Parity Dependent Level Densities and its
Applications to Nuclear Reaction Rates
Darko Mocelj, T. Rauscher, G. Martı́nez-Pinedo, K. Langanke and
F.K. Thielemann
Universität Basel
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Motivation
Ingredients for Nucleosynthesis Calculations:
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Nuclear Reaction Networks
Include terms for:
• neutron captures
• neutron-induced fission
• photodisintegration
• β -decays
• β delayed neutron emission/fission
• ν ’s
Ẏ (Z, A) = nn YZ,A−1 hσvi(n,γ) + YZ,A+1 λZ,A+1 −
Z,A
+
λ
− YZ,A (nn hσvi(n,γ) + λZ,A + λZ,A
β
βn )
+ YZ−1,A λβZ−1,A +YZ−1,A+1 λZ−1,A+1
βn
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Nuclear Reaction Networks
Ẏ (Z, A) = nn YZ,A−1 hσvi(n,γ) + YZ,A+1 λZ,A+1 −
Z,A
+
λ
− YZ,A (nn hσvi(n,γ) + λZ,A + λZ,A
β
βn )
+ YZ−1,A λβZ−1,A +YZ−1,A+1 λZ−1,A+1
βn
(n,γ)
(γ,n)
Z,A
β−
β− + 1n
Z
A
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Calculation of Thermonuclear Rates: iµ (j, o)mν
Nuclear reaction rate per particle pair at a given stellar
temperature T is determined by folding the reaction cross
section with the Maxwell-Boltzmann velocity distribution
of the projectiles:
hσvi =
8
πµ
1/2
1
(kT )3/2
Z
∞
0
E
σ(E)E exp(− )dE
kT
Detailed knowledge of the cross-section is important!
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Calculation of Thermonuclear Rates: iµ (j, o)mν
Theoretical Cross-Sections are calculated within the
Hauser-Feshbach Framework:
σiµν (j, o)
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=
π~2 /(2µ
(2Jiµ
ij Eij )
+ 1)(2Jj + 1)
X
J,π
(2J+1)
Tjµ (E, J, π)Toν (E, J, π)
Ttot (E, J, π)
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Calculation of Thermonuclear Rates: iµ (j, o)mν
Target States µ in an astrophysical plasma are
thermally populated:
σi∗ (j, o) =
P
µ
(2J
µ
i
P µν
µ
+ 1) exp(−Ei /kT ) ν σi (j, o)
P
µ
µ
(2J
+
1)
exp(−E
µ
i
i /kT )
with
To (E, J, π) =
νm
X
ν=0
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Toν +
Z
E−Sm,o
νm
Em
X
To ρ(Em , Jm , πm )dEm
Jm ,πm
Darko Mocelj VISTARS Winter School in Russbach, March 14-19th 2004 – p.4/13
Calculation of Thermonuclear Rates: iµ (j, o)mν
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Level Density: Back-Shifted Fermi-Gas Model
1
ρ(U, J, π) = F (U, J)ρ(U )
2
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Level Density: Back-Shifted Fermi-Gas Model
1
ρ(U, J, π) = F (U, J)ρ(U )
2
Can we drop the assumption of equally distributed
parities and determine an energy dependent ratio of the
number of odd- and even-parity states ?
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Level Density: Back-Shifted Fermi-Gas Model
1
ρ(U, J, π) = F (U, J)ρ(U )
2
Can we drop the assumption of equally distributed
parities and determine an energy dependent ratio of the
number of odd- and even-parity states ?
ρ(U, J, π) = Π(U )F (U, J)ρ(U )
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QMC Calculations
Y.Alhassid et. al Phys. Rev. Letters 84, 4313
Parity properties can in principle be calculated within
the interacting shell model
. . . requires large model spaces, → QMC
1
56
Fe
Ni
68
Zn
60
Z−/Z+
0.8
0.6
0.4
0.2
0
0.0
0.5
1.0
1.5
2.0
β (MeV )
−1
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Distribution of the Π-Parity Group
Assume Poisson-Distribution:
f n −f
e
P (n) =
n!
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Distribution of the Π-Parity Group
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Distribution of the Π-Parity Group
P− =
X
P (n) = e−f sinh f
n, odd
P+ =
X
P (n) = e−f cosh f
n, even
P−
Z− (β)
=
= tanh(f )
P+
Z+ (β)
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What is f ?
Reminder:
Z− (β)
Z+ (β)
= tanh(f ).
Below critical βc : Fermi-Dirac
f=
X
a∈Π
1
1 + eβ(a −µ)
Above critcal βc : Solve BCS - Equations
f=
X
a∈Π
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1+e
β
√
1
(a −µ)2 +∆2
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How does one get ρ− /ρ+ ?
Use BBF: → Z(β) = ρ(Ex )e−βEx dEx.
With Z− + Z+ = Z and Z− /Z+ = tanh f
R
Z± = Z(1 + tanh± f )
Saddle-Point Approximation:
1
ρ± (E) = p
exp(βE± + ln Z± (β))
−2
2πβ C±
where
2
∂
∂
E± = −
ln Z(β) , C± = β 2 2 ln Z(β)
∂β
∂β
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Results
2
2
58
Fe
Fe
Fe
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
ρ−/ρ+
2
56
54
0
10
20
30
40
2
0
0
10
20
30
40
2
0
62
Fe
1.5
1
1
1
0.5
0.5
0.5
20
30
40
0
30
40
20
30
40
Fe
1.5
10
20
64
Fe
1.5
0
10
2
60
0
0
0
10
20
30
40
0
0
10
Ex [MeV]
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Influence on the reaction rates
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Influence on the reaction rates
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Outlook
Recalculate nuclear rates with the improved level
densities
Update Reaclib95 to Reaclib200(4)
Apply the new rates to an astrophysical scenario,
e.g. r-process, rp-process ...
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TOC
Motivation
Nuclear Reaction Networks
Thermonuclear Rates
Level Density BBF
QMC Calculations
Parity Distribution
What is f ?
ρ− /ρ+
Results
Influence on the Reaction Rates
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