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Quasi-Cherenkov radiation in crystals
Nuclear Instruments and Methods in Physics Research A248 (1986) 29-30 North-Holland, Amsterdam
29
QUASI-CHERENKOV RADIATION IN CRYSTALS G.M . GARIBIAN and C. YANG Yerevan Physics Institute, Armenia, USSR
The physical nature of the quasi-Cherenkov radiation generated by fast charged particles passing through a monocrystal is discussed.
The theory of Vavilov-Cherenkov radiation was originally developed by Tamm and Frank [1] for a uniform medium . Later it was found [2-41 that if the medium is a periodically nonuniform one, VavilovCherenkov radiation has highly interesting new features . Particularly, if in a uniform medium the radition at a given frequency (or in a given narrow frequency range) takes place only at an angle 0, where cos 0, =1/N e(w),
N=V/c,
(N
E(w) > 1),
then in a periodically nonuniform medium the radiation may arise at many angles simultaneously . For example, if the medium is a layered one, i .e . it consists of different layers with thickness a and b and dielectric permeabilities ei and ez, the angles 0 at which the radiation arises satisfy a transcendental equation cos
ca (a+b) U
= cos
co au, c
cos
1 ( e2ul -2 ei uZ ui,z
r = Vet,,
wbu2
+
c El U 2
ez ui
) sin vaut sin c
wbu,
c
,
- sin z8 .
This radiation was called by Fainberg and Hizhniak [21 "parametric Cherenkov radiation" . Parametric Cherenkov radiation may also arise if ,ß el ,z < 1, i .e . if the Vavilov-Cherenkov radiation in the ordinary meaning does not take place in the layers taken separately. The point is, as is well known (see, e.g . ref [5]), Vavilov-Cherenkov radiation results from interference between secondary waves emitted by the atoms of the medium under the influence of a passing charged particle . In the case of a uniform medium, if the particle's velocity is less than the phase velocity of the electromagnetic wave in the medium, the abovemenlioned secondary waves cancel each other and so Vavilov-Cherenkov radiation is not formed . But in a nonuniform medium such total canceling of the secondary waves does not take place, so a residual radiation arises . In this sense one may regard the "parametric Cheren0168-9002/86/$03 .50 © Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)
kov radiation" in a layered medium as a result of interference of transition radiation [6] formed at every layer boundary . In a monocrystal the atoms are located regularly, i.e . there is a three-dimensional periodically nonuniform space distribution of electron density with periods of the order of an X-ray wavelength. It was found [7-9] that while a relativistic charged particle passes through a monocrystal, radiation is expected at frequencies and angles satisfying the Bragg relation 2w -n .K=K2, c
where n is a unit vector in the radiation direction, and K the reciprocal lattice vector (multiplied by 21r) of the crystal . The origin of this radiation, as mentioned above, is the interference of the secondary waves emitted by the atoms of the crystal under the influence of a passing charged particle . Originally, it was called by different names. We think that it is more appropriate to call this radiation "quasi -Cherenkov radiation" . Thus a relativistic charged particle forms a kind of "Laue pattern" in a monocrystal . The central spot is the X-ray transition radiation [10] generated at the boundaries of the crystal with a continuous frequency spectrum. On the other hand, the lateral spots of the quasi-Cherenkov radiation generated over the whole thickness of the crystal are almost monochromatic in each spot . The intensity of the quasi-Cherenkov radiation depends on the atomic composition of the crystal, the crystal thickness and the reciprocal lattice vector (i .e . the family of crystalline planes at which the radiation is generated) . If the crystal thickness is less than the extinction length and the absorption length for the radiation frequency, the intensity of the quasi-Cherenkov radiation is proportional to the thickness, as for ordinary Vavilov-Cherenkov radiation in a uniform medium . II . THEORETICAL ASPECTS
30
G. M . Gambian, C. Yang / Quast-Cherenkov radiation in crystals
However, the intensity of quasi-Cherenkov radiation from crystals of larger thickness saturates and becomes independent of the crystal thickness. The intensity of quasi-Cherenkov radiation (as well as that of Vavilov-Cherenkov radiation) does not depend on the particle's Lorentz-factor y if
experiments for detecting and studying this radiation are being carried out now, but no data have been published yet.
