CNR2011, Prague, Czech Republic, Sep. 19–23, 2011

Monte Carlo Simulation forStatistical Decay of Compound Nucleus

T. Kawano, P. Talou, M.B ChadwickLos Alamos National Laboratory

Compound Nuclear Reaction, and Related Models

Resonance and Hauser-Feshbach Theoris are CentralNuclear Databasemass, structure, discrete levels,ground state deformation,fission barrier

Modelsoptical model, level density,photo strength function,fission

Non-CN Contributionsdirect reaction, DSD capture,pre-equilibrium emission

All physical quantities can be evaluatedby microscopic, phenomenological, orexperimental approache

Sta

tistic

alH

ause

r-F

eshb

ach

Width FluctuationCorrectionGOE, Moldauer, HRTW, KKM

Off-DiagonalMatrix ElementsKKM, NWY,generalized transmission,EW transformation,detailed barance

However, accurate knowledge about nucleus is crucial.

HF Theory: Significance in Nuclear Data World

Neutron capture on 89YCalculated cross sections often reasonable from keV to 150 MeV

0.0001

0.001

0.01

0.1

0.001 0.01 0.1 1 10

Cap

ture

Cro

ss S

ectio

n [b

]

Neutron Incident Energy [MeV]

ENDF/B-VII.0 70groupBoldeman

ENDF/B-VII.1 70group

Nowadays, the HF codes play a central role in the nuclear data evaluation above theresonance regions.

Inferred Cross Section

Explore Unknown C.S. By Combining Theory and Experiments

•• prompt γ-rays from a decay of compound nucleus•• this is partial information — requires supplemental theoretical calculations

73.04180.22

138.89

357.69

521.90

180.06

361.86

516.57

478.98

299.35

563.35

469.38

3/2[402]

1/2[400]11/2[505]

3/2+

5/2+

7/2+

9/2+

1/2+

3/2+

5/2+

7/2+

11/2-

13/2-

15/2-

7/2-

9/2-

0

219keV

483keV

399keV

389keV

Ir193

IT 100%

10.53 d

0

0.5

1

1.5

2

0 5 10 15 20

Pro

duct

ion

Cro

ss S

ectio

n [b

]

Neutron Energy [MeV]

GEANIE 4 Gammas sumBayhurst (1975)

CN spin-dist. (4 Gammas)CN spin-dist. (Total)

FKK spin-dist. (4 Gammas)FKK spin-dist. (Total)

Isomeric state production cross section for 193Ir.

Many aspects involved in Compound Reactions

•• foramlism of compound nuclearreactions

•• experimental technique, includingdirect / indirect methods

•• surrogate reaction technique•• microscopic descriptions of

nuclear properties•• time-dependent simulations for

dynamical compound nuclearreaction process

•• non-equilibrium process•• strong connection with applications•• and more ...

CNR* 2007(+2n): an ideal place for exchanging our expertise

HF Theory: Challenges for Future Development

Digging into better modeling / parameters in Hauser-Feshbach Modelmicroscopic descriptions•• neutron capture off-stability, fission, reactions on excited state

improved systematics•• reduce uncertainties•• better prediction for unknown reaction cross section

Beyond Cross Sectionsapplication to other nuclear processes•• new approach — Monte Carlo (this talk)

•• more sensitive to nuclear structure•• coincidence, correlation

•• new applications•• gamma-ray strength function•• level density•• fission neutron•• event generator in transport simulations

Two Implementations for MCHF at LANL

Computer Programs — CoH3 and CGMCoH3 + ECLIPSE (talks at CNR2009 and SNA&MC 2010)

•• general Hauser-Feshbach and pre-equilibrium calculation code•• generate decay probabilities P for all reaction channels first,

P (cn, kn, cm, km) =T (cmkm → cnkn)∑

cmkm T (cmkm → cnkn)

•• then, Monte Carlo calculations are done with another code, ECLIPSE•• calculation fast, but angular momenta conserved only in an average sense

TK, P. Talou, et al. J. Nucl. Sci. Technol. 47, 462 (2010)

CGM (present work)

•• calculate compound nucleus decay by using both deterministic and stochastic (MC)methods, with very fine energy grid

•• include neutron and γ-ray channels only, but conserve spin and parity•• multi-neutron emission, with non-constant energy grid

(just technical, but will be important)

CGM — Cascading Gamma-ray and Multiplicity

CGMGamma-ray cascade simulationBeta-decay calculation

Nuclear MassesAW & FRDM

Statistical DecayParticle TransmissionGamma-ray Transmission

Entrance ChannelOptical Model(transmission generator)

RIPL-3Discrete Levels

Level DensityGilbert-Cameronparameters

Spectra of gamma-ray, neutron,electron, and neutrinofor beta-decay

QRPAGamow-TellerStrength

ENSDFbeta-decay tolow-lying states

Spectra of gamma-ray and neutron,and multiplicities from a given state deterministic, or Monte Carlo

CGM (about 70% of the code are from CoH3) was developed

(a) for studying β-delayed neutron and γ emission,(b) as an event generator in a transport code (MCNP6), and(c) a Monte Carlo approach to the prompt fission neutron spectrum (talk by P. Talou).

