FIRST FORBIDDEN β-DECAY MATRIX ELEMENTS ON THE PARAMETERS IN WOODS-SAXON POTENTIAL

Abstract-In this paper, 0→0 first forbidden β-decay matrix elements have been calculated. For Pb→Bi transitions relativistic M(ρA,λ=0) and nonrelativistic M(jA,κ =1,λ=0) matrix elements have been calculated. In this calculations, the eigenfunctions and eigenvalues of Schrodinger equation solved by the more realistic Woods-Saxon potential have been used. The dependence of the radial parts in the matrix elements on the parameters (r0: the radii of nucleus, η: isovector parameter, rc: Coulomb radius and χ diffusion parameter) of Woods-Saxon potentialhas been investigated.


INTRODUCTION
First forbidden beta decay process have been studied "for single-particle configurations in the region of 208 Pb" by Bohr [1].First forbidden resonance (FFR) has been studied to confirm that the spin-isospin vibration mode of (p, n) reactions exists [2][3][4].Descriptions of the Gamow-Teller Resonance (GTR) and FFR have been reported for deformed nuclei [5,6].Civitarese & et al. studied the Collective Effects of Charge-Exchange Vibrational modes on 0 -→0 + and 2 -→0 + first-forbidden β-decay [3,7].As known, the theoretical description of nuclear double beta decay processes is one of the open questions in the field of nuclear structure theories.The use of intermediate virtual excitations other than the allowed ones has been advocated by some authors [7][8][9] in spite of the fact [10] that the leptonic wave functions include terms which are proportional to the product of the electron or neutrino velocity with the nuclear radius.
Many researchers preferred "the use of virtual excitations" (which include additional terms) instead of "allowed ones".Generally, the calculations for 0 -→0 + First Forbidden β-Decay have been made in the base of harmonic oscillator [3][4][5].
In this study, 0 -→0 + first forbidden β-decay transitions for the nucleus 206- 214 Pb→ 206-214 Bi have been studied.The calculations have been made in the Woods-Saxon potential base.M ± (ρ A ,λ=0) matrix element has been calculated analytically.Dependency of the Radial parts of the matrix element on the Woods-Saxon potential has been examined.

FORMALISM
In the "ξ approximation" the decay rates for 0 -→0 + transition can be written in the following form [1]. where, Relativistic matrix element is: and nonrelativistic matrix element is: In these notes the upper and lower signs refer to β -, β + decays, respectively.g A and g v refer to axial and vector coupling constant respectively.It can be shown that reduced matrix elements of the operators given in the formulas (2), (3) are proportional to the expressions given below [11][12][13].
( ) ), Where, l n ψ is the radial part of wave function, n is the main quantum number and l is the orbital angular momentum quantum number.

RESULTS AND DISCUSSIONS
In our calculations, the eigenfunctions and eigenvalues of Schrodinger equation solved by Woods-Saxon potential have been used.The matrix elements which are calculated with the formulas (2) and (3) depend on the Woods-Saxon potential parameter [13].The dependency of the radial section on the potential parameter has been examined as that dependency occurred with radial part.
The reduced matrix elements determining the transition probability from the ground state (0 + ) of 210 Pb isotope to the first 0 -state of 210 Bi isotope have been calculated.Since the first 0 -state of 210 Bi isotope is not collective the structure of this state is assumed to be ( ).Therefore, the reduced matrix elements of the operators given in the formulas (2) and (3) for the transitions between the 0 + ground state of 210 Pb and 0 -state of 210 Bi have been calculated and the dependence of their radial parts on Woods-Saxon potential parameter have been investigated.The calculation results are shown in Figs.1-3.
The dependence of the radial integrals in relativistic and nonrelativistic matrix elements on the radii of nucleus (r 0 ) which is one of the parameters in the average field potential is shown in Figs.1a and 1b.It is clearly seen that there is no change in the radial parts of these matrix elements even though the radii of nucleus varies in the range of (1.24-1.34)fm.

Fig. 1a :Fig. 1b :
Fig.1a: For 210 Pb → 210 Bi transition, the dependence of the radial part of 0 + → 0 -first forbidden beta decay relativistic matrix elements on the radii of nucleus