Theoretical Determination of Level Spins of Superdeformed Bands for Nuclei in the Mass Region A = 80 – 104

The bandhead spins of seventeen superdefomed bands in A = 80 – 104 region (38Sr, 39Y, 40Zr, 41Nb,42Mo, 43Tc, 46Pd) have assigned by an indirect method. The dynamical moment of inertia J(2) as a function of rotational frequency ђω are extracted from Harris expansion and fitted to the experimental values by using a computer simulated search program. The calculated dynamic moment of inertia with the best optimized parameters are integrated to give the spins. The intrinsic aligned angular momentum (the integration constant) is assumed to be zero. The values of the spins resulting from our approach are consistent with all spin assignments of other approaches, and have been used to determine the kinematic moment of inertia J(1). The systematic variation of J (2) and J (1) with rotational frequency ђω is investigated, which turns out to be helpful in the spin prediction. Most SD bands in this mass region exhibits decreasing in J(1) and J(2) with increasing ђω. The bandhead moment of inertia J0 which occur at J(2) = J (1) has been sensitive guideline parameter to spin proposition. The relationship between the Harris expansion three parameter model and the four parameter Bohr-Mottelson formula is derived.


Introduction
Study of superdeformed (SD) nuclei has been one of the most exciting fields in nuclear spectroscopy since the discovery of SD states at high spins in 152 Dy [1].More than 330 SD bands have now been observed in the mass regions A ≈ 30, 60, 80, 130, 150 and 190 [2,3].Many theoretical and experimental efforts were devoted to explore the nature of SD states in nuclei.The SD mass region A ≈ 80 -100 is very interesting region because they exhibit highest rotational frequencies.
In SD bands, gamma ray transition energies are the only spectroscopic information available till now.However, level spins, parities and excitation energies in most of these bands were not determined experimentally because linking transitions between the SD states and the normal deformed (ND) states were not observed.In the past few years several empirical and semiempirical approaches were proposed for the spin assignments in SD bands [4][5][6][7].All these available approaches obtained mainly from the comparison of the calculated gamma transition energies or dynamical moments of inertia with experimental results.In previous papers we have used Harris ω 2 expansion [8][9][10][11][12], Bohr-Mottelson I (I+1) expansion [13], ab expression [14,15] and variable moment of inertia (VMI) model [16,17] to assign spin.
The main purpose of the present work is to determine the spins of energy levels of some SD bands in the mass region 80 ≤ A ≤ 104 and examine the behaviors of moments of inertia.We will use the Harris expansion and its relationship with the Bohr-Mottelson formula.The paper is arranged as follows: Following this introduction, the Harris and Bohr-Mottelson expansions employed to assign spins are presented and discussed in the next section (2).Numerical calculations and discussion are performed in section (3) for even-even and odd -A SD nuclei in the mass region A = 80 -104.The data set include 17 SD bands in Sr / Y / Zr / Nb / Mo / Tc / Pd nuclei.Conclusion and remarks are given in section (4).
In this section, we will fit the experimental dynamical moment of inertia values with the Harris power series formula [18].The expansions parameters obtained from the fitting will be used to assign the spins.In such parameterize the spin may be expressed as an expression in the rotational frequency.Also the relation between Harris expansion which depend on the rotational frequency and the Bohr-Mottelson formula which depends on the spin will be derived.
The nuclear energy E of the nucleus can be expanded in powers of angular velocity ω by Harris expansion [18] as an extension of cranking model: Where: only even powers of ω are present in systems invariant with respect to time reversal.
In general, the above Harris expansion converges faster than the Bohr-Mottelson expansion [19] in powers of I (I+1): =    + 1 +    + 1 2 +    + 1 3 +    + 1 4   (2 where A is the rotational constant parameter and B, C and D are the corresponding higher order constant parameters. In framework of nuclear collective rotational model, two types of moments of inertia are usually discussed, which are related to the first and second order derivatives of the excitation energy with respect to the angular momentum.We define the second order derivative dynamical moment of inertia by:  (2)    The corresponding expression for formulae (1) and ( 2) are: The parameter α corresponds to the bandhead moment of inertia.
For SD bands experimentally, the rotational frequency ђω, the dynamic J (2) and kinematic J (1) moments of inertia are usually extracted from the observed transition energies Eγ between two consecutive quadruple transition within a band from the following finite difference approximations, ℏ = where, the experimental γtransition energies of the SD band is in MeV.
It is seen that while the extracted J (1) depends on the spin I proposition, J (2) does not (see equations 27, 26).Thus, if the dynamic moments of inertia J (2) were a constant, the transition energy difference would be the same for all values of spin.Often this is not the true and J (2) is found to change with increasing spin.The two moments of inertia can related as follows: (2)   ℏ 2 =  (1)  = 1 ℏ 2  (1) +   (1)    (28) Solving for J (1) , yield: (1) =  (2) +  .

