a Si crystal
*,
Y. Matsuda a, T. Ikeda a, H. Nitta b, H. Minowa ‘, Y.H. Ohtsuki
a
a Department of Physics, Waseda University Ohkubo 3-4-1, Shinjuku-ku. Tokyo, 169, Japan b Department of Physics. Tokyo Gakugei University Nukuikita-machi, Koganei, Tokyo, 184, Japan ’ Department of Physics, Teikyou Heisei University Ohoyatsu. Uruido, Ichihara, Chiba, 290-Ol( Japan Abstract Recently, a new type of radiation called “parametric X-ray radiation,” is of special interest and several experimental studies have been reported. By using the kinematical theory, we calculate differential scattering cross section of parametric X-ray radiation (PXR) and diffracted channeling radiation (CR) by (111) planes in silicon for 15-50 MeV electron beams channeled along (110) planes.
1. Introduction The phenomenon of parametric X-ray radiation (PXR) is that the X-ray radiation is observed near the Bragg angle from relativistic electron beams in a single crystal. In 1972, this phenomenon was predicted by Ter-Mikaelian [l]. The process of PXR is considered to be coherent polarization radiation from atoms polarized by the field of the incident electron. However, the origin of PXR is not yet very well understood. PXR is monochromatic and has a narrow angle distribution theoretically. In 1985, PXR was observed in Tomsk, for the first time. On the other hand, there are coherent radiations in a crystal due to the periodic modulation of the trajectories of incident electron, i.e., channeling radiation (CR). When the energy of the incident electron is under 50 MeV, the energy of CR is under 100 keV. Thus, CR is diffracted in a crystal. Therefore, in this paper, we calculate the angular distribution of PXR under the channeling condition (we call PXRC) when the Lorentz factor of the incident electrons: y 5 100. In this case, as the quantum effects appear remarkably about the electron motion, we use a semi-quantum approximation. To solve the wave equation of a channeling electron, we use a continuous approximation for the longitudinal motion parallel to the planes and use a many wave approximation for the transverse motion. We show that the angular distribution of PXRC of electron beams has an interesting shape, and a very strong peak compared to PXR
author. Tel.: + 8 1 3 5286 3440; fax: + 8 1 3 * Corresponding 3200 2567; e-mail: [email protected]. 0168-583X/%/$15.00 Copyright PII SO168-583X(96)00286-8
is obtained at the Bragg angle at a certain energy of the incident electron. We compare the intensity of PXRC in the case of a positron with that in the case of an electron.
2. Theory Not considered about spinor, the interaction Hamiltonian is given by H’=
-
&(A .P)*
and the transition probability per unit time is given by
where p is the momentum of the incident electron, u is the polarization-direction, Jl(r) is the wave function for the incident electron written as
where L is the length of the crystal. The transverse motion of channeled particles can be described by a S&r&linger equation with the relativistic mass.
(-&VT 1
0 1996 Elsevier Science B.V. All rights reserved
+U(r,) cp,(r.)-4,cpn(r,).
(4)
Y. Matsuda et al. / Nucl. Instr. and Meth. in Phys. Res. B I15 (1996) 396-400
Since the planar potential is periodic, it may be expanded in a Fourier series and the eigenfunctions become Bloch waves: IJ(r,)=
CG, h,
exp(ih,
cp,(rl)
=exp(k,
.r,)>
.ri)CC?j h,
(5) exp(h,
.r,).
Summing up the polarization of emitted X-rays, the number of photons per unit time:
dx,,12w dfi dNk= 27rc’2(1 -lVp;)
‘(
(6)
&(X1+q2 +
c u, I _
cp;
c$y’ = 0.
h’,
x [k;
1
- E’;‘(k,)
InX
[z,l(w-ni()h-(w/c*)2v,l]
+[(kh.z_)h-(W/C*>2z,,)] I’)
The band index n distinguishes the solutions of Eq. (4). which are assumed to have the many-beam matrix form
[
397
- (w/c*)‘]
-2
(13)
where I,,=((cpr, IexP(ikh.r.)lcpi,),
(7)
h’,
Z,=(q+,
(14)
lexp(-ik,.r,)(-ih)V,
IcPi.)
