Signature of a Pairing Transition in the Heat Capacity of Finite Nuclei

The heat capacity of iron isotopes is calculated within the interacting shell model using the complete (pf + 0g9/2)-shell. We identify a signature of the pairing transition in the heat capacity that is correlated with the suppression of the number of spin-zero neutron pairs as the temperature increases. Our results are obtained by a novel method that significantly reduces the statistical errors in the heat capacity calculated by the shell model Monte Carlo approach. The Monte Carlo results are compared with finite-temperature Fermi gas and BCS calculations. Typeset using REVTEX 1 Pairing effects in finite nuclei are well known; examples include the energy gap in the spectra of even-even nuclei and an odd-even effect observed in nuclear masses. However, less is known about the thermal signatures of the pairing interaction in nuclei. In a macroscopic conductor, pairing leads to a phase transition from a normal metal to a superconductor below a certain critical temperature, and in the BCS theory [1] the heat capacity is characterized by a finite discontinuity at the transition temperature. As the linear dimension of the system decreases below the pair coherence length, fluctuations in the order parameter become important and lead to a smooth transition. The effects of both static fluctuations [2,3] and small quantal fluctuations [4] have been explored in studies of small metallic grains. A pronounced peak in the heat capacity is observed for a large number of electrons, but for less than ∼ 100 electrons the peak in the heat capacity all but disappears. In the nucleus, the pair coherence length is always much larger than the nuclear radius, and large fluctuations are expected to suppress any singularity in the heat capacity. An interesting question is whether any signature of the pairing transition still exists in the heat capacity of the nucleus despite the large fluctuations. When only static and small-amplitude quantal fluctuations are taken into account, a shallow ‘kink’ could still be seen in the heat capacity of an even-even nucleus [5]. This calculation, however, was limited to a schematic pairing model. Canonical heat capacities were recently extracted from level density measurements in rare-earth nuclei [6] and were found to have an S-shape that is interpreted to represent the suppression of pairing correlations with increasing temperature. The calculation of the heat capacity of the finite interacting nuclear system beyond the mean-field and static-path approximations is a difficult problem. Correlation effects due to residual interactions can be accounted for in the framework of the interacting nuclear shell model. However, at finite temperature a large number of excited states contribute to the heat capacity and very large model spaces are necessary to obtain reliable results. The shell model Monte Carlo (SMMC) method [7,8] enables zeroand finite-temperature calculations in large spaces. In particular, the thermal energy E(T ) can be computed versus temperature T and the heat capacity can be obtained by taking a numerical derivative C = dE/dT . However, 2 the finite statistical errors in E(T ) lead to large statistical errors in the heat capacity at low temperatures (even for good-sign interactions). Such large errors occur already around the pairing transition temperature and thus no definite signatures of the pairing transition could be identified. Furthermore, the large errors often lead to spurious structure in the calculated heat capacity. Presumably, a more accurate heat capacity can be obtained by a direct calculation of the variance of the Hamiltonian, but in SMMC such a calculation is impractical since it involves a four-body operator. The variance of the Hamiltonian has been calculated using a different Monte Carlo algorithm [9], but that method is presently limited to a schematic pairing interaction. Here we report a novel method for calculating the heat capacity within SMMC that takes into account correlated errors and leads to much smaller statistical errors. Using this method we are able to identify a signature of the pairing transition in realistic calculations of the heat capacity of finite nuclei. The signature is well correlated with the suppression in the number of spin-zero pairs across the transition temperature. The Monte Carlo approach is based on the Hubbard-Stratonovich (HS) representation of the many-body imaginary-time propagator, e = ∫ D[σ]GσUσ, where β is the inverse temperature, Gσ is a Gaussian weight and Uσ is a one-body propagator that describes non-interacting nucleons moving in fluctuating time-dependent auxiliary fields σ. The canonical thermal expectation value of an observable O can be written as 〈O〉 =