Numerical Calculations and Discussion
For SD bands, gamma-transition energies Eγ are the only spectroscopic information universally available.The information about Eγ are commonly translated into values of rotational frequency ђω equation ( 25) and dynamical moment of inertia J (2) equation ( 26).One of the most supervising characteristic of data on SD bands is the different behavior of J (2) as a function of ђω.
The optimized expansion parameters α, β, γ of J (2) values in the Harris parameterization for each SD band have been calculated from best fit method [12] to the experimental J (2) values extracted from Eγ.The quality of the fit is indicated by the common χ quantity in order to obtain a minimum rootmean square (rms) deviation.N is the total number of experimental points entering into the fitting procedure.It was found that the rms deviation of the calculated results with experiments, χ, depends on the number of transitions involved, and in some cases χ is insensitive to the suggested spin, that is the rms deviations may be close to each other for two or more spin propositions in this case, it is difficult to make a unique spin proposition.
The best adopted optimized parameters α, β and γ obtained from the fitting procedure have been used to determine the spins with the help of equation (5).The constant of integration i0 which represent the aligned angular momentum at zero frequency has been taken to be zero.The resulting best parameters α, β and γ and values of the lowest bandhead spin I0 and the bandhead moment of inertia J0 = α are listed in Table (1).The data set include 17 SD bands in A = 80-104 mass region for Strontium (38Sr), Yttrium (39Y), Zirconium (40Zr), Niobium (41Nb), Molybdenum (42Mo), Technetium (43Tc) and December 12 , 2014 Palladium (46Pd).The experimental data of transition energies are taken from references [1,2].Table (2) lists the optimized parameters A,B,C and D of the Bohr -Mottelson four expansion.
Using our assigned spin values, the Kinematic moment of inertia J (1) of the SD bands can be consequently determined.The evolution of dynamic J (2) moments of inertia as a function of rotational frequency ђω are illustrated in Figure (1).It is seen that most SD bands exhibits decreasing J (2) with increasing ђω.
Table (1 moments of inertia are platted a function of rotational frequency ћω.The experimental J (2) are labeled by closed circles.

Conclusion
The main conclusion of the present work can be summarized as follows: The transition energies of SD nuclei in the mass region A = 80 -104 can be quantitatively described excellently by Harris expansion to third term.The dynamical moment of inertia J (2) has been derived in terms of Harris parameters.The optimized parameters have been adjusted by using a computer simulated search program to fit the calculated theoretical J (2) with the corresponding experimental values.The bandhead spins have been assigned by integrating J (2) and using the best optimized parameters.The bandhead spins of our selected SD bands from the present study are excellent consistent with all spin assignments of other approaches.The calculated transition energies, level spins, rotational frequencies, kinematic and dynamic moments of inertia and bandhead moments of inertia are analyzed as a function of rotational frequency.It was found that the bandhead moments of inertia are helpful guide line in the spin prediction.
( + 1)rather than angular momentum I provide the proper limiting case for an ideal rotor with energy proportional to the quintal square I (I+1) rather than I 2 .