@let@token statistical approach to nuclear level density...

Statistical Approach to Nuclear Level Density

Roman Sen’kov

Physics DepartmentCentral Michigan University

East Lansing, May 7, 2014

♦ Support from DOE grant DE-SC0008529 and NSF grant PHY-1068217 is acknowledged

Outline:

I From Shell Model to Nuclear Level Density

I Moments Method: Possible Applications

Thanks:

I Vladimir Zelevinsky (Michigan State University)

I Mihai Horoi (Central Michigan University)

I Jagjit Kaur (Western Michigan University)

Publications:

I M. Horoi, J. Kaiser, and V. Zelevinsky, Phys. Rev. C 67, 054309 (2003).I M. Horoi, M. Ghita, and V. Zelevinsky, Phys. Rev. C 69, 041307(R) (2004).I M. Horoi, M. Ghita, and V. Zelevinsky, Nucl. Phys. A785, 142c (2005).I M. Horoi and V. Zelevinsky, Phys. Rev. Lett. 98, 262503 (2007).I M. Scott and M. Horoi, EPL 91, 52001 (2010).I R.A. Sen’kov and M. Horoi, Phys. Rev. C 82, 024304 (2010).I R.A. Sen’kov, M. Horoi, and V. Zelevinsky, Phys. Lett. B702, 413 (2011).I R.A. Sen’kov, M. Horoi, and V. Zelevinsky, Comp. Phys. Comm. 184, 215 (2013).I V. Zelevinsky, M. Horoi, and R. Sen’kov, proceedings of the conference “Nuclear Physics in Astrophysics

VI”, Lisbon, Portugal, May 19–24, 2013.

Nuclear Reactions and Nuclear Level Density

I Nucleosynthesis

I Nuclear structure

I Nuclear reactors

I Medical applications

For nuclear reactions we need energy level density!

For application to nuclear physics, level densities should be computedby means of microscopic models, able to reproduce experimentalinformation.

Challenge: there is no good microscopic theory which can provideus energy-, spin-, and parity-dependent nuclear level densities.

Energy distribution of the excited states

Nuclear level density (NLD)

ρ(E) =∆N∆E

.

– number of energy levels per unitof energy

28Si

8

0

2

4

6

8

10

E HMeVL

DE

Theoretical models:

I Non-interacting Fermi-particles (H.A. Bethe, 1937):

ρ(E) ∼ exp (S) = exp(

2√

aE).

E = aT 2

dSdE = 1

T

}⇒ S(E) = 2

√aE , where a =

π2

6g.



ρ(E) =∆N∆E

.


28Si

8

0

2

4

6

8

10

E HMeVL

DE

Theoretical models:


ρ(E) =

√π

12a1/4E5/4 exp(

2√

aE).

– still need to add spin and parity dependence.



ρ(E) =∆N∆E

.


28Si

8

0

2

4

6

8

10

E HMeVL

DE

Theoretical models:


ρ(E ,M) =

√π

12a1/4E5/4

1√2πσ2

exp(

2√

aE − M2

2σ2

).

I Mean field based models (S. Goriely, S. Hilaire, et al)

I Shell Model

Microscopic description of Nuclear Level Density

Shell model:

I Restricted model spaceDim(sd) ∼ 106

Dim(fp) ∼ 1010

I Need effective interaction

I Numerical diagonalizationcore

28Si

d5�2

s1�2

d3�2

sd

fp

ps

I High accuracy: δE ∼ ±200KeV

How it works:

Many-body states in Shell Model: |α〉 =Dim∑k=1

Cαk |k〉.

Schrodinger equation: H|α〉 = Eα|α〉 ⇒ H~Cα = Eα~Cα.

Statistical approach to Nuclear Level Density

There are so many states, are all of them unique?

I Any interaction will mixstates strongly

|〈m|V |n〉| � |Em −En|.

I States “forget” their“initial conditions”.

Information entropy – the degree of complexity of the state |α〉

Sα = −∑

k

Wαk ln Wα

k , where Wαk = |Cα

k |2.

Lα = exp Sα – Information length: the degree of delocalization of thestate |α〉, if Wα = const = N−1 then Sα = ln N and Lα = N.

Statistical approach to Nuclear Level Density (cont.)

