Application of the collective model to determine some rotational bands of 239 U nucleus

U material is component in fuels of nuclear reactor core. Understanding properties and structure of U nucleus is necessary before simulating and designing nuclear reactor. Besides that, the study of nuclear reaction is necessary to identify the specific characteristics of nucleus, it is the most effective experimental method up to now. However, in order to explain the properties of nuclear structure, in addition to study of the nuclear reaction, nuclear structure models and its theory must be used. There are many nuclear structure models to solve those properties of nucleus. This paper presents application of the Collective Model to determine some rotational bands of U nucleus, using Prompt gamma neutron activation analysis method (PGNAA). Experiment is performed at channel No.2 of Dalat Research Reactor (DRR), using Filtered Thermal Neutron Beam and Compton Suppression Spectroscopy with High – Purity Germanium detector (HPGe). The results have found 11 rotational bands of U nucleus. This work is very necessary for the research of nuclear structure which controls material technology by itself. Index Terms – Collective model, U, rotational bands


INTRODUCTION
ollective Model was developed in the 1950s by Reynolds, A. Bohr and Mottelson, Hill and Wheele [1]. The Collective Model emphasizes the coherent behavior of all nucleons in heavy nuclei. The spherical symmetric potential of the nucleus with full shell is exceptionally stable to the effects of additional nucleons. Therefore, it still remains spherical symmetric form. The excited state of a nucleus is defined by single-particle levels in the spherical symmetric potential and the quadrupole field of a spherical symmetric nucleus.
By the increase of external nucleons of the full shell of nucleus, the individual motion effect of nucleons on the potential field increases and the centrifugal pressure of nucleons appears.
Collective motion increases rapidly and impact the core of full shell nucleus leading to the decrease of the potential field stability, it means that the nucleus has spherical asymmetric form. According to quantum mechanics, spherical asymmetric nuclei can rotate.
The Collective Model has been very successful in describing variety of nuclear properties, especially energy levels in nuclei that the Shell Model and the Liquid Drop Model cannot be applied. These energy levels show the characteristics of rotating or vibrating systems. The properties of these nuclei, including excited state energies, angular momentum, magnetic moments, nuclear shapes, etc. can be understood by using the Collective Model.
In 1969, Larry Shelton Varnell [2] applied the Collective Model to determine the rotational band of some deformed nuclei, using the vacuum chamber of the Si(Li) electron spectrometer with Lithium Drifted Silicon detector. The result had determined 12 rotational bands of 152 Sm, 14 rotational bands of 154 Gd, and 12 rotational bands of 166 Er.
So far, there are many researches on 239 U nucleus. In 1959, the smoothed gamma-ray spectrum due to neutron capture by 238 U obtained by Campion et al [3]. The gamma rays in the energy region between 0.14 MeV and 3.4 MeV were examined with a crystal spectrometer. The gamma-rays in the 3.4 MeV to 4.2 MeV range were studied using a pair spectrometer with a resolution of about 1%. In the 1972s, Booth et al [4] had found 21 discrete energy levels and the spin-parity assignments of the excited 239 U nucleus up to 0.950 MeV. In the 1970s, John et al [5] studied about neutron capture gamma radiation from neutron capture in 238 U. The results had found 16 energy gamma-rays from 1 MeV to 4.75 MeV, with the intensity was reported on a number per one hundred capture events, etc. But there have C NATURAL SCIENCES, VOL 2, NO 1, 2018 no research before about the rotational band of 239 U nucleus.
Prompt gamma neutron activation analysis (PGNAA) is a rapid, nondestructive sample technique which is very effective in nuclear structure research [6]. 238 U exists in nature, its abundance is about 99.27%, which is component in fuels of nuclear reactor core. 238 U is heavy eveneven nucleus which has 92 protons and 146 neutrons. 239 U is heavy even-odd nucleus which has 92 protons, 146 neutrons and 1 added neutron. In this experiment, PGNAA method is used to acquire the prompt gamma-rays emitted from 238 U (n, ) 239 U reaction. 1 238 Where n is incident neutron, 238 U is the target nucleus, ( 239 U) * is compound nucleus, ( 239 U) is product nucleus and prompt is prompt gamma-rays.

