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OCR for page 37
Nuclear Structure ant!
Dynamics
The modern era of nuclear physics began with the surprising
revelation that, despite the violent forces that are present in the
nucleus, the nucleons can for the most part be considered to be moving
independently in a single, smoothly varying force field. This is the
conceptual basis of the shell model, which is the foundation for much
of our quantitative understanding of nuclear energy levels and their
properties. In this model, individual nucleons are considered to fill
energy states successively, forming a series of nuclear shells that are
analogous to the shells formed by electrons in the atom.
At the simplest level, the shell model predicts that nuclei having
closed (completely occupied) shells of protons or neutrons should be
unusually stable- as is, in fact, observed. (The chemical analogy is the
noble gases, in which all the electrons are in closed shells.) If a nucleus
has one nucleon beyond the closed shells, many of the properties of the
nucleus can be attributed to that one nucleon just as the chemistry of
sodium can be explained largely in terms of the sodium atom's single
valence electron.
The shell model has been developed to incorporate the residual
. . · . . .
forces among the nucleons that are not included in the smooth field.
This has evolved to a valuable tool for understanding and predicting
many of the energy levels and their properties, such as electromagnetic
interactions and decay rates. However, the shell model with interac-
tions can be computationally difficult or impossible, depending on the
37
OCR for page 38
38 NUCLEAR PHYSICS
number of nucleons and the number of shells that the nucleons move
in.
Under such circumstances, or when a simpler description is needed,
other models have enjoyed considerable success. The liquid-drop
model depicts the nucleus as a drop of liquid having such familiar
properties as pressure and surface tension. This model has been useful
in systematizing the data on binding energies and in providing useful
qualitative pictures of vibrations and the process of nuclear fission. An
important feature of the liquid-drop model is the collective motion of
many particles, which is often observed in the properties of nuclear
levels.
Another simplified model is the interacting boson model. Here
nucleons spanning many shells are thought to combine to form
even-numbered nucleon clusters (which have integral values of spin
and can therefore be regarded as bosons), which can be studied by the
application of symmetry principles. For many of these models, it is
possible to make the connection with the more fundamental but more
complicated shell-model description.
Experimentalists study nuclear structure by determining what en-
ergy states appear in a given nucleus and what states play a role in
particular nuclear reactions. In the early days of nuclear physics,
experiments were restricted to the states involved in the decay of
naturally occurring radioactive nuclides or in a few low-energy reac-
tions that could be carried out with alpha particles emitted by radio-
active minerals. The advent of accelerators greatly increased the
number of nuclear states that could be excited, by making available
new projectile species having a wide range of precisely controllable
bombarding energies. Electrons, protons, light ions, and heavy ions
can be supplied by acceleration acting on the projectile's electric
charge. Furthermore, secondary beams of neutral (uncharged) projec-
tiles-for instance, photons and neutrons can be produced in primary
reactions, a technique that can also supply exotic projectiles such as
plans and even neutrinos. In fact, intense pion beams have become a
standard tool of nuclear-physics research during the past decade.
A great many nuclear states have thus become accessible, partly
because the number of excited states increases with increasing energy
above the ground state and partly because the interactions of different
projectiles cause different types of internal nuclear motions to be
excited. For example, highly charged heavy-ion projectiles can exert
powerful Coulomb (electric) forces on the protons of a target nucleus
(a process called Coulomb excitation) while remaining well outside the
OCR for page 39
NUCLEAR STR UCTURE AND D YNAMICS 39
range of nuclear forces. Thus, the effects of Coulomb excitation can be
studied with no interference from unwanted nuclear interactions.
The ability to excite certain types of nuclear motion selectively has
become an even more important tool in nuclear-structure studies over
the past decade. The following sections discuss some excitation modes
of current interest and the kinds of information that they provide on
nuclear structure and dynamics.
ELEMENTARY MODES OF EXCITATION
Extreme limiting cases, in which one type of behavior overshadows
all competing effects, are often the easiest to deal with in physics.
Nuclear physicists have therefore concentrated much of their attention
on excited states corresponding either to the shell model, at one
extreme, or to the liquid-drop model, at the other. In the first case, the
excitation is designed to alter the motion of only one nucleon, while the
remaining core nucleons remain essentially unaffected, so that the
excited states generated can be related to the motion of just the one
nucleon. In the second case, the excitation requires all the nucleons to
"forget" their individual motions and to participate in an overall
coherent motion, much as a milling school of fish, when frightened,
suddenly darts away in a single direction. Both of these modes of
excitation are amenable to experiment and theory and give unique
views of the behavior of the nuclear many-body system.
The collective motions of nuclei include rotations and internal
vibrations. Collective rotations occur only in deformed, nonspherical
nuclei and entail the coherent swirling of some nucleons around a
motionless inner core. Collective vibrations can occur in any nucleus
and are somewhat akin to the comnle.x healing Of ~ uu~tPr_fill-A hell
that is being shaken.
