Lectures on the Theory of the Nucleus

309 Pages·1975·17.97 MB·English

by  A. G. Sitenko and V. K. Tartakovskii (Auth.)

LECTURES
ON THE
THEORY OF THE NUCLEUS
by
A. G. SITENKO AND V. K. TARTAKOVSKII
(Theoretical Physics Institute,
Academy of Sciences of the Ukrainian SSR)
Translated and Edited by
P. J. SHEPHERD
(Department of Physics, University of Exeter)
PERGAMON PRESS
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SYDNEY . BRAUNSCHWEIG
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First Edition 1975
Library of Congress Cataloging in Publication Data
Sitenko, Alekseï Grigor'evich.
Lectures on the theory of the nucleus.
(International series of monographs in natural philosophy, v. 74)
Translation of Lektsii po teorii iadra.
Includes bibliographies.
1. Nuclear physics. I. Tartakovskiï, Viktor Konstantinovich,
joint author. II. Title.
QC173.S46213 1975 539.7'23 74-10827
ISBN 0-08-017876-6
Printed in Hungary
FOREWORD
THE lectures of A. G. Sitenko and V. K. Tartakovskiï give a very full account of modern
ideas about the structure of nuclei. All of the material in the lectures is presented in a clear
and intelligible form, and this is undoubtedly a great merit of the book as a student text.
Notwithstanding the high theoretical level of the book, it is also accessible to experimental
physicists, inasmuch as it contains all the necessary information and concepts that fall out­
side the scope of the usual quantum mechanics courses. The book successfully combines
the features of a serious scientific monograph with the qualities of an accessible textbook,
and is undoubtedly one of the best books on the theory of the nucleus available at the pres­
ent time. The lectures will be very useful to a wide circle of theoretical and experimental
nuclear physicists. They will also be of great benefit to teachers and research students work­
ing in the field of nuclear physics.
Acad. N. N. BOGOLYUBOV
PREFACE
THIS book is an expanded account of a course of lectures on nuclear theory given by
the authors over a number of years at Kiev State University. The fundamentals of modern
ideas on the structure of atomic nuclei are described in the lectures.
It is well known that the most important problem of nuclear physics is the study of nuclear
structure and the explanation of the properties of nuclei on the basis of the nuclear interac­
tion between the nucléons, which are the simplest constituent particles of nuclei. The diffi­
culties in modern nuclear theory are connected both with our inadequate knowledge of the
nature of nuclear interactions and with the many-body character of nuclear systems. The
experimental data on nuclear interactions available at the present time do not contradict
the hypothesis that the nuclear forces are two-body in character. However, the absence of
rigorous calculational methods for many-body systems with strong interaction obliges us
to confine ourselves to treating the phenomenological models of the nucleus introduced to
describe particular properties of nuclei. The development of models of the nucleus has
proceeded in different directions, and the models themselves have been modified and have
become more complicated. Despite their apparent contradictions, the different models of the
nucleus complement each other to a considerable extent. As a result of the development of
model concepts considerable progress has been possible in our understanding of the structure
of atomic nuclei. In the lectures different models of the nucleus are examined in detail and
the interrelations between them are also discussed.
The lecture course offered consists of five Chapters: 1. Nuclear forces, 2. Nuclear matter,
3. Shell structure of nuclei, 4. Rotation and vibrations of nuclei, and 5. Pair correlations in
nuclei. Although in the book most attention is given to the basic principles, we have also
endeavoured to describe some of the most recent achievements in nuclear theory. A number
of the problems discussed in the lectures have appeared only in original papers and have
not been described in textbooks before. Since the lectures are intended as a student textbook,
we have limited the references (with a few exceptions) to literature that is widely accessible.
The list of References given at the end of the Preface includes books on nuclear theory in
which the reader can find more detailed accounts of a number of questions and also refer­
ences to the original papers. The reader is assumed to be familiar with the quantum mecha­
nics taught in university courses (as covered by, for example, Davydov (1963), Landau and
Lifshitz (1963), and Sitenko (1971)).
The idea of writing this course of lectures arose out of joint work with Dr. A. I. Akhiezer
and we feel it our duty to express our sincere gratitude to him. We also express our
sincere thanks to Dr. P. V. Skorobogatov for substantial assistance in the writing of
Chapter 4, and to Dr. P. J. Shepherd for translating and editing the book.
xi
xii PREFACE
References
AKHTEZER, A. I. and POMERANCHUK, I. Ya. (1950) Topics in the Theory of the Nucleus, Gostekhteorizdat,
Moscow.