ρ(E , β) =∑κ

Dβκ ·G(E − Eβκ, σβκ)

G(x , σ) - Gaussian distribution

β = {n, J,Tz , π} - quantum numbers

κ - configurations0 20 40 60 80

Excitation energy (MeV)

0

50

100

Nu

clea

r le

vel

den

sity

(M

eV-1

)

28Si, J

π=0

+

Eκ

κ d 52 s 1

2 d 32

1 6 0 02 5 1 03 5 0 14 4 2 0· · · · · · · · · · · ·15 0 2 4

Dβκ - number of many-body states with givenβ that can be built for a given configuration κ

Moments of H for each configuration κ:

Eβκ = Tr(βκ)[H]/Dβκ

σ2βκ = Tr(βκ)[H2]/Dβκ −

(Tr(βκ)[H]/Dβκ

)2

M. Horoi, M. Ghita, and V. Zelevinsky, PRC 69 (2004) 041307(R)

28Si, parity=+1, some J, sd-shellShell Model (solid line) vs. Moments Method (dashed line).

20 40 60 80Excitation energy (MeV)

0

50

100

Nucl

ear

level

den

sity

(M

eV-1

)

J=0


0

100

200

300

Nucl

ear

level

den

sity

(M

eV-1

)

J=1


0

200

400

Nucl

ear

level

den

sity

(M

eV-1

)

J=2


0

200

400

Nucl

ear

level

den

sity

(M

eV-1

)

J=3

52Fe, 52Cr, parity=+1, some J, pf -shellShell Model (solid line), Moments Method (dashed line), andHF+BCS method (dotted line).

0 2 4 6 8 10


0

5

10

15

20

25

30

Nu

clea

r le

vel

den

sity

(M

eV-1

)

exact SM (pf-shell, gx1a)

moments methodmodel of Goriely et al.

52Fe, J

π= 0

+

0 2 4 6 8 10


0

10

20

30

Nu

clea

r le

vel

den

sity

(M

eV-1

)



52Fe, J

π= 1

+

0 2 4 6 8 10


0

5

10

15

20

25

30

35

40

Nu

clea

r le

vel

den

sity

(M

eV-1

)



52Cr, J

π= 0

+

0 2 4 6 8 10


0

20

40

60

Nu

clea

r le

vel

den

sity

(M

eV-1

)



52Cr, J

π= 1

+

64Ge, 68Se, parity=+1, J = 0,2, pf + g 92 -model space

Moments Method: pf -shell (solid curve), pf + g 92 -shell (dashed line),

and HF+BCS method (dotted line).

2 4 6 8 10Excitation energy (MeV)

0

20

40

60

80

Nucl

ear

level

den

sity

(M

eV-1

)

Moments, pf-shell

Moments, pf+g9/2-shell

Hilaire’s table

64Ge, J

π

=0+


0

100

200

300

Nu

clea

r le

vel

den

sity

(M

eV-1

)

Moments, pf-shell


Hilaire’s table

64Ge, J

π

=2+


0

20

40

60

80

Nucl

ear

level

den

sity

(M

eV-1

)

Moments, pf-shell


Hilaire’s table

68Se, J

π

=0+


0

100

200

300

Nu

clea

r le

vel

den

sity

(M

eV-1

)

Moments, pf-shell


Hilaire’s table

68Se, J

π

=2+

Removal of the center-of-mass spurious states

Harmonic oscillator:

Nspur (K~ω) ∼K∑

K ′=1

Npure((K − K ′)~ω),

where K ′ presents how many times we actwith A†cm

P. Van Isacker, Phys. Rev. Lett. 89, 262502 (2002)

A+ �

Acm+ �

Nuclear level density. Recursive method:

ρpure(E , J,K ) = ρ(E , J,K )−K∑

K ′=1

K ,step 2∑JK ′=Jmin

J+JK ′∑J′=|J−JK ′ |

ρpure(E , J ′,K − K ′)

M. Horoi and V. Zelevinsky, Phys. Rev. Lett. 98, 262503 (2007)

22Mg, (s-p-sd-pf)-model space, 1~ω, β · A = 110 MeV


0

200

400

600

Nu

clea

r le

vel

den

sity

(M

eV-1

)

0 4 8 12 160

20

40

Jπ=1

−


0

200

400

600

800

Nucl

ear

level

den

sity

(M

eV-1

)

0 4 8 12 160

20

40

60Jπ=2

−

– Shell Model. Density with spurious states.– Shell Model. Density with shifted spurious states.

– Moments Method. Density with spurious states.– Moments Method. Density without spurious states.– Moments Method. Spurious states.