Tom ;_ ~.~1~
~4~^ ~^ ~MA BAIT V~IVV11
1 11~ lilV[l~ll Q1 1lU~l~Ullb in three-dimensional space, however, is not
the only way collective modes can arise. The direction of the spin axes
of several nucleons may flip back and forth in concert after an
excitation. Because a nucleon's magnetic field lies parallel to its spin
axis (similar to the alignment of the Earth's magnetic field with the
polar axis), a spin-ffip collective mode gives the nucleus an oscillating
spin direction and therefore an oscillating magnetic field. In a related
collective mode called the Gamow-Teller resonance, the excitation
flips the isospin (causing a proton to change to a neutron, or vice versa)
as well as the spin. These spin-flipping and isospin-flipping modes have
both recently been observed unambiguously in actual nuclei, as
discussed later in this chapter. These modes make up a new class of
OCR for page 40
Giant Electric Resonances
40 NUCLEAR PHYSICS
excited states that gives some insight into how the interaction between
two nucleons is modified by the presence of neighboring nucleons. The
discovery of these modes has stimulated the development of nuclear-
structure theory.
.
In the late 1940s, physicists studying neutron-emission reactions
caused by bombarding nuclei with gamma rays were startled to find a
large peak- a resonance in the curve of the reaction cross section
(the probability of reaction) when it was measured over a wide range of
gamma-ray energies. This peak represented a value typically 50 to 100
times greater than those of the cross sections for neighboring ener-
gies truly a giant resonance. The gamma-ray energy of the peak was
found to decrease systematically with increasing thass number, from 23
MeV in carbon to 14 MeV in lead.
The giant resonance is a general characteristic of the nuclear
many-body system and does not depend on the detailed structure of a
particular nuclide. It is now recognized as a giant electric dipole
vibration caused by collective motion in the nucleus: the oscillating
electric field associated with the gamma ray induces the protons in the
nucleus to oscillate. The neutrons, being uncharged, do not respond to
an electric field, so a vibration is set up in which the center of electric
charge (due to the protons) oscillates with respect to the center of
mass, as shown schematically in Figure 2.1. Classically, this type of
linear charge oscillation is described as an oscillating electric dipole-
hence the name of the phenomenon. The peak in the cross-sectional
curve is caused by an amplifying resonance between the oscillation
frequency of the gamma ray's electric field and the natural frequency of
the dipole oscillation in the target nucleus.
The maximum possible probability for a nucleus to absorb a gamma
ray can be calculated from very general considerations and is ex-
pressed as a sum rule involving a sum over all the nuclear charges and
masses. The observed probability for absorption of the gamma rays at
resonance energies is nearly equal to the theoretical maximum from the
sum rule for electric dipole oscillations strong evidence that essen-
tially all of the protons take part in the collective motion.
The giant electric dipole resonance peak extends over a width of 3 to
7 MeV in energy, depending on the nucleus. This is a relatively wide
peak, and wide peaks generally correspond to short lifetimes. The giant
electric dipole oscillation is estimated to go through only a few
OCR for page 41
NUCLEAR STR UCTURE AND D YNAMICS 41
FIGURE 2.1 The giant electric dipole vibration, as described in the text. The relative
motions of the protons (dark circles) and neutrons (light circles) during the intermediate
stages of the vibration are indicated by the arrows. (After G. F. Bertsch, Scientific
American, May 1983, p. 62.)
complete cycles before it dissipates, corresponding to a lifetime of
roughly 10-2~ second.
For about 25 years, the electric dipole resonance remained the only
known giant vibrational mode. As the above description implies,
gamma rays are efficient at exciting only linear dipole vibrations;
vibrations corresponding to more complex patterns (multipoles) are
best studied with other means of excitation. Experimentalists therefore
turned to the inelastic scattering of charged particles from nuclei, in
which the projectile retains its identity but deposits some of its energy
in the target. In the early 1970s, a group in Darmstadt, West Germany,
using inelastic electron scattering, and a group at Oak Ridge National
Laboratory, using inelastic proton scattering, both found clear evi-
dence for a giant electric quadrupole resonance. Here the protons and
neutrons move together in a quadrupole vibration, in which the center
OCR for page 42
42 NUCLEAR PHYSICS
of charge and the center of mass do not move, but the distributions of
charge and mass change rhythmically as the nucleus oscillates between
a prolate (football) shape and an ablate (doorknob) shape.
Later, the inelastic scattering of alpha particles was found to be
particularly efficient at exciting the giant quadrupole vibration. This
technique provides a particularly handy tool, because the necessary
100- to 150-MeV alpha-particle beams are available at many cyclotrons
and because the scattered alpha particles are easy to detect. Use of the
alpha-particle excitation has established the energy peak, the energy
width, the strength, and some of the decay modes of the giant electric
quadrupole resonance for a wide range of nuclei. The resonance tends
to appear at 10 to 20 MeV above the ground state and has a width
between 2 and 8 MeV, depending on the nuclide. The sum rule
appropriate to quadrupole vibrations indicates that nearly all of the
nucleons in heavy nuclei take part in the collective motion.
Unlike gamma-ray absorption, which excites dipole vibrations se-
lectively, the inelastic scattering of charged particles can excite several
vibrational modes. To disentangle the individual vibrational patterns
from the measured angular intensities of the scattered particles,
physicists exploit the fact that each multipole is associated with a
definite integer value L of angular momentum (L = 1 for dipole, L =
2 for quadrupole). Thus, the particles scattered during the excitation of
a particular multipole vibration show an angular pattern characteristic
of the L value; the experimental data usually have to be analyzed as a
sum of several different angular patterns from different resonances.
