54
E. D. Davis
kurtosis. Second, the integral representation of C ab (ε) derived by Verbaarschot et
al. [6] permits, in principle, the exact evaluation of R tot (ε) within the stochastic CN
model introduced by these authors (the VWZ model).
It is the VWZ model which is the starting point for the majority of the
investigations into the origin of deviations from the PTD. De facto, it is the
capacity of the VWZ model to describe CN phenomena which is under scrutiny.
A comparison of its predictions for R tot (ε) with data in the unresolved (but weakly
overlapping) resonance regime would constitute another test, and any discrepancies
found could not be attributed to the method of evaluation of R tot (ε).
Is there any reason to expect that data on R tot (ε) for unresolved but weakly
overlapping resonances may display sensitivity to non-generic dynamics? Previous
investigations [7, 8], in which results deduced from the VWZ model have been
compared with microwave resonator and CN data in the regime of weakly overlapping resonances, do not address this point. To this end, it is helpful to consider
the approximation of C ab (ε) in the statistical Breit–Wigner (SBW) model, using the
scheme of calculation laid out in [9].
In the SBW model, guided by the empirical characterization of data on partial
widths [10], it can be assumed that partial widths are drawn from a χ 2 distribution
of ν degrees of freedom. In the weakly overlapping resonance regime, the dominant
contribution to C ab (0) is then
C
(d)
ab (0) =
1 +
2
ν
δ ab
T a T b I
(ν)
ab ,
(2)
where the transmission coefficients T c = 1 − ||S cc | 2 , and
I
(ν)
ab =
∞
0
c
1 +
2
ν T c τ
−ν/2
1 +
2
ν T a τ
1 +
2
ν T b τ
dτ.
(3)
(The product in the integrand above is over all open channels c.)
The ν dependence in (2) is encouraging. Figure 1 displays the relative change
δ ≡ C
(d)
aa (0)
C
(d)
aa (0) [ν = 1] − 1 in the dominant contribution to C aa (0) as ν
ranges from its value in the Porter-Thomas limit (ν = 1) through values implied
by the analysis of Pt neutron width data (ν ≈
1
2 ). [In δ, the value of the dominant
contribution to C aa (0) for arbitrary ν is divided by its value for ν = 1.] In generating
Fig. 1, all transmission coefficients have, for simplicity, been taken to be equal in all
open channels, meaning that δ is a function of only ν and .
For values of ν comparable to those found in the statistical analysis of reduced
neutron widths in [10], Fig. 1 suggests that R tot (0) could deviate from its value
in the VWZ model by more than 20%. This should be a large enough signal to
warrant determination of R tot (0) with high quality total cross section data for weakly
overlapping resonances in the unresolved resonance regime.
E. D. Davis
kurtosis. Second, the integral representation of C ab (ε) derived by Verbaarschot et
al. [6] permits, in principle, the exact evaluation of R tot (ε) within the stochastic CN
model introduced by these authors (the VWZ model).
It is the VWZ model which is the starting point for the majority of the
investigations into the origin of deviations from the PTD. De facto, it is the
capacity of the VWZ model to describe CN phenomena which is under scrutiny.
A comparison of its predictions for R tot (ε) with data in the unresolved (but weakly
overlapping) resonance regime would constitute another test, and any discrepancies
found could not be attributed to the method of evaluation of R tot (ε).
Is there any reason to expect that data on R tot (ε) for unresolved but weakly
overlapping resonances may display sensitivity to non-generic dynamics? Previous
investigations [7, 8], in which results deduced from the VWZ model have been
compared with microwave resonator and CN data in the regime of weakly overlapping resonances, do not address this point. To this end, it is helpful to consider
the approximation of C ab (ε) in the statistical Breit–Wigner (SBW) model, using the
scheme of calculation laid out in [9].
In the SBW model, guided by the empirical characterization of data on partial
widths [10], it can be assumed that partial widths are drawn from a χ 2 distribution
of ν degrees of freedom. In the weakly overlapping resonance regime, the dominant
contribution to C ab (0) is then
C
(d)
ab (0) =
1 +
2
ν
δ ab
T a T b I
(ν)
ab ,
(2)
where the transmission coefficients T c = 1 − ||S cc | 2 , and
I
(ν)
ab =
∞
0
c
1 +
2
ν T c τ
−ν/2
1 +
2
ν T a τ
1 +
2
ν T b τ
dτ.
(3)
(The product in the integrand above is over all open channels c.)
The ν dependence in (2) is encouraging. Figure 1 displays the relative change
δ ≡ C
(d)
aa (0)
C
(d)
aa (0) [ν = 1] − 1 in the dominant contribution to C aa (0) as ν
ranges from its value in the Porter-Thomas limit (ν = 1) through values implied
by the analysis of Pt neutron width data (ν ≈
1
2 ). [In δ, the value of the dominant
contribution to C aa (0) for arbitrary ν is divided by its value for ν = 1.] In generating
Fig. 1, all transmission coefficients have, for simplicity, been taken to be equal in all
open channels, meaning that δ is a function of only ν and .
For values of ν comparable to those found in the statistical analysis of reduced
neutron widths in [10], Fig. 1 suggests that R tot (0) could deviate from its value
in the VWZ model by more than 20%. This should be a large enough signal to
warrant determination of R tot (0) with high quality total cross section data for weakly
overlapping resonances in the unresolved resonance regime.