Calculation of statistical properties of nuclei in a finite-temperature mean-field theory requires projection onto good particle number, since the theory is formulated in the grand canonical ensemble. This projection is usually carried out in a saddle-point approximation. Here we derive formulas for an exact particle-number projection of the finite-temperature mean-field solution. We consider both deformed nuclei, in which the pairing condensate is weak and the Hartree-Fock (HF) approximation is the appropriate mean-field theory, and nuclei with strong pairing condensates, in which the appropriate theory is the Hartree-Fock-Bogoliubov (HFB) approximation, a method that explicitly violates particle-number conservation. For the HFB approximation, we present a general projection formula for a condensate that is time-reversal invariant and a simpler formula for the BardeenCooper-Schrieffer (BCS) limit, which is realized in nuclei with spherical condensates. We apply the method to three heavy nuclei: a typical deformed nucleus 162 Dy, a typical spherical nucleus 148 Sm, and a transitional nucleus 150 Sm in which the pairing condensate is deformed. We compare the results of this projection with results from the saddle-point approximation and exact shell model Monte Carlo calculations. We find that the approximate canonical HF entropy in the particlenumber projection decreases monotonically to zero in the limit when the temperature goes to zero. However, in a nucleus with a strong pairing condensate, the approximate canonical HFB entropy in the particle-number projection decreases monotonically to a negative value, reflecting the violation of particle-number conservation. Computationally, the exact particle-number projection is more efficient than calculating the derivatives required in the saddle-point approximation.

The statistical model of compound-nucleus reactions predicts that the fluctuations of the partial gamma decay widths for a compound-nucleus resonance are governed by the Porter-Thomas distribution (PTD), and that consequently the distribution of total gamma-decay widths is very narrow. However, a recent experiment [1] reported large fluctuations of the total gamma-decay widths in the 95 Mo(n, γ) 96 Mo* reaction, contrary to this expectation. Furthermore, in recent theoretical works it was argued that sufficiently strong channel couplings can cause deviations of the partial width distributions from PTD. Here, we investigate whether the combined influence of a large number of nonequivalent gamma-decay channels, each of which couples weakly to the compound-nucleus resonances, can modify the statistics of the partial widths. We study this effect in neutron scattering off 95 Mo within a random-matrix model that includes coupling to the entrance neutron channel and to the large number of gamma channels. Using realistic coupling parameters obtained from empirical models for the level density and the gamma strength function, we find that the PTD describes well the distribution of partial widths for all decay channels, in agreement with the statistical-model expectation. Furthermore, we find that the width of the distribution of the total gamma-decay widths is insensitive to wide variations in the parameters of the gamma strength function, as well as to deviations of the partial-width distributions from the PTD. Our results rule out an explanation of the recent experimental data within a statistical-model description of the compound nucleus. arXiv:1909.06250v1 [nucl-th]

A basic prediction of the statistical model of compound nucleus reactions is that the partial widths for decay into any open channel channel fluctuate according to the Porter-Thomas distribution (PTD). A recent experiment on sand p-wave neutron scattering from platinum isotopes found that the experimental s-wave partial neutron width distributions deviated substantially from the PTD. Several explanations for this finding have been proposed within the statistical model, but none has resolved this issue. Here, we review the application of a realistic resonance-reaction model to s-wave neutron scattering from 194 Pt. Our main conclusion is that the PTD provides an excellent description of the partial neutron width distribution, provided that the secular energy dependence of the average neutron width is correctly described. Within a realistic range of model parameters, there can be a near-threshold bound or virtual state of the neutron channel that changes this secular dependence from the usual √ E dependence, as proposed by Weidenmüller [1]. In this case, the use of the √ E dependence to analyze the data will lead to apparent deviations from the PTD. We discuss the limited parameter range where such a near threshold state can have a significant effect.

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