The giant monopole vibration L = 0 is a breathing mode in which the
nuclear volume expands and contracts symmetrically, as Figure 2.2
illustrates. Discovering the giant monopole resonance experimentally
was not easy. It is generally masked by the quadrupole resonance
except at very small scattering angles, where the detector system must
be carefully designed to avoid false counts from the intense beam of
undeflected projectiles. In 1977, a group at Texas A&M University
identified the giant monopole resonance with certainty by studying
inelastic alpha scattering at angles as small as 3° from the projectile
beam direction. The monopole mode was recognized by its unique
small-angle scattering pattern. Further evidence came from the mono-
pole sum rule, which was satisfied essentially fully by the observed
scattering intensity, as would be expected for a collective mode in
which all the nucleons are taking part.
The monopole vibration is particularly important because its fre-
quency is directly related to the compressibility of nuclear matter, a
heretofore unmeasured property. The value for the compressibility
OCR for page 43
NUCLEAR STRUCTURE AND DYNAMICS 43
FIGURE 2.2 The giant monopole vibration, as described in the text. As the protons
(dark circles) and neutrons (light circles) move in and out from their equilibrium
positions, the nucleus "breathes," and its density oscillates. (After G. F. Bertsch,
Scientific American, May 1983, p. 62.)
derived from measured monopole vibration frequencies turns out to be
in good agreement with values predicted by various theoretical models.
To gain an appreciation of the extraordinary differences between
nuclear matter and ordinary atomic matter, it is worth noting that the
latter is about 1022 times more compressible, i.e., all ordinary matter is
almost infinitely soft by comparison.
Preliminary experimental evidence exists for giant multipole reso-
nances of higher L values, such as the pear-shaped octupole vibration
L = 3. Heavy ions might be especially suitable projectiles for exciting
vibrations with large L values, because such massive ions can transfer
a large amount of angular momentum to a target nucleus. Also,
variations on monopole or quadrupole vibrations are possible in which
the neutrons and protons move in opposition rather than together.
Such out-of-phase vibrations have not yet been explored systemati
OCR for page 44
44 NUCLEAR PHYSICS
cally, but there is recent evidence that the monopole mode is selec-
tively excited in reactions that transfer charge between a projectile
pion and the target nucleus.
In fact, the pion has turned out to be an efficient indicator of the
relative roles of protons and neutrons in nuclear excitations. Both
positive and negative pion beams can be focused on a target. Positive
plans in a certain energy range interact with target protons almost ten
times more strongly than with target neutrons; the reverse is true for
negative pions, which interact much more strongly with target neu-
trons. Direct comparison of the results obtained with these two probes
thus yields a measure of the relative importance of the protons and
neutrons in a particular nuclear vibration. Some excited states in light
nuclei, for example, have been shown to be essentially pure proton or
pure neutron excitations. Even when the differences between the target
protons and neutrons are much smaller, as in the giant quadrupole
vibrations in heavy nuclei, they can be detected through positive and
negative pion scattering. This technique thus provides a sensitive test
of the microscopic theory of nuclear vibrations.
/
Giant Spin Vibrations
In addition to vibrations involving the motion of nucleons, nucleon
spins can also exhibit collective behavior. A nucleon has a built-in "bar
magnet" along its spin axis, so a collective mode for spin is also a
collective mode for magnetism. Nucleons have spin 1/2, and, according
to quantum mechanics, the nucleon spin measured along a coordinate
axis can be only + 1/2 (spin oriented parallel to the axis) or -1/2 (spin
antiparallel). Under certain conditions, the spin of a nucleon can flip
between + 1/2 and -1/2, simultaneously reversing the direction of the
magnetic field that it produces.
Researchers at the Indiana University Cyclotron Facility, using
proton beams of 100 to 200 MeV, were recently able to flip the spin and
isospin of nucleons in the nucleus without upsetting the spatial
arrangements of the nucleons. Thus, they were able to excite Gamow-
Teller resonances without obscuring them with other forms of excita-
tion. The trick is to observe a neutron coming out of the nucleus in
exactly the same direction in which the proton entered. The neutron
has nearly the same velocity as the proton, so the law of conservation
of momentum tells us that hardly any momentum was transferred to the
nucleus. Hence, the only change inside the nucleus is that a neutron
changed to a proton, and possibly its spin flipped. In experiments now
OCR for page 45
NUCLEAR STRUCTURE AND D YNAMICS 45
being carried out, the spins of the proton and the neutron are actually
measured.
It is a simple matter to count the number of neutrons that are
available to be changed into protons in the nucleus. Then the total
probability of the Gamow-Teller process for a nucleus relative to the
process for a free neutron can be calculated with great accuracy. A
surprising result of the measurements is that the actual total probability
is only 50 to 75 percent of the calculated probability. One possible
explanation for the strength shortfall is that the transition from a
neutron to a proton is not the elementary process. Rather, we should
consider that the nucleons are made of quarks and that the elementary
Gamow-Teller process is a spin-isospin flip of one of the constituent
quarks. The quark flip can indeed change a neutron into a proton, but
it can also change a neutron to a higher-energy configuration called a
delta resonance (which is a baryon resonance). In this model, the delta
states must also be counted in the total transition probability. Then,
possibly, the strength will come out right. Complete calculations on
this model have not yet been done, and the missing strength problem
has not been resolved.
