Table Of ContentENCYCLOPEDIA OF PHYSICS
EDITED BY
S. FLUCCE
VOLUME XL
NUCLEAR REACTIONS I
WITH 280 FIGURES
SPRINGER-VERLAG
BERLIN· GOTTINGEN . HEIDELBERG
1957
HANDBUCH DER PHYSIK
HERAUSGEGEBEN VON
s.
FLOGGE
BAND XL
KERNREAKTIONEN I
MIT 280 FIGUREN
SPRINGER-VERLAG
BERLIN· GOTTINGEN . HEIDELBERG
1957
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0
Softcover reprint of the hardcover 1st edition 1957
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Contents.
Page
Nuclear Reactions Levels, and Spectra of Light Nuclei. By W. E. BURCHAM, Ph. D.,
Professor of Physics, Physics Department, The University of Birmingham, Birming
ham (Great Britain). (With 102 Figures)
Introduction. . . . . . . . . . . . . . .
A. Nuclear level schemes for the light nuclei. 2
B. General features of nuclear reactions. . . 13
C. Experimental methods for studying energy levels 23
I. General techniques . 23
II. Elastic scattering . . . . . . . . . . . . 35
III. Radiative transitions . . . . . . . . . . 41
IV. Angular distribution and correlation experiments 54
D. Experimental evidence on nuclear reactions 62
I. Reactions of protons with light nuclei . 62
II. Reactions of neutrons with light nuclei. 97
III. Reactions of alpha particles with light nuclei 106
IV. Reactions of deuterons with light nuclei 116
V. Other reactions. 143
E. Energy level schemes . . . . . . . 152
I. The odd isobars. . . . . . . . 154
+
II. The even isobars of mass 4n 2 172
III. The even isobars of mass 4n 182
General references. . . . . . . 198
Nuclear Reactions, Levels, and Spectra of Heavy Nuclei. By Dr. B. B. KINSEY, Atomic
Energy Research Establishment, Harwell, Berkshire (Great Britain). (With 81 Fi-
gures). . . . . . . . . . 202
A. Introduction . . . . . . . . . . . 202
B. The compound nucleus. . . . . . . 203
C. Statistical theory of nuclear reactions 212
D. Direct interactions. . . . . 220
E. Total neutron cross sections 226
F. Elastic scattering . . . 232
G. Fast neutron reactions. 242
H. Proton reactions 260
1. Deuteron reactions 267
J. Photonuclear reactions. 281
K. Level densities . . . . 296
L. Slow neutron reactions. 302
M. Systematics of low energy states 319
N. Low energy states in deformed nuclei with N > 90 334
O. Energy levels in closed shell nuclei. 356
P. Energy levels of nuclei with N < 90 368
General references. . . . . . . . 372
VI Contents.
Page
Resonance Processes by Neutrons. By JAMES RAINWATER, Ph. D., Professor of Physics,
Columbia University, Hastings-on-Hudson, New York (USA). (With 63 Figures) 373
I. Introduction . . . . . . . . . 373
II. Theory and experimental results 403
References. . . . . . . . . . . . . 444
Nuclear Reactions at High Energies. By A. WATTENBERG, Ph. D., Research Physicist
of the Laboratory for Nuclear Science, Massachussetts Institute of Technology,
Cambridge, Mass. (USA). (With 34 Figures) 450
A. Resume . . . . . . . . . . . . . . . . . . . . . . . . 450
I. Introduction . . . . . . . . . . . . . . . . . . . . 450
II. Information obtainable from high energy nuclear physics. 450
B. Theoretical concepts useful for high energy reactions. 456
C. Experimental techniques of high energy physics. 468
I. High energy machines 468
II. Sources of particles . 469
III. Detection techniques 473
D. Reactions of nucleons . . 481
E. Electron reactions 505
F. Nuclear reactions of X-rays. 510
G. Reactions of mesons 524
General references 537
Sachverzeichnis (Deutsch-Englisch) 538
SUbject Index (English-German) . . 546
Nuclear Reactions, Levels, and Spectra of Light Nuclei.
By
W. E. BURCHAM.
With 102 Figures.
Introd uction.
