THE INVESTIGATION OF SUPERALLOWED BETA DECAY BY PYATOV METHOD D.I.Salamov (1) , A.E.Calik (2) , M.Gerceklioglu (2) , N.Cakmak (1) (1) Anadolu University, Science Faculty, Physics Department, ESKISEHIR, TURKEY
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D.I.Salamov(1), A.E.Calik(2), M.Gerceklioglu(2), N.Cakmak(1)
(1) Anadolu University, Science Faculty, Physics Department,ESKISEHIR, TURKEY
(2) Ege University, Science Faculty, Physics Department, Bornova – IZMIR 35100, TURKEY
In this study, the superallowed beta decay properties have been investigated byPyatov’s method . According to method, Fermi effective interaction strength parameter has been obtained in a self-consistent way. The details of method can be found in references  and . The calculated ft values for superallowed beta decay have been shown in the table.
The establishment of the new proton-rich nuclei with the development of heavy mass ion technology allows us to study the isospin mixing effects. The isospin mixing effects are more important in determination of the effective vector coupling constant based on the superallowed Fermi transitions, for the description of energies and widths of analog states, for energies of isospin multiplets, etc. [4-7]. The isospin mixing is basically caused by the Coulomb potential, or more accurately by that part of the Coulomb potential which changes over the nuclear volume. Thus, isospin remains a good quantum number for the low-lying states of pratically all nuclei . Coulomb forces are more dominant in nuclei for which the proton and the neutron number are close to each other.
The isospin admixtures in the ground states have been calculated in various models. The two liquid hydrodynamic model has been used to estimate the energies of the collective isovector monopole excitation with an isospin of T=T0+1 . Quantative estimates performed by using shell model [10-12] are approximately an order of magnitude larger than the estimate of Bohr and Mottelson.
In the present study, based on the spherical single-partical wavefunctions and energies with the pairing interaction and the residual isovector interaction treated by QRPA, the ft values of the superallowed beta decay were calculated. In these calculations, considering the restoration of the isotopic invariance for the nuclear part of the total Hamiltonian [1,13], the effective interaction strength (γρ) in the quasiparticle space has been obtained in such a way that it is self-consistent with the Woods-Saxon form of the shell-model potential.
However, at most of the previous studies the effect of the pairing correlations have not been considered. There are two ways to consider the pairing forces in this problem. At the first one, the work what should be done is to restore the breaking of the isospin invariance of the Hamiltonian in the particle space and later is to add the pairing forces to the Hamiltonian in a isospin invariant manner. At the second one, the breaking in question is restored in the quasiparticle space. In this study, following the second avenue. This method makes our model free of any adjustable parameters.
2. Pyatov’s Method
Let us consider a system of the nucleons in aspherical symmetric average field interacting via pairing forces. In this case, the corresponding single quasiparticle Hamiltonian of the system is given by
We obtain the following orthonormalization condition for the amplitudes;
The eigenvalues and the eigenfunctions of the restored Hamiltonian can be obtained by solving the equation of motion in QRPA
Here the ωi are the energies of the isobaric 0+ states. Employing the conventional procedure of QRPA, we obtain the dispersion equation for the excitation energy of the isobaric 0+ states as follows;
The amplitudes for the neutron-proton quasiparticle pair can then be expressed analytically in the following form:
The quantity Z(ωi) is determined from the normalization condition given above.
4. Fermi Beta Transitions
The isobaric 0+ states in the neighbour odd-odd nuclei (N-1,Z+1 and N+1,Z-1) are characterized by the Fermi transition matrix elements between these states and the ground state of the neighbour even-even nuclei . One could obtain the following Fermi transition matrix elements by using the wavefunctions above:
a) for the transitions (N,Z) → (N-1,Z+1);
Where the εj is the single quasiparticle energy of the nucleons and the α+(α) is the quasiparticle creation(annihilation) operator. The shell model potential contains the isovector terms in nuclei having different proton and neutron numbers in addition to the Coulomb potential and the neutron-proton mass difference which break the isotopic invariance
b) for the transitions (N,Z) → (N+1,Z-1);
where VC is the Coulomb potential given by the following expression:
It is possible to show that the transitions in question obey the Fermi sum rule
The isospin operators Tρ are defined in the following way:
5. Numeric Calculations
ft values calculated for superallowed beta decay from the method mentioned above have been compared with experimental and reference  and  results given in the table below. The second column shows the ft calculation without pairing and the third column shows with pairing.
The broken symmetry should be restored by the residual interaction h which obeys the following equation [1,13]:
where h is defined by Pyatov and Salamov  as;
Table.ft values for superallowed Fermi transitions in considering nuclei.
γρ is an average of double commutator in the ground state;
3. Isobar Analogue States
We shall consider the isobaric 0+ excitations in odd-odd nuclei generated from the correlated ground state of the parent even-even nucleus by the charge-exchange forces and use the eigenstates of the single quasiparticle Hamiltonian Hsqp as a basis. The basis set forthe neutron-proton quasiparticle pair creation A+pn and annihilation Apn operators is defined as
The bosonic commutation relations of these operators in quasiboson approximation are given by
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The form of h and γρ in quasipartical space is given
In QRPA, the collective 0+ states are considered as one phonon excitations described as follows;
Ψnp and φnp are the amplitudes for the neutron-proton quasiparticle pair. The Q+i is the phonon creation operator. The |0 is the phonon vacuum which corresponds to the ground state of the even-even nucleus;