Journal of Nuclear Science, Engineering and Technology (JONSAT)
In cooperation with the Iranian Nuclear Society

Calculation of the level density parameter and nuclear level density for 160-162Dy nuclei using the BCS model

Document Type : Research Paper

Author

Kh. Benam

Department of Physics, Faculty of Basic Sciences, Shahrekord University, P.O. Box: 115, Shahrekord - Iran

https://doi.org/10.24200/nst.2024.1555.2012
Abstract
Studying the structure of the nucleus through the consideration of pairing correlation between nucleons plays a crucial role in determining the thermodynamic properties of the nucleus. In this study, we examined the thermodynamic characteristics of the nucleus using the BCS model, which incorporates pairing correlation. The gap parameter serves as the symbol of pairing correlation in this model. By calculating the gap parameter at various temperatures using the BCS model, we were able to determine the excitation energy, temperature-dependent level density parameter, and nuclear level density. Additionally, we calculated the nuclear level density using the BSFG model and the temperature-dependent level density parameter. Finally, we compared the results of the nuclear level density obtained through these methods with each other and experimental data.

Highlights

  1. Bardeen J, Cooper L.N, Schrieer J.R. Theory of Superconductivity. Phys. Rev. 1957;108(5):1175. https://doi.org/10.1103/PhysRev.108.1175.

 

  1. Dinh Dang N. Influence of particle number fluctuations and vibrational modes on thermodynamic characteristics of a hot nucleus. Z. Phys. A. 1990(3);335:253-264. https://doi.org/10.1007/BF01304703.

 

  1. Arve P, Bertsch G, Lauritzen B, Puddu G. Static

path approximation for the nuclear partition

function. Ann. Phys. 1988;183(2):309-319. https://doi.org/10.1016/0003-4916(88)90235-7.

 

  1. Lauritzen B, Anselmino A, Bortignon P.F, Broglia R.A. Pairing Phase Transition in Small Particles. Ann. Phys. 1993;223(2):216-228. https://doi.org/10.1006/aphy. 1993.1032.

 

  1. Rossignoli R, Canosa N, Ring P. Thermal and quantal fluctuations for fixed particle number in finite superfluid systems. Phys. Rev. Lett. 1998;80(9):1853. https://doi.org/10.1103/PhysRevLett.80.1853.

 

  1. Mühlschlegel B, Scalapino D.J, Denton R. Thermodynamic properties of small superconducting particles. Phys. Rev. B. 1972;6(5):1767. https://doi.org/10.1103/PhysRevB.6.1767.

 

  1. Mohammadi P, Dehghani V, Mehmandoost-Khajeh-Dad A.A. Applying modified Ginzburg-Landau theory to nuclei. Phys. Rev. C. 2014;90(5):054304. https://doi.org/10.1103/PhysRevC.90.054304.

 

  1. Dehghani V, Forozani Gh, Benam Kh, Calculating the thermal properties of 93,94,95Mo using the BCS model with an average value gap parameter. Nucl. Sci. Tech. 2017;28(128):1-6. https://doi.org/10.1007/s41365-017-0284-x.

 

  1. Moretto L.G. Statistical description of a paired nucleus with the inclusion of angular momentum. Nucl. Phys. A. 1972;185(1):145-165. https://doi.org/10.1016/0375-9474(72)90556-8.

 

  1. Behkami A.N, Hulzenga J.R. Comparison of experimental level densities and spin cut of factors with microscopic theory for nuclei near A=60. Nucl. Phys. A. 1973;217(1):78-92. https://doi.org/10.1016/0375-9474(73) 90624-6.

 

  1. Moretto L.G. Pairing fluctuations in excited nuclei and the absence of a second order phase transition. Phys. Lett. B. 1972;40(1):1-4. https://doi.org/10.1016/0370-2693(72)90265-1.

 

  1. Razavi R, Behkami A.N, Dehghani V. Pairing phase transition and thermodynamical quantities in 148,149Sm. Nucl. Phys. A. 2014;930:57-62. https://doi.org/10.1016/ j.nuclphysa.2014.07.016.

 

  1. Razavi R, Behkami A.N, Mohammadi S, Gholami M, Ratio of neutron and proton entropy excess in 121Sn compared to 122Sn. Phys. Rev. C. 2012;86(4):047303. https://doi.org/10.1103/PhysRevC.86.047303.

 

  1. Benam Kh, Dehghani V, Alavi S.A. Role of magic numbers in thermodynamic quantities of 206Pb and 138Ba using BCS and Lipkin-Nogami models. Eur. Phys. J. A. 2019;55(105):1-14. https://doi.org/10.1140/epja/i2019-12785-3.