The study of the nuclear reactions of the light nuclei is not at present one of
the basic investigations of nuclear physics. Already in 1935 attention was being
concentrated on the simpler scattering problems, which would lead to some
knowledge of the forces between nucleons. At present, emphasis is placed in
creasingly on attempts to interpret the inter-nucleon force in terms of a quantised
field theory in which ;7/;-mesons are the quanta. The behaviour of the meson
nucleon system and the role played by new unstable particles in field theory
have therefore become the central questions of nuclear physics. At first sight
it might appear that the interactions of relatively complex systems of nucleons,
moving with non-relativistic velocities, could contribute little to an answer to
these questions. This is however not so, a[ least in so far as the force between
nucleons is concerned. Although important properties of this force can be inferred
from the two-body system and from the knowledge that nuclear forces show
saturation, some properties, such as charge independence, do seem to be best
exhibited in the existence of nuclear charge multiplets. The existence of non
central couplings is perhaps basically a matter for high energy experiments,
but such properties are also revealed in nuclear level splittings and in the se
quence of single particle states of the nuclear shell model, and the possibility
of many-body forces can only be investigated through a detailed knowledge of
many-body systems. These reasons, apart from the general interest of the subject
as a field for experimental endeavour, justify the effort still being expended in
investigating the behaviour of light nuclei, and in tabulating their excited states.
Both the theoretical and the experimental study of the excited states of
nuclei have been much helped by progress in the understanding of atomic struc
ture. Although the nucleus and the atom are held together by forces of a fun
damentally different character, there has been no reason to doubt that quantum
mechanics provides a satisfactory description of most of the properties of both
these systems. For this reason there are notable resemblances between atoms
with a size of 10-8 cm and nuclei, whose extent is 105 times smaller. The boundary
conditions for the two systems both lead to a set of discrete energy levels above
the ground state, in which atoms or nuclei may exist for a finite time. Transitions
of the atom or nucleus between two of its quantised energy levels may be accom
panied by emission or absorption of radiation, and the radiation field is coupled
to the multipole moment of the radiating system in the same way, despite the
great disparity in size. It is true that the difference in origin of the fields of
electric and nuclear force leads to corrections to such properties as magnetic
moments which are small for electrons and large for protons, but the general
similarity between more complex atoms and nuclei as radiating systems exists.
Handbuch der Physik, Ed. XL.
2 W. E. BURCHAM: Nuclear Reactions, Levels, and Spectra of Light Nuclei. Sect. 1.
The periodic classification of the elements, based on the quantum theory
of the atom and the PAULI exclusion principle has clearly demonstrated the
existence of a shell structure in atoms. Although the relatively simple atomic
type of structure with electrons moving round an attracting centre does not
exist in nuclei it has recently become clear that many nuclear properties do
exhibit that marked periodicity with atomic weight which suggests such a struc
ture. .The reconciliation of these discoveries with the earlier ideas of BOHR,
which envisaged the nucleus as a closely packed system of strongly interacting
particles has been a major concern of theoretical nuclear physics since 1950.
It is a matter to which the detailed experimental study of nuclear states can
contribute.
The experimental methods used in the study of atomic states are only in
frequently of direct use in nuclear physics owing to the different scale of energies
involved, but the general collision theory developed to describe the bombardment
of atoms by electrons is of basic importance in nuclear dynamics. The classi
fication of such collision processes into elastic and inelastic types is also directly
relevant. Since however the apparatus of nuclear physics is generally more
complex and often different in conception from the equipment used in atomic
spectroscopy, and since it often influences very markedly the type of information
which may be sought, the proper appreciation of experimental results demands
an account of the methods by which they have been obtained. This is given
in this article in Part C.
In Part A an attempt is made to pick out those general properties of inter
nucleon forces which may be expected to influence the structure of complex
nuclei in a way that is susceptible of experimental study. A brief account is
given of the theoretical approach to an interpretation of discrete level schemes
and of the consequences of particular coupling schemes for angular momentum
vectors in a complex system. Part B reviews the properties of the excited states
of a complex nucleus with particular reference to their formation and decay in
the course of a nuclear reaction. Part C is devoted to the experimental methods
which have been most successful in revealing the properties of nuclear states.
In Part D the experimental results obtained from nuclear reactions (up to July
1955) are reviewed according to reaction type. Only the main results are given,
with emphasis on those which may be interpreted theoretically. Fuller details
are given in [32J and [33J. Nuclear energy level systems and reaction constants
for particular levels are reviewed according to mass number in Part E. The
review extends to the end of the nuclear d-shell at mass number A = 40.
Similar treatments of the general subject of the excited states of light nuclei,
including both experimental and theoretical aspects, are given in [1] to [4].
