No abstract

Several m e t h o d s for the d e t e r m i n a t i o n of the trace of the p r o d u c t of an a r b i t r a r y n u mber of Pauli matrices are established. F o r m u l a e are derived lar the evaluation of various types of products of two traces when terms ~~f theŸ a i l a i 2 . . . ~in occur in b o t h o f them. Expressions are found for the p r o d u c t of two different traces and the square of the trace of an a r b i t r a r y n u m b e r of Pauli matrices. Similar formulae are obtained when Dirac matrices occurring as ,~x~ 9,iAiare considered instead o f P a u l i matrices. F r o m this all previous results in which ~~5 has i_l been considered separately ate recovered. A useful identity lar traces involving either Pauli of Dirac matrices is given.

In this paper several methods for the evaluation of the spur of the product of F5 matrix and any number of Dirac matrices have been established. The formulae for the determination of the various types of products of two spurs when Fi, Ff/# 9'i~'j~'k, Fffj~k~'l and these terms associated with ~5 matrix occur in both of the spurs, hav› been deriv› It has been shown that the product of two spurs each involving ~5 matrix, can be equated to ah expression involving no 75 matrix. One useful identity for spurs involving 75 matrix has been deduced.

In this paper we evaluate some higher-order terms in the eikonal expansion of the T matrix for potential scattering. We calculate eikonal correction terms which are of second order in potential strength V, through seventh order in inverse momentum K -I , which leads us to conjecture two plausible formulas for general eikonal correction terms of O(K -2"Vo2) and of O(K -2n-'Vo2). This may enable us to obtain an eikonal expansion of the second Born amplitude for any type of potential. We further evaluate eikonal correction terms which are of third order in potential strength and fourth and fifth order in K -'. With the help of eikonal corrections evaluated in this paper, a general formula based on a WKB phase function, and a relation existing between the amplitude term and the eikonal phase of the impact-parameter representation of the scattering T matrix, we are able to determine the T matrix completely through fourth order in inverse momentum. We may also note that the formula for the eikonal correction term of O(K -' " ' V O 2 ) when generalized in a suitable manner surprisingly yields the expression for the explicitly evaluated eikonal correction term of O (K -'vO3). It is further shown in this paper that the previously mentioned conjectured formula for the general eikonal phase correction term of O(K ' " ' V o 2 ) is somewhat supported by the form of the corresponding correction term to the usual WKB phase-shift result. All the higher-order eikonal correction terms evaluated in this paper vanish for the Coulomb interaction.

In this work we have studied the properties of asymmetric nuclear matter at zero and finite temperature in a wide range of density and asymmetry parameter using recently proposed generalized hybrid derivative coupling model. The temperature and asymmetry dependence of the bulk quantities such as compressibility, binding energy per nucleon, saturation density, chemical potential, entropy per nucleon, etc. have been explored. This is useful to understand the dynamics of supernova explosion. We have studied the density and temperature dependence of symmetry energy and briefly discussed its role in the evaluation of proton fraction at T = 0, in equilibrium with neutron rich matter. We have also studied proton fraction for T =1= 0 by free energy minimization. Dependence of bulk properties on the hybridization parameter of our model has been studied. The asymmetry and temperature dependence of the characteristics of liquid-gas phase transition have also been studied. We have compared our results with those of other investigators. §1. IntroductionThe study of the properties of cold and hot asymmetric nuclear matter is very important to understand the dynamics of iron-core collapse of some massive stars which produce type II supernova, the mechanism of supernova explosion, the structure of neutron-star remnants and also unstable neutron rich nuclei produced in the laboratory.!) For this purpose we need an equation of state (EOS) which is reliable at high density and/or non-zero temperature. The later stage of the collapsing presupernova core involves neutron rich matter with N ~ 2Z (or proton concentration (Z/(N + Z) of about 1/3) at moderate temperature T = 1 -10 MeV and densities up to about 4P02) where Po is the saturation density of nuclear matter. Neutron star structure involves almost pure neutron matter with N » Z, essentially at zero temperature and densities up to about (8 -1O)Po. 2 )The problem of asymmetric nuclear matter for zero and non-zero temperature has been investigated by several investigators who used non-relativistic BruecknerBethe-Goldstone approach. 2 ),3) They assumed potential theory description. In this problem several workers 4 ) have performed relativistic Dirac-Brueckner calculations, based on a one-meson exchange interaction only. Prakash and Ainsworth 5 ) applied field theoretic method based on chiral sigma model to study asymmetric nuclear matter properties at zero temperature. We intend to study the properties of asymmetric nuclear matter using a recently proposed general form of hybrid derivative coupling model 6 ) considering different values of asymmetric parameter (3( = (N -Z) / (N + Z)) and hybridization parameter Q. It may be noted that several investigators 7 ) have at