Structure of Complex Nuclei / Struktura Slozhnykh Yader / CTPYKTYPA CЛOЖHЫX ЯдEP: Lectures presented at an International Summer School for Physicists, Organized by the Joint Institute for Nuclear Research and Tiflis State University in Telavi, Georgian SS

215 Pages·1969·8.177 MB·English

by  I. M. Ulehla (auth.), N. N. Bogolyubov (eds.)

STRUCTURE OF COMPLEX NUCLEI
STRUKTURA SLOZHNYKH YADER
CTPYKTYPA CJIOJKHHX HtiEP
STRUCTURE OF
COMPLEX NUCLEI
Lectures presented at an International Summer School for
Physicists, Organized by the Joint Institute for Nuclear Research
and Tiflis State University in Telavi, Georgian SSR
Edited by
Academician N. N. Bogolyubov
Academy of Sciences of the USSR
Translated from Russian
@
Springer Science+Business Media, LLC 1969
Library of Congress Catalog Card Number 69-12510
The original Russian text was published by Atomizdat in Moscow in 1966
ISBN 978-1-4899-4835-9 ISBN 978-1-4899-4833-5 (eBook)
DOI 10.1 007/978-1-4899-4833-5
© 1969 Springer Science+Business Media New York
Originally published by Consultants Bureau in 1969.
Ali rights reserved
No part of this publication may be reproduced in any
form without written permission from the publisher
PREFACE
The International Summer School on the Structure of Complex Nuclei was held from
August 11 to28, 1965, in Telavi, Georgian SSR. Organized by the Joint Institute for Nuclear Re
search together with the Tbilisi (Tiflis) State University, it was attended by 146 physicists from
many different countries, including Poland, Rumania, Bulgaria, East Germany, North Vietnam,
and Mongolia.
The Telavi Summer School dealt with one of the most important questions of physics-the
study of atomic nuclei. In recent years, considerable'progress has been recorded in the experi
mental study of the properties of atomic nuclei, and new favorable possibilities have appeared
for determining the most important characteristics of the ground and excited states of both light
and heavy nuclei. Significant advances have been made in the theory of atomic nuclei, mainly
in the application of mathematical methods developed in the many-body problem. Work on the
theory of nuclear matter has also been important. Appreciable progress has resulted from the
application of mathematical methods developed in superconductivity theory to nuclear theory
and in the development of the superfluidity model of the nucleus. Research on the theory of
finite Fermi systems as applied to the nucleus has been interesting. The trend due to the ap
plication of group methods to the study of light nuclei has also proved important.
It was the aim of the Summer School to acquaint the audience with the principal achieve
ments in the study of the structure of complex nuclei. The lectures given at the School covered
three main sections: the theory of heavy nuclei (including many-body nuclear problems), the
classification and analysis of experimental data on heavy nuclei, and the theory of light nuclei.
The activities of the School included seminars, at which original communications were pre
sented.
The School owes its success to the considerable assistance provided by our Georgian
colleagues and the authorities of the Republic and of the town of Telavi. Those who took part
in the School are grateful to the people of Telavi for their cordiality and hospitality.
