Bức xạ Vavilov-Cherenkov và Synchrotron: Nền tảng và Ứng dụng (G. Afanasiev)

Vavilov-Cherenkov and Synchrotron Radiation www.com 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. Editorial Advisory Board: GIANCARLO GHIRARDI, University of Trieste, Italy LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, U.

CLIVE KILMISTER, University of London, U. 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. HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der Wissenschaften, Germany Volume 142 www.com Vavilov-Cherenkov and Synchrotron Radiation Foundations and Applications by G. Afanasiev Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region, Russia KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW www.com 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.com and the Springer Global Website Online at: http://www.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 .3 Original Tamm problem .4 Comparison of the Tamm and exact solutions .5 Spatial distribution of shock waves .6 Time evolution of the electromagnetic field on the surface of a sphere .7 Comparison with the Tamm vector potential .2 Spatial distribution of Fourier components .1 Quasi-classical approximation .3 Quantum analysis of the Tamm formula .4 Back to the original Tamm problem .2 Restoring vector potential in the spectral represen- tation .3 The Tamm approximate solution .4 Concrete example showing that the CSW is not al- ways reduced to the interference of BS shock waves 77 2.5 Schwinger’s approach to the Tamm problem .1 Instantaneous power frequency spectrum .2 Instantaneous angular-frequency distribution of the power spectrum .3 Angular-frequency distribution of the radiated en- ergy for a finite time interval .4 Frequency distribution of the radiated energy .6 The Tamm problem in the spherical basis .com vi CONTENTS 2.1 Expansion of the Tamm problem in terms of the Leg- endre polynomials .7 Short résumé of this chapter. 97 3 NON-UNIFORM CHARGE MOTION IN A DISPERSION- FREE MEDIUM 99 3.2 Statement of the physical problem .1 Simplest accelerated and decelerated motions [9] .2 Completely relativistic accelerated and decelerated motions [11] .3 Smooth Tamm problem in the time representation .1 Moving singularities of electromagnetic field .4 Concluding remarks for this chapter. 124 chapter4 CHERENKOV RADIATION IN A DISPERSIVE MEDIUM127 4.3 Electromagnetic potentials and field strengths .4 Time-dependent polarization of the medium .1 Another choice of polarization .5 On the Krönig-Kramers dispersion relations .6 The energy flux and the number of photons .1 Charge velocity exceeds the critical velocity .2 Charge velocity is smaller than the critical velocity 160 4.1 Estimation of non-radiation terms .9 The influence of the imaginary part of
.10 Application to concrete substances .11 Cherenkov radiation without use of the spectral representation188 4.12 Short résumé of this Chapter. 204 5 INFLUENCE OF FINITE OBSERVATIONAL DISTANCES AND CHARGE DECELERATION 209 5.2 Finite observational distances and small acceleration .1 The original Tamm approach .com CONTENTS vii 5.2 Exact electromagnetic field strengths and angular- frequency distribution of the radiated energy .4 Decelerated charge motion .3 Motion in a finite spatial interval with arbitrary acceleration 233 5.2 Main mathematical formulae .5 The absolutely continuous charge motion.6 Superposition of uniform and accelerated motions .7 Short discussion of the smoothed Tamm problem .8 Historical remarks on the VC radiation and bremsstrahlung .4 Short résumé of Chapter 5.

279 6 RADIATION OF ELECTRIC, MAGNETIC AND TOROIDAL DIPOLES MOVING IN A MEDIUM 283 6.2 Mathematical preliminaries: equivalent sources of the elec- tromagnetic field .1 A pedagogical example: circular current.2 The elementary toroidal solenoid.3 Electromagnetic field of electric, magnetic, and toroidal dipoles in time representation.1 Electromagnetic field of a moving point-like current loop .2 Electromagnetic field of a moving point-like toroidal solenoid .3 Electromagnetic field of a moving point-like electric dipole .4 Electromagnetic field of induced dipole moments .4 Electromagnetic field of electric, magnetic, and toroidal dipoles in the spectral representation .1 Unbounded motion of magnetic, toroidal, and electric dipoles in medium .2 The Tamm problem for electric charge, magnetic, electric, and toroidal dipoles .5 Electromagnetic field of a precessing magnetic dipole .6 Discussion and Conclusion .com viii CONTENTS 7 QUESTIONS CONCERNING OBSERVATION OF THE VAVILOV-CHERENKOV RADIATION 341 7.2 Cherenkov radiation from a charge of finite dimensions .1 Cherenkov radiation as the origin of the charge de- celeration .3 Cherenkov radiation in dispersive medium .4 Radiation of a charge moving in a cylindrical dielectric sample .1 Radial energy flux .2 Energy flux along the motion axis .5 Vavilov-Cherenkov and transition radiations for a spherical sample .6 Discussion on the transition radiation .1 Comment on the transition radiation .2 Comment on the Tamm problem. 390 8 SELECTED PROBLEMS OF THE SYNCHROTRON RADIATION 397 8.2 Synchrotron radiation in vacuum.2 Energy radiated for the period of motion .3 Instantaneous distribution of synchrotron radiation .3 Synchrotron radiation in medium .2 Electromagnetic field strengths .3 Singularities of electromagnetic field .4 Digression on the Cherenkov radiation .5 Electromagnetic field in the wave zone .6 Numerical results for synchrotron motion in a medium434 8. 444 9 SOME EXPERIMENTAL TRENDS IN THE VAVILOV- CHERENKOV RADIATION THEORY 447 9.1 Fine structure of the Vavilov-Cherenkov radiation .1 Simple experiments with 657 MeV protons .2 Main computational formulae .com CONTENTS ix 9.5 Concluding remarks to this section .2 Observation of anomalous Cherenkov rings .3 Two-quantum Cherenkov effect .1 Pedagogical example: the kinematics of the one-photon Cherenkov effect .2 The kinematics of the two-photon Cherenkov effect .3 Back to the general two-photon Cherenkov effect .4 Relation to the classical Cherenkov effect .4 Discussion and Conclusion on the Two-Photon Cherenkov Effect .com This page intentionally left blank www.com PREFACE The importance of the Vavilov-Cherenkov radiation stems from the property that a charge moving uniformly in a medium emits γ quanta at the angle uniquely related to its energy. This has numerous applications.

