60
D. Brown et al.
10
0
10
1
10
2
10
3
10
4
0
0.25
0.5
0.75
1
Transmission coeff.
Averaged RRR
ECIS
SR
MS
SPRT
weak (1
st order)
weak (2
nd order)
10
1
10
2
10
3
10
4
0
0.25
0.5
0.75
10
1
10
2
10
3
10
4
Incident Neutron Energy (keV)
(a) L=0, J=1/2
(c) L=1,
J=3/2
(b) L=1,
J=1/2
(d) L=2, J=3/2
(e) L=2, J=5/2
Fig. 1 Neutron transmission coefficients of 90 Zr, computed using ECIS and RIPL optical model
potential #612 [11] and computed directly from the resolved and unresolved resonance parameters
in the ENDF/B-VIII.0 file. Figure from Ref. [5]
3 Superradiance
The WFC in Eq. (2) was derived in the weak coupling limit (x c = π Γ c /D 1) but
it is regularly applied outside its region of validity. Both Eqs. (1) and the WFC (2)
contain factors Γ b /
c Γ c which, by substituting the sum rule parameterization in
Table 1, give
Γ b
c Γ c
SR
=
T b /
√
1 − T b
c T c /
√
1 − T c
.
(4)
There is an additional factor of Γ a in σ abs
a which we will return to. If instead one
used the Moldauer–Simonius parameterization, we find a similar expression. Both
of these substitutions reduce to the one shown in Eq. (3) in the weak coupling
limit. The SPRT parameterization does not provide a unique mapping between
x c = π Γ c /D and T c due to its behavior at large x c , so we do not know how to
make an equivalent substitution for it.
Both Eq. (4) and the equivalent Moldauer–Simonius parameterization have
potentially dramatic implications. When we reach the strong coupling limit in only
one channel (so T c → 1), that channel dominates the cross section, an effect
known as superradiance [13]. This might happen if there are many close resonances
D. Brown et al.
10
0
10
1
10
2
10
3
10
4
0
0.25
0.5
0.75
1
Transmission coeff.
Averaged RRR
ECIS
SR
MS
SPRT
weak (1
st order)
weak (2
nd order)
10
1
10
2
10
3
10
4
0
0.25
0.5
0.75
10
1
10
2
10
3
10
4
Incident Neutron Energy (keV)
(a) L=0, J=1/2
(c) L=1,
J=3/2
(b) L=1,
J=1/2
(d) L=2, J=3/2
(e) L=2, J=5/2
Fig. 1 Neutron transmission coefficients of 90 Zr, computed using ECIS and RIPL optical model
potential #612 [11] and computed directly from the resolved and unresolved resonance parameters
in the ENDF/B-VIII.0 file. Figure from Ref. [5]
3 Superradiance
The WFC in Eq. (2) was derived in the weak coupling limit (x c = π Γ c /D 1) but
it is regularly applied outside its region of validity. Both Eqs. (1) and the WFC (2)
contain factors Γ b /
c Γ c which, by substituting the sum rule parameterization in
Table 1, give
Γ b
c Γ c
SR
=
T b /
√
1 − T b
c T c /
√
1 − T c
.
(4)
There is an additional factor of Γ a in σ abs
a which we will return to. If instead one
used the Moldauer–Simonius parameterization, we find a similar expression. Both
of these substitutions reduce to the one shown in Eq. (3) in the weak coupling
limit. The SPRT parameterization does not provide a unique mapping between
x c = π Γ c /D and T c due to its behavior at large x c , so we do not know how to
make an equivalent substitution for it.
Both Eq. (4) and the equivalent Moldauer–Simonius parameterization have
potentially dramatic implications. When we reach the strong coupling limit in only
one channel (so T c → 1), that channel dominates the cross section, an effect
known as superradiance [13]. This might happen if there are many close resonances