The Lebedev Physics Intitute Series 44 Academician D. V. Skobel'tsyn Nuclear Physics and Interaction of Particles with Matter NUCLEAR PHYSICS AND INTERACTION OF PARTICLES WITH MATTER YADERNAYA FIZIKA I VZAIMODEISTVIE CHASTITS S VESHCHESTVOM lt;lEI'IIAll <1>11311KA ll B3AHMO).\Ei1CTBJ.1E 4ACTJ.1~ C BElll,ECTBOM The Lebedev Physics Institute Series Editor: Academician D. V. Skobel'tsyn Director, P. N. Lebedev Physics Institute, Academy of Seiences of the USSR Volume 25 Optical Methods of Investigating Solid Borlies Volume 26 Cosmic Rays Volume 27 Research in Molecular Spectroscopy Volume 28 Radio Telescopes Volume 29 Quantum Field Theory and Hydrodynamics Volume 30 Physical Optics Volume 31 Quantum Electronics in Lasers and Masers Volume 32 Plasma Physics Volume 33 Sturlies of Nuclear Reactions Volume 34 Photomesonie and Photonuclear Processes Volume 35 Electronic and Vibrational Spectra of Molecules Volume 36 Photodisintegration of Nuclei in the Giant Resonance Region Volume 37 Electrical and Optical Properties of Semiconductors Volume 38 Wideband Cruciform Radio Telescope Research Volume 39 Optical Sturlies in Liquids and Solids Volume 40 Experimental Physics: Methods and Apparatus Volume 41 The Nucleon Compton Effect at Low and Medium Energies Volume 42 Electronics fn Experimental Physics Volume 43 Nonlinear Optics Volume 44 Nuclear Physics and Interaction of Particles with Matter Volume 45 Programming and Computer Techniques in Experimental Physics Volume 47 Radio Astronomy: Instruments and Obse~ations Volume 48 Surface Properties of Semiconductors and Dynamics of Ionic Crystals Volume 49 Quantum Electronics and ParamagneUe Resonance In preparation Volume 46 Cosmic Rays and Interaction of High-Energy Particles Volume 50 Electroluminescence Volume 51 Physics of Atomic Collisions Proceedings (Trudy) of the P. N. Lebedev Physics Institute Volume 44 NUCLEAR PHYSICS AND INTERACTION OF PART ICLES WITH MATTER Edited by Academician D. V. Skobel'tsyn Director, P. N. Lebedev Physics Institute Academy of Seiences of the USSR, Moscow Translated from Russian SPRINGER SCIENCE+BUSINESS MEDIA, LLC 1971 The Russian textwas published by Nauka Press in Moscow in 1969 for the Academy of Seiences of the USSR as Volume 44 of the Proceedings (Trudy) of the P. N. Lebedev Physic:s Institute. The present translation is pub lished under an agreement with Mezhdunarodnaya Kniga, the Soviet book ex- port agency. Library of Congress Catalog Card Number 7Q-120025 ISBN 978-1-4757-6034-7 ISBN 978-1-4757-6032-3 (eBook) DOI 10.1007/978-1-4757-6032-3 11:> 1971 Springer Science+Business Media New Yo rk Originally published by Plenum Publishing Corporation, N ew Y ork in 1971 All rights reserved No part of this publication may be reproduced in any form without written permission from the publisher CONTENTS Analysis of the Resonance Reactions H3(d,n)He4 and He3(d,p)He4 in the Effective Interaction Radius Approximation I. Ya. Barit and V. A. Sergeev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Basic Relations of the Effective Radius Approximation . . . . . . . . . . . . . . . . . 2 2. Determination of the Parameters of the Effective Radius Approximation from Experimental Data for the Reactions H3(d,n)He4 and He3(d,p)He4• 6 3. Results of an Analysis of the H3(d,n)He4 and He3(d,p)He4 Reactions Using the Formulas of the R-Matrix Theory and the Effective Radius Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . 9 4. Particular Features of lsolated Resonances in the Various Theories . . . . . . . . 11 Conclusion . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . 12 References . . • . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • • . . . . . . 13 Single-Particle Vibrational Input States in Interactions between Neutrons and Nuclei V. I. Popov. . . . • . • . . • . . • . • . . . . . . • . . • • . . . . . . . . . . . . . . . . . . . . . • • • 14 Introduction . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • 14 1. Intermediate States of the Phonon/Particle Type for Low-Energy Neutrons. . . . 16 2. Neutron Force Functions . . . . . . . . • . . . . . • . • . . . . . . . . . . . . . . . . • . . . 18 3. Radiative Neutron Capture by Intermediate States . . . . . . . . . . . . . . . . . . . . 20 Conclusion . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 References . . . . . . • • . . . . . . . • . . . . . . . . . . . . . • • • . • . . . . . . . . . . . . • . 23 Radiation of a Charged Particle in the Presence of a Separating Boundary V. E. Pafomov. . . . . . . • . