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

A Unified Theory of Nuclear Reactions, II

The statistical properties of the characteristic values of a matrix the elements of which show a normal (Gaussian) distribution are well known (cf. [6] Chapter XI) and have been derived, rather recently, in a particularly elegant fashion.1 The present problem arose from the consideration of the properties of the wave functions of quantum mechanical systems which are assumed to be so complicated that statistical considerations can be applied to them. Since the physical problem has been given rather recently in some detail in another journal [3], it will not be reviewed here. Actually, the model which underlies the present calculations shows only a limited similarity to the model which is believed to be correct. Nevertheless, the calculation which follows may have some independent interest; it certainly provided the encouragement for a detailed investigation of the model which may reproduce some features of the actual behavior of atomic nuclei.

Characteristic Vectors of Bordered Matrices with Infinite Dimensions II

https://escholarship.org/content/qt2dj9p3h4/qt2dj9p3h4.pdf?t=pmi3bw

ENDF/B-VIII.0: The 8th Major Release of the Nuclear Reaction Data Library with CIELO-project Cross Sections, New Standards and Thermal Scattering Data

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

In the consideration of the independent-particle model, a distinction can be made between the spacing $D$ of the levels of the whole nucleus and the spacing $d$ of the levels of individual nucleons. Except in the immediate neighborhood of the normal state of closed-shell nuclei, $d\ensuremath{\gg}D$. In the "giant-resonance" interpretation considered here, the deviations from the independent-particle model are strong enough to mix many states of the whole nucleus, but the mixing is restricted to an energy range which is less than the order of $d$. According to this interpretation, the reduced particle widths of the levels of the compound nucleus are, on the average, anomalously large close to the energy values of those states of the independent-particle model which correspond to an unexcited target nucleus and a virtual level of the incident particle. As a consequence, the nuclear cross sections have a gross structure which is similar to a giant resonance, such as is implied by the complex square well representation of the nucleon-nucleus interaction. The position, width, and height of these maxima in the average cross sections are expressed in terms of the parameters of the independent-particle model and the departure of the actual nuclear potential which are responsible for the inaccuracy of this model. It is shown, however, that the conventional nuclear potential gives far too large values for the widths of the giant resonances (that is, for the imaginary part of the presentative complex square well potential).

Giant Resonance Interpretation of the Nucleon-Nucleus Interaction

The collision matrix found by Newton to satisfy at all excitation energies the requirement that it describe an excited nucleus whose decay is independent of the mode of formation is shown to imply the vanishing of the absorption cross sections at high energies where the levels overlap and therefore does not describe a compound nucleus in the usual sense. The essential characteristic of this matrix is the high degree of the correlations of the signs of the square roots ${\ensuremath{\gamma}}_{\ensuremath{\lambda}c}$ of the reduced level widths for the various levels $\ensuremath{\lambda}$ and channels $c$. On the other hand, a collision matrix, which is similar to one first considered by Bethe and involves ${\ensuremath{\gamma}}_{\ensuremath{\lambda}c}$ whose signs are uncorrelated, implies energy average decays that are independent of the formation mode and absorption cross sections that are of the order of magnitude of the nuclear area at high energies. These matrices are derived and discussed by using the $R$-matrix theory of Wigner, Eisenbud, and Teichmann. It is shown that the Bethe form of the collision matrix, which is valid only if all of the partial level widths are much less than the spacings, may be modified by means of the Teichmann-Wigner channel elimination procedure so that it is also valid in situations where some of the partial widths exceed the spacings. The form of the compound-nucleus collision matrix thus obtained is similar to one deduced by Weisskopf by considerations involving the compound-nucleus hypothesis and the reciprocity theorem. The pole strength functions ${s}_{c}$, which are the averages of the ratios of reduced widths ${{\ensuremath{\gamma}}_{\ensuremath{\lambda}c}}^{2}$ to level spacings, and their Stieltjes energy transforms are decisive in the determination of the behavior of these collision matrices and their associated cross sections. The $s$ functions and their transforms are presented and discussed in the cases of the strong-coupling and complex square well potential representations of the particle-nuclei interactions. The latter representation with an additional surface absorption is also considered. The present theory indicates that the imaginary part of the complex square well potential should increase with the absorption width, and it suggests a "giant resonance" interpretation of the average cross-section behaviors. The effect of the compound-nucleus on non-compound-nucleus processes such as stripping and pickup is also mentioned.

Collision Matrices for the Compound Nucleus

The contributions to ($n, d$), ($p, d$) reactions and their inverses from the pickup and stripping mechanisms are considered as corrections to the compound-nucleus or $R$-matrix theory of nuclear reactions. In an ($n, d$) reaction, for example, the $R$-matrix theory neglects the interaction of the incident neutron with the target-nucleus proton "tails" which extend beyond the nuclear radius. The pickup correction to the collision-matrix component, or reaction amplitude, appears as the matrix element of the neglected interaction involving an exact wave function and the approximate wave function of the compound-nucleus system not having the interaction; a distorted-wave Born approximation is used in which the former exact wave function is replaced by one of the latter type with the appropriate radiation condition. An explicit expression is given for the collision-matrix component which, together with the compound-nucleus contribution, can be substituted directly into the formulas of Blatt and Biedenharn for total reaction cross sections and angular distributions. In general, the angular distributions contain interference terms in addition to the straight pickup and compound-nucleus contributions. If the distorted neutron and deuteron spherical partial waves are assumed to depend only on the angular momenta, and not explicitly on the total spin and the channel spins, the formula of Tobocman is obtained for the pickup contribution, while Butler's formula is obtained if plane waves are used instead of distorted waves. There are discussions of the various approximations, the exchange terms, and the question of the nuclear radius.

COLLISION MATRIX FOR (n,d) AND (p,d) REACTIONS

If the energy spread of the neutron source used in a transmission type of measurement of total neutron cross sections is large compared with the mean spacing of the resonance levels of the sample, these levels may effect a deviation from the simple exponential dependence on sample thickness. The correction to this dependence is derived in the case where capture, inelastic scattering, and Doppler broadening may be neglected. The derivation does not depend upon any special assumption regarding the distributions of the widths and spacings of the levels other than that they have suitable averages, and the correction is found to be proportional to the ratio of these averages, as one would expect. It is noted that by measurement of this ratio for certain elements, it may be possible to distinguish between the predictions of the strong-coupling and complex square-well representations of the neutron-nucleus interaction.

R. G. Thomas

Papers

Fluctuations of Nuclear Reaction Widths

Giant Resonance Interpretation of the Nucleon-Nucleus Interaction

Collision Matrices for the Compound Nucleus

COLLISION MATRIX FOR (n,d) AND (p,d) REACTIONS

Correction to the Exponential Dependence of Neutron Transmissions