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Parametric X-ray radiation by relativistic channeled particles
Nuclear Instruments and Methods in Physics Research B I 1.5(19%) 380-383 ~0~
B
6eamtn&waetions with Materials & Atoms
ELSEVIER
Parametric X-ray radiation by relativistic channeled particles T. lkeda a, Y. Matsuda a, H. Nitta b, Y.H. Ohtsuki a a Deparfmenr of Physics. Waseda Uniu.. Ohkubo 3-4-1, Shinjuku-ku. Tokyo, 169, Japan b Department of Physics, Tokyo Gakugei Univ., ~uku~ki~a-machi, Koganei, Tokyo, 184, Japan Abstract In this paper, the radiation intensity around the Bragg angle (with respect to a target crystal plane) by relativistic channeled particles is calculated based on the kinematical theory of parametric X-ray radiation (PXR). It is predicted that, in consequence, not only PXR but also Bragg diffraction of channeling radiation will be observed. The latter process is divided into two types by the incident energy. Ordinary photon diffraction occurs in the high energy region and virtual photon diffraction in the low energy region.
1. Int~uction A relativistic charged particle under channeling condition emits X-ray photons mainly in the direction of the particle’s motion. This radiation is called channeling radiation (CR). CR was correctly considered by Kumakhov [I] for the first time. Subsequently, there have been many experimental and theoretical investigations on CR [2,3]. Theories of CR are divided into two approaches by the energy of an incident particle. Classical theory is used at high energy, and quantum theory at low energy. For an electron or a positron, quantum effects am essential in the MeV-energy range (up to tens of MeV). On the other hand, parametric X-ray radiation (PXR) is observed around the direction of the angle satisfying the Bragg condition for X-ray diffraction, when a relativistic charged particle passes through crystal planes. TerMikaelian first calculated the PXR intensity using classical electrodynamics [4]. Afterwards, Baryshevsky and Feranchuk have developed a dynamical theory of PXR [S]. However, Nitta showed the mechanism of PXR more clearly by using a kinematical theory [6]. Actually, according to several experimental studies [7], the dynamical effect is not essential to explain the PXR mechanism. After all, the kinematical theory is more suitable for understanding the PXR physical process than the dynamical one, at the sacrifice of a little accuracy. The purpose of this paper is to calculate the radiation intensity around the Bragg angle from a channeled particle (PXRC) and to clarify the elementary process of this radiation as well. To tell the truth, Baryshevsky and Dubovskaya calculated the PXRC intensity using the dy-
namical theory in 1982 181, before PXR was confirmed ex~~men~lly. However, their formulae are too complicated. That is why, we derive the PXRC formula using the kinematical theory and, according to our formula, one can understand the physical process of PXRC more clearly. In Section 2, first, the general formula for the PXRC intensity is derived with the quantum theory of CR. Next, it is modified to a more useful form for comparison with experiments; incidentally, PXRC has not been observed yet. In Section 3 we discuss the PXRC process and possibility of observation.
2.1. General forrnuia
The transition probability of PXRC per unit time is given by
where pF is the density of final states, w is the frequency of the emitted photon, and EiFf are the initial and final energy of the incident particle, respectively. To be exact, the initial and final wavefunction of the particle, qi and To should satisfy the Dirac equation. However, assuming the influence of spin is negligible, we can apply scalar wavefunctions satisfying the much simpler Klein-Gordon equation approximately. In addition, a basic separation into longitudinal ( 11)and transverse (I) motion with respect to a channeling plane (or axis) is introduced and the gauge
0168-583X/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved PI/ SO168-583X(96)00167-X
T. Ikeda et al./Nucl.
Instr. and Meth. in Phys. Res. B 115 (1996) 380-383
381
condition is set as 0 = 0, where 4 is the scalar potential. Then the wavefunctions and the interaction Hamiltonian H’ are described by 1 = -‘Pi fi
vi(r)
i
I Cri > ev XPill ’ rll 1 (
1
(3) with
where a=e’/iic,
H’= -+.A, ymc
-E,.)/fi,
(4)
(5) where E,, (n = i, f) means the transverse energy of the particle, U, is the planar (or axial) potential and V: is the Laplacian which operates on r I . Because of the target crystal’s periodic structure, the electromagnetic field A in Eq. (4) is written in a form of the Bloch function, = cc
u) exp(ikA.r)
x[A,(k,
(6)
+c.c.],
Ala
0) XI &(h.
