Nuclear Models
The central theoretical problem of nuclear physics is the derivation of the properties of nuclei from the laws that govern the interactions among nucleons. The central problem in theoretical chemistry is entirely analogous: the derivation of the properties of chemical compounds from the laws (electromagnetic and quantum-mechanical) that determine the interactions among electrons and nuclei. The chemical problem is complicated by the lack of mathematical techniques, other than approximate ones, for analyzing the properties of systems that contain more than two particles. The nuclear problem also suffers from this difficulty, but in addition it has two others:
1. The law that describes the force between two free nucleons is not completely known.
2. There is reason to believe that the force exerted by one nucleon on another when they are both also interacting with other nucleons is not identical to that which they exert on each other when they are free; in other words, there apparently are many-body forces.
Under these circumstances there is no alternative but to make simplifying assumptions that provide approximate solutions of the fundamental problem. These assumptions lead to the various models employed; or, more usually, a model for a nucleus or an atom is suggested by experimental results, and subsequently the assumptions consistent with the model are worked out. Consequently, several different models may exist for the description of the same physical situation; each model is used to describe a different aspect of the problem. For example, the Fermi-Thomas model of the atom is particularly useful for calculating quantities such as atomic form factors, which depend mainly on the spatial distribution of electron charge within the atom, but is less good than Hartree's self-consistent field approximation when questions of chemical binding are under analysis.
In the following sections we describe the models that have been found useful in codifying a large array of nuclear data, in particular, the energies, spins, and parities of nuclear states, as well as nuclear magnetic and quadrupole moments. First we sketch what is known about nuclear forces and their implications for the properties of complex nuclei.
Information about the forces that exist between two free nucleons may be obtained most directly from observations on the scattering of one nucleon by another and from the properties of the deuteron. The quantity that is immediately useful for calculation is not the force between two nucleons, but rather the potential energy as a function of the coordinates (space, spin and nucleon type) of the system. The quantity that we seek, therefore, plays a role similar to that of the Coulomb potential in the analysis of atomic and molecular properties and of the gravitational potential in the analysis of the motion of planets and satellites. The nuclear potential, though, seems to be considerably more complex than either the Coulomb or the gravitational potential. Although it is not yet possible to write down a unique expression for the nuclear potential, several of its properties are well known.
Characteristics of Nuclear Potential
The potential energy of two nucleons shows great similarity to the potential-energy function that describes the stretching of a chemical bond.
1. It is not spherically symmetrical. For the chemical system this is simply a statement of the directional character of the chemical bond, the direction being determined by the other atoms in the molecule. For the nuclear interaction the direction is determined by the angles between the
spin axis of each nucleon and the vector that connects the two nucleons.
The quadrupole moment of the deuteron gives unambiguous evidence that the ground state of the deuteron lacks spherical symmetry, hence the potential cannot be a purely central one. The spherically symmetric part of the potential is called a central potential; the asymmetric part is the tensor interaction.
2. It has a finite range and becomes large and repulsive at small distances. The potential energy involved in the stretching of a chemical bond is adequately described by the well-known Morse potential, which is large and repulsive for the small distances at which electron clouds start to overlap, goes through a minimum several electron volts deep at distances of a few angstroms, and then essentially vanishes at distances of several angstroms. The nuclear potential behaves in much the same way, except that the distances are about 105 times smaller and the energies about 107 times larger. The nuclear potential becomes repulsive at distances smaller than about 0.5 fm and has essentially vanished when the internucleon separation is between 2 and 3 fm.
The detailed knowledge of the potential energy of the chemical bond comes mainly from information about excited vibrational states and from the determination of bond lengths from either diffraction studies or rotational spectra. The range and depth of the nuclear potential are derived from the binding energy of the only bound state of the deuteron (there are no excited states of the deuteron that are stable with respect to decomposition) and from studies of the collisions between nucleons. The size and binding energy of the deuteron are reasonably consistent with an attractive square-well potential about 25 MeV deep with a range of about 2.4 fm. More detailed information on the nuclear potential at smaller distances comes from the angular distributions of nucleon-nucleon scattering at several hundred MeV. These data require a repulsive core at a distance of about 0.5 fm and an attractive potential of about 200 MeV just before the repulsive potential sets in. At larger distances, rather than resembling a square well, the potential approaches zero in an approximately exponential fashion. The potential energy diagram for these two cases is given in figure 1.
Figure 1 -Schematic diagram of nucleon-nucleon potential energy as a function of separation. The solid curve is the central potential for parallel spins and even relative angular momentum. The dashed curve is the effective potential that can describe the deuteron.
The factor of 107 in the relative strengths of the nuclear and chemical forces is the source of the usual remark that nuclear forces are very strong; nevertheless, in view of their short range, nuclear forces behave, in point of fact, as if they were very weak. This apparently paradoxical statement can be easily understood when it is recalled that, if two particles are to be confined within a distance R of each other, they must have a de Broglie wavelength in the center-of-mass system that is no larger than 2R. If μ=m1m2/(m1+m2) is the reduced mass of the two-particle system and v is relative velocity, this condition can be written as
The kinetic energy of two nucleons that are to remain within the range of nuclear forces (2.4 fm) must be at least
which is greater than the depth of the potential well that is meant to hold them together. Thus the absence of excited states of the deuteron, its low binding energy (~2.2 MeV), and its large size (the proton and neutron spend about one half the time outside the range of the nuclear force) result from the weakness of the nuclear force when viewed in the context of its small range.