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . • . . . . . . . 25 Introduction . . • . • • • . • . . . . . . • . . . . . . . . . . . . . . • . . . . . . . . • . . . . • . . • 25 C hap te r 1. The Image-Representation Technique . . . . . . . . . . . . • . . . . . • • 26 1. Introduction. • . . • . • . . . . • . . . . . • . . . . . . . . . . • . . . . . . . . . • • . • • . 26 2. Dipole Images . . . . . . • . • . • . • • • . . . . . . . . . • • . • • • . . . • . • . • . • • . 27 3. Dipole Field and a Moving Charged Particle in the Presence of a Boundary. • 30 4. Spherical Waves . . . . . . . . • . . . • • . . • . • . . . • . • . . . . . . • . • . . • . • . . 35 C h a p t er 2 . Radiation in the Presence of a Separating Boundary . . . . . . . . . • 38 5. Time and Path of Coherence . . . . . . . . . . . . . . • . . . • . . . . . . • . . . . . . . 38 6. The Radiation Field Generated over a Finite Path of a Uniformly Moving Charge in a Ferrodielectric Medium. . . . . . . . . . . . • . . . • . . . . . . . . . 39 7. Cerenkov Radiation in the Frequency Range Involving Negative Group Velocities . . • . . . . . . • . . . . . . . . . . . . • . . . • . . . . . . . . . • . . . . . . 42 8. Radiation from a Charged Particle Moving Perpendicular to the Boundary 45 V vi CONTENTS' 9. Charge Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 52 10. Transition Radiation of a Relativistic Charged Partide in the Optical Frequency Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 11. Transition Radiation at Inclined Incidence ...•.............•.... 56 12. Radiation of an Oscillator Situated near the Boundary ............. . 63 13. Vacuum Radiation of an Oscillator Situated in a Medium Separated by a Plane Boundary from the Vacuum .... , .................... . 65 14. Transition Radiation of a Fixed Oscillator ..................... . 66 15. Luminescence Radiation ................................. . 72 C h a p t e r 3 . Radiation in the Presence of Two Boundaries ............ . 73 16. Spherical Waves Generated by an Arbitrarily Moving Charged Particle .. . 73 17. Radiation from an Oscillator Situated in a Thin Film .............. . 78 18. Radiation of a Charged Partide Moving Uniformly along a Straight Line through Two Boundaries ............................... . 80 19. Radiation of a Partide Passing in the Normal Direction through a Plate .. 87 20. Energy Losses of a Charged Relativistic Partide in a Thin Dielectric Plate ....•........................................ 93 C hap te r 4. Radiation of a Partide Moving in a Layered Medium ....... . 98 21. Radiation from a Charged Partide Moving in Some Fashion in a Layered Medium .......................................... . 98 22. Radiation Generated by a Charged Partide in a Stack of Plates ....... . 101 23. Radiation Generated by a Partide Passing through a Stack of Transparent Plates .................................. . 105 C h a p t er 5 . Radiation with Scattering ..................... . 107 24. Radiation with Single Scattering of a Nonrelativistic Particle near a Surface ..........•............................ 107 25. Effect of Multiple Scattering upon the Transition Radiation ........ . 114 26. Bremsstrahlung at Frequencies Exceeding Optical Frequencies ....... . 122 27. Optical Bremsstrahlung in an Absorbing Medium ................ . 127 C h a p t e r 6 . E xcitation of Surface Waves ....................... . 139 28. Field of a Surface Wave Generated by a Moving Partide ........... . 139 29. Surface Waves in the Case of Normal Incidence ................. . 142 30. Coherent Generation of a Surface Wave ......•................. 145 31. Excitation of Surface Waves in the Case of Inclined Partide Incidence ..• 149 32. Work of the Braking Force ............................... . 153 References . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ . 155 Introduction of Heavy and Light Radioactive Nuclei into Nudear Emulsions T. A. Romanova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ . 158 Introduction ................................................ . 159 I. Introduction of Uranium into Emulsions for Nuclear Research .•.....•... 159 1. Loading and Development of Nudear Emulsions Sensitive to Protons with Energies of up to 50 MeV ..............•. ·. ......•. · · . · · · 160 2. Introduction of Uranium into 250- to 300-~-t-thick Nuclear Emulsions Sensitive to Particles at the Ionization Minimum ...............• 162 3. Desensitizing Effect of Uranium Salts ........................ . 