1 INTRODUCTION 1
2 THE TAMM PROBLEM IN THE VAVILOV-CHERENKOV
RADIATION THEORY 15
2.1 Vavilov-Cherenkov radiation in a finite region of space . 15
2.1.1 Mathematical preliminaries .............. 15
2.1.2 Particular cases.................... . 16
2.1.3 Original Tamm problem ................ .. 32
2.1.4 Comparison of tle Tamm and exact solutions . . 35
2.1.5 Spatial distribution of shock waves .......... . 38
2.1.6 Time evolution cf the electromagnetic field on the
surface of a sphere . ....... .......... 41
2.1.7 Comparison with the Tamm vector potential . . 46
2.2 Spatial distribution of Fourier components .......... 50
2.2.1 Quasi-classical a proximation . .......... .. 51
2.2.2 Numerical calculations ..... . . . . . ..52
2.3 Quantum analysis of the Tamm formula . . . . . . . . . . . 58
2.4 Back to the original Tamm problem . . . . . . . . . . . . . 63
2.4.1 Exact solution ................... . . 63
2.4.2 Restoring vectorI potential in the spectral represen-
tation ....... .. ... .... ....... .70
2.4.3 The Tamm approximate solution . . . . . . . . . . . 74
2.4.4 Concrete example showing that the CSW is not al-
ways reduced to the interference of BS shock waves 76
2.5 Schwinger's approach to the Tamm problem . . . . . . . . . 78
2.5.1 Instantaneous pdwer frequency spectrum . . . . . 80
2.5.2 Instantaneous angular-frequency distribution of the
power spectrum ..................... 84
2.5.3 Angular-frequency distribution of the radiated en-
ergy for a finite time interval. . . . . . . . . . . . 84
2.5.4 Frequency distribution of the radiated energy . . . . 86
2.6 The Tamm problem in the spherical basis . . . . . ..... 93
2.6.1 Expansion of the Tarhm problem in terms of the Leg-
endre polynomials ................... 93
2.7 Short resume of this chapte . . . . . . . . . . . . . ..... 97
3 NON-UNIFORM CHARGE MOTION IN A DISPERSION-
FREE MEDIUM 99
3.1 Introduction. .............. .. ... ....... . 99
3.2 Statement of the physical prpblem .............. 100
3.2.1 Simplest accelerated and decelerated motions [9] . .101
3.2.2 Completely relativistic accelerated and decelerated
motions [11] .. ............. ...... 107
3.3 Smooth Tamm problem in the time representation . . . . . 115
3.3.1 Moving singularities of electromagnetic field . . . . . 115
3.4 Concluding remarks for this chapter . . . . . . . . . . 124
chapter4 CHERENKOV RADIATION
IN A DISPERSIVE MEDIUM127
4.1 Introduction ................. ....... . . 127
4.2 Mathematical preliminaries. . . . . . . . . . . . . . . . . . 128
4.3 Electromagnetic potentials and field strengths . . . . . . . . 131
4.4 Time-dependent polarization of the medium . . . . . . . . . 141
4.4.1 Another choice of polarization . . . . . . . . . . . . 144
4.5 On the Krinig-Kramers dispersion relations . . . . . . . . . 148
4.6 The energy flux and the number of photons . . . . . . . . . 149
4.7 WKB estimates .... ... . .............. . 155
4.7.1 Charge velocity exceeds the critical velocity . . . .158
4.7.2 Charge velocity is smaller than the critical velocity 160
4.8 Numerical results . ................... . . 162
4.8.1 Estimation of non-radiation terms . . . . . . . . . . 164
4.9 The influence of the imaginary part of . . . . . . . . . . . 167
4.10 Application to concrete substances . . . . . . . . . . . . 175
4.10.1 Dielectric permittivity (4.7) . . . . . . . . . . . . . . 179
4.10.2 Dielectric permittivity (4.45) . . . . . . . . . . . . . 184
4.11 Cherenkov radiation without Use of the spectral representation188
4.11.1 Particular cases .......... . ... ... . 191
4.11.2 Numerical Results...................... 196
4.12 Short resume of this Chapter. . . . . . . . . . ........ 204
5 INFLUENCE OF FINITE OBSERVATIONAL DISTANCES
AND CHARGE DECELERATION 209
5.1 Introduction ................. ......... . 209
5.2 Finite observational distances and small acceleration . . .. 210
5.2.1 The original Tamm approach . . . . . . . . . . . . . 210
5.2.2 Exact electromagnetic field strengths and angular-
frequency distribution of the radiated energy . . . 212
5.2.3 Approximations.. ..... ............. 214
5.2.4 Decelerated charge motion. . . . . . . . . . . . . . 216
5.2.5 Numerical results .................. 219
5.3 Motion in a finite spatial interval with arbitrary acceleration 232
5.3.1 Introduction . .................. . . 233
5.3.2 Main mathematical formulae . . . . . . . . . . . . . 235
5.3.3 Particular cases ...... ....... ... 238
5.3.4 Analytic estimates ..... .............. 257
5.3.5 The absolutely continuous charge motion. . . . . . . 261