A Michigan State University-Orsay collaboration working at Orsay,
France, has identified a component of the Gamow-Teller excitation in
which the charge of the nucleus remains the same; according to isospin
symmetry arguments, such an excitation should exist. The measure-
ment had to be made as close to the beam direction as possible, with
the best possible discrimination between the beam and the scattered
particles, which had similar energies. The experimental solution was to
use an extremely precise magnetic spectrometer that could identify the
scattered protons and operate close to the beam.
Deltas in Nuclei
One interesting aspect of the Gamow-Teller resonance arises from
the possible importance of the delta resonance in this low-energy
phenomenon. Deltas are high-energy excited states of the baryon. The
first (lowest-level) such state has a mass of 1.23 GeV, compared with
0.94 GeV for a nucleon, and this great excess of mass-energy causes it
to decay (into a pion and a nucleon) even before it has traversed the
diameter of the nucleus. With such a short lifetime, the delta is not
regarded as a true particle, and yet it can play a crucial role in nuclear
phenomena.
The importance of the delta in nuclear physics has become clear
during the last decade, mostly in experiments with pions. When a pion
OCR for page 46
46 NUCLEAR PHYSICS
with an energy of several hundred MeV collides with a nucleus, one of
the nucleons may absorb the pion to become a delta. This transforma-
tion creates a vacancy, or hole, in the energy state originally occupied
by the nucleon. The progress of the reaction is then determined by the
dynamics of the delta-hole system as it propagates through the nucleus.
A comparison of predictions based on this mechanism with experi-
ments on pion-nucleus reactions (carried out at meson factories such as
the Los Alamos Meson Physics Facility) casts light on several phe-
nomena of current interest, e.g., modification of the delta lifetime and
mass by the nuclear environment, the nature of pion absorption by
nucleons, and the nature of the delta-nucleon interaction. It is surpris-
ing that one can even think about the average potential seen by such a
short-lived particle inside the nucleus. And yet experiments can be
interpreted to show that the delta is substantially less bound than a
nucleon in the center of a nucleus, whereas the elective spin-
dependent potential for a delta is comparable with that for the nucleon.
Study of the propagation of other baryon resonances in nuclei is just
· . .
beginning.
Electron-Scattering Results
There are several reasons why the scattering of high-energy elec-
trons is a powerful tool for studying nuclear structure. First, the
interaction is electromagnetic and thus more readily understood. (The
weak part of the electroweak interaction plays a significant role only if
one looks directly at its unique effects, for example, in an experiment
that exhibits parity violation.) This implies that the experimental
results have a direct interpretation in terms of the quantum-mechanical
structure of the nuclear target. (By contrast, it is often difficult to
separate the reaction mechanism from the target structure in hadronic
scattering of strongly interacting particles.) Of course, these comments
also apply to photon scattering, but a second great advantage of
electron scattering is that, for afixed nuclear excitation energy, one can
vary the momentum transferred by the scattered electron to the
nucleus and map out the charge and current densities, even in the deep
interior of the nucleus. Thus an electron accelerator is, in effect, a huge
microscope for studying the spatial distributions of charges and cur-
rents inside a nucleus, which has a typical diameter of 10- 13 cm. To see
smaller and smaller distances, we require higher and higher momentum
transfer, which implies higher and higher electron energies.
The charge density in the nucleus arises from the proton distribution.
One part of the current arises because of the motion of the protons.
OCR for page 47
NUCLEAR STRUCTURE AND DYNAMICS 47
Both the neutron and proton have a small magnetic moment, and hence
each behaves like a small magnet. This intrinsic magnetization also
contributes to the electromagnetic interaction of electrons with the
nucleus. In addition, there are exchange currents present in the
nucleus due to the fleeting presence of virtual pions and other charged
mesons.
Another feature of electron scattering allows us to obtain a nuclear
excitation energy prof le by varying the momentum transferred to the
target. At low momentum transfer, the spectrum is dominated by
electric dipole transitions. At high momentum transfer, however,
transitions that require a high angular momentum may take place, and
it becomes possible to investigate high-spin states. Furthermore,
because the interaction of the electron with the intrinsic magnetization
is enhanced at high momentum transfer and large electron scattering
angles, it is possible to examine high-spin states of a magnetic
character.
Finally, at the very high energy and momentum transfers that are
obtainable at the Stanford Linear Accelerator Center (SLAC), it has
been possible to study small distances in the nuclear system and to see
the pointlike quarks inside the protons and neutrons.
We clearly cannot touch on all the recent advances In electron
scattering from nuclei. Instead, we will briefly discuss two examples.
Elastic charge scattering of electrons from nuclei makes it possible
to measure the detailed spatial distribution of the charge inside the
nucleus in its ground state. Our most precise knowledge of the sizes
and shapes of nuclei comes from such experiments. The basic process
is analogous to what is observed when light passes through a small
circular aperture: the waveless from each part of the aperture interfere
with each other and produce a diffraction pattern consisting of rings of
varying light intensity that can be observed on a screen. Since a basic
hypothesis of quantum mechanics is that electrons also possess wave
properties, a diffraction pattern (of a somewhat different kind) is
observed when an electron is scattered by a nuclear charge distribu-
tion.
.