ed
- (w/c* 1%
k; - (W/C*)’
where V is the volume of the crystal, ec = 1 + mean value of the dielectric function, c* = c/ \/”.sc sponds to the phase velocity of the X-ray and e,, polarization vector. xI is the Fourier component electric susceptibility and, for X-rays, we use
’
(7)
is the correis the of the
-
(8)
r?df t
where pr is the Fourier coefficient of the local density of the crystal electrons. Using these equations, we can calculate Eq. (1). Consequently, the number of emitted photons per unit time by an incident particle can be written as dN;= -
(1’) and u,, is the parallel component of the incident particle’s velocity. In addition, the dispersion relation which photons emitted as PXRC should satisfy is f&f + h . u,, w =
i-2, + k, ,, . u,, =
(‘2)
1-p;;.n
In Eq. (I), the effect of channeling is only in the wavefunctions. Therefore, if the plane wave solutions are substituted into Eq. (1) instead of Eqs. (2) and (3), one can obtain the ordinary PXR intensity. 2.2. Dipole approximation and modification to more useful
~4~~1~da 2lrc*2&;‘2(1
-p,;
.n)
Here we confine the following calculations to the case that if f. That is because Eq. (9) corresponds to the formula for PXR, at the case that i = f. Now we modify Eq. (9) to obtain a more useful form for comparison with experiments. For PXR, the form derived by Feranchuk and Ivashin [9] is convenient. Therefore, we derive a form similar to it, since the form is also useful for PXRC. First we use the dipole approximation. Accordingly, except in the case that i = f, Eqs. (10) and (11) become I,,== I,=
47r e2pi XI=
Ri,=(Ei,
form
where h is the reciprocal lattice vector of the crystal, k is the wave-vector of a photon, kL = k + h is that of a diffracted wave and a represents the polarization direction. A, is obtained by solving the classical Maxwell equation with perturbation theory [6], and written as ‘%(k,
n=k/Ikl,
( ‘0)
where L is the crystal length, y is the Lorentz factor, c is the velocity of light, and e, m and p are the electric charge, the mass and the momentum of the incident particle, respectively. The transverse wavefunction cp*(rl) satisfies the following Schriidinger equation, modified by the Lorentz factor y:
A(r)
PIi =v,,/c*,
-ikL.rLi,,
(‘3)
dr, (cp,, I -IcPi.)= dr
-ifiifrijf.
(14)
where
r~ifE((‘PflIr~
I9il)
(‘5)
To get the last form of Eq. (14). we employed the Heisenberg equation. Secondly we think of a planar channeling case for simplicity. Especially we assume that channeling planes are perpendicular to diffraction planes. The coordinate system is defined as in Fig. 1. A photon with wave vector k is emitted near the direction of the z axis, and its frequency satisfies the relation w = oa, where wa is the V. CRYSTALASSISTEDPROCESSES
T. Ikeda et al./Nucl.
382
Instr. and Meth. in Phys. Res. B I15 (1996) 380-383
3. Discussion charged beam
charged beam
. Y*Y
Z
3
20, 2.
x
x
Fig. 1. Charged particles penetrate along the Z-axis. The Y-axis is perpendicular to channeling planes as in (a). The (x, y, z) system is defined by a rotation of the (X, Y, Z) system through the angle
29, about the Y-axis (b). Therefore, X-ray photons are emitted near the direction of the r-axis.
