Moldauer’s Sum Rule Implies
Superradiance in Compound Nuclear
Reactions
David Brown, Mike Herman, and Gustavo Nobre
1 Background
For neutron-induced reactions, the unresolved resonance region (URR) interpolates
between the fast neutron range and the resolved resonance region (RRR). The
fluctuations in the neutron cross sections in the URR are not fully resolvable yet the
URR in a typical nucleus is in the 100 keV–2 MeV window, where fission spectra
peak, therefore introducing substantial uncertainty in practical applications. As the
cross section fluctuates strongly in the URR, at best we can describe the probability
distribution of the cross section in terms of the average resonance spacing D, the
average channel widths Γ c , and the number of degrees of freedom ν c for each
channel c. Here “channel” denotes the two incoming/outgoing particles and all the
quantum numbers needed to specify their state (for our purposes only the orbital
angular momentum L and total angular momentum J ).
We focus on the energy average cross sections in the URR as a first step toward
determining the full probability distribution. The Gaussian Orthogonal Ensemble
(GOE) triple integral result of Verbaarschot, Weidenmüller, and Zirnbauer [1] is
believed to provide an exact solution for the energy averaged cross section, but it is
both difficult to interpret physically and numerically expensive to use in practice.
Given this, we consider the Hauser-Feshbach equation with Moldauer’s Width
Fluctuation Correction [2] as it is both easier to use and simpler to interpret. The
Hauser-Feshbach equation is
D. Brown () · G. Nobre
National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY, USA
e-mail: dbrown@bnl.gov
M. Herman
National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY, USA
Los Alamos National Laboratory, Los Alamos, NM, USA
© This is a U.S. government work and not under copyright protection
in the U.S.; foreign copyright protection may apply 2021
J. Escher et al. (eds.), Compound-Nuclear Reactions, Springer Proceedings in
Physics 254, https://doi.org/10.1007/978-3-030-58082-7_7
57
Superradiance in Compound Nuclear
Reactions
David Brown, Mike Herman, and Gustavo Nobre
1 Background
For neutron-induced reactions, the unresolved resonance region (URR) interpolates
between the fast neutron range and the resolved resonance region (RRR). The
fluctuations in the neutron cross sections in the URR are not fully resolvable yet the
URR in a typical nucleus is in the 100 keV–2 MeV window, where fission spectra
peak, therefore introducing substantial uncertainty in practical applications. As the
cross section fluctuates strongly in the URR, at best we can describe the probability
distribution of the cross section in terms of the average resonance spacing D, the
average channel widths Γ c , and the number of degrees of freedom ν c for each
channel c. Here “channel” denotes the two incoming/outgoing particles and all the
quantum numbers needed to specify their state (for our purposes only the orbital
angular momentum L and total angular momentum J ).
We focus on the energy average cross sections in the URR as a first step toward
determining the full probability distribution. The Gaussian Orthogonal Ensemble
(GOE) triple integral result of Verbaarschot, Weidenmüller, and Zirnbauer [1] is
believed to provide an exact solution for the energy averaged cross section, but it is
both difficult to interpret physically and numerically expensive to use in practice.
Given this, we consider the Hauser-Feshbach equation with Moldauer’s Width
Fluctuation Correction [2] as it is both easier to use and simpler to interpret. The
Hauser-Feshbach equation is
D. Brown () · G. Nobre
National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY, USA
e-mail: dbrown@bnl.gov
M. Herman
National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY, USA
Los Alamos National Laboratory, Los Alamos, NM, USA
© This is a U.S. government work and not under copyright protection
in the U.S.; foreign copyright protection may apply 2021
J. Escher et al. (eds.), Compound-Nuclear Reactions, Springer Proceedings in
Physics 254, https://doi.org/10.1007/978-3-030-58082-7_7
57