Resonances to Continuum
47
where T c is the transmission coefficient. Here, we have neglected the width
fluctuation correction, which is unimportant for the case we consider below.
The Monte Carlo simulations of the reaction cross section of interest are based on
discrete levels sampled from distributions consistent with the HF parametrization.
This approach builds on the ideas Mohr et al. [8] who have estimated the
18 Ne(α, p) 21 Na reaction rate using experimental data for level positions and some
spectroscopic properties, while using Monte Carlo for the unknown spins, parities,
and partial widths. Although not performed for reaction rate calculations, there have
been several previous Monte Carlo studies of compound-nuclear reactions; see, e.g.,
Moldauer [9] and the recent work of Kawano et al. [10].
The transmission coefficient T c for channel c in the HF formula is related to the
mean partial width c and level density ρ via [11, 12]
T c = 1 − exp(−2π c ρ) ,
(3)
which reduces to T c = 2π c ρ when T c 1. If the T c and ρ are assumed to be
known, the level positions, spins, and parities can be sampled from the level density
allowing for the Wigner distribution of level spacings [10]. Likewise, the reduced
level widths can be sampled from the Porter-Thomas distribution [8, 10]. Finally, the
reaction rate can be calculated by taking the usual Maxwell–Boltzmann average. By
repeating the Monte Carlo process, the statistical uncertainty in the reaction rate can
be determined.
4 Details of the Calculation
We limit our consideration to E c.m. < 5 MeV, where α p and in the HF
approach only the α transmission factor is important. Since 34 Ar and the α particle
both have J π = 0 + , the entrance channel label may be identified by c = J and the
statistical factor is ω J = 2J + 1. Since the total with is dominated by the outgoing
proton widths, the cross section for 34 Ar(α, p) 37 K resulting from Eq. (2) is
σ =
π
k 2
∞
J =0
(2J + 1)T J ,
(4)
where in practice only the few lowest J values contribute appreciably. Following previous work [13–15], we utilize the McFadden and Satchler α optical
potential [16]. Our result for the 34 Ar(α, p) 37 K cross section, presented as an
astrophysical S factor, is shown in Fig. 1. We find that the cross section at low
energies is sensitive to the tail of the Woods Saxon potential out to a distance of
about 18 fm. We note that this part of the potential is certainly not constrained by any
of the higher-energy elastic scattering data considered in Ref. [16]. Also shown in
Fig. 1 is a calculation using the Non-Smoker code [13–15] using the same α optical
47
where T c is the transmission coefficient. Here, we have neglected the width
fluctuation correction, which is unimportant for the case we consider below.
The Monte Carlo simulations of the reaction cross section of interest are based on
discrete levels sampled from distributions consistent with the HF parametrization.
This approach builds on the ideas Mohr et al. [8] who have estimated the
18 Ne(α, p) 21 Na reaction rate using experimental data for level positions and some
spectroscopic properties, while using Monte Carlo for the unknown spins, parities,
and partial widths. Although not performed for reaction rate calculations, there have
been several previous Monte Carlo studies of compound-nuclear reactions; see, e.g.,
Moldauer [9] and the recent work of Kawano et al. [10].
The transmission coefficient T c for channel c in the HF formula is related to the
mean partial width c and level density ρ via [11, 12]
T c = 1 − exp(−2π c ρ) ,
(3)
which reduces to T c = 2π c ρ when T c 1. If the T c and ρ are assumed to be
known, the level positions, spins, and parities can be sampled from the level density
allowing for the Wigner distribution of level spacings [10]. Likewise, the reduced
level widths can be sampled from the Porter-Thomas distribution [8, 10]. Finally, the
reaction rate can be calculated by taking the usual Maxwell–Boltzmann average. By
repeating the Monte Carlo process, the statistical uncertainty in the reaction rate can
be determined.
4 Details of the Calculation
We limit our consideration to E c.m. < 5 MeV, where α p and in the HF
approach only the α transmission factor is important. Since 34 Ar and the α particle
both have J π = 0 + , the entrance channel label may be identified by c = J and the
statistical factor is ω J = 2J + 1. Since the total with is dominated by the outgoing
proton widths, the cross section for 34 Ar(α, p) 37 K resulting from Eq. (2) is
σ =
π
k 2
∞
J =0
(2J + 1)T J ,
(4)
where in practice only the few lowest J values contribute appreciably. Following previous work [13–15], we utilize the McFadden and Satchler α optical
potential [16]. Our result for the 34 Ar(α, p) 37 K cross section, presented as an
astrophysical S factor, is shown in Fig. 1. We find that the cross section at low
energies is sensitive to the tail of the Woods Saxon potential out to a distance of
about 18 fm. We note that this part of the potential is certainly not constrained by any
of the higher-energy elastic scattering data considered in Ref. [16]. Also shown in
Fig. 1 is a calculation using the Non-Smoker code [13–15] using the same α optical