Futoshi Minato

JAEA Nuclear Data Center, Tokai

Theoretical calculations of beta-delayed neutrons and

sensitivity analyses

1

2

1. Delayed Neutron (DN) Emission Probabilities by Skyrme-HF+QRPA plus Hauser-Feshbach Statistical Model (HFSM)

2. Incident Neutron Energy Dependence of DN Yields

3. Sensitivity Analysis of DN with JENDL evaluated libraries

4. Important Precursors in r-process Nucleosynthesis

Contents of This Talk

3

1. Delayed Neutron (DN) Emission Probabilities by Skyrme-HF+QRPA plus Hauser-Feshbach Statistical Model

2. Incident Neutron Energy Dependence of DN Yields

3. Sensitivity Analysis of DN with JENDL evaluated libraries

4. Important Precursors in r-process Nucleosynthesis

Contents of This Talk

4

1. DN Emission Probabilities by SHF+QRPA plus HFSM

Calculations of DN Emission Probability

β-

Parent/Precursor

(Z,N)

Daughter(Z+1,N-1)

(Z+1,N-2)

ng.s.

g.s.

g.s.

γ

5

1. DN Emission Probabilities by SHF+QRPA plus HFSM

β-

Parent/Precursor

(Z,N)

Daughter(Z+1,N-1)

(Z+1,N-2)

n

QRPA HFSM

g.s.

g.s.

g.s.

T1/2

Strength FunctionPn (P1n, P2n, P3n)

Neutron Spectrum

γ

Calculations of DN Emission Probability

6

1. DN Emission Probabilities by SHF+QRPA plus HFSM

QRPA •On top of Skyrme-Hartree-Fock+BCS•Deformation (cylind. coordinate space)•Volume-type Pairing force in BCS•Residual Interaction : Same as G.S. Include All Terms self-consistently

SkO SAMiStrength of Pairing (p,n)

Odd Nuclei

Skyrme Effective Force

p or ncore

(330,323)(256,258)

Valence Particle is assumed to follow Indep. Particle Model

Blocking Effect in QRPA

7

1. DN Emission Probabilities by SHF+QRPA plus HFSM

QRPA

Prescription to determine T=0 pairing strength Vpp1. Search appropriate Vpp reproducing T1/2 of E-E Nuclei2. Calculate Average Vpp

ave(Z) from Vpp of same Z3. Calculate T1/2 of isotope chains systematically with Vpp

ave(Z)

SkO

Atomic Number Z

Isospin T=0 pairing Attractive in GT channelStrong pairing Low 1+ state in daughter Shorter T1/2

V pp(Z

)

Daughterg.s.

1+

1+

1+

8

1. DN Emission Probabilities by SHF+QRPA plus HFSM

233 nucleirms=5.09

T 1/2(c

alc)

/ T 1/

2(exp

) SKO

9

1. DN Emission Probabilities by SHF+QRPA plus HFSM

HF Hauser-Feshbach Models implemented in the nuclearreaction calculation code, “CCONE”.

Neutron Transmission Coefficient : Koning-Delaloche Optical Pot.Gamma-ray Transmission Coefficient : Kopecky-Uhl’s EGLOLevel-Density: Fermi Gas Model with Mengoni-Nakajima parameter set at high excitation energy

(Z+1,N-2)

n

g.s.

g.s.

Daughter(Z+1,N-1)

γn

γ

β-

Parent(Z,N)

Qβ=1.SHF+BCS 2.Experiment

DN Emission Prob. for Z=35-44 (Qβ : SHF+BCS)

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1. DN Emission Probabilities by SHF+QRPA plus HFSM

SKO

SKO

SKO

SKO

11

1. DN Emission Probabilities by SHF+QRPA plus HFSM

DN Emission Prob. for Z=27-30 (Qβ : exp. or KTUY)

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1. DN Emission Probabilities by SHF+QRPA plus HFSM

P2n & P3n (Qβ : exp. or KTUY)

A (Co isotopes)

P2n

or P

3n

SAMi(P2n)

SkO(P2n)SAMi(P3n)

SkO(P3n)

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DN Spectra

1. DN Emission Probabilities by SHF+QRPA plus HFSM

Cu-81Zn-83

g.s.Daughter(Z+1,N-1)

β-

Parent(Z,N)