To see the details of this charge density due to nuclear orbits and
shells requires measuring the scattered electron energies to better than
1 part in 20,000, a precision unattainable 10 years ago. Today,
spectrometers with the necessary energy discrimination are in use,
notably at CEN Saclay (FranceJ and at the MIT Bates Accelerator
Laboratory. In Figure 2.3 we show an example of a diffraction pattern
of scattered electrons obtained with a calcium-40 target. Such data can
be used to make accurate maps of the spatial distributions of charge in
OCR for page 56
56
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OCR for page 57
NUCLEAR STR UCTURE AND D YNAMICS 57
outer part of the nuclear force just balanced the repulsive inner part of
the nuclear force, the repulsive Coulomb force, and the repulsive
centrifugal force that arises when two nuclei revolve around each
other. Because of the way the forces vary with distance, such a balance
may not be possible for most nuclei, and even if it were achieved, it
would not be expected to last long. If the attractive force outweighed
the repulsive forces, the nuclei would crash together; if the attractive
force were too weak, they would fly apart.
According to the uncertainty principle, the narrowness of the
resonances in the reaction of two carbon-12 nuclei suggests lifetimes
between 10-2~ and 10-22 second for these states. Although this is
unimaginably brief on the macroscopic time scale of the everyday
world, it is several times longer than the interaction time in ordinary
nuclear reactions- long enough for a nuclear molecule to make many
rotations about its center of mass.
Deep-Inelastic Collisions
The compound-nucleus picture of reactions has been used success-
fully in nuclear physics for a long time, because compound-nucleus
formation is a common mode of reaction when the projectiles are
low-energy nucleons or alpha particles. The approximately head-on
collision of heavy ions at low energy is also liable to produce a
compound nucleus. But when the impact parameter lies between the
grazing and head-on limits, the interaction between low-energy heavy
ions is likely to result in a deep-inelastic collision instead (see Figure
2.7).
Deep-inelastic collisions display surprising new phenomena not seen
in compound-nucleus reactions, and they have therefore received
much attention in heavy-ion physics. They involve some of the same
reaction mechanisms that occur in fission, but in deep-inelastic colli-
sions, these can be studied in a controlled way by the suitable choice
of projectile, target, and energy, for example.
In a deep-inelastic collision, the projectile nucleus can lose most of
its energy as it plows into the target nucleus; the energy loss is often so
great that the emerging reaction fragments are initially nearly at rest,
and they fly apart mainly because of the repulsive Coulomb force
between them. But unlike reactions that proceed by compound-nucleus
formation, a deep-inelastic collision retains a "memory" of the initial
conditions, so that the reaction fragments are closely related to the
original colliding nuclei.
OCR for page 58
58 NUCLEAR PHYSICS
A deep-inelastic collision presents seemingly contradictory proper-
ties: the substantial energy loss might appear to indicate a violent
collision, yet the retention of identity of the products suggests a
relatively gentle collision. The most successful approach to under-
standing this paradox views the original nuclei as starting with values
of the basic parameters, such as neutron-to-proton ratio, energy,
angular momentum, and mass, that are suited only to the stable
equilibrium of two nuclei far apart. The new stable equilibrium in the
collision environment requires different values of these parameters,
however, and during the collision, each of the properties begins to shift
toward the new values.
The value of a property cannot change, however, without some
driving mechanism. In general, the mechanisms for different properties
operate at different rates, so some properties move more rapidly than
others toward their new equilibrium values. The pertinent rates in a
deep-inelastic collision can be sorted out experimentally by using a
built-in "clock" for the reaction. The off-center nature of the collision
starts the system rotating, so that the angle of rotation increases with
time; fragments given oE at small rotation angles therefore correspond
to an early stage in the reaction. Analysis of the reaction fragments
shows that the neutron-to-proton ratio reaches its equilibrium value
very quickly, in 10-22 second or so. Energy equilibrates next, followed
by angular momentum. The masses of the fragments take so long to
reach equilibrium (roughly 50 times longer than for the neutron-to-
proton ratio) that the collision is over before the masses are able to
change much from their original values. Providing accurate models for
the various driving mechanisms has been a challenge to nuclear
theorists combining, as it does, collective motion with the statistical
nature of the approach to equilibrium.
The nuclear matter in a low-energy, deep-inelastic collision is not
highly excited, and relatively few excited states are accessible to the
nucleons. Under these conditions, the Pauli exclusion principle still
diminishes the effects of the nuclear force, and a given nucleon can
move fairly freely through the nuclear interiors. Interactions among
nucleons occur mainly near the nuclear surface, where the average
force on a nucleon is no longer constant. Simple models therefore
describe deep-inelastic collisions as the exchange of freely moving
nucleons between two nuclei, including the ejects of surface `'fric-
tion" at the contact region between the fragments. Such models have
had considerable success in describing the experimental data. A more
fundamental description is based on a time-dependent generalization of
the shell model, where now the average potential experienced by each
OCR for page 59
NUCLEAR STRUCTURE AND DYNAMICS 59
nucleon changes rapidly as the colliding system evolves toward a new
equilibrium.
Despite the progress that has been made in understanding deep-
inelastic, heavy-ion collisions, much remains to be done, such as
identifying the mechanism responsible for dissipating excess energy.
On the theoretical side, the successful models need to be related to
more fundamental theories, and the time-dependent average potential
calculations need to be extended to higher bombarding energies.