DE BENEDETTI, S. (1964) Nuclear Interactions, Wiley, N.Y.
BETHE, H. A. and MORRISON, P. (1956) Elementary Nuclear Theory, Chapman & Hall, London.
BLATT, J. M. and WEISSKOPF, V. F. (1952) Theoretical Nuclear Physics, Wiley, N.Y.
DAVYDOV, A. S. (1958) Theory of the Atomic Nucleus, Fizmatgiz, Moscow.
DAVYDOV, A. S. (1963) Quantum Mechanics, Fizmatgiz, Moscow (translation published by Pergamon Press,
Oxford, 1965).
EISENBUD, L. and WIGNER, E. P. (1958) Nuclear Structure, Oxford University Press.
LANDAU, L. D. and LIFSHTTZ, E. M. (1963) Quantum Mechanics, Fizmatgiz, Moscow (translation published
by Pergamon Press, Oxford, 1965).
LANDAU, L. D. and SMORODINSKII, Ya. A. (1955) Lectures on Nuclear Theory, Gostekhteorizdat, Moscow
(translation published by Plenum Press, N.Y., 1959).
PRESTON, M. A. (1962) Physics of the Nucleus, Addison-Wesley, Reading, Mass.
SITENKO, A. G. (1971) Lectures in Scattering Theory, "Vysshaya Shkola", Kiev (translation published by
Pergamon Press, Oxford, 1971).
Chapter 1
NUCLEAR FORCES
1.1. Composition of Nuclei and Properties of Nucléons
Principal characteristics of nuclei
Atomic nuclei are characterized by definite masses and electric charges. The charge q of an
atomic nucleus is opposite in sign to the electron charge — e and equal in magnitude to a
multiple of the electron charge:
q = Ze.
The integer Z determines the position of the atom in the periodic table and is called the
atomic number. Nuclei with the same charge but different masses are called isotopes. If the
mass of the most common isotope of carbon is taken equal to 12, the masses of all nuclei are
close to integers. The integer A closest to the value of the mass is called the nuclear mass
number.
According to modern ideas, atomic nuclei consist of protons and neutrons—elementary
particles with approximately equal masses. Nuclear forces act between these particles in
nuclei. The proton has a positive electric charge, equal in magnitude to the electron charge*
and the neutron is electrically neutral. Despite the difference in their electrical properties,
protons and neutrons behave in the same way in nuclear interactions and so are often
brought together under a single name and called nucléons.
An atomic nucleus with charge q — Ze and mass number A consists of Z protons and
N = A—Z neutrons. The mass number A gives the total number of nucléons in the nucleus.
Isotopes are nuclei with the same number Z of protons but with different numbers N of
neutrons. Nuclei with the same number N of neutrons but different numbers Z of protons
are called isotones. Nuclei consisting of the same number of nucléons (that is, having the
same mass number A) are called isobars.
Properties of neutrons and protons
The properties of the individual nucléons have been well studied. The proton mass is
equal to 1836-12 electron masses, or 1-007276 atomic units of mass (a.u.m.); the neutron
mass is approximately 2-5 electron masses greater than the proton mass (the neutron mass
2 LECTURES ON THE THEORY OF THE NUCLEUS [1.1
is 1-008665 a.u.m.). The proton and neutron have spins of the same magnitude, equal to
|-, and obey Fermi-Dirac statistics. The magnetic moments of the proton and neutron
(expressed in nuclear magnetons) are, respectively,
μ = 2-7927, μ„=-1-9131. (1.1)
ρ
These values differ substantially from the values 1 and 0 predicted by the Dirac equation
for particles with spin \. The anomalous values (1.1) of the magnetic moments indicate that
the Dirac description is incomplete for nucléons.
A free neutron is unstable and decays into a proton, an electron and an anti-neutrino :
n -* p+e~ + v. In the decay of the neutron, the excess mass, equivalent to approximately
1-3 MeV, is expended on the formation of the electron (0-5 MeV) and on the kinetic energy
of the particles formed (0-8 MeV). The mean lifetime of a neutron is about 12 min. Although
neutrons are unstable in the free state, together with protons they can form stable nuclei.
On the other hand, the proton in the free state is a stable particle, whereas in a bound state
inside a nucleus it can decay into a neutron, a positron and a neutrino: p -+ n+e+ + v.
Nuclei are stable only for certain relationships between the numbers of neutrons and protons.
If these relationships are not satisfied, neutron or proton decay is possible and proceeds
until the nucleus becomes stable.
