Nuclear Level Densities
105
7 Configuration-Interaction Shell Model Methods
The CI shell model includes shell effects and correlations beyond the mean-field
approximation, and thus can in principle provide the most precise microscopic calculation of level densities. However, the combinatorial growth of the dimensionality
of the many-particle model space with the number of valence nucleons and/or the
number of valence orbitals has hindered its application in mid-mass and heavy
nuclei.
7.1 Spectral Averaging Theory (Moment Method)
The spectral averaging theory, also known as the moment method, describes the
density as a superposition of Gaussian densities for various partitions of the singleparticle orbitals with centroids and widths that are determined by the first two
moments of the Hamiltonian [23–26].
The method requires a reliable calculation of the ground-state energy, which is
required for determining the excitation energy. The calculation of second moments
is time consuming in large model spaces, and so far the method has been applied
to light and mid-mass nuclei, where it provides good agreement with experimental
data and with exact CI shell model calculations (in sd-shell nuclei) [27]. For more
details of the method and its applications see Refs. [28, 29].
7.2 Auxiliary-Field Quantum Monte Carlo Method
The auxiliary-field quantum Monte Carlo (AFMC) method, also known in nuclear
physics as the shell model Monte Carlo (SMMC) [30–34], is based on the Hubbard–
Stratonovich (HS) transformation [35], in which Gibbs ensemble e −β ˆ
H is written as
a superposition of ensembles ˆ
U σ describing non-interacting nucleons moving in
external auxiliary fields σ (τ )
e
−β ˆ
H
=
D[σ ]G σ ˆ
U σ ,
(23)
where G σ is a Gaussian weight. The calculation of the integrand for a given
configuration of the auxiliary fields σ reduces to matrix algebra in the single-particle
space of typical dimension ∼50−100. The integration over the large number of
auxiliary fields is carried out using Monte Carlo methods.
The AFMC state density is calculated using the thermodynamic approach of
Sect. 2.1 [36–38]. The canonical thermal energy E(β) is calculated as a function
105
7 Configuration-Interaction Shell Model Methods
The CI shell model includes shell effects and correlations beyond the mean-field
approximation, and thus can in principle provide the most precise microscopic calculation of level densities. However, the combinatorial growth of the dimensionality
of the many-particle model space with the number of valence nucleons and/or the
number of valence orbitals has hindered its application in mid-mass and heavy
nuclei.
7.1 Spectral Averaging Theory (Moment Method)
The spectral averaging theory, also known as the moment method, describes the
density as a superposition of Gaussian densities for various partitions of the singleparticle orbitals with centroids and widths that are determined by the first two
moments of the Hamiltonian [23–26].
The method requires a reliable calculation of the ground-state energy, which is
required for determining the excitation energy. The calculation of second moments
is time consuming in large model spaces, and so far the method has been applied
to light and mid-mass nuclei, where it provides good agreement with experimental
data and with exact CI shell model calculations (in sd-shell nuclei) [27]. For more
details of the method and its applications see Refs. [28, 29].
7.2 Auxiliary-Field Quantum Monte Carlo Method
The auxiliary-field quantum Monte Carlo (AFMC) method, also known in nuclear
physics as the shell model Monte Carlo (SMMC) [30–34], is based on the Hubbard–
Stratonovich (HS) transformation [35], in which Gibbs ensemble e −β ˆ
H is written as
a superposition of ensembles ˆ
U σ describing non-interacting nucleons moving in
external auxiliary fields σ (τ )
e
−β ˆ
H
=
D[σ ]G σ ˆ
U σ ,
(23)
where G σ is a Gaussian weight. The calculation of the integrand for a given
configuration of the auxiliary fields σ reduces to matrix algebra in the single-particle
space of typical dimension ∼50−100. The integration over the large number of
auxiliary fields is carried out using Monte Carlo methods.
The AFMC state density is calculated using the thermodynamic approach of
Sect. 2.1 [36–38]. The canonical thermal energy E(β) is calculated as a function