26Al and 28Si, (s-p-sd-pf)-model space, both parities, all J


0

10

20

30

40

Nucl

ear

level

den

sity

(M

eV-1

) 26Al, π=+1


0

10

20

30

40

Nucl

ear

level

den

sity

(M

eV-1

) 26Al, π=−1

0 2 4 6 8 10 12 14Excitation energy (MeV)

0

10

20

30

40

Nucl

ear

level

den

sity

(M

eV-1

) 28Si, π=+1

0 2 4 6 8 10 12 14Excitation energy (MeV)

0

10

20

30

40

Nucl

ear

level

den

sity

(M

eV-1

) 28Si, π=−1

Moments Method Code scaling (NERSC/Hopper)

Speedup = T1/Tn,n - number of coresTn - calculation time with n cores

Domain decomposition: many-body configurations

Algorithm: Dynamically Load Balancing0 1000 2000 3000 4000

Number of cores

0

1000

2000

3000

4000

Spee

dup

ideal speedup68

Se, pf+g9/2, 0.6 106 configurations

64Ge, pf+g9/2, 0.5 10

6 configurations

at number of cores ∼ 2000 the calculation time T is about 1 min!68Se and 64Ge:

J-Dim ∼ 108 for pf∼ 1012 for pf + g 9

2

Egs(pf , 68Se) = −353.1 MeVEgs(pf , 64Ge) = −304.3 MeV

0 1000 2000 3000 4000 5000 6000

Number of cores

0

1000

2000

3000

4000

5000

6000

Spee

dup

ideal speedup28

Si, spsdpf, 4hw, 7.1 106, "Calc"

28Si, spsdpf, 4hw, 7.1 10

6, "All"

Applications: ground state energy in a larger model space

Egs(pf + g 92) = Egs(pf )−∆E


0

20

40

60

80

Nucl

ear

level

den

sity

(M

eV-1

)

pf+g9/2, ∆E = 0.0 MeV

pf+g9/2, ∆E = 1.55 MeVpf+g9/2, ∆E = 2.45 MeVpf

64Ge, J

π= 0

+


0

20

40

60

80

Nucl

ear

level

den

sity

(M

eV-1

)

pf+g9/2, ∆E = 0.0 MeV

pf+g9/2, ∆E = 3.4 MeVpf

68Se, J

π= 0

+


0

100

200

300

Nu

clea

r le

vel

den

sity

(M

eV-1

)

pf+g9/2, ∆E = 0.0 MeV


64Ge, J

π= 2

+


0

100

200

300

Nu

clea

r le

vel

den

sity

(M

eV-1

)

pf+g9/2, ∆E = 0.0 MeV

pf+g9/2, ∆E = 3.4 MeVpf

68Se, J

π= 2

+

Egs(pf + g 92 ,

64Ge) = −306.7 MeV, Egs(pf + g 92 ,

68Se) = −356.5 MeV

Applications: ground state energy in a larger model space

Egs(pf + g 92) = Egs(pf )−∆E


0

20

40

60

80

Nucl

ear

level

den

sity

(M

eV-1

)

pf+g9/2, ∆E = 0.0 MeV


64Ge, J

π= 0

+


0

20

40

60

80

Nucl

ear

level

den

sity

(M

eV-1

)

pf+g9/2, ∆E = 0.0 MeV

pf+g9/2, ∆E = 3.4 MeVpf

68Se, J

π= 0

+


0

100

200

300

Nu

clea

r le

vel

den

sity

(M

eV-1

)

pf+g9/2, ∆E = 0.0 MeV


64Ge, J

π= 2

+


0

100

200

300

Nu

clea

r le

vel

den

sity

(M

eV-1

)

pf+g9/2, ∆E = 0.0 MeV

pf+g9/2, ∆E = 3.4 MeVpf

68Se, J

π= 2

+

Egs(pf + g 92 ,

64 Ge) = −306.7 MeV, Egs(pf + g 92 ,

68 Se) = −356.5 MeV

Applications: playing with the interaction

H = h + k1V (pairing) + k2V (non-pairing).

0 10 20 30 40 50 60 70 800

2000

4000

6000

8000

1000028

Si, J=all, k1=k

2={0.1-1.0}

0 10 20 30 40 50 60 70 800

1e+06

2e+06

3e+06

4e+06

5e+06 52Fe, J=all, k

1=k

2={0.1-1.0}

0 10 20 30 40 50 60 70 800

2000

4000

6000

8000

1000028

Si, J=all, k2=0.1, k

1={0.1-1.5}

0 10 20 30 40 50 60 70 800

1e+06

2e+06

3e+06

4e+06

5e+06 52Fe, J=all, k

2=0.1, k

1={0.1-1.5}

Applications: playing with the interaction (cont.)


