3
The Structure of Nuclei

Introduction

The nucleus, the core and center of the atom, is a quantal many-body system governed by the strong interaction. Just as hadrons are composed of quarks and gluons, the nucleus is composed of the most stable of these hadrons—neutrons and protons. The question of how the strong force binds these nucleons together in nuclei is fundamental to the very existence of the universe. A few minutes after the Big Bang, the mutual interactions between nucleons led to the formation of light nuclei. These, and the subsequent nuclear process synthesizing heavier nuclei during stellar evolution and in violent events like supernovae, have been crucial in shaping the world we live in.

One of the central goals of nuclear physics is to come to a basic understanding of the structure and dynamics of nuclei. In approaching this goal, nuclear physicists address a broad range of questions, from the origin of the complex nuclear force to the origin of the elements. Among the key issues still to be resolved are the following:

  • How do the interactions between quarks and gluons generate the forces responsible for nuclear binding?
  • What is the microscopic structure of nuclei at length scales of the size of the nucleon? Is this structure best understood by including quarks and gluons explicitly in the treatment of nuclei?
  • How are the different approximate symmetries that are apparent in nuclear structure related to the underlying interaction and how can they be derived from many-body theory?


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--> 3 The Structure of Nuclei Introduction The nucleus, the core and center of the atom, is a quantal many-body system governed by the strong interaction. Just as hadrons are composed of quarks and gluons, the nucleus is composed of the most stable of these hadrons—neutrons and protons. The question of how the strong force binds these nucleons together in nuclei is fundamental to the very existence of the universe. A few minutes after the Big Bang, the mutual interactions between nucleons led to the formation of light nuclei. These, and the subsequent nuclear process synthesizing heavier nuclei during stellar evolution and in violent events like supernovae, have been crucial in shaping the world we live in. One of the central goals of nuclear physics is to come to a basic understanding of the structure and dynamics of nuclei. In approaching this goal, nuclear physicists address a broad range of questions, from the origin of the complex nuclear force to the origin of the elements. Among the key issues still to be resolved are the following: How do the interactions between quarks and gluons generate the forces responsible for nuclear binding? What is the microscopic structure of nuclei at length scales of the size of the nucleon? Is this structure best understood by including quarks and gluons explicitly in the treatment of nuclei? How are the different approximate symmetries that are apparent in nuclear structure related to the underlying interaction and how can they be derived from many-body theory?

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--> What are the limiting conditions under which nuclei can remain bound, and what new structure features emerge near these limits? What is the origin of the naturally occurring elements of our world? Quantitative answers to these questions are essential to our understanding of nuclei; they also have a potential impact far beyond nuclear structure physics. Probes of short-range structures in nuclei can illuminate the nature of quark confinement, by exposing the extent to which quarks either remain confined to their particular neutrons or protons within nuclear matter or are shared among nucleons as electrons are shared in molecules. As yet poorly understood properties of medium-mass nuclei and of very neutron-rich nuclei critically affect the collapse and explosion of supernovae. In creating the heaviest nuclei in the laboratory, nuclear physicists are extending the periodic table of the elements and revealing deviations from chemical periodicity. Among the new isotopes they have produced in approaching the limits of nuclear stability are ones whose radioactive decay will provide crucial new tests of fundamental symmetry principles. Progress in all these areas relies on technical advances in theoretical and computational approaches, as well as in accelerator and detector design. For example, investigations of short-range structures in nuclei have been spurred by novel developments in proton accelerators and, especially, by the advent of continuous high-energy electron beams. The role of quarks and gluons in such structures is most likely to be revealed in the lightest nuclei, for which experimental maps can now be compared to essentially exact theoretical calculations based on the picture of interacting nucleons. These calculations have been made possible by adapting the latest quantum Monte Carlo computing methods to the unique aspects of nuclear forces. On the other hand, it is well known in all branches of physics that a direct approach to the dynamics of complex many-body systems, based on the elementary interactions between their constituents, is not always useful. For example, many properties of heavier nuclei can be accurately described using simpler approximations that retain some, but not all, essential microscopic ingredients. Deep insight into the crucial features of nuclear structure can be gained from an understanding of why such approximations work well, and of where they break down. Particular challenges are to understand the variety of collective motions of nucleons in heavy nuclei, and the fascinating phenomenon of nuclear superconductivity. Significant progress in our understanding of heavy nuclei is expected to come from advances in experimental capabilities. Another major advance is provided by facilities producing beams of short-lived nuclei. Current understanding of both nuclear structure and nucleosynthesis is largely based on what is known of the properties of stable and long-lived, near-stable nuclei. Between these nuclei and the drip lines, where nuclear binding comes to an end, lies an unexplored landscape containing more than 90

