Extended Thomas - Fermi expansion from nuclear linear response

Physical Review A, 1990

Atoms at finite temperature and with a nucleus of finite extension are analyzed by a modification of the Thomas-Fermi model for TWO. Applications to strange-matter atoms are included.

Physical Review, 1961

The Hartree-Fock to Fermi-Thomas reduction is carried through for finite nuclei, starting with the iT-matrix formulation. The resulting expression represents the nuclear energy to good accuracy in terms of the density and its first derivatives only; it differs in detail from the semiempirical expressions previously proposed for this purpose. This improved expression shows the inadequacy of the "semi-infinite" approximation used often in earlier studies.

Physics Letters B, 1976

We examine for finite nuclei the Thomas-Fermi (TF) approximation for the kinetic energy density and its correc-

Communications in Mathematical Physics, 1981

Let E(R\ respectively e(R\ denote the total energy, respectively the electronic contribution to the energy, in the Thomas-Fermi theory for a system of two fixed nuclei a distance R apart. We prove that e(R) and-E(R) increase as R does. For the case of N fixed nuclei, we prove the monotonicity of e and E under certain displacements of the coordinates of the nuclei. The analogous result for the electronic contribution to the Born-Oppenheimer energy is proved.

Mathematically rigorous versions of Thomas-Fermi theory and its generalizations were developed in the 1970's and 1980's by Lieb, Simon, Bdnilan, Brezis and others. At issue is the ground state electron density for an N electron quantum mechanical system. Von Weizs&cker and Dirac corrections to Thomas-Fermi theory amended the kinetic energy term by expanding it in powers of the density or the gradient of the density. In this paper the kinetic energy is viewed as a functional of position and density. This paper also generalizes the existing theory to d (_> 3) dimensions. Existence (even for some negative ions), nonexistence and uniqueness results are obtained, as well as qualitative properties of the solutions. 1. Introduction. The quantum mechanical Hamiltonian for a system of N electrons (or, more generally, fermiohs) in R d is (1) H=T+V,•+V• where (2) T =-h2 N 2m • Aj j=l is the kinetic energy operator, V•e = ]•-]•v= 1V(xj) is the nuclear-electron potential energy, and Ve• is the electron-electron repulsion energy. We assume d) 3, x = (Xl,... ,xN) E R d and xj E R d represents the position of the jth electron. Henceforth we set most of the physical constants, i.e., m, e, equal to unity, and we ignore spin. The most important special case occurs when we place a nucleus with Zj protons at the fixed point Rj • R •. The nuclear-electron attraction term then becomes (3) M V(y)-ly-j=l , y•R d 1<7<d-2. 407 4O8 GISJ•LE RUIZ RIEDER For the electron-electron repulsion term we take 1 (4) V** = • E Ix-/-z• 2-d. j• Note that T is a differential operator, and V,, and V** are multiplication operators. The operator H acts on Lea(RaN), the space of antisymmetric square-integrable functions, i.e., f ß Le•(R aN) iff f ß Le(R aN) and f(x•,...,x•N) = sgn•rf(xl,...,XN) for all Xl,...,XN ß R a and all permutations •r of {1,..., N}. For generality we allow V to be a general realvalued function on •t a. We define the ground state energy E of H on Le•(R a•) by E = infer(H). In most applications we expect that E is a simple eigenvalue. We define the quantum mechanical energy functional oeQM : T)(oeQM) C_ Lea(R aN)-• R by oeQM(½) = (H½,½). Then E is the solution of the variational problem (5) E= min oeQM(½). The Euler-Lagrange equation which arises from (5) is (6) = where A is the Lagrange multiplier corresponding to the constraint ]]ll-1. The unit vector which solves (6) is the ground state wavefunction; however, in general we cannot explicitly solve equation (6). In fact, in cases of bulk matter, N is of the order of 1026; thus, so many variables are involved that the ground state wavefunction (even if it were explicitly describable in a few million pages) would not provide much recognizable information about the properties of the system. Let qb ß Lea(R aN) be a unit vector and define (7) P(x) = N fI• "' fR IqS(x'xe"'"XN) 2 dx2'"dxN d d for x ß R a. Note that p ß L•(Ra), p •_ 0 and f•t•P(x)dx-N. We call such a function p a density. We write • •-• p if p is obtained from • via (7).