(‘5)
A good description of many experimental results is given by the following Doyle-Turner expression for a single plane, which is obtained by fitting the electron-scattering factor determined by a Hartree-Fock calculation with three parameters to experimental results:
and R ;,= ( Ej
U(r,)=
where Fh is the structure factor, v, is the volume of unit cell, A is the wavelength of the X-ray. We obtain the angular distribution of photons near the Bragg angle from a channeled electron in the case that the final state is different from the initial state,
(8)
I --IT, I )/fL, OL is the fine structure constant, n = k/] k I, xh is the Fourier component of the electric susceptibility. For X-rays, we have
Xh= _i
--
(‘6)
IT
aN
a
afw, -zc*2
E,=E,=\l(p,~)*+(rnc~)*, (9)
tq+(e2
co?
28, -2(n,,/oB))e:
[e:+et-2(R,,/o,)+Y-?+
where N is the atomic density in the crystal, d, is the interplanar distance, ai and bj = (2rr)‘B, are the coe.fficients tabulated by Doyle and Turner, pt is the one-dimensional mean square vibrational amplitude, Ei and Er are the initial and the final energy of the incident electron. The vector-potential A(r) is explained by Bloch waves:
+(n,,/oe)2
X
E,-E,+iio=/~+h.
Ixo/]2
('7) PX is radiated without any relations to a transverse state and the angular distribution of photons is already explained [S] as: aN
a
oB
-=~-+h12
A(r) = c h
c k’
c [Ah(k’,
ae,ae,
a’)
d
Xexp(ikL
(COG 20,)@-8; . r,)a,,,,
+ c.c.1.
(‘0) (11)
kh(
h.AO(k))
k:
-
(w/c*
)2Ao(k)
,
(W/c*)Z
(18) x(2e,+8~+y-2+lxol)2' where 8, and 8, are measured from the direction of the Bragg angle, rlif = (‘p, I ri I cpi>. (i.e., the final state corresponds to the initial state.) As a result, near the Bragg condition, the angular distribution of photons is obtained by adding Eqs. (17) and (18).
(‘2)
is the phase velocity of light in the where c * = c/G crystal, l k. is the polarization-vector, V is the volume of the crystal, k is the wave-vector of a photon, k, = k + h corresponds to that of a diffracted wave.
3.Calculations
and discussion
If the value of the denominator is zero in Eq. (13) (i.e., the value of Eq. (13) is infinity), then “a real photon” is V.
CRYSTALASSISTED
PROCESSES
Y. Matsuda et al./Nucl.
398
Insrr. and Meth. in Phys. Res. B II.5 (1996) 396-400
not diffracted. Thus, from the intensity of CR, the intensity of diffracted X-rays is obtained. The angular dis~bution of CR is written as [7]: -dwif = dO
e=r= iiic):,[(l -(l
- /&,12) sin’ 6 cos’ +]
X[2nhc3(1-
-30 -2
I
y,
,
n=O .
0
j 2
a.u. Fig. 1. The (1 IO)planar potential evaluated from Eq. (7) with the contributions from neighboring planes included. The vibrational amplitude for room temperature 300 K, as obtained from the Lkbye model with To =L625 K, is p, = 0.126 a.u. (0.067 A>. Energy levels for bound states of y = 100 (51.2 MeV) electrons are indicated, and the intensity dist~butions across the plane are shown.
diffracted. Therefore, the value has an upper limit for this condition. According to two-wave approximation of dynamical diffraction theory for X-rays, the mean intensity of diffracted X-rays is equal to the intensity of X-rays that is
- lP,,lcos a)*
lp,,lcos
a)“]-‘.
(19)
Assuming that the z axis is directed along the longi~dinal motion and x axis is perpendicular to the channeling planes and where 6 is the angle between k and the z axis, + is the polar angle of the vector k in the x, y plane. When the electron is under channeling conditions, the angular dis~bution of photons near the Bragg angle is different from PXR. We calculate the angular distribution under the following conditions. 15-50 MeV electron beams are channeled along (110) planes and CR photons are diffracted by (111) planes in silicon. The Bragg angle 0,=7.9” (1.379X 10-l r-ad), dllO>1.92 I;i, d(lll)= 3.135 A, wu = 14.38 keV, x0 i=i4.625 X 10e6, x,, = 2.479 X 10p6. Fig. 1 shows the potential of an incident electron, corrected for the contribution from neighboring planes, for a (110) plane in silicon; the squares of the wave-function from Pq. (4) are also shown. In the case of y = 100, photons forming four concentric circles are obtained from diffracted CR in addition to PXR (Fig. 2). However, the
(rad.) O-O3 ” Fig. 2 The angular distribution of PXRC forms four concentric circles near the Bragg angle for y = 100. From inside to outside, each circle represents the intensity of diffracted radiation for each transition (4 --f 3). (3 + 21, (2 --) 1) and ( 1 -+ 0).
