Moment and Fermi gas methods for modeling nuclear state densities

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Moment (and Fermi gas) methodsfor modeling
nuclear state densities

• Calvin W. Johnson (PI)
• Edgar Teran (former postdoc)
• San Diego State University
• supported by grants US DOE-NNSA

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We all know that Art is not truth. Art is a lie
that makes us realize the truth, at least the
truth that is given to us to understand.- Pablo
Picasso
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Tools mean-fields and moments
3 approaches to calculating nuclear level/state
density
an incomplete list of names
Fermi gas / combinatorial Goriely, Hilaire,
Hillman/Grover, Cerf, Uhrenholt
Monte Carlo shell model Alhassid, Nakada
Spectral distribution/ moment methods French,
Kota, Grimes, Massey, Horoi, Zelevinsky,
Johnson/Teran
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Overview of Talk
Theme Testing approximate schemes for
level/state densities against (tedious)
full-scale CI shell model diagonalization
Part I Fermi gas model vs. exact shell model
-- Single particle energies from Hartree-Fock --
Adding in rotation (new!)
Part II Moment (spectral distribution) methods
-- mean-field (centroids or first moments) --
residual interaction (spreading widths or second
moments) -- collective interaction (third
moments)
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Comparison against exact results
How reliable are calculations? If we put in the
right microphysics, do we get out reasonably good
densities?
To answer this, we compare against exact
calculations from full configuration-interaction
(CI) diagonalization of realistic Hamiltonians
in a finite shell-model basis.
We can then look at an approximate method
(Fermi gas, spectral distribution) using the
same input and compare.
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What an interacting shell-model code does
Input into shell model set of
single-particle states (1s1/2,0d5/2, 0f7/2 etc)
many-body configurations constructed from
s.p. states (f7/2)8, (f7/2)6(p3/2)2, etc.
two-body matrix elements to determine
Hamiltonian between many-body states lt(f7/2)2
J2, T0 V (f5/2 p3/2) J2, T0gt (assume
someone else has already done the integrals)
Output eigenenergies and wavefunctions (vectors
in basis of many-body Slater determinants)
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Mean-field Level densities
Result exact state density from CI shell-model
diagonalization
32S
Okay, lets start comparing with approximations!
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Fermi Gas Models
Start with (equally-spaced) single-particle
levels and fill them like a Fermi gas (Bethe,
1936)
Some modern version use realistic
single-particle levels derived from Hartree-Fock
(Goriely)
The parameter a reflects the density of
single-particle states near the Fermi surface
The single-particle levels arise from a mean
field!
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Fermi Gas Models
The thermodynamic method centers around the
partition function
(1) Construct the partition function either from
single-particle density of states or (later)
from Monte Carlo evaluation of a path integral
(2) Invert the Laplace transform through the
saddle-point approximation
approximate integrand by a Gaussian
the saddle-point condition fixes the value of
ß0 for a given energy E
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Fermi Gas Models
Of course, one needs the partition function!
Traditionally one derives it from the
single-particle density of states g(e), either
from Fermi gas or from Hartree-Fock (Bogoliubov)
corrections for rotation, shell structure, etc.
Single-particle energies from Hartree-Fock
mean-field eip,n
Single-particle density of states
Partition function
Then apply saddle-point method...
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Mean-field Level densities
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Mean-field Level densities
These are both spherical nuclei... what about
deformed nuclides?
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Mean-field Level densities
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Mean-field Level densities
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Mean-field Level densities
Difference is due to fragmentation of
Hartree-Fock single-particle energies in
deformed mean-field
0d3/2
1s1/2
Fermi surface
Fermi surface
0d5/2
Smaller s.p. level density
smaller nuclear level density
deformed
spherical
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Adding collective motion (NEW)
Deformed HF state as an intrinsic state
(get moment of inertia I from cranked HF)
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Adding collective motion (NEW)
Z(ß) Zp(sp) ? Z?(sp)
?(1 Zrot)
All of the parameters derived directly from HF
calculation (SHERPA code by Stetcu and
Johnson) using CI shell-model interaction
Computationally very cheap a matter of a few
seconds
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Adding collective motion (NEW)
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Adding collective motion (NEW)
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Adding collective motion (NEW)
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Adding collective motion (NEW)
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Adding collective motion (NEW)
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Adding collective motion (NEW)
Cranking is problematic for odd-odd (I used real
wfns and should allow complex)
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Introduction toStatistical Spectroscopy
(also known as spectral distribution theory)
Pioneered by J. Bruce French 1960s-1980s other
luminaries include J. P. Draayer, J. Ginocchio,
S. Grimes, V. Kota, S.S.M. Wong, A.P. Zuker
many others...
Problem diagonalization is too hard and gives
too much detailed information
Solution instead of diagonalizing H, find
moments tr Hn
Key question how many moments do we need?
Rather than many moments (over the entire space)
tr H n, n 1,2,3,4,5,6,7... compute low moments
(n 1,2,3,4) on subspaces
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How we do ita detailed version
The important configuration moments
Dimension
Centroid
Width
Higher central moments
Scaled moments
Asymmetry (or skewness) m3(a)
Excess m4(a) - 3
0 for Gaussian
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Introduction toStatistical Spectroscopy
Primer on moments
Interpretation of moments centroid spherical
HF energy width avg spreading width of
residual interaction asymmetry measure of
collectivity
centroid
centroid
asymmetric
width
width
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Introduction toStatistical Spectroscopy
Then we consider the level density as being the
sum of individual configuration densities
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Level densities as a sum of configuration
densities
We model the level density as a sum of partial
(configuration) densities, each of which are
modeled as Gaussians
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Level densities as a sum of configuration
densities
What can we do to improve our model?
Go to third moments asymmetries
Not satisfactory!
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Level densities as a sum of configuration
densities
It is (often) important to include much better
than 3rd and 4th moments using only second
moments
collective states difficult to get
starting energy also difficult to control
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Comparison with experiments
NB computed parity states and multiplied 2
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Obligatory Summary
Fermi gas model can work surprisingly well and is
computationally cheap
Deformed nuclei need to have rotation put in.
One can use the single-particle energies and
moment of inertia from (cranked) Hartree-Fock
compares well to full CI calculation
For larger model spaces may need pairing, shell
effects, etc.
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Obligatory Summary
View nuclear many-body Hamiltonian through
lens of moment methods
1st (configuration) moments mean-field
2nd moments spreading widths of residual
interaction
3rd moments collectivity of residual interaction