Moldauer’s Sum Rule and Superradiance
61
10
-4
10
-3
10
-2
1
2
3
4
5
6
7
8
9
90 Zr(n,γ)
(a)
Cross Section (b)
Incident Neutron Energy (MeV)
EXFOR
No Superradiance
Superradiance (SR)
Superradiance (MS)
0.0
0.5
1.0
1.5
1
2
3
4
5
6
7
8
9
90
Zr(n,inel)
(b)
Cross Section (b)
Incident Neutron Energy (MeV)
No Superradiance
Superradiance (SR)
Superradiance (MS)
Fig. 2 Plots of 90 Zr cross sections, computed with and without the superradiance-modified
Hauser-Feshbach equation. Cross sections computed with the sum rule and Modauer-Simonius
parameterizations are labeled “SR” and “MS,” respectively. Experimental data from the EXFOR
library are also shown [16]. Figure from Ref. [5]
(causing D → 0) or a few very strong resonances (causing Γ c → ∞) acting
incoherently [14]. The superradiant effect has been seen in many other mesoscopic
systems [13, 14] and because compound nuclear reactions are only treated in the
weak coupling limit, the effect is neglected.
The peaks in the p-wave transmission coefficients suggest that we might see
superradiance in 90 Zr cross sections. We calculated the 90 Zr cross sections using
the EMPIRE [15] reaction code, modified with the substitution in Eq. (4). The
results are shown in Fig. 2 for the capture and total inelastic cross sections. The
effects of superradiance are not obvious either at low energy (where we are in the
weak coupling limit) or at high energy (where there are a large number of open
channels and the effects of pre-equilibrium emission become evident). In the region
around 2–4 MeV, we see noticeable differences between the cross sections computed
with and without superradiance. While the most dramatic changes are in the total
inelastic and capture cross sections, the elastic cross section shows an effect as
well. Superradiance appears to cause an interesting modification to the shape of the
inelastic cross section just above threshold. A measurement of 90 Zr(n, n γ ) between
2 and 4 MeV would be very helpful by providing experimental evidence for (or lack
of) superradiance.
4 Conclusion and Outlook
In Ref. [5], we investigated the consequences of different formulations of the
neutron transmission coefficient, enabling a rigorous connection between the RRR,
URR, and fast neutron regions. This work suggests that predictions of the mean
level spacing coupled with an optical model potential can allow one to predict the
average neutron widths even in the strong coupling regime. This provides a tool for
predicting neutron widths far off stability.
61
10
-4
10
-3
10
-2
1
2
3
4
5
6
7
8
9
90 Zr(n,γ)
(a)
Cross Section (b)
Incident Neutron Energy (MeV)
EXFOR
No Superradiance
Superradiance (SR)
Superradiance (MS)
0.0
0.5
1.0
1.5
1
2
3
4
5
6
7
8
9
90
Zr(n,inel)
(b)
Cross Section (b)
Incident Neutron Energy (MeV)
No Superradiance
Superradiance (SR)
Superradiance (MS)
Fig. 2 Plots of 90 Zr cross sections, computed with and without the superradiance-modified
Hauser-Feshbach equation. Cross sections computed with the sum rule and Modauer-Simonius
parameterizations are labeled “SR” and “MS,” respectively. Experimental data from the EXFOR
library are also shown [16]. Figure from Ref. [5]
(causing D → 0) or a few very strong resonances (causing Γ c → ∞) acting
incoherently [14]. The superradiant effect has been seen in many other mesoscopic
systems [13, 14] and because compound nuclear reactions are only treated in the
weak coupling limit, the effect is neglected.
The peaks in the p-wave transmission coefficients suggest that we might see
superradiance in 90 Zr cross sections. We calculated the 90 Zr cross sections using
the EMPIRE [15] reaction code, modified with the substitution in Eq. (4). The
results are shown in Fig. 2 for the capture and total inelastic cross sections. The
effects of superradiance are not obvious either at low energy (where we are in the
weak coupling limit) or at high energy (where there are a large number of open
channels and the effects of pre-equilibrium emission become evident). In the region
around 2–4 MeV, we see noticeable differences between the cross sections computed
with and without superradiance. While the most dramatic changes are in the total
inelastic and capture cross sections, the elastic cross section shows an effect as
well. Superradiance appears to cause an interesting modification to the shape of the
inelastic cross section just above threshold. A measurement of 90 Zr(n, n γ ) between
2 and 4 MeV would be very helpful by providing experimental evidence for (or lack
of) superradiance.
4 Conclusion and Outlook
In Ref. [5], we investigated the consequences of different formulations of the
neutron transmission coefficient, enabling a rigorous connection between the RRR,
URR, and fast neutron regions. This work suggests that predictions of the mean
level spacing coupled with an optical model potential can allow one to predict the
average neutron widths even in the strong coupling regime. This provides a tool for
predicting neutron widths far off stability.