New Approaches for Spin- and Parity-Dependent Shell Model Nuclear Level Density
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Transcript of New Approaches for Spin- and Parity-Dependent Shell Model Nuclear Level Density
INT October 28, 2004 Mihai Horoi - Central Michigan Univ 1
New Approaches for Spin- and Parity-Dependent Shell Model
Nuclear Level Density
Mihai Horoi,
Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA
Support from NSF grant PHY-02-44453 is acknowledged
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Plan of the Talk
• Part I: Methods for Shell Model NLD– Motivation– Sum on partitions vs moments of the whole
density– Exponential Convergence Method– Fixed-J Configuration Centroids and Widths– Energy-Dependent Cutoff Description– PRC 67, 054309(2003), PRC 69, 041307(2004)
• Part II: Methods of Removal of the Center-of-Mass Spurious Contribution
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Hauser and Feshbach, Phys. Rev 87, 366 (1952)
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The Back-Shifted Fermi Gas Model for Nuclear Level Density
),( NZ
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A.Adams, G.Mitchell, J.F. Shriner Phys.Lett, B422, 13(1998)
26Al
sd-shell model, USD interaction
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Data: Table of Isotopes
Theory: sd-shell model + USD interaction28Si: positive parity
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p - 102 -1960’ssd - 105 - 1980’spf - 109 - 1990’spf5/2-g9/2- 1010 - 2006
Example: 76Sr
PRL 92, 232501
pf5/2-g9/2 dimension
11,090,052,440
CMichSM code
- m-scheme dimension 250,000,000 on one-processor machine
- 150 Lanczos iterations/week
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12 particles in sd model space
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Nuclear Shell Model
pmmmm :,...],,[
.
)31(
)20(
1
0
321
d = 2 (2 j + 1)
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Sum on Partitions vs Moments of the Whole Distribution
6 particles in pf5/2 -g9/2
New interaction A. Lisetskiy et al. PRC 2004
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12 particles in sd model space
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))(),((),(
),()(),(
0 JJEEEGJEG
JEGJdJE
ppp
P
ppp
12 particles in sd model space
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6 particles in p-sd model space
))(),((),(
),()(),(
0 JJEEEGJEG
JEGJdJE
ppp
P
ppp
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Exponential Convergence Method
CbnAEn )/2exp(
))(),()((),()(
),()()(),(
0 JJEEEFRGJEFRG
JEFRGJdJE
ppp
P
ppp
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Exponential Convergence Method for fp-nuclei
nBeAnE )(
)2002(027303,65.Re.,,, CvPhysZelevinskyBrownHoroi
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Exponential Convergence Method for fp-nuclei
Central Michigan Shell Model (CMichSM) code
Exact: -203.196 MeV
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r,s,.. – orbits, not states
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Fixed J Configura
tion Centroids
and Widths
C. Jacquemin, Z. Phys. A 303, 135 (1981)
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Shell Model vs Fixed-J Centroids and Widths Density of States28Si:
12 particles in sd, Tz=0
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Shell Model vs Fixed-J Centroids and Widths Density of States28Si:
12 particles in sd, Tz=0
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Spin Cutoff Factor
)()(
)(2
1)(
)(8
)12()(
22
2
)2/()2/1(
3
222
EME
EE
EeJ
E
lev
JJ
mmm
m MEFRGJdE
EM
2
2
)()()(
1
)(
Zeroth-Order:
S.S.M. Wong,
Nuclear Spectroscopy,
Oxford 1986, p. 45,171
28Si: 12 particles in sd, Tz=0
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Shell Model <M2>
28Si:
12 particles in sd, Tz=0
)()(
)(2
1)(
)(8
)12()(
22
2
)2/()2/1(
3
222
EME
EE
EeJ
E
lev
JJ
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Shell Model <M2>
)()(
)(2
1)(
)(8
)12()(
22
2
)2/()2/1(
3
222
EME
EE
EeJ
E
lev
JJ
28Si:
12 particles in sd, Tz=0
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Zeroth-Order <M2>
)()(
)(2
1)(
)(8
)12()(
22
2
)2/()2/1(
3
222
EME
EE
EeJ
E
lev
JJ
28Si:
12 particles in sd, Tz=0
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Zeroth-Order <M2>
)()(
)(2
1)(
)(8
)12()(
22
2
)2/()2/1(
3
222
EME
EE
EeJ
E
lev
JJ
28Si:
12 particles in sd, Tz=0
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Summary of Part I• Shell Model NLD look very promising, at least up to the particle
emission threshold. More comparison with experimental data necessary.
