Nuclear Chemistry

468 Pages·1968·9.699 MB·English

by  L. Yaffe (Eds.)

CONTRIBUTORS TO VOLUME I
JOHN M. ALEXANDER
JACK M. HOLLANDER
J. HUDIS
T. D. NEWTON
NORBERT T. PORILE
K. S. TOTH
A. ZUCKER
NUCLEAR
CHEMISTRY
Edited hyL. Y AFFE
DEPARTMENT OF CHEMISTRY
MC GILL UNIVERSITY
MONTREAL, QUEBEC, CANADA
Volume I
ACADEMIC PRESS New York and London 1968
COPYRIGHT © 1968, BY ACADEMIC PRESS INC.
ALL RIGHTS RESERVED.
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
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LIST OF CONTRIBUTORS
Numbers in parentheses indicate the pages on which the authors contributions begin.
JOHN M. ALEXANDER (273), Department of Chemistry, State University
of New York at Stony Brook, Stony Brook, New York
JACK M. HOLLANDER (359), Lawrence Radiation Laboratory, University
of California, Berkeley, California
J. HUDIS (169), Chemistry Department, Brookhaven National Laboratory,
Upton, Long Island, New York
T. D. NEWTON (1), Atomic Energy of Canada Limited, Chalk River Nuclear
Laboratories, Chalk River, Ontario, Canada
NORBERT T. PORILE (57), Department of Chemistry, Purdue University,
Lafayette, Indiana
K. S. TOTH (409), Oak Ridge National Laboratory, Oak Ridge, Tennessee
A. ZUCKER (409), Oak Ridge National Laboratory, Oak Ridge, Tennessee
PREFACE
The aim of these volumes is to bring to interested readers areas of topical
interest in nuclear chemistry, written by experts who are actively engaged in
working in these fields. The concept differs from the sequential and historical
approach found in several very good nuclear chemistry texts.
The material found in these chapters should be useful to research workers
and graduate students and should serve as a focus of interest for those whose
areas interact with nuclear chemistry.
In general, each chapter is meant to be self-sufficient and referral to other
chapters, while obviously unavoidable, has been kept to a minimum. Con­
versely application of the above principles has meant that a small unavoidable
amount of repetition has been found necessary. The difference in approach
used by the authors of the chapters has even made, it seems to the editor,
this small amount of duplication useful.
Montreal, Quebec L. YAFFE
November, 1967
CONTENTS OF VOLUME II
7. NUCLEAR FISSION, J. E. Gindler and J. R. Huizenga
8. THE CHEMICAL EFFECTS OF NUCLEAR TRANSFORMATIONS,
A. G. Maddock and R. Wolfgang
9. MODERN RAPID RADIOCHEMICAL SEPARATIONS, Saadia
Amiel
10. ELECTROMAGNETIC SEPARATOR AND ASSOCIATED TECH­
NIQUES, F. Brown
11. COMPUTERS APPLIED TO NUCLEAR CHEMISTRY, David L.
Morrison
12. GEO- AND COSMOCHEMISTRY, Oliver A. Schaeffer
Author Index—Subject Index
XI
Chapter i
NUCLEAR MODELS
T. D. NEWTON1
Atomic Energy of Canada Limited
Chalk River Nuclear Laboratories
Chalk River, Ontario, Canada
I. Simple Properties of Nuclei 1
II. Why Use Nuclear Models? 4
III. Relations between Nuclear Models 5
IV. The Wigner Method 7
V. The Nuclear Shell Model 13
VI. The Unified Model 25
Appendix : Transformations of Collective Coordinates 35
VII. The Optical Model and the Statistical Method 42
References 53
I. Simple Properties of Nuclei
In the Rutherford-Bohr model of an atom the nucleus was simply a particle
containing most of the atomic mass and having a positive charge equal to the
atomic number. This simple concept was used successfully to describe the
early data on scattering of a-particles by nuclei. The scattering results implied
that the volume of a nucleus was proportional to its mass. After the discovery
of the neutron a nucleus was pictured as a spherical collection of neutrons
and protons with a radius R = r A1/3, where r is a constant length and A is
0 0
the nuclear mass number. This no longer suffices.
1 Present address: Department of Mathematics and Statistics, University of Guelph,
Guelph, Ontario, Canada.
1
2 T. D. NEWTON
Scattering of high-energy electrons has provided us with detailed informa­
tion about the distribution of charge in a nucleus (Ehrenberg et al, 1959;
Meyer-Berkhout et al, 1959; Hofstadter, 1963). If we assume that the ratio
of neutron to proton densities is the same throughout a nucleus then we can
determine the mass distribution from the charge distribution. It appears that
for all nuclei larger than 6Li the central density has the same value; one which
implies that the centers of nucléons are, on the average, (1.08 ± 0.01)10~13
cm apart. There is a surface region in which the density decreases from the
central value to zero. The "surface thickness" is usually specified to be the
radial distance in which the density decreases from 90 to 10% of the central
value. The surface thickness is about 2 x 10"13 cm.
In a nucleus lighter than carbon the density decreases more or less smoothly
from the central maximum value to zero. In heavier nuclei, beginning al­
ready with oxygen, the uniform central density extends over a significant
volume.
If one computes a root mean square (rms) radius for these systems one
finds that R JA1/3 is not constant but decreases with A, changing rapidly for