Experimentally, many questions need to be answered. How is angular
momentum transferred in the colliding system? What is the mechanism
for ejecting prompt light particles? How does the behavior of the
reacting system change as the bombarding energy becomes comparable
with the internal energy of nucleons in a nucleus? Can collisions just on
the border between fusion and deep-inelastic collisions be used to
probe the long-term dynamics of nearly unstable nuclear systems?
THE NUCLEAR MANY-BODY PROBLEM
A long-standing goal of nuclear physics has been to develop a
microscopic many-body theory that can account quantitatively for the
structure and interactions of nuclei in terms of the cumulative effects of
individual nucleon-nucleon (NN) forces. There are many roadblocks
on the way toward achieving this ambitious goal. First, the NN force
itself is not known in sufficient detail. The scattering of nucleons
provides much information, but only for a situation characterized by a
constant total energy of the two colliding nucleons; in a nucleus, where
nearby nucleons can transfer energy, other aspects of the NN force can
come into play. Furthermore, even if the NN force were completely
understood, available mathematical techniques cannot readily handle
the complexities of many closely spaced, strongly interacting nucleons
in a nucleus.
Great progress has nevertheless been made in microscopic nuclear
theory during the past decade, thanks to the steadily increasing
knowledge of the NN force, improved calculational techniques, and
more precise data on nuclear structure and interactions. A broad
conclusion from this work is that the traditional picture of interacting
nucleons alone cannot explain the detailed behavior of nuclear matter.
Necessary corrections appear to involve many-body forces, the rela-
tivistic description of nucleon motion, the presence of virtual mesons
in nuclei, and, ultimately, the nucleon's internal quark-gluon structure.
Progress in incorporating these corrections into many-body calcula
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60 NUCLEAR PHYSICS
lions will be hastened if experiments can be devised with specific
sensitivity to the effects in question.
The following sections summarize the status, successes, and short-
comings of the traditional nucleon picture of nuclear matter and discuss
briefly the seemingly essential corrections to that picture.
The Three-Nucleon Nucleus and Infinite Nuclear Matter
Advances in many-body calculations are usually tested first on two
limiting cases, to see if an extension to more complicated systems is
warranted. Two such cases often employed are the three-nucleon
nucleus and an infinite nuclear matter consisting of neutrons and
protons filling all space uniformly at a given density. For simplicity, the
neutron and proton masses are taken to be equal in infinite nuclear
matter; the Coulomb repulsion between protons is assumed to be
inoperative, so that only the strong interaction is operative.
The three-nucleon nucleus is the simplest possible many-nucleon
system. Nature provides two actual examples: hydrogen-3 (tritium;
one proton, two neutrons) and helium-3 (two protons, one neutron). A
wealth of experimental data for testing theories is available, including
the binding energy (the minimum energy required to separate the three
nucleons completely), the charge and mass distribution (nuclear ra-
dius), the nuclear magnetism, and the ways in which the nuclei react
with photons, nucleons, muons, and pions. With the aid of a new
mathematical technique, the properties of hydrogen-3 and helium-3 can
now be calculated numerically in great detail, once the form of the NN
force is chosen.
In practice, popular choices assume that the force acts only between
pairs of nucleons (two-body forces). Various parameters specifying the
force are adjusted to give good agreement with measured nucleon-
nucleon scattering and with the properties of the bound neutron-proton
system (the deuteron). A number of admissible forms satisfy these mild
constraints, but in general, all admissible two-body forces give a
three-body binding energy that is too small by 1 to 2 MeV (out of 8
MeV) and a nuclear radius too large by 9 percent or so. The accuracy
of the binding-energy prediction is better than might at first appear,
however, because binding energy is the relatively small difference
between two large, nearly equal terms: the energy of motion of the
nucleons and the energy content of the NN forces. Nevertheless, the
discrepancies appear to be greater than the accuracy of the calcula-
tions, and they must be taken seriously as indicative of shortcomings in
the assumed interactions.
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NUCLEAR STR UCTURE AND D YNAMICS 61
Infinite nuclear matter exists in nature in neutron stars. It is a useful
system to consider because it avoids the complications that arise from
having to take into account the properties of a nuclear surface.
Although it does not exist on Earth, its supposed properties can be
inferred from measurements on real nuclei. Of particular interest are
the nucleon density of nuclear matter, 0.16 nucleon per cubic fermi,
and the average binding energy per nucleon, inferred to be 15.8 MeV
per nucleon. A third property, the compressibility, has recently been
derived from giant monopole resonances in real nuclei, as described
earlier; the compressibility tells how the binding energy per nucleon
changes when the nucleon density is varied.
During the 1970s, major advances in mathematical techniques and in
the development of powerful computers spurred a vast amount of
theoretical work that largely eliminated earlier inconsistencies among
various techniques for calculating the properties of nuclear matter. The
discrepancies between theoretical predictions and experimental facts
still remain, however. A major long-term challenge for nuclear physi-
cists is to expand the traditional many-body theory of nuclear matter in
ways that will remove these discrepancies. How this goal might be
achieved is discussed at the end of this chapter.
Properties of Finite Nuclei
Although more effective computational techniques are under devel-
opment, most calculations of the properties of real nuclei are carried
out at present using a modification of the Hartree-Fock method, which
was originally invented to calculate the electronic structures of atoms
and molecules. In this method, each nucleon is assumed to move
according to the average force exerted by the other nucleons. But the
average force itself depends on how the nucleons move, so the
calculations are carried out iteratively until the computed nucleon
motion and the assumed average force are consistent with each other.