The small differences in the masses, the fact that the spins and certain other properties
are the same, and also the possibility of interconversion, permit us to consider the neutron
and proton as two different states of the same particle—the nucléon. We can then assign to
the nucléon, along with its space and spin coordinates, an additional internal degree of
freedom—the so-called charge coordinate or isospin coordinate.
In the non-relativistic energy region, neutrons and protons can be regarded as point
particles; at high enough energies, however, it is necessary to take the spatial structure of
the nucléons into account. The effects of the latter appear, for example, in scattering experi­
ments at sufficiently high energies. The spatial structure of nucléons, like their anomalous
magnetic moments, can be studied quantitatively only within the framework of field theory.
1.2. Nuclear Interaction between Nucléons
Short-range nature of nuclear forces
Between neutrons and protons, the constituent particles of atomic nuclei, specific nuclear
forces act. These forces cannot be reduced to electromagnetic forces, since they appear
irrespective of whether the nuclear particles possess electric charge or are neutral.
The most detailed information on the nature of the nuclear forces can be obtained by
studying a system of two nucléons, since the two-body problem is the simplest and permits
an exact solution. The nuclear forces in a two-nucleon system cause the particles to scatter
each other and also lead to the formation of a complex particle, the deuteron, consisting of a
neutron and a proton bound together. The existence of a bound state of the neutron-
proton system indicates that the nuclear interaction between a neutron and a proton corre­
sponds to attraction between the particles.
The most typical property of the nuclear interaction is the fact that the nuclear forces
1.2] NUCLEAR FORCES 3
are characterized by extremely short range and very great magnitude within this range.
That the nuclear forces are short-range was first postulated by Wigner. On the basis of an
analysis of the small magnitude (2-23 MeV) of the binding energy of the deuteron and of the
large magnitude (28 MeV) of the binding energy of the 4He nucleus, he showed that the
rangeof the nuclear forces must be of the order of 2 X 10~~13cm. If we characterize the nuclear
interaction between a neutron and proton by some potential well of width 2X10~ 13 cm,
then, because of the small binding energy of the deuteron, the effective depth of the well
will be found to be approximately equal to 30 MeV. For comparison, we point out that the
energy of the Coulomb interaction between two protons separated by a distance of 2X
10"13 cm amounts only to 0-7 MeV in order of magnitude.
Other features of nuclear forces
In the non-relativistic limit, when the speeds of the particles are considerably less than
the speed of light, the nuclear interaction does not depend on the velocities of the interacting
particles and can be described by a potential. A distinctive feature of nuclear forces is that
the nuclear interaction potential depends not only on the distance between the particles but
also on the mutual orientation of the spins of the interacting particles. The dependence of
the nuclear interaction on the spin follows directly from experiments on the scattering of
slow neutrons by molecular hydrogen. The fact that the deuteron has an electric quadrupole
moment indicates that the nuclear interaction is non-central, that is, depends on the mutual
orientation of the total spin and the relative position vector of the interacting particles.
Finally, the appearance of polarization of the particles in the scattering points to the fact that
spin-orbit forces play an important role in the nuclear interaction.
It follows from the existence of stable nuclei that an important part of the nuclear interac­
tion between nucléons corresponds to forces of attraction. However, the experimental data
available at the present time on the scattering of nucléons at sufficiently high energies
indicate that the nuclear forces of attraction give way to forces of repulsion at very short
distances (less than 0-4X 10~13 cm). It follows from the data on the scattering of nucléons
at high energies that the nuclear forces also have a partially exchange character, that is, the
nucléons can exchange certain properties (spatial coordinates, spins or charges) in the
interaction.
It follows from the properties of mirror nuclei (nuclei in which the neutrons are replaced
by protons and the protons by neutrons) that the forces acting between two protons are
equivalent to the forces acting between two neutrons (if we exclude the Coulomb interaction
from consideration). This property was given the name of the charge symmetry of nuclear
forces. It was found later that the charge symmetry of nuclear forces is a manifestation of a
deeper symmetry of the nuclear interaction, known as the charge invariance or isotopic
invariance of nuclear forces. The isotopic invariance of nuclear forces means that the inter­
action in any two pairs of nucléons is the same if these pairs of nucléons are in the same
states. All the presently available experimental data on the interaction of nucléons (at both
low and high energies) are in agreement with the postulate of the isotopic invariance of
nuclear forces.
As long ago as 1935 Yukawa, starting from the short-range nature of the nuclear inter-
4 LECTURES ON THE THEORY OF THE NUCLEUS [1.2
action, proposed a field theory of nuclear forces which predicted the existence of particles
with intermediate mass called mesons; these were afterwards discovered experimentally.
