The effect of angular momentum on the thermodynamic properties of 146-147 Nd isotopes

Document Type : Original Article

Author
Shahab Afrookhteh
Department of Physics, Education department of Kangan, Bushehr, Iran
10.22034/jmrph.2026.137958.1004
Abstract
In this paper, the effect of angular momentum on the thermodynamic properties of 146-147Nd isotopes is investigated. To this end, the Mean-gap BCS (MEGBCS) model has been employed to consider thermal fluctuations. In this model, the thermodynamic quantities of the nucleus are calculated using the average pairing gap parameter at various angular momenta. Results from the standard BCS model are also presented for comparison. The obtained results show that, in contrast to the standard BCS model where pairing correlations vanish at the critical temperature, in MEGBCS model, with increasing temperature, pairing does not vanish at the critical temperature but instead tends toward a constant value. Also, the pairing gap increases with increasing temperature at high angular momentum, which is referred to as thermally assisted pairing correlation. Moreover, the inclusion of thermal fluctuations leads to an S-shaped behavior in the heat capacity versus temperature, which can be interpreted as the pairing phase transition. Furthermore, it is observed that as angular momentum increases, the S-shaped structure in the heat capacity disappears, and the nuclear phase transition occurs at lower temperatures. The nuclear level density was calculated as a function of excitation energy for various angular momenta and compared with the experimental data and results of Shell Model Monte Carlo (SMMC) approach. It is observed that the nuclear level density increases with excitation energy, whereas it decreases as the angular momentum increases. The calculations performed using the MEGBCS approach for 146-147Nd isotopes show excellent agreement with both experimental data and SMMC model results.
Keywords
Angular momentum
Thermal fluctuations
Pairing correlation
Heat capacity
Nuclear level density
Subjects
[1] Z. Kargar and S. Afrookhteh, "Deformation dependence of thermal properties of hot rotating 184Re with inclusion of nuclear pairing fluctuations". Results in Physics, vol. 52, pp. 106899, 2023.
[2] K. Van Houcke, S. M. A. Rombouts, K. Heyde and Y. Alhassid, "Microscopic calculation of symmetry projected nuclear level densities", Physical Review C, vol. 79, no. 2, pp. 024302, 2009.
[3] A. N. Behkami and Z. Kargar, "Nuclear level densities and spin cut-off factors deduced from microscopic theory for nuclei from 20F to 244Am", Journal of Physics G: Nuclear and Particle Physics, vol. 18, no. 6, pp. 1023-1037, 1992.
[4] Z. Kargar, N. Ghahramani and A. Moradi, "Microscopic Study of Nuclear Level Densities and Thermal Properties in 56, 57Fe", Communications in Theoretical Physics, vol. 48, no. 3, pp. 531-533, 2007.
[5] A. N. Behkami and Z. Kargar, "Statistical treatment of nuclear level densities", Physica Scripta, vol. 66, no. 2, pp. 22-31, 2002.
[6] A. N. Behkami and Z. Kargar, "Nuclear Level Density at High Spin and Excitation Energy", Communications in Theoretical Physics, vol. 36, no. 3, pp. 305-312, 2001.
[7] J. R. Huizenga, A. N. Behkami, I. M. Govil, W. U. Schröder and J. Toke, "Influence of rotation-induced nuclear deformation on α-particle evaporation spectra", Physical Review C, vol. 40, no. 2, pp. 668, 1989.
[8] R. Rossignoli, N. Canosa and P. Ring, "Microscopic description of hot nuclei: the SPA+RPA approach", Nuclear Physics A, vol. 654, no. 1, pp. 719c-722c, 1999. 
[9] K. Kaneko and A. Schiller, "Particle-number conservation in static-path approximation for thermal superfluid systems", Physical Review C—Nuclear Physics, vol. 76, no. 6, pp. 064306, 2007.
[10] Y. Nogami, "Improved superconductivity approximation for the pairing interaction in nuclei", Physical Review, vol. 134, no. 2B, pp. B313, 1964.
[11] H, C. Pradhan, Y. Nogami and J. Law, "Study of approximations in the nuclear pairing-force problem", Nuclear Physics A, vol. 201, no. 2, pp. 357-368, 1973.