 

  1. Kargar Z. Pairing correlations and thermodynamical quantities in 96,97Mo. Phys. Rev. C. 2007;75(6):064319. https://doi.org/10.1103/PhysRevC.75.064319.

 

  1. Koning A.J, Hilaire S, Goriely S, Global and local level density models. Nucl. Phys. A. 2008;810(1):13-76. https://doi.org/10.1016/j.nuclphysa.2008.06.005.

 

  1. Demetriou P, Goriely S. Microscopic nuclear level densities for practical applications. Nucl. Phys. A. 2001;695(1):95-108. https://doi.org/10.1016/S0375-9474(01)01095-8.

 

  1. Toke J, Swiatecki W.J. Surface-layer corrections to the level-density formula for a diffuse Fermi Gas.

Nucl. Phys. A. 1981;372(1):141-150. https://doi.org/10.1016/0375-9474(81)90092-0.

 

  1. Prakash M, Wambach J, Ma Z.Y. Effective mass in nuclei and the level density parameter. Phys. Lett. B. 1983;128(3):141-146. https://doi.org/10.1016/0370-2693(83)90377-5.

 

  1. Egidy T.V, Bucurescu D. Systematics of nuclear level density parameters. Phys. Rev. C. 2005;72(4): 044311. https://doi.org/10.1103/PhysRevC.73.049901.

 

  1. Lestone J.P. Determination of the time evolution of fission from particle emission. Phys. Rev. Lett. 1993;70(15):2245. https://doi.org/10.1103/PhysRevLett. 70.2245.

 

  1. Lestone J.P. Temperature dependence of the level density parameter. Phys. Rev. C. 1995;52(2):1118. https://doi.org/10.1103/PhysRevC.52.1118.

 

  1. Shlomo S, Natowitz J.B. Level density parameter in hot nuclei. Phys. Lett. B. 1990;252(2):187-191. https://doi. org/10.1016/0370-2693(90)90859-5.

 

  1. Shlomo S, Natowitz J.B. Temperature and mass dependence of level density parameter. Phys. Rev. C. 1991;44(6):2878. https://doi.org/10.1103/PhysRevC.44. 2878.

 

  1. Benam Kh, Mousavi S.Z, Dehghani V, Alavi S.A. Calculating the thermodynamic quantities of nucleus using the temperature dependence of level density parameter. Journal of Nuclear Science and Technology. 2024;44(4):12-19 [In Persian]. https://doi.org/10.24200/ nst.2022. 1209.1785.

 

  1. Benam Kh, Dehghani V, Alavi S.A. Thermal properties of 97Mo and 90Y nuclei using temperature dependent level density parameter. Eur. Phys. J. A. 2023;59(221):1-8. https://doi.org/10.1140/epja/s10050-023-01130-4.

 

  1. Dehghani V, Alavi S.A. Nuclear level density of even-even nuclei with temperature- dependent pairing energy. Eur. Phys. J. A. 2016;52(306):1. https://doi.org/10.1140/epja/i2016-16306-8.

 

  1. Canbula B, Bulur R, Canbula D, Babacan H. A Laplace-like formula for the energy dependence of the nuclear level density parameter. Nucl. Phys. A. 2014;929:54-70. https://doi.org/10.1016/j.nuclphysa.2014.05.020.

 

  1. Ignatyuk A.V, Smirenkin G.N, Tishin A.S. Phenomenological description of energy dependence of the level density parameter. Sov. J. Nucl. Phys. 1975;21(6):485-490.

 

  1. Dwivedi N.R, Monga S, Kaur H, Sudhir R.J. Ignatyuk damping factor: A semiclassical formula. Int. J. Mod. Phys. E. 2019;28(8):1950061. https://doi.org/10.1142/ S0218301319500617.

 

  1. Moller P, Sierk A.J, Ichikawa T, Sagawa H. Nuclear ground-state masses and deformations: FRDM(2012). Atom. Data. Nucl. Data. 2016;109-110:1-204. https://doi.org/10.1016/j.adt.2015.10.002.

 

  1. Damgaard J, Pauli H.C, Pashkevich V.V, Strutinsky V.M. A method for solving the independent particle Schrodinger equation with a deformed average field. Nucl. Phys. A. 1969;135(2):432-444. https://doi.org/10.1016/0375-9474(69)90174-2.

 

  1. Cwiok S, Dudek J, Nazarewicz W, Skalski J, Werner T. Single-particle energies, wave functions, quadrupole moments and g-factors in an axially deformed woods-saxon potential with applications to the twocentre-type nuclear problems. Comput. Phys. Commun. 1987;46(3):379-399. https://doi.org/10.1016/ 00104655(87)90093-2.