φn (

MeV

-1)/

1 fi

ssio

n

E (MeV) E (MeV)

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1. Delayed Neutron (DN) Emission Probabilities by Skyrme-HF+QRPA plus Hauser-Feshbach Statistical Model

2. Incident Neutron Energy Dependence of DN Yields

3. Sensitivity Analysis of DN with JENDL

4. Important Precursors in r-process Nucleosynthesis

Contents of This Talk

15

2. Incident Neutron Energy Dependence of Delayed Neutron Yields

Energy Dependence of DN Yields in Nuclear Data𝜈𝑡𝑜𝑡

16

𝜈𝑡𝑜𝑡=∫𝜈𝑑(𝑡)𝑑𝑡𝜈𝑑(𝑡)=∑

𝑖

𝜆𝑃𝑛(𝑖)𝑦 𝑖(𝑡) Fission Yield Data

Decay DataActivity of DN

DN Yield

2. Incident Neutron Energy Dependence of Delayed Neutron Yields

Detail can be found in [2].Calculation is performed with Code [3].

Bateman Equation𝑑𝑛1(𝑡)𝑑𝑡

=− 𝜆1𝑛1(𝑡)

𝑑𝑛𝑘(𝑡)𝑑𝑡

=− 𝜆𝑘𝑛𝑘 (𝑡 )+𝜆𝑘−1𝑛𝑘−1 (𝑡 )(2≤𝑘)

𝑛𝑖 (𝑡 )=𝑦 𝑖∑𝑗=1

𝑖

𝑑 𝑗𝑒−𝜆 𝑗 𝑡

𝑑 𝑗=1 (𝑖= 𝑗=1 )

𝑑 𝑗=∏𝑘=1

𝑖−1

𝜆𝑘

∏𝑘=1 ,𝑘≠ 𝑗

𝑗

(𝜆¿¿𝑘−𝜆 𝑗)(2≤ 𝑗≤ 𝑖 ) ¿

Energy Dependent

137Te 137I 137Xe 137Cs 137Ba

137mBa

炉物理の研究 第 64 号（ 2012 年 3 月）吉田先生

β- β- β- β-

β-0.4% 0.1%3.2%2.7%

Indep. FY

2.6m IT

2.5s 24.5s 3.8m 30.1yβn=2.9% βn=7.1%

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2. Incident Neutron Energy Dependence of Delayed Neutron Yields

Decay Data ： JENDL/FPD-2011 J. Katakura, JAEA-DATA/Code2011-025(2011)

Bad Reproduction

Fission Yields ： 5 Gaussian Model

1. Mass Distribution & Prompt Neutron: J. Katakura, JAERI-Research2003-004(2003)2. Charge Distribution: T.F. England and B.F.Rider, LA-UR-94-3106,ENDF-349(1994).3. Isomer states: J. Katakura, JAEA-DATA/Code2011-025(2011)

Wahl, IAEA-TECDOC-1168(2000)

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2. Incident Neutron Energy Dependence of Delayed Neutron Yields

Correction in A) D.R. Nethaway, Lawrence Livermore Laboratory Report No. UCRL-51538, (1974). 　 (see also D.R. Alexander and M.S. Krick, Nucl. Sci. Eng. 62, 627 (1977) )B) V.A. Roshchenko, V.M. Piksaikin et al., Phys. Rev. C74, 014607 (2006)

Energy Dependence of charge distribution:

E (eV)

DN

Yie

ld/fi

ssio

n

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1. Delayed Neutron (DN) Emission Probabilities by Skyrme-HF+QRPA plus Hauser-Feshbach Statistical Model

2. Incident Neutron Energy Dependence of DN Yields

3. Sensitivity Analysis of DN with JENDL

4. Important Precursors in r-process Nucleosynthesis

Contents of This Talk

20

𝑆𝑌 𝑖=

(∆𝜈¿¿𝑑¿𝜈𝑑)(∆𝑌 𝑖/𝑌 𝑖)

¿

𝑆𝑃 𝑛𝑖=

(∆𝜈 ¿¿𝑑 ¿𝜈𝑑)(∆ 𝑃𝑛𝑖 /𝑃𝑛𝑖)

¿

Sensitivity Test

Fission Yields

DN Emission Prob.

