Reaction Matrix Elements Research Articles

Relation between various methods used in calculating nuclear reaction matrix elements

Various methods for calculating nuclear reaction matrix elements from phase shifts are examined. Calculations are made for the singlet S channel using these various methods and the results are compared. It is shown that a close analogy exists between the auxiliary potential method of calculating nuclear reaction matrix elements and the Scott-Moszkowski separation method.

Nuclear energy levels of 51V and 59Ni

The reaction matrix elements of Kuo and Brown are tested by calculating the nuclear energy levels of 51V and 59Ni. Good agreement between calculated and experimental energies is obtained for 51V. For 59Ni, the agreement is unsatisfactory but shows a little bit of improvement over the previously published theoretical results.

Particle-hole multiplets in 208Bi

The energy levels of 208Bi are calculated with the reaction matrix elements deduced from the Hamada-Johnston potential, and compared with the recent experimental data of Alford, Schiffer and Schwartz. The calculated wave functions for most low-energy states show very little configuration admixture. The energy levels have been calculated with and without the inclusion of the core-polarization effects, and the results of the two calculations show no significant difference. Remarkable agreement with experiment is obtained for all 36 experimentally identified low energy levels. But a large discrepancy is observed for the isobaric analogue state of 208Pb; the calculated position being about 4 MeV too low. A multipole analysis of the effective particle-hole interactions has been carried out.

Off-Energy-Shellt-Matrix Elements for Local Potentials Containing Hard Cores

A method is presented for calculating the off-energy-shell two-body reaction matrix elements for a local potential that contains a hard core. An analytic expression is presented for the matrix elements, $t({k}^{\ensuremath{'}}, k; s)$, for the special case of a hard-core potential with no other interaction. Graphs are presented showing the effects of the hard core for negative values of the energy parameter $s$ and $l=0$ relative-angular-momentum states, using a potential of exponential shape and parameters chosen to approximate the nucleon-nucleon interaction.

Reaction matrix elements for the 0f-1p shell nuclei

The shell-model reaction matrix elements for the 0f-1p shell are calculated and tabulated. The matrix elements consist of two parts; the bare reaction matrix elements and the renormalizations due to core polarizations which are both calculated with the Hamada-Johnston nucleon-nucleon potential. The core polarizations are evaluated for two cases; first for a 40Ca core and then for a 48Ca core. As a preliminary test, these matrix elements have been compared with available empirical effective interactions and used to calculate (i) the spectra of the valence particles of 42Ca, 42Sc, 50Ca, 50Sc and 50Ti, (ii) the changes of single-particle (hole) levels such as the changes of single-neutron levels from 41Ca to 49Ca and (iii) the 0 f 7 2 −0 d 3 2 particle-particle states of 38Cl and the corresponding particle-hole states of 40K. In general, the calculated results are in good agreement with the experimental values, and the core polarization effects are usually of great importance.

Reaction matrix elements for closed-shell nuclei using realistic forces

A method for calculating the reaction matrix elements of closed-shell nuclei is discussed. An improved treatment of the Pauli projection operator is proposed, and accurate solutions of the Bethe-Goldstone equation are obtained for the 16O nucleus using the Hamada-Johnston potential. Solutions for the coupled and uncoupled integro-differential equations are used to obtain the first-order binding energy of the 16O nucleus. The reaction matrix is examined as a function of the single-particle energies. It is found that the 3S 1 states are very sensitive to the choice of single-particle energies; whereas, the 1S 0 states are relatively insensitive to this choice. Calculated singularities of the reaction matrix are also reported for both the 1S 0 and 3S 1 states.

Three-Body Correlations in Reaction-Matrix Calculations

An improved method of treating three-body cluster correlations in reaction-matrix calculations is presented and applied to nuclear matter. A three-particle reaction matrix is defined which allows three particles to interact with each other in all possible ways after specifying the particles involved in the initial and terminal interactions. The method is similar to that proposed by Bethe but, since all off-energy-shell terms can be treated separately, full advantage can be taken of the Faddeev formalism as applied to the three-nucleon system. The Pauli exclusion operator is included in calculating on-energy-shell reaction-matrix elements but is neglected in the off-energy-shell terms. Numerical calculations are presented for two spin-independent, $s$-state, separable potentials, one of which contains a hard-shell repulsive term. Higher-order terms are found to be important for the hard-shell potential but not for the simple attractive potential. In both cases the total three-body correlation energy is found to be small.