We mention only the neutrino experiments in which the neutrino energy is estimated by the angle at which the electron originating from the decay of neutrino is observed. This book is intended for students of the third year and higher, for postgraduates, and professional scientists, both experimentalists and theo- reticians. The Landau and Lifschitz treatises Quantum Mechanics, Classi- cal Field Theory and Electrodynamics of Continuous Media are more than enough for the understanding of the text. There are three monographs devoted to the Vavilov-Cherenkov radia- tion.

Jelly’s book Cherenkov Radiation and its Applications published in 1958 contains a short theoretical review of the Vavilov-Cherenkov radi- ation and a rather extensive description of experimental technique. Ten years later, the two-volume Zrelov monograph Vavilov-Cherenkov Radia- tion and Its Application in High-Energy Physics appeared. Its first volume is a quite extensive review of experimental and theoretical results known up to 1968. The second volume is devoted to the construction of the Cherenkov counters.

In 1988, the Frank monograph Vavilov-Cherenkov Radiation. The- oretical Aspects was published. It presents mainly a collection of Frank’s papers with valuable short commentaries describing their present status. It is highly desirable to translate this book into English.

The main goal of this book is to present new developments in the theory of the Vavilov-Cherenkov effect for the 15 years following the appearance of Frank’s monograph. We briefly mention the main questions treated: 1) The Vavilov-Cherenkov radiation for the unbounded charge motion in a medium (the so-called Tamm-Frank problem); 2) Exact solutions for semi-infinite and finite charge motions in a non- dispersive medium. Their study allows one to identify how the Cherenkov shock waves and the bremsstrahlung shock waves are distributed in space; 3) Accelerated and decelerated charge motions in a medium. Their study allows one to observe the formation and time evolution of the singular shock xi www.com xii PREFACE waves (including the finite Cherenkov shock wave) arising when the charge velocity coincides with the velocity of light in a medium; 4) The consideration of the Vavilov-Cherenkov radiation in dispersive media with and without damping supports Fermi’s claim that a charge moving uniformly in a dispersive medium radiates at each velocity.

It turns out that the position and magnitude of the maximum of the frequency distribution depend crucially on the damping parameter value; 5) The measurement of the radiation intensities at finite observational distances leads to the appearance of plateau in some angular interval. The linear (not angular) dimensions of this plateau on the observational sphere do not depend on the sphere radius. Inside this plateau the radiation inten- sity is not described by the Tamm formula at any observational distance; 6) The taking into account of the finite dimensions of a moving charge or the medium dispersion leads to the finite energy radiated by a moving charge for the entire time of its motion. This in turn allows one to determine how a charge should move if all its energy losses were owed to the Cherenkov radiation; 7) The Vavilov-Cherenkov radiation for a charge moving in a finite medium interval.

This includes the consideration of the original Tamm problem (having instantaneous velocity jumps at the beginning and the end of the charge motion), the smooth Tamm problem (in which there are no discontinuities of the charge velocity) and the absolutely continuous charge motion (for which the charge velocity and all its time derivatives are continuous functions of time) in a finite spatial interval. This permits one to relate the asymptotic behaviour of the radiation intensities to the discontinuities of the charge trajectory; 8) It is studied how the radiation intensity changes when a charge moves in one medium while the observations are made in another, with different dielectric properties (in fact, this is a typical experimental situation); 9) The Vavilov-Cherenkov and transition radiations for the spherical interface between two media (previously, only the plane interface was con- sidered in the physical literature); 10) The radiation of electric, magnetic, and toroidal dipoles moving in a medium. This allows one to study the radiation arising from the moving neutral particles (e., neutrons, neutrinos, etc.); 11) The fine structure of the Cherenkov rings is studied. We mean un- der this term the plateau in the radiation intensity (which is due to the Cherenkov shock wave), sharp maxima at the ends of this plateau (we as- sociate them with bremsstrahlung shock waves arising at the accelerated and decelerated parts of the charge trajectory) and small oscillations in- side this plateau (they are due to the interference of the Cherenkov and bremsstrahlung shock waves); www.com PREFACE xiii 12) The kinematics of the two-photon simultaneous emission for a charge moving uniformly in medium.

It turns out that under certain circumstances the photon emission angles are fixed. The radiation intensity should have sharp maxima at these angles (similarly to the single-photon Cherenkov emission). This creates favourable conditions for the observation of the two-photon Cherenkov effect. The importance of the synchrotron radiation is because it is extensively used for the study of nuclear and particle reactions, astrophysical prob- lems, and has a variety of biological and medical applications.

There are a few books of the Moscow State University School, and the recently (2002) published book Radiation Theory of Relativistic Particles (Ed. Bor- dovitsyn) which, in fact, presents a collection of papers of various authors devoted to the questions related to the synchrotron radiation. In the present monograph we study the synchrotron radiation in a medium, and the syn- chrotron radiation in vacuum, in the near zone. These questions were not considered in the references just mentioned.