ENERGY AND STRUCTURE STATES OF LOW-LYING BANDS IN Gd

The experimental results of the literary and electronic nuclear database for Gd were summarized and analyzed. Inertial parameters of rotating core were determined using the Harris method. The theoretical values of energy and wave functions were calculated within the framework of a phenomenological model that takes into account Coriolis mixing of state rotational bands. The calculated energy values were compared with existing experimental data, which were in good agreement. ABSTRAK: Hasil dapatan kajian melalui eksperimen pangkalan data nuklear dan elektronik bagi Gd diringkaskan dan dianalisis. Parameter inersia putaran berputar ditentukan menggunakan kaedah Harris. Nilai teori fungsi tenaga dan gelombang dikira dalam kerangka model fenomenologi yang mengambil kira campuran Coriolis pada band putaran keadaan. Nilai pengiraan tenaga dibandingkan dengan data eksperimen memberikan persetujuan yang baik.


INTRODUCTION
The gadolinium isotope with a mass A=156 is one of the most studied nuclei. The main reason is the large cross section ( , γ ) n -reaction in 156 Gd , which provides big opportunities for studying the emission spectra in this reaction. Full results on this nucleus are given in references [1,2]. Several other nuclear processes supplement information on the levels and rotation bands in 156 Gd . In the reaction of (α, 2 ) n data were obtained on the states of rotational bands with A wealth of experimental information was obtained as a result of studying the reaction ( , γ ) n n  [3]. Based on the totality of the experimental data, one can think that for 156 Gd all or almost all levels were found up to excitation energy of 2 MeV. In 156 Gd five rotational bands with π 0 K + = are known. Therefore, systematic study of the properties of these levels is very important to search the corresponding levels in the neighboring nuclei. 156 Gd and 158 Gd are the first nuclei in which a new type of excitation was discovered -collective states with π 1 I + = [4]. At present, there have been several collective states experimentally observed in 156 Gd with π 1 I + = . Their excitation energies and reduced probabilities ( 1) B M  are determined [5]. Nonadiabaticity is observed in the energies and especially in the electromagnetic characteristics of the excited states of rotational bands with π 0 K + = and π 2 K + = of these nuclei [6][7][8][9]. In the present work, to explain these nonadiabatics, we use a phenomenological model that takes into account Coriolis mixing of low-lying rotational bands [10,11]. The energy levels and wave functions of the rotational band states are calculated. The nonadiabatics observed in the energies and wave functions of rotational states are discussed.
More details of the phenomenological model used are given in Ref. [10] and this model was also successfully applied to study the mixing of state bands of both positive and negative parity in [11][12][13][14].

DESCRIPTION OF THE MODEL
According to the phenomenological model of the nucleus [10], the Hamilton operator has the following appearance: -angular frequency of rotation of the core; К  − the head energies of the rotational bands; ˆx K j K    -matrix element of Coriolis interaction between the states of rotational bands; χ( , ) I K -the coefficients are as follows: The Eigen wave function of the Hamilton operator (1) has the following appearance: The total energies of the states can be found by the following formula: There are different methods for determining the energy of the core rotational motion of the nucleus, for example, Harris proposed to determine the following two-parameter formula [15]: https://doi.org/10.31436/iiumej.v22i1.1497  and 1  -inertia parameters of the rotational core.

CALCULATION AND RESULTS
According to Bohr-Mottelson and Bengtsson-Frauendorf, at small values of the spin of the nucleus, the rotational energy of the nucleus corresponds to the ground rotational states energy [16,17]. Therefore, the inertial parameters of the core were determined by the Harris method using the experimental energy states of ground band up to spin 8 I  . Table 1: Values of the model parameters used in calculating level energies   ( 1) ( ) ω ( ) It can be seen from the comparison that there is a difference between them at high values of angular rotation frequency. I.e. at the large values of the total angular momentum I , deviations from the adiabatic theory are observed in the energies of the ground band. Such nonadiabaticity is more pronounced in the energies of the vibration bands with 0 To determine the causes of nonadiabaticity in the energies of the states of lowlying rotational bands, we use the phenomenological model presented above.
The energy of the rotational motion of the nucleus, for the bands included in the basis of the Hamiltonian operator (1), is considered the same and its values are determined by formula (6). The rotational motion at low frequencies is not related to its internal structure, i.e., the rotation of the nucleus does not affect the Hamiltonian basic model (1)  The selected numerical values of the model parameters are given in Table 1.  The results of the comparison show that at small values of spin, the compatibility between them is good, and the difference increases with increasing spin. This difference suggests that additional effects must be taken into account when studying the properties of high-spin states. For example, it may be necessary to take into account the effect of the rotational motion of the nucleus on the intrinsic energy when the rotation of the nucleus is high.
The wave functions of the states π In Table 2, it can be seen that in the band with

CONCLUSION
Theoretical calculations were carried out for 156 Gd nucleus in the framework of the phenomenological model taking into account Coriolis mixing of low-lying rotation bands with positive parity.
Nonadiabaticities observed in the energies is explained by the Coriolis mixing of low excited rotational states. To describe all the adiabatic rotational bands, the same moments of inertia have been used. The energy spectra of the positive parity states have been calculated. The results of calculation of energy spectra for ground band states is compared with the existing experimental data correspondingly and their compatibility is given. The mixing effects of the lower bands have been shown to be significant even at small spins.
In the high spin states of ground band, the difference between the theory and experiment was observed. This may be due to the fact that for large values of the angular frequency of rotation of the nucleus, it is necessary to take into account the effects of rotation on intrinsic energy.
The wave function of states of the rotational bands is calculated. The regularities of the change in the state components of the mixing bands are studied depending on the total angular momentum.