Unified theory of nuclear reactions

https://digital.library.unt.edu/ark:/67531/metadc888955/m2/1/high_res_d/900147.pdf

ENDF/B-VII.0: Next Generation Evaluated Nuclear Data Library for Nuclear Science and Technology

New kinds of statistical ensemble are defined, representing a mathematical idealization of the notion of ``all physical systems with equal probability.'' Three such ensembles are studied in detail, based mathematically upon the orthogonal, unitary, and symplectic groups. The orthogonal ensemble is relevant in most practical circumstances, the unitary ensemble applies only when time‐reversal invariance is violated, and the symplectic ensemble applies only to odd‐spin systems without rotational symmetry. The probability‐distributions for the energy levels are calculated in the three cases. Repulsion between neighboring levels is strongest in the symplectic ensemble and weakest in the orthogonal ensemble. An exact mathematical correspondence is found between these eigenvalue distributions and the statistical mechanics of a one‐dimensional classical Coulomb gas at three different temperatures. An unproved conjecture is put forward, expressing the thermodynamic variables of the Coulomb gas in closed analytic form as functions of temperature. By means of general group‐theoretical arguments, the conjecture is proved for the three temperatures which are directly relevant to the eigenvalue distribution problem. The electrostatic analog is exploited in order to deduce precise statements concerning the entropy, or degree of irregularity, of the eigenvalue distributions. Comparison of the theory with experimental data will be made in a subsequent paper.

Statistical Theory of the Energy Levels of Complex Systems. I

In recent years, notions drawn from non-Hermitian physics and parity–time (PT) symmetry have attracted considerable attention. In particular, the realization that the interplay between gain and loss can lead to entirely new and unexpected features has initiated an intense research effort to explore non-Hermitian systems both theoretically and experimentally. Here we review recent progress in this emerging field, and provide an outlook to future directions and developments. This Review Article outlines the exploration of the interplay between parity–time symmetry and non-Hermitian physics in optics, plasmonics and optomechanics.

Non-Hermitian physics and PT symmetry

Random-matrix theories in quantum physics : common concepts

The fluctuations of the neutron reduced widths from the resonance region of intermediate and heavy nuclei have been analyzed by a statistical procedure which is based on the method of maximum likelihood. It is found that a chi-squared distribution with one degree of freedom is quite consistent with the data while a chi-squared distribution with two degrees of freedom (an exponential distribution) is not. The former distribution corresponds to a Gaussian distribution for the reduced-width amplitude, and a plausibility argument is given for it which is based on the consideration of the matrix elements for neutron emission from the compound nucleus and of the central limit theorem of statistics. This argument also suggests that within the framework of the compound-nucleus theory all reduced-width amplitudes have Gaussian distributions, and that many of the distributions for the various channels may be independent. One consequence of the latter suggestion is that the total radiation width for a given spin state which is formed in neutron capture will be essentially constant, in agreement with some observations, because it is the sum of many partial radiation widths. The fluctuations of the provisional fission widths of ${\mathrm{U}}^{235}$ are best described by a chisquared distribution with about 2\textonehalf{} degrees of freedom, indicating that there are effectively only a few independently contributing fission channels.

Fluctuations of Nuclear Reaction Widths

Model for Nuclear Reactions with Neutrons

It is shown that "repulsion of energy levels" of the same symmetry type occurs in complex atomic spectra. Thus, for the elements Hf, Ta, W, Re, Os, and Ir, for which the spin-dependent forces are relatively strong, the spacings between neighboring odd-parity levels of the same $J$ value follow the Wigner distribution (approximately). For the elements Sc, Ti, V, Cr, Mn, Fe, Co, and Ni, for which the spin-dependent forces are relatively weak, a similar distribution is obtained for the odd-parity levels having fixed values for $S$, $L$, and $J$. (When the quantum numbers $S$ and $L$ are disregarded, the same levels give rise to a distribution of spacings which is approximated by a random superposition of a number of appropriately weighted Wigner distributions.) For the elements Y, Zr, Nb, Mo, Ru, Rh, and Pd, for which the spin-dependent forces are of intermediate strength, the empirical distribution of spacings between the odd levels of the same $J$ value has a character which is intermediate between the Wigner and exponential distributions. All of these observations are explained in terms of a statistical model for the Hamiltonian matrix in the $S$, $L$, $J$, $\ensuremath{\pi}$ representation. Quantitative results are obtained for a relatively simple form of the model which depends on only two parameters, viz., the dimensionality $N$ and the ratio $\ensuremath{\mu}$ of the dispersions of the normal distributions for the off-diagonal and diagonal matrix elements. The transition from the exponential to the Wigner distribution occurs, roughly speaking, in the range of $N{\ensuremath{\mu}}^{2}$ from 0 to 1. The results of this work suggest that an empirical study of the distribution of the spacing between the energy levels of a complex quantum system may be capable of pointing to the existence of constants of the motion beyond those which are already known.

"Repulsion of Energy Levels" in Complex Atomic Spectra

The formation of a compound nucleus in neutron reactions

A brief survey was presented of the experimental data on the distributions of the neutron widths of nuclear finestructure levels and of the nearest-neighbor spacings for both atomic and nuclear energy levels. Random- matrix mcdels to predict these data were discussed both analytically (in all possible cases) and via Monte-Carlo machine calculations. Evidence from explicit atomic and nuclear model calculations was shown to support the statistical hypotheses suggested for the Hamiltonian matrix. On this basis it was found that it was possible to achieve a unified discussion of the underlying statistical properties of atomic and nuclear spectra, excluding collective phenomena of which the statistical behavior of the fission widths was the most prominent example. (auth)

Charles E. Porter

Papers

Fluctuations of Nuclear Reaction Widths

Model for Nuclear Reactions with Neutrons

"Repulsion of Energy Levels" in Complex Atomic Spectra

The formation of a compound nucleus in neutron reactions

Statistical properties of atomic and nuclear spectra