State dependence of shell-model reaction matrix elements

Shell-model reaction matrix elements are calculated with a combination of the Moszkowski-Scott separation method and the reference-spectrum method of Bethe, Brandow and Petschek. The reaction matrix G is expanded in terms of G s and V L, where G s is the reaction matrix for the short-range potential V s and V L the long-range potential. The contribution from G s is then calculated with the reference-spectrum method, where the main criterion for choosing the separation distance is that G s is best approximated by G s R, the reference-spectrum approximation of G s. The second-order tensor term V TL ( Q/ e) V TL is calculated with the closure approximation of Kuo and Brown, however the effective energy denominator is now state-dependent, which means that it has a dependence on binding energy, the local density and the centre-of-mass quantum number. The contributions from V TL ( Q/ e) V TL vary almost linearly with ϱ 2 3 , where ϱ is the local density. The reaction matrix elements so calculated have a fairly strong state-dependence which comes in predominantly through G s and V TL ( Q/ e) V TL. The state dependence can be approximated quite accurately in a simple way, and thus the application to finite nuclei is convenient. Shell-model applications have been made for nuclei 18O and 18F and we find that the matrix elements are generally weaker than those of Kuo and Brown, especially for those of T = 1, J = 0 + and T = 0, J = 1 +. This is desirable, because the Kuo and Brown matrix elements are often somewhat too strong.

The self-consistent structure of light nuclei

Nuclear single-particle wave functions are expanded in sets of harmonic oscillator functions. The special properties of these functions allow the derivation of a relatively simple expression for the nuclear reaction matrix elements in the reference spectrum approximation. The same expansion allows a reduction of the Brueckner-Hartree-Fock equations to a set of simple radial equations. The two calculations are coupled together in a straightforward iterative procedure for obtaining self-consistency. The formalism allows a unique definition of a local shell-model potential for light nuclei. The structure of this potential is examined and some explicit expressions for the nuclei 4He and 16O are presented.

Reaction matrix elements for shell-model studies of nuclear spectra

Two aspects of the evaluation of reaction matrix elements for shell-model studies of nuclear spectra are considered. It is shown that the use of Blatt-Jackson low-energy potentials has approximate validity in studies of nuclear spectra. For a range of nuclei estimates are made of the parameters of harmonic-oscillator single-particle wave functions to be used in the calculation of reaction matrix elements.

Application of the reaction matrix theory (I). The calculation of nuclear reaction matrix elements

The reference-spectrum expansion of Bethe, Brandow and Petschek is used to express the nuclear reaction matrix as a perturbation series in terms of a state-independent reference matrix. A simple treatment of the projection (or Pauli) operator in the intermediate-state sum is obtained, in the harmonic-oscillator representation of shell-model states by making certain well-defined averages. The same operator is also examined from the point of view that they are made up of localized operators of the nuclear-matter type corresponding to local effective densities which are position-dependent. These two distinct points of view, called respectively global and local, are investigated numerically in the 1S 0, 3S 1 and other states for the Hamada-Johnston potential in an A = 18 nucleus. It is found that the simple local-density approximation for treating the Pauli operator gives results very close to the global treatment, which is considered exact here. In particular, the local approximation reproduces the global results if the position-dependent effective density is allowed to extend, by some 0.9 fm, beyond the local density at the half-density point. By using state-dependent average effective Fermi momenta based on such density distributions, it is shown that the calculation of nuclear reaction matrix elements is now made as simple (and probably as accurate) as that for nuclear matter. The reliable calculation methods thus obtained make it possible to examine the properties of these matrix elements. It is found that the local interactions (in local treatments) are enhanced in the nuclear surface region and that the reduced matrix elements are particularly sensitive to the single-particle energies of the initial state. The 3S 1 matrix elements are shown to decrease rapidly in strength as the amount of tensor force in the nucleon-nucleon potential increases. The importance of using correct single-particle wave functions is emphasized by showing that in light nuclei improvement of the simple harmonic-oscillator shell model leads to drastic corrections.

Structure of finite nuclei and the free nucleon-nucleon interaction: An application to 18O and 18F

The intention of this work is to investigate the applicability of the free nucleon-nucleon potential determined by the scattering data in the shell-model description of finite nuclei. The potential is chosen to be that of Hamada and Johnston. We have chosen 18O and 18F as our first numerical calculations. A major part of the work reported here concerns the evaluation of the shell-model reaction matrix elements. They are evaluated using the separation method for the singlet-even and triplet-even states and the reference spectrum method for the singlet-odd and triplet-odd states. The second-order Born term for the triplet-even tensor force is found to be very important. It can be calculated conveniently and with good accuracy using the closure approximation with an energy denomiator of ≈ 220 MeV. Assuming the singlet-particle wave functions to be those of a harmonic oscillator well, the single-particle energy levels of 0 d 5 2 , 1 s 1 2 and 0 d 3 2 for 170 are calculated by letting the valence nucleon interact with 16O core. The results are very encouraring. The 0 d 5 2 −0 d 3 2 splitting is found to come mostly from the triplet-odd 1 · s force through the reaction matrix Hartree-Fock process. The spectra of 18O and 18F are satisfactorily reproduced by diagonalizing the model interaction GΘ 3 p1 h in the sd shell, except for the states which are presumed to be largely deformed; G is the reaction matrix and Ω 3p1h is the wave operator which takes care of the corrections arising from the one-particle-one-hole excitation of the 16O core. It is found that the conventional shell-model effective interactions are well reproduced by the model interaction deduced from the free nucleon-nucleon potential.

Effective interaction for 1P shell nuclei calculated from the Hamada-Johnston potential

Reaction matrix elements with respect to harmonic oscillator orbitals for use in shell model calculations of the spectra of the 1p shell nuclei have been calculated in the extended Eden-Emery approximation employing the Hamada-Johnston potential.