where wo is the plasma frequency of the crystal. At smaller y values the radiation intensity falls rapidly with decreasing y. For example, the intensity of the quasi-Cherenkov radiation corresponding to the crystalline planes (220) and a radiation quantum energy hw = 37 keV from a silicon crystal of thickness a > 5 mm for y > 1000 is of the order of (6-7) X 10 -4 quanta per particle . In this case the Bragg angle 8B = 5°, i.e . the quasi-Cherenkov radiation is emitted under an angle 20 B = 10° with respect to the particle velocity . The angular width of the spot is of the order of a mrad or less and the relative frequency width dw/w is of the order of 10 -2 . Quasi-Cherenkov radiation may serve as a good source of quasimonochromatic and highly directional X-rays emitted under a large angle (» y -1 ) with respect to the particle velocity. Besides, one may use quasi-Cherenkov radiation for the investigation of the properties of crystals and charged particles. As we know,
[1] I.E. Tamm and LM. Frank, Dokl . Akad . Nauk SSSR [Sov . Phys . Dokl] 14 (1937) 107. [2] Ya .B . Fainberg and N.A . Hizhmak, Zh. Eksp . Teor. Fiz. [Sov. Phys. JETP] 32 (1957) 883. [3] M.L . Ter-Mikaelian, Dokl . Akad. Nauk SSSR [Sov . Phys . Dokl .] 134 (1960) 318. [4] K.F. Casey, C. Yeh and Z.A . Kaprielian, Phys. Rev. B140 (1965) 768. [5] G.M . Garibian and C. Yang, X-ray Transition Radiation (Acad. Sciences of Arm. SSR, Yerevan, 1983). [6] V.L . Ginzburg and I.M . Frank, J. Phys. (USSR) 9 (1945) 353. [71 M.L . Ter-Mikaelian, High-Energy Electromagnetic Processes in Condensed Media (Wiley, New York, 1972). [8] G.M . Gambian and C. Yang, Zh. Eksp . Teor. Fiz. [Sov . Phys . JETP] 61 (1971) 930; 63 (1972) 1198 . [9] V.G . Banshevski and I.D . Feranchuk, Zh. Eksp. Teor . Fiz. [Sov . Phys . JETP] 61 (1971) 944; Izv . Akad . Nauk Bel. SSR, ser. Phys . Math . Sci. no. 2 (1973) 102. [10] G.M . Garibian, Zh. Eksp . Teor. Fiz. [Sov . Phys . JETP] 37 (1959) 527.
y » w/wo,
References
29
QUASI-CHERENKOV RADIATION IN CRYSTALS G.M . GARIBIAN and C. YANG Yerevan Physics Institute, Armenia, USSR
The physical nature of the quasi-Cherenkov radiation generated by fast charged particles passing through a monocrystal is discussed.
The theory of Vavilov-Cherenkov radiation was originally developed by Tamm and Frank [1] for a uniform medium . Later it was found [2-41 that if the medium is a periodically nonuniform one, VavilovCherenkov radiation has highly interesting new features . Particularly, if in a uniform medium the radition at a given frequency (or in a given narrow frequency range) takes place only at an angle 0, where cos 0, =1/N e(w),
N=V/c,
(N
E(w) > 1),
then in a periodically nonuniform medium the radiation may arise at many angles simultaneously . For example, if the medium is a layered one, i .e . it consists of different layers with thickness a and b and dielectric permeabilities ei and ez, the angles 0 at which the radiation arises satisfy a transcendental equation cos
ca (a+b) U
= cos
co au, c
cos
1 ( e2ul -2 ei uZ ui,z
r = Vet,,
wbu2
+
c El U 2
ez ui
) sin vaut sin c
wbu,
c
,
- sin z8 .