 

  1. Patyk Z, Sobiczewski A. Ground-state properties of the heaviest nuclei analyzed in a multidimensional deformation space. Nucl .Phys. A. 1991;533(1):132-152. https://doi.org/10.1016/0375-9474(91)90823-O.

 

  1. Guttormsen M, Bagheri A, Chankova R, Rekstad J, Siem S, Schiller A, Voinov A. Thermal properties and radiative strengths in 160,161,162Dy. Phys. Rev. C. 2003;68(6):064306.

Keywords

Nuclear level density
Temperature-dependent level density parameter
Pairing correlation
BCS model
BSFG model
  1. Bardeen J, Cooper L.N, Schrieer J.R. Theory of Superconductivity. Phys. Rev. 1957;108(5):1175. https://doi.org/10.1103/PhysRev.108.1175.

 

  1. Dinh Dang N. Influence of particle number fluctuations and vibrational modes on thermodynamic characteristics of a hot nucleus. Z. Phys. A. 1990(3);335:253-264. https://doi.org/10.1007/BF01304703.

 

  1. Arve P, Bertsch G, Lauritzen B, Puddu G. Static

path approximation for the nuclear partition

function. Ann. Phys. 1988;183(2):309-319. https://doi.org/10.1016/0003-4916(88)90235-7.

 

  1. Lauritzen B, Anselmino A, Bortignon P.F, Broglia R.A. Pairing Phase Transition in Small Particles. Ann. Phys. 1993;223(2):216-228. https://doi.org/10.1006/aphy. 1993.1032.

 

  1. Rossignoli R, Canosa N, Ring P. Thermal and quantal fluctuations for fixed particle number in finite superfluid systems. Phys. Rev. Lett. 1998;80(9):1853. https://doi.org/10.1103/PhysRevLett.80.1853.

 

  1. Mühlschlegel B, Scalapino D.J, Denton R. Thermodynamic properties of small superconducting particles. Phys. Rev. B. 1972;6(5):1767. https://doi.org/10.1103/PhysRevB.6.1767.

 

  1. Mohammadi P, Dehghani V, Mehmandoost-Khajeh-Dad A.A. Applying modified Ginzburg-Landau theory to nuclei. Phys. Rev. C. 2014;90(5):054304. https://doi.org/10.1103/PhysRevC.90.054304.

 

  1. Dehghani V, Forozani Gh, Benam Kh, Calculating the thermal properties of 93,94,95Mo using the BCS model with an average value gap parameter. Nucl. Sci. Tech. 2017;28(128):1-6. https://doi.org/10.1007/s41365-017-0284-x.

 

  1. Moretto L.G. Statistical description of a paired nucleus with the inclusion of angular momentum. Nucl. Phys. A. 1972;185(1):145-165. https://doi.org/10.1016/0375-9474(72)90556-8.

 

  1. Behkami A.N, Hulzenga J.R. Comparison of experimental level densities and spin cut of factors with microscopic theory for nuclei near A=60. Nucl. Phys. A. 1973;217(1):78-92. https://doi.org/10.1016/0375-9474(73) 90624-6.

 

  1. Moretto L.G. Pairing fluctuations in excited nuclei and the absence of a second order phase transition. Phys. Lett. B. 1972;40(1):1-4. https://doi.org/10.1016/0370-2693(72)90265-1.

 

  1. Razavi R, Behkami A.N, Dehghani V. Pairing phase transition and thermodynamical quantities in 148,149Sm. Nucl. Phys. A. 2014;930:57-62. https://doi.org/10.1016/ j.nuclphysa.2014.07.016.

 

  1. Razavi R, Behkami A.N, Mohammadi S, Gholami M, Ratio of neutron and proton entropy excess in 121Sn compared to 122Sn. Phys. Rev. C. 2012;86(4):047303. https://doi.org/10.1103/PhysRevC.86.047303.

 

  1. Benam Kh, Dehghani V, Alavi S.A. Role of magic numbers in thermodynamic quantities of 206Pb and 138Ba using BCS and Lipkin-Nogami models. Eur. Phys. J. A. 2019;55(105):1-14. https://doi.org/10.1140/epja/i2019-12785-3.

 

  1. Kargar Z. Pairing correlations and thermodynamical quantities in 96,97Mo. Phys. Rev. C. 2007;75(6):064319. https://doi.org/10.1103/PhysRevC.75.064319.

 

  1. Koning A.J, Hilaire S, Goriely S, Global and local level density models. Nucl. Phys. A. 2008;810(1):13-76. https://doi.org/10.1016/j.nuclphysa.2008.06.005.