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--> percent of all expected bound nuclear systems, a region where many new nuclear phenomena are anticipated. As is evident from the map of the nuclear terrain in Figure 3.1, the limits of nuclear binding are poorly known at present; often, those limits are close to the regions where the processes that form the elements in stars must proceed. In the 1996 Long Range Plan, a new experimental facility to explore nuclei near the limits of nuclear binding was identified as the choice for the next major construction project in nuclear science. Recommendation II of the present report is the construction of such a facility. Beams of short-lived nuclei will be produced and accelerated at this facility, and their reactions with target nuclei will be used to synthesize new nuclear species in uncharted territory. By elucidating the properties of these new exotic species, and enabling their use in reactions of astrophysical interest and in tests of fundamental symmetries, this new facility will provide answers to some of the most profound nuclear structure questions identified above. Nuclear Forces and Simple Nuclei Measuring various properties of nuclear forces and tracing their origins to the fundamental interactions between quarts and gluons has been one of the major recent goals of nuclear physics. The long-range part of the nuclear force is known to be mediated by pions, the lightest of the mesons. However, our knowledge of the short-range parts is still incomplete. When two nucleons are separated by subfemtometer distances, their internal quark-gluon structures overlap. In such cases, description in terms of the quark-gluon exchange becomes necessary. The force between two nucleons has been studied extensively over the years by scattering one nucleon from another, and the data have been used to constrain parameters in models of the force. In the past decade, a few successful parameterizations of the low-energy nucleon-nucleon force have emerged; they offer descriptions that differ in their assumptions about short-range behavior. It is an important challenge to experiment and theory to find ways to better understand this aspect of the nuclear force, where the interface with QCD is the most critical. Such information is provided, for instance, in experiments measuring meson production in nucleon-nucleon collisions. In reactions at threshold energies, the two colliding nucleons must come essentially to rest, giving up all of their kinetic energy to produce the meson's mass. The rate of such reactions is sensitive to the strong, short-range parts of nuclear forces. New experiments aim to obtain additional information on pion production by using spin-polarized beams, and to search for the threshold production of heavier mesons. These experiments also probe meson-nucleon interactions at very low energies and provide crucial tests of QCD-based techniques for deriving the effective nucleon-nucleon interaction.

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--> FIGURE 3.1 The bound nuclear systems are shown as a function of the proton number Z (vertical axis) and the neutron number N (horizontal axis). The black squares represent the nuclei that are stable, in the sense that they have survived long enough since their formation in stars to appear on Earth; these form the "valley of stability." The yellow color indicates man-made nuclei that have been produced in laboratories and live a shorter time. By adding either protons or neutrons, one moves away from the valley of stability, finally reaching the drip lines where nuclear binding ends because the forces between neutrons and protons are no longer strong enough to hold these particles together. Many exotic nuclei with very small or very large N/Z ratios are yet to be made and explored: they are indicated by the green color. The proton drip line is established by experiments up to Z = 83. In contrast, the neutron drip line is considerably further from the valley of stability and harder to approach. Except for the lightest nuclei where it has been reached, the position of the neutron drip line is estimated on the basis of nuclear models; it is uncertain due to the large extrapolations involved. Green and purple lines indicate the paths along which nuclei are believed to form in stars; only some of the dominant processes are shown. While these processes often pass near the drip lines, the nuclei decay rapidly within the star into more stable ones. One important exception to this stability plot occurs in extremely massive and compact aggregations of neutrons, neutron stars, under the combined influences of the nuclear forces and gravity.

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--> Unique information on the strong force between hadrons can be obtained by comparing the forces between two nucleons and between a nucleon and a lambda particle in which one of the quarks is a heavier strange quark. Any difference between these forces is entirely due to the change in a single quark. The force between the lambda particle and the nucleon is being mapped with the improved experimental capabilities at CEBAF, as well as through the investigation of bound nuclear systems called hypernuclei, in which a nucleon is replaced by a lambda particle. Even the best available parameterization of the nucleon-nucleon force cannot accurately explain nuclear binding. In order to reproduce the binding energies of the simplest light nuclei, it is essential to add three-body forces to the pairwise interactions determined from nucleon-nucleon scattering. Such three-nucleon forces are expected because the nucleons are themselves composite objects whose constituents can be distorted by an external force. A more familiar example of such a three-body force is known from the analysis of orbits of artificial satellites. In the Earth-moon satellite system, the tides induced by the moon in oceans in turn alter Earth's pull on the satellite. The nuclear three-body forces are believed to be rather weak, and it has not been possible yet to measure their small effects on the scattering of three nucleons. For now, the strengths of three-body forces have been adjusted to reproduce the binding energies of light nuclei. However, a satisfactory microscopic picture of the three-body force between nucleons is still lacking. Advances and Challenges in Understanding Light Nuclei An important ongoing research effort is devoted to measuring various properties of light nuclei having up to eight nucleons. These are the simplest of all nuclei, and the first quantitative comparisons between experimental and theoretical maps of their global and short-range structure have been made. These nuclei are ideal for probing the microscopic aspects of nuclear structure, especially those related to quarks and gluons. The light nuclei also have important roles in astrophysics, elementary particle physics, and energy production. For example, most of the matter in the visible universe is in the form of these light nuclei. The nuclear physics of the Big Bang and of conventional stars like our Sun is primarily governed by the reactions between light nuclei. Nuclear fusion reactors would use some of these reactions as their energy source. Free neutrons are unstable to radioactive decay. Deuterium (2H) and helium-3 (3He) are the best available surrogates for neutron targets, needed for comparative measurements of the internal structure of neutrons versus protons. A detailed understanding of the structure of these nuclei is necessary for interpreting the results of such experiments. A direct way to probe the structure of nuclei is again through electron scattering.