However, up to the present time, the meson theory has been unable to explain satisfactorily
all the characteristic properties of the nuclear interaction. The absence of a consistent
theory of nuclear forces prevents the construction of a systematic theory of the nucleus and
nuclear processes. Therefore, the following account will be phenomenological, that is,
will be based essentially on those properties of the nuclear interaction that can be extracted
directly from experimental data.
1.3. Ground State of the Deuteron
Principal characteristics of the deuteron
As we have already observed, a system consisting of a neutron and a proton can be found
in a bound state. The bound state of the neutron-proton system is called the deuteron and is
the simplest compound atomic nucleus. Despite the fact that one of the particles (the
neutron) constituting the deuteron is unstable, the deuteron is characterized by an infinite
lifetime, that is, is a stable system.
The most important characteristic of the deuteron is its binding energy, which is deter­
mined by the difference in mass between the deuteron and the particles forming it. The
deuteron binding energy can be measured from the threshold of the photodisintegration
reaction. The value of the deuteron binding energy found by this method is
s = 2·226±0·003 MeV. (1.2)
Other important characteristics of the deuteron are its spin, magnetic moment and electric
quadrupole moment. The spin of the deuteron is equal to 1. The magnitude of the deuteron
spin can be determined from the band intensities of the spectrum of molecular deuterium.
The magnetic moment of the deuteron is close to the sum of the magnetic moments of the
neutron and proton. The magnitude of the deuteron magnetic moment, expressed in nuclear
magnetons, is
μ = 0-8574. (1.3)
α
The magnetic moment of the deuteron is measured most accurately in experiments on the
deflection of a molecular beam in an inhomogeneous magnetic field.
The electric quadrupole moment of the deuteron is
Q = 2-82Xl0-27cm2. (1.4)
This quantity was found from the fine structure in the radio-frequency magnetic resonance
spectrum of deuterium. Note the relative smallness of the deuteron electric quadrupole
moment. In fact, the magnitude of the electric quadrupole moment must be compared with
the transverse dimensions of the system. If we assume that the dimensions of the deuteron
coincide with the range of the nuclear forces, then even in this case the quadrupole moment
is approximately 50 times smaller than the estimated cross-sectional area of the deuteron.
The small magnitude of the quadrupole moment of the deuteron and the approximate
1.3] NUCLEAR FORCES 5
additivity of the magnetic moments of the neutron and proton in the deuteron imply
that the ground state of the deuteron is almost spherically symmetric. But the ground state
of a system is characterized by spherical symmetry in cases where the interaction is central.
Therefore, it follows from the spherical symmetry of the deuteron ground state that the
nuclear interaction between the neutron and the proton is also almost central.
Ground state of the deuteron in the case of central forces*
We shall study the properties of the ground state of the deuteron, assuming that the
nuclear interaction between the neutron and proton is described by a potential V(r), where
r is the magnitude of the distance between the particles. The Schrödinger equation for the
wave function ip(r) describing the relative motion of the neutron and proton has the form
-^V2+F(r)-^U(f) = 0, (1.5)
where μ = M/2 is the reduced mass (for simplicity, we assume that the masses of the neutron
and proton are the same and are equal to M) and E is the energy of the relative motion.
For the ground state of the system, the energy E is negative and equal in magnitude to
the binding energy, E = — ε. The wave function \p(r) should vanish at infinity and be finite
at r = 0.
In the case of a central interaction, the ground state of the system is characterized by a
zero value of the relative orbital angular momentum, 1 = 0 (S-state). The wave function
for this state is spherically symmetric and can be written in the form
V(r) = u(r)/r, (1.6)
where u{r) depends only on the magnitude of the distance between the particles. Substituting
(1.6) into (1.5), we obtain the following equation for the function u{r):
u"-(M/h2) [V(r)+e]u = 0, (1.7)
where the function u(r) must satisfy the boundary conditions w(0) = w(°°) = 0. The nuclear
interaction is characterized by a finite range r, so that outside the range of the nuclear
0
forces, eqn. (1.7) is simplified:
"-a2w = 0, r >/-(). (1.8)
w
Here we have introduced the notation α2 = Με/ή2.
The solution of (1.8) that vanishes at infinity has the form
= Ce-ar, r>r , (1.9)
M 0
where C is a normalization constant. Since the wave function and its derivative must be
continuous, the solution of eqn. (1.8) in the outer region (r => r ) must match with the
0
solution of eqn. (1.7) in the inner region (r =s r), which means that the logarithmic deri-
0
t The interaction of two nucléons at low energies is treated in detail in the review by Hulthén and Suga-
wara (1957), which contains a detailed bibliography.
LNT 2