 

  1. Demetriou P, Goriely S. Microscopic nuclear level densities for practical applications. Nucl. Phys. A. 2001;695(1):95-108. https://doi.org/10.1016/S0375-9474(01)01095-8.

 

  1. Toke J, Swiatecki W.J. Surface-layer corrections to the level-density formula for a diffuse Fermi Gas.

Nucl. Phys. A. 1981;372(1):141-150. https://doi.org/10.1016/0375-9474(81)90092-0.

 

  1. Prakash M, Wambach J, Ma Z.Y. Effective mass in nuclei and the level density parameter. Phys. Lett. B. 1983;128(3):141-146. https://doi.org/10.1016/0370-2693(83)90377-5.

 

  1. Egidy T.V, Bucurescu D. Systematics of nuclear level density parameters. Phys. Rev. C. 2005;72(4): 044311. https://doi.org/10.1103/PhysRevC.73.049901.

 

  1. Lestone J.P. Determination of the time evolution of fission from particle emission. Phys. Rev. Lett. 1993;70(15):2245. https://doi.org/10.1103/PhysRevLett. 70.2245.

 

  1. Lestone J.P. Temperature dependence of the level density parameter. Phys. Rev. C. 1995;52(2):1118. https://doi.org/10.1103/PhysRevC.52.1118.

 

  1. Shlomo S, Natowitz J.B. Level density parameter in hot nuclei. Phys. Lett. B. 1990;252(2):187-191. https://doi. org/10.1016/0370-2693(90)90859-5.

 

  1. Shlomo S, Natowitz J.B. Temperature and mass dependence of level density parameter. Phys. Rev. C. 1991;44(6):2878. https://doi.org/10.1103/PhysRevC.44. 2878.

 

  1. Benam Kh, Mousavi S.Z, Dehghani V, Alavi S.A. Calculating the thermodynamic quantities of nucleus using the temperature dependence of level density parameter. Journal of Nuclear Science and Technology. 2024;44(4):12-19 [In Persian]. https://doi.org/10.24200/ nst.2022. 1209.1785.

 

  1. Benam Kh, Dehghani V, Alavi S.A. Thermal properties of 97Mo and 90Y nuclei using temperature dependent level density parameter. Eur. Phys. J. A. 2023;59(221):1-8. https://doi.org/10.1140/epja/s10050-023-01130-4.

 

  1. Dehghani V, Alavi S.A. Nuclear level density of even-even nuclei with temperature- dependent pairing energy. Eur. Phys. J. A. 2016;52(306):1. https://doi.org/10.1140/epja/i2016-16306-8.

 

  1. Canbula B, Bulur R, Canbula D, Babacan H. A Laplace-like formula for the energy dependence of the nuclear level density parameter. Nucl. Phys. A. 2014;929:54-70. https://doi.org/10.1016/j.nuclphysa.2014.05.020.

 

  1. Ignatyuk A.V, Smirenkin G.N, Tishin A.S. Phenomenological description of energy dependence of the level density parameter. Sov. J. Nucl. Phys. 1975;21(6):485-490.

 

  1. Dwivedi N.R, Monga S, Kaur H, Sudhir R.J. Ignatyuk damping factor: A semiclassical formula. Int. J. Mod. Phys. E. 2019;28(8):1950061. https://doi.org/10.1142/ S0218301319500617.

 

  1. Moller P, Sierk A.J, Ichikawa T, Sagawa H. Nuclear ground-state masses and deformations: FRDM(2012). Atom. Data. Nucl. Data. 2016;109-110:1-204. https://doi.org/10.1016/j.adt.2015.10.002.

 

  1. Damgaard J, Pauli H.C, Pashkevich V.V, Strutinsky V.M. A method for solving the independent particle Schrodinger equation with a deformed average field. Nucl. Phys. A. 1969;135(2):432-444. https://doi.org/10.1016/0375-9474(69)90174-2.

 

  1. Cwiok S, Dudek J, Nazarewicz W, Skalski J, Werner T. Single-particle energies, wave functions, quadrupole moments and g-factors in an axially deformed woods-saxon potential with applications to the twocentre-type nuclear problems. Comput. Phys. Commun. 1987;46(3):379-399. https://doi.org/10.1016/ 00104655(87)90093-2.

 

  1. Patyk Z, Sobiczewski A. Ground-state properties of the heaviest nuclei analyzed in a multidimensional deformation space. Nucl .Phys. A. 1991;533(1):132-152. https://doi.org/10.1016/0375-9474(91)90823-O.

 

  1. Guttormsen M, Bagheri A, Chankova R, Rekstad J, Siem S, Schiller A, Voinov A. Thermal properties and radiative strengths in 160,161,162Dy. Phys. Rev. C. 2003;68(6):064306.

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