Comparisons between various width fluctuation correction factors for compound nucleus reactions
Introduction
In the original Hauser–Feshbach formula [1], the compound nucleus cross section σab for an entrance channel a and an outgoing channel b readswhere ka is the wave number of the entrance channel and Ti is the transmission coefficient for the channel i. It is however known that this expression is incorrect and should be multiplied by a factor Wab called the width fluctuation correction factor (WFCF). The WFCF accounts for the correlations that exist between the incident and outgoing waves. From a qualitative point of view, these correlations enhance the elastic channel and accordingly decrease the other open channels. Above a few MeV of projectile energy, when many competing channels are open, the WFCF can be neglected and the simple Hauser–Feshbach is adequate to describe the compound nucleus decay.
This WFCF has been widely studied in the past [2], [3], [4], [5], [6], [7], [8], [9], [10], [11] and various simple expressions are currently employed to calculate Wab. One of the most recent expressions is the one derived within the framework of the Gaussian orthogonal ensemble (GOE) of Hamiltonian matrices [9]. This approach does not explicitly give Wab but rather σab as function of the S-matrix elements aswhich after complicated algebraic transformations reduces to a complicated triple integral.
Upon looking at the hypothesis necessary to derive this triple integral compared to what is assumed to obtain the approximate expressions of Refs. [2], [3], [4], [5], [6], [7], it is clear that the GOE result should be considered as the reference value for compound nucleus decay. However, in practice, such a triple integral may require too much calculation time to be used extensively for cross sections calculations. It is therefore important to see whether a more simple expression can provide similar results. The goal of the present work is thus to compare the results yielded by two well-known expressions for Wab, namely those of Hoffmann–Richert–Tepel–Weidenmüller (HRTW) [5], [6], [7] and Moldauer [3], [4], with those yielded by the GOE method, considered here as the reference result. For this purpose, all these expressions have been implemented in a statistical nuclear reaction code, TALYS [12]. Similar studies have already been performed comparing the GOE results either with the results obtained using the Moldauer expression [13], [14] or those yielded by the HRTW method [15], [16]. However, these studies do not consistently take into account all possible reaction channels (including fission), nor do they compare the three expressions at the same time and thus do not enable one to select between the HRTW and the Moldauer approach for practical applications.
We first give in Section 2 the general compound nucleus formula. We then review in Section 3 both the HRTW and the Moldauer expressions and give some details about the way we calculated them. Section 4 contains a considerable mathematical outline of the calculation of the triple integral. Finally in Section 5, we study neutron-induced reactions on both non-fissile and fissile nuclei, to discuss the various expressions for the WFCF and we conclude on which of the latter can be used with enough confidence for practical applications.
Section snippets
Compound nucleus formula
In the compound nucleus picture of binary reactions, we consider a projectile a with incident energy Ea and a target A with incident energy EA forming a compound nucleus with energy Etot. This system then decays to produce an ejectile b with excitation energy Eb and a residual nucleus B with excitation energy EB such thatwhere Sa (resp. Sb) is the projectile (resp. ejectile) separation energy in the compound nucleus. The cross section σab of such a process reads [17]
Moldauer and HRTW approaches
We first recall the two approximate methods usually considered to calculate the WFCF’s, namely the iterative HRTW method and the Moldauer method. For convenience, we simplify the notation of Eq. (2) and write the compound nucleus cross section for an entrance channel a and an outgoing channel b as
Numerical evaluation of the GOE triple integral
As mentioned in Section 1, the GOE method does not explicitly define the WFCF Wab but rather σab as function of the S-matrix elements,After complicated algebraic transformations [9] one obtains the complicated triple integral
Discussions
We now compare the various expressions described in 3 Moldauer and HRTW approaches, 4 Numerical evaluation of the GOE triple integral for some practical cases. We first consider a non-fissile nucleus, and then a fissile nucleus.
Conclusions
We have implemented the most exact expression available in the litterature for the width fluctuation corrections, that based on the GOE method, and compared it with the two usually employed approximate expressions of Moldauer and HRTW. Since the differences between the methods are generally smaller than experimental uncertainties, we have used the exact GOE approach as a reference to determine which of the practicable methods deserves to be used in routine calculations. This comparison has been
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