3. Sensitivity Analysis of Delayed Neutron with JENDL

Δ 𝑦 𝑖 /𝑦 𝑖=¿ Δ 𝑃𝑛(𝑖)/𝑃𝑛 (𝑖 )=0.1¿

235U: 86Ge,89Br,90Br,94Rb, 137I

𝑆 𝑦 𝑖

𝑆𝑃 𝑛(𝑖)

Thermal Neutron Fission

239Pu: 89Br,90Br,94Rb,98mY,137I,138I

Remarkable Nuclei

235U: 86As, 88Br,89Br,90Br,94Rb, 137I

239Pu: 88Br,89Br,90Br,94Rb,98mY,137I,138I

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3. Sensitivity Analysis of Delayed Neutron with JENDL

Nucl. Yield 　 error Err./YieldGe-86 0.6278 0.1005 (16%)As-85 0.1212 0.0775 (63%)As-86 0.0199 0.0127 (64%)Br-89 1.0379 0.0415 (4%)Br-90 0.5518 0.0331 (6%)Rb-94 1.5644 0.0438 (2.8%)Y-98m 1.8739 0.5996 (32%)Sb-135 0.1449 0.0116 (8%)Te-137 0.3919 0.0313 (8%)Te-138 0.0661 0.0423 (64%)I-137 2.6189 0.1048 (4%)I-138 1.4222 0.0398 (2.8%)

Nucl 　 Pn(%) err. 　 err./PnGe-86 6 N/A (----)As-85 59.4 29 (48.8%)As-86 33.0 4.0 (12%)Br-89 13.8 0.4 (2.9%)Br-90 25.2 0.9 (3.6%)Rb-94 10.5 0.4 (3.8%)Y-98m 3.4 1.0 (29%)Sb-135 22 3 (13.6%)Te-137 2.99 0.16 (5.4%)Te-138 6.3 2.1 (33%)I-137 7.14 0.23 (3.22%)I-138 5.56 0.22 (3.96%)

Uncertainties in JENDL/FPY & FPD-2011Indep. Fission Yields Pn(%)

Red represents uncertainty > 5% 22

3. Sensitivity Analysis of Delayed Neutron with JENDL

23

1. Delayed Neutron (DN) Emission Probabilities by Skyrme-HF+QRPA plus Hauser-Feshbach Statistical Model

2. Incident Neutron Energy Dependence of DN Yields

3. Sensitivity Analysis of DN with JENDL

4. Important Precursors in r-process Nucleosynthesis

Contents of This Talk

24

4. Important Precursors in r-process Nucleosynthesis

𝑆=∫𝑡 𝑓 .𝑜 .

∞

𝑃𝜈(𝑖)𝑌 𝑖

∑𝑖∫𝑡 𝑓 . 𝑜 .

∞

𝑃𝜈 (𝑖)𝑌 𝑖

This Work is performed with T. Kajino & Shibagaki at NAOJ

Important DN precursor after freeze-out(f.o.) in r-process

We define

Tells information which nucleus emits neutron efficiently after f.o.

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4. Important Precursors in r-process Nucleosynthesis

1. Ag-129 4.79E-012. Rh-127 1.54E-013. Pd-128 7.33E-024. Cd-130 6.16E-025. Rh-125 5.59E-026. In-131 2.76E-027. Ru-126 1.68E-028. Pd-126 1.67E-029. Al-35 1.64E-0210. Nb-109 1.56E-021. Sb-137 9.63E-022. Sb-135 8.06E-023. Ag-129 6.75E-024. P-41 6.72E-025. I-141 6.27E-026. Cl-46 5.67E-027. Cd-130 4.12E-028. Sn-136 3.14E-029. I-143 3.07E-0210. Sn-134 2.76E-02

Ye=0.3, τ=16.6ms, s/k=105

Ye=0.3, τ=16.6ms, s/k=155

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4. Important Precursors in r-process Nucleosynthesis

1. Sb-137 1.28E-012. Cl-46 7.14E-023. P-41 7.12E-024. Sb-135 6.29E-025. I-143 3.10E-026. I-141 2.99E-027. Sn-136 2.43E-028. La-157 2.41E-029. Pr-161 2.08E-0210. La-155 1.93E-02

Ye=0.3, τ=16.6ms, s/k=205

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Thank you for you attention