The Fifth Conference “ Nuclear Science and Its Application”, 14-17 October 2008
EMPIRIC UNIVERSAL FUNCTION OF b(A) FOR WIGNER MASS
FORMULA
A.M. NURMUKHAMEDOV
Institute o f Applied Physics under the National University o f Uzbekistan,
Aiming on analytical description of empirical universal function of b(A) for Wigner mass formula the analysis of nuclear masses was performed in the framework of Wigner mass formula which has an appearance as:
M (A, Z ) = a(A) + b(A)C2 + E Coul (A, Z ) + Esl (Z, N ) + E paır (Z ,N ), (I)
where a(A) b(A) are Wigner empiric universal functions, C2 is a Casimir operator, ECoul (A, Z ) is a Coulomb energy of nuclei, EsI(Z ,N ) is an energy of spin-orbit interaction and Epair (Z> V) is a pair energy. To simplify a task there were considered only nuclei with odd mass
number ranged in 1 < A < 257 and where E pmr (Z ,N ) = 0 by definition. As a selection criteria the certain the features of b(A) have been used e.g. universality for the given isobar and its smooth dependency from the mass number. Universal functions b( A) have been calculated for different
A by the formula:
b(A)
=^mıcl
^ ~^ nud
^ ~ ^ +^Eçpui (A, Z)
^ (T +1.5) . . .
The expression (2) obtained in assumption of universality of the function a( A) . For the
(2)
expression (2) A E ( A ,Z ) = 703 ,2 (2 Z + l)A ~ l/3 (1 - 1 .2 8 A~2n ) ± 4.5 (keV) is a difference of Coulomb energies o f neighboring nuclei of isobar [1]. For (2) A mic/(A ,Z ) the surplus of the nuclear mass was recalculated from the surplus of the mass of neutral atom as a difference of the surplus of mass of neutral atom and the mass of Z electrons taking into account summative binding
energy of electrons: A mcl ( A ,Z ) ~ A M (A ,Z ) - Z m c '- + Y 1E„ (3)
M
where A algm( A ,Z )- i$ a surplus of the mass of neutral atom taken from [2], m - is a
z
rest mass of electron and 'Y : El - is a summative binding energy of electrons for atom with order number Z . Calculating b('l) by formula (2) it was not taken into account the difference o f energies of spin-orbit interaction of neighboring nuclei of the same isobar.
In total the analysis used masses of about 900 nuclei with odd mass number. Values of b(A) were selected in a view of compliance with the requirements of universality and smoothness of the function b(A) . The amount o f “selected” nuclides is 101.
The analysis of universal function b(A) for the “selected” nuclei using the method of the least squares allowed describing experimental values of b(A) by the empirical function:
b ( A ) = bt e x p (b2A ) + b3 e x p (b4A]( 1 — e x p j— (Tz — 0 .5 ) /0 .5 ] } . (4)
Numeric values o f factors in (4) are: bx =1522(47) keV, b2 = -0.0029(1), b3 = 4123(200)
Section II. Basic Problems O f Nuclear Physics
The Fifth Conference “ Nuclear Science and Its Application”, 14-17 October 2008
keVand b4 =-0.0238(8). References:
1. Yu.V. Gaponov, N.B. Shulgina, D.M. Vladimirov, Nucl.Phis. A391, 93 (1982). 2. G. Audi, A.H. Wapstra and C. Thibault, Nucl.Phys. A729, 337 (2003).
Section II. Basic Problems Of Nuclear Physics
