INT October 28, 2004Mihai Horoi - Central Michigan Univ1 New Approaches for Spin- and...

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


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


Hauser and Feshbach, Phys. Rev 87, 366 (1952)


The Back-Shifted Fermi Gas Model for Nuclear Level Density

),( NZ


A.Adams, G.Mitchell, J.F. Shriner Phys.Lett, B422, 13(1998)

26Al

sd-shell model, USD interaction


Data: Table of Isotopes

Theory: sd-shell model + USD interaction28Si: positive parity


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


12 particles in sd model space


Nuclear Shell Model

pmmmm :,...],,[

.

)31(

)20(

1

0

321

d = 2 (2 j + 1)


Sum on Partitions vs Moments of the Whole Distribution

6 particles in pf5/2 -g9/2

New interaction A. Lisetskiy et al. PRC 2004


))(),((),(

),()(),(

0 JJEEEGJEG

JEGJdJE

ppp

P

ppp



6 particles in p-sd model space

))(),((),(

),()(),(

0 JJEEEGJEG

JEGJdJE

ppp

P

ppp


Exponential Convergence Method

CbnAEn )/2exp(

))(),()((),()(

),()()(),(

0 JJEEEFRGJEFRG

JEFRGJdJE

ppp

P

ppp


Exponential Convergence Method for fp-nuclei

nBeAnE )(

)2002(027303,65.Re.,,, CvPhysZelevinskyBrownHoroi


Exponential Convergence Method for fp-nuclei

Central Michigan Shell Model (CMichSM) code

Exact: -203.196 MeV


r,s,.. – orbits, not states


Fixed J Configura

tion Centroids

and Widths

C. Jacquemin, Z. Phys. A 303, 135 (1981)


Shell Model vs Fixed-J Centroids and Widths Density of States28Si:

12 particles in sd, Tz=0


Shell Model vs Fixed-J Centroids and Widths Density of States28Si:



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


Shell Model <M2>

28Si:


)()(

)(2

1)(

)(8

)12()(

22

2

)2/()2/1(

3

222

EME

EE

EeJ

E

lev

JJ


Shell Model <M2>

)()(

)(2

1)(

)(8

)12()(

22

2

)2/()2/1(

3

222

EME

EE

EeJ

E

lev

JJ

28Si:



Zeroth-Order <M2>

)()(

)(2

1)(

)(8

)12()(

22

2

)2/()2/1(

3

222

EME

EE

EeJ

E

lev

JJ

28Si:



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.


The Center-of-Mass Problemnucl-th/0111068


Nuclear Shell Model

,...],,[

.

)31(

)20(

1

0

321 mmmm

N


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


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


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


C. Jacquemin, Z. Phys. A 303, 135 (1981)

Fixed J Restricted Configura

tion Widths

srjirjiji DDD ,,


]/)[2/3(" AHHH CM 1


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


20Ne: 20 particles in s-p-sd-pf shell model space

),(),( JEJEEE xgsx



),(),( JEJEEE xgsx


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


10B: 10 particles in s-p-sd-pf shell model space


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()(,,[


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

INT October 28, 2004Mihai Horoi - Central Michigan Univ1 New Approaches for Spin- and...

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