R-matrix Theory Of Nuclear Reactions Research Articles

Theory of multistep statistical decays in chains of genetically related nuclei

Systems of kinetic equations for decay chains of genetically related nuclei, the R-matrix theory of nuclear reactions with formation of unstable particles, and a diagram technique with Green’s function are used to study the general form for the widths of n-step statistical decays of parent nuclei that are in quasi-stationary resonance states. Conditions under which sequential and virtual decays occur are analyzed. Cases where interactions between particles emitted at different stages of processes must be allowed for are considered. Processes related to fissile nuclei that simultaneously emit three or more particles are discussed.

A consistent description of the successive two-proton decay of nuclei using the diagram technique, the R-matrix theory of nuclear reactions, and kinetic equations for the decay of radioactive nuclei

Using the results from the many-particle theory of deep subbarrier single-proton decays (SPDs) of nuclei, which is based on integral formulas for SPD widths, a general theory of successive two-stage two-proton decays (TSTPDs) from the ground and excited states of both spherical and deformed nuclei is developed. Such decays represent a pair of actual sequential SPDs of the parent (Z, A) nucleus and the intermediate (Z − 1, A − 1) nucleus that are associated by Green’s function G(Z − 1, A − 1) for an intermediate nucleus. It is demonstrated that virtual TSTPDs with widths at positive but fairly low heat values Q 1 of the parent nucleus SPD (potentially being much greater than the corresponding widths of successive TSTPDs) appear when the virtual states (lying outside the mass surface of the decaying system) of the intermediate nucleus in the above Green’s function are considered. It is shown that the formulas for the widths of successive TSTPDs obtained using this theory agree with those for similar widths constructed in the theory of nuclear reactions, which includes the production of unstable particles, and with systems of kinetic equations for a chain of nuclei undergoing radioactive transformation. The conditions under which these systems of kinetic equations are generalized by including processes related not only to successive decay but to virtual decays of nuclei as well are studied.

Two-step mechanisms of two-proton decays of nuclei

A formalism for describing two-step two-proton decays of nuclei is developed on the basis of the multiparticle theory of deep-subbarrier one-proton decays of nuclei that employs integral expression for the decay widths in question. This formalism relies on the idea that the interaction between the emitted protons has but a slight effect on the widths with respect to the two-proton decays being considered. It is shown that such a decay is naturally broken down into the sequential one-proton decays of an (A, Z) parent nucleus and an (A − 1, Z − 1) intermediate nucleus, these decays being related by the Green’s function G(A − 1, Z − 1) that describes the intermediate nucleus with allowance for its real and virtual states, which give rise to, respectively, the sequential and the virtual two-step two-proton decay of the parent nucleus. It is also shown that the widths with respect to sequential two-step two-proton decays coincide with the analogous widths constructed within the R-matrix theory of nuclear reactions leading to the production of unstable particles and with their counterparts obtained with the aid of solving the set of kinetic equations for the chain of nuclei undergoing radioactive decays. It is found that the widths with respect to virtual two-step two-proton decays are close in structure to the widths constructed for the simultaneous two-proton decays of nuclei by using integrated formulas within a simplified model of the method of three-particle hyperspherical polynomials.

R-Matrix Analysis of Structures in Economic Indices: from Nuclear Reactions to High-Frequency Trading

It is shown that the R-matrix theory of nuclear reactions is a viable mathematical theory for the description of the fine, intermediate and gross structure observed in the time-dependence of economic indices in general, and the daily Dow Jones Industrial Average in particular. A Lorentzian approximation to R-matrix theory is used to analyze the complex structures observed in the Dow Jones Industrial Average on a typical trading day. Resonant structures in excited nuclei are characterized by the values of their fundamental strength function, (average total width of the states)/(average spacing between adjacent states). Here, values of the ratios (average lifetime of individual states of a given component of the daily Dow Jones Industrial Average)/(average interval between the adjacent states) are determined. The ratios for the observed fine and intermediate structure of the index are found to be essentially constant throughout the trading day. These quantitative findings are characteristic of the highly statistical nature of many-body, strongly interacting systems, typified by daily trading. It is therefore proposed that the values of these ratios, determined in the first hour-or-so of trading, be used to provide valuable information concerning the likely performance of the fine and intermediate components of the index for the remainder of the trading day.

Widths of fission and cluster decays of nuclei with regard to the channel coupling

Generalization of the R-matrix theory of nuclear reactions with the use of the method of projection operators has been performed to describe the fission and cluster decays of atomic nuclei with the emission of two or three fragments with successive quantum-mechanical consideration of the fission and decay channel coupling.

