Multistep Compound Research Articles

Statistical theory of multistep compound reactions.

We generalize a simple version of the optical background representation for the transition amplitude to the case of well-nested doorways. Using this simple generalization of the optical background representation an alternate statistical theory of multistep compound reactions is suggested, which uses the idea of nested averaging in the case of well-nested doorways and is valid under very general conditions. Simple connections are established between the present approach of multistep compound reactions and that of Feshbach, Kerman, and Koonin, of Agassi, Weidenm\uller, and Mantzouranis, and of Hussein and McVoy. The present formulation exhibits the chaining condition of Feshbach, Kerman, and Koonin in the extreme low energy limit, is time reversal symmetric, and yields the Bohr description of compound nuclei in the appropriate limit.

Particle-hole state densities for calculation of the multi-step compound emission

Analytical formulae have been derived for the density of particle-hole states having all particles bound. Such states are involved in the statistical multi-step compound MSC emission. Results of calculations are compared with predictions of other theoretical approaches.

Statistical multi-step compound emission in (n, 2n) reactions

The multi-step compound-emission (MSC) theory has been used to analyse the preequilibrium effects in the neutron-emission spectra and excitation curves of the (n, 2n) reactions. Cross sections for the 191Ir(n, 2n) and 193Ir(n, 2n) reactions are reported in the neutron-energy range from 13 MeV to 18 MeV and analysed in terms of combined compound-nucleus and MSC theory.

The average angular distribution of emitted particles in multi-step compound processes

A simple model for the differential cross section that describes the angular distribution of emitted particles in heavy-ion induced multi-step compound reactions, is constructed. It is suggested that a careful analysis of the deviations of the experimental data from the pure Hauser-Feshbach behaviour may shed light on the physical nature of the pre-compound, heavy-ion configuration.

Statistical Theory of Multistep Compound Reactions Revisited

With use of a simple version of the optical background representation for the transition amplitude in the case of well nested doorways, an alternate statistical theory of multistep compound reactions is suggested, which is valid under very general conditions. Simple connections are estalished between the present approach and that of Feshbach, Kerman, and Koonin and of Agassi, Weidenm\"uller, and Mantzouranis.

Doorways and multistep compound reactions

The "optical background" representation of Kawai, Kerman, and McVoy for the fluctuation amplitude is written in an alternate form which has certain advantages in formulating the theory of doorway states and of multistep compound reactions. We reconsider the doorway state approach of Feshbach, Kerman, and Lemmer using the present representation of the fluctuation amplitude. Finally, we use the present representation of the fluctuation amplitude for developing a theory of multistep compound reactions following the idea of chaining due to Feshbach, Kerman, and Koonin. The advantages of the present approach are emphasized in both the cases.NUCLEAR REACTIONS Optical background representation of the fluctuation amplitude, doorways, preequilibrium reactions, multistep compound reactions.

Emission of alpha particles in statistical multistep compound reactions

Excitation functions and angular distributions of alpha particles emitted from the $^{27}\mathrm{Al}(^{3}\mathrm{He}, \ensuremath{\alpha})$ and $^{27}\mathrm{Al}(\mathrm{p}, \ensuremath{\alpha})$ reactions are measured at a composite nucleus excitation energy of about 30 MeV. The experimental results together with the widely varying correlation widths extracted from the fluctuation analysis are successfully interpreted on the basis of the statistical multistep compound emission mechanism. Confirmation is found of the interpretation of each experimental correlation width in terms of a coherent sum of the widths of the relevant stages of the reactions, previously proposed in the case of proton channels. Moreover, the $\ensuremath{\alpha}$-spectrum shape and the correlation width values are shown to be consistent with an $\ensuremath{\alpha}$ preformation picture of the target nuclei considered.NUCLEAR REACTIONS $^{27}\mathrm{Al}(^{3}\mathrm{He}, \ensuremath{\alpha})$ $E=8.2\ensuremath{-}13$ MeV; $^{27}\mathrm{Al}(\mathrm{p}, \ensuremath{\alpha})$, $E=18.5\ensuremath{-}25.6$ MeV; measured $\ensuremath{\sigma}(E;\ensuremath{\vartheta})$; statistical multistep compound emission, calculated $\ensuremath{\sigma}(E)$, $\ensuremath{\sigma}({E}_{\ensuremath{\alpha}})$; generalized autocorrelation analysis, extracted coherence widths; suggested $\ensuremath{\alpha}$ preformation.