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--> BOX 3.1 A Microscope to Measure the Distributions of Protons and Neutrons in the Nucleus By bombarding nuclei with electrons from the new generation of electron accelerators and using pairs of spectrometers to detect both the scattered electron and a proton ejected from the nucleus, valuable new insight is gained into how protons and neutrons are distributed in the nucleus. The energies and angles of the electron and proton are measured with the spectrometers, and from this the energy E and the momentum k of the recoiling excited nucleus are deduced. The two-dimensional energy-momentum map yields precision information on the nucleus. As shown in Figure 3.1.1, at small values of E (the so-called valence knockout region, where the recoiling nucleus is in or near its ground state), sharp spikes appear. When E is fixed to be on one of these spikes and the distribution in momentum k is examined, the resulting pattern gives a clear picture of how the least-bound proton orbital is distributed in the nucleus, providing a powerful high-energy electron microscope for studying nuclear structure. At high excitation energies (hundreds of MeV), the picture is much less clear. From work at intermediate energies, it is known that there is a significant probability for such reactions to occur. The violence of these reactions breaks the recoiling nucleus into fragments; theoretical studies lead us to expect that the short-range part of the nuclear force must play a major role in the process. An important goal of experimental and theoretical studies of such (e,e'p) reactions at high energies is to explore this new territory, that is, the highE, high-k part of the excitation map. Studies of (e,e'2p) reactions, where an energetic electron knocks out two protons, offer an additional, promising tool for finding two protons close together inside the nucleus, an excellent measure of the short-range correlations.

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--> CEBAF was constructed in part with these kinds of studies in mind. It is capable of the required precision measurements and exceeds the capabilities of previous high-intensity accelerators in this energy range by orders of magnitude. CEBAF is now starting to open a fascinating new window for studies at short distance scales. There are additional new knobs to turn in electron-induced studies of nuclear structure. These involve using spin to filter out special features of the reaction. The new generation of electron accelerators all have polarized electron beams, where the spin of the electron is pointed in a direction that is controlled in the experiment. For one of the spectrometers shown in the photograph on the left in Figure 3.1.1, it is possible to measure the direction of the ejected proton's spin; in other cases, the nuclear target may be polarized, having its spin pointed in some specific direction. The control of these spins amounts to having selective knobs at the experimenters' fingertips; in favorable cases, it allows the construction of three-dimensional views of nuclear structure. FIGURE 3.1.1 Experiments in which electrons knock out protons bound in the nucleus and both are detected in coincidence are being used to probe the distributions of nucleons in the nucleus. The photograph on the left shows an example of the equipment required for these experiments at high energies—the high-resolution spectrometers used in Hall A at CEBAF. On the right, results from previous studies with intermediate energy electrons are shown. (Courtesy Thomas Jefferson National Accelerator Facility.)

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--> The essentially structureless electron, possessing both an electric charge and a magnetic moment, is used as a probe to map the distribution of charge and magnetism within the nucleus. As the energy of the electrons is raised, the quantum mechanical wavelength of the electron is reduced and the resolution of the maps increases. High-energy electrons are also used to knock out the constituents of nuclei and gain valuable information. For example, by detecting the knocked-out proton or neutron along with the scattered electron, one can assess the momentum and energy distribution of nucleons in the nucleus. In particular, the data at especially large values of nucleon momenta provide crucial information on the strong forces at short distances. New insights can also be obtained by knocking out mesons from nuclei. Under certain kinematic conditions the electron can penetrate deeply within a nucleus, striking one of the virtual pions or heavier mesons being exchanged between nucleons and making it detectable. Of particular importance to these measurements are the recently developed continuous electron beams. Previously, electron beams were largely delivered in short bursts containing many electrons. In a single burst of the beam, many particles are produced by interaction of different electrons with different target nuclei. When it is necessary to detect more than one particle produced from a single electron-nucleus interaction, the background produced in a burst often swamps signals of interest. With a continuous beam, the background can be dramatically reduced, resulting in clear identification of the desired signal (see Box 3.1). The new experimental effort to probe light nuclei aims to fully utilize the capabilities of the recently completed continuous-beam electron accelerators and large-acceptance detectors. The ground states of most light nuclei have a nonzero spin, and many experiments propose to use spin-polarized targets to obtain three-dimensional maps of their structure. Measurements over the next 5 to 10 years will provide a detailed picture of the quantum-mechanical wave functions of two-and three-nucleon systems. Key goals are to get a quantitative measure of the small components in the wave function to pin down the role of quarks in hadrons other than nucleons and the distribution of high-momentum nucleons. On the theoretical side, due to recent progress in computational techniques used in nuclear physics, all the bound states of these light nuclei can now be calculated, essentially exactly, from realistic nuclear forces that can interchange protons and neutrons and may flip their spins. In particular, quantum Monte Carlo methods have been adapted to study nuclei, using massively parallel computing platforms. The method is being extended to take into account leading relativistic effects in nuclear structure. Correlations (e.g., how the separations between nucleons are distributed in nuclei) are strongly influenced by the nature of nuclear forces. Present calculations predict that neutron-proton pairs in nuclei have substantial probability to form intricate structures of toroidal (doughnut-like) and dumbbell shapes of

