Thermal pairing treatment within the path integral formalism

A method for the treatment of the pairing correlations at finite temperature is proposed within the path integral formalism. It is based on the square root extraction of the pairing term in the Hamiltonian of the system. Gap equations, as well as expressions of the pairing gap parameter Δ, the energy E and the heat capacity C are established. The formalism is first tested using the Richardson model which enables comparison with exact solution. The results obtained using the present formalism are also compared to the Finite Temperature BCS (FTBCS) ones. An improvement compared to the FTBCS model is noted especially at low temperature. Indeed, it is shown that the agreement between the Δ values of the present work and the exact ones is very good at low temperature. This leads to a better agreement between the values of E and C of the present model and the exact values than with the FTBCS values. However, a critical value of the temperature still exists. Realistic cases are then considered using single-particle energies of a deformed Woods-Saxon mean-field for the nuclei ¹⁶²Dy and ¹⁷²Yb. It is shown that in the framework of the present approach, the pairing effects persist beyond the FTBCS critical temperature. Moreover, at low temperature, a good agreement between the present model results and semiexperimental values of the heat capacity is observed. A clear improvement compared to the FTBCS method is noted. It is no more the case at higher temperature.