The nucleons

In this section we will give a brief description of nuclear matter and its constituents. We begin with the nucleons and their properties. Then follows a discussion of the properties of the atomic nucleus as a whole, where we try to relate, in a qualitative way at least, the nuclear properties with nucleonic properties. Finally, the electric and magnetic fields generated from stationary states of the nucleus are discussed briefly. 

Two types of particles may be considered as main constituents of atomic nuclei, namely the protons (p) and the neutrons (n), jointly called the nucleons. Together with the electrons (e) they form the atoms in ordinary matter. The masses of the nucleons, and m , are roughly two thousand times the mass of an electron (mg). Only the proton carries a total charge 

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Mn is the mass of the nucleon, jis Planck s constant divided by 2ti, m. is the mass of the electron. This expression omits some temis such as those involving relativistic interactions, but captures the essential features for most condensed matter phases. 

R. Hofstadter (Stanford) pioneering studies of electron scattering in atomic nuclei and discoveries concerning the structure of the nucleons. 

We can use this idea of the relation of mass to energy in several ways. The mass of a 3iU nucleus is less than the sum of the masses of the 92 protons and 143 neutrons postulated to lie in it. The diirercnce in mass represents the binding energy which holds the nucleons together in... 

By comparing these two answers we can see that the repulsive force between two protons in the nucleus is about ten billion times as great as the repulsive force between two protons bound together in a hydrogen molecule. In order to overcome these enormous intranuclear coulomb repulsions and hold the nucleus together there must exist some very strong attractive forces between the nucleons. The nature of these forces is not understood and remains a very important problem in physics. 

Atomic nuclei are extraordinary particles. They contain all the protons in the atom crammed together in a tiny volume, despite their positive charges (Fig. 17.1). Yet most nuclei survive indefinitely despite the immense repulsive forces between their protons. In some nuclei, though, the repulsions that protons exert on one another overcome the force that holds the nucleons together. Fragments of the nucleus are then ejected, and the nucleus is said to decay. ... 

FIGURE 17.7 When a nucleus ejects an a particle, the atomic number of the atom decreases by 2 and the mass number decreases by 4. The nucleons ejected from the upper nucleus are indicated by the blue boundary. 

I have found that the assumption that in atomic nuclei the nucleons are in large part aggregated into clusters arranged in closest packing leads to simple explanations of many properties of nuclei. Some aspects of the closest-packing theory of nuclear structure are presented in the following paragraphs.1... 

The Relation between the Shell Model and Layers of Spherons.—In the customary nomenclature for nucleon orbitals the principal quantum number n is taken to be nr + 1, where nr> the radial quantum number, is the number of nodes in the radial wave function. (For electrons n is taken to be nT + l + 1.) The nucleon distribution function for n = 1 corresponds to a single shell (for Is a ball) about the origin. For n = 2 the wave function has a small negative value inside the nodal surface, that is, in the region where the wave function for n = 1 and the same value of l is large, and a large value in the region just beyond this surface. 

I assume that in nuclei the nucleons may. as a first approximation, he described as occupying localized 1. orbitals to form small clusters. These small clusters, called spherons. arc usually hclions, tritons, and dincutrons in nuclei containing an odd number of neutrons, an Hc i cluster or a deuteron may serve as a spheron. The localized l.v orbitals may be described as hybrids of the central-field orbitals of the shell model. 

Binding energy A measure of the strength of the force holding the nucleons together in the nucleus of an atom. The term is sometimes applied to the force holding an electron in an atom. 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

Electrons of 2s orbitals have lower energy than those of 2p orbitals because electrons in 2s orbitals tend, on the average, to be much closer to the nucleons than electrons in 2p orbitals. 

Nucleons are the particles comprising the nucleus, i.e., it is a collective term for the protons and neutrons in a nucleus. The number of protons is the atomic number the sum of the protons and the neutrons (the nucleons) is the mass number. 

Figure 8. Histograms of the nearest-neighbor spacing distribution for the nucleon (left plots) and the delta (right plots). The data is for Goldstone-boson exchange and for one-gluon exchange compared to a pure linear confinement potential of the same strength. Curves represent the Poisson and the GOE-Wigner distributions.

A description of nuclear matter as an ideal mixture of protons and neutrons, possibly in (5 equilibrium with electrons and neutrinos, is not sufficient to give a realistic description of dense matter. The account of the interaction between the nucleons can be performed in different ways. For instance we have effective nucleon-nucleon interactions, which reproduce empirical two-nucleon data, e.g. the PARIS and the BONN potential. On the other hand we have effective interactions like the Skyrme interaction, which are able to reproduce nuclear data within the mean-field approximation. The most advanced description is given by the Walecka model, which is based on a relativistic Lagrangian and models the nucleon-nucleon interactions by coupling to effective meson fields. Within the relativistic mean-field approximation, quasi-particles are introduced, which can be parameterized by a self-energy shift and an effective mass. 

We will use the so called TM1 model which is given by the following Lagrangian, describing coupling of the nucleon field to the non-linear sigma, omega and rho meson fields (index i p, n denotes protons or neutrons),... 

For a temperature T and chemical potentials fit (relative to the nucleon masses) the nucleon occupation probability reads... 

The antinucleon distribution functions follow by changing the sign of the effective chemical potential [11], Since we are interested in the low density region the contribution of the antinucleons can be neglected. Then, in mean-field approximation the chemical potential is related to the nucleon number density according to... 

Expressions for the medium modifications of the cluster distribution functions can be derived in a quantum statistical approach to the few-body states, starting from a Hamiltonian describing the nucleon-nucleon interaction by the potential V"(12, l/2/) (1 denoting momentum, spin and isospin). We first discuss the two-particle correlations which have been considered extensively in the literature [5,7], Results for different quantities such as the spectral function, the deuteron binding energy and wave function as well as the two-nucleon scattering phase shifts in the isospin singlet and triplet channel have been evaluated for different temperatures and densities. The composition as well as the phase instability was calculated. 

To evaluate the linear term of the bound state energy shift within perturbation theory we need the bound state wave function. For this, we have to specify the interaction. We will adopt the following parameterization of the nucleon-nucleon interaction. 

Relativistic corrections. Before the possible effects of TBF are examined, one should introduce relativistic corrections in the preceding nonrelativistic BHF predictions. This is done in the Dirac-Brueckner approach [4], where the nucleons, instead of propagating as plane waves, propagate as spinors in... 

In the right panel of Fig. 4 we display the symmetry energy as a function of the nucleon density p for different choices of the TBF. We observe results in agreement with the characteristics of the EOS shown in the left panel. Namely, the stiffest equation of state, i.e., the one calculated with the microscopic TBF,... 

Figure 6. The single-particle potentials of nucleons n, p and hyperons , A in baryonic matter of fixed nucleonic density pN = 0.4 fm-3, proton density pp/pN = 0.2, and varying density pz/pN = 0.0, 0.2, 0.5. The vertical lines represent the corresponding Fermi momenta of n, p, and . For the nucleonic curves, the thick lines represent the complete single-particle potentials Un, whereas the thin lines show the values excluding the contribution, i.e., U + uffi.

The different single-particle potentials involved in the previous equations are illustrated in Fig. 6, where neutron and proton densities are fixed, given by pN = 0.4 fm-3 and pp/pN = 0.2, and the density is varied. Under these conditions the T, single-particle potential is sizeably repulsive, while Ua is still attractive (see also Ref. [15]) and the nucleons are both strongly bound. The single-particle potential has a particular shape with an effective mass m /m close to 1, whereas the... 

Lambda effective mass is typically about 0.8 and the nucleon effective masses are much smaller. 

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