Part of the success of the Hartree-Fock method stems from the
exclusion principle, which inhibits strong short-range nucleon colli-
sions in a nucleus, thus allowing two-body interactions to be replaced
by a smoothly varying average force through the nuclear interior.
An important recent advance in the theoretical treatment of finite
nuclei has been the density-dependent Hartree-Fock (DDHF) method,
which takes into account the effect of the density of surrounding
nucleons on the NN force. The DDHF method is well adapted for
calculating charge and matter distributions in nuclei, because self-
consistency is achieved only when the nucleon motion, average force,
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62 NUCLEAR PHYSICS
and local density are in accord. The repulsive short-distance part of the
NN 'force is particularly important in finite nucleus calculations, to
keep the nucleons the correct distance apart. To obtain agreement of
theory with experiment, the NN interaction in the DDHF method must
be augmented by suitable empirical terms.
Electron-scattering experiments have provided exquisitely detailed
pictures of nuclear charge distributions, all the way to the centers of
nuclei and over the full range of the chemical elements. The detail of
the measurements is sufficient to show the varying proton densities'
associated with the nuclear shell structure, providing a good test of
DDHF methods. The general agreement with theoretical predictions is
good, but some small systematic discrepancies remain.
Electron-scattering experiments do not yield the distribution of
matter in a nucleus, however, because electrons interact primarily with
the electric charge of the protons and do not "see" the neutrons.
Protons interact with all nucleons, and many of the data on matter
distributions come from the elastic scattering of protons on nuclei.
When the projectile's energy is much higher than the energies of the
bound nucleons (800-MeV protons are available at the Los Alamos
Meson Physics Facility, for instance), the erects of the binding become
less important, and the NN force derived from the scattering of free
nucleons becomes a good approximation. The proton-nucleus scatter-
ing data can then be understood with the help of these factors to derive
the unknown neutron distribution. DDHF calculations generally repro-
duce the measured distributions quite well, but they are more accurate
for the differences among neighboring nuclear species than for absolute
neutron densities.
Calculations of finite nuclei can now also be tested in favorable cases
by the measured distribution of an individual nucleon in atnucleus a
major advance in the field during the past decade. One method makes
use of electron scattering to measure the proton distributions in nuclei
differing by only one proton for example, thallium-205 (81 protons,
124 neutrons) and lead-206 (82 protons, 124 neutrons); the comparison
yields a one-proton distribution. Neutrons in a nucleus associate in
pairs with their spins antiparallel, effectively canceling their intrinsic
magnetism. If a nucleus has an odd (unpaired) neutron, this neutron's
magnetism and hence its distribution in the nucleus-can be seen by
electron scattering, especially for scattering at large angles in collisions
that transfer a large amount of momentum from the electron projectile.
DDHF calculations also generally reproduce the measured single-
nucleon distributions well, as in the case of overall charge and matter
distributions. The remaining discrepancies, however, seem to indicate
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NUCLEAR STR UCTURE AND D YNAMICS 63
the need for small but significant corrections arising, for example, from
relativistic effects or electromagnetic contributions due to meson
exchange between nucleons in the nucleus.
The Effective NN Interaction at Intermediate Energies
For the properties of finite nuclei to be calculated properly, many-
body theory must evaluate how the interaction between two given
nucleons in a nucleus is modified by the presence of the other nucleons.
The attractive gravitational force between a planet and the Sun, or the
repulsive Coulomb force between two electrons in an atom, can be
described in terms of the separation distance alone. The effective
nucleon-nucleon force is more complicated, depending not only on
distance but also on momentum, spin, and isospin and all of these
factors are modified in a nucleus by the inhibiting effect of the Pauli
. .
prlnclp e.
With so many factors involved at once, it would obviously benefit
the development of nuclear theory to have experiments that signifi-
cantly test only one specific factor at a time. A suitable type of
experiment for this purpose is the reaction that involves the interaction
of a projectile nucleon with only one nucleon in the target nucleus. A
typical example is the charge-exchange reaction of a fast proton with
carbon-14, in which the projectile proton changes to a neutron while a
target neutron becomes a proton, leaving a nitrogen-14 nucleus as the
reaction product. This type of reaction (discussed earlier from another
perspective) involves the transfer of a charged pion from the proton to
the target neutron and is of special interest because of its sensitivity to
the pion field inside a nucleus. The target, bombarding energy, reaction
type, and especially the specific state in which the product nucleus is
left can be chosen so as to make a particular factor in the NN
interaction dominant. Progress in developing such selective filters has
been rapid in recent years, with the availability of high-quality proton
(and electron) beams at intermediate energies.
Intermediate projectile energies from 100 to 400 MeV are employed
because it is at these energies that the NN interaction is weakest; this
makes it more likely that the projectile nucleon will interact mainly
with only one target nucleon. Also, modifications of the NN force
induced by other nucleons are not too large at intermediate energies,
thus simplifying the interpretation of the data. Further information on
the properties of the target-nucleon state can sometimes be obtained
from electron inelastic scattering or from other nuclear processes, such
as beta decay.
