Vavilov-Cherenkov and Synchrotron Radiation Fundamental Theories of Physics An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application Editor: ALWYN VAN DER MERWE, University of Denver, U.S.A. Editorial Advisory Board: GIANCARLO GHIRARDI, University of Trieste, Italy LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, U.K. CLIVE KILMISTER, University of London, U.K. PEKKA J. LAHTI, University of Turku, Finland ASHER PERES, Israel Institute of Technology, Israel EDUARD PRUGOVECKI, University of Toronto, Canada FRANCO SELLERI, Università di Bara, Italy TONY SUDBURY, University of York, U.K. HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der Wissenschaften, Germany Volume 142 Vavilov-Cherenkov and Synchrotron Radiation Foundations and Applications by G.N. Afanasiev Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: 1-4020-2411-8 Print ISBN: 1-4020-2410-X ©2005 Springer Science + Business Media, Inc. Print ©2004 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: http://ebooks.springerlink.com and the Springer Global Website Online at: http://www.springeronline.com CONTENTS PREFACE xi 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 Mathematicalpreliminaries.............. 15 2.1.2 Particular cases. .................... 16 2.1.3 OriginalTammproblem................ 32 2.1.4 Comparison of the Tamm and exact solutions . 36 2.1.5 Spatialdistributionofshockwaves.......... 38 2.1.6 Time evolution of the electromagnetic field on the surfaceofasphere................... 41 2.1.7 Comparison with the Tamm vector potential . 46 2.2SpatialdistributionofFouriercomponents.......... 51 2.2.1 Quasi-classicalapproximation............. 51 2.2.2 Numericalcalculations................. 53 2.3QuantumanalysisoftheTammformula........... 58 2.4BacktotheoriginalTammproblem............. 63 2.4.1 Exactsolution..................... 64 2.4.2 Restoring vector potential in the spectral represen- tation .......................... 70 2.4.3 TheTammapproximatesolution........... 74 2.4.4 Concrete example showing that the CSW is not al- ways reduced to the interference of BS shock waves 77 2.5Schwinger’sapproachtotheTammproblem......... 78 2.5.1 Instantaneouspowerfrequencyspectrum...... 80 2.5.2 Instantaneous angular-frequency distribution of the powerspectrum..................... 84 2.5.3 Angular-frequency distribution of the radiated en- ergyforafinitetimeinterval............. 84 2.5.4 Frequency distribution of the radiated energy . 86 2.6TheTammprobleminthesphericalbasis.......... 93 v vi CONTENTS 2.6.1 Expansion of the Tamm problem in terms of the Leg- endrepolynomials................... 93 2.7 Short r´esum´eofthischapter.................. 97 3 NON-UNIFORM CHARGE MOTION IN A DISPERSION- FREE MEDIUM 99 3.1Introduction........................... 99 3.2 Statement of the physical problem . ............. 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 Movingsingularitiesofelectromagneticfield..... 115 3.4Concludingremarksforthischapter............. 124 chapter4 CHERENKOV RADIATION IN A DISPERSIVE MEDIUM127 4.1Introduction........................... 127 4.2Mathematicalpreliminaries.................. 129 4.3Electromagneticpotentialsandfieldstrengths........ 131 4.4Time-dependentpolarizationofthemedium......... 141 4.4.1 Anotherchoiceofpolarization............ 144 4.5 On the Kr¨onig-Kramersdispersionrelations......... 148 4.6Theenergyfluxandthenumberofphotons......... 149 4.7WKBestimates......................... 155 4.7.1 Charge velocity exceeds the critical velocity . 158 4.7.2 Charge velocity is smaller than the critical velocity 160 4.8Numericalresults........................ 162 4.8.1 Estimationofnon-radiationterms.......... 164 4.9 The influence of the imaginary part of ........... 167 4.10Applicationtoconcretesubstances.............. 175 4.10.1 Dielectric permittivity (4.7) . ............. 179 4.10.2 Dielectric permittivity (4.45) ............. 185 4.11 Cherenkov radiation without use of the spectral representation188 4.11.1Particularcases..................... 191 4.11.2NumericalResults.................... 