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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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

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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.

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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

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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.