Validation of the Characteristic Function Model for the Unresolved Resonance Region

There is a significant extension of the region of resolved resonances (RRR) for some nuclei to higher energies, and this has repercussions on the last updated versions of the evaluated nuclear data libraries. This energy extension covers intervals of the resonant cross-section structure, previously treated as an unresolved resonance region (URR). The reality of this situation provides an opportunity to verify a new statistical model of the resonant cross-section structure in the URR based on the characteristic function F of the R-matrix element distribution. For this purpose, the average cross sections and self-shielding factors obtained by the characteristic function model are compared with the corresponding quantities calculated by the Reich-Moore formalism of the R-matrix theory of nuclear reactions with the evaluated resonance parameters in the RRR. The ENDF/B-VI and JENDL 3.2 resolved resonance parameters of 56Fe and 238U are used for the cross-section calculations.

Monte Carlo method for the many-body scattering problem

The non-relativistic many-body scattering problem is cast in a form suitable for treatment by Monte Carlo methods. The formulation is based on resonant standing waves of the compound system, similar to those used in the R-matrix theory of nuclear reactions. Integral expressions are derived from which the scattering matrix elements at low energies can be evaluated. The method is demonstrated by application to simple one- and two-channel one-dimensional scattering problems.

A program calculating the formulae for polarization effects in nuclear reactions

In the R-matrix theory of nuclear reactions, properly normalized observables can be expanded in terms of rotation matrices or Legendre functions. The corresponding expansion coefficients are then bilinear functions of the reaction matrix elements in the (1,s,J) coupling scheme. The numerical factor for each bilinear combination of matrix elements in a given coefficient is a function involving Clebsch-- Gordan, Racah, and X coefficients. By obtaining numerical values for the angular momentum functions and taking into account all relevant symmetries, the program FATSO computes the explicit formulas for any observable quantity for a given set of (1,s,J)-matrix elements. A set of 40 matrix elements can be accommodated, with up to 20 expansion coefficients. Approximate running time for cross sections involving vector- and tensorpolarized deuteron beams for a set of 20 matrix elements is 2 minutes. The program, in FORTRAN IV for the UNIVAC 1108 and the operating system EXEC 8, requires 44030 36- bit words of computer memory. (RWR)

Isospin Mixing in 4¯ States of 12C

Fits are made to measured 12C(n�,no:)12C cross sections in terms of an isospin-mixed doublet of 4- 12C levels near 19�5 MeV, using the R-matrix theory of nuclear reactions. Parameter values are restricted by information from other reactions and from bound-state shell model calculations.

S- and p-wave neutron spectroscopy. Xe. Intermediate structure in 90Y and the generalized R-matrix theory

The resonance structure observed in the 89Y( n, n) 89Y total cross section measurements in the range of 0.3 to 1.2 MeV incident energy was investigated using the generalized R-matrix theory of nuclear reactions and the doorway interpretation of intermediate structure. The energies and wave functions of the doorway resonances were calculated in a 2-particle and 3 p-1 h basis of the shell model. The model space and the parameters of the model calculation chosen were consistent with other shell model calculations in the mass-90 region. Several strong p-wave doorways with J π = 0 +, 1 +, and 2 + were predicted by the model in the energy range studied. This is due to proximity of p-wave giant resonance. The escape widths Γ ↑ and the spreading widths Γ ↓ for these states were evaluated using the model wave functions and the R-matrix formalism. The calculated energy dependence of the total cross section shows that most of the predicted doorways are in general agreement with the observed anamolies with similar relative strength. More significantly, the underlying p-wave gross structure representing a grand average is of very similar shape in both theory and experiment. As expected in the mass 90-region, the s- and d-wave doorways contribute less significantly to the calculated resonance structure.

The description of off-shell resonances by R-matrix theory

The R-matrix theory of nuclear reactions is extended off the energy shell. An isolated resonance in the off-shell reaction amplitude is described by a simple generalization of the Breit-Wigner formula. In going from on- to off-shell, a resonance generally changes its size and shape. In particular, when the single-particle widths are small, an on-shell resonance of arbitrary shape due to interference with the background, will always become symmetric off the energy shell.

The Boundary Condition Parameter in R-Matrix Theory

In the general R-matrix theory of nuclear reactions, all measurable quantities such as the cross sections scc' should be independent of the choice of the boundary condition parameters Bc although the values of the level parameters E? and ??c depend on the Bc. For applications involving only a finite number of levels ?, it is not obvious that this is still the case as completeness arguments can no longer be used. It is shown here that scc' can be made independent of the choice Bc of for any finite number of levels and any number of channels, and the resulting formulae giving the dependence of E? and ??c on Bc are derived. An application is made to the pair of 5/2- levels of 7Li near 7 MeV excitation.