frequency corresponding to the Bragg angle 0,. In addition, we assume that u,, = u. Therefore, after averaging Eq. (9) over the polarization of the emitted photon, we can obtain from Bq. (9) d2N’
o
y-2+
IXoI-2nir/wB=0
w; Yi:IXJ2
4n ce2
sin’ 8,
x (e; + (e; cos2 28, - 2ni&Je; + (nif/wB>2)
x([e:+e;+y2+ where sent *e e ;I;ffd;f;o;
Ix~l-2fii~/~B]‘)-‘, (16) ;B;yllf I; i.e.exand 0, rep=-
Yif”(‘Pfl(Y)lYIcPi.(Y)).
(‘7)
when 8, = 8, = 0. The energy of CR emitted into forward direction depends on the incident energy and is given by [21 fLW#-a= 2y%R,,.
(18)
Considering Bq. (18) and neglecting X0, we can easily obtain from Eq. (17), oa = Oc..
de, de, r----_
In Section 2 we derived the formula of PXRC, but it has singularities. The denominator of Eq. (9) looks of the same form as the formula of PXR. However, the PXR formula cannot have singularities, because the PXR process is ‘virtual photon diffraction’ [6]. In other words, 4 represents a virtual photon. On the contrary, for PXRC, k/ can satisfy the relation kj = (o/c * j2 and, therefore, represents a real photon. In this case, first, a real photon whose wave vector is kd is emitted as CR. Secondly it is diffracted by crystal planes and observed around the Bragg angle. The incident energy decides whether Eq. (16) diverges or not. If the energy of a photon emitted as CR is smaller than the energy satisfying the Bragg condition, Eq. (16) does not diverge and ‘virtual photon diffraction’ occurs similarly to PXR. On the other hand, if the CR energy is larger, Eq. (16) diverges and ordinary photon diffraction occurs. The condition of divergence can be written as
( ‘9)
When this condition is satisfied, the ratio of PXRC to PXR will become maximum. Therefore, the PXRC main process is the diffraction of CR, and its intensity will be as strong as CR. Recently some experiments of PXR under channeling condition were carried out. However, PXRC was not observed, because the incident energy was not appropriate for the condition of Eq. (19). If an appropriate setting of experiment is chosen carefully, PXRC will be observed.
References
Now
we must take notice of the denominator of Eq. (16). Because of the last term -2&/w,, Bq. (16) has a possibility of divergence. To compare with PXR, we also take the case that i = f. This case corresponds to PXR, and leads to [9] d2N’
(Y
de, de, = GoB
e; c0s22e,+e; lXI12 sin’ e B [e:+8:+y-2+IXo1]2'
Ill M.A. Kumakhov, Phys. Lett. A 57 (19761 17.
01 A.W. SLnz and H. ijberall, Coherent Radiation Sources (Springer-Vedag. 1985). [31V.V. Beloshitsky and F.F.
Komarov, Phys. Rep. 93 (1982) 117. [41 M.L. Ter-Mikaelian, High-EnergyElectromagneticProcesses in Condensed Media (Wiley-Interscience, New York, 1972) p. 332.
T. Ikeda et al./ Nucl. fnstr. and Meth. in Phys. Res. B 115 (I 9961380-383 [5] V.G. Baryshevsky and I.D. Feranchuk, J. Phys. (Paris) 44
(1983) 913. [6] H. Nitta, Phys. Lett. A 158 (1991) 270. [7] A.V. Shchagin, V.I. Pristupa and N.A. Khizhnyak, Phys. Lett. A 148 (1990) 485. [81 V.G. Baryshevsky and I.Ya. Dubovskaya, J. Phys. C: Solid State Phys. 16( 1983) 3663.
383
[9] I.D. Femnchuk and A.V. Ivashin, J. Phys. (Paris) 46 (1985) 1981. [IO] Y.H. Ohtsuki, Charged Beam Interaction with Solids (Taylor & Francis, 1983). [I 11 C. Kittel, Introduction to Solid State Physics, 6th ed. (John Wiley, 1986).