167 4. Reducing Effect of Triethanolamine ................•......... 169 5. The Hardening Effect of Uranium ........................... . 171 Basic Conclusions of Beetion I ........................ · · · · · · · · · · 173 li. Introduction of Hydrogen and Deuterium into Emulsions Sensitive to Particles at the Ionization Minimum ......................... . 174 III. Introduction of Lithium into Emulsion Plates .................... . 178 CONTENTS vii IV. Dilution of NIKFI-R Emulsions with Gelatin . . . . . . . . . . . . . . . . . . . . . . 180 V. Using the Results of the Methodological Studies in Physics Experiments . . . 182 Heferences . . . . . . . . . . . . . . . . . . . . . . . . . • . . . • . . . . . . . . . . . . . • . . . 190 Neutron-Transfer Theory for Inhomogeneaus Media A. V. Stepanov. . . . . . . • . . . . . . . . . . • . . . . . . • . . . . . . . . . . . . . . . . . . . • 193 1. lntroduction . • • . . . . . . . • . • • • . . • . . • • • . . • . . . . . . . . . . . . . . . . . . 193 2. Derivation of General Formulas • • • • • • • • • • . . . . . • • . . . • . . . . . . . . • 195 3. Gorrelation Functions Kij(r) . • . . • • . . . • • . • • • . . . . . • • . . . . . . . . . . . 200 4. Stationary Diffusion ofThermal Neutrons • • • . . • . • . . . . . . . . . . . . . . . . 209 5. Stationary Single-Velocity Kinetic Equation . . . • . • . • . . . . . . • . . • . . . . . 217 6. Nonstationary Diffusion ofThermal Neutrons. . . . . . . . . . . . . • . . . . . . . . 223 7. Energy Dependence of the Neutron Distribution Function . . • . . • . . . . • . . . 226 8. Applicability Range of the M1 Approximation . . • • • • • . • . . • • • . . • . • . . . 230 Conclusion . . . . . . . . • . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . 239 Appendix A. . . . • . • • . . . . . . • . . . . . . • • . • • . • . . • . . . . . . . . . . . . . . . • 239 Appendix B. . . . . • • . . . . . . . . . • . • . . . • . • • . . . • . . . . . . . . . . . . . . . . . 243 Appendix C . • . • . . . • . • . . • . • . • • . • . . . • • . • . • . • . . . . . . . . . . . . . . • • 245 Heferences . • . • . . . • . . • . • . . . . . • . . . . . . . . • . . . . . . . . . . . . . . . . . • . 246 Model of a Heavy Moderator and an Analytical Solution of the Neutron Thermalization Problem M. V. Kazarnovskii. • . . . . . • • . . . . • . . . . . . • • • • • . • • . • • . . . • . . • . . • . . 248 lntroduction . . . • . . . • . . . • . . • • • . . . • . • . • • • • . • • • • • . . . . . . . • . . . • 248 1. Neutron-Transfer Equation and General Properties of the Scattering Operator . . . . . • . • . . . • . . . . . . . . . . . • . . . . . . . . • . • . . . . . . . . . 249 2. Model of a Scattering Operatorfora Heavy Moderator. . . . . . . • . . . . . • . • 253 3. Selection of the Functions w, J.l., and a . • . . . . • • . • • . • . • • • . • • • • • . . . • 256 4. Solution of the Neutron-Transfer Equation in the Case of an Infinite, Homogeneaus Moderator. . . . • • . • • . • . . . • . . . . . . . . . . . . . . . . . . • 264 Conclusion . • . • . . . . . • . . . . . . . . . . . • . • • . • . . • . . . . . . . . . . . • . . . . . 268 Heferences . . . . . . . . . . . . . . . . . . . . . . . . • . • • • . . . . . . • . • . . . . • . . . . 269 ANALYSIS OF THE RESONANCE REACTIONS H (d,n)He AND He (d,p)He IN THE EFFECTIVE 3 4 3 4 INTERACTION RADIUSAPPROXIMATION I. Ya. Barit and V. A. Sergeev Introduction Rather comprehenBive experimental information on the reactionB W(d,n)He4 and He3(d,p)· He4 haB become available in the low-energy region in which reBonanceB are obBerved at Ed = 0.064 MeV* in the (d,n) reaction and at Ed = 0.255 MeV in the (d,p) reaction. It could be eBtab liBhed that the reBonanceB reBult from the channel with i" = 3/2+. The energy dependence of the croBs Beetions oftheBe reactions near the reBonances (0.006 MeV< Ed < 0.250 MeV for the (d,n) reaction and 0.024 MeV < Ed < 0.350 MeV for the (d,p) reactioil) haB been conBidered in numerouB papers (see, e.g., [1] and [2]). The well-known formula for a single level in the Wigner-Aizenbad theory [3] was employed in thoBe papers: dadN Prl r~PNr~ (1) ~ qkd2 -SN&~-Sdr~ + (Pdr~ + PNT~)2 = (E,_ -Ed)2 The notations are interpreted aB followB: q = (2J + 1) / (2i1 + 1) (2i2 + I) = 2/3 iB a BtatiBtical fac tor; Pd= Pt(kdll,1J), ~ = sri"<l5.Ja,1)), ~ = Pi"(kNa,O), SN= llt<kNa,O); and N denoteB the neutron or proton channel. lt iB a characteriBtic feature of the reBearch performed so far that the energy dependence of the resonance cross Beetions can be