Professor V. G. Solov1ev
Chairman of the School Organizing Committee
v
CONTENTS
Atomic Nuclear Theory
I. M. Ulehla ................ . 1
Pair Correlations and Structure of Deformed Nuclei
V. G. Solov'ev ................. . 23
The Quasiparticle Method in Nuclear Theory
A. B. Migdal .................. . 53
Light Nuclei Calculations by the Self-Consistent Field Method
D. M. Brink ..................................... . 69
Group Theory Methods in Nuclear Physics
G. Flach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Investigation of the Structure of the Nucleus by Means of Reactions Produced by H3 Nuclei
0. Nathan .................................................. 101
Deformed States of Nuclei
B. S. Dzhelepov ........................... . 119
Radioactive Properties of the Nuclei of the Heaviest Elements
G. N. Flerov, V. A. Druin ....................................... 153
Magnetic Properties of Strongly Deformed Atomic Nuclei
A. Z. Hrynkiewicz, S. Ogaza ...................................... 167
Some Properties of Deformed Odd-A Nuclei in the Rare-Earth Element Region (Review of
Experimental Data)
K; Ya. Gromov .......................... . 185
Some Problems in the Study of Spherical Odd-Odd Nuclei
L. K. Peker ................................................. 201
vii
ATOMIC NUCLEAR THEORY
I. M. Ulehla
joint Institute for Nuclear Research
Introduction
The real atomic nuclear theory cannot be based on model concepts, but should start from
the general laws of motion and interaction of the particles forming the atomic nucleus. In ad
dition to explaining the fundamental physical properties of nuclei, binding energy, nucleon dis
tribution, charge distribution, spin, magnetic and quadrupole moments, etc., this theory ought to
1
predict the spectrum of atomic nuclei and nuclear reactions. One of the main problems is also
the explanation of one model or the other.
This is a complex problem and the physics of the atomic nucleus is at the beginning of its
solution. In the course of the last 15 years, theory has been provided with a basis which starts
with the following assumptions.
1. The nucleus may be regarded as a many-particle system of nucleons.
2. The laws of nonrelativistic quantum theory suffice to describe this system.
3. The interaction between nucleons in the nucleus has the character of a two-particle
force which may be derived from the potential.
Each of the assumptions enumerated is open to criticism. From the point of view of field
theory, to regard the nucleus as a system of nucleons is too limited. It is not clear how large
is the relativistic effect of nuclear forces which vary rapidly with distance. It has not been
demonstrated that the forces acting between the nucleons may be described by means of a two
particle potential, and that many-particle forces are not active between the nucleons.
However, the assumptions expressed are the only ones of their kind, and they already make
it possible at the present time to formulate a nuclear theory. From the fundamental point of
view, it is reasonable to verify such a theory also as a function of the results obtained or to
develop it or to exclude certain basic assumptions and adopt others.
A relatively simple verification of nuclear theory may be given by the two extreme cases,
the application of this theory to the examination of the properties of nuclear matter and to rela
tively light nuclei.
By nuclear matter is understood the infinite system of nucleons, between which electro
magnetic forces do not act. This system is uniform and isotropic. The density of nuclear mat
ter is defined by a finite quantity. Extrapolation of the data for heavy nuclei gives a binding
energy of a nucleon of nuclear matter of the order of 16 MeV and r0 -1f, the radius characteriz
ing the elementary volume %1T"r~ of one nucleon. The advantage of nuclear matter, which may
1
2 I. M. ULEHLA
perhaps exist in heavy nuclei or in some astronomical objects, is defined by the statement that
in nuclear matter the nucleons are described by plane waves, since the laws of translational in
variance are then operative.
From the point of view of theory, all nuclei up to Ca40 are assumed to be light nuclei.
They are suitable because the number of particles is not too large to prevent the system from
being examined within the framework of modern mathematical methods.
As shown by the two extreme cases, the results of theory depend to a great extent on the
form of the two-particle potential. It is well known that up to the present there has been no law
of two-nucleon interaction, such as the Coulomb law or the law of gravitation. Experiment pro
vides us with data which are not completely unambiguous and from which it is difficult to derive
an analytical form of the two-nuclear potential. The most successful potentials which describe
two-nuclear data up to energies of 300 MeV are the potentials of Gammel and Thaler [1),
the Yale group potentials [2) and the potentials of Hamada and Johnson [3). It is typical
of these potentials that they have a hard core and different parameters for singlet and
triplet states and for even and odd l.