Nuclear Chew-Frautschi Trajectory for Nuclides with Z = 1 - 20

A study of the Chew-Frautschi for low-energy nuclear physics in the Z= 1-20 region is examined in the light of the current nuclear model. Seven examples for nucleon-nucleus and deuteron-nucleus scattering are discussed as an illustration. It is too early to draw conclusion on the definite form of the nuclear trajectory by empirics. Since the R-matrix theory of nuclear reactions is independent of the nuclear model, one can infer that the Reggeized S-matrix theory, and hence the nuclear Chew-Frautschi trajec­ . tory, may be defined without referring to the specific nuclear model by showing the equiva­ lence of two approaches. The idea developed seems to be completely new for some of nuclear physicists so that the physical interpretation of the nuclear trajectory is discussed in comparison with the cur­ rent nuclear model wherever possible.

Tests of the extended R-matrix theory of nuclear reactions

The extended R-matrix theory of nuclear reactions is a generalization of the R-matrix formalism that makes it possible to calculate the collision matrix from the compound nucleus eignestates provided by any nuclear structure calculation. In addition, optical potentials for each channel may be used to represent the nonresonant part of the collision matrix. To test the extended R-matrix formalism (ERF) it was applied to simple, soluble, two channel, S-wave scattering models. The ERF was found to be superior to the standard R-matrix formalism (SRF). Like the SRF, the ERF results are very sensitive to the choice of the channel radius parameters. In addition the ERF is plagued by the formalism (XF) completely eliminates the dependence on the matching radius. The XF is found to give fairly good results in our model calculations and reduces the false resonance difficulty. However, the XF has the drawback of being sensitive to a set of boundary condition parameters whose values will be somewhat ambiguous in practical calculations. This shortcoming is eliminated in the modified X-matrix formalism (MXF) which is found to give excellent results for our two channel model and shows no false resonances. We conclude that the MXF is the version of choice among the various versions of the ERF.

A model for nuclear threshold levels

The observation of some levels in light nuclei closer to thresholds than one might expect from average level densities suggests the investigation of models which might produce such an effect. One model based on the R-matrix theory of nuclear reactions is shown to predict that near thresholds of channels with small barriers there should be an increased density of levels with large reduced widths. The experimental evidence for such an effect is meagre.

Threshold Effects in Average Cross Sections According toR-Matrix Theory

This paper attempts to place on a firm basis certain expressions for effects in elastic and total cross sections caused by, and in the neighborhood of, a reaction threshold. Explicit expressions are derived for the collision matrix near a reaction threshold. These expressions are based on the R-matrix theory of nuclear reactions and extend slightly work by Wigner and by Breit on threshold effects. The expressions are quite general, allowing for the presence of compound resonances. Both the channel matrix and the ievel matrix formulations of R-matrix theory are used. The former turns out to be convenient for formulating the general expressions of the collision matrix and the cross sections. The latter is more convenient for performing energy averages. It is shown that certain formulas for energy-averaged total and elastic cross sections, which were made plausible in a previous paper by the author, follow from the above-mentioned general expressions by performing suitable energy averages. Consequences of the random-sign approximation of the value quantities'' gamma lambda and of a partial breakdown of this assumption are examined and related to the assumptions of the optical model. It is shown that under the assumption of random signs, the total cross section should showmore » no threshold effects, whereas if this assumption is relaxed threshold effects appear. Hence it is, in principle, possible to decide by experiment which situation obtains. Finally, cross section threshold effects under the single-level approximation are given; with a slight generalization of the phase shift, these are identical to expressions derived by Baz and by Newton. (auth)« less

Analysis of Overlapping Resonances in Proton?Carbon Scattering

The R-matrix theory of nuclear reactions is discussed for the situation where distant levels must be included and is successfully applied to the elastic scattering of protons by l2C in the energy range ELAB = 10-11� 5 MeV, where the resonance levels are overlapping. Definite spin and parity assignments are made for three levels in this region by comparing the predicted excitation functions at several angles of scattering with experiment. Previously unexplained polarization data are explained by this analysis. The optical.model potential is found to be parity dependent for ELAB=8.5 MeV and yields a close fit to the available data at ELAB = 8�5 MeV.

A Formulation of the Theory of Alpha-particle Decay from Time-independent Equations

The theory of alpha-particle decay is formulated from the time-independent R-matrix theory of nuclear reactions. It is shown that the most appropriate proper solutions are the ones with radial nodes near the channel radii, and therefore the Teichmann-Wigner sum rules involving the alpha-particle kinetic energy may be applied. Exact and approximate relations are given for the dependence on the angular momentum of the decay rate for the square well model. Barrier penetration formulas are given which include estimates of the effects of an external nuclear exponential potential and the atomic electron screening.