V. CRYSTAL ASSISTED PROCESSES
B
6eamtn&waetions with Materials & Atoms
ELSEVIER
Parametric X-ray radiation by relativistic channeled particles T. lkeda a, Y. Matsuda a, H. Nitta b, Y.H. Ohtsuki a a Deparfmenr of Physics. Waseda Uniu.. Ohkubo 3-4-1, Shinjuku-ku. Tokyo, 169, Japan b Department of Physics, Tokyo Gakugei Univ., ~uku~ki~a-machi, Koganei, Tokyo, 184, Japan Abstract In this paper, the radiation intensity around the Bragg angle (with respect to a target crystal plane) by relativistic channeled particles is calculated based on the kinematical theory of parametric X-ray radiation (PXR). It is predicted that, in consequence, not only PXR but also Bragg diffraction of channeling radiation will be observed. The latter process is divided into two types by the incident energy. Ordinary photon diffraction occurs in the high energy region and virtual photon diffraction in the low energy region.
1. Int~uction A relativistic charged particle under channeling condition emits X-ray photons mainly in the direction of the particle’s motion. This radiation is called channeling radiation (CR). CR was correctly considered by Kumakhov [I] for the first time. Subsequently, there have been many experimental and theoretical investigations on CR [2,3]. Theories of CR are divided into two approaches by the energy of an incident particle. Classical theory is used at high energy, and quantum theory at low energy. For an electron or a positron, quantum effects am essential in the MeV-energy range (up to tens of MeV). On the other hand, parametric X-ray radiation (PXR) is observed around the direction of the angle satisfying the Bragg condition for X-ray diffraction, when a relativistic charged particle passes through crystal planes. TerMikaelian first calculated the PXR intensity using classical electrodynamics [4]. Afterwards, Baryshevsky and Feranchuk have developed a dynamical theory of PXR [S]. However, Nitta showed the mechanism of PXR more clearly by using a kinematical theory [6]. Actually, according to several experimental studies [7], the dynamical effect is not essential to explain the PXR mechanism. After all, the kinematical theory is more suitable for understanding the PXR physical process than the dynamical one, at the sacrifice of a little accuracy. The purpose of this paper is to calculate the radiation intensity around the Bragg angle from a channeled particle (PXRC) and to clarify the elementary process of this radiation as well. To tell the truth, Baryshevsky and Dubovskaya calculated the PXRC intensity using the dy-
namical theory in 1982 181, before PXR was confirmed ex~~men~lly. However, their formulae are too complicated. That is why, we derive the PXRC formula using the kinematical theory and, according to our formula, one can understand the physical process of PXRC more clearly. In Section 2, first, the general formula for the PXRC intensity is derived with the quantum theory of CR. Next, it is modified to a more useful form for comparison with experiments; incidentally, PXRC has not been observed yet. In Section 3 we discuss the PXRC process and possibility of observation.
2.1. General forrnuia
The transition probability of PXRC per unit time is given by
where pF is the density of final states, w is the frequency of the emitted photon, and EiFf are the initial and final energy of the incident particle, respectively. To be exact, the initial and final wavefunction of the particle, qi and To should satisfy the Dirac equation. However, assuming the influence of spin is negligible, we can apply scalar wavefunctions satisfying the much simpler Klein-Gordon equation approximately. In addition, a basic separation into longitudinal ( 11)and transverse (I) motion with respect to a channeling plane (or axis) is introduced and the gauge
0168-583X/96/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved PI/ SO168-583X(96)00167-X
T. Ikeda et al./Nucl.
Instr. and Meth. in Phys. Res. B 115 (1996) 380-383
381
condition is set as 0 = 0, where 4 is the scalar potential. Then the wavefunctions and the interaction Hamiltonian H’ are described by 1 = -‘Pi fi
vi(r)
i
I Cri > ev XPill ’ rll 1 (
1
(3) with
where a=e’/iic,
H’= -+.A, ymc
-E,.)/fi,
(4)
(5) where E,, (n = i, f) means the transverse energy of the particle, U, is the planar (or axial) potential and V: is the Laplacian which operates on r I . Because of the target crystal’s periodic structure, the electromagnetic field A in Eq. (4) is written in a form of the Bloch function, = cc
u) exp(ikA.r)
x[A,(k,
(6)
+c.c.],
Ala
0) XI &(h.