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--> femtometer size, as small as a single isolated nucleon. Such shapes are formed by the joint action of the short-range repulsive force and the anisotropic pion-exchange force between nucleons. A clear view of these structures can be obtained in the only nucleus with just two nucleons, the deuteron. When the deuteron is spin-polarized in one of its three possible quantum states, it has a toroidal shape; in the other two states, it resembles a dumbbell, as illustrated in Box 3.2. The difference between the distribution of electric charge within the different deuteron spin-states has been measured only in recent years. The measurements confirm that the density in the toroidal distribution peaks at the predicted radius of about half a femtometer, making the deuteron the smallest ring-type structure known. Similar experiments being carried out at higher energies will probe the thickness of this toroidal distribution. Since the two nucleons forming the dense part of the toroidal structure are so close, their quark substructures probably overlap. Indeed, the results of another early experiment at the TJNAF indicate such an overlap. The scattering of electrons by nuclei also provides information about the carriers of the nuclear force. This is possible because the nuclear force is mediated in part by the exchange of charged mesons, which results in a so-called exchange current contribution to the total electric current in a nucleus. In contrast, there is no comparable contribution to the electric current in an atom, because the electromagnetic force that binds electrons to nuclei is mediated by the exchange of electrically neutral photons. Important exchange current contributions have been revealed, for example, in beta decay, in the disintegration of deuterons by energetic electrons, and in the elastic scattering of high-energy electrons from 2H and 3He nuclei, but a comprehensive understanding of the quantitative basis of this exchange current in terms of QCD waits upon data from the new facilities. Significant advances have been made in methods used to compute rates of reactions involving three and four nucleons from realistic models of nuclear forces. One example is the extension of exact calculations for nucleon-deuteron reactions; at higher energies, the dynamical effects of three-nucleon forces on reaction processes may become accessible to experimental delineation for the first time. A second example is that of low-energy fusion reactions in which two light nuclei fuse, with the excess energy radiated away in the form of photons. New experiments are being carried out to study these processes at very low energy with spin-polarized beams of protons and deuterons. Progress in understanding these reactions is relevant for related fusion processes in the Sun, affected by the weak, rather than the electromagnetic, interaction. These weak capture reactions contribute significantly to energy production at solar temperatures and densities, but they occur at too low a rate to be studied directly in the laboratory.

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--> BOX 3.2 Measuring the Shape of the Deuteron The deuteron, with just one proton and one neutron, is the simplest of nuclei beyond the proton. The nucleon density distribution in the deuteron, illustrated in Figure 3.2.1, is determined by nuclear forces; the density is large where the forces are attractive, and small where they are repulsive. It is different for the deuteron states with different spin projection because the pion exchange forces depend upon the nucleon spin orientations. The dumbbell-shaped distribution in the state with spin projection ± 1 is formed by rotating the toroidal distribution in the spin projection zero state. These basic properties of the deuteron can be measured with a conceptually simple but technologically demanding experiment. Deuterons are bombarded with high-energy electrons, and the spin projection of the struck deuterons along the direction of their recoil is measured. Apart from small corrections, when the quantum wavelength of the recoiling deuterons is twice the diameter of the doughnut, the recoiling deuterons have only zero spin projection, and the anisotropy shown in Figure 3.2.2 has its minimum possible value of -1.4. The measured value of this wavelength then is a clear signal that the radius of the torus in the core of the deuteron is about half a femtometer. These experiments, initiated in 1984, require high-intensity, high-energy electron beams. The most recent data, taken with 4-GeV electrons, correspond to wavelengths of about one femtometer. In this region one expects to observe maximum anisotropy when the wavelength becomes equal to the thickness of the toroidal distribution. When the nucleons are in the densest part of the deuteron, their quark structures overlap. The effect of this overlap can be seen in the cross section for disintegrating the deuteron with a high-energy photon. There are two possible processes. In the nucleon process, three strongly interacting quarks within one nucleon absorb the photon's energy. This energetic nucleon then collides against the other, causing the deuteron to break up. In the quark process, which can only occur when the quarks in the two nucleons overlap in space, the photon's energy is absorbed by all of the six strongly interacting quarks, which break apart into two energetic nucleons. The observed cross section shown in Figure 3.2.3 has been scaled such that the quantity plotted is flat for the quark process. It appears that FIGURE 3.2.1 Theoretical predictions of surfaces of equal density in the deuteron in states with the projection of the spin being either one or zero along the symmetry axis. The surfaces shown are for half the maximum density of the deuteron.

