Ground-state bands of doubly even ¹⁶⁶Hf Nucleus

This study was carried out to investigate the rotational structure of even-even ¹⁶⁶Hf isotopes using the phenomenological fitting, Sood’s semi-empirical formula. The rotational energies from the calculated values were compared to the experimental spectrum. The result shows that in ¹⁶⁶Hf, calculated energies fit the experimental values to a remarkable degree of accuracy.

In the collective model proposed by Bohr and Mottelson [1], a new parameter called nuclear deformation is introduced, in which the surface of the nucleus may undergo oscillations in a rotating nucleus. These results in the prediction of rotational energy levels being restricted to even values due to oscillating symmetry, and it is easily observed with nuclei having a number of nucleons far from closed shells. Experimental data comes to a good agreement with these predictions, and applications such as probability for B (E2) transitions, magnetic moments, quadruple moments and isomeric transitions are so far successful.

In the macroscopic view, the common simplest starting point of modeling the atomic nucleus is based on semi-classical liquid drop model [2,3] where the nucleus is assumed to behave a drop of dense incompressible liquid where the spherical shape is the result of attractive forces between all the particles contained in the nucleus. This model is then expanded to describe quantum mechanical collective motion, such as rotation and vibrations, leading to the Bohr and Mottelson collective model [1]. In the collective model, a new parameter called nuclear deformation is introduced, in which the surface of the nucleus may undergo oscillations in a rotating nucleus. These results in the prediction of rotational energy levels being restricted to even values due to oscillating symmetry, and it is easily observed with nuclei having a number of nucleons far from closed shells. In current study, we contribute the ground state bands of rotational spectra of even-even Hafnium ¹⁶⁶Hf nucleus with spin state up to 18⁺ by the phenomenological fitting of Sood’s semi-empirical formula (SSEF).

Theoretical calculations

The shape of atomic nuclei with a large number of nucleons outside of closed shells is axially symmetric and permanently deformed [1]. The rotational energy spectrum of such an axially symmetric rotator shows the characteristic J(J+1) dependence if we take adiabatic approximations for the motions of all the particles in the nucleus. The low-lying energy levels follows spin sequence Jπ = 0⁺, 2⁺, 4⁺, 6⁺, ... , for all of even parity rotational nuclei with each energy level given by the expression [4]:

EJ = AJ(J+1) (1)

Where rotational parameter A is inversely proportional to the nuclear moment of inertia ℑ, such that A = ħ²/2ℑ. The moment of inertia is also assumed to be constant following an approximation where the energy ratio E (4⁺)/E (2⁺) = 10/3, E (6⁺)/E (2⁺) = 7, E (8⁺)/E (2⁺) = 12. Deviations by a few percent from these ratios are thought to be a coupling of vibrational and rotational modes of the nuclei. This coupling depends on J² and constitutes changes in rotational energy from the first-order perturbation theory. However, for fitting the first three or four excited states, the first-order correction term can be added to equation (1) where the rotational energy [4-6]:

EJ = AJ(J ⁺ 1) – BJ²(J ⁺ 1)2 (2)

Stephens, et al. [7] and Singh, et al. [8] added second-order CJ³(J⁺1)3 and third-order DJ⁴(J⁺1)⁴ correction terms respectively to account the deviations in energy bands for even-even actinide nuclei. The rotational energy equation according to an infinite power series as [9].

EJ = AJ(J ⁺ 1) – BJ²(J ⁺ 1)2 ⁺ CJ³(J ⁺ 1)3 – DJ⁴(J ⁺ 1)4 (3)

Assuming that the coefficient parameters are constant through each successive order of correction terms, where C/B = D/C = N (B/A) Sood derived a semi-empirical formulation for rotational energy by summing of infinite series in equation (3):

$E_J=AJ\left(J+1\right)\left[\frac{1+\left(N-1\right)\left(\frac{B}{A}\right)J\left(J+1\right)}{1+N\left(\frac{B}{A}\right)J\left(J+1\right)}\right]$ (4)

Where ratio B/A = (10-3R)/ (200-18R) and R = E (4⁺) /E (2⁺). The constant N was obtained by Sood [9] via empirical method was:

N = 2.85 – 0.05J

In this study, Sood’s semi-empirical formula (SSEF) has been utilized to calculate the ground state bands of even-even hafnium nucleus of A = 166 with spin state up to 18⁺. Hafnium nucleus was chosen because it lies at the beginning of the transition region between well deformed rare-earth nuclei and the doubly-closed spherical ²⁰⁸Pb nucleus [10].

The calculated values of energy levels based on Sood’s semi-empirical formula are presented with experimental values of low-lying rotational energy levels where Kπ = 0⁺ and Jπ take even values from 0⁺ up to 18⁺. We note that the percentage of deviation between calculated and experimental values is very small. The discrepancy increases in higher spin states of even-even ¹⁶⁶Hf nucleus.

To ease the presentation of data, numerical results extracted from nuclear data sheets are presented as red dots in a graph of the spin state against excitation energy in the KeV scale. The black squares represent calculated energy values based on Sood’s semi-empirical formula. The purpose of lines connecting the points is only to guide the eyes because the spin state takes integer values. The error bars in experimental values are not visible due to relatively small values compared to observed energy scales.

From figure 1, we notice that for Hafnium isotopes with mass number 166, the phenomenological Sood’s semi-empirical formula fits the observed energies to the extent of remarkable agreement up to spin state 18⁺ which relate to data in table 1. However, in higher spin states, deviations are observed between calculated and experimental energies. For ¹⁶⁶Hf, the deviation is 0.52, 5.05, 6.94, 8.76, 11.05 and 14.08% at 2⁺, 4⁺, 6⁺, 8⁺, 10⁺, 12⁺ and 14⁺ states respectively. The deviations from J(J⁺1) rule in rotational energy bands are interpreted as the compression of energy levels, or in other words, an increase in nuclear moment of inertia as the nucleus undergoes changing rotational angular velocity. The calculated results fit the experimental values to a remarkable degree of accuracy. The results of our calculation for ¹⁶⁶Hf show good agreement with data in comparison with other calculation [9].

Sood’s semi-empirical formula is successfully applied to calculate the rotational spectrum of even-even ¹⁶⁶Hf isotopes. The results align well with experimental values within the expectations of the simple phenomenological fitting. The calculated data, as well as experimental data, indicated that ¹⁶⁶Hf nucleus is rotational spectrum. The deviations from J(J⁺1) rule in rotational energy bands are interpreted as the compression of energy levels.

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