Neutron, Gamma-ray Emission Probability

Z, A Z, A-1

Sn

E1Ex

E0

gamma-ray emission

P (�γ)dE0 =Tγ(Ex − E0)ρ(Z, A, E0)

NdE0

neutron emission

P (�n)dE1 =Tn(Ex − Sn − E1)ρ(Z, A − 1, E1)

NdE1

where Tn,γ are the transmission coefficients, ρ(Z, A, E) isthe level density, and the normalization N is given by

N =∫ Ex0

Tγ(Ex − E0)ρ(Z, A, E0)dE0

+∫ Ex−Sn0

Tn(Ex − Sn − E1)ρ(Z, A − 1, E1)dE1

•• integration performed only for spin and parityconserved states

•• at low excitation energies, discrete level data are used(taken from RIPL-3)

Monte Carlo Hauser-Feshbach Method

Z, A Z, A-1

S (A)n

(c)

(b)

(a)

Z, A-2

S (A-1)n

(d)

Tot

al E

xcita

tion

Ene

rgy

Algorithm in CGM•• starting at (Z, A, E0), P (�n) and P (�γ)

are calculated•• choose a next state (Z, A − 1, E1)

by a random sampling method•• repeat this until the state reaches

at a discrete level•• each time P ’s are re-calculated

•• it is faster if all the P ’s are calculatedat the beginning, but the memory sizecan be ∼ GByte

•• at a discrete level, do Monte Carlogamma-ray cascade based onbranching ratios in RIPL-3

Gamma-Ray Energy and Multiplicity

238U + n (Eth), γ-ray production probabilities

•• E1, M1, and E2 are included•• m = 4.77•• �γ = 1.01 MeV

•• M1 added•• Eγ = 2 MeV, Γγ = 0.6,

σ0 = 1.2 mb (assumed)•• m = 4.52•• �γ = 1.06 MeV

Gamma-Ray Energy Spectra for n+U238

Looking for pygmy resonance / scissors mode

0.001

0.01

0.1

1

10

100

0 1 2 3 4 5

Gam

ma-

Ray

Spe

ctra

[1/M

eV]

Gamma-Ray Energy [MeV]

without scissors modewith scissors mode

0.001

0.01

0.1

0 1 2 3 4 5

Gam

ma-

Ray

Spe

ctra

[1/M

eV]

Gamma-Ray Energy [MeV]

without scissors modewith scissors mode

Total Energy Spectra Spectra for m = 2

4π-calorimeter experiments like DANCE, and high-intensity γ-ray source like HIγS atTUNL are able to identify these dipole resonances(priv. comm. M. Krtička, J. Ullmann, A. Tonchev).

Gamma-Ray Spectra w/o Neutron Competition154Gd below and above Sn

Jπ = 1−,2− Jπ = 5−,6−

0.001

0.01

0.1

1

10

100

0 0.2 0.4 0.6 0.8 1

Gam

ma-

Ray

Spe

ctra

[/M

eV d

ecay

]

Gamma-Ray Energy [MeV]

100 keV above Snbelow Sn

0.001

0.01

0.1

1

10

100

0 0.2 0.4 0.6 0.8 1G

amm

a-R

ay S

pect

ra [/

MeV

dec

ay]

Gamma-Ray Energy [MeV]

100 keV above Snbelow Sn

Gamma-Ray Spectra Depends on Parity Distribution

0.001

0.01

0.1

1

10

100

0 0.2 0.4 0.6 0.8 1

Gam

ma-

Ray

Spe

ctra

[/M

eV d

ecay

]

Gamma-Ray Energy [MeV]