196 4.12 Short r´esum´eofthisChapter................. 204 5 INFLUENCE OF FINITE OBSERVATIONAL DISTANCES AND CHARGE DECELERATION 209 5.1Introduction........................... 209 5.2 Finite observational distances and small acceleration . 210 5.2.1 TheoriginalTammapproach............. 210 CONTENTS vii 5.2.2 Exact electromagnetic field strengths and angular- frequency distribution of the radiated energy . 213 5.2.3 Approximations..................... 214 5.2.4 Decelerated charge motion . ............. 216 5.2.5 Numericalresults.................... 219 5.3 Motion in a finite spatial interval with arbitrary acceleration 233 5.3.1 Introduction...................... 233 5.3.2 Mainmathematicalformulae............. 235 5.3.3 Particularcases..................... 238 5.3.4 Analyticestimates................... 257 5.3.5 Theabsolutelycontinuouschargemotion....... 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..................... 277 5.4 Short r´esum´eofChapter5................... 279 6 RADIATION OF ELECTRIC, MAGNETIC AND TOROIDAL DIPOLES MOVING IN A MEDIUM 283 6.1Introduction........................... 283 6.2 Mathematical preliminaries: equivalent sources of the elec- tromagneticfield........................ 285 6.2.1 A pedagogical example: circular current. ..... 285 6.2.2 Theelementarytoroidalsolenoid............ 287 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 toroidalsolenoid.................... 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, andelectricdipolesinmedium............ 313 6.4.2 The Tamm problem for electric charge, magnetic, electric,andtoroidaldipoles.............. 327 6.5 Electromagnetic field of a precessing magnetic dipole . 334 6.6DiscussionandConclusion................... 337 viii CONTENTS 7 QUESTIONS CONCERNING OBSERVATION OF THE VAVILOV-CHERENKOV RADIATION 341 7.1Introduction........................... 341 7.2 Cherenkov radiation from a charge of finite dimensions . 344 7.2.1 Cherenkov radiation as the origin of the charge de- celeration........................ 349 7.3Cherenkovradiationindispersivemedium.......... 350 7.4 Radiation of a charge moving in a cylindrical dielectricsample........................ 355 7.4.1 Radialenergyflux................... 356 7.4.2 Energyfluxalongthemotionaxis.......... 357 7.4.3 Opticalinterpretation................. 358 7.5 Vavilov-Cherenkov and transition radiations forasphericalsample..................... 360 7.5.1 Opticalinterpretation................. 360 7.5.2 Exactsolution..................... 362 7.5.3 Metallic sphere . ................... 376 7.6Discussiononthetransitionradiation............ 382 7.6.1 Commentonthetransitionradiation......... 385 7.6.2 CommentontheTammproblem........... 390 8 SELECTED PROBLEMS OF THE SYNCHROTRON RADIATION 397 8.1Introduction........................... 397 8.2Synchrotronradiationinvacuum................ 399 8.2.1 Introductoryremarks................. 399 8.2.2 Energy radiated for the period of motion . 404 8.2.3 Instantaneous distribution of synchrotron radiation . 407 8.3Synchrotronradiationinmedium............... 422 8.3.1 Mathematicalpreliminaries.............. 422 8.3.2 Electromagneticfieldstrengths............ 423 8.3.3 Singularitiesofelectromagneticfield......... 424 8.3.4 Digression on the Cherenkov radiation ........ 426 8.3.5 Electromagneticfieldinthewavezone........ 428 8.3.6 Numerical results for synchrotron motion in a medium434 8.4Conclusion........................... 444 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 Maincomputationalformulae............. 453 CONTENTS ix 9.1.3 Numericalresults.................... 462 9.1.4 Discussion........................ 463 9.1.5 Concluding remarks to this section .......... 472 9.2ObservationofanomalousCherenkovrings.......... 473 9.3Two-quantumCherenkoveffect................ 473 9.3.1 Pedagogical example: the kinematics of the one-photon Cherenkoveffect.................... 474 9.3.2 The kinematics of the two-photon Cherenkov effect . 476 9.3.3 Back to the general two-photon Cherenkov effect . 483 9.3.4 Relation