arXiv subjects
Publications and source records attributed to G. F. Bertsch.
A recent article (D.A. Brown, et al., Phys. Rev. C98 024616 (2018)) rproposed a modification of the cross section formula used in practical calculations of compound nucleus reactions. We discuss the main concepts and approximations of statistical reaction theory and conclude that the standard practical implementations of the cross section formula remain the preferred choice.
The strong quadrupolar component of the Coulomb field between newly formed fission fragments can affect the internal energy of the fragments and their angular momentum. Previous estimates of these effects gave contradictory conclusions about their magnitude. Here we calculate them by solving the time-dependent Hamiltonian equation for the daughter $^{100}$Zr produced in the fission reaction n + $^{235}$U $\rightarrow$ $^{136}$Te + $^{100}$Zr. The Hamiltonian was constructed from the projected angular momentum eigenstates of an aligned deformed mean-field configuration. For typical initial conditions, the average angular momentum of the lighter fragment increases by 1.5 - 3 units.
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 $s$- and $p$-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 $\sqrt{E}$ dependence, as proposed by Weidenmüller [1]. In this case, the use of the $\sqrt{E}$ 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.
The strong deformation present immediately after scission has consequences for the angular momentum population of the fragments as well as the angular distribution of their decay radiation. We find that the usual spin-cutoff parameterization describes very well the angular momentum distribution associated with the deformation of the fragments at the scission point. Depending on the deformation, its contribution can comparable to the thermal contribution to the angular momentum of the newly formed fragments. The $M$-distribution of the angular momentum is highly polarized and gives rise to large anisotropies in the subsequent gamma cascade. We treat in detail a typical gamma cascade in a daughter nucleus, following usual model assumptions except for the anisotropy of the initial state. In principle, the observed anisotropy can provide information on the relative amounts of deformation and thermal energy present at the scission point.
We study the feasibility of applying the Generator Coordinate Method (GCM) of self-consistent mean-field theory to calculate decay widths of composite particles to composite-particle final states. The main question is how well the GCM can approximate continuum wave functions in the decay channels. The analysis is straightforward under the assumption that the GCM wave functions are separable into internal and Gaussian center-of-mass wave functions. Two methods are examined for calculating decays widths. In one method, the density of final states is computed entirely in the GCM framework. In the other method, it is determined by matching the GCM wave function to an asymptotic scattering wave function. Both methods are applied to a numerical example and are found to agree within their determined uncertainties.
A recent experiment on s-wave neutron scattering from $^{192,194,196}$Pt found that the reduced neutron width distributions deviate significantly from the expected Porter-Thomas distribution (PTD), and several explanations have been proposed within the statistical model of compound nucleus reactions. Here, we study the statistics of reduced neutron widths in the reaction $n+ ^{194}$Pt within a model that combines the standard statistical model with a realistic treatment of the neutron channel. We find that, if the correct secular energy dependence of the average neutron widths is used, then the reduced neutron width distribution is in excellent agreement with the PTD for a reasonable range of the neutron-nucleus coupling strength and depth of the neutron channel potential. Within our parameter range, there can be a near-threshold bound or virtual state of the neutron channel potential that modifies the energy dependence of the average width from the $\sqrt{E}$ dependence, commonly assumed in experimental analysis, in agreement with the proposal of H. A. Weidenmüller [1]. In these cases, the reduced neutron width distributions extracted using the $\sqrt{E}$ dependence are significantly broader than the PTD. We identify a relatively narrow range of parameters where this effect is significant.
We propose a framework to calculate the dynamics at the scission point of nuclear fission, based as far as possible on a discrete representation of orthogonal many-body configurations. Assuming axially symmetric scission shapes, we use the $K$ orbital quantum number to build a basis of wave functions. Pre-scission configurations are stable under mean-field dynamics while post-scission configurations evolve to separated fragments. In this first exploratory study, we analyze a typical fission trajectory through to scission in terms of these configurations. We find that there is a major rearrangement of the $K$ occupancy factors at scission. Interestingly, very different fragment shapes occur in the post-scission configurations, even starting from the same pre-scission configuration.
The dependence of the nuclear level density on intrinsic deformation is an important input to dynamical nuclear processes such as fission. Auxiliary-field Monte Carlo (AFMC) method is a powerful method for computing nuclear level densities. However, the statistical distribution of intrinsic shapes is not readily accessible due to the formulation of AFMC in a spherical configuration-interaction shell-model approach. Instead, theory of deformation up to now has largely relied on a mean-field approximation which breaks rotational symmetry. We show here how the distributions of the intrinsic quadrupole deformation parameters can be calculated within the AFMC method, and present results for a chain of even-mass samarium nuclei ($^{148}$Sm, $^{150}$Sm, $^{152}$Sm, $^{154}$Sm) which includes spherical, transitional, and strongly deformed isotopes. The method relies on a Landau-like expansion of the Helmholtz free energy in invariant polynomials of the quadrupole tensor. We find that an expansion to fourth order provides an excellent description of the AFMC results.
