The thermal properties of Tin isotopes such as 68≤N≤78, are studied. The calculations are performed by means of the modified Lipkin–Nogami (MLN) approach that takes into account the thermal and quantal fluctuations. The obtained results are compared to the conventional finite-temperature Bardeen–Cooper–Schrieffer (FTBCS) approach and to the modified Bardeen–Cooper–Schrieffer (MBCS) method. The numerical results illustrate the effect of the statistical and quantal fluctuations on the description of the phenomenon of pairing phase transition.

The pairing phase transition is investigated in hot even-even and even-odd P d isotopes such as 106 ≤ A ≤ 108 in a framework of a microscopic approach that takes into account the statistical fluctuations. In this aim, one considers the Modified BCS method (MBCS). The latter is extended to the odd system case, where the blocking effect is included. The model was applied to the evaluation of the thermal properties such as the excitation energy, entropy and heat capacity. The obtained results are compared on one hand to the usual finite temperature BCS (FTBCS) method and on the other hand to the experimental data. The obtained results allow to show that the thermal fluctuations smooth out the superfluid-normal (SN) phase transition observed in the usual FTBCS results. Moreover, in the region where the pairing phase transition occurs, the experimental data of thermal properties are better reproduced when statistical fluctuations are considered in MBCS method instead of the FTBCS approach.

The S-shaped heat capacity, which has recently been claimed to be a fingerprint of the superfluid-to-normal phase transition in nuclei, is studied in the case of 166 Er element. The calculations are performed by means of the modified Bardeen–Cooper–Schrieffer (MBCS) approach that takes into account the thermal fluctuation. The obtained results are compared to the experimental data and to the FT-BCS theoretical predictions. The present study illustrates the effect of the fluctuations of the quasiparticle (QP) number on the heat capacity.

At finite temperature, very limited information exists on nuclear level density and thermodynamic properties for such nuclei. So, one proposes in the present work to study the phenomenon of pairing phase transition, by evaluating the heat capacity, in the case of Tin isotopes. So, as a first step, our study will include the ordinary Sn isotopes such as 116 ≤ A ≤ 119, it will then be extended to the neutron-rich nuclei such as 126 ≤ A ≤ 129. Theoretically, we use the modified Lipkin-Nogami method (MLN) to eliminate the quantal and statistical fluctuations inherent in the FTBCS approach. The obtained results are compared to the conventional FTBCS results and to the MBCS predictions as well as to the experimental data when available. The inclusion of quantal and statistical fluctuations induces S-shape in the heat capacity curves, which is in a good agreement with experimental data.

We propose to study the thermal properties of the odd isotopes of Tin: [Formula: see text]Sn. To this end, one used two methods to evaluate the properties of these elements. The first theoretical consideration uses a simple prescription to perform the calculation of these properties based on those of even–even neighboring isotopes, assuming the quasi-particle entropy extensivity. The even–even elements are treated as part of the Modified Lipkin–Nogami (MLN) method that allows to take into account the quantal and statistical fluctuations. The second theoretical approach consists of the generalization of the MLN formalism in the case of odd systems, by using the blocking technique. Then, this approach is applied to evaluate the thermal properties of the considered elements. The obtained results by both theoretical approaches are compared to the experimental data. The latter are deduced from the experimental level density within the canonical ensemble. It appears that the assumption of quasi-particle entropy extensivity at low excitation energy allows a simple and an effective treatment of thermal properties of odd nuclei. Indeed, this approach allows to give a good reproduction of experimental data in the particular in the region where the pairing transition occurs.