Statistical Hauser-Feshbach theory with width-fluctuation correction including direct reaction channels for neutron-induced reactions at low energies

A model to calculate particle-induced reaction cross sections with statistical Hauser-Feshbach theory including direct reactions is given. The energy average of the scattering matrix from the coupled-channels optical model is diagonalized by the transformation proposed by Engelbrecht and Weidenmüller [C. A. Engelbrecht and H. A. Weidenmüller, Phys. Rev. C 8, 859 (1973)]. The ensemble average of $S$-matrix elements in the diagonalized channel space is approximated by a model of Moldauer [P. A. Moldauer, Phys. Rev. C 12, 744 (1975)] using the newly parametrized channel degree-of-freedom ${\nu }_a$ to better describe the Gaussian orthogonal ensemble (GOE) reference calculations. The Moldauer approximation is confirmed by a Monte Carlo study using a randomly generated $S$ matrix, as well as the GOE threefold integration formula. The method proposed is applied to the ${}^{238}\mathrm{U}\left(n,n^{\prime }\right)$ cross-section calculation in the fast-energy range, showing an enhancement in the inelastic scattering cross sections.

Figure 1

Ratio of calculated cross sections using the randomly generated $S$ matrix, as a function of the direct reaction strength. The ratio is that of generalized transmission coefficient calculations to the EW transformation case. Panel (a) is for a number of channels of $\mathrm{\Lambda }=2$ and 3, panel (b) is for $\mathrm{\Lambda }=4$ and 5, and panel (c) is for $\mathrm{\Lambda }=6$ and 7.

Figure 2

Ratio of calculated inelastic scattering cross section with the generalized transmission coefficient calculations to the EW transformation case, as a function of the elastic enhancement factor $W_a$.

Figure 3

Ratio of the cross sections calculated with the generalized transmission coefficient calculations to the cross sections calculated with the EW transformation case, for $\mathrm{\Lambda }=3$ and the third channel is uncoupled.

Figure 4

Calculated $|\overline{S_{aa}{S}_{bb}^{*}}|$ with the GOE triple-integral formula for randomly generated $S$ matrix and number of channels. The results are shown as a function of ${\sum }_c{T}_c$.

Figure 5

Comparison of Moldauer's estimate for $|\overline{S_{aa}{S}_{bb}^{*}|}$ given by Eq. (19) for various $T_a$ values and channels, shown by the ratios to the GOE calculation. Two different estimates for the channel degree-of-freedom $\nu$, Refs. [26] and [20], are used; panel (a) is for the smaller ${\sum }_c{T}_c$ case, and panel (b) is for the larger ${\sum }_c{T}_c$ case.

Figure 6

Compound elastic and inelastic cross sections calculated with the randomly sampled $S$ matrix as well as using Moldauer's estimate for $|\overline{{\tilde{S}}_{\alpha \alpha }{\tilde{S}}_{\beta \beta }^{*}|}$, as a function of the dimensionless total cross section. The results are shown by the deviation from the GOE results. The top panel is for the two-channel case, and the bottom panel is for the five-channel case.

Figure 7

Calculated ${}^{238}\mathrm{U}$($n,n^{\prime }$) reaction cross sections with the EW transformation (solid curves) compared with the modified transmission calculation (dashed curves), as well as with the evaluated cross sections in JENDL-4 (dot-dashed curves).

Figure 8

Ratios of calculated capture, total inelastic, and fission cross sections without the EW transformation to the EW cases.