Nuclear Properties of ν g 9 / 2 + Isomers in Odd 59 Cr to 69 Se Nuclei for N = 35

We have reported the systematic magnetic quadrupole reduced transition probabilities B (M2) between 9/2+ and 5/2in odd 59Cr to 69Se nuclei for neutron N=35. At present work the calculated B(M2) values are 0.079 W.U., 0.067 W.U., 0.230 W.U., 0.412 W.U., 0.060 W.U., and 0.044 W.U. for 59Cr, 61Fe, 63Ni, 65Zn, 67Ge, and 69Se nuclei respectively. The theoretical calculations of the reduced transition probabilities were compared to the previous experimental values. The width of isomeric levels, mean-life and Weisskopf hindrance factors in the odd 59Cr to 69Se nuclei were also calculated. The isomeric levels 9/2+ were systematically studied as a function of even atomic number from 24 to 34. It is shown that the isomeric levels are increases from Z= 24 to Z= 28 and then decreases monotonically up to Z= 34.


Introduction
Understanding the nuclear properties and its structure has been a crucial study.A theoretical study of nuclear structure can be predicted through several different nuclear models but some of them can explain some limited features of the nucleus.Nuclear shell model has explained about the properties of nuclear isomers in the vicinity of closed shells excited near to the magic numbers.The low-lying excited states in nuclei are widely spaced and can be explained by the shell model by considering several potential.A long-lived excited state of a nucleus which has been called isomers is near to shell closure or magic numbers (Baglin, 2002;Bhat, 1999;Bhat & Tuli, 2000).Broda et al. found sub-shell closure in 68 Ni for N=40 experimentally (Broda et al., 1995).This nucleus containing double magic nucleons (Z=28, N=40) and 2 + level in 68 Ni is very high compare to neighbor nucleus.Usually isomers belong to shell closure.Even-odd nuclei 59 Cr, 61 Fe, 63 Ni, 65 Zn, 67 Ge, and 69 Se belong near to 68 Ni.The M2 transitions of those nuclei were established due to νg (9/2) n configuration (Baglin, 2002;Bhat, 1999;Bhat and Tuli, 2000;Broda et al., 1995;Brown, 2010;Erjun, 2001;Junde et al., 2005;Hossain et al., 2011).
The isomeric properties of odd-even Arsenic (As) nuclei of πg (9/2) n configuration raised the possibility to calculate the isomerism of even-odd nuclei νg (9/2) n configuration.Single particle nuclear transition is governed when the nuclear excitation is due to only one nucleon.A study of multipolarity transition in neutron rich nuclei has been made by using shell model calculation (Hossain et al., 1998).It showed that proton motion and neutron motion are collective for quadrupole transition (Umeya et al., 2006).Quadrupole magnetic (M2) transition between the state 9/2 + to 5/2 -of odd As isotopes with A = 67-79 were observed (Hossain et al., 1998).Moreover, the systematic mean lives, reduced transition probabilities, width of isomeric levels, and Weisskopf hindrance factors in odd 59 Cr to 69 Se nuclei were not investigated yet.In the present work we investigate the properties of isomers of those nuclei systematically by theoretical calculation.

Reduce Transition Probabilities B(M2)
In order to obtain the total transition probabilities that determine the lifetime of the initial state we have to sum over all the possible substate μ and m f .Now the total transition probabilities are where the quantity is called the "reduced transition probability".While the transition probability depends on the transition energy, being related to E 2L +1 , the reduced transition probability does not depend on energy, but is the square of transition matrix element.Theoretical evaluation of reduced transition probability involves nuclear quantizes and depends on specific models of the nucleus.
By taking the assumption that the nucleus is spherical and the gamma emission is the result of a transition of a single particle from one state to another, Weisskopf estimated for the reduced transion probabilities: The reduced transition probabilities B(M2) are determine the structure and the wave functions of nuclei, comparison between experimental and theoretical of B(M2) is a good investigation.The equation below is used to find the B(M2) of the isomers, B ( M 2 ; I i → I f ) = 7.381 x 10 -8 Eγ -5 Pγ (M2; where, I i and I f are the spin of the initial and final states respectively, Eγ = gamma energy The Weisskopf estimate of B w (M2) is given by B w (M2) = 1.65A 2/3 μ fm So, the magnetic reduce transition probability for M2 transition is where, T 1/2 = half-life So, = / .

