Advances in R-matrix Data Analysis
41
unitarity defect) could (for example) be fed into a Hauser-Feshbach decay model
built only on the missing physics channels.
But if the total width of a damped resonance is large, then the flux will be missing
at lower energies, even below the known threshold for the excluded channels! In that
case, absorption would still be present below the threshold of the missing channels,
and that would be unphysical.
I therefore consider energy-dependent damping widths, which allows me to
describe the energy dependence of flux going to an excluded channel with known
threshold E 0 . This makes the damping width energy-dependent, Γ α (E). Ideally,
we would like the energy dependence to mimic a set of missing level widths, each
behaving as the formal R-matrix widths Γ = 2γ 2 P L (E−E 0 ). So I used, for each Rmatrix level p above threshold, a formula which cuts off the width below threshold:
Γ α (E) = ˜
Γ α
P L (E − E 0 )
P L (e p − E 0 )
,
(4)
for penetrability functions P L (E − E 0 ). This cuts off the damping for E < E 0 , and
gives Γ α (e p ) = ˜
Γ α as a parameter to be fitted. Making this work depends on having
good experimental data for angular distributions above the E 0 threshold. We may
also need to choose the e p energy in the Brune basis in order to keep it at the right
energy above the threshold.
If we know the physics of missing channels we can estimate L and Coulomb
barriers in the penetrability functions. This would even allow many-body exit
channels, in particular three-body (M = 3) channels such as (p, pn). If these are
described by hyper-spherical harmonics, then there is a new quantum number K ≥ 0
that describes the ratios of the new three-body coordinates for given moment of
inertia ρ. For each value of K there is a centrifugal barrier L(L + 1)/ρ 2 where
L = K + (3M − 6)/2. For 3-body breakup channels this gives L = K + 3/2. If
a particular K dominates in an exit channel, then the corresponding L-value should
be used in Eq. (4).
As an example fit with energy-dependent damping, I refitted 4 He+ 3 He data from
Tombrello [15] with only the elastic channel, and no explicit p+ 6 Li channel which
should open above 10 MeV. The effect of the missing channel is to be represented by
the new fitted damping parameters. I fitted the e p energy in the Brune basis, keeping
L = 0. The result is shown in Fig. 3, with a fit quality of χ 2 /df= 4.25 compared with
2.63 in the full R-matrix fit.
This first attempt at least gives (blue line on the right) transfer cross-sections that
are close to the average of the more complete model (black line). It has no absorption
below 10 MeV, unlike what we would get from fixed damping widths (red line).
This kind of treatment is reminiscent of optical models for elastic scattering,
where energy-dependent imaginary terms are added even though the total Hamiltonian is no longer Hermitian or even energy-dependent. It is available as a resort
above the energy range of a strict Lane and Thomas model, by generalizing the
Reich–Moore approximation to particle channels.
41
unitarity defect) could (for example) be fed into a Hauser-Feshbach decay model
built only on the missing physics channels.
But if the total width of a damped resonance is large, then the flux will be missing
at lower energies, even below the known threshold for the excluded channels! In that
case, absorption would still be present below the threshold of the missing channels,
and that would be unphysical.
I therefore consider energy-dependent damping widths, which allows me to
describe the energy dependence of flux going to an excluded channel with known
threshold E 0 . This makes the damping width energy-dependent, Γ α (E). Ideally,
we would like the energy dependence to mimic a set of missing level widths, each
behaving as the formal R-matrix widths Γ = 2γ 2 P L (E−E 0 ). So I used, for each Rmatrix level p above threshold, a formula which cuts off the width below threshold:
Γ α (E) = ˜
Γ α
P L (E − E 0 )
P L (e p − E 0 )
,
(4)
for penetrability functions P L (E − E 0 ). This cuts off the damping for E < E 0 , and
gives Γ α (e p ) = ˜
Γ α as a parameter to be fitted. Making this work depends on having
good experimental data for angular distributions above the E 0 threshold. We may
also need to choose the e p energy in the Brune basis in order to keep it at the right
energy above the threshold.
If we know the physics of missing channels we can estimate L and Coulomb
barriers in the penetrability functions. This would even allow many-body exit
channels, in particular three-body (M = 3) channels such as (p, pn). If these are
described by hyper-spherical harmonics, then there is a new quantum number K ≥ 0
that describes the ratios of the new three-body coordinates for given moment of
inertia ρ. For each value of K there is a centrifugal barrier L(L + 1)/ρ 2 where
L = K + (3M − 6)/2. For 3-body breakup channels this gives L = K + 3/2. If
a particular K dominates in an exit channel, then the corresponding L-value should
be used in Eq. (4).
As an example fit with energy-dependent damping, I refitted 4 He+ 3 He data from
Tombrello [15] with only the elastic channel, and no explicit p+ 6 Li channel which
should open above 10 MeV. The effect of the missing channel is to be represented by
the new fitted damping parameters. I fitted the e p energy in the Brune basis, keeping
L = 0. The result is shown in Fig. 3, with a fit quality of χ 2 /df= 4.25 compared with
2.63 in the full R-matrix fit.
This first attempt at least gives (blue line on the right) transfer cross-sections that
are close to the average of the more complete model (black line). It has no absorption
below 10 MeV, unlike what we would get from fixed damping widths (red line).
This kind of treatment is reminiscent of optical models for elastic scattering,
where energy-dependent imaginary terms are added even though the total Hamiltonian is no longer Hermitian or even energy-dependent. It is available as a resort
above the energy range of a strict Lane and Thomas model, by generalizing the
Reich–Moore approximation to particle channels.