aa r X i v : . [ a s t r o - ph ] A p r Neutral nuclear core vs super charged one
M. Rotondo, R. Ruffini and S.-S. Xue
ICRA and Physics Department, University of Rome“La Sapienz”, P.le A. Moro 5, 00185 Rome, ItalyICRANeT Piazzale della Repubblica, 10 -65122, Pescara, Italy
Based on the Thomas-Fermi approach, we describe and distinguish the electrondistributions around extended nuclear cores: (i) in the case that cores are neutralfor electrons bound by protons inside cores and proton and electron numbers arethe same; (ii) in the case that super charged cores are bare, electrons (positrons)produced by vacuum polarization are bound by (fly into) cores (infinity).
Equilibrium of electron distribution in neutral cores.
In Refs. [1, 2, 3], the Thomas-Fermiapproach was used to study the electrostatic equilibrium of electron distributions n e ( r )around extended nuclear cores, where total proton and electron numbers are the same N p = N e . Proton’s density n p ( r ) is constant inside core r ≤ R c and vanishes outside the core r > R c , n p ( r ) = n p θ ( R c − r ) , (1)where R c is the core radius and n p proton density. Degenerate electron density, n e ( r ) = 13 π ¯ h ( P Fe ) , (2)where electron Fermi momentum P Fe , Fermi-energy E e ( P Fe ) and Coulomb potential energy V coul ( r ) are related by, E e ( P Fe ) = [( P Fe c ) + m e c ] / − m e c − V coul ( r ) . (3)The electrostatic equilibrium of electron distributions is determined by E e ( P Fe ) = 0 , (4)which means the balance of electron’s kinetic and potential energies in Eq. (3) and degenerateelectrons occupy energy-levels up to + m e c . Eqs. (2,3,4) give the relationships: P Fe = 1 c h V ( r ) + 2 m e c V coul ( r ) i / ; (5) n e ( r ) = 13 π ( c ¯ h ) h V ( r ) + 2 m e c V coul ( r ) i / . (6)The Gauss law leads the following Poisson equation and boundary conditions,∆ V coul ( r ) = 4 πα [ n p ( r ) − n e ( r )] ; V coul ( ∞ ) = 0 , V coul (0) = finite . (7)These equations describe a Thomas-Fermi model for neutral nuclear cores, and have numer-ically solved together with the empirical formula [1, 2] and β -equilibrium equation [3] forthe proton number N p and mass number A = N p + N n , where N n is the neutron number. Equilibrium of electron distribution in super charged cores
In Ref. [4, 5], assuming thatsuper charged cores of proton density (1) are bare, electrons (positrons) produced by vacuumpolarization fall (fly) into cores (infinity), one studied the equilibrium of electron distributionwhen vacuum polarization process stop. When the proton density is about nuclear density,super charged core creates a negative Coulomb potential well − V coul ( r ), whose depth ismuch more profound than − m e c (see Fig. [1]), production of electron-positron pairs takeplaces, and electrons bound by the core and screen down its charge. Since the phase spaceof negative energy-levels ǫ ( p ) ǫ ( p ) = [( pc ) + m e c ] / − V coul ( r ) , (8)below − m e c for accommodating electrons is limited, vacuum polarization process com-pletely stops when electrons fully occupy all negative energy-levels up to − m e c , even electricfield is still critical. Therefore an equilibrium of degenerate electron distribution is expectedwhen the following condition is satisfied, ǫ ( p ) = [( pc ) + m e c ] / − V coul ( r ) = − m e c , p = P Fe , (9)and Fermi-energy E e ( P Fe ) = ǫ ( P Fe ) − m e c = − m e c , (10)which is rather different from Eq. (4). This equilibrium condition (10) leads to electron’sFermi-momentum and number-density (2), P Fe = 1 c h V ( r ) − m e c V coul ( r ) i / ; (11) n e ( r ) = 13 π ( c ¯ h ) h V ( r ) − m e c V coul ( r ) i / . (12)which have a different sign contracting to Eqs. (5,6). Eq. (7) remains the same. However,contracting to the neutrality condition N e = N p and n e ( r ) | r →∞ → N ion e = Z r πr drn e ( r ) < N p , (13)where r is the finite radius at which electron distribution n e ( r ) (12) vanishes: n e ( r ) = 0, i.e., V coul ( r ) = 2 m e c , and n e ( r ) ≡ r > r . N ion < N p indicates thatsuch configuration is not neutral. These equations describe a Thomas-Fermi model forsuper charged cores, and have numerically [4] and analytically [5] solved with assumption N p = A/ Ultra-relativistic solution