Statistical Theory of Nuclear Collision Cross Sections

A formalism is developed for a statistical treatment of the energy variations of nuclear scattering and reaction cross sections. A statistical collision matrix ${\mathrm{U}}^{S}$ is defined which has the form of an energy-independent direct-transition matrix plus a fixed simple resonance-pole expansion, the matrix residues of which are products of complex channel-width amplitudes. By direct comparison with the Wigner-Eisenbud and Kapur-Peierls collision matrices it is found that under widely applicable conditions the statistical collision matrix may be used to calculate averages of observables over energy intervals containing many resonances and many total widths. The problem of determining the statistical properties of the parameters of ${\mathrm{U}}^{S}$ is defined and is solved for several special cases by relating it to the statistics of $R$-matrix parameters. Using these methods averages and mean-square fluctuations of total and reaction cross sections are calculated under general conditions admitting direct and compound processes and arbitrary average values of the total widths $\ensuremath{\Gamma}$ and the resonance spacings $D$. The results are expressed in terms of the direct-reaction matrix elements and the statistical properties of resonance parameters appropriate to the energy region under consideration and are related to the locally applicable optical-model phase shifts and transmission coefficients. Simplifications are obtained under special assumptions such as uncorrelated width amplitudes, small and large $\frac{\ensuremath{\Gamma}}{D}$, pure compound-nucleus reactions, many competing open channels, and many competing direct processes. In the limit of small $\frac{\ensuremath{\Gamma}}{D}$ one obtains the leading terms of an expansion of the average cross section which had previously been derived from $R$-matrix theory directly. In the limit of large $\frac{\ensuremath{\Gamma}}{D}$, many competing channels, but no direct reactions, the nonelastic fluctuation (or average compound nucleus) cross sections approach the Hauser-Feshbach formula. Except in this limit, corrections due to partial-width fluctuations and resonance-resonance interference are applicable. The former are sensitive to the magnitudes of direct reaction matrix elements, the latter to the correlations of resonance energies. Competing direct reactions are shown to require reductions of the transmission coefficients. The mean-square fluctuations of cross sections are found to approach Ericson's results in the limit of large $\frac{\ensuremath{\Gamma}}{D}$ and many competing channels, but are in general much larger for moderate $\frac{\ensuremath{\Gamma}}{D}$ and few channels. They are also sensitive to the details of resonance parameter statistics.

Three-Nucleon Clusters in Nuclear Matter

The effect of three-body clusters on the binding energy of nuclear matter is studied. An attempt is made to treat all such clusters of the third order as seif-energy diagrams and to inciude their effect into the single- particle energies, thereby improving the convergence of the Goldstone expansion considerably. For this purpose, an expression for offdiagonal reaction matrix elements is derived. One of the consequences of this procedure is that, to a good approximation, the contribution of all the third-order diagrarns together is equal to that arising from just the even l'' states of the simplest of these diagrams. Many of the arguments and approximations used are based on the previous paper of this issue by Bethe et al. (auth)

Nonsingular Formulation of the Brueckner Approximation for an Infinite Fermi System

In the approximation neglecting any but single pair correlations, singularities of the reaction matrix render Brueckner's integral for the average energy per particle a singular integral. Several attempts to overcome this difficulty have been unsuccessful. By considering the infinite Fermi system to be a limit of finite systems, it is shown that the correct result merely involves replacing Brueckner's ordinary integral over diagonal reaction matrix elements by a principal value integral. In a finite system the level shift of a Bethe-Goldstone state differs from the diagonal reaction matrix element by a normalization factor which does not approach unity uniformly in the integration variable as the volume becomes infinite. In the neighborhood of a singularity the expression for the two-particle energy shift takes the form $\frac{\mathrm{cy}}{({y}^{2}+c{\mathcal{U}}^{\ensuremath{-}1})}$, where $y$ is the unperturbed energy measured from the singularity, $c$ is the square of a matrix element, and $\mathcal{U}$ is the quantization volume. Hence as $\mathcal{U}\ensuremath{\rightarrow}\ensuremath{\infty}$ the sum over the energy $y$ indeed approaches a principal value integral. An alternative derivation, employing a modified reaction matrix for which there is no difference between level shift and matrix element, leads to the same result. The general derivations are preceded by a soluble example.The connection of the Brueckner approximation with a phase-shift approximation for low-density systems is discussed. Some corrections to the higher order terms in existing derivations of the "separation method" expansion of the Brueckner reaction matrix are given.

The use of polarised deuterons to study the D+D reactions

It is shown that further information on the spin transitions involved in the D+D reactions at low energies ( E D ≲ 1 MeV) may be obtained by the use of polarised deuteron beams, but that a complete determination of the reaction matrix elements requires two double polarisation measurements. The selection rules on the spin transitions are discussed in terms of the internucleon forces and the earlier formulae giving the energy dependence of the cross sections and polarisation are slightly modified to give a further degree of freedom in fitting the experimental data.