The chemical bond, on the other hand, has a range of about 105 times that of a nuclear force, and so the kinetic energy requirement is 1010 times smaller, or only 10 2 eV, which is but a small fraction of the depth of the potential. This large difference between the "real" strengths of the interatomic and internucleon forces is of great importance to our understanding of the properties of nuclear matter.
3. It depends on the quantum state of the system. The potential-energy curve that describes the stretching of a chemical bond depends on the electronic state of the molecule. For example, the stable H2 molecule is one in which the two electrons have opposed spin (singlet state); when the electrons have parallel spin (triplet state) the molecule is unstable with respect to dissociation into two atoms.
The stable state of the deuteron is the one in which neutron and proton have parallel spins (triplet state); the potential energy of the singlet state is sufficiently different from that of the triplet so that there are no bound states of the isolated system consisting of one neutron and one proton with opposed spins. In addition to this spin dependence of the nucleon-nucleon potential, scattering experiments show that the potential also depends on the relative angular momentum of the two particles as well as on the orientation of this relative angular momentum with respect to the intrinsic spins of the nucleons. This latter term represents spin-orbit coupling which can lead to a partly polarized beam of scattered nucleons arising from an initially unpolarized beam.
4. It has exchange character. Our understanding of the chemical bond
entails the exchange of electrons between the bonded atoms. If, fw
example, a beam of hydrogen ions were incident on a target of hydrogen
atoms and many hydrogen atoms were observed to be ejected in the same
direction as the incident beam, any analysis of the problem would have to
include the process in which a hydrogen atom in the target merely handed
an electron over to a passing hydrogen ion. The formal result would be that
a hydrogen ion and a hydrogen atom would have exchanged coordinates
It has been observed that the interaction between a beam of high-energy neutrons and a target of protons leads to many events (more than can be explained by head-on collisions), in which a high-energy proton is emitted in the direction of the incident neutron beam. The analysis of the observation entails the idea that the neutron and proton, when within the range of nuclear forces, may exchange roles. The observation is an excellent example of what is meant by the exchange character of the nuclear potential. The exchange character of the potential in conjunction with the requirement that the wave function describing the two-nucleon system be antisymmetric can give rise to the type of force described in (3).
5. It can be described by semiempirical formulas. Despite the com
plexity of the potential between two nucleons it has been possible to construct semiempirical formulas for it that do reasonably well at describing the scattering of one nucleon by another up to energies of several
hundred MeV. These potentials are rather complex as they must contain at
least a purely central part with four components to account for the effect
of parallel or antiparallel spins as well as the evenness or oddness of the
relative angular momentum. In the most general form these potentials must
also contain two components each of a tensor force and spin-orbit force
that occur only for parallel spins but with either odd or even relative
angular momentum, and four components of a second-order spin-orbit
force that can occur for both parallel and antiparallel spins. An example of
the central force for parallel spins and even relative angular momentum is
given in figure 1 where it is compared with the effective potential that
can describe the deuteron.
Charge Symmetry and Charge Independence
So far we have not distinguished among neutron-neutron forces, proton-proton forces, and neutron-proton forces. The first evident difference is the Coulomb repulsion that must exist between two protons. At distances of the order of 1 fm this is much smaller than the attractive nuclear potential. Second, since the neutron and proton have differing magnetic moments, there will be different potential energies because of the magnetic interaction; this effect is even smaller than the Coulomb repulsion and is generally neglected.
m the observation that the difference in properties of a pair of mirror clei (nuclei in which the number of neutrons and the number of protons ° interchanged, for example, and ) can be accounted for by the differing Coulomb interactions in the two nuclei, the purely nuclear part of the proton-proton interaction in a given quantum state has been taken to be identical to that of two neutrons in the same quantum state as the protons. This identity is known as charge symmetry. A more powerful generalization arises from the similarity between neutron-proton scattering and proton-proton scattering when the two systems are in the same spin state and have equal momenta and angular momenta. This docs not mean that the scattering of neutrons by protons is identical to that of protons by protons. The Pauli exclusion principle makes certain states inaccessible to the two protons that may be quite important in the neutron-proton scattering. For example, in low-energy scattering that takes place in states without orbital angular momentum (s states), the two isotons must have opposite spins (), whereas the neutron and proton may have either opposite spins () or parallel spins (). This similarity leads to the assumption of charge independence, which asserts that the interaction of two nucleons depends only on their quantum state and not at all on their type, except, of course, for the Coulomb repulsion between two protons. So far there is no sizable divergence between this assertion and experimental results, but the search for small deviations continues to be actively pursued.
Isospin
The charge independence of nuclear forces leads to the idea that the proton and neutron can be considered as two different quantum states of a single particle, the nucleon. Since only two states occur, the situation is analogous to that of the two spin states an electron may exhibit, and thus the whole quantum-mechanical formalism developed for a system of electron spins has been taken over for the description of the charge state of a group of nucleons. The physical property involved is called variously isospin, isotopic spin, or isobaric spin (T). Each nucleon has a total isospin of J just as the electron has a total spin of ½. The z component of the isospin Tz may be either +½ or -½; in nuclear physics the +½ state is taken to correspond to a neutron and the -½ state to a proton. In elementary-particle physics, the opposite convention is used. For example, 9Be with 5 neutrons and 4 protons has Tz=+½. The concept of isospin for individual nucleons approximately carries over to complex nuclei, where the corresponding quantity is the vector sum of the isospins of the constituent nucleons, which is nearly a good quantum number and thus nearly a conserved quantity. The concept of nearly good quantum numbers is well known in quantum mechanics. Small deviations from rigorously conserved quantities are treated by perturbation theory in terms of small parameters. The mixing of isospin states results from the force that makes nucleon-nucleon interactions not really independent of nucleon type: the Coulomb force.