Resonances to Continuum
49
0
2
4
6
8
1 0
1 2
U (MeV)
10
-1
10
0
10
1
10
2
10
3
ρ[
38
Ar] (MeV
-1
)
Beckerman
von Egidy and Bucurescu, Constant Temperature
Fig. 2 The level density for the nucleus 38 Ar, as a function of excitation energy. The solid black
curve is from Beckerman [17] and the red dashed curve is the constant temperature result from
von Egidy and Bucurescu [18]
The Monte Carlo sampling of the energy levels assumes the distribution of level
spacings for a given J π is given by the Wigner distribution
P W (s) =
π
2
s exp
−
πs 2
4
,
(8)
where 0 ≤ s < ∞ and s is the ratio of the actual spacing to the average spacing
defined by the level density. The widths were sampled from the Porter-Thomas
distribution
P P T (t) =
1
√
2π
exp
−
t 2
2
,
(9)
where −∞ < t < ∞ and c = t 2 c . The average partial width c is determined
from the transmission coefficient and level density using Eq. (3).
The two-body thermonuclear reaction rate is in general given by
σ v =
8π
μ
1/2
(kT )
−3/2
∞
0
E σ (E) exp
−
E
kT
dE ,
(10)
where μ is the reduced mass of the reactants and kT is the temperature in units of
energy. This formula may be used together with the HF cross section to determine
the HF reaction rate. Alternatively, the cross section may be represented by narrow