A. Nuclear level schemes for the light nuclei.
1. The interaction between nucleonsl, Detailed calculations of nuclear struc
ture and in particular the estimation of the energies of excited states, must be
based on knowledge of the force between nucleons in a nucleus. Since no ex
periment gives directly the properties of this force, it is customary to make cer
tain definite assumptions and to see how far experimental results can be so ex
plained 2. The first assumptions, against which there is no evidence, are that the
1 Details on this problem are given in HULTHEN and SUGAWARA'S article, Vol. XXXIX
of this Encyclopedia.
2 R. E. PEIERLS: Sixth RUTHERFORD Lecture: The atomic nucleus and its constituents.
Proc. Phys. Soc. Land. A 66, 313 (1953).
Sect. 1. The interaction between nucleons. 3
force is of a two-body type and attractive, that it is static, i.e. that no terms depend
ing on nucleon velocities enter into the potential energy, and that it is mainly
of a central character, so that the potential energy can be written
(1.1)
where r is the distance between nucleons 1 and 2. The main evidence as a result
12
of which these assumptions may be modified and supplemented comes from a
study of the two body problem and from observation of regularities in nuclear
structure. In the first place it is known from scattering experiments that the
force is of short range; in contrast with the long range COULOMB forces familiar
in the atom, the force between nucleons vanishes if r is greater than a few
12
times 10-13 cm. Since the force is assumed to be attractive 1 all the nucleons of
a complex nucleus might be expected to come so close together that the binding
energy of a nucleon to a nucleus would increase with the number of particles.
This is not so, since the binding energy and nuclear density seem to be nearly
independent of mass number. The generally accepted explanation of this phenome
non of saturation is that the force between two nucleons is partly of an exchange
character, and that attraction only arises when the pair of interacting particles
can exchange position, or spin, or both. Since the PAULI principle limits the
number of particles with which exchanges can take place, such forces will show
saturation. The exchange nature of the force also provides an explanation of
the angular distribution observed in the scattering of high energy neutrons by
protons.
The exchange properties are included in the nucleon-nucleon interaction by
writing
(1.2)
where the operators J!., Fa, ~ interchange respectively position, spin and charge
co-ordinates in nucleon wave functions on which l-i2 may operate. The coeffi
cients W, M, B, H (WIGNER, MA]ORANA, BARTLETT, HEISENBERG) determine
the precise nature of the force; a frequently used set of values, chosen to ensure
saturation, is given in [8], p.234.
The inclusion of the spin exchange terms Band H in the two-body inter
action is sufficient to interpret the difference in energy between the 35 and 15
states of the deuteron, which is clear evidence of the spin dependence of the
neutron-proton force. This property is of considerable importance in determining
the spin of the ground state of more complex nuclei, particularly of mass number
4n+2. The general interaction (1.2) is charge-independent in the sense that it
postulates the same interaction between all pairs of nucleons in corresponding
states. In such states it must be possible for a neutron to be changed into a
proton or vice versa without violating the PAULI principle. It follows that the
IS state is possible for the (PP) (nn) and (np) systems, while the 35 state is only
possible for the (np) interaction. It is customary to label states of motion of
the two nucleon system in which all three pairs of nucleons may take part as
states of isotopic spin T = 1, and those describing only the neutron-proton
system as states of isotopic spin T = O. This rather formal classification 2 has been
found useful in comparing the energy levels of isobaric nuclei. The charge inde
pendence of nuclear forces receives considerable direct support from low energy
nucleon scattering experiments, according to which the singlet intrinsic range
1 There is evidence from high energy experiments that the internucleon force becomes
repulsive at very small distances.
2 The term isobaric spin is also frequently used.
1*
4 W. E. BURCHAM: Nuclear Reactions. Levels. and Spectra of.Light Nuclei. Sect. 2.
for both the (np) and (PP) scattering appear to be about the same. Charge
independence follows very readily from the symmetrical meson theory of nuclear
forces. Deviations of nuclear behaviour from the consequences of strict charge
independence can probably be ascribed to the effect of the charge dependent
COULOMB forces.
The central potential ~ 2 is known to be only a first approximation because
of the existence of an electric quadrupole moment in the deuteron. This is most
easily explained by a tensor interaction according to which the force between
nucleons depends on the relative orientation of the spin and orbital vectors.
Such a force may considerably modify the wave functions obtained with a pure
central potential and the consequences for nuclear theory in general have not yet
been fully worked out. It may be found that the phenomenological spin-orbit
interaction
H = a(l.s)
which is found necessary to explain doublet splitting in nuclear spectra, and
possibly also the 15 -35 splitting in the deuteron are consequences of the tensor
force.
Any general theory of nuclear structure suitable for comparison with experi
ment should include the properties of nuclear forces so far discussed in successive
approximations. The first general discussion along these lines of the level systems
of complex nuclei was given by WIGNER and his collaborators.