If a complete spin of two nucleons is denoted by S, the isospin operator (scalar in iso
space) by T, having the value +1 for the isotriplet and -3for the isosinglet, the orbital moment
by L, and the relative coordinates of the two nucleons by r, the general two-nucleon potential
then has the form
(1)
where
If Vi increases unlimitedly in a certain finite region, we are dealing with a potential con
taining a hard core. If Vi contains the operator ajar, we have a potential which is dependent on
velocity.t
In the potentials [1-3], the vis are independent of the velocities. Potentials which are de
pendent on velocity and do not contain a hard core are used relatively little, and principally only
for tentative calculations. Potentials containing a hard core have, in the outer region, the form:
Recently, potentials with a soft core are being used. These potentials in the vicinity of
r = 0 are repulsive and in the extreme case have a singular point at r = 0. As shown by re
searches at the Joint Institute for Nuclear Research, these potentials also are not completely
exact, and therefore the results obtained in calculations using them must be regarded critically.
The values of the isospin operator T, spin operator s2 and parity 1r(l) are interdependent:
sz 'rr(l)
~
~}singlet -3
I +
~}triplet -3 +
1
tTerms containing (La), (Ls)2 are also said to be terms dependent on velocity.
ATOMIC NUCLEAR THEORY 3
Here, in the singlet spin state, the operat.ors (Ls) (spin-orbital) and (rs)2/r2 (tensor) are equal to
0. In all the calculations in nuclear theory, based on the above-mentioned assumptions, the per
turbation theory is used as the basic method. An important part of this theory is the determina
tion of the two-particle reaction matrix from which, as shown by Watson [4], the optical poten
tial may be derived.
Perturbation Theory
We shall examine the perturbation theory in the second-quantization concept. We have a
system of fermions, described by the Hamiltonian
+
=H+ = H0 J!,
H 0= ~En'll~'11n• V ~ (n1n2lvln3n4)'1!~'1/t'll•,'IJn,. (2)
n1nsn3n,
En are the eigenvalues of the Hamiltonian of the single-particle problem. The subscript n sym
bolizes all the quantum numbers determining the corresponding single-particle state, N =
{nx1ny1nz1s1T}. The quantity (n1n21vln311,j) is defined as follows:
where C'fJni (xi) are the wave functions of the single-particle problem; v(x1x2) is the potential (1),
acting between the first and second particles; 7]~, 7Jn are the creation and annihilation operators
of fermions in the n state satisfying the known commutation relations
The system of fermions described by the Hamiltonian H is regarded as the unperturbed
0
system. We denote the vacuum of the unperturbed system by I> . In the ground state of the un
perturbed system -I 0>, which (as is assumed) is nondegenerate, all the states whose energy is
less than the Fermi energy are occupied and the other states are free.
For the occupied states we use the subscript k, for the free states l. We determine the
new operators
(3)
Evidently
because the operators b* create particles and operators b annihilate them, while the operators
a*, a create and annihilate holes.
The expression for the Hamiltonian (2) may be transformed by means of the operators a,
b. For example,
where
4 I. M. ULEHLA
is the energy of the ground state of the unperturbed system. Similarly, V may be transformed
so that all the products of the operators will be written in the form of normal products. This
means that the creation operators will be on the left and the annihilation operators will be on
the right.
A detailed deduction of the perturbation theory for a Hamiltonian of type (1) is given in
[5, 6].
For the ground state II/!> of the Hamiltonian H we get
I~>= ( 1+E0--1H- V0 +E-0--1H V0 -E-0-V1H 0+ ···) L [0>. (4)
"'
The state I~> is normalized, such that
<O[<f>=l.
The energy corresponding to this state is equal to
E = £0 + !:J.E, )
!:J.E= < Of(V +V-1-V + .. ·) [0>-
E,-H, L (5)
The subscript L means that in all the expressions only those diagrams are consid!=!red which do
not contain vacuum parts, i.e., parts having no external lines and which are not connected with
other parts of the diagram. The permissible diagrams are called coupled diagrams. Since ex
pressions (4), (5) contain only coupled diagrams, the denominators in these equations cannot be
equal to zero.
Using the Wick theory, an exact expression for the energy in any approximation may be
derived from expression (5). Putting
llE=~flEU>,
i=I
the following expressions or the corresponding diagrams will be contained in the individual
terms of the energy:
8
k,
!:J.£(1)=+ ~(k1k2\V[k1k2) (6a)
k!l<, Kz
(6b)