We examine the autocorrelation function of the $^{235}$U(n,f) reaction with a view to quantify the presence of intermediate structure in the cross section. Fluctuations due to compound nucleus resonances on the eV energy scale are clearly visible up to $\sim100$ keV neutron energies. Structure on the one-keV energy scale is not present as a systematic feature of the correlation function, although it is present in the data covering the region around 20 keV.
We review a method that we recently introduced to calculate the finite-temperature distribution of the axial quadrupole operator in the laboratory frame using the auxiliary-field Monte Carlo technique in the framework of the configuration-interaction shell model. We also discuss recent work to determine the probability distribution of the quadrupole shape tensor as a function of intrinsic deformation $β,γ$ by expanding its logarithm in quadrupole invariants. We demonstrate our method for an isotope chain of samarium nuclei whose ground states describe a crossover from spherical to deformed shapes.
Statistical reaction theories such as Hauser-Feshbach assume that branching ratios follow Bohr's compound nucleus hypothesis by factorizing into independent probabilities for different channels. Corrections to the factorization hypothesis are known in both nuclear theory and quantum transport theory, particularly an enhanced memory of the entrance channel. We apply the Gaussian orthogonal ensemble to study a complementary suppression of exit channel branching ratios. The effect can be large enough to provide information on the number of exit channels. The effect is demonstrated for the branching ratios of neutron-induced reactions on a $^{235}$U target.
We develop a formalism for calculating the distribution of the axial quadrupole operator in the laboratory frame within the rotationally invariant framework of the configuration-interaction shell model. The calculation is carried out using a finite-temperature auxiliary-field quantum Monte Carlo method. We apply this formalism to isotope chains of even-mass samarium and neodymium nuclei, and show that the quadrupole distribution provides a model-independent signature of nuclear deformation. Two technical advances are described that greatly facilitate the calculations. The first is to exploit the rotational invariance of the underlying Hamiltonian to reduce the statistical fluctuations in the Monte Carlo calculations. The second is to determine quadruple invariants from the distribution of the axial quadrupole operator. This allows us to extract effective values of the intrinsic quadrupole shape parameters without invoking an intrinsic frame or a mean-field approximation.
The nuclear matrix elements for the momentum quadrupole operator are important for the interpretation of precision atomic physics experiments that search for violations of local Lorentz and CPT symmetry and for new spin-dependent forces. We use the configuration-interaction nuclear shell model and self-consistent mean field theory to calculate these matrix elements in $^{21}$Ne, $^{23}$Na, $^{131}$Xe, $^{173}$Yb and $^{201}$Hg. These are the first microscopic calculations of the nuclear matrix elements for the momentum quadrupole tensor that go beyond the single-particle estimate. We show that they are strongly suppressed by the many-body correlations, in contrast to the well known enhancement of the spatial quadrupole nuclear matrix elements.
We propose using the frequency-domain bootstrap (FDB) to estimate errors of modeling parameters when the modeling error is itself a major source of uncertainty. Unlike the usual bootstrap or the simple $χ^2$ analysis, the FDB can take into account correlations between errors. It is also very fast compared to the the Gaussian process Bayesian estimate as often implemented for computer model calibration. The method is illustrated drop model of nuclear binding energies. We find that the FDB gives a more conservative estimate of the uncertainty in liquid drop parameters in better accord with more empirical estimates. For the nuclear physics application, there no apparent obstacle to apply the method to the more accurate and detailed models based on density-functional theory.
Gerry Brown initiated some early studies on the coexistence of different nuclear shapes. The subject has continued to be of interest and is crucial for understanding nuclear fission. We now have a very good picture of the potential energy surface with respect to shape degrees of freedom in heavy nuclei, but the dynamics remain problematic. In contrast, the early studies on light nuclei were quite successful in describing the mixing between shapes. Perhaps a new approach in the spirit of the old calculations could better elucidate the character of the fission dynamics and explain phenomena that current theory does not model well.
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 Bardeen-Cooper-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 particle-number 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 codes HFgradZ.py and HFgradT.py find axially symmetric minima of a Hartree-Fock energy functional for a Hamiltonian supplied in a shell-model basis. The functional to be minimized is the Hartree-Fock energy for zero-temperature properties or the Hartree-Fock grand potential for finite-temperature properties. Various constraints may be imposed on the minima.
The configuration-interaction shell model approach provides an attractive framework for the calculation of nuclear level densities in the presence of correlations, but the large dimensionality of the model space has hindered its application in mid-mass and heavy nuclei. The shell model Monte Carlo (SMMC) method permits calculations in model spaces that are many orders of magnitude larger than spaces that can be treated by conventional diagonalization methods. We discuss recent progress in the SMMC approach to level densities, and in particular the calculation of level densities in heavy nuclei. We calculate the distribution of the axial quadrupole operator in the laboratory frame at finite temperature and demonstrate that it is a model-independent signature of deformation in the rotational invariant framework of the shell model. We propose a method to use these distributions for calculating level densities as a function of intrinsic deformation.