ed
- (w/c* 1%
k; - (W/C*)’
where V is the volume of the crystal, ec = 1 + mean value of the dielectric function, c* = c/ \/”.sc sponds to the phase velocity of the X-ray and e,, polarization vector. xI is the Fourier component electric susceptibility and, for X-rays, we use
’
(7)
is the correis the of the
-
(8)
r?df t
where pr is the Fourier coefficient of the local density of the crystal electrons. Using these equations, we can calculate Eq. (1). Consequently, the number of emitted photons per unit time by an incident particle can be written as dN;= -
(1’) and u,, is the parallel component of the incident particle’s velocity. In addition, the dispersion relation which photons emitted as PXRC should satisfy is f&f + h . u,, w =
i-2, + k, ,, . u,, =
(‘2)
1-p;;.n
In Eq. (I), the effect of channeling is only in the wavefunctions. Therefore, if the plane wave solutions are substituted into Eq. (1) instead of Eqs. (2) and (3), one can obtain the ordinary PXR intensity. 2.2. Dipole approximation and modification to more useful
~4~~1~da 2lrc*2&;‘2(1
-p,;
.n)
Here we confine the following calculations to the case that if f. That is because Eq. (9) corresponds to the formula for PXR, at the case that i = f. Now we modify Eq. (9) to obtain a more useful form for comparison with experiments. For PXR, the form derived by Feranchuk and Ivashin [9] is convenient. Therefore, we derive a form similar to it, since the form is also useful for PXRC. First we use the dipole approximation. Accordingly, except in the case that i = f, Eqs. (10) and (11) become I,,== I,=
47r e2pi XI=
Ri,=(Ei,
form
where h is the reciprocal lattice vector of the crystal, k is the wave-vector of a photon, kL = k + h is that of a diffracted wave and a represents the polarization direction. A, is obtained by solving the classical Maxwell equation with perturbation theory [6], and written as ‘%(k,
n=k/Ikl,
( ‘0)
where L is the crystal length, y is the Lorentz factor, c is the velocity of light, and e, m and p are the electric charge, the mass and the momentum of the incident particle, respectively. The transverse wavefunction cp*(rl) satisfies the following Schriidinger equation, modified by the Lorentz factor y:
A(r)
PIi =v,,/c*,
-ikL.rLi,,
(‘3)
dr, (cp,, I -IcPi.)= dr
-ifiifrijf.
(14)
where
r~ifE((‘PflIr~
I9il)
(‘5)
To get the last form of Eq. (14). we employed the Heisenberg equation. Secondly we think of a planar channeling case for simplicity. Especially we assume that channeling planes are perpendicular to diffraction planes. The coordinate system is defined as in Fig. 1. A photon with wave vector k is emitted near the direction of the z axis, and its frequency satisfies the relation w = oa, where wa is the V. CRYSTALASSISTEDPROCESSES
T. Ikeda et al./Nucl.
382
Instr. and Meth. in Phys. Res. B I15 (1996) 380-383
3. Discussion charged beam
charged beam
. Y*Y
Z
3
20, 2.
x
x
Fig. 1. Charged particles penetrate along the Z-axis. The Y-axis is perpendicular to channeling planes as in (a). The (x, y, z) system is defined by a rotation of the (X, Y, Z) system through the angle
29, about the Y-axis (b). Therefore, X-ray photons are emitted near the direction of the r-axis.
frequency corresponding to the Bragg angle 0,. In addition, we assume that u,, = u. Therefore, after averaging Eq. (9) over the polarization of the emitted photon, we can obtain from Bq. (9) d2N’
o
y-2+
IXoI-2nir/wB=0
w; Yi:IXJ2
4n ce2
sin’ 8,
x (e; + (e; cos2 28, - 2ni&Je; + (nif/wB>2)
x([e:+e;+y2+ where sent *e e ;I;ffd;f;o;
Ix~l-2fii~/~B]‘)-‘, (16) ;B;yllf I; i.e.exand 0, rep=-
Yif”(‘Pfl(Y)lYIcPi.(Y)).