[12] N. Dang, N. Dinh, P. Ring and R. Rossignoli, "Lipkin-Nogami method at finite temperature in the static-path approximation", Physical Review C, vol. 47, no. 2, pp. 606, 1993.
[13] J. Bardeen, L. N. Cooper and J. R. Schrieffer, "Theory of superconductivity", Physical review, vol. 108, no. 5, pp. 1175, 1957.
[14] L. G. Moretto, "Statistical description of a paired nucleus with the inclusion of angular momentum", Nuclear Physics A, vol. 185, no. 1, pp. 145-165, 1972.
[15] L. N. Cooper, "Bound electron pairs in a degenerate Fermi gas", Physical Review, vol. 104, no. 4, pp. 1189-1190, 1956.
[16] D. G. Madland and J. R. Nix, "New model of the average neutron and proton pairing gaps", Nuclear Physics A, vol. 476, no. 1, pp. 1-38, 1988. 
[17] W. Chongshi and Z. Jinyan, "Nuclear Pairing Phase Transition", Chinese Physics C, vol. 12, no. 1, pp. 95-101, 1988.
[18] P. Cejnar, J. Jolie and R. F. Casten, "Quantum phase transitions in the shapes of atomic nuclei", Reviews of Modern Physics, vol. 82, no. 3, pp. 2155-2212, 2010.
[19] E. Yüksel, F. Mercier, J. P. Ebran and E. Khan, "Clustering in nuclei at finite temperature", Physical Review C, vol. 106, no. 5, pp. 054309, 2022.
[20] P. S. Deo, J. P. Pekola and M. Manninen, "Heat capacity of mesoscopic superconducting disks", EPL (Europhysics Letters), vol. 50, no. 1, pp. 649-655, 2000.
[21] H. S. Røising, G. Wagner, M. Roig, A. T. Rømer and B. M. Andersen, "Heat capacity double transitions in time-reversal symmetry broken superconductors", Physical Review B, vol. 106, no. 17, pp. 174518, 2022.
[22] N. Q. Hung and N. D. Dang, "Pairing reentrance in hot rotating nuclei", Physical Review C—Nuclear Physics, vol. 84, no. 5, pp. 054324, 2011
[23] L. Liu, Z. H. Zhang and P. W. Zhao, "Thermodynamics of pairing transition in hot nuclei", Physical Review C, vol. 92, no. 4, pp. 044304, 2015.
[24] H. Le, D. X. Tran, N. A. Nguyen, M. H. Nguyen, C. D. Tran and H. Nguyen, "Updated heat capacities of 161-164Dy nuclei", Communications in Physics, vol. 33, no. 2, pp. 133-142, 2023.
[25] M. Guttormsen, A. Yoram, W. Ryssens, K. O. Ay, M. Ozgur, E. M. Algin, et al., "Strong enhancement of level densities in the crossover from spherical to deformed neodymium isotopes", Physics Letters B, vol. 816, pp. 136206, 2021.
[26] L. G. Moretto, "Statistical description of a paired nucleus with the inclusion of angular momentum", Nuclear Physics A, vol. 185, no. 1, pp.145-165, 1972.
[27] Z. Kargar and R. N. Gashtaseb, "Nuclear pairing fluctuations and angular momentum dependence of superconductivity in 94Mo nucleus", International Journal of Modern Physics E, vol. 27, no. 11, pp. 1850096, 2018.
[28] M. Sano and S. Yamasaki, "Phase transition and level density of atomic nuclei", Progress of Theoretical Physics, vol. 29, no. 3, pp. 397-414, 1963.
[29] Z. Kargar and S. Afrookhteh, "Deformation dependence of superconductivity in neodymium isotopes", The European Physical Journal Plus, vol. 139, no. 9, pp. 862-869, 2024.
[30] O. Civitarese, M. Reboiro and P. Vogel, "Neutron-proton pairing in the BCS approach", Physical Review C, vol. 56, pp. 1840-1843, 1997.
[31] S. G. Nilsson, C. F. Tsang, A.  Sobiczewski, Z. Szymański, S. Wycech, C. Gustafson, et al., (1969), "On the nuclear structure and stability of heavy and superheavy elements", Nuclear Physics A, vol. 131, no. 1, pp. 1-66, 1969.
Journal Of Modern Research Physics
Volume 11, Issue 1 - Serial Number 20
September 2026
Pages 42-58