rm
A < 16. On the other hand if one defines the nuclear size by the radius R(£)
at which the density reaches \ the central value then R{\)jA113 increases with
A, again rapidly for A < 16 (see Fig. 1). This merely reflects the fact that a
greater fraction of the mass of a light nucleus lies in the region R > R{\).
The final break with the simple picture of the nucleus occurred when we
found that the nucleus was not spherical. The angular dependence of electron
scattering from nB and 14N indicates that these nuclei have nonspherical
shapes (Meyer-Berkhout et al, 1959). The deformation of nuclei has been
demonstrated in many ways, most commonly by the occurrence of rotational
bands in the energy level spectra.
To summarize, we now have a picture of an assemblage of nucléons,
usually not spherical, containing a central region in which the nucléons are
so closely spaced that their charge distributions overlap and a surface region
in which this density falls gradually to zero. This description arises almost
directly from observation. Some mathematics is needed to interpret the
electron scattering results but no theory of the nucleus need be involved.
Another set of simple nuclear properties can be deduced from the masses
of nuclei in their ground states. The Bethe-Weizsacher formula expresses the
nuclear binding energy E as a simple sum of terms representing the volume,
B
surface, Coulomb, and symmetry energies
E = -aA + a A2/3 + a Z(Z - 1)/T1/3 + 0.25a (N - Z)2A~l. (1)
B x 2 3 4
Green and Engler (1953) obtained a = 16 MeV, a = 18 MeV, a = 100 MeV,
x 2 4
and the fit was good enough to predict at so early a date that in the radius
1. NUCLEAR MODELS 3
formula R = r A1/3, one required r less than the then fashionable value
0 0
1.4 x 10~13 cm (cf. value 1.08 from electron experiments).
The data from ground states of nuclei, as summarized in (1) must be kept
in mind during all nuclear calculations. For example, for a long time shell model
calculations were unable to predict reasonable binding energies. Also formula
1.8
-1/3
A R
1/3
Fig. 1. The equivalent uniform nuclear radius R= [5<r2>/3]1/2 and the radius R. at
05
which the nuclear density is half the central value, each divided by Al/2, are shown as
functions of Al/3. Some of the errors of estimation are shown.
4 T. D. NEWTON
(1) is frequently corrected by a term representing the odd-even mass difference.
This was the first, and very strong, evidence for the effect of short-range
correlations now frequently represented in nuclear theory by the "pairing
force." Departures from the Bethe-Weizsacher formula show the effects of
nuclear shells and of nuclear deformations.
In order to study collections of nucléons without these perturbing effects,
the concept of nuclear matter, i.e., an extended medium supposed to have the
properties of matter in the central regions of nuclei, has been extensively
discussed. These discussions test whether or not various assumed inter­
nucleon forces (including those which fit the two-body data) provide a
(volume) binding energy of 16 MeV at a density corresponding to r = 1.08
0
l(T13cm.
These calculations have been quite successful, and will undoubtedly con­
tinue until our understanding of internucleon forces and our methods of
calculating with them are considerably improved. So far, however, the
methods used for nuclear matter have had only limited success for finite
nuclei, so they cannot yet be said to supply a theory of nuclei.
II. Why Use Nuclear Models?
Experiment has shown what the nucleus looks like and experiment also
tells us quite precisely how pairs of nucléons interact, at least in those energy
regions of interest for the purposes of this chapter, i.e., from the bound state
of the deuteron to, say, 200 MeV. While our knowledge of the interaction
between two nucléons is not yet complete it is established that the data below
200 MeV can be quite adequately represented by a potential function
(Signell and Marshak, 1958; Gammell and Thaler, 1957; Hamada and
Johnston, 1962). The potential is complicated, but can be used.
We believe that for energies of interest for problems of nuclear structure
the equations of ordinary nonrelativistic quantum theory should be valid.
Despite all the information available this chapter is called "Nuclear
Models" and not "Theory of Nuclear Structure." Why do we continue to
use models? The answer is, most simply, that the problem is too hard. All of
our theories avoid the difficulty of a complete solution of the nuclear structure
problem by one means or another. In most cases one makes a set of simplifying
assumptions which defines a model; i.e., an imaginary object which is not a
nucleus, but which we hope may display properties of interest which are the
same as those of a nucleus. We seek to avoid the situation that every calcula­
tion uses a different model.
One fundamental difficulty is that a complex nucleus may not be describ-
able as an assemblage of neutrons and protons, at least if a nucléon is defined