Journal of Physics G: Nuclear and Particle Physics, 2000

We generalise the finite range momentum and density dependent Seyler-Blanchard nucleon-nucleon effective interaction to the case of interaction between two baryons. This effective interaction is then used to describe dense hadronic matter relevant to neutron stars in the nonrelativistic Thomas-Fermi approach. We investigate the behaviour of nuclear symmetry energy in dense nuclear and hyperon matter relevant to neutron stars. It is found that the nuclear symmetry energy always increases with density in hyperon matter unlike the situation in nuclear matter. This rising characteristic of the symmetry energy in presence of hyperons may have significant implications on the mass-radius relationship and the cooling properties of neutron stars. We have also noted that with the appearance of hyperons, the equation of state calculated in this model remains causal at high density.

arXiv (Cornell University), 2009

Using the recently established scaling laws for the solutions of the relativistic Thomas-Fermi equation we consider the two limiting cases of compressed atoms and compressed massive nuclear density cores. The Feynman, Metropolis and Teller treatment of compressed atoms is extended to the relativistic regimes. Each configuration is confined by a Wigner-Seitz cell and is characterized by a positive electron Fermi energy. There exists a limiting configuration with a maximum value of the electrons Fermi energy $(E_e^F)_{max}$ reached when the Wigner--Seitz cell radius equals the radius of the nucleus, and it can be expressed analytically in the ultra-relativistic approximation. The results are compared and contrasted to approximate treatments in the literature. This treatment is then extrapolated to compressed massive nuclear density cores with $A\simeq (m_{\rm Planck}/m_n)^3 \sim 10^{57}$. Again an entire family of equilibrium configurations exist for selected values of the electron Fermi energy varying in the range $0 < E_e^F \leq (E_e^F)_{max}$. The configuration with $E_e^F=(E_e^F)_{max}$ has global and local charge neutrality and no electrodynamical structure. The remaining configurations have electric fields on the core surface, increasing for decreasing values of the electron Fermi energy reaching values much larger than the critical value $E_c = m_e^2c^3/(e\hbar)$ for $E_e^F=0$. We compare and contrast our results with the ones of Thomas-Fermi model in strange stars. In both, the case of atoms and the massive nuclear density cores, the configuration with $E_e^F=0$, reached when the Wigner-Seitz cell radius tends to infinity corresponds to the ground state of the system.

Il Nuovo Cimento A, 1970

ricevuto il 60ttobre 1972)

Nuclear Physics A, 1995

The Bethe-Brueckner-Goldstone many-body theory of the Nuclear Equation of State is reviewed in some details. In the theory, one performs an expansion in terms of the Brueckner two-body scattering matrix and an ordering of the corresponding many-body diagrams according to the number of their hole-lines. Recent results are reported, both for symmetric and for pure neutron matter, based on realistic twonucleon interactions. It is shown that there is strong evidence of convergence in the expansion. Once three-body forces are introduced, the phenomenological saturation point is reproduced and the theory is applied to the study of neutron star properties. One finds that in the interior of neutron stars the onset of hyperons strongly softens the Nuclear Equation of State. As a consequence, the maximum mass of neutron stars turns out to be at the lower limit of the present phenomenological observation.

Physical Review C, 1998

The density matrix expansion is used to derive a local energy density functional for finite range interactions with a realistic meson exchange structure. Exchange contributions are treated in a local momentum approximation. A generalized Slater approximation is used for the density matrix where an effective local Fermi momentum is chosen such that the next to leading order off-diagonal term is canceled. Hartree-Fock equations are derived incorporating the momentum structure of the underlying finite range interaction. For applications a density dependent effective interaction is determined from a G-matrix which is renormalized such that the saturation properties of symmetric nuclear matter are reproduced. Intending applications to systems far off stability special attention is paid to the low density regime and asymmetric nuclear matter. Results are compared to predictions obtained from Skyrme interactions. The ground state properties of stable nuclei are well reproduced without further adjustments of parameters. The potential of the approach is further exemplified in calculations for A=100.. .140 tin isotopes. Rather extended neutron skins are found beyond 130 Sn corresponding to solid layers of neutron matter surrounding a core of normal composition.