0 10 20 30 40 50 60 70 800

50

100

150

200

250

300

35028

Si, J=0, k1=k

2={0.1-1.0}

0 10 20 30 40 50 60 70 800

20000

40000

60000

80000

52Fe, J=0, k

1=k

2={0.1-1.0}

0 10 20 30 40 50 60 70 800

50

100

150

200

250

300

35028

Si, J=0, k2=0.1, k

1={0.1-1.5}

0 10 20 30 40 50 60 70 800

20000

40000

60000

80000

52Fe, J=0, k

2=0.1, k

1={0.1-1.5}

Applications: playing with the interaction (cont.)


0 10 20 30 40 50 60 70 800

50

100

150

200

250

300

35028

Si, J=0, k2=0.1, k

1={0.1-1.0}

0 10 20 30 40 50 60 70 800

50

100

150

200

250

300

35028

Si, J=0, k1=1.0, k

2={0.1-1.0}

0 5 10 15 200

20

40

60

80

100

12028

Si, J=0, k2=0.1, k

1={0.1-1.0}

0 5 10 15 200

20

40

60

80

100

12028

Si, J=0, k1=1.0, k

2={0.1-1.0}

Applications: Fermi Gas Model

log[ρ(E ,M2)

]= log [ρ(E ,0)]−M2/2σ2

0 20 40 60 80 100

-5

0

5

10

Lo

g[ρ

(Ε,Μ

)]

E=5 MeVE=10 MeVE=15 MeVE=20 MeVE=25 MeV

28Si

0 20 40 60 80 100

0

5

10

15

Lo

g[ρ

(Ε,Μ

)]


52Fe

0 20 40 60 80 100

M2

-4

-2

0

2

4

6

Log[ρ

(Ε,Μ

)]


44Ca

0 20 40 60 80 100

M2

-4

-2

0

2

4

6

Log[ρ

(Ε,Μ

)]


64Cr

Applications: Fermi Gas Model (cont.)

Interpolation: σ2 =√

E(α + βE), for E ∈ [5,25]

20 25 30 35 40 45 50 55 60 65 70 75 800

5

10

15α

-par

amet

er All othersZ=20 isotopes

Z=21 isotopes

Z=22 isotopes

N=40 isotones

20 25 30 35 40 45 50 55 60 65 70 75 80

mass number, A

-0.2

-0.1

0

0.1

0.2

β-p

aram

eter

Fermi gas model (Ericson, 1960): σ2 = gT 〈M2〉


Interpolation: σ2 =√

E(α + βE), for E ∈ [5,25]

20 25 30 35 40 45 50 55 60 65 70 75 800

5

10

15α

-par

amet

er Moments MethodStatistical calcRigid sphere

20 25 30 35 40 45 50 55 60 65 70 75 80

mass number, A

-0.2

-0.1

0

0.1

0.2

β-p

aram

eter

Fermi gas model (Ericson, 1960): σ2 = gT 〈M2〉


Interpolation: log [ρ(E ,0)] = 2√

aE − 5/4 log(E) + constant

20 30 40 50 60 70 800

5

10a-

par

amet

er All othersZ=20 Isotopes

Z=21 Isotopes

Z=22 Isotopes

N=40 Isotones

20 30 40 50 60 70 80

mass number, A

-5

0

5

const

ant

G. Rohr, Z. Phys. A 318, 299 (1984).

S.I. Al-Quraishi, S.M. Grimes, T.N. Massey, and D.A. Resler, Phys. Rev. C 63, 065803 (2001).


Interpolation: log [ρ(E ,0)] = 2√

aE − 5/4 log(E) + constant

20 30 40 50 60 70 800

5

10a-

par

amet

er Moments MethodFG: RohrFG: Al-Quraishi

20 30 40 50 60 70 80

mass number, A

-5

0

5

const

ant

G. Rohr, Z. Phys. A 318, 299 (1984).

S.I. Al-Quraishi, S.M. Grimes, T.N. Massey, and D.A. Resler, Phys. Rev. C 63, 065803 (2001).

Conclusion and Future plans

I Moments Method code is finalized, published, and ready to use.

I Calculation of reaction rates and cross sections (use talys code).

I Use the Moments Method to predict the ground state energies.

I Use Moments Method to “play” with the interaction.

I Use Moments Method to verify/study the Fermi-Gas model.

I Removal of spurious states for incomplete shells.


Overall performance, 28Si:

α = 2.37± 0.09, β = −0.023± 0.006a = 1.78± 0.03, constant = −2.92± 0.13

0 5 10 15 20 25 300

500

1000

1500

2000

interp, M=0

interp, M=1

interp, M=2

interp, M=3

interp, M=4

interp, M=5

real, M=0real, M=1real, M=2real, M=3real, M=4real, M=5

5 10 15 200

100

200

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