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--> FIGURE 3.2.2 The observed difference in the way high-energy electrons scatter from the deuteron depending on the deuteron's spin is shown plotted against the recoil momentum or wavelength of the recoiling deuteron. This difference is characterized by the quantity t20, which is -1.4 when all recoiling deuterons have a spin projection of zero along the direction of recoil; the other extreme is t20 = 0.7, when all deuterons have spin projections of one in this direction. FIGURE 3.2.3 The scaled probability for the disintegration of the deuteron by a high-energy photon into a neutron and proton of equal energy is plotted against the energy of the photon. The probability is scaled such that in the quark picture it would be constant, independent of the photon energy. the nucleon process is dominant at small photon energies, while at higher energies the quark process dominates. These experiments also need high-energy, high-intensity electron beams. The earlier data went up to 2.7-GeV photons, and in one of the first experiments carried out at the Thomas Jefferson National Accelerator Facility in 1996, these data were extended up to 4 GeV. A further extension to 6 GeV is planned.

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--> bands indicating very elongated, superdeformed shapes having the longer axis twice as long as the shorter (see Box 3.3). This simple 2:1 ratio gives rise to a set of magic numbers that are different from those in spherical nuclei, and their existence is necessary for the stability of the observed superdeformed states. There is an ongoing search for even more deformed, hyperdeformed states with a 3:1 axis ratio, which are expected at the largest spins (just before the nucleus breaks up). In a state of a very high angular momentum, the nucleus behaves like a solid in a strong magnetic field. Hence, many magnetic phenomena may occur, similar to effects already known from condensed-matter physics. One of the main challenges for the theory of rotating nuclei is to understand the nature of the dramatic spin polarization induced by fast rotation. The spacing of energy levels in a nucleus changes from the regular, widely spaced pattern near the ground state to the more dense and random or chaotic pattern higher up. Gamma-ray spectroscopy with the new generation of detector systems, such as Gammasphere, is a unique probe of quantum chaos, roughly defined as a regime where the pattern of quantum numbers that may be used to characterize low-lying states of a many-body system is gone. Here, the important issues under study are these: At what energy does chaos set in? What are the unique fingerprints of the transition from regular to chaotic motion? The signatures for the onset of chaos are observed not only in the distribution of energy levels, but also in the properties of electromagnetic transition intensities. The increased precision of spectroscopic tools has allowed nuclear physicists to unveil some subtle, but startling, new phenomena: Identical Bands. Identical sequences of ten or more photons are observed, associated with rotational bands in different nuclei. This comes as a great surprise: it has long been believed that the gamma-ray emission spectrum for a specific nucleus represents a unique fingerprint. Explanation of these identical patterns in a wide variety of nuclei is still lacking. Magnetic Rotation. In nearly spherical nuclei, sequences of gamma rays are reminiscent of collective rotational bands, but with a quite different character: namely, each photon carries off only one unit of angular momentum and couples to the magnetic, rather than electric, properties of the nucleons. This is a new form of quantal rotor. ΔI = 4 Bifurcation. Extremely small but regular fluctuations are seen in the energies of photons emitted from some superdeformed nuclei. They could be driven by quantum tunneling motion at high angular momentum. Band Termination. Large rotational velocities can induce a gradual shape transition from the deformed state, which can rotate to the nearly spherical configuration incapable of rotational motion. Such a “death of the rotational band" has been seen in a number of nuclei. Nuclear Meissner Effect. Quenching, a total disappearance of nuclear

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--> FIGURE 3.5 The existence of nuclei with stable deformed shapes was known early in the history of nuclear physics. The observation of large quadrupole moments led to the suggestion that some nuclei might have spheroidal shapes, which was confirmed by observing rotational band structures and measurements of their properties. For most deformed nuclei, such as 238U, a description as an elongated sphere (i.e., a spheroid) is adequate to describe the band's spectroscopy. Because such a shape is symmetric, all members of the rotational band will have the same parity. However, it has since been found that some nuclei might have a shape more like that of a pear, which is asymmetric under reflection. The rotational band of 220Ra consists of levels of both parities, hence the name parity doublet. Another signature of such a parity doublet is the enhanced electric-dipole radiation in 220Ra due to a nonzero electric-dipole moment. Advances in high-resolution, gamma-ray detector systems are also responsible for a revolution in our study of low-spin nuclear behavior. Here, new insights have been gained on the nature of collective nuclear vibrations. In particular, vibrational multiphonon states long searched for were found experimentally. The existence of these new collective modes, both at low and high frequencies, is intimately connected with the effects of the Pauli exclusion principle.

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--> superconductivity, is observed at high rotational frequencies. The angular momentum behaves like an external magnetic field: it tries to align the angular momenta of nucleons along the axis of rotation, and this destroys Cooper pairs of nucleons responsible for superconductivity. Symmetry Scars. These highly excited states, such as those in a superdeformed band, represent order in chaos. The motion in these states is ordered (i.e., it is well characterized by a number of quantum numbers). The symmetry scars are embedded in the sea of many other states that can be described in terms of chaotic motion. Advances in high-resolution, gamma-ray detector systems are also responsible for a revolution in our studies of low-spin nuclear behavior carried out at low-energy accelerators. Here, new insights have been gained into the long unresolved, but basic, questions about the mechanisms of nuclear vibrations, about nuclear superconductivity, the statistical properties of excited nuclei, and the approximate symmetries of the many-body system. For instance, long-searched-for, vibrational multiphonon states have been discovered. The very existence of these new collective modes of nuclei, both at low and high frequencies, is intimately connected with the effects of the Pauli exclusion principle. In nuclei, nature displays the enormous diversity found in the behavior of many-body systems (see Box 3.4). Exploring this diversity in experiments, discovering its new facets, and finding effective theoretical approaches to account for them remains a continuing challenge.

