ISSN 1063-7788, Physics of Atomic Nuclei, 2016, Vol. 79, No. 1, pp. 1–10. c  Pleiades Publishing, Ltd., 2016. NUCLEI Theory Nuclear Symmetry Energy in Terms of Single-Nucleon Potential and Its Effect on the Proton Fraction of β-Stable npeμ npeμ npeμ Matter ∗ Babita Sahoo 1)** , Suparna Chakraborty 2)*** , and Sukadev Sahoo 3)**** Received August 24, 2015 Abstract—Momentum and density dependence of single-nucleon potential u τ (k, ρ, β) is analyzed using a density dependent finite range effective interaction of the Yukawa form. Depending on the choice of the strength parameters of exchange interaction, two different trends of the momentum dependence of nuclear symmetry potential are noticed which lead to two opposite types of neutron and proton effective mass splitting. The 2nd-order and 4th-order symmetry energy of isospin asymmetric nuclear matter are expressed analytically in terms of the single-nucleon potential. Two distinct behavior of the density dependence of 2nd-order and 4th-order symmetry energy are observed depending on neutron and proton effective mass splitting. It is also found that the 4th-order symmetry energy has a significant contribution towards the proton fraction of β-stable npeμ matter at high densities. DOI: 10.1134/S1063778816010178 1. INTRODUCTION The nuclear equation of state (EOS) of isospin- asymmetric nuclear matter (ANM) plays a key role for the better understanding of the structure of radioac- tive nuclei, the reaction dynamics induced by rare isotopes, and the liquid gas phase transition in ANM. It has also got importance due to its implications in certain areas beyond standard nuclear physics, such as astrophysical phenomena like the structure of neu- tron stars and the dynamics of supernova collisions [1–5]. The equation of state (EOS) of nuclear matter is generally defined as the binding energy per nucleon as a function of density. At zero temperature, the binding energy per nucleon of asymmetric nuclear matter (ANM) can be expressed as a power series of isospin asymmetry β = ρn−ρp ρ , where ρ n and ρ p are the neutron and proton densities, respectively, and the total density ρ = ρ n + ρ p . Up to the 4th-order of the isospin asymmetry β it can be written as [6] E(ρ, β)= E(ρ)+ E sym,2 (ρ)β 2 + E sym,4 (ρ)β 4 , ∗ The text was submitted by the authors in English. 1) Department of Applied Sciences, Durgapur Institute of Ad- vanced Technology & Management, West Bengal, India. 2) Department of Physics, Michael Madhusudan Memorial College, West Bengal, Durgapur, India. 3) Department of Physics, National Institute of Technology, West Bengal, Durgapur, India. ** E-mail: patra_babita@rediffmail.com *** E-mail: banerjee.suparna@hotmail.com **** E-mail: sukadevsahoo@yahoo.com where E(ρ)= E(ρ, β = 0) represents the binding en- ergy per nucleon of symmetric nuclear matter (SNM), E sym,2 (ρ) the 2nd-order nuclear matter symmetry energy, and E sym,4 (ρ) the 4th-order nuclear matter symmetry energy. At normal nuclear matter den- sity ρ 0 the 2nd-order symmetry energy E sym,2 (ρ) is known to be around 30 MeV from the analysis of nu- clear masses within liquid-drop models, the 4th-order symmetry energy E sym,4 (ρ 0 ) has been estimated to be less than 1 MeV [7, 8]. At supra-saturation densities the higher order terms are found to have a significant contribution to the EOS [9]. Unfortunately, we have very limited knowledge about the density dependence of E sym,2 (ρ) and E sym,4 (ρ). During last few decades significant progress has been made experimentally and theoretically for constraining E sym,2 (ρ) around and below normal nuclear matter density [2–5, 10, 11] but at super-normal density its behavior is largely controversial [12–14], whereas the behavior of E sym,4 (ρ) is found to be model dependent at these densities [15–21]. Theoretically, almost all many- body theory calculations discussed in the literature so far revealed that the 2nd-order nuclear symmetry energy E sym,2 (ρ) positively characterizes the isospin- dependent part of the EOS of ANM and the higher- order terms in the isospin asymmetry are not so important, at least for moderate values of densities [6]. It may be a good approximation to the EOS of ANM, but at the same time it may cause large errors when it is applied to determine some special conditions. For example, the higher order terms in the isospin asymmetry presented in the EOS of ANM 1