Two body residual interaction in the statistical multistep compound and direct theories

It is shown that within the multistep compound theory the use of matrix elements calculated with a density dependent two-body residual interaction of the Yukawa form makes possible reproduction of the experimental data of multistep compound reactions with an interaction strength ${V}_{0}$ which turns out to be consistent with the one used in the analysis of multistep direct processes.NUCLEAR REACTIONS Statistical multistep compound and direct analysis, deduced two-body residual interaction strength.

Angular distributions in pre-equilibrium reactions

A new model is proposed for calculating angular distributions in preequilibrium reactions. In this model, as in the model of Feshbach et al. the system consisting of target plus projectile initially branches into two sets of states with either no particle in the continuum (multistep compound states) or with at least one particle in the continuum (multistep direct states). The multistep compound emission is assumed to be isotropic while the angular distribution of the multistep direct emission is described using the fast particle model of Mantzouranis et al. A similar master equation is used for both chains of states differing only in the angular dependence of the emission rates. The two chains of states are treated independantly neglecting inter-branch transitions. The angular distributions for 14.6 MeV neutrons calculated using this model are found to be in better agreement with the data than the fast particle model.

Random-walk model of precompound decay II: Stochastic uncertainties in the lifetimes and cross-sections

The uncertainties arising from the stochastic nature of precompound decay nuclear reactions are analyzed in the framework of the preequilibrium exciton and random-walk models. It is demonstrated that the standard deviations and the mean values of the exciton-state lifetimes are of the same order of magnitude. Their correlations are weakly positive, except for exciton states near the equilibrium number, where the correlations are significant. The usefulness and the limitations of the never-come-back approximation are discussed. A general proof is presented of the conditions under which the master-equation and random-walk approaches to Markov processes are equivalent. Connections between different preequilibrium models, e.g. the multi-step compound model and the microscopic statistical theory of precompound decay, are pointed out. It is shown that the waiting time between subsequent collisions is governed by a Poisson process, suggesting that the variance associated with the nucleon mean free path in nuclear matter, as estimated from preequilibrium models, is considerable. The stochastic uncertainties in the emission cross-sections correspond to those of a Bernoulli process.

Interpretation of some (p, n), (n, p), and (He3,p) reactions by means of the statistical multistep compound emission theory

A number of ($p, n$), ($n, p$), and ($^{3}\mathrm{He}$, $p$) reactions have been interpreted on the basis of the statistical multistep compound emission mechanism. Good agreement with experiment is found both in spectrum shape and in the value of the coherence widths.NUCLEAR REACTIONS $^{25}\mathrm{Mg}(^{3}\mathrm{He}, p)$, $^{27}\mathrm{Al}(^{3}\mathrm{He}, p)$, $^{40}\mathrm{Ca}(n, p)$, $^{59}\mathrm{Co}(p, n)$, $^{89}\mathrm{Y}(p, n)$, $^{103}\mathrm{Rh}(p, n)$, ${E}_{\mathrm{inc}}=13, 13, 14.5, 14.7, 14.8, 18$ MeV. Statistical multistep compound emission mechanism. Calculated differential cross sections and coherence widths.

Eigenclass sum rules in multi-step compound reactions

In a doorway model of pre-compound reactions (such as the exciton model), the fine-structure levels of Bohr's compound-nucleus model are generalized to include one or more classes of doorway states, through which flux percolates during the relaxation of the system toward equilibrium. In the 1-class model (i.e., normal compound-nucleus model) with Γ ⪢ D, Wiedenmuller and collaborators have noted the unitarity sum rule Σ cT c= 2πΓ corr D . W show that, in the N-class case, N sum rules of analogous form exist, one for each eigenclass.

Variance of Ericson fluctuations in multistep compound processes

It is pointed out that within the nested doorway model, the approximate Ericson's formula for the variance of cross section fluctuations is valid independently of the number of doorway classes present. Comparison is presented with the experimental values of the variance for the recently analyzed $^{3}\mathrm{He}$+$^{25}\mathrm{Mg}$ reactions, in which multistep compound processes have been found to be dominant.NUCLEAR REACTIONS $^{25}\mathrm{Mg}(^{3}\mathrm{He}, p)$, $^{25}\mathrm{Mg}(^{3}\mathrm{He}, \ensuremath{\alpha})$; multistep compound; Ericson fluctuations; extracted and calculated reduced variance.

Multistep compound processes in heavy-ion-induced reactions

We suggest the possible contribution of multistep compound processes to light heavyion reactions at above-barrier energies. The presence of several correlation widths that characterize the fluctuation in energy of the excitation functions for different incidentoutgoing channel combinations is indicated and demonstrated explicitly for the system $^{12}\mathrm{C}$($^{15}\mathrm{N}$, $\ensuremath{\alpha}$) at ${E}_{\mathrm{c}.\mathrm{m}.}=9.51\ensuremath{-}17.33$ MeV. The importance of multistep compound processes to heavy-ion fusion studies is pointed out.NUCLEAR REACTIONS Multistep compound processes in heavy ion reactions discussed.