5.3.6 Superposition of Uniform and accelerated motions 272
5.3.7 Short discussion of the smoothed Tamm problem 275
5.3.8 Historical remarks on the VC radiation and
bremsstrahlung . ...... .. ..... ... . 276
5.4 Short resume of Chapter 5 ................... 279
6 RADIATION OF ELECTRIC, MAGNETIC AND
TOROIDAL DIPOLES MOVING IN A MEDIUM 283
6.1 Introduction. . .............. . . ....... . 283
6.2 Mathematical preliminaries: equivalent sources of the elec-
tromagnetic field ... . .... ............. 285
6.2.1 A pedagogical example: circular current. . . . . . . 285
6.2.2 The elementary toroidal solenoid . . . . . ..... . . 286
6.3 Electromagnetic field of electric, magnetic, and
toroidal dipoles in time representation. . . . . . . . . . . . . 293
6.3.1 Electromagnetic field of a moving point-like current
loop ............. . ... ........... 293
6.3.2 Electromagnetic field of a moving point-like
toroidal solenoid ................. . .. 300
6.3.3 Electromagnetic field of a moving point-like electric
dipole . . . .. . . . . . .... . . . ... .. . 307
6.3.4 Electromagnetic field of induced dipole moments . 310
6.4 Electromagnetic field of electric, magnetic,
and toroidal dipoles in the spectral representation . . . . . 313
6.4.1 Unbounded motion of magnetic, toroidal,
and electric dipoles in medium . . . . . . . .... . 313
6.4.2 The Tamm problen for electric charge, magnetic,
electric, and toroidal dipoles . . . . . . . . . . . . . . 327
6.5 Electromagnetic field of a precessing magnetic dipole . . . 334
6.6 Discussion and Conclusion . . . . . . . . . . . . ....... 337
7 QUESTIONS CONCERNING OBSERVATION
OF THE VAVILOV-CHERENKOV RADIATION 341
7.1 Introduction ............. ........... . 341
7.2 Cherenkov radiation from a charge of finite dimensions . . 343
7.2.1 Cherenkov radiation as the origin of the charge de-
celeration ....... .. ..... ......... 349
7.3 Cherenkov radiation in dispersive medium ........ . . 350
7.4 Radiation of a charge moving in a cylindrical
dielectric sample ........................ 355
7.4.1 Radial energy flux .................. 356
7.4.2 Energy flux along the motion axis . . . . ..... . . . 357
7.4.3 Optical interpretation . . . . . . . . . . . . .. . 358
7.5 Vavilov-Cherenkov and transition radiations
for a spherical sample .... ........ . .. ..360
7.5.1 Optical interpretation . . . . . . . . . . . . . . . . . 360
7.5.2 Exact solution ................. . . 362
7.5.3 Metallic sphere ................... .. . 376
7.6 Discussion on the transition radiation . . . . . . . . . . . . 382
7.6.1 Comment on the transition radiation . . . . . . . . 382
7.6.2 Comment on the Tanmm problem . . . . . . . . . . . 390
8 SELECTED PROBLEMS OF THE
SYNCHROTRON RADIATION 397
8.1 Introduction. .................. ....... 397
8.2 Synchrotron radiation in vacuum. . . . . . . . . . . . . . . . 398
8.2.1 Introductory remarks . . . . . . . . . . . . . . . . . 399
8.2.2 Energy radiated for the period of motion . . . . . . 404
8.2.3 Instantaneous distribution of synchrotron radiation . 407
8.3 Synchrotron radiation in medium . . . . . . . . . . . . . . . 422
8.3.1 Mathematical preliminaries . . . . . . . . . . . . . . 422
8.3.2 Electromagnetic field strengths . . . . . . . . . . . . 423
8.3.3 Singularities of electromagnetic field . . . . . . . . . 424
8.3.4 Digression on the Cherenkov radiation . . . . . . . . 426
8.3.5 Electromagnetic field in the wave zone . . . . . . . . 428
8.3.6 Numerical results for synchrotron motion in a medium434
8.4 Conclusion .................. ......... 442
9 SOME EXPERIMENTAL TRENDS IN THE VAVILOV-
CHERENKOV RADIATION THEORY 447
9.1 Fine structure of the Vavilov-Cherenkov radiation. .. . . . 447
9.1.1 Simple experiments With 657 MeV protons . . . . . 451
9.1.2 Main computational formulae . . . . . . . . . . . . . 453
9.1.3 Numerical results . . . . . . . . . . . . . . . . . . . 462
9.1.4 Discussion . . . . . . . . . . . . . . . . 462
9.1.5 Concluding remarks to this section . . . . . . . . . . 470
9.2 Observation of anomalous Cherenkov rings. . . . . . . . . . 471
9.3 Two-quantum Cherenkov effect . . . . . . . . . . . . . . . . 471
9.3.1 Pedagogical example: the kinematics of the one-photon
Cherenkov effect ............... 472
9.3.2 The kinematics of the two-photon Cherenkov effect . 474
9.3.3 Back to the general two-photon Cherenkov effect . . 480
9.3.4 Relation to the classical Cherenkov effect . . . . . . 483
9.4 Discussion and Conclusion on the Two-Photon Cherenkov
Effect . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . 483