Transcript Slide 1

Computational
Nuclear Structure
Moment (and Fermi gas) methods
for 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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Computational
Nuclear Structure
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
Computational
Nuclear Structure
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
Computational
Nuclear Structure
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
Computational
Nuclear Structure
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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Computational
Nuclear Structure
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:
<(f7/2)2 J=2, T=0| V| (f5/2 p3/2) J=2, T=0>
(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
Computational
Nuclear Structure
Result: “exact” state density from CI shell-model diagonalization
32S
Okay, let’s start
comparing with
approximations!
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Fermi Gas Models
Computational
Nuclear Structure
Start with (equally-spaced) single-particle levels and fill
them like a Fermi gas (Bethe, 1936):
2 exp 4a ( E   )
 BBFG ( E ) 
12 ( 4a )1/ 4 ( E   ) 5 / 4
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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Computational
Nuclear Structure
Fermi Gas Models
The “thermodynamic method”
Z ( ) 
centers around the partition function

 E
dE

e
 (E)

0
(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
 ( E )  21
i
E
d


e
Z ( )

i
approximate integrand by a Gaussian
Z (  0 )e  0 E
 ln Z (  0 )
 2 ln Z (  0 )
 (E) 
,E  
,D 
2




2D
the “saddle-point condition” fixes
the value of β0 for a given energy E
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Computational
Nuclear Structure
Fermi Gas Models
Of course, one needs the partition function!
Traditionally one derives it from the single-particle density of states g(ε),
either from Fermi gas or from Hartree-Fock (Bogoliubov)

ln Z ( ,  )   d  g ( ) ln 1  e  

+ corrections for rotation,
shell structure, etc.
Single-particle energies from Hartree-Fock mean-field: εip,n
Single-particle density of states:
g ( )    (   i )
i
Partition function :
ln Z ( ,  )   ln 1  exp(  i ) 
i
Then apply saddle-point method...
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Mean-field Level densities
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Computational
Nuclear Structure
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Mean-field Level densities
Computational
Nuclear Structure
These are both
spherical nuclei...
what about
deformed nuclides?
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Mean-field Level densities
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Computational
Nuclear Structure
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Mean-field Level densities
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Computational
Nuclear Structure
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Mean-field Level densities
Computational
Nuclear Structure
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
spherical
deformed
2 exp 4a ( E   )
 BBFG ( E ) 
12 ( 4a )1/ 4 ( E   ) 5 / 4
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Smaller s.p. level density
smaller
nuclear level density
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Computational
Nuclear Structure
Adding collective motion (NEW)
Deformed HF state as an intrinsic state:
HF   aJ J
J
1 2 

(
J

(
2
J

1
)
2
2)

aJ 
exp 
2
2
2
2 

J ( J  1)
EJ 
2I
Z rot  
J
2  HF
2
2
ˆ
J HF
(get moment of inertia I from cranked HF)
J

,J 
(2 J  1)a exp(EJ )
3/ 2
1  Erot 
2
J
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J ( J  1) , Erot
J2

2I
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Adding collective motion (NEW)
Computational
Nuclear Structure
Z(β) = Zπ(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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Computational
Nuclear Structure
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Adding collective motion (NEW)
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Computational
Nuclear Structure
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Adding collective motion (NEW)
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Computational
Nuclear Structure
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Adding collective motion (NEW)
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Computational
Nuclear Structure
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Adding collective motion (NEW)
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Computational
Nuclear Structure
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Adding collective motion (NEW)
Computational
Nuclear Structure
Cranking is problematic for
odd-odd (I used real wfns
and should allow complex)
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Introduction to
Statistical Spectroscopy
Computational
Nuclear Structure
(also known as “spectral distribution theory”)
Pioneered by J. Bruce French 1960’s-1980’s
other luminaries include: J. P. Draayer, J. Ginocchio, S.
S.S.M. Wong, A.P. Zuker + many others...
Grimes, V. Kota,
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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Computational
Nuclear Structure
How we do it:
a detailed version
The important configuration moments
Dimension
Centroid:
d  TrP
E 
1
d
TrP H
Higher central
moments
Width:
 n ( ) 
Scaled moments
1
d
 
TrP H  E 
2
1
d
TrP H  E 
n
mn ( )  n ( ) / 
n
Asymmetry (or skewness): m3(α)
Excess
: m4(α) - 3
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= 0 for Gaussian
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Introduction to
Statistical Spectroscopy
Primer on moments
Computational
Nuclear Structure
Interpretation of moments:
centroid
centroid = spherical HF energy
centroid
width = avg spreading width
of residual interaction
asymmetric
asymmetry = measure of collectivity
width
width
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Introduction to
Statistical Spectroscopy
Computational
Nuclear Structure
Then we consider the
level density as being
the sum of individual
configuration densities
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Computational
Nuclear Structure
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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Computational
Nuclear Structure
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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Computational
Nuclear Structure
Level densities as a sum of
configuration densities
It is (often) important to include
3rd and 4th moments
much better than
using only second moments
“starting energy” also difficult to control
collective states difficult to get
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Computational
Nuclear Structure
Comparison with experiments
NB: computed
+ parity states
and multiplied × 2
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Obligatory Summary
Computational
Nuclear Structure
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
Computational
Nuclear Structure
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
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