BatiBfactorily described by greatly different Bets of reB onance parameterB a, EA, ~, y~. Even when one Bimultaneously analyzes data on the above reactionB and data on the elaBtic reBonance scattering of deuterons in this energy range [2], the indeterminacy in the parameter Belection cannot be completely removed. On the other hand, it Yl is a feature common to all setB of parameterB that the reduced deuteron width iB large: » I E~ j, y~ » rN/2, t where y~ ~ 'l'i 2/llda2 (l'i 2/p.da2 denotes the characteriBtic reduced single-particle width). The quantity E~ = EA- yJ~ and rN/2 = yJPN can be conBidered conBtant because PN and ~ are practically independent of the energy: kN = J(QdN+ ~)2/Lwtth the energy liberated in the reactions (Qdn= 17.58 MeV and Qd =18.34 MeV) exceed conBiderably the energy interval under inspection (t.Ed z 0.35 MeV). P *Throughout our discussion we uBe the center of maBB ByBtem. tMore preciBely, yJifo<Po » E~- yf~0;lp0 and "Y~o<Po » rN;2, where ~o~t, f/Jo denotes the coeffi cientB of the expansion in Coulomb functions [Bee below, EqB. (14)]. 1 2 I. YA. BARIT AND V. A. SERGEEV However, we have for the minimum wavelength of the deuteron ~min = h/v'2J..Lct .1.Ect > ä, where ä denotes the approximate interaction radius assumed to be the sum of the radii of the colliding nuclei. As has been shown in [4), in such a situation it is convenient to use the effective interac tion radius approximation for the analysis of resonances. Three parameters with a simple physical meaning are used in this approximation and the analytical fonns of the energy depen dence of both reaction cross section and scattering cross section are greatly simplified since the parameters in the effective radius theory are more directly related to the scattering ampli tudes and the reaction cross sections than the resonance parameters of the R-matrix theory. The effective radius approximation has been successfully employed in several cases in which resonance effects were observed in a particular energy interval, e.g., in the analysis of nucleon/nucleon scattering [5), in p/H3 scattering below the threshold of the (p,n) reaction [6), and in He4/ He4 scattering (7). The present article shows that the energy dependence of the cross sections of the reac tions H3(d,n)He4 and He3(d,p)He4 in the resonance region can be described by the effective radius approximation using three parameters. The various types of undeterminedness which occur in the analysis of the reaction cross sections are discussed. The results are compared with an analysis of experimental data interpreted with the R-matrix theory. The question is discussed to what extent the single level approximation in the Humblet-Rosenfeld theory (8) resembles the two methods mentioned above for the description of resonance effects. §1. BasicRelations of the Effective Radius Approximation The effective radius approximation, which is limited to the low-energy region, is based on extremely general properties of the scattering amplitudes. This approximation is therefore more convenient than the single level approximation of the R-matrix theory in which definite assumptions regarding the energy dependence of the logarithrnic derivative of the internal wave function at the boundary of the nucleus are made. In the multichannel effective radius theory [9), one considers the energy dependence of a symmetric real matrix M which is related to the scattering matrix S and the amplitude f by the following matrix relations: M = ik1+'1•(S -1f1 (S + 1) k1+'\ (2) I= k1 (M-ik'1•' f' k1• (3) The notations are interpreted as follows: k1 is a diagonal matrix with (k1)ab = kaOab; the ampli tude fab defines the partial cross section of an a - b process [5): (4) The energy dependence of the M-matrix elements has a simple form at low energies, and when certain conditions are satisfied, the energy dependence can be simply related to the inter action radius of the systems considered 15, 9, 10). When only one channel exists, we have S = exp(2io1), M = ~Z+1ctgc5z, and f = (!.{ ctg c5z-ikt1• In the following discussion we will consider the case of two coupled channels: d + H3(d + He3) and n + He4(p + He4), where Zct = 0, lct = 3/.2, ZN= 2, IN= 1/.2, and J = 3/.2 corresponding to the mo mentum and parity of the level involved. It follows from the matrix equation (3) that the partial amplitudes of the deuteron scattering fdct = (d, lct, Ictl i ld, lct, lct) and of the (d, N) reaction, fctN = (d, Zd, Id I !1 I N, ZN, IJ, can be represented in the form