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.
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) =
√π
12a1/4E5/4 exp(
2√
aE).
– still need to add spin and parity dependence.
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 ,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)
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
20 40 60 80Excitation energy (MeV)
0
100
200
300
Nucl
ear
level
den
sity
(M
eV-1
)
J=1
20 40 60 80Excitation energy (MeV)
0
200
400
Nucl
ear
level
den
sity
(M
eV-1
)
J=2
20 40 60 80Excitation energy (MeV)
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
Excitation energy (MeV)
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
Excitation energy (MeV)
0
10
20
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= 1
+
0 2 4 6 8 10
Excitation energy (MeV)
0
5
10
15
20
25
30
35
40
Nu
clea
r le
vel
den
sity
(M
eV-1
)
exact SM (pf-shell, gx1a)
moments methodmodel of Goriely et al.
52Cr, J
π= 0
+
0 2 4 6 8 10
Excitation energy (MeV)
0
20
40
60
Nu
clea
r le
vel
den
sity
(M
eV-1
)
exact SM (pf-shell, gx1a)
moments methodmodel of Goriely et al.
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+
2 4 6 8 10Excitation energy (MeV)
0
100
200
300
Nu
clea
r le
vel
den
sity
(M
eV-1
)
Moments, pf-shell
Moments, pf+g9/2-shell
Hilaire’s table
64Ge, J
π
=2+
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
68Se, J
π
=0+
2 4 6 8 10Excitation energy (MeV)
0
100
200
300
Nu
clea
r le
vel
den
sity
(M
eV-1
)
Moments, pf-shell
Moments, pf+g9/2-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 50 100 150Excitation energy (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 50 100 150Excitation energy (MeV)
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 2 4 6 8Excitation energy (MeV)
0
10
20
30
40
Nucl
ear
level
den
sity
(M
eV-1
) 26Al, π=+1
0 2 4 6 8Excitation energy (MeV)
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
2 4 6 8 10Excitation energy (MeV)
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
+
2 4 6 8 10Excitation energy (MeV)
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
+
2 4 6 8 10Excitation energy (MeV)
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 = 1.55 MeVpf+g9/2, ∆E = 2.45 MeVpf
64Ge, J
π= 2
+
2 4 6 8 10Excitation energy (MeV)
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
2 4 6 8 10Excitation energy (MeV)
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
+
2 4 6 8 10Excitation energy (MeV)
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
+
2 4 6 8 10Excitation energy (MeV)
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 = 1.55 MeVpf+g9/2, ∆E = 2.45 MeVpf
64Ge, J
π= 2
+
2 4 6 8 10Excitation energy (MeV)
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.)
H = h + k1V (pairing) + k2V (non-pairing).
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.)
H = h + k1V (pairing) + k2V (non-pairing).
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
)]
E=5 MeVE=10 MeVE=15 MeVE=20 MeVE=25 MeV
52Fe
0 20 40 60 80 100
M2
-4
-2
0
2
4
6
Log[ρ
(Ε,Μ
)]
E=5 MeVE=10 MeVE=15 MeVE=20 MeVE=25 MeV
44Ca
0 20 40 60 80 100
M2
-4
-2
0
2
4
6
Log[ρ
(Ε,Μ
)]
E=5 MeVE=10 MeVE=15 MeVE=20 MeVE=25 MeV
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〉
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 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〉
Applications: Fermi Gas Model (cont.)
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).
Applications: Fermi Gas Model (cont.)
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.
Applications: Fermi Gas Model (cont.)
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