S tructure of Superdeformed Rotational Bands in A ~ 150 Mass Region

The structure of superdeformed rotational bands (SDRB's) in A ~ 150 mass region are studied by using the Harris three – parameter expansion and the incremental alignment. The bandhead spins I o have been determined with best fit procedure in order to obtain a minimum root mean square deviation between the calculated and the experimental dynamical moments of inertia. The kinematic moment of inertia has been calculated as a function of rotational frequency and compared to the corresponding experimental ones by assuming three spin values I o - 2 , I o , I o + 2. The transition energies and the variation of the moments of inertia as a function of rotational frequency have been calculated. The agreement between theory and experiment are excellent. The identical bands of SDRB's with ΔI = 2 staggering in 148 Gd (SD6) and 149 Gd (SD1) are investigated. Also the presence of ΔI = 2 staggering effect in the yrast bands of 147 Eu and 150 Tb has been examined.


Introduction
The study of superdeformed rotational bands (SDRB's) is one of the most exciting areas in nuclear structure.More than 335 superdeformed (SD) bands were observed in various mass regions [1,2].They are associated with extremely large quadrupole deformation.The difference between the SDRB's in various mass regions is manifested through the behavior of the moments of inertia.The exact excitation energies, spins and parities of SDRB's remain unknown.Spin best fit extrapolation procedures were used to predict the spins of these SD states [3][4][5][6][7][8][9][10][11][12][13][14][15][16].It is noted that all the available approaches gain from the comparison of the calculated transition energies or the dynamical moments of inertia with the experimental results.
One of the most interesting phenomena observed in SD bands is the identical bands (IB's) [17][18][19] whereby SD bands with very nearly identical energies were observed in different nuclei.The incremental alignment depending only on γtransition energies was introduced [20][21][22] to compare the SD bands in neighboring nuclei.The ΔI = 2 energy staggering in gammaray transitions [23][24][25][26] is one of the few mechanisms which not predicted theoretically and up to now is poorly understood.This phenomenon was first observed in 149 Gd [23], where the dynamical moment of inertia of the yrast SD band exhibits a small oscillation when plotted versus the rotational frequency of the nucleus.This effect is commonly called ΔI = 4 bifurcation because the band is consequently divided into two sequences with levels I, I+4, I+8, … and I+2, I+6, I+10, … differing in angular momentum by four units.Despite several theoretical attempts [16,19,22,[27][28][29][30][31] search for the physical origin of the staggering, there is no general agreement.Some discussions connect this effect with the presence of a C4 symmetry of nuclear Hamiltonian [27][28][29].Other studies [30,31] argue that the staggering could be related to band crossing.
The purpose of the present paper is to report results of the structure of SDRB's in mass region A~ 150 by using the three parameters Harris formula and the incremental alignment which has the important advantage that it does not require knowledge of the spins of the states.The paper is arranged as follows: In section 2 , we outline the concept of Harris parameterization for SDRB's.The properties of the incremental alignment of SD bands are discussed in section 3.In section 4 we review the concept of Δ I = 2 staggering.In section 5, we presented a numerical calculations and obtained results and discussions for five SDRB's in Eu / Gd / Tb nuclei in mass region A = 150.

Outline of the Theory
A power series expansion with improved convergence properties was first suggested by Harris [32], as an extension of cranked model.In the Harris formulation, the nuclear excitation energy E is given in terms of even powers of the angular frequencies ω.The expansion up to ω 6 is: where the expansion coefficients α, β and γ have the dimensions, ℏ 2  −1 , ℏ 4  −3 and ℏ 6  −5 respectively.The angular frequency ω is not directly observed quantity, but is derived from the observed rotational spectrum according to the canonical relation with  = ( + 1) 1 2 is the intermediate nuclear spin.
The corresponding expression of dynamical moment of inertia J (2) for the Harris expansion equation ( 1) is which leads to expression for the intermediate nuclear spin I as a function of ω, by integrating equation ( 3) with respect to ω: where i o is the constant of integration (aligned spin) The expression for the kinematic moment of inertia J (1) for Harris expansion reads: One can extract the rotational frequency, kinematic and dynamical moments of inertia by using the experimental interaband E2 transition energies as: 2  = 4    + 2 →  −    →  − 2 ℏ 2  −1 (7) J a n u a r y 21, 2 0 1 5 It is seen that, while J (1) depends on the spin proposition, J (2) does not.

Transition Energies Based on the Incremental Alignment
The incremental alignment Δi between two bands A and B defined as [20]: where ΔEγ is obtained by subtracting the transition energy in a band of interest A from the closest transition energy in the reference SD band B and ∆   is calculated as the energy difference between the two closest transitions in the SD band of the reference B. The reference nuclei involves either the same proton (6  ) or neutron (7  ) intruder configuration.The incremental alignment has the important advantage that it dose not require knowledge of the spins of the states.To compare directly the transition energies of SD bands in given mass region to reference band one must study the evaluation of Δi with rotational frequency.Δi is linked to the total alignment i through the relation i = Δi + ΔI where ΔI is the difference between the angular momenta associated with the transitions.
If the incremental alignment ΔiCD between unknown band D and reference band C has the same value of the incremental alignment ΔiAB of another pair A and B, that is Then one can calculate the gamma transition energies of band D from the relation