(‘7)
when 8, = 8, = 0. The energy of CR emitted into forward direction depends on the incident energy and is given by [21 fLW#-a= 2y%R,,.
(18)
Considering Bq. (18) and neglecting X0, we can easily obtain from Eq. (17), oa = Oc..
de, de, r----_
In Section 2 we derived the formula of PXRC, but it has singularities. The denominator of Eq. (9) looks of the same form as the formula of PXR. However, the PXR formula cannot have singularities, because the PXR process is ‘virtual photon diffraction’ [6]. In other words, 4 represents a virtual photon. On the contrary, for PXRC, k/ can satisfy the relation kj = (o/c * j2 and, therefore, represents a real photon. In this case, first, a real photon whose wave vector is kd is emitted as CR. Secondly it is diffracted by crystal planes and observed around the Bragg angle. The incident energy decides whether Eq. (16) diverges or not. If the energy of a photon emitted as CR is smaller than the energy satisfying the Bragg condition, Eq. (16) does not diverge and ‘virtual photon diffraction’ occurs similarly to PXR. On the other hand, if the CR energy is larger, Eq. (16) diverges and ordinary photon diffraction occurs. The condition of divergence can be written as
( ‘9)
When this condition is satisfied, the ratio of PXRC to PXR will become maximum. Therefore, the PXRC main process is the diffraction of CR, and its intensity will be as strong as CR. Recently some experiments of PXR under channeling condition were carried out. However, PXRC was not observed, because the incident energy was not appropriate for the condition of Eq. (19). If an appropriate setting of experiment is chosen carefully, PXRC will be observed.
References
Now
we must take notice of the denominator of Eq. (16). Because of the last term -2&/w,, Bq. (16) has a possibility of divergence. To compare with PXR, we also take the case that i = f. This case corresponds to PXR, and leads to [9] d2N’
(Y
de, de, = GoB
e; c0s22e,+e; lXI12 sin’ e B [e:+8:+y-2+IXo1]2'
Ill M.A. Kumakhov, Phys. Lett. A 57 (19761 17.
01 A.W. SLnz and H. ijberall, Coherent Radiation Sources (Springer-Vedag. 1985). [31V.V. Beloshitsky and F.F.
Komarov, Phys. Rep. 93 (1982) 117. [41 M.L. Ter-Mikaelian, High-EnergyElectromagneticProcesses in Condensed Media (Wiley-Interscience, New York, 1972) p. 332.
T. Ikeda et al./ Nucl. fnstr. and Meth. in Phys. Res. B 115 (I 9961380-383 [5] V.G. Baryshevsky and I.D. Feranchuk, J. Phys. (Paris) 44
(1983) 913. [6] H. Nitta, Phys. Lett. A 158 (1991) 270. [7] A.V. Shchagin, V.I. Pristupa and N.A. Khizhnyak, Phys. Lett. A 148 (1990) 485. [81 V.G. Baryshevsky and I.Ya. Dubovskaya, J. Phys. C: Solid State Phys. 16( 1983) 3663.
383
[9] I.D. Femnchuk and A.V. Ivashin, J. Phys. (Paris) 46 (1985) 1981. [IO] Y.H. Ohtsuki, Charged Beam Interaction with Solids (Taylor & Francis, 1983). [I 11 C. Kittel, Introduction to Solid State Physics, 6th ed. (John Wiley, 1986).
V. CRYSTAL ASSISTED PROCESSES