Nuclear Shell Model and Level Density
125
This was qualitatively understood long ago, see, for example, [14–16]. In the
book [17] it was suggested to describe the nuclear level density in the restricted
orbital space by the sequence of statistical moments corresponding to Gaussian
distributions of the observables. Later it became possible to formulate a practical
“moments method,” see [9] and the references therein, which works very well
reproducing exact shell-model results and avoiding the diagonalization of huge
matrices. The computational aspects of the moments method are formulated in [18].
Another possible way to avoid the full diagonalization is currently under way [19]
based on the Lanczos algorithm.
2 Moments Method
In the moments method we start from the shell-model Hamiltonian in a truncated
space using the mean-field basis of many-body states. The basis states are divided
into partitions p by distributing the particles over the mean-field levels. The first
moment is just the mean-field energy for a given partition p with the dimension
D αp and a given set of exact quantum numbers α,
E αp = =H αp ≡
1
D αp
Tr
(αp) (H ).
(1)
This trace includes the matrix elements of the interactions diagonal with respect to
the occupancies of the given partition. The second moment,
H
2
αp ≡
1
D αp
Tr
(αp) (H
2 ),
(2)
includes the bilinear combinations of mean-field energies and interaction matrix
elements forming the two-step processes based on the partition p with the inclusion
of transitions to the configurations outside this partition and back. These moments
can be read just from the Hamiltonian matrix.
The first two moments allow one to construct the effective Gaussian G αp (E) with
the centroid (1) and the width defined as its variance by the second moment (2). This
determines the effective contribution of a partition p to the level density of the class
of states α,
ρ α (E) =
p
D αp G αp (E).
(3)
There are important technical details (renormalization of the unphysical tails of the
Gaussians, the fit of the ground state energy, removal of spurious center-of-mass
excitations, and so on) which can be found in [18] and other publications. As a
125
This was qualitatively understood long ago, see, for example, [14–16]. In the
book [17] it was suggested to describe the nuclear level density in the restricted
orbital space by the sequence of statistical moments corresponding to Gaussian
distributions of the observables. Later it became possible to formulate a practical
“moments method,” see [9] and the references therein, which works very well
reproducing exact shell-model results and avoiding the diagonalization of huge
matrices. The computational aspects of the moments method are formulated in [18].
Another possible way to avoid the full diagonalization is currently under way [19]
based on the Lanczos algorithm.
2 Moments Method
In the moments method we start from the shell-model Hamiltonian in a truncated
space using the mean-field basis of many-body states. The basis states are divided
into partitions p by distributing the particles over the mean-field levels. The first
moment is just the mean-field energy for a given partition p with the dimension
D αp and a given set of exact quantum numbers α,
E αp = =H αp ≡
1
D αp
Tr
(αp) (H ).
(1)
This trace includes the matrix elements of the interactions diagonal with respect to
the occupancies of the given partition. The second moment,
H
2
αp ≡
1
D αp
Tr
(αp) (H
2 ),
(2)
includes the bilinear combinations of mean-field energies and interaction matrix
elements forming the two-step processes based on the partition p with the inclusion
of transitions to the configurations outside this partition and back. These moments
can be read just from the Hamiltonian matrix.
The first two moments allow one to construct the effective Gaussian G αp (E) with
the centroid (1) and the width defined as its variance by the second moment (2). This
determines the effective contribution of a partition p to the level density of the class
of states α,
ρ α (E) =
p
D αp G αp (E).
(3)
There are important technical details (renormalization of the unphysical tails of the
Gaussians, the fit of the ground state energy, removal of spurious center-of-mass
excitations, and so on) which can be found in [18] and other publications. As a