In analytical approach [5, 6], the ultra-relativistic approximationis adopted for V coul ( r ) ≫ m e c , the term 2 m e c V coul ( r ) in Eqs. (5,6,11,12) is neglected. Itturns out that approximated Thomas-Fermi equations are the same for both cases of neutraland charged cores, and solution V coul ( r ) = ¯ hc (3 π n p ) / φ ( x ), φ ( x ) = − h − / sinh(3 . − √ x ) i − , for x < , √ x +1 . , for x > , , (14)where x = 2( π/ / α / n / p ( r − R c ) ∼ . r − R c ) /λ π and the pion Compton length λ π =¯ h/ ( m π c ). At the core center r = 0( x → −∞ ), V coul (0) = ¯ hc (3 π n p ) / ∼ m π c . On coresurface r = R c ( x = 0), V coul ( R c ) = 3 / V coul (0) ≫ m e c , indicating that the ultra-relativisticapproximation is applicable for r < ∼ R c . This approximation breaks down at r > ∼ r . Clearly,it is impossible to determine the value r out of ultra-relativistically approximated equation,and full Thomas-Fermi equation (7) with source terms Eq. (6) for the neutral case, andEq. (12) for the charged case have to be solved.For r < r where V coul ( r ) > m e c , we treat the term 2 m e c V coul ( r ) in Eqs. (6,12) as asmall correction term, and find the following inequality is always true n neutral e ( r ) > n charged e ( r ) , r < r , (15)where n neutral e ( r ) and n charged e ( r ) stand for electron densities of neutral and super chargedcores. For the range r > r , n charged e ( r ) ≡ n neutral e ( r ) → V ( r ) in Eq. (6) is neglected.In conclusion, the physical scenarios and Thomas-Fermi equations of neutral and supercharged cores are slightly different. When the proton density n p of cores is about nucleardensity, ultra-relativistic approximation applies for the Coulomb potential energy V coul ( r ) ≫ m e c in 0 < r < r and r > R c , and approximate equations and solutions for electrondistributions inside and around cores are the same. As relativistic regime r ∼ r and non-relativistic regime r > r (only applied to neutral case) are approached, solutions in twocases are somewhat different, and need direct integrations. -40 -20 20 40 H r - R c L(cid:144)H Λ Π L -250-200-150-100-50V (cid:144) m e m e c - m e c m e c - V - m e c - V r ® FIG. 1: Potential energy-gap ± m e c − V coul ( r ) and electron mass-gap ± m e c in the unit of m e c are plotted as a function of ( r − R c ) / (10 λ π ). The potential depth inside core ( r < R c ) is about pionmass m π c ≫ m e c and potential energy-gap and electron mass-gap are indicated. The radius r where electron distribution n e ( r ) vanishes in super charged core case is indicated as r − , since itis out of plotting range.[1] J. Ferreirinho, R. Ruffini and L. Stella, Phys. Lett. B 91, (1980) 314.[2] R. Ruffini and L. Stella, Phys. Lett. B 102 (1981) 442.[3] R. Ruffini, M. Rotondo and S.-S. Xue, Int. Journal of Modern Phys. D Vol. 16, No. 1 (2007)1-9.[4] B. Muller and J. Rafelski, Phys. Rev. Lett., Vol. 34, (1975) 349.[5] A. B. Migdal, D. N. Voskresenskii and V. S. Popov, JETP Letters, Vol. 24, No. 3 (1976) 186,Sov. Phys. JETP 45 (3), (1977) 436., since itis out of plotting range.[1] J. Ferreirinho, R. Ruffini and L. Stella, Phys. Lett. B 91, (1980) 314.[2] R. Ruffini and L. Stella, Phys. Lett. B 102 (1981) 442.[3] R. Ruffini, M. Rotondo and S.-S. Xue, Int. Journal of Modern Phys. D Vol. 16, No. 1 (2007)1-9.[4] B. Muller and J. Rafelski, Phys. Rev. Lett., Vol. 34, (1975) 349.[5] A. B. Migdal, D. N. Voskresenskii and V. S. Popov, JETP Letters, Vol. 24, No. 3 (1976) 186,Sov. Phys. JETP 45 (3), (1977) 436.