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--> BOX 3.3 The Champion of Fast Rotation Rotation is a common phenomenon in nature—most objects in the universe, from the very small to the very large, rotate (Figure 3.3.1). The largest and slowest rotors are galaxy clusters. The rotation of the Andromeda Galaxy, the nearest major galaxy to our Milky Way, can be inferred from its giant spiral-shaped disk containing some hundred billion stars. Saturn is an excellent example of a deformed oblate (flattened sphere) rotator; its shape deformation is caused by a large centrifugal force. Among stellar bodies, pulsars are by far the fastest rotors: the Crab pulsar makes one revolution every 0.033 seconds! Among the dizziest mechanical man-made objects are ultracentrifuges used for isotope separation. With some modifications, the concept of rotation can be applied to small microscopic systems, such as molecules, nuclei, and even hadrons, viewed as quark-gluon systems. Atomic nuclei, with their typical dimensions of several femtometers and rotation periods ranging from 10-20 to 10-21 sec, are among the giddiest systems in nature. What makes the nuclear rotation special and interesting are quantal effects due to the nuclear shell structure and superconducting correlations. Most rotating celestial bodies show a common behavior: at low angular momenta they acquire axial shapes depressed at the poles, like our Earth or Saturn, but at sufficiently rapid rotation, a shape transition to ellipsoidal forms having three different axes takes place. At even faster rotations, the body becomes so elongated FIGURE 3.3.1

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--> that it fissions into two fragments. Interestingly, the shape changes found in hot nuclei, where the shell effects and superconductivity can be ignored, seem to resemble this behavior. The best nuclear rotators have elongated shapes resembling a football. The signature for such superdeformed states is a "picket fence" spectrum of gamma rays. The experimental spectrum of superdeformed 150Gd shown in the upper part of Figure 3.3.2 was obtained in 1989 in England. The bottom, much improved, spectrum was taken at the Lawrence Berkeley National Laboratory using the new-generation germanium-array Gammasphere with 55 detectors. The increased precision of experimental tools of gamma-ray spectroscopy has made it possible to probe new effects on the scale of a 1/100,000 of the transition energies. FIGURE 3.3.2

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--> BOX 3.4 The Nucleus: A Finite Many-Body System While the number of degrees of freedom in heavy nuclei is large, it is still very small compared to the number of electrons in a solid or atoms in a mole of gas, and as such the nucleus presents one of the most challenging many-body problems. Many fundamental concepts and tools of nuclear theory, such as the treatment of nuclear superconductivity and of nuclear collective modes, were brought to nuclear physics from other fields. Today, because of its wide arsenal of methods, nuclear theory contributes significantly to the interdisciplinary field of finite many-body systems. From Nuclei to Molecules, Clusters, and Solids: Shells and Collective Phenomena The existence of shells and magic numbers is a consequence of independent particle motion. The way the energy bunching of this shell structure occurs depends on the form and the shape of the average potential in which particles are moving. The electromagnetic force acting on electrons in an atom is different from the force acting between nucleons in a nucleus; this is why atoms and nuclei have different magic numbers. Small clusters of metal atoms (typically made up of thousands of atoms or fewer) represent an intermediate form of matter between molecules and bulk systems. Such clusters have recently received much attention—not least because of their striking similarity to nuclei. When the first experimental data on the structure of such clusters was obtained, it was immediately realized that the shell-model description could be applied to valence electrons in clusters. The nuclear shell energy and the shell energy for small sodium clusters are shown in Figure 3.4.1. In both cases, the same technique of extracting the shell correction has been used. The sharp minima in the shell energy correspond to the shell gaps. Nuclei and clusters that do not have all their shells fully occupied have nonspherical shapes. In Figure 3.4.1, the deformation effect is manifested through the reduction of the shell energy for particle numbers that lie between magic numbers. The deformation of the clusters can be deduced by studying a collective vibration that is a direct analog of the nuclear giant dipole resonance. As in nuclei, the occurrence of the deformation of the cluster implies that the dipole frequency will be split. This and many other properties of metallic clusters have been initially predicted by nuclear theorists, and their existence has been subsequently confirmed experimentally. Interacting electrons moving on a thin surface under the influence of a strong magnetic field exhibit unusual collective behavior known as the fractional quantum Hall effect. At special densities, the electron gas condenses into a remarkable state, an incompressible liquid, while the resistance of the system becomes accurately quantized. Although the fractional quantum Hall effect seems quite different from those in nuclear many-body physics—the electron-electron interaction is long range and repulsive rather than short range and attractive—it has been shown recently that these incompressible states have a shell structure similar to that of light nuclei. Nuclear and atomic physicists recognized early that the behavior of complex many-body systems is often governed by symmetries. Some of those symmetries reflect the invariance of the system with respect to fundamental operations, such as translations, rotations, inversions, exchange of particles, and so forth. Other

