Thermal pairing treatment within the path integral formalism

Mohamed Fellah, N. Allal, M. R. Oudih
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Abstract

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.
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路径积分形式中的热配对处理
在路径积分形式主义中提出了一种在有限温度下处理配对相关性的方法。该方法基于系统哈密顿中配对项的平方根提取。建立了配对间隙参数Δ、能量 E 和热容量 C 的间隙方程和表达式。首先使用理查森模型对形式主义进行了测试,以便与精确解进行比较。使用本形式主义得到的结果还与有限温度 BCS(FTBCS)的结果进行了比较。与 FTBCS 模型相比,特别是在低温条件下,结果有了明显改善。事实上,在低温条件下,本研究的 Δ 值与精确值的一致性非常好。这使得本模型的 E 值和 C 值与精确值之间的一致性比与 FTBCS 值之间的一致性更好。不过,温度临界值仍然存在。然后,我们使用¹⁶²Dy 核和¹⁷²Yb 核的变形伍兹-撒克逊平均场的单粒子能量来考虑实际情况。研究表明,在本方法的框架内,配对效应会持续到 FTBCS 临界温度之后。此外,在低温条件下,本模型的结果与热容量的半实验值之间具有良好的一致性。与 FTBCS 方法相比,有明显的改进。但在较高温度下,情况就不一样了。
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