0→0 FIRST FORBIDDEN β-DECAY MATRIX ELEMENTS IN SPHERICAL NUCLEI

In this study, the first forbidden beta decay matrix elements have been directly calculated within Saxon-Woods potential. Procedures for calculating the relevant matrix elements and combining them to form the decay rate are described. Calculations have been performed by two different methods. The values of our single particle matrix elements have been compared with the calculated values in the two different tables. KeywordsBeta decay matrix element. 1.INTRODUCTION The first forbidden beta decay matrix elements have been studied "for singleparticle configurations in the region of Pb" by Bohr [1]. The use of intermediate virtual excitations, other than the allowed ones has been advocated by some authors [2-9] in spite of the fact that the leptonic wave functions would include terms which are proportional to the product of the electron or neutrino velocity and the nuclear radius. Many researchers preferred "the use of virtual excitations which include additional terms instead of "allowed ones" [10,11]. In Ref. [11], the matrix element of the relativistic beta moment M(ρA,λ=0) has not been calculated analytically, but assumed to be proportional to matrix element of non-relativistic beta moment iM(jA,k=1,λ=0). Generally, calculations have been made in the base of harmonic oscillator [11,12]. In our study, the first Forbidden β-Decay Matrix Elements 0→0 for some nuclei have been studied. Our calculations have been done by two different methods. In the first method, the relativistic beta moment has been calculated directly without any assumption. In the second method, it was assumed that the relativistic beta moment is proportional to the non-relativistic one. Accordingly, our calculations have been done in the Saxon-Woods potential base. M(ρA,λ=0) matrix element has been calculated analytically. The theoretical approach for our calculations is given in Sect.2. Dependency of the radial parts of the matrix element on the Saxon-Woods potential and some conclusions are discussed in sect.3. 2. FORMALISM In the “ξ approximation” the decay rates for 0→0 transition can be written in form [1]. π λπ 4 / ) 0 ( 2 0 V Dg B t f = = ⋅ − (1) where İ. Kenar, C. Selam and A. Küçükbursa 180 = = − ) 0 (λπ B 1 2 1 + i I 2 ) 0 , 1 , ( . ) 0 , ( i A e A f I j c m i I = = Μ − = ± λ κ ξ λ ρ m m h M . “First” matrix element is: ( ) ( ) ∑ ⋅ = = − k k A A v k k t c g M ) ) ( )( ( 4 0 , 2 1 σ π λ ρ m m . (2) “Second” matrix element is: ( ) ∑ = = = k k k A A k r Y r k t g j M 0 1 )) ( ) ( ( ) ( 0 , 1 , σ λ κ ) m m , (3) 3 / 1 . 2 . 1 − ≈ A Z ξ , D=6250sec. In these equations the upper and lower signs refer to − β and + β decays, and V g , A g are vector and axial vector coupling constant, respectively. The following formulas have been used in the calculations of the matrix elements of the operators that are given in Eqs. (2) and (3) [13]. ( ) np p n p n n J k k F J J n J k r Y r p J n 0 010 1 2 1 2 1 4 ) 1 2 )( 1 2 ( 6 . ) 1 ( , ) ( ) ( , 2 / 1 0 1 l l l l l ) l


1.INTRODUCTION
The first forbidden beta decay matrix elements have been studied "for singleparticle configurations in the region of 208 Pb" by Bohr [1].
The use of intermediate virtual excitations, other than the allowed ones has been advocated by some authors [2][3][4][5][6][7][8][9] in spite of the fact that the leptonic wave functions would include terms which are proportional to the product of the electron or neutrino velocity and the nuclear radius.Many researchers preferred "the use of virtual excitations which include additional terms instead of "allowed ones" [10,11].
In Ref. [11], the matrix element of the relativistic beta moment M ± (ρ A ,λ=0) has not been calculated analytically, but assumed to be proportional to matrix element of non-relativistic beta moment iM ± (j A ,k=1,λ=0).Generally, calculations have been made in the base of harmonic oscillator [11,12].
In our study, the first Forbidden β-Decay Matrix Elements 0 -→0 + for some nuclei have been studied.Our calculations have been done by two different methods.In the first method, the relativistic beta moment has been calculated directly without any assumption.In the second method, it was assumed that the relativistic beta moment is proportional to the non-relativistic one.Accordingly, our calculations have been done in the Saxon-Woods potential base.M ± (ρ A ,λ=0) matrix element has been calculated analytically.
The theoretical approach for our calculations is given in Sect.2.Dependency of the radial parts of the matrix element on the Saxon-Woods potential and some conclusions are discussed in sect.3.

FORMALISM
In the "ξ approximation" the decay rates for 0 -→0 + transition can be written in form [1].
"First" matrix element is: "Second" matrix element is: In these equations the upper and lower signs refer to − β and + β decays, and V g , A g are vector and axial vector coupling constant, respectively.The following formulas have been used in the calculations of the matrix elements of the operators that are given in Eqs. ( 2) and (3) [13].
( ) where F np is the neutron-proton overlap integral: , R nl is the radial part of single particle wave function, and with U l = rR nl .

RESULTS AND DISCUSSIONS
In the present calculations, it has been used the eigenfunctions and eigenvalues of Schrödinger equation which is solved to Saxon-Woods Potential [14].
The calculations have been done for transition, namely In the following tables, values for the single particle of the first and second matrix element calculated to a few nuclei with Saxon Woods potential by using the Chepurnov parameters [14] are given.As seen from the tables, the values we have calculated are different from calculations of Ref. [11], i.e. approximately 1.2-1.4times larger.The present calculations are very close to the calculations of Bohr and Mottelson.Although our calculations are 2.5 times lower than the ones given in the Ref. [11] for the transition of 3P 1/2 (n) → 3S 1/2 (p), they are 1.5 times larger for transition 2G 9/2 (n) → 1H 9/2 (p).However, the present calculations are also very close to the calculations of Bohr and Mottelson model for the first matrix element.

Table-1. The values of reduced single particle matrix elements for the
According the present calculations, the following conclusions can be stated.The values of first matrix element, which we have calculated directly, are different from the values calculated in the Ref. [11].
The approximate calculations of Bohr and Mottelson for neighboring nucleus to the 208 Pb nucleus are very close to our calculations.

Fig. 1 .Fig. 2 .
Fig. 1.The wave functions at the radial integral of the first matrix element

Table - 2
. The values of reduced single particle matrix elements for the