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--> FIGURE 3.4.1 Top—Experimental and calculated nuclear shell energy as a function of the neutron number. The sharp minima at 20, 28, 50, 82, and 126 are due to the presence of nucleonic magic gaps. (From Nuclear Data Tables 59, 185, 1995 and Peter Moller.) Bottom—Experimental and calculated shell energy of sodium clusters as a function of the electron number. Here, the magic gaps correspond to electron numbers 58, 92, 138, and 198. In both cases, the shell energy has been calculated by means of the same nuclear physics technique, assuming the individual single particle motion (of nucleons or electrons) in an average potential. The reduction of the shell energy for particle numbers that lie between the magic numbers is due to deformation (the Jahn-Teller effect). (From Stefan Frauendorf and V.V. Paskevich, Annalen der Physik 5, 36, 1986.) symmetries can be attributed to the features of the effective interaction acting in the system. These dynamical symmetries can often dramatically simplify the description of otherwise complicated systems. Symmetry-based nuclear methods have made it possible to describe the structure and dynamics of molecules in a much more accurate and detailed way than before. In particular, it has been possible to describe the rotational and vibrational motion of large molecules, such as buckyballs, hollow spheres made of 60 carbon atoms. Chaotic Phenomena: From Compound Nuclei to Quantum Dots With modern and powerful computers quickly processing satellite data, weather forecasts should be simple. However, the outcome of long-range weather modeling depends on seemingly trivial assumptions, and changes in the initial conditions assumed, even by a minute amount, can result in a different outcome. Even the soft flapping of butterfly wings may end in a hurricane. Predicting the final result of any chaotic system is impossible, simply because it is impossible to know the initial conditions with sufficient precision. Corresponding to classical chaos in quantum systems, a subfield known as quantum chaos has developed. Its origins are in nuclear physics theory—specifically in the random matrix theory that was developed in the 1950s and 1960s, initially by Nobel Prize-winning Eugene Wigner, to explain the statistical properties

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--> of the compound nucleus in the regime of neutron resonances. Today, the random matrix theory is the basic tool of the interdisciplinary field of quantum chaos, and the atomic nucleus is still a wonderful laboratory of chaotic phenomena. Remarkable recent advances in materials science permit the fabrication of new systems with small dimensions, typically in the nanometer-to-micrometer range Mesoscopic physics (meso originates from the Greek mesos, middle) describes an intermediate realm between the microscopic world of nuclei and atoms, and the macroscopic world of bulk matter. Quantum dots are an example of mesoscopic microstructures that have been under intensive investigation in recent years; they are small enough that quantum and finite-size effects are significant, but large enough to be amenable to statistical analysis. Quantum dots are formed in the interface of semiconductor layers where applied electrostatic potentials confine a few hundred electrons to small, isolated regions. For "closed" dots, the movement of electrons at the dot interfaces is forbidden classically but allowed quantum mechanically by a process known as tunneling. Tunneling is enhanced when the energy of an electron outside the dot matches one of the resonance energies of an electron inside the dot, leading to rapid variations in the conductance of quantum dots as a function of the energy of the electrons entering the dot (Figure 3.4.2). These fluctuations are aperiodic and have been explained by analogy to a similar chaotic phenomenon in nuclear reactions (known as Ericson fluctuations). Recently, the same nuclear random matrix theory that was originally invoked to explain the fluctuation properties of neutron resonances was used to develop a statistical theory of the conductance peaks in quantum dots. That is, a quantum dot can be viewed as a nanometer-scale compound nucleus! FIGURE 3.4.2 Measured neutron resonances in a compound nucleus 233Th (top) and conductance peaks in GaAs quantum dots (bottom). The presence of peaks in both spectra can be explained in terms of a quantum-mechanical tunneling: whenever the energy of an incoming neutron (electron) matches one of the resonance energies inside a compound nucleus (dot), the probability for capture increases. Distributions P(x) of neutron resonance widths and conductance peak heights are shown on the right. These are well described using the statistical random matrix theory, which gives the same universal probability distribution (red solid line) in both cases. (Courtesy of Yoram Alhassid, Yale University.)