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64 NUCLEAR PHYSICS
Complementary proton and electron inelastic-scattering experiments
have been carried out with narrow energy resolution (smaller than one
part per thousand) for a number of nuclei. The results have demon-
strated for the first time the real possibility of attaining a quantitative
microscopic understanding of nucleon-nucleus collisions. The density
of surrounding nucleons seems to have an especially important effect
on the part of the NN interaction that is independent of spin or isospin.
Some small discrepancies between theory and experiment remain in
the study of the spin-independent interactions, but their relationship to
the known shortcomings of nuclear theory is not yet clear.
The spin-dependent parts of the NN interaction are currently a
subject of great experimental and theoretical interest. As an example of
how nucleon-induced reactions can act as a selective filter, consider
the proton/carbon-14 charge-exchange reaction described earlier,
which flips the isospin of a target neutron, changing it to a proton. If the
reaction does not simultaneously flip the spin, the nitrogen-14 product
nucleus is left in an excited state with the same spin as the target
nucleus. If, however, the reaction also flips the neutron's spin (this is
the Gamow-Teller transition described earlier), the product nucleus is
left in an even higher excited state. Experimental results show that as
the bombarding energy is increased from 60 to 200 MeV, the isospin-
flipping reaction (without spin flip) diminishes in importance while the
Gamow-Teller reaction increases; this implies different energy depen-
dences for the spin-dependent and spin-independent parts of the NN
interaction. The NN force between free nucleons displays a similar
trend in the relative strengths, but predictions based on it are not in
quantitative agreement with these experiments; the nuclear environ-
ment can dramatically modify pion-exchange processes, as various
many-body calculations have suggested.
The results to date have demonstrated that nucleon-induced transi-
tions at intermediate bombarding energies can indeed act as a selective
filter for various components of the nucleon-nucleon force in nuclei.
This program is likely to have its real payoff in the future, with a more
systematic application of state-of-the-art many-body techniques to a
wider variety of reactions' nuclear excitations, bombarding energies,
and measured properties Especially spin-dependent observables).
Expanding the Traditional Many-Body Theory
Traditional nuclear theory considers only structureless, non-
relativistic nucleons interacting through two-body forces. The persis-
tent discrepancies between the best traditional calculations and exper
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NUCLEAR STR UCTURE AND D YNAMICS 65
intent are widely attributed to the oversimplifications of the traditional
picture, and serious efforts have been made recently to improve the
theory by including some of our modern understanding of strong
interactions.
The main direction of the effort is to incorporate mathematically the
effects of additional hadrons beyond the traditional proton and neu-
tron an approach that might descriptively be called quantum
hadrodynamics (QHD). (Hadrons interact through the strong force and
encompass all the baryons and all the mesons.) Much as the electro-
magnetic force between charged particles can be viewed as arising
from the exchange of virtual photons, the strong force between
hadrons can be viewed as arising from the exchange of virtual mesons
(which are themselves hadrons). Pions are the mesons of lightest mass,
and since the mass of the virtual particle is inversely related to the
range of the force, single-pion exchange is responsible for the longest-
range part of the nuclear force. The shorter-range part is due to
multipion exchange and to the exchange of heavier mesons, such as the
sigma, rho, and omega mesons.
The existence of baryon resonances in nuclei leads to the possibility
of new phenomena omitted in traditional theory. For instance, one
nucleon could excite a second nucleon to the delta state, and the delta
could then interact with a third nucleon. Invoking such three-body
forces may enable theorists to remove the discrepancies that currently
exist between experiment and the theories of three-nucleon systems
and of nuclear matter, as discussed above. For example, this approach
has been suggested in an attempt to explain the unexpected dip in the
central region of the charge distribution of the helium-3 nucleus
inferred from electron-scattering measurements. However, three-body
forces have not yet been fully incorporated into many-body calcula-
tions, nor have their effects been clearly identified experimentally.
A quantum field theory of the hadronic interactions in nuclei
combines relativity and quantum mechanics. These are essential
features of any reliable extrapolation of the properties of nuclear
matter to extreme conditions of temperature (average nuclear energy)
and density. One advantage of relativistic theories is that spin interac-
tions are naturally present in the fundamental equations and need not
be included as additional terms. Such theories also predict that the
apparent mass of a nucleon in a nucleus is altered, a possibly significant
influence on the origin of the repulsive forces that keep the nucleus
from collapsing. Although there are as yet few experiments or calcu-
lations bearing on a fully relativistic field theory of hadronic interac-
tions in nuclei, the description of nuclei within such a framework will
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66 NUCLEAR PHYSICS
be a major future objective. One recent attempt at constructing a
meson-baryon field theory starts from only a few mesons (pi, rho,
sigma, omega) and a few baryons (proton, neutron), but it has already
had significant success in treating both nuclear structure and nucleon-
nucleus reactions.
Although mesons and baryons represent an efficient and appropriate
language for describing much of nuclear structure, we know that these
hadrons are themselves made up of quarks and gluons, whose behavior
is described by quantum chromodynamics (QCD). Ultimately, QCD
must reproduce the known meson-exchange currents between any two
baryons at large internucleon separation. The central issues for under-
standing the nuclear many-body problem are thus to identify unambig-
uously the quark and color contributions to the description of nuclear
systems, to establish the theoretical relationship between the quantum
chromodynamic and quantum hadrodynamic pictures of nuclear struc-
ture, and to develop a description of nuclei entirely within the
framework of quantum chromodynamics.
Representative terms from entire chapter:
nuclear matter