100 keV above Sneven parity 80%even parity 20%

154Gd above Snparity distribution important•• odd (negative) parity at

neutron capture state•• parity flips by E1 transition•• γ-ray multiplicity = 2 – 3•• fewer even parity states

suppress γ branching•• an exact parity distribution in

the continuum unknown

Variable Bin Width Calculation

Behavior of low energy neutrons•• constant energy-bin

•• calculations faster•• no information on neutrons when energies are

less than ∆E•• variable energy-bin

•• slower, algorithm becomes complicated•• gives correct spectrum shape at low energies

neutron and gamma-rays from 137Xe at 10 MeV

0.0001

0.001

0.01

0.1

1

10

0.001 0.01 0.1 1 10

Spe

ctru

m [/

MeV

]

Secondary Energy [MeV]

gamma-ray (x1000)neutron

0.0001

0.001

0.01

0.1

1

10

0 10

Spe

ctru

m [/

MeV

]

Secondary Energy [MeV]

gamma-ray (x1000)neutron

Z, A Z, A-1

S (A)n

Z, A-2

S (A-1)nTot

al E

xcita

tion

Ene

rgy

Z, A-3

S (A-2)n

low energy neutrons comefrom all the every compounddecay stages

Evaporation (Weisskopf) or Maxwellian ?

Asymptotic form at very low energies

•• Evaporation: fE(�) = A� exp(−�/T )•• f ′E(� → 0) = 1•• fE(�) ∼ � for � → 0•• Maxwellian: fM(�) = A√� exp(−�/T )•• f ′M(� → 0) = 12√�•• fM(�) ∼ √�/2 for � → 0•• Watt: fW (�) = A sinh(√B�) exp(−�/T )•• f ′W (� → 0) =

√B

2√

�•• fW (�) ∼ √�/2 for � → 0 0.001

0.01

0.1

1

0.01 0.1 1 10

Spe

ctra

[arb

. uni

t]

Emission Energy [arb. unit]

MaxwellianEvaporation

Watt

from Hauser-Feshbach

•• s-wave neutron transmission coefficient T0 = 2πS0 ∝ √�•• level density is assumed to be constant within a small energy width•• fHF (�) ∝ T0ρ(Ex) = C√� for � → 0

Comparison with CGM No-Cascade Mode

0.0001

0.001

0.01

0.1

1

10

0.001 0.01 0.1 1 10

Neu

tron

Spe

ctru

m [/

MeV

]

Secondary Neutron Energy [MeV]

10 MeV

CGMMaxwellian

EvaporationWatt

0.0001

0.001

0.01

0.1

1

10

0.001 0.01 0.1 1 10

Neu

tron

Spe

ctru

m [/

MeV

]

Secondary Neutron Energy [MeV]

15 MeV

CGMMaxwellian

EvaporationWatt

0.0001

0.001

0.01

0.1

1

10

0.001 0.01 0.1 1 10

Neu

tron

Spe

ctru

m [/

MeV

]

Secondary Neutron Energy [MeV]

20 MeV

CGMMaxwellian

EvaporationWatt

0.0001

0.001

0.01

0.1

1

10

0 2 4 6 8 10

Neu

tron

Spe

ctru

m [/

MeV

]

Secondary Neutron Energy [MeV]

10 MeV

CGMMaxwellian

EvaporationWatt

0.0001

0.001

0.01

0.1

1

10

0 2 4 6 8 10

Neu

tron

Spe

ctru

m [/

MeV

]

Secondary Neutron Energy [MeV]

15 MeV

CGMMaxwellian

EvaporationWatt

0.0001

0.001

0.01

0.1

1

10

0 2 4 6 8 10

Neu

tron

Spe

ctru

m [/

MeV

]

Secondary Neutron Energy [MeV]

20 MeV

CGMMaxwellian

EvaporationWatt

The evaporation spectrum does not give a correct spectrum shape in the low energy region.

The Watt spectrum better describes the Hauser-Feshbach spectrum (but in CMS).

Watt Spectrum?

Asymptotic Gamma-Ray Energy Spectra

•• from Hauser-Feshbach•• transmission coefficient — E1 assumed

T (�) = C�3

•• level density — constant temperatureρ(Ex) =

1

Texp

(Ex

T

)•• spectrum will be

f(�) = T (�)ρ(Ex) = C′�3 exp

(−

�

T

)•• Lemaire et al. (Phys. Rev. C 73, 014602 (2006))

f(�) =�2

2T2exp

(−

�

T

) 0.0001

0.001

0.01

0.1

1

10

0.01 0.1 1 10

Gam

ma-

Ray

Spe

ctra

[arb

. uni

t]

Gamma-Ray Energy [MeV]

E2 exp(-E/T)E3 exp(-E/T)

CGM

Sequential Neutron Emission

Neutron and gamma-ray emission spectra from excited 140Xe

•• initial spin distribution by the level density•• 100,000 events — 1 ∼ 2 hours on a laptop computer