• J-dependent SM NLD are very accurately described by a sum of finite range Gaussians with fixed-J centroids and widths, if one knows with good precision the energy of g.s. and yrast states. We derived explicit expression to calculate fixed-J centroids and widths.
• Exponential Convergence Method (ECM) proves to be a very powerful tool for finding yrast and non-yrast energies, by doing shell model calculations in truncated model spaces.
• J-dependent SM NLD are reasonably well described by spin cutoff formula with exact cutoff factor, except for higher J’s, but not very well described by spin cutoff formula with zeroth-order cutoff factor. Improvement in estimating cutoff factor requires knowledge of higher order moments.
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The Center-of-Mass Problemnucl-th/0111068
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Nuclear Shell Model
,...],,[
.
)31(
)20(
1
0
321 mmmm
N
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The Center-of-Mass Problem
),(2
3),('
...,),'(),(),,()0,0(:)( intint
NJNNJHHHH
KNJKJNJNJN
CMCMCMCM
JKCMCMJ
iii
A
i
A
jijijiii
A
jijiji
A
iii
A
ii
A
iiCM
rmipm
aaaaaaaA
rrmppAm
rmpmA
rA
mApmA
H
2
1:
2
3
2
1
2
11
1
2
1
2
1
1 1)(
1)(
22
1
222
2
1
2
2
1
),(),(
2
3"
NJNANJH
AHHHHH
CMCMCM
CMCM
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No E (MeV) Ex (MeV) J T
1 -40.0000 0.0000 0.0000 0.0000
2 -39.6000 0.4000 1.0000 0.0000
3 -39.0000 1.0000 1.0000 1.0000
4 -38.8000 1.2000 2.0000 0.0000
5 -38.8000 1.2000 2.0000 0.0000
6 -37.6000 2.4000 3.0000 0.0000
7 -37.0000 3.0000 3.0000 1.0000
8 -36.0000 4.0000 4.0000 0.0000
9 -29.6000 10.4000 1.0000 0.0000
10 -29.4000 10.6000 0.0000 1.0000
11 -29.0000 11.0000 1.0000 1.0000
12 -29.0000 11.0000 1.0000 1.0000
13 -28.8000 11.2000 2.0000 0.0000
14 -28.2000 11.8000 2.0000 1.0000
15 -28.2000 11.8000 2.0000 1.0000
16 -27.0000 13.0000 3.0000 1.0000
17 -10.0000 30.0000 0.0000 0.0000
18 -9.6000 30.4000 1.0000 0.0000
19 -9.6000 30.4000 1.0000 0.0000
20 -9.0000 31.0000 1.0000 1.0000
21 -8.8000 31.2000 2.0000 0.0000
22 -8.8000 31.2000 2.0000 0.0000
23 -8.2000 31.8000 2.0000 1.0000
24 -7.6000 32.4000 3.0000 0.0000
25 0.4000 40.4000 1.0000 0.0000
No E (MeV) Ex (MeV) J T
1 -60.0000 0.0000 0.0000 0.0000
2 -60.0000 0.0000 0.0000 0.0000
3 -60.0000 0.0000 0.0000 0.0000
4 -59.6000 0.4000 1.0000 0.0000
5 -59.6000 0.4000 1.0000 0.0000
6 -59.6000 0.4000 1.0000 0.0000
7 -59.6000 0.4000 1.0000 0.0000
8 -59.4000 0.6000 0.0000 1.0000
9 -59.0000 1.0000 1.0000 1.0000
10 -59.0000 1.0000 1.0000 1.0000
11 -59.0000 1.0000 1.0000 1.0000
12 -59.0000 1.0000 1.0000 1.0000
13 -59.0000 1.0000 1.0000 1.0000
14 -58.8000 1.2000 2.0000 0.0000
15 -58.8000 1.2000 2.0000 0.0000
16 -58.8000 1.2000 2.0000 0.0000
17 -58.8000 1.2000 2.0000 0.0000
18 -58.8000 1.2000 2.0000 0.0000
19 -58.2000 1.8000 2.0000 1.0000
20 -58.2000 1.8000 2.0000 1.0000
21 -58.2000 1.8000 2.0000 1.0000
22 -57.6000 2.4000 3.0000 0.0000
23 -57.6000 2.4000 3.0000 0.0000
24 -57.0000 3.0000 3.0000 1.0000
25 -57.0000 3.0000 3.0000 1.0000
26 -56.0000 4.0000 4.0000 0.0000