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--> Nuclear Matter While heavy and even superheavy nuclei have been discussed above, nature also provides nuclei of virtually infinite size in the dense cores of neutron stars, where hadronic matter exists as a uniform medium, rather than clumped into individual nuclei. Much of what is learned from finite nuclei—the nature of the nucleon-nucleon force, the role of many-nucleon interactions, collective excitations, and so forth—is crucial in explaining the properties of infinite matter. But many new issues, such as matter composed almost entirely of neutrons, have no counterparts in ordinary nuclei. In the absence of Coulomb forces, the ground state of hadronic matter is a uniform liquid having equal numbers of protons and neutrons. The density of this liquid is about 2.5 × 1014 g/cc or, equivalently, 0.16 nucleons per cubic femtometer. In reality, such a liquid has a prohibitively large Coulomb energy, and thus does not exist. However, nuclei can often be regarded as relatively stable, small drops of cold nuclear matter, and some of their properties can be related to those of uniform nuclear matter. In the cosmos, extended nuclear matter, having a large excess of neutrons over protons and containing electrons to neutralize the electric charge of the protons, occurs in the interiors of neutron stars, and briefly in massive stars collapsing and then exploding as supernovae. Studies of nuclear matter received a large impetus after the observational discovery of neutron stars in 1968, and the subsequent interest in understanding supernovae. The dependence of pressure on density and temperature, the equation of state, is one of the most basic properties of matter. At small values of neutron excess (i.e., when the numbers of neutrons and protons are similar), cold nuclear matter is a quantum liquid. It expands on heating, and it undergoes a liquid-gas phase transition with an estimated critical temperature of approximately 18 MeV. In contrast, matter made up of neutrons alone is believed to be a gas even at zero temperature. The binding energies and density distributions of medium and large nuclei provide information on the equation of state of cold nuclear matter at densities up to its equilibrium density and at values of the neutron-to-proton ratio up to 1.5. Currently available data are inadequate to determine the equation of state of matter with the large neutron excess that is of interest in astrophysics. Hence, extrapolations rely on theoretical predictions of the equation of state of pure neutron matter. In the next decade, a large effort will be made to study the unstable neutron-rich nuclei, using radioactive beams as discussed in the last section. This effort will provide significant additional information. Nuclei and even large chunks of nuclear matter can undergo global vibrations in which the neutrons move against protons, or, alternatively, neutrons and protons move in unison to produce density and shape oscillations. These vibrations, conceptually similar to the fundamental vibrations of a bell or of a liquid

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--> drop, are called giant resonances. Instead of the sound waves of a bell, they emit gamma rays. These modes can be excited easily by inelastically scattering energetic particles from the nucleus, or by the shock of fusing two nuclei together. The frequencies of these vibrations are directly related to the strength of the restoring forces and to the basic properties of nuclear matter. The simplest mode, the monopole vibration, directly yields the incompressibility of nuclei and by extension of nuclear matter. Such studies have been done for some time on cold nuclei, but such new effects as the compression of the thin skin of nuclei (versus the dense inner core) are still being explored. Recently, it has become possible to excite vibrations in hot nuclei. The short-range aspects of nuclear structure, related to nuclear forces and binding, are similar in all nuclei, and thus regarded as properties of nuclear matter. They can be studied by using high-energy electrons to knock nucleons and mesons out of nuclei from deep inside the nuclear interior. The coincidence experiments, now starting at the new generation of electron accelerators, will provide exciting new information about nuclear matter. Specifically, it should be possible to refine current understanding about the nature of nucleons bound in nuclear matter. How do they differ from free nucleons? They will also improve our understanding of the boundary between the descriptions of nuclei as bound states of nucleons or quarks. It is known from earlier, noncoincidence experiments that when the electrons are softly scattered, one obtains a relatively successful picture by assuming that they simply knock nucleons out of the nucleus. Whereas for scattering with large loss of energy and momentum, a quark-based picture appears to work. How does one go from one picture to the other? Do the descriptions coexist under some conditions? Future studies with high-energy electrons will help to elucidate these fascinating issues, especially by detecting the specific particles knocked out of the nucleus under conditions where at present only the fact that an electron was scattered is known. Outlook The study of the structure of the atomic nucleus provides us with many insights into systems made of many strongly interacting particles. Many features of such systems are well described by amazingly simple models. The way these models emerge from the basic theory of the strong interaction is the subject of continuing study. The coming decade promises substantial progress in our understanding, by extending the study of nuclei into new domains, to the limits of their existence as bound systems. New experimental facilities and the next generation of computers are essential ingredients in this quest. New facilities to provide exotic short-lived nuclear beams for research will open new opportunities in exploring these limits. Theoretical descriptions of nuclei far from the line of stability suggest that their structure is different from what has been seen in stable nuclei. Nuclei far from stability also play an

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--> important role in the way the universe works and how elements are synthesized in the cosmos. The properties of such nuclei are essential to a quantitative understanding of these processes. Advances in detector technology promise a wealth of new information for nuclear structure studies. New capabilities in electron scattering will provide key information on the short-distance aspects of nuclear structure. Links in the chain that connect the fundamental theory of strong interactions of quarks and gluons to the properties of actual nuclei need to be better understood. Advances in computer technology and theoretical many-body techniques will make it possible to derive nuclear forces from the dynamics of the quarks and gluons that are confined within nucleons and mesons. New calculations will reveal the links between nuclear forces and effective nuclear interactions acting in complex nuclei. Fundamental questions concerning nuclear dynamics will be answered about the microscopic mechanism governing the large amplitude collective motion, the manifestations of short-range correlations, and the impact of the Pauli exclusion principle on nuclear collective modes. The interdisciplinary character of these studies, common frontiers with condensed matter physics and atomic physics, will be of increasing interest.