0.001

0.01

0.1

1

10

0 1 2 3 4 5 6 7 8 9 10

Spe

ctru

m [/

MeV

]

Secondary Energy [MeV]

10 MeV first neutronsecond neutron

gamma-ray

0.001

0.01

0.1

1

10

0 1 2 3 4 5 6 7 8 9 10

Spe

ctru

m [/

MeV

]

Secondary Energy [MeV]

15 MeV first neutronsecond neutron

gamma-ray

0.001

0.01

0.1

1

10

0 1 2 3 4 5 6 7 8 9 10

Spe

ctru

m [/

MeV

]

Secondary Energy [MeV]

20 MeV first neutronsecond neutron

third neutrongamma-ray

10 MeV 15 MeV 20 MeV�γ = 0.87 MeV 0.89 MeV 1.06 MeV�n = 1.37 MeV 1.44 MeV 1.48 MeV

Correlation Between First and Second Neutrons

Energy correlation in the emitted neutrons from excited 140Xe

10 MeV 15 MeV 20 MeV

•• the joint probability normalized to [decay, n-MeV, γ-MeV]−1•• patterns shown are due to discrete levels in the residual nucleus

Correlation Between Gamma Energy and Neutrons

Energy correlation between total γ-ray energy and neutrons fromexcited 140Xe

10 MeV 15 MeV 20 MeV

•• the joint probability normalized to [decay, n-MeV, γ-MeV]−1

Neutrons and Gamma-Ray Multiplicity Correlation

Neutron spectra from 140Xe∗ for each gamma-ray multiplicity

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8 9 10

Pro

babi

lity

Multiplicity

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8 9 10

Pro

babi

lity

Multiplicity

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8 9 10

Pro

babi

lity

Multiplicity

Initial Spin Distribution Important

Multiplicity depends on average spin in CN (σ2 doubled case)

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8 9 10

Pro

babi

lity

Multiplicity

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8 9 10

Pro

babi

lity

Multiplicity

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5 6 7 8 9 10

Pro

babi

lity

Multiplicity

Conclusion

MCHF: Monte Carlo Hauser-Feshbach Method

•• In this study we performed Monte Carlo simulationsfor neutron and γ-ray emissions.

•• CGM: Monte Carlo Hauser-Feshbach code developed at LANL•• The evaporation spectrum does not give a correct asymptotic shape at low energies,

which should be√

�.•• Correlated neutron - gamma-ray emission from excited nucleus;

with the MCHF technique it is possible to calculate:

•• correlated neutron and γ-ray emissions•• neutron energy spectra for individual gamma-ray multiplicity

Perspective

•• Neutron and γ-ray generator in a transport simulation•• radiation shielding, detector efficiency simulation, etc.

•• More detailed comparison with experimental data•• MCHF method sensitive to nuclear structure

Level Density Parameter Systematics

0

10

20

30

40

50

0 50 100 150 200 250 300

Leve

l Den

sity

Par

amet

er [M

eV-1

]

Mass Number

a(from D0 data)a(asymptotic)

Least-Squares Fit

Washing-out of shell effects•• shell correction (δW ) and pairing

energies (∆) taken fromKTUY05 mass formula

a = a∗{1 +

δW

U

(1 − e−γU

)}

a∗ = 0.126A + 7.52 × 10−5A2

•• at low excitation energies, the constanttemperature model is used with

T = 47.1A−0.89√

1 − 0.1δW

•• obtained from discrete level data ofmore than 1000 nuclei

TK, S. Chiba, H. Koura, J. Nucl. Sci. Technol., 43, 1 (2006)

and updated parameters by TK in 2009

Gamma-Ray Strength Function and Transmission

•• Standard LorentzianfE1(�γ) = Cσ0Γ0

�γΓ0(�2γ − E20)2 + �2γΓ

20

•• Generalized Lorentzian, finite value at low energies, energy dependent widthfE1(�γ) = Cσ0Γ0

�γΓ(�γ, T )(�2γ − E20)2 + �2γΓ2(�γ, T ) + 0.7Γ(�γ = 0, T )

E30

where C = 8.68 × 10−8 mb−1MeV−2

•• in CGM, E1, M1, and E2 are considered•• pygmy resonance and scissors mode can beincluded if necessary•• γ-ray transmission coefficient is given by

Tγ(�γ) = 2π∑m

E2L+1γ fm(�γ)

10-10

10-9

10-8

10-7

10-6

0 5 10 15 20 25 30

Str

engt

h F

unct

ion,

f(E

)

E [MeV]

Generalized LorentzianStandard Lorenzian