ΔI = 2 Staggering in SDRB's
To explore the ΔI = 2 staggering in a band, one must subtract from the difference between two consecutive transitions in the band ΔEγ(I)= Eγ(I+2)-Eγ(I) a smooth reference    () calculated with the help of the finite difference approximation to the norder derivatives of the transition energies with respect to the spin d n Eγ(I)/dI n .This smooth difference is given by (i) The Flibotte definition Flibotte et al [23] described the deviation by means of a function of four consecutive transition energies which is denoted as the 4 point formula (ii) The Cederwall definition In this case, a function of five consecutive Eγ value is used the 5-point formula [24] where Eγ(I) is the transition energy from a spin state with I to I-2.It is worth while to point out that Δ ' Eγ(I) is proportional to the inverse of the dynamical moment of inertia J (2) .
In order to see the variation in the experimental transition energies, we subtract from the 4-point and 5-point formulae the calculated ones.The corresponding staggering parameters are S (3) (I) and S (4) (I) respectively

Numerical Calculations and Discussion
The optimized best parameters α, β, γ of Harris expansion for our selected SDRB's have been calculated by using a computer simulated search program in order to minimize the common definition of the root mean square (rms) deviation χ, given by J a n u a r y 21, 2 0 1 5 where N is the number of experimental data points entering into the fitting procedure.The assigned spin values of the bandhead are extracted from equation (4).It has been argued that at zero frequency the aligned spin io is equel to zero or half for our selected SDRB's.
Using these assigned spin values Io, the kinematic J (1) (I) is plotted versus rotational frequency ℏ and compared to the J (1) value obtained from the experimental transition energies by assuming three different spins Io-2, Io, Io+2 for the lowest SD states.Figure (1) represent an example for 148 Gd(SD6) at Io = 36, 38, 40, we see that the best agreement is obtained for bandhead spin Io= 38 which is the predicted value from the theory.From the figure one, notice that the absolute value of J (1) (I) and also the slope are sensitive to the spin assignment.Table (1) lists the calculated transition energies Eγ(I), the bandhead spin Io and model parameters α, β, γ resulting from the best fitting procedure for 148 Gd(SD1) and 149 Gd(SD1).Agreement between theory and experiment are excellent (the experimental data are taken from Refs [1,2]).Now, we will use the incremental alignment as a tool to predict the transition energies of 147 Eu(SD1), 147 Eu(SD5) and 150 Tb(SD1).The incremental alignment ΔiAB of the SDRB's in nucleus A relative to nucleus B is calculated and used to predict the transition energies in nucleus D belonging to another pair CD has the same incremental alignment of ΔiAB.The nucleus D is an isotope or isotone to the nucleus C which involves either the same proton or neutron intruder configuration.In the A~ 150 mass region, there is a neutron gap at N = 85, this means that for 148 Eu85 excited neutron configurations can only be created via energetically costly excitations across the gap, while for 147 Eu84 several low energy neutron excitations are possible.This is analogous to the structure in the isotope 148 Gd84 and 149 Gd85.For proton, there is a gap at Z = 64, thus we expect that many of low-lying SD bands in A~ 150 nuclei will be identical to bands in the neighboring Z+1 Gadolinium nuclei.The positive signature (α = + 1/2) 1/2 [301] orbital is also close to Fermi surface, and it may generate identical bands (IB's).The intruder orbital SD band configurations of our reference nuclei B and C are: The calculated transition energies of 147 Eu(SD1), 147 Eu(SD5) and 150 Tb(SD1) deduced from the incremental alignment and reference SDRB's are shown in Table (2a, 2b, 2c), these values are in good agreement with experimental values [1,2].
The behavior of moments of inertia seems to be very useful to understand the properties of SDRB's, because of J (2) is related to the curvature of the excitation energy as a function of spin and can be derived from the energy difference between two consecutive transitions in the band, therefore, J (2) does not depend on the knowledge of the spin I but only on measured γray energies.The calculated results of the dynamic J (2) and kinematic J (1) moments of inertia as a function of rotational frequency ℏ are plotted in Figure (2) for our selected SDRB's.the general trends of the evolution of J (2) , shows considerable variation from one nucleus to another depending on the occupancy of high-N intruder orbitals.

Conclusion
The three parameters Harris formula for energy levels is proposed in this paper to parameterize the E2 transition γray energies and the dynamical moment of inertia in five SD bands in the A = 150 mass region.The incremental alignment which depends on the occupation of specific single particle orbitals and not depends on the knowledge of the spin has been also used to predict the transition energies of SD bands of 147 Eu and 150 Tb.The role of occupation of highj intruder orbitals in the structure of the SD bands has been investigated.Our results indicate that the N = 80 gap is considerable more stable than that Z = 64 gap.In all our selected SDRB's the calculated results agree with experimental data very well, this indicate that Harris formula and incremental alignment can describe both the yrast and the excited SD bands.Finally our results suggest that the identical bands 148 Gd (SD6), 149 Gd (SD1) and also the yrast SD bands of 147 Eu and 150 Tb exhibits ΔI = 2 staggering pattern.
while the SD configurations in the neigh boring nuclei are: J a n u a r y 21, 2 Fig. (2) calculated results of the dynamic moment of inertia J(2) (closed circle) and kinematic moment of inertia J(1)