Width of Isomeric Levels, Гγ
Width of isomeric level can be calculated by using equation below which indicates the thickness of gamma ray produced from the transition.

Weisskopf Hindrance Factor, F w
Hindrance factor is useful to give a test with which different γ-ray transition probabilities can be compared and others that might perturb the lifetime of the nuclear state.Weiskopf Hindrance Factor can be determined by; where, ( ) = 1.65

Results and Discussion
Table 1 shows the calculations of reduced transition probabilities B(M2), M2 gamma transition between 9/2 + to 5/2 -, isomeric levels, mean life, width of isomeric levels and Weiskopf hindrance factor of odd 59 Cr to 69 Se for N=35 which were presented at the conference (Ghani et al., 2012).

Systematic Reduced Transition Probabilities
Experimental data have shown the reduced transition probabilities of odd 59 Cr to 69 Se nuclei with neutron N=35 (Baglin, 2002;Bhat, 1999;Bhat and Tuli, 2000;Brown, 2010;Erjun, 2001;Junde et al., 2005;Hossain et al., 2011).All transitions between 9/2 + to 5/2 -have been assigned to M2-type based on the selection rule.At present work the B(M2) values are 0.079 W.U., 0.067 W.U., 0.230 W.U., 0.412 W.U., 0.060 W.U., and 0.044 W.U. for 59 Cr, 61 Fe, 63 Ni, 65 Zn, 67 Ge, and 69 Se nuclei respectively.Figure 2 shows the comparison between the experimental and theoretical data.The calculated data showed that B (M2) values of the isomers are good agreement with the previous experimental results except for 65 Zn.Because branching ratio of gamma ray in 65 Zn is very small (10%), which showed strong deviation from experimental values.The maximum magnetic quadrupole reduced transition probability is at 65 Zn.The B(M2) values as well as the isomeric level 9/2 + increases towards magic number Z=28, and then M2 strength and isomeric level decreases as neutron number increases except for 65 Zn.The B(M2) values in 65 Zn seem to relate to the configuration of ground state of 5/2 - states as well as to the deformation of the nuclei.

Width of Isomeric Levels (Γγ)
The widths of isomeric levels were calculated using the Equation ( 8).The width indicates the thickness of gamma rays produced from the transition.The value of width of isomeric levels are 47.64 eV, 18.30 eV, 1.3 x 10 3 eV, 7.9 x 10 3 eV, 31.The Weisskopf hindrance Factor, F w was calculated according to Equation ( 9).The values of hindrance factors are 12.62, 15.13, 4.34, 2.41, 16.57, and 22.83 of 59 Cr, 61 Fe, 63 Ni, 65 Zn, 67 Ge, and 69 Se nuclei, respectively.The minimum value and maximum value of hindrance factor is 2.41 and 22.83 for 63 Ni and 69 Se, respectively.

Conclusion
The systematic reduced transition probabilities B(M2), width of isomeric level, mean-life and Weisskopf factor hindrance were calculated in odd 59 Cr to 69 Se nuclei for N=35.The theoretical calculation of reduced transition probabilities B (M2) is good agreement with the previous experimental values (Baglin, 2002;Bhat, 1999;Bhat and Tuli, 2000;Brown, 2010;Erjun, 2001;Junde et al., 2005;Hossain et al., 2011).The isomeric level and B(M2) values are increases with proton towards magic number Z=28, and then M2 strength and isomeric level decreases as proton number increases except for 65 Zn.

Table 1 .
33 eV and 4.76 eV for 59 Cr, 61 Fe,63Ni, 65 Zn, 67 Ge, and 69 Se nuclei respectively.It indicates that width of isomeric levels have no correlation with increase of neutrons of 59 Cr, 61 Fe,63Ni, 65 Zn, 67 Ge, and 69 Se nuclei.Isomeric properties of odd 59 Cr to 69 Se nuclei for N=35