Erratum: Cross sections of neutron-induced reactions [Phys. Rev. C82, 044613 (2010)]

Abstract

The first equation of Eqs.(3) in [1] was used to describe the mass number and energy dependence of experimental total neutron cross sections for the first time in [2], while the second and third ones were used for scattering and reaction cross sections in [3]. The omissions of these two references were unintended. We derived these equations and Eq. (4) of Ref.[4] (Ref.[12] of our paper [1]) as follows. From partial wave analysis of scattering theory, we know the standard expressions for scattering σ sc and reaction σ r cross sections as, where the quantity η l = e 2iδ l . With the assumption that the phase shift δ l is independent of l and the summation over partial waves l is up to kR only, it follows that σ sc = π (R + λ -) 2 (1 + α 2 − 2α cos β), σ r = π (R + λ -) 2 (1 − α 2 ), and σ tot = σ sc + σ r = 2π (R + λ -) 2 (1 − α cos β), where λ -= 1/k, R is the channel radius beyond which partial waves do not contribute, β = 2Reδ l = 2Reδ, α = e −2Imδ l = e −2Imδ , and summing over l from 0 to kR yielded l (2l + 1) = (kR + 1) 2 .We used the name "nuclear Ramsauer model" from Ref.[12] of our paper [1]. Carpenter and Wilson [5] were the first to call the structure found in total neutron cross * tkm@veccal.ernet.in † joy@veccal.ernet.in ‡ dnb@veccal.ernet.in sections the nuclear Ramsauer effect. This name was adopted by subsequent authors, although the nature of the oscillation in fast-neutron cross sections is essentially different from that observed for slow electrons by Ramsauer. In other works the names "semiclassical optical model" [3] or "diffraction effect" [6] were used, which are more appropriate. From the model of [1] one cannot expect the accuracy of a complete quantummechanical optical model. However, the simple semiclassical optical model [1] obtained to calculate cross sections up to 600 MeV are of relevance as phenomenological optical model potentials are limited up to 150-200 MeV.In fact, the radius of the potential well is just r 0 A 1 3 = r 1 A 1 3 +γ and r 1 = constant. The parameter γ is a very small number (0.00793) compared to the 1 3 needed for fine tuning. It should, therefore, be emphasized that, as mentioned in our paper [1], it is r 0 which is used for fixing β 0 . It is the channel radius which is energy dependent. Channel radius is the radius [appearing in Eqs. (3) of our paper] beyond which no partial waves contribute. It is well known from R-matrix theory that the channel radius is less than the nuclear (potential) radius, which is precisely the case here.Obviously, these omissions do not affect the results and conclusion of the original manuscript [1].We thank Drs. I. Angeli and J. Csikai for bringing this matter to our attention.[1] T.

Theoretical exploration of S-factors for nuclear reactions of astrophysical importance

“…where E is the centre-of-mass energy, S(E) is the astrophysical S-factor and η is the Sommerfeld parameter, defined by η = Z1Z2e 2 hv where Z 1 and Z 2 are the charges of the interacting nuclei in units of elementary charge e. In case of a narrow resonance, the resonant cross section is approximated generally by a Breit-Wigner expression, whereas the neutron induced reaction cross sections at low energies can be given by σ(E) = R(E) v [12] where R(E) is a slowly varying function of energy [13] and is similar to S-factor. The astrophysical S-factor S(E) which is thus a rescaled variant of the total cross section σ(E) is required for many astrophysical applications particularly at energies below the Coulomb barrier.…”

Astrophysical S-factor for the deep sub-barrier fusion reactions of light nuclei

“…where S(E) is the astrophysical S-factor and ζ is the Sommerfeld parameter, defined by ζ = Z1Z2e 2 hv where Z 1 and Z 2 are the charges of the reacting nuclei in units of elementary charge e. Except for narrow resonances, the S-factor S(E) is a smooth function of energy, which is convenient for extrapolating measured cross sections down to astrophysical energies. In the case of a narrow resonance, the resonant cross section is generally approximated by a Breit-Wigner expression whereas the neutron induced reaction cross sections at low energies can be given by σ(E) = R(E) v [5] facilitating extrapolation of the measured cross sections down to astrophysical energies, where R(E) is a slowly varying function of energy [6] and is similar to S-factor.…”

Angular distributions of neutron-nucleus collisions

“…Often, these cross sections are evaluated using phenomenological optical potentials, and much effort has gone into defining global sets of parameter values for those optical potentials with which to estimate cross sections as yet unmeasured. In this context it would be useful to study the systematics of neutron absorption and scattering cross sections on various nuclei well approximated by a simple convenient functional form [10,11]. The present model is an easy-to-use analytical parametrization that may be adequate to produce the scattering cross sections needed in Monte Carlo neutron transport calculations at energies higher than 150-200 MeV.…”

“…The radius of the nuclear potential is given by R = r 0 A 1 3 , whereas the channel radius can be parametrized as R ch = r 0 A [11]. The value of α 0 is kept fixed at 0.2929, and the nonlinear least-squares fits yield the value for the imaginary potential W 0 = 5.293 MeV and its energy dependence W E = 33.88 × 10 −2 .…”

“…The nuclear radius parameter r 0 is also fitted reasonably well to 1.378A γ fm, which means that the nuclear potential radius R = r 0 A 1 3 +γ where γ = 7.93 × 10 −3 is a very small number (needed for fine tuning) compared to 1 3 . In our earlier work [11] the effect of the imaginary potential was included in the parameter α in a somewhat ad hoc manner lacking justification for it to be weakly mass dependent. The present work is an improvement over our earlier work, since the imaginary part of the nuclear potential is now treated appropriately with the explicit appearance of the imaginary part of the nuclear potential.…”