THOMAS-FERMI MODEL WITH SPIN ORBIT COUPLING ENERGY AND MOMENTUM-DEPENDENT POTENTIAL-ENERGY AND ITS APPLICATION TO THE ATOMIC NUCLEUS
Authors: LIU HAN-CHAO  ,   南开大学物理系
Published: November 2020

The Thomas-Fermi model is generalized for an approximately spherically symmetric system, to include the spin orbit coupling energy G(r)L.S and the momentum-dependent potential energy V(r, p2)=V0(r) + pV2(r)p + p2V4(r)p2 +……. The latter form fol-lows in the nuclear case as a result of the combination of the Thomas-Fermi approximation and the Brueckner's theory of infinite nucleus. The generalized Thomas-Fermi model may be considered in the nuclear case as an approximate generalization of Brueckner's theory to finite nucleus. The Hamiltonians H+ and H- of the j = l + 1/2 and the j=l-1/2 particles respectively are sufficiently approximately given by where pr and p(?) denote respectively the radial and the perpendicular components of momentum P. In the ground state, the particles occupy the whole volume H±≤ λof the μ-phase space, where A denotes the largest energy of the particles. As a first approximation, sufficient for some problems in the nuclear case, letthe bar denoting the average of momentumover all particles in the vincinity of r. The corresponding density-potential relation iswhere p(r) denotes the particle density of the+ 1), R1 and R2 are positive roots of λ - A(r) = 0 and a=0 respectively. For second and higher approximations of V(r, p2), the corresponding density-potential relations are obtained as a sum of terms of decreasing magnitude, the first term being just the -expression (3). The generalized form of the total energy as a function of Vi(r) (i=0, 2, 4......) and G(r) may be similarly obtained. By applying the variational method, thebinding energy, the functions G(r), V(r, p2) and p±r) may be obtained. The detailedsteps of determining the level density on the basis of the generalized Thomas-Fermimodel are given.The generalized form including G(r)L.S and V(r, p2) given by equation (2), ofthe Fermi equation[2] applied as an integral equation to the case of nucleus by severalauthors[3-7], is given by the two coupled integral equations,where N±ldenotes the number of particles supplied by the shell model, of orbital angular momentum l and j=l±/2, N±L2) representing number of particles with j=l?/2 and orbital angular momentum≥L, are known functions of G(r) and (λ- V1(r)) (the function V2(r) of equation (2) being determined from (± - V1(r)) and the "reduced mass" in Brueckner's theory of infinite nucleus). Equations (4) with the normalization conditionswhere n+,n- are found by the shell model, determine G(r) and (λ - μ(r)) except for one range parameter which in the nuclear case, may be determined by the energy difference 2p3/2-Is of the μ-mesonic X-ray or more simply by the relation,with r0=(1.15±.03) X 10-13 cm or=(1.18 ±.03) X 10-13 cm for proton distributions near the trapezoidal type or near the Fermi type respectively.In order to estimate the accuracy of the method, equations (4), (5), (6) are solved by neglecting the spin orbit coupling energy and the momentum dependence of potential energy. The results so obtained should not change appreciably when these factors are taken into account. The allowable variation of the central density is similar to that of Hahn from the experiment of high-energy election scattering. At the present stage two and only two parameters e.g. the radial parameter e and the surface thickness t defined by Hahn , can be determined for density distributions with nearly flat central parts. The errors for the proton system of Au197 due to the uncertainties of the data and the statistical procedures used, are estimated to be cm for c and cm for t. The effect of taking into account the momentum dependence of potential energy is to increase the surface thickness t from 1.6 X 10-13 cm to 2.1 X 10-13 cm while the radial parameter c remains unchanged ( 6.5 X 10-13 cm). The latter values of t and also c are in agreement with those of Hahn to within the estimated errors. The influence on the values of c and t due to the spin orbit coupling eneigy may probably be very small, and underlying reasons are given.