Nuclear Level Densities
109
results are compared with empirical distributions determined from the analysis of
complete sets of experimentally known nuclear energy levels [51, 52]. A staggering
effect in spin can be seen in 56 Fe.
The spin-cutoff parameter can be related to the thermal moment of inertia through
Eq. (13). For even–even nuclei, the moment of inertia is found to be suppressed
below the pairing transition [40, 44].
Exact parity projection was also implemented in AFMC [8, 36]. The resulting
parity distributions in mid-mass nuclei were found to be well described by Eq. (17)
when, below the pairing transition temperature, f is taken to be the average
occupation of the quasi-particle states with parity π .
7.2.5 The Deformation Dependence of Level Densities
Modeling of shape dynamics, e.g., fission, requires knowledge of the level density
as a function of intrinsic deformation. The theory of deformation has mostly relied
on mean-field approximation that breaks rotational invariance.
In Ref. [53] a model-independent method was developed to calculate distributions of intrinsic deformation within the rotationally invariant framework of the
CI shell model without invoking a mean-field approximation. The method uses a
projection on the axial quadrupole operator in the laboratory frame [54, 55], and
is based on a Landau-like expansion of the logarithm of the quadrupole shape
distribution in quadrupole invariants [56, 57] up to fourth order. We note that this
expansion is similar to the Landau expansion of the free energy used to describe
shape transitions in nuclei with the quadrupole deformation playing the role of the
order parameter [58, 59].
The method of Ref. [53] enables the calculation of shape-dependent state
densities ρ(E x , β, γ ) as a function of excitation energy E x and intrinsic quadrupole
deformation parameters β, γ . To facilitate the presentation of the shape-dependent
densities, the β − γ plane is divided into three regions: spherical, prolate, and oblate
as shown in Fig. 6, and ρ(E x , β, γ ) is integrated over each one of these regions
using the metric 4π 2 β 4 | sin 3γ | dβ dγ to obtain ρ shape (E x ). In Fig. 7, the fraction
ρ shape (E x )/ρ(E x ) of the state density in each of these three regions is shown as
a function of excitation energy for spherical ( 148 Sm), transitional ( 150 Sm), and
deformed ( 152 Sm, 154 Sm) nuclei.
As is seen in Fig. 7, the spherical region dominates the state density in 148 Sm.
In the deformed 152 Sm and 154 Sm nuclei, the prolate region dominates the state
density at lower excitation energies but the spherical density becomes comparable
and exceeds the prolate density at higher excitations where a shape transition occurs
in the mean-field approximation.
109
results are compared with empirical distributions determined from the analysis of
complete sets of experimentally known nuclear energy levels [51, 52]. A staggering
effect in spin can be seen in 56 Fe.
The spin-cutoff parameter can be related to the thermal moment of inertia through
Eq. (13). For even–even nuclei, the moment of inertia is found to be suppressed
below the pairing transition [40, 44].
Exact parity projection was also implemented in AFMC [8, 36]. The resulting
parity distributions in mid-mass nuclei were found to be well described by Eq. (17)
when, below the pairing transition temperature, f is taken to be the average
occupation of the quasi-particle states with parity π .
7.2.5 The Deformation Dependence of Level Densities
Modeling of shape dynamics, e.g., fission, requires knowledge of the level density
as a function of intrinsic deformation. The theory of deformation has mostly relied
on mean-field approximation that breaks rotational invariance.
In Ref. [53] a model-independent method was developed to calculate distributions of intrinsic deformation within the rotationally invariant framework of the
CI shell model without invoking a mean-field approximation. The method uses a
projection on the axial quadrupole operator in the laboratory frame [54, 55], and
is based on a Landau-like expansion of the logarithm of the quadrupole shape
distribution in quadrupole invariants [56, 57] up to fourth order. We note that this
expansion is similar to the Landau expansion of the free energy used to describe
shape transitions in nuclei with the quadrupole deformation playing the role of the
order parameter [58, 59].
The method of Ref. [53] enables the calculation of shape-dependent state
densities ρ(E x , β, γ ) as a function of excitation energy E x and intrinsic quadrupole
deformation parameters β, γ . To facilitate the presentation of the shape-dependent
densities, the β − γ plane is divided into three regions: spherical, prolate, and oblate
as shown in Fig. 6, and ρ(E x , β, γ ) is integrated over each one of these regions
using the metric 4π 2 β 4 | sin 3γ | dβ dγ to obtain ρ shape (E x ). In Fig. 7, the fraction
ρ shape (E x )/ρ(E x ) of the state density in each of these three regions is shown as
a function of excitation energy for spherical ( 148 Sm), transitional ( 150 Sm), and
deformed ( 152 Sm, 154 Sm) nuclei.
As is seen in Fig. 7, the spherical region dominates the state density in 148 Sm.
In the deformed 152 Sm and 154 Sm nuclei, the prolate region dominates the state
density at lower excitation energies but the spherical density becomes comparable
and exceeds the prolate density at higher excitations where a shape transition occurs
in the mean-field approximation.