Table Of ContentThe 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