This radiation was called by Fainberg and Hizhniak [21 "parametric Cherenkov radiation" . Parametric Cherenkov radiation may also arise if ,ß el ,z < 1, i .e . if the Vavilov-Cherenkov radiation in the ordinary meaning does not take place in the layers taken separately. The point is, as is well known (see, e.g . ref [5]), Vavilov-Cherenkov radiation results from interference between secondary waves emitted by the atoms of the medium under the influence of a passing charged particle . In the case of a uniform medium, if the particle's velocity is less than the phase velocity of the electromagnetic wave in the medium, the abovemenlioned secondary waves cancel each other and so Vavilov-Cherenkov radiation is not formed . But in a nonuniform medium such total canceling of the secondary waves does not take place, so a residual radiation arises . In this sense one may regard the "parametric Cheren0168-9002/86/$03 .50 © Elsevier Science Publishers B.V . (North-Holland Physics Publishing Division)
kov radiation" in a layered medium as a result of interference of transition radiation [6] formed at every layer boundary . In a monocrystal the atoms are located regularly, i.e . there is a three-dimensional periodically nonuniform space distribution of electron density with periods of the order of an X-ray wavelength. It was found [7-9] that while a relativistic charged particle passes through a monocrystal, radiation is expected at frequencies and angles satisfying the Bragg relation 2w -n .K=K2, c
where n is a unit vector in the radiation direction, and K the reciprocal lattice vector (multiplied by 21r) of the crystal . The origin of this radiation, as mentioned above, is the interference of the secondary waves emitted by the atoms of the crystal under the influence of a passing charged particle . Originally, it was called by different names. We think that it is more appropriate to call this radiation "quasi -Cherenkov radiation" . Thus a relativistic charged particle forms a kind of "Laue pattern" in a monocrystal . The central spot is the X-ray transition radiation [10] generated at the boundaries of the crystal with a continuous frequency spectrum. On the other hand, the lateral spots of the quasi-Cherenkov radiation generated over the whole thickness of the crystal are almost monochromatic in each spot . The intensity of the quasi-Cherenkov radiation depends on the atomic composition of the crystal, the crystal thickness and the reciprocal lattice vector (i .e . the family of crystalline planes at which the radiation is generated) . If the crystal thickness is less than the extinction length and the absorption length for the radiation frequency, the intensity of the quasi-Cherenkov radiation is proportional to the thickness, as for ordinary Vavilov-Cherenkov radiation in a uniform medium . II . THEORETICAL ASPECTS
30
G. M . Gambian, C. Yang / Quast-Cherenkov radiation in crystals
However, the intensity of quasi-Cherenkov radiation from crystals of larger thickness saturates and becomes independent of the crystal thickness. The intensity of quasi-Cherenkov radiation (as well as that of Vavilov-Cherenkov radiation) does not depend on the particle's Lorentz-factor y if
experiments for detecting and studying this radiation are being carried out now, but no data have been published yet.
where wo is the plasma frequency of the crystal. At smaller y values the radiation intensity falls rapidly with decreasing y. For example, the intensity of the quasi-Cherenkov radiation corresponding to the crystalline planes (220) and a radiation quantum energy hw = 37 keV from a silicon crystal of thickness a > 5 mm for y > 1000 is of the order of (6-7) X 10 -4 quanta per particle . In this case the Bragg angle 8B = 5°, i.e . the quasi-Cherenkov radiation is emitted under an angle 20 B = 10° with respect to the particle velocity . The angular width of the spot is of the order of a mrad or less and the relative frequency width dw/w is of the order of 10 -2 . Quasi-Cherenkov radiation may serve as a good source of quasimonochromatic and highly directional X-rays emitted under a large angle (» y -1 ) with respect to the particle velocity. Besides, one may use quasi-Cherenkov radiation for the investigation of the properties of crystals and charged particles. As we know,
[1] I.E. Tamm and LM. Frank, Dokl . Akad . Nauk SSSR [Sov . Phys . Dokl] 14 (1937) 107. [2] Ya .B . Fainberg and N.A . Hizhmak, Zh. Eksp . Teor. Fiz. [Sov. Phys. JETP] 32 (1957) 883. [3] M.L . Ter-Mikaelian, Dokl . Akad. Nauk SSSR [Sov . Phys . Dokl .] 134 (1960) 318. [4] K.F. Casey, C. Yeh and Z.A . Kaprielian, Phys. Rev. B140 (1965) 768. [5] G.M . Garibian and C. Yang, X-ray Transition Radiation (Acad. Sciences of Arm. SSR, Yerevan, 1983). [6] V.L . Ginzburg and I.M . Frank, J. Phys. (USSR) 9 (1945) 353. [71 M.L . Ter-Mikaelian, High-Energy Electromagnetic Processes in Condensed Media (Wiley, New York, 1972). [8] G.M . Gambian and C. Yang, Zh. Eksp . Teor. Fiz. [Sov . Phys . JETP] 61 (1971) 930; 63 (1972) 1198 . [9] V.G . Banshevski and I.D . Feranchuk, Zh. Eksp. Teor . Fiz. [Sov . Phys . JETP] 61 (1971) 944; Izv . Akad . Nauk Bel. SSR, ser. Phys . Math . Sci. no. 2 (1973) 102. [10] G.M . Garibian, Zh. Eksp . Teor. Fiz. [Sov . Phys . JETP] 37 (1959) 527.
y » w/wo,
References