27 0.0000 60.0000 0.0000 0.000022 ˆ3.0ˆ2.010 TJHH CM
p-sd s-p-sd
2 particles 6 particles
N = 1 N = 1
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Dimensions of Nonspurious Spaces
2/)1(1
),'(),(),(
min
12
'min
KK
N
K
K
stepJJ
JJ
JJJnspnsp
J
KNJDNJDNJDKK
K
K
onsoandK
K
K
K
J K
44,2,0
33,1
22,0
11
1
1
1
1 '
)0,'()1,(K J
JJ
JJJnspsp
K
K
K
JDJD
Example: s-p-sd, 6 particles
J N=1(K=1) N=0
0 4 =4 2
1 2+4+3=9 4
2 4+3+1=8 3
3 3+1 =4 1
4 1 =1 0
Total 26
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C. Jacquemin, Z. Phys. A 303, 135 (1981)
Fixed J Restricted Configura
tion Widths
srjirjiji DDD ,,
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]/)[2/3(" AHHH CM 1
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N Nonspurious Level Density
N
K
K
stepJJ
JJ
JJJnspnsp
KK
K
K
KNJDNJDNJD1
2'min
),'(),(),(
N
K
K
stepJJ
JJ
JJJnspnsp
KK
K
K
KNJENJENJE1
2'min
),',(),,(),,(
)0,()2,0(,)0,'()2,2(
,)1,'()1,1(),2,()0,0(:)2(
intint
intint
JJ
JJNJN
CMJCM
JCMCMJ
intint)( HHVVVTH CMCMklowrel
3
1'
)0,',()1,2,()1,2,(:J
nspnsp JEEJEExample
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20Ne: 20 particles in s-p-sd-pf shell model space
),(),( JEJEEE xgsx
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20Ne: 20 particles in s-p-sd-pf shell model space
),(),( JEJEEE xgsx
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20Ne: 20 particles in s-p-sd-pf shell model space
),(),( JEJEEE xgsx
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Nonspurious Level Density: (0+2)
2
1
2
2'min
)2,',()20,,()20,,(K
stepJJ
JJ
JJJnspnsp
KK
K
K
KJEJEJE
),0,()2,0(,)0,'()2,2(
,)1,'()1,1(),2,()0,0(
)0,()0,0(:)20(
intint
intint
int
JJ
JJNJ
JNJN
CMJCM
JCMCM
CMJ
intint)( HHVVVTH CMCMklowrel
3
1'
2
20
2
2'
)1,',()0,',(
)20,2,()20,2,(:
Jnsp
stepJ
J
JJnsp
nsp
JEJE
EJEExample
K
K
K
)2,,()0,,()20,,( JEJEJE nspnspnsp
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10B: 10 particles in s-p-sd-pf shell model space
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10B: 10 particles in s-p-sd-pf shell model space
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Nonspurious Level Density: General
2
1
2
20 '
2
1
2
2'
)]2(,',[]20,,[
)]2()0(,',[]20,,[]20,,[min
KstepJ
JJ
JJJnsp
Kstep
JJ
JJ
JJJnspnsp
K
K
K
KK
K
K
KJEJE
KKJEJEJE
)]3(,',[)]3()1(,',[)1(
)]3()1(,',[]31,,[]31,,[3
1
3
2'min
KJEKKJEKif
KKJEJEJE
nspnsp
Kstep
JJ
JJ
JJJnspnsp
KK
K
K
mn
K
mn
stepJJ
JJ
JJJnsp
nsp
KK
K
K
KmnKnKnJE
mnnnJEmnnnJE
2
1
2
2'min
)]2()2()(,',[
)]2()2()(,,[)]2()2()(,,[
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Summary • We derived explicit expressions to calculate fixed-
J centroids and widths for restricted set of configurations, such Nconfigurations
• We found recursive formulae to calculate the dimensions of nospurious spaces
• We found recursive formulae for calculating exactly the nonspurious level density when one knows the level density for a restricted set of configurations, such Nconfigurations
• Using our method of calculating the level density for restricted set of configurations we can calculate very accurately the nonspurious level density