The determination of the bottom edge of the crust based on the CLDM of nuclei requires a highly precise calculation of the finite-size contribution, En ,surf + ECoul, in Ecell. One has to construct a CLDM representation of the ground state of the inner crust and then find the crust-core transition from the condition of thermodynamic phase equilibrium. One should use the same nuclear Hamiltonian for the crust and the core phases. One needs also a very precise many-body method to describe nuclear structures at the crust bottom, which is a very difficult task (see Figs. 3.10 and 3.11). Luckily, the crust-core phase transition is very weakly first-order (see above). Therefore, one can locate this transition using a completely different method well known in the theory of phase transitions in condensed matter. This will serve as an independent test for the precision of the CLDM calculation of pcc, described above. We will locate the crust edge by checking the stability of the uniform npe matter. We start from high densities, where the homogeneous phase is certainly stable with respect to the formation of spatial inhomogeneities (BBP, Pethick et al. 1995). By lowering the density, we will eventually find the threshold density, at which the uniform npe matter becomes unstable. This threshold gives a very good approximation of the actual pcc.

For a given nb, the ground state of the uniform npe matter minimizes the energy density E (nn,npjne) = E0 at a fixed baryon number density (nn+np = nb) and under the condition of electric charge neutrality (ne = np). This implies beta equilibrium between the matter constituents and ensures vanishing of the first variation of E due to small perturbations 5nj (r) (where j = n,p, e) of the equilibrium solution (at a fixed total nucleon number and a global charge neutrality of the system). However, this does not guarantee the stability of the spatially homogeneous state of the npe matter; the stability requires the second variation of E, quadratic in 5nj, be positive.

The energy functional of a slightly nonuniform matter can be calculated using the semi-classical ETF treatment of the kinetic and the spin-gradient terms in the nucleon contribution to E (Brack et al., 1985). Assuming small spatial gradients, we keep only the quadratic gradient terms in the ETF expressions. This approximation is justified because characteristic wavelengths of periodic perturbations are indeed much larger than internucleon distances. In this case the change of the energy (per unit volume) induced by density perturbations can be expressed (BBP, Pethick et al. 1995) as

E — Eo = 2/ (2^3 £ Fjk(q) 5n (q) 5nk (q)* , (3.44) where we use the Fourier representation n (r) = /(2nF n (q)eiqr. (3.45)

The Hermitian matrix Fik (q) determines the stability of the uniform npe matter with respect to spatially periodic perturbations with a wave vector q. Due to the isotropy of the uniform npe matter, Fik depends only on q = |q|. The matrix elements Fik can be calculated from the second variation of the microscopic energy functional E[nj (r), Vnj (r)] (BBP, Pethick et al. 1995).

The condition for the Fj matrix to be positive-definite is equivalent to the requirement that the matrix determinant is positive (Pethick et al., 1995). At any density nb, one has thus to check whether det[Fj(q)] > 0. Let us start with some nb, at which det[Fj(q)] > 0 for any q. By decreasing nb, we find eventually a wave-number Q at which the stability condition is violated for the first time; let it happen at a density nQ. For nb < nQ the homogeneous state ceases be the ground state of the npe matter.

Calculations performed using several effective nuclear Hamiltonians indicate that nQ ~ ncc, within a percent or better (Pethick et al., 1995; Douchin & Haensel, 2000). For the ETF approximation to be correct, the wavelength Xq = 2n/Q of critical density perturbations must be significantly larger than the mean inter-nucleon distance. The critical wave numbers Q are typically ~ 0.3 fm-1 so that Xq ~ 20 fm. At nQ ~ 0.1 fm-3 the fraction of protons is only about 3-4% but Xq ~ 20 fm is still 2-3 times larger than an inter-proton distance. The ratio of Xq to the inter-neutron distance is typically about eight. Calculations of nQ show that the actual precision of the ETF approximation is much better than guessed from the ratio of Xq to the inter-nucleon distance. This feature is well known from the ETF calculations of the energy of terrestrial atomic nuclei (Ring & Schuck, 1980).

The instability at nQ signals a phase transition with a loss of translational symmetry of the npe matter, producing nuclear structures. The agreement of nQ and ncc is a good test for the precision of calculated values of ncc. It indicates that the approximation of the spherical unit cell for 3N or 3B phases remains valid at p & pcc. It means also that the linear approximation of the curvature correction in a is sufficiently precise. Finally, it is a convincing argument for the validity of the CLDM at very large neutron excesses.

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