Y. Matsuda et al./Nucl.
Table I The intensities
Instr. and Meth. in Phys. Res. B I15
of PXRC and PXR arc shown when CR satisfies
the Bragg condition Ya
Transition b
Number of photons
PXR (max)
PXRC 60 50 41 29
4.58X 4.90x 4.05x 2.84X
4-3 3-2 2-l I-O
lO-4 10-4 lo-4 1O-4
9.02X 6.27x 4.66X 2.18X
lo-’ IO- * IO-’ lo-*
a Lorentz factor of incident electron. b Transition state (initial-fmal).
intensity is weak compared with PXR. As the value of the Lorentz factor y decreases, the radius of the photon circles decreases. In the case of y = 60, 50, 41 and 29, the
f 1996) 396-400
399
photons radiated in the direction of the longitudinal motion of the incident electron satisfy the Bragg condition. (The radiated photons for the each transition of (4 + 31, (3 --f 2). (2 --$ 1) and (1 -+ 0) satisfy this condition corresponding with each value of r.> In this case, the intensity of PXRC is some thousands times stronger than PXR under the same conditions (see Table 1 and Fig. 3). When the value of y is little less than these values, these peaks are much smaller than that of PXR. These peaks are explained as diffracted virtual photons, kinematically. Incidentally, we compare the angular distribution of the radiation in the case of positron incidence with that of electron incidence. As the potential for a positron is almost harmonica1 in the crystal, the difference of the energy level is almost the same. Thus, the angular distribution of PXRC makes one thick circle. The transition probability is more than that in the case of electron. According to our calcula-
b-ad.1 _o,/l
0.4
N
0.2
0
-i
?ad.)
Fig. 3. The angular distributions satisfies the Bragg condition.
of PXRC (a) and PXR (b) in the case of y = 29. In this case, the radiated photon for (1 -+ 0) transition
V. CRYSTAL
ASSISTED
PROCESSES
400
Y. Matsuda
et al./Nucl.
Instr.
and Meth. in Phys. Res. B II5
tions, the intensity of PXRC by positrons is one order higher than that by electrons. As a result, we show the angular distribution of PXRC, but a kinematical theory does not predict the intensity of diffracted CR. To estimate the intensity, a dynamical theory must be used. However, the kinematical formula can easily explain the places where the intensity of PXRC is most strong.
References
[II M.L. Ter-Mikaelian, High-energy Electromagnetic Processes in Condensed Media (Wiley-Interscience, New York, 1972) p. 332. M.A. Kumakhov, Phys. Lett. A 57 (1976) 17. L-21 131V.G. Baryshevsky and I.D. Feranchuk, J. Phys. (Paris) 44 (1983).
(1996) 396-400
V.G. Baryshevsky and I. Ya. Dubovskaya, J. Phys. C 16 (1983) 3663. (51 I.D. Feranchuk and A.V. Ivashin, J. Phys. (Paris) 46 (1985) 1981. [6] H. Nina, Phys. Lett. A 158 (1991) 270. [7] V.V. Beloshitsky and F.F. Komarov, Phys. Rep. 93 (1982) 117. [8] Y.H. Ohtsuki, Charged Beam Interaction with Solids (Taylor and Francis, 1983). [9] A.W. Sknz and H. ijberall, Coherent Radiation Sources (Springer-Verlag. 1985). [lo] C. Kittel, Introduction to Solid State Physics, 6th ed. (Wiley, 1986). [ 1l] M.A. Kumakhov and R. Wedell, Radiation of Relativistic Light Particles during Interaction with Single Crystals: Spectrum (Akad. Verl., 1991). [ 121 J.A. lhers and W.C. Hamilton, International Tables for X-my Crystallography, (Vol. 4) (The International Union of Crystallography, 1974). [4]