Theory
The Nilsson model is a shell model for a deformed nucleus. It provides a description of single-particle motion in a spherical asymmetric potential. An appropriate single-particle Hamiltonian for a nucleus with the symmetry axis z is given by [7]: where x, y and z are one-dimensional oscillator frequencies in the x, y, and z direction. C and D are constant. The l 2 and ls terms ensure the proper order and energies of the single-particle levels in the spherical limit.
where I is a total angular momentum,  is rotational angular momentum and K is the spin of nucleus.
where J0 is the moment of the inertia of nucleus, R is radius of nucleus and R is a deformation parameter of nucleus.

Equipments
The experiment is performed at channel No.2 of DRR, which using Filtered Thermal Neutron Beam, and HPGe detector with PGNAA method. Configuration of the system is shown in Fig. 1. The thermal neutron flux at the sample position is 1.6 x 10 6 n/cm 2 xs, and the Cd ratio is 420 [8]. Inside the channel No.2, a chamber with the internal high density polyethylene (HDPE) is set up, it also has 5% Li to shield the scattered neutrons. In the mid-core of this chamber, a holder is made of PTFE (Teflon plastic) material which fixed the sample during the acquisition process.
Due to the large number of gamma-rays incident on the main detector, the Compton continuum is a significant hindrance for low background. The Compton continuum causes the difficult search of low-intensity peaks and increases the uncertainty of the measured activities. Therefore, a Compton suppression spectroscopy has been set-up and installed at DRR 500 kW for neutron activation analysis and nuclear data measurement. The central detector is a GR7023 Canberra n-type coaxial HPGe detector. Its FWHM is 2.36 keV for the 1.33 MeV of 60 Co peak. The relative efficiency is 72%. There are 12 Bismuth Germanium (BGO) guard detectors shielded by a lead of 10 cm thickness. A lead-stepped collimator is located in the front of the opening of the guard detectors. The length and inner diameter of the lead collimator are 180 mm and 40 mm, respectively. The reduction of the Compton continuum has been achieved by surrounding the HPGe detector with the BGO detectors whose signals are used for the anti-coincidence gating in the analog-to-digital converter (ADC). The Compton continuum is reduced about 1.5 to 2 times, up to 1 MeV region of energy [9]. The detectors and shielding system are configured as Fig. 2. The electronic modules are manufactured by Canberra except the high voltage module for BGO detectors, which were produced by Fast Comptec. They include 2026 main amplifiers (AMP), 3106D high voltage power supply, multiport II with ADC 16K and multichannel analyzer (MCA), using the Genie 2000 software. Its configuration is shown in Fig. 3. Fig. 3. The block schema of the gamma acquisition system 238 U natural metal is used. Its diameter, thickness and weight are 1.2 cm, 0.5 cm and 23.68586 g respectively. Geometric form of 238 U sample is cylinder form, which is shown in Fig. 4.
The 238 U sample is placed in the holder at the irradiation position, the angle between the neutron flux and the sample is 45°, the distance from the sample to the detector is 38.5 cm.  where x is the channel number and y is the gamma energy (keV) Prompt gamma spectrum of 239 U acquired at channel No.2 of DRR after eliminating the effect of background is shown in Fig. 5. The statistical count of the spectrum is 1.92 x 108 counts.
Experimental data are shown in Table 1. There are 36 prompt gamma-rays emitted from 238 U (n, ) 239 U reaction. Determination of rotational bands of 239 U is calculated by Equation (5) and (6). Results compared between experimental data and theoretical calculation are shown in Table 2.  [10] Results in Table 2 show that 239 U nucleus has 11 rotational bands, which are 96.74 keV; 160.07 keV; 252.32 keV; 326.12 keV; 431.34 keV; 533.01 keV; 695.63 keV; 810.82 keV; 974.87 keV; 1137.44 keV and 1311.97 keV. Among 36 energy peaks from the prompt gamma spectrum of 239 U nucleus, 25 another peaks are from 235 U (n, ) 236 U reaction (abundance of 235 U nucleus in the sample is about 0.73%) and from the background spectrum. However, we can't find the 42.53 keV peak from the spectrum which is the first excited state of 239 U nucleus [10]. It's the limitation of experimental procedure.
Howerer the determination of 11 peak energies of the experimental spectrum is very closed to the theoretical calculation of rotational bands by using Equation (6). Therefore we can conclude that the 42.53 keV energy is a part of rotational bands.

CONCLUSION
From prompt gamma spectra acquired at the channel No.2 of DRR using application of Collective Model in nuclear structure research, some rotational bands of 239 U deformed nucleus are identified.
The results are quite relevant to the theory of the Collective Model when studying about the heavy nucleus, which has large different between the neutron and proton numbers. These results have shown that the 239 U deformed nucleus has spherical asymmetric structure.