Fluctuations due to statistical multistep compound emission in theMg25(He3,p) andMg25(He3,α) reactions

Excitation functions and angular distributions of protons and alpha particles emitted from the $^{3}\mathrm{He}$ + $^{25}\mathrm{Mg}$ reaction were measured in the $^{3}\mathrm{He}$ energy range from 8.05 to 20.425 MeV. The excitation functions show large, uncorrelated structures. Analysis on the basis of the autocorrelation function and the spectral density method yields largely varying values of the coherence energy, which can be attributed to the different steps of the statistical multistep compound emission chain. This process explains the average shape of the excitation functions, and also the absolute value of the emitted proton spectrum, measured at 13 MeV incident $^{3}\mathrm{He}$ beam.NUCLEAR REACTIONS $^{25}\mathrm{Mg}$ ($^{3}\mathrm{He}$, $p$), ($^{3}\mathrm{He}$, $\ensuremath{\alpha}$), $E=8.05\ensuremath{-}20.425$ MeV; measured $\ensuremath{\sigma}(E;\ensuremath{\theta})$; calculated $\ensuremath{\sigma}(E)$, $\ensuremath{\sigma}({E}_{p})$; deduced coherence widths.

“Sum rules” for preequilibrium reactions

I present evidence that suggests that the correct relationship between the optical transmission matrix, P, and the several correlation widths, Γ n , found in multistep compound (preequilibrium) nuclear reactions is Σ n2πΓ n/D n = tr P . A s econd sum rule is also derived within the shell model approach to nuclear reactions. Indications of the potential usefulness of the sum rules in preequilibrium studies are given.

Unchained multi-step compound reactions

The chaining constraint employed by Feshbach, Kerman and Koonin (FKK) in multi-step compound reactions is removed and S-matrix elements are computed exactly. The cross sections have the same form as in the work of FKK — except that the probability flow is modified to take into account all possible paths for the depletion of the flux.

Calculation of the lifetime of the multistep compound process in the reaction27Al(3He, p)

Calculation of the lifetime of the multistep compound process in the reaction27Al(3He, p)

Phenomenology of continuum angular distributions. I. Systematics and parametrization

A large number of experimental angular distributions have been studied for particles emitted into the continuum in preequilibrium nuclear reactions in order to study their systematics. For pure multistep direct reactions it has been found that to first order the shapes of these angular distributions are determined by the energy of the outgoing particle. In addition, there is evidence that the division of the cross section into multistep direct and multistep compound components with the latter having a symmetric angular distribution is a meaningful one. The shapes of the angular distributions can be described in terms of Legendre polynomials. The polynomial coefficients are given by simple phenomenological relations involving the energy of the outgoing particle. The same reduced coefficients are used for multistep direct and multistep compound processes except that only even order polynomials are considered in the latter case. The present formulation has been shown to have significant predictive ability for light ion reactions.NUCLEAR REACTIONS $A(a,b)$, $A=^{12}\mathrm{C} \mathrm{to} ^{232}\mathrm{Th}$, $a=p,d,^{3}\mathrm{He},^{4}\mathrm{He}$, $b=n,p,d,t,^{3}\mathrm{He},^{4}\mathrm{He}$, $E=18\ensuremath{-}80$ MeV, ${E}_{b}=4\ensuremath{-}60$ MeV. Deduced systematics of $\frac{\ensuremath{\sigma}({E}_{b},\ensuremath{\theta})}{\ensuremath{\sigma}({E}_{b})}$ in continuum. Parametrized $\frac{\ensuremath{\sigma}({E}_{b},\ensuremath{\theta})}{\ensuremath{\sigma}({E}_{b})}$ in continuum with energy dependent Legendre polynomial coefficients.

Phenomenology of continuum angular distributions. II. Griffin preequilibrium model

The Griffin or exciton model for preequilibrium nuclear reactions has been extended so that particle unbound states in the equilibrating system are distinguished from bound states. This permits the preequilibrium cross section to be divided into components due to multistep direct and multistep compound processes for calculation of emitted particle angular distributions. Preequilibrium models traditionally assume that all states of a given particle-hole number are equally likely whereas present work indicates that unbound states are preferentially populated. Nevertheless, the emission energy spectra calculated are not significantly changed from earlier versions of the model.NUCLEAR REACTIONS Extended Griffin preequilibrium statistical model to calculate multistep direct and multistep compound cross sections separately. Tested equal a priori probabilities assumption.