P(nn,v) = -mr" I *») = 3^ ZL dk ^ + („n - ,)2
One can easily see, that the model contains only two free parameters, ga/ma and gw/mw. They can be adjusted to reproduce two selected experimental parameters of the saturated nuclear matter,22 for instance, the binding energy per nucleon and the saturated nucleon density, B0 and no. No wonder that in this case the values of remaining parameters of the saturated nuclear matter, such as the incompressibility K0 and the symmetry energy So, strongly deviate from the experimental values (§ 5.4): K0 — 2 K0xp and S0 — 0-5 S0xp. Moreover, the EOS of pure neutron matter predicts a non-existent bound state at nb — n0.
The problem of incorrect incompressibility and fictitious bound state was eliminated by Boguta & Bodmer (1977) who extended the ,-w model by including self-interactions of the ,-field. These interactions were introduced via the term
where b and c are constants; U(,) is added to the Hamiltonian density (and subtracted from the Lagrangian density). The argument for including U(,) is that it makes the quantum version of the model renormalizable (Boguta & Bodmer, 1977). Two additional free parameters b and c can be adjusted to reproduce correct values of K0 and m* (§ 5.4). Still there remains the problem of the correct value of S0. It is solved by introducing the charge-triplet of the vector p-meson fields, which couple to the neutron-proton number density difference and, therefore, contribute to the symmetry energy. In this way one gets a modern version of the RMF model, used in neutron-star calculations since the mid-1980s and described in detail by Glendenning (2000).
The RMF model involves the nucleon bispinor fields ^n and and five meson fields. The meson fields are: the neutral scalar ,, the neutral vector
22Interestingly, Duerr (1956) demonstrated the existence of a strong spin-orbit coupling of nucleons moving in a nuclear potential well. This important feature is typical of relativistic mean field models, where the strong spin-orbit term is not postulated separately (in contrast to the non-relativistic mean-field theories) but is rigorously derived. However, the spin-orbit contribution to the energy averages out to zero in the mean-field approximation for a spatially uniform nuclear matter.
and the charge-triplet vector p^ (q = —1,0, +1). Each meson field 0 of mass m$ is coupled to an appropriate nucleon current (with a coupling constant g$) and yields a scalar interaction term in the Lagrangian LN$. Within the p triplet, gp does not depend on the meson charge, which reflects the charge independence of the strong interactions. The crucial assumption which makes the RMF model so easy in use (and therefore so attractive) is that nucleon currents are treated as uniform (constant) in space and that the Fock terms in the energy expectation value are neglected. These assumptions do not come from first principles; they are strictly valid only in the limit of nb —> x>, as we have already mentioned in the context of the Duerr (1956) paper [see Eq. (5.65)], so that the model is essentially phenomenological.
Within the RMF model, all three spatial components of nucleon currents and meson fields, as well as of charged p-fields, vanish in the ground state Non-vanishing time-like components of vector fields describe the meson potentials in which nucleons move. These time-like components are constant in space, as a result of the assumed constancy of field sources. The ground-state momentum distribution for nucleons is that of a free Fermi gas.
The constant values of the meson fields are determined from the Euler-Lagrange equations derived for the RMF model Lagrangian. The solutions for the w and p fields are particularly simple, because the Lagrangian is linear in these fields, guwo = ( — ) (nn + np) , gppO = 2 ( — ) (np — n„) . (5.72) V mu J 2 V mp J
The equation for the sigma field is much more complicated and reads gaa = ( *L V -1 V fkFN dk /(m — *^ g VmJ N=npJo Vk2 + (m — gaa)2
The nucleon energy spectrum is obtained from the Dirac equations for nucleon bispinors ^n and ^p:
en(k) = \/k2 + (mN — ga a)2 + gu wo — gppO , (5.74a) ep(k) = \Jk2 + (mN — ga a)2 + gu wo + gppO . (5.74b)
While the charge independence of e(k) is broken, the spin degeneracy still remains, because the matter is spin-unpolarized. We can now express the energy density in the ground state, E = (^o|T °°|^o),by summing energies of nucleon states up to the Fermi level and adding energies of meson fields:
+2 (mr) 2 ^ + 2( m) 2 (9P-0)2 - -( ma) 2 <9" a)2
The time-like components of meson fields should be expressed through nn and np by solving Eqs. (5.72) and (5.73).
The RMF model described above contains five free parameters: ga/ma, 9w/mw, gp/mp, b, and c. These parameters can be adjusted to reproduce the five experimental parameters of nuclear matter at the saturation point: n0, B0, S0, K0, and the nucleon effective mass m* at the Fermi surface (§ 5.4).
The effective mass m* in the RMF model deserves an additional comment. The effective mass can be defined in various ways. Two important definitions are: (1) the Dirac effective mass relevant for the nucleon Dirac equation in the RMF model, mD = m - gaa; it comes from the nucleon kinetic energy ekin(k) = \J(mD)2 + k2; and (2) the Landau effective mass m*L = (1/kF)(de(k)/dk)-=fcp . It is easy to show that these effective masses are related by (m^)2 = (mD)2 + kF. An effective mass in nuclear matter, measured experimentally (see § 5.4), should be identified with m£.
As we have mentioned before, the p-field is crucial for fitting the experimental nuclear symmetry energy at the saturation, S0xp ^ 30 MeV. To show this, let us consider a weakly asymmetric nuclear matter. This can be done by introducing the familiar variables nb = nn + np and 5 = (nn - np)/nb in the limit of 5 ^ 1. At a fixed nb, we have
5(nb) = \(9A2 nb + . ^ _ , (5.77b) 6V mP' ^k2 + (mN - 9aa)2
Table 5.1. Examples of parameter sets of the RMF model adjusted to different sets of parameters of nuclear matter at the saturation point (Glendenning & Moszkowski, 1991; Glendenning, 2000).
gl /ml gl/ml g2p/m2p 103& 103c (fm2) (fm2) (fm2)
Was this article helpful?