Calculating the thermal properties of 93,94,95 Mo using the BCS model with an average value gap parameter V. Dehghani 1 • Gh. Forozani 2 • Kh. Benam 2 Received: 6 February 2017 / Revised: 18 April 2017 / Accepted: 26 April 2017 Ó Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Chinese Nuclear Society, Science Press China and Springer Nature Singapore Pte Ltd. 2017 Abstract The gap parameter of the standard BCS model is replaced by the order parameter of the modified Ginzburg– Landau theory. Using this new form of the BCS model, the energy, entropy, and heat capacity of 93;94;95 Mo nuclei are calculated. The results are compared with the experimental data and standard BCS results. Since the order parameter does not drop to zero at a critical temperature, our results for thermal properties are free of singularities. We have shown that the heat capacity as a function of temperature behaves smoothly and it is highly in agreement with the experimental heat capacity, while heat capacity behaves singularly and discontinuously in the standard BCS model. A smooth peak in the heat capacity is observed which is interpreted as a signature of the transition from the super- fluid to the normal phase. Keywords BCS  Ginzburg–Landau  Statistical fluctuations 1 Introduction Systems of paired fermions are present in different fields of physics, and their size ranges from astronomical objects, such as neutron stars, to small systems, such as nuclei. Regardless of the origin of the pairing potential between fermions, the BCS [1–8] model and it’s number projected versions [9–12] are the most popular tools used in investigations of paired systems. An important step in deriving BCS equations is finding the gap parameter. This parameter is the measure to find whether the system is in the paired phase or not. In the standard BCS model, we choose the most probable values of gap parameter, which are the values of gap parameter that minimize the free energy [2, 13]. This choice seems to be relevant when the number of constituents of the system are from the order of Avogadro’s number. But when the number of particles decreases by orders of magnitude and we are dealing with a finite system such as nuclei, this choice is not the best one. In a finite system, the probability which the system remains in the states that are not the free energy mini- mum can be comparable with the probability of being in the free energy minimum [13]. Based on this fact, some authors used the mean value of gap parameter in place of the most probable value of it [13, 14]. In this method, the grand potential of the BCS model was used as the dis- tribution function that reveals the importance of different values of the gap parameter at each temperature, and finally the mean value of the gap parameter can be cal- culated by integrating different values of the gap param- eter weighted by this distribution function. Since calculating the mean value of the gap parameter is mathematically complicated, we are interested in the replacement method that can do the same job. Static path approximation(SPA) [15, 16] and it’s modified versions [17–19] are the methods that systematically take the effect of statistical fluctuations into account and can be used in investigation of the paired finite systems. But & V. Dehghani vdehghani@phys.usb.ac.ir 1 Physics Department, Faculty of Sciences, University of Sistan and Baluchestan, Zahedan, Iran 2 Physics Department, Bu-Ali Sina University, Hamedan, Iran 123 NUCL SCI TECH (2017)28:128 DOI 10.1007/s41365-017-0284-x