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THE ATOMIC CHEMICAL ENVIRONMENT AND BETA ELECTRON FINAL STATE INTERACTION EFFECT ON BETA DECAY PROBABILITIES
Odessa National Polytechnical University, a/c 108, Odessa-9, 65009
Contributions of the atomic chemical environment effect and final states interaction of the beta electrons into the decay characteristics are calculated within a new theoretical, optimized gauge invariant Dirac-Fock approach. The results are presented for the non-unique transition of the first forbidding (241Pu241Am) and the super allowed transitions O+ O+.
Paper is devoted to calculating contributions of the chemical environment and final states interactions of the beta electrons into the decay characteristics on the basis of the new theoretical, optimized gauge invariant Dirac-Fock (GIDF) approaches [1-4]. In present time calculating the decay processes attracts a great interest especially due to the new experimental studies of the decay for a number of nuclei [5-12]. It is known that the disagreement between different experimental data regarding the -decay in heavy radioactive nuclei is provided by different chemical environment of the radioactive nucleus. More over, the contribution of cited effect over in the different chemical compounds differs from each other. In some papers such disagreement in data on the half-decay period for a number of isotopes (c.f. [10,11]) is explained by existence of the special beta decay channel. The beta particle in this channel does not transit into free state, but it occupies the external free atomic level. The differences in population of these levels are to be a reason of an influence of the atomic chemical environment on the beta decay. Usually one has to consider the following effects, connected with effect of the chemical bond: i.). the changing electron wave functions because of the changing atomic electric field due to the difference in the valence shells occupation numbers in different chemical substances; ii). a changing up limit of integration under calculating the standard Fermi integral function in different chemical substances. Due to the changing the nuclear charge Z on unite during the beta decay, this entire energy of electron shell of an atom changes in different chemical compounds by different way; iii). together with beta decay and ejection of the beta particle out atom it is possible additional channel when the beta electron occupies non-occupied place on the bonded external orbitals of atom. More detailed description of the chemical bond effect is given in ref. [1,9-11]. Another effect, which is usually not taken into account, is connected with the final state interaction of the beta electrons and related phenomena. In ref.  it has been calculated an effect of interaction in the final state between beta electron and atomic electrons with an accuracy to (Z/v)2 in the first non-vanishing approximation. In fact, this is a parameter of the Coulomb interaction between beta electron and atomic bound electrons and v is the beta electron velocity. It is not seldom, that this contribution can be quite essential.
Standard approach to calculating the beta decay characteristics is based on the usual non-relativistic Hartree-Fock or Hartree-Fock-Slater approach with account of the finite nuclear size. More correct calculation uses the relativistic approaches, in particular, the well known DF method Though the DF approach is one of the most wide-spread calculation methods, but, as a rule, the corresponding DF orbitals basis’s are not optimized. It often results to quite large mistakes in calculation of atomic characteristics. Besides, some problems are connected with correct definition of the nuclear size effects, QED corrections etc. In ref. [1,2] we have developed new ab initio schemes (GIDF and GI Dirac-Kohn-Sham schemes) to calculating spectra, wave functions basis’s of the heavy elements with account of relativistic, correlation, nuclear, QED effects, based on the gauge-invariant QED perturbation theory [15-17]. In ref.[3,4] we used these methods as basic in calculating an influence of the chemical environment on the decay characteristics for a number of the beta decays (33P33S, 63Ni63Cu, 35S35Cl).
Here we study contributions of the atomic chemical environment effect and final states interactions of the beta electrons into the decay characteristics. The new theoretical, optimized gauge invariant GIDF is used in calculation. The results are presented for the non-unique transition of the first forbidding (241Pu241Am) and the super allowed transitions O+ O+. Note that such a calculation provides a new methodology to studying the chemical bond and electron structure of different solid state compounds.
A distribution of the beta particles on energy in the permitted transitions is as follows:
Here G is the weak interaction constant; Е and р=(Е2-1)1/2 are an entire energy and pulse of beta particle; Е0=1+(Еbn /mec2) , Еbn is the boundary energy of β-spectrum; |M| is a matrix element, which is not dependent upon an energy in a case of the permitted β- transitions. As usually for calculation of the decay shape and decay half period one should use tables of the Fermi function and integral Fermi function. The Fermi function F and integral Fermi function f are defined as:
Here f+1 and g-1 are the relativistic electron radial functions; the indexes l=, where =(l-j)/(2j+1). Two schemes of calculation are usually used: i). the relativistic electron radial wave functions are calculated on the boundary of the spherical nucleus with radius R0 (it has done in ref. ); ii). values of these functions in the zero are used (see ref.). The half decay period can be defined as follows:
T1/2= 2π3 ln2/[G2 |M|2f(E0, Z)].
The normalisation of electron radial functions f and g provides the behaviour of these functions for large values of radial valuable as follows:
g (r)→r -1[(E+1)/E]1/2 sin(pr +), (3а)
f (r)→r -1(/||) [(E-1)/E]1/2 cos (pr+). (3б)
An effect of interaction in the final state between beta electron and atomic electrons with an accuracy to (Z/v)2 in the first non-vanishing approximation is described by diagrams, presented on
fig.1 . This contribution changes the energy distribution of the beta electron on value:
where in – the wave function of atom initial state, z is a number of electrons, aB is the Bohr radius.
Fig.1. The diagrams, which give a contribution into the final state interaction
in the first non-vanishing approximation
To calculate the relativistic atomic fields and electron wave functions, we have used the GIDF scheme. Its detailed description is given in ref.[1-5]. The GIDF equations for N-electron system are written and contain the potential: V(r)=V(r|nlj)+Vex+V(r|R), which includes the electrical and polarization potentials of the nucleus. The part accounts for exchange inter-electron interaction. Note that a procedure of the exchange account in the GIDF scheme is similar to one in the usual DF approach. The rest of the exchange-correlation effects are accounted for in the first two QED perturbation theory orders by the total inter-electron interaction. The optimization of the orbital basises is realized by iteration algorithm within gauge invariant QED procedure . Numerical calculation and analysis shows that used methods allow getting the results, which are more precise in comparison with analogous data, obtained with using non-optimized basises. Approach allows calculating the continuum wave functions with the entire exchange account of the continuum electron with electrons of the atom.
We have calculated the atomic chemical environment and final state interaction effects on the decays for the transition 241Pu241Am (it is non-unique of the first forbidding) and the super allowed transitions O+ O+. Calculation has shown the significant changing the beta decay probability and value of T1/2 for the 241Pu. According to our calculation, the Fermi function was less for decay of the doubly ionized Pu (on 0,71%) and, as result, decay runs more slowly and, other versa, the Fermi function is more for neutral Pu and its decay runs more quickly. The typical example of the Pu compounds is pair PuX4 and PuX . In table 1 we present our results for corrections ft to probability of the super allowed transitions O+ O+ due to the final state interaction effect (FSI). There are also presented the compilated data (without accounting for the final state interaction effect) and the estimates from ref. within the Durand approach. The FSI correction may not be directly defined in the heavy nuclei due to the great indefiniteness in nuclear matrix elements. Exception is the super allowed transition O+ O+. These transitions are used for definition of the weak interaction constant GV (ft~ GV2). As it has been seen there is quite significant difference between presented values for different nuclei. One can suppose that there is systematic defect of existing theory of accounting corrections, which increase with the growth of Z.
As an analysis shows, our data can be considered as the most correct ones.
Corrections to probability of the super allowed transitions O+ O+ due to the final state interaction effect (FSI)