Moldauer’s Sum Rule and Superradiance
59
Table 1 Summary of transmission coefficients T c under consideration in this contribution
Parameterization
Equation
Derivation
SPRT [6]
T SPRT
c
=
2x c
(1 + x c /2) 2 + (P c R ∞
c ) 2
Replace R-matrix with energy
average R
Sum rule [8]
T SR
c = 2x c
x 2
c + 1 − x c
Compute energy average S
using Moldauer’s sum rule of
the S-matrix
Moldauer–
Simonius [7]
T MS
c
= 1 − exp(−2x c )
Phenomenological
Weak coupling limit
T weak
c
= 2x c (1 − x c )
The other three parameterizations reduce to this in limit
x c 1
Here x c = π Γ c /D = 2πP c s c
To better visualize the different parameterizations, we turn to 90 Zr, recently reevaluated by S.F. Mughabghab [9]. Using the URR parameters of 90 Zr we computed
the neutron transmission coefficients using the prescriptions in Table 1. We also
computed the transmission coefficients using the coupled-channel code ECIS [10]
and a Lane consistent dispersive soft rotor coupled-channel optical model potential
(RIPL OMP #612) [11]. In Fig. 1 we show these transmission coefficients. For s−,
p−, and d− wave neutrons impinging on the 0 + ground state of 90 Zr, only the given
J shown in Fig. 1 are possible.
In Fig. 1, all of the transmission coefficient parameterizations are consistent at
low energies but the two weak coupling approximations diverge from the rest above
500 keV. The other three parameterizations (SPRT, Moldauer–Simonius (MS), and
sum rule (SR)) agree over the entire range of the URR and with the RRR at low
energy. The transmission coefficients computed by ECIS are roughly consistent
with the resolved and unresolved resonances, but disagree in detail. The spin orbit
coupling in the optical model potential generates a J dependence which is clearly
visible in the plots, especially in the p-wave (L = 1) channels. We note that the
neutrons in Fig. 1 approach the strong coupling limit already at 1 MeV in the p-wave
channels. A coupled-channel calculation with a realistic optical model potential
should not allow T = 1 as this would violate unitarity when combined with the
other channels in the problem.
Above 1 MeV, the optical model potential predicts a turnover in T c . Were we
to extend the URR parameters to higher energies, we would see this behavior
in the SPRT parameterization (see the SPRT equation in Table 1), but not in
the sum rule or Moldauer-Simonius parameterizations. We speculate that this is a
result of an implicit neglect of interference effects in the sum rule and Moldauer–
Simonius parameterizations. We note that Ref. [12] describes a numerical study
using stochastically generated scattering matrices which strongly supports the SPRT
parameterization over either the sum rule or Moldauer–Simonius parameterizations.
59
Table 1 Summary of transmission coefficients T c under consideration in this contribution
Parameterization
Equation
Derivation
SPRT [6]
T SPRT
c
=
2x c
(1 + x c /2) 2 + (P c R ∞
c ) 2
Replace R-matrix with energy
average R
Sum rule [8]
T SR
c = 2x c
x 2
c + 1 − x c
Compute energy average S
using Moldauer’s sum rule of
the S-matrix
Moldauer–
Simonius [7]
T MS
c
= 1 − exp(−2x c )
Phenomenological
Weak coupling limit
T weak
c
= 2x c (1 − x c )
The other three parameterizations reduce to this in limit
x c 1
Here x c = π Γ c /D = 2πP c s c
To better visualize the different parameterizations, we turn to 90 Zr, recently reevaluated by S.F. Mughabghab [9]. Using the URR parameters of 90 Zr we computed
the neutron transmission coefficients using the prescriptions in Table 1. We also
computed the transmission coefficients using the coupled-channel code ECIS [10]
and a Lane consistent dispersive soft rotor coupled-channel optical model potential
(RIPL OMP #612) [11]. In Fig. 1 we show these transmission coefficients. For s−,
p−, and d− wave neutrons impinging on the 0 + ground state of 90 Zr, only the given
J shown in Fig. 1 are possible.
In Fig. 1, all of the transmission coefficient parameterizations are consistent at
low energies but the two weak coupling approximations diverge from the rest above
500 keV. The other three parameterizations (SPRT, Moldauer–Simonius (MS), and
sum rule (SR)) agree over the entire range of the URR and with the RRR at low
energy. The transmission coefficients computed by ECIS are roughly consistent
with the resolved and unresolved resonances, but disagree in detail. The spin orbit
coupling in the optical model potential generates a J dependence which is clearly
visible in the plots, especially in the p-wave (L = 1) channels. We note that the
neutrons in Fig. 1 approach the strong coupling limit already at 1 MeV in the p-wave
channels. A coupled-channel calculation with a realistic optical model potential
should not allow T = 1 as this would violate unitarity when combined with the
other channels in the problem.
Above 1 MeV, the optical model potential predicts a turnover in T c . Were we
to extend the URR parameters to higher energies, we would see this behavior
in the SPRT parameterization (see the SPRT equation in Table 1), but not in
the sum rule or Moldauer-Simonius parameterizations. We speculate that this is a
result of an implicit neglect of interference effects in the sum rule and Moldauer–
Simonius parameterizations. We note that Ref. [12] describes a numerical study
using stochastically generated scattering matrices which strongly supports the SPRT
parameterization over either the sum rule or Moldauer–Simonius parameterizations.