Nuclear Level Densities
101
The spin distribution is calculated from
ρ J = ρ M=J − ρ M=J +1 ≈ −
dρ M
dM
M=J +1/2
.
(14)
Using Eq. (12), we find for the spin-cutoff model
ρ J
ρ
=
2J + 1
2
√
2πσ 3
e
−J (J +1)/2σ 2 .
(15)
3.2 Parity Distribution
A simple model for the parity distribution of level densities is obtained by assuming
the particles occupy the single-particle states independently and randomly [3]. We
divide the single-particle levels into two groups of positive and negative parities,
and denote by π the parity of the group with the smaller occupation probability p π .
The probability to have n particles in this group is then a binomial distribution
P (n) =
A
n
p
n
π (1 − p π )
A−n ,
(16)
where A is the total number of excited particles. For an even-particle system,
a negative (positive) parity many-particle state corresponds to odd (even) values
of n, and the total probability to have a negative (positive) parity is obtained by
summing P (n) over all odd (even) values of n. For small p π and large A, we can
approximate (16) by a Poisson distribution f π =
f n
n! e −f π , which depends on a
single parameter f = Ap π , the total occupation of the π -parity orbitals. For an
even-particle system, the ratio of negative- to positive-parity partition functions at a
given temperature is then given by Alhassid et al. [8]
Z −
Z +
=
n odd
P (n)/
n even
P (n) = tanh f.
(17)
Equation (17) holds more generally for an even–even nucleus with f = f p + f n
being the total average occupation of the π -parity orbitals for both protons and
neutrons. For an even–even nucleus, the positive-parity states dominate at low
excitations, but equilibration of both parities is achieved above a certain excitation
energy. In practical applications, it is often assumed that the parity distribution is
already equilibrated at the neutron resonance energy.
101
The spin distribution is calculated from
ρ J = ρ M=J − ρ M=J +1 ≈ −
dρ M
dM
M=J +1/2
.
(14)
Using Eq. (12), we find for the spin-cutoff model
ρ J
ρ
=
2J + 1
2
√
2πσ 3
e
−J (J +1)/2σ 2 .
(15)
3.2 Parity Distribution
A simple model for the parity distribution of level densities is obtained by assuming
the particles occupy the single-particle states independently and randomly [3]. We
divide the single-particle levels into two groups of positive and negative parities,
and denote by π the parity of the group with the smaller occupation probability p π .
The probability to have n particles in this group is then a binomial distribution
P (n) =
A
n
p
n
π (1 − p π )
A−n ,
(16)
where A is the total number of excited particles. For an even-particle system,
a negative (positive) parity many-particle state corresponds to odd (even) values
of n, and the total probability to have a negative (positive) parity is obtained by
summing P (n) over all odd (even) values of n. For small p π and large A, we can
approximate (16) by a Poisson distribution f π =
f n
n! e −f π , which depends on a
single parameter f = Ap π , the total occupation of the π -parity orbitals. For an
even-particle system, the ratio of negative- to positive-parity partition functions at a
given temperature is then given by Alhassid et al. [8]
Z −
Z +
=
n odd
P (n)/
n even
P (n) = tanh f.
(17)
Equation (17) holds more generally for an even–even nucleus with f = f p + f n
being the total average occupation of the π -parity orbitals for both protons and
neutrons. For an even–even nucleus, the positive-parity states dominate at low
excitations, but equilibration of both parities is achieved above a certain excitation
energy. In practical applications, it is often assumed that the parity distribution is
already equilibrated at the neutron resonance energy.