Random-matrix approach to the statistical compound nuclear reaction at low energies using the Monte Carlo technique

Using a random-matrix approach and Monte Carlo simulations, we generate scattering matrices and cross sections for compound-nucleus reactions. In the absence of direct reactions we compare the average cross sections with the analytic solution given by the Gaussian orthogonal ensemble (GOE) triple integral, and with predictions of statistical approaches such as the ones from Moldauer; Hofmann, Richert, Tepel, and Weidenmüller; and Kawai, Kerman, and McVoy. We find perfect agreement with the GOE triple integral and display the limits of validity of the latter approaches. We establish a criterion for the width of the energy-averaging interval such that the relative difference between the ensemble-averaged and the energy-averaged scattering matrices lies below a given bound. Direct reactions are simulated in terms of an energy-independent background matrix. In that case, cross sections averaged over the ensemble of Monte Carlo simulations fully agree with results from the Engelbrecht-Weidenmüller transformation. The limits of other approximate approaches are displayed.

Figure 1

An example of an elastic-scattering cross section generated with the statistical $R$-matrix method (bottom panel), compared with the real cross section in ENDF/B-VII.1 (top panel), for a neutron-induced reaction on ${}^{56}\mathrm{Fe}$ in the 1- to 2-MeV energy range.

Figure 2

Generated elastic- $|1-S_{aa}{|}^2$ and inelastic- $|S_{ab}{|}^2$ scattering cross sections with the GOE $S$-matrix for three different transmission coefficients $T_a$ of 0.1, 0.5, and 0.99. The solid curves are the elastic, and the dot-dashed curves are the inelastic cross sections. The left column is for $N=20$ and the right is for the $N=100$ case.

Figure 3

Distribution of elastic-scattering cross section at $E=0$ for many GOE $S$-matrix realizations. Three cases, $T_a=T_b=0.1$, 0.5, and 0.99 are shown. The arrows show the actual average values for each distribution.

Figure 4

$L(T_a,\mathrm{\Lambda },I)$ as defined in Eq. (45) versus the Lorentzian width $I$ for $\mathrm{\Lambda }=10$ channels and for different values of the transmission coefficient $T_a$. From the lowest to the highest curve $T_a$ changes from 0.1 to 0.9 in steps of width 0.1.

Figure 5

$L(0.5,\mathrm{\Lambda },I)$ as defined in Eq. (45) versus the Lorentzian width $I$ for $T_a=0.5$ and for different channel numbers $\mathrm{\Lambda }$. From the lowest to the highest curve the values of $\mathrm{\Lambda }$ are 2, 5, 10, 20, 30, 40, and 50.

Figure 6

$L(0.99,\mathrm{\Lambda },I)$ as defined in Eq. (45) versus the Lorentzian width $I$ for $T_a=0.99$ and for different channel numbers $\mathrm{\Lambda }$. From the lowest to the highest curve the values are $\mathrm{\Lambda }=2$, 3, 4, 5, 6, 8, 10, 12, and 15. The horizontal line shows $L=0.1$.

Figure 7

The symbols show for which values of the Lorentzian width $I$ and channel number $\mathrm{\Lambda }$ the function $L(0.99,\mathrm{\Lambda },I)$ attains the value $0.1$. The line is a least-squares fit to the symbols; $I(L=0.1)\simeq 2.2\mathrm{\Lambda }+1.9.$

Figure 8

Distribution of decay amplitudes ${\gamma }_{a\sigma }$, when the GOE $S$ matrix is written in the $K$-matrix form. The histograms correspond to $T_a=0.1$, 0.5, and 0.99, respectively. For $T_a=0.99$, we compare the Gaussian distribution with the one obtained from the standard deviation in Eq. (59).

Figure 9

Standard deviation of GOE decay amplitude distribution as a function of transmission coefficient. The dashed curves are Eq. (52) and the solid curve is Eq. (59).

Figure 10

Channel degree-of-freedom values ${\nu }_a$ as functions of ${\sum }_a{T}_a$. The symbols are the Monte Carlo simulation results, see text. The solid curves in the top panel are from Eq. (A17), and the dotted curves are from Eq. (A10) for $T_a=0.25$ and 0.75. The curves in the bottom panel show Moldauer's systematics given by Eq. (A7) for the same set of $T_a$.

Figure 11

Ratio of the elastic cross section to the Hauser-Feshbach prediction as a function of the ratio $T_a/T_b$ of transmission coefficients. The symbols are the MC simulation results. The curves are predictions by various statistical models.

Figure 12

Simulated elastic- (top panel) and inelastic- (bottom panel) scattering cross sections as functions of the background parameter $k_0$.

Figure 13

Ratio ${\sigma }_{\mathrm{CI}}/{\sigma }_{\mathrm{R}}$ of the compound inelastic-scattering cross section to the reaction cross section as a function of the ratio ${\sigma }_{\mathrm{DI}}/{\sigma }_{\mathrm{R}}$ of direct reaction cross section to the reaction cross section. The symbols show the ensemble average of the MC simulation, the dotted lines are cross sections calculated with the modified transmission coefficients $T_a^{\prime }$, and the solid lines are the result of the EW transformation. The top panel is for $T_a=0.5$, and the bottom panel is for $T_a=0.9$.