The properties of nuclear matter

Any many-body theory of neutron-star matter has to reproduce empirical data on bulk nuclear matter. Roughly speaking, the nuclear matter is what the heavy atomic nuclei are built of. Strictly speaking, it is an idealized infinite uniform system of nucleons, where the Coulomb interaction is switched off. The notion of the nuclear matter appears naturally within the Liquid Drop Model of nuclei, if we put ECoul = 0 and pass to the limit of A —> x>. In this limit, the energy per nucleon, E, depends only on the neutron and proton densities. It is convenient to express this dependence in terms of the nucleon density nb and the asymmetry parameter 5 = (nn — np)/nb, so that nn = (1 + 5)nb/2, np = (1 — 5)nb/2. Charge symmetry of nuclear forces (see, e.g., Preston & Bhaduri 1975) implies that E(nb, 5) = E(nb, —5), i.e., E does not change if protons are replaced by neutrons and vice versa. The case of 5 = 0 corresponds to symmetric nuclear matter, while for 5 = 1 we are dealing with neutron matter. The case of the symmetric nuclear matter is especially simple: in view of charge symmetry of nuclear forces this matter can be treated as a many-body system composed of one kind of particles - nucleons. Small effects of charge-symmetry breaking, like neutron-proton mass difference or charge-symmetry breaking terms in the NN interaction can be neglected (see Haensel 1977, for a detailed discussion of these effects). The symmetric nuclear matter is the simplest approximation of the bulk nuclear matter in heavy atomic nuclei. The effects of small S > 0 (quadratic in 5 because of charge symmetry of nuclear forces) can be considered as corrections to the leading 5 = 0 term. Typically, one has 52 < 0.04, for terrestrial nuclei, and the symmetric nuclear matter is then a reasonable approximation. On the other hand, a pure neutron matter, which is a one-component system, is the simplest approximation of the matter in a neutron star core. However, in general case one should deal with a two-component system.

Since the end of the 1950s, nuclear matter calculations represent the testing ground of nuclear many-body theories (the present status of these theories is described in Baldo & Burgio 2001). Calculations yield the energy per nucleon, E, versus the nucleon number density nb. In this section, we will not include the nucleon rest energy into E. Some examples of E(nb) are shown in Fig. 5.1. They are calculated for a specific model of nuclear matter, but their qualitative features are generic.

The minimum of the E(nb) curve for symmetric nuclear matter (5 = 0) corresponds to a bound equilibrium state at zero pressure. The values of E and nb at this minimum will be denoted by E0 and n0. Since P = nb dE/ dnb, the dotted segment corresponds to negative pressure and is therefore not interesting. The solid segment gives E(nb) for symmetric nuclear matter compressed to a density nb > n0. As clear from Fig. 5.1, B0 = —E0 is the maximum binding energy per nucleon in nuclear matter. The binding energy per nucleon B(A, 5) in a self-bound (i.e., bound under zero pressure) system of A nucleons with a nonzero neutron excess parameter 5 will be smaller than B0. The value of B(A, 5) will tend to B0 from below, if A —> x>, 5 —> 0, and the Coulomb forces are switched off. Simultaneously, the mean number density of the system will tend to n0. This property, resulting from the interplay of the short-distance repulsion and the long-distance attraction in the NN interaction, is called saturation; B0 = —E0 is called the binding energy at saturation, and n0 is the saturation density.

First let us consider the case of small 5 and small (nb — n0)/n0, characteristic of terrestrial nuclei. Keeping only the quadratic terms,11 we get

9 n0

11 The linear term Ea& resulting from charge-symmetry breaking in NN interaction can be neglected because of the smallness of Ea (Haensel 1977)

Figure 5.1. Energy per nucleon versus baryon number density for symmetric nuclear matter (S = 0), asymmetric nuclear matter with S = 0.4 (such an asymmetry corresponds to the neutron-drip point in a neutron star crust and to a central core of a newly born protoneutron star), and pure neutron matter (S = 1). Minima of the E(nb) curves are indicated by filled dots. Dotted segments correspond to negative pressure. Calculations are performed for the SLy4 model of effective nuclear Hamiltonian, which was used to calculate the SLy EOS by Douchin & Haensel (2001). It yields no = 0.16 fm~3 and Eo = -16.0 MeV.

where S0 and K0 are, respectively, the nuclear symmetry energy and incom-pressibility at the saturation point,12

The symmetry energy S0 determines the increase in the energy per nucleon due to a small asymmetry 5; the incompressibility K0 gives the curvature of the

12A traditional factor of nine in the definition of Ko is introduced for historical reasons. In the original definition of Ko the energy per nucleon in the symmetric nuclear matter was treated as a function of a common Fermi momentum (in units of ft) for neutrons and protons, fcp, related to nb via nb = 2fcp/(3n2). This resulted in Ko = (fcpdE/ dfcp)fcF=fcF0 and produced a factor of nine while replacing the derivative with respect to fcp by the derivative with respect to nb.

E(nb) curve at nb = n0 and the associated increase of the energy per nucleon of the symmetric nuclear matter due to a small compression or rarefaction.

The values of B0, n0, S0, and K0 can be extracted from experimentally measured nuclear masses. The uncertainties in the empirical values of these parameters result from the ambiguities of semi-empirical mass formulae, used to reproduce thousands of nuclear masses (e.g, Myers 1976; Groote et al. 1976; Seeger & Havard 1976; Bauer 1976; Janecke 1976). One obtains n0 = 0.16 ± 0.01 fm-3 and B0 = 16.0 ± 1.0 MeV. It is more difficult to extract S0, because of the ambiguity in separating the bulk and surface symmetry terms in binding energies. Thus, the uncertainty of the empirical value is rather large, S0 = 32 ± 6 MeV. The extraction of K0 from experimental data is even more complicated. Analyses of isoscalar giant monopole modes in heavy nuclei, summarized by Blaizot (1980), suggested K0 = 210 ± 30 MeV. A more recent analysis, based on precise measurements of the properties of giant monopole resonances in 90Zr, 116 Sn, 144 Sm, and 208Pb (excited by inelastic scattering of a particles), yields K0 = 231 ± 5 MeV (Cavedon et al., 1987; Youngblood et al., 1999). This result is consistent with another recent determination, K0 & 234 MeV, by Myers (1998) using a phenomenological Thomas-Fermi model fitted to measured nuclear masses and diffusenesses of the nuclear surface.

The last parameter discussed in this section is the nucleon effective mass, m*, calculated at the Fermi surface in saturated symmetric nuclear matter. It can be evaluated theoretically from the momentum dependence of the nucleon quasiparticle energy ek via

The notion of a nucleon quasiparticle is naturally introduced while considering low-lying excitations of nuclear matter. Such excitations can be treated in terms of quasiparticles (with number density ^ nb) of energy ek, where hk is the quasiparticle momentum. The value of m* enters the density of quasiparticle states (per unit energy and volume) at the Fermi surface (taking into account spin and isospin degeneracies),

(where N£ is the number of states with the energies below e).

A determination of m* from nuclear physics experiments is a complicated task, because of the coupling of quasiparticle states with nuclear surface vibration modes. The nuclear matter value of m* at saturation is evaluated as m* ~ 0.8m (see, e.g., Onsi & Pearson 2002 and references therein).

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