AN ANALYTICAL SOLUTION OF BOHR ’ S COLLECTIVE HAMILTONIAN

The collective states of 126Xe, 128Xe, 130Xe and 130Ba 52 < Z, N < 116 are studied using Bohr's Collective Hamiltonian and a simple analytic form for the potential. The energy levels are calculated for this model and the results are

and and also on Euler's angles specifying the orientation of the intrinsic system.Depending on the shape of the potential energy of deformation, this Hamiltonian can describe both the brational limit and the rotational limit.In both cases, the solution of the col- lectlve Schrdinger equation is simple.However, there are many cases which cannot be fitted using the above treatment.Therefore, many attempts [2,3,4,5,6,7] have been made in order to relax some of the requirements characterizing the rotational and vibrational descriptions.Usually, the problem is approximated by a simple model in which the potential is replaced by a simple analytic function where the Schrdinger equation can be easily solved.
The present work represents one such trial in which some even-even nuclei in the region 52 < Z, N < 116, susceptible to gamma deformation, are studied.These 126Xe 12  130Xe 130Ba nuclei are: 8Xe, and The collective states of these nuclei are studied using Bohr's collective Hamiltonian.

THEORETICAL CONSIDERATIONS.
Different microscopic computations [5,8,9,10,11] which have been performed for the potentials of nuclei belonging to the investigated region indicated a weak dependence of their potentials on .

/CB
The general solution of (3.2) can be written in the form y F(Z) exp-Z2/2 1 where It is evident that F(b,d;x) is a polynomial iff b is a non-positive integer; therefore, the requirement that the solution (3.10) be bounded at is expressed by the condition where n is a non-negative integer.The energy difference dE is given by i AE ECaLC" h/ [n8 + (Al Ao)] (3.14) This result was used to calculate the energy levels for 126Xe, 128Xe, 130Xe, and 130Ba in the region 52 < Z, N <_ 116.The calculations were carried out at n8 0 and I 1,2,3,4,5,6.The mass parameter B was taken equal to 50 MeV -I as given by Wilets [5] and the parameters 80 and C were fitted to the experimental energy levels.For greater than 2, the present model was found to give a better fitting to the energy levels when a correction term was introduced with the llmlta- tion of (2-3) splittlngs of the levels.The comparison between the experimental and the calculated values without splittlngs for each of the nuclei enabled us to find the correction term in the form + i__ / n% where n% 0,1,2, (l-2)
The results are given in Tables (1-4).The results showed that: l) Some characteristics of the nuclei in the region 52 < Z, N <_ 116 can be described by a model in which the potential is expressed in the form of a simple analytic function, such as the one used in the present study.
2) The effect of the perturbation term can be replaced by a correction term.The correction term in the present study is related to the quantum number with a limitation of (2%-3) splittings for % > 2.
The method in the present study has an advantage over numerical solutions, since it gives a simple physical picture by classifying the states according to known quantum numbers.The degree of accuracy as shown from the tables is satis- factory for the ordinary scope of the studies.
Substituting for P,and A in (3.11), we get E ,%:h + i i