2. The WIGNER approximation. IX) Classification of states. In the general
classification of nuclear states proposed by WIGNER ([8J, p. 199, and [9J) it
assumed that nuclear forces are independent of both charge and spin. The
coefficients Hand B in Eq. (1.2) are put equal to zero, so that the exchange
part of the force is of the MAIORANA type calling for exchange of spatial positions
of nucleons. The quantum numbers used to specify a nuclear state are based
on conservation laws. The most general are (d. [8J, p. 199).
(a) J, the total angular momentum quantum number, related to the total
angular momentum vector J by the equation
(2.1)
±
(b) 'Tt, the parity, which must be 1, i.e. even or odd, because of the invariance
of nuclear structure with respect to reflection of the wave function through the
origin of coordinates.
(c) T. the third component of total isotopic spin which is defined as
T.=~LT! =t(Z -N) = tA-N (2.2)
•
and measures the excess of protons over neutrons in the nucleus. Z is the atomic
number, A the mass number and N the neutron number. T. is the third component
of the isotopic variable T which is used, following HEISENBERG, to label the charge
state of a nucleon. Here 7:. = 1 is taken to mean a proton and 7:. = -1 a neutron.
These quantum numbers are related directly to the properties of space and
time and to the conservation of charge, and have definite values for a nuclear
state (J, ±, T.). Other quantum numbers which are valid under special circum
stances are:
(d) T, the total isotopic spin quantum number which may be defined if nuclear
LT'
forces are charge independent, and is related to a formal vector T = ~ of
total isotopic spin by the equation
+
I TI = VT(T 1). (2.3)
Sect. 2. The WIGNER approximation. 5
+
For a given T, T. has 2 T 1 values each corresponding to the same binding
energy and defining a set of isobaric nuclei of different charge in each of which
this particular energy state may be identified. This group of nuclear states is a
charge-multiplet.
(e) Land 5, the quantum numbers for total orbital and total intrinsic angular
momentum. The corresponding vectors add to make the total angular momentum
J of a nuclear state:
J=L+S. (2.4)
If the interaction between spin and orbital motion may be neglected, Land 5
are good quantum numbers and J may have 25+ 1 values, between IL - 5 I and
+
L 5. These states form a spin multiplet. The total spin 5 is compounded from
the spins of individual nucleons by the summation S = Ii2:..'.: :":; '-(J; and L is compounded
•
from the orbital vectors. These orbital vectors are conveniently obtained from
a simple shell model for a given number of nucleons and the directions of the spin
vectors are then determined by application of the PAULI principle.
In the WIGNER theory, the assumption of charge and spin independence of
the nuclear forces implies that the states of charge and spin multiplets do not
differ in energy. A group of degenerate nuclear states, including perhaps several
charge and spin multiplets is known as a supermultiplet. The states of a super
multiplet have all the same orbital angular momentum and parity, since these
are related to the spatial part of the wave function. The charge and spin multi
plets are inter-related through the exclusion principle, and together determine the
symmetry of the spatial part of the wave function with respect to exchange
of nucleons.
The symmetry properties of the spatial wave functions of a WIGNER super
multiplet cannot be simply described in the many-body case, because the function
may be symmetric for spatial exchange of some pairs of particles and antisym
metric for exchange of other pairs. The supermultiplet may be characterised
by four numbers AI' A2, la, A4, with L ),;=A the mass number, representing
;
respectively the number of neutrons and protons wIith spin "up" and spin
" down". An equivalent representation is in terms of the numbers
+
P = ~ (AI - A2 Aa - A4),
P' = t (AI + A2 - A3 - A4) , (2.5)
pIt = ! (AI - A2 - A3 + A4)
which are respectively the maximum values of 5., T., and y. (=! L a~"l"~) in the
supermultiplet. ;
The individual states of a given supermultiplet (P, P', pIt) are thus classified
by writing down the permitted spin-charge multiplets (5, T) or (T, 5) and
combining with these the orbital momentum allowed for the component particles;
the individual multiplet symbol is then (2T+1l(2S+1lL.
The states of n equivalent particles with orbital angular momentum l may be
classified by group theoretical methods. The technique and results for the s,
p, d and 1 shells are reviewed by FLOWERS [11].
(J) Calculation 01 binding energies. In the WIGNER approximation the binding
energy of a nuclear state for a given nuclear interaction depends on the symmetry
properties of the spatial part of the wave function P of the supermultiplet to
which the (degenerate) state belongs. The potential energy jP*VPdv for s
