Challenges beyond Hauser-Feshbach for Nuclear …...FRDM HF-BCS T. Kawano (T-2, LANL) Challenges beyond Hauser-Feshbach for Nuclear Reaction ModelingOct. 14–17, 2014 13 / 23 Combining

LA-UR-14-27974

Challenges beyond Hauser-Feshbachfor Nuclear Reaction Modeling

T. Kawano

T-2, LANL

Oct. 14–17, 2014

in collaboration withP. Talou (LANL/T2), H. Weidenmuller (MPI),

L. Bonneau (CENBG), and S. Kunieda (JAEA)

T. Kawano (T-2, LANL) Challenges beyond Hauser-Feshbach for Nuclear Reaction ModelingOct. 14–17, 2014 1 / 23

Introduction

Introduction

Cross section calculations in the keV to a few tens MeV range with theHauser-Feshbach codes

Optical and statistical Hauser-Feshbach models with width fluctuationand pre-equilibrium emission play a central role.

width fluctuation models implemented in HF codes give somedifference in calculated cross section.

In general, we may say, the model works pretty well.the Hauser-Feshbach codes, like GNASH, TALYS, Empire, CCONE,CoH3, etc., have been successfully utilized in nuclear data evaluationfor many years.although many phenomenological or empirical treatments of modelparameters are involved.


Introduction

Issues Partially Solved, or Remain

Indeed, we calculate cross sections, but in a proper way?

Deformed system needs more attentiondirect inelastic scattering process in HF not so well studied

Uncertainty in the photon and fission channels could be largephoton transmission coefficient from the Giant Dipole Resonancemodelfission transmission with a simple WKB approximation

Phenomenological model for pre-equilibrium process still widely usedlong discussions on quantum mechanical models for PE, which werevery active in 1990s, basically disappeared, becauseThe exciton model reasonably works in nuclear data evaluation.

... and more


Introduction

Table of Contents

This talk (or showcase) includes:

Applicability of the Hauser-Feshbach theory itselfwidth fluctuation correction with direct reaction

Model parameters for practical calculationsincorporating nuclear structure models into the reaction calculations

Perturbative contributions from the other reaction mechanismspre-equilibrium, direct/semidirect capture calculation


Statistical Theory

Width Fluctuation Correction for Spherical Nuclei

The problematic assumption in the Hauser-Feshbach formula⟨ΓµaΓµb

Γµ

⟩=

⟨Γµa⟩ ⟨

Γµb⟩⟨

Γµ⟩ (1)

This leads to the width fluctuation correction (WFC)

〈σflab〉 =

TaTb∑c Tc

Wab (2)

Rigorously speaking, Wab should be separated into two parts

the elastic enhancement factor Wa

the width fluctuation correction factor


GOE with Direct Reaction

Stochastic Scattering Matrix

Generate Resonances with the Random Matrix

S ab(E) = δab − 2πi∑µν

WaµD−1Wνb (3)

Dµν = Eδµν − HGOEµν + πi

∑c

WµcWcν (4)

HGOEµν HGOE

ρσ =1N

(δµρδνσ + δµσδνρ) (5)

Ta given by eigenvalues of WWT

poles are distributed in [−2, 2]

energy average 〈|S aa|2〉 replaced by ensemble average |S aa|

2.



Monte Carlo Generated Cross Sections

Physical Mathematical

0

5

10

1 1.2 1.4 1.6 1.8 2

Elas

tic S

catte

ring

[b]

Incident Neutron Energy [MeV]

(a)ENDF/B-VII.1

0.001

0.01

0.1

1

10

-2 -1 0 1 2

|1-S

|2

Energy [in unit of lambda]

(b)

0

5

10

1 1.2 1.4 1.6 1.8 2

Elas

tic S

catte

ring

[b]

Incident Neutron Energy [MeV]

(b)Statistical R-matrix

0.001

0.01

0.1

1

10

-2 -1 0 1 2|1

-S|2

Energy [in unit of lambda]

(c)

Statistical R-matrix includes distributions of d and γ,while GOE has a random matrix in the propagator.



Inclusion of Direct Channel

Approximated Methodcalculate transmissions from Coupled-Channels S-matrix

Ta = 1 −∑

c

|〈S ac〉〈S ∗ac〉|2

eliminate flux going to the direct reaction channelsat least

∑a Ta gives correct compound formation cross section

HF performed in the direct-eliminated cross-section space

Engelbrecht-Weidenmuller transformationdiagonalize S-matrix to eliminate the direct channelsHF performed in the channel spacetransform back to the cross section space

Kawai-Kerman-McVoycorrect at the limit of channel d-o-f ν = 2.0



Implementation of Direct Channel in Stochastic S-Matrix

S ab = S (0)ab − i

∑µ

γµaγµb

E − Eµ − (i/2)Γµ(6)

Since S (0)ab is unitary, it can be diagonalized by the orthogonal

transformation. However, making a unitary matrix S (0)ab including

off-diagonal elements is not so easy.Instead, we employ a K-matrix method.

Kab(E) = K(0) +∑µ

WaµWµb

E − Eµ. (7)

where the background term K(0) is a model parameter. When K is real andsymmetric, unitarity of S is automatically fulfilled.



Generated Elastic/Inelastic Cross Sections

Fixed resonances, background component Kab changed from 0 to 2N = 100, Λ = 2

Elastic Inelastic

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2Energy [in unit of lambda] 0

0.5

1

1.5

2

Background k 0

0.5 1

1.5 2

2.5 3

3.5 4

|1-S|2 0 0.5 1 1.5 2 2.5 3 3.5 4

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2Energy [in unit of lambda] 0

0.5

1

1.5

2

Background k 0

0.2 0.4 0.6 0.8

1

|S|2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Inelastic scattering cross sections affected by the direct reaction stronglydue to the interference between the resonances and the background term.



Inelastic Scattering Enhancement

Compound inelastic scattering crosssection as a function of σDI/σR

0

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15

Com

poun

d In

elas

tic C

.S.

Direct Inelastic C.S. / Total Reaction C.S.

Λ=2

Λ=10

(a) s = 1

MC simulationmodified transmission

EW transformation

The approximation methodusing the modifiedtransmission coefficientsdoes not work when thedirect channels are strong,

since the compoundinelastic scattering crosssections will be largelyunderestimated.This happens when

direct cross section isstrongthe number of openchannels small


Combining Nuclear Structure Models

Limited Nuclear Structure Information

Nuclear structure information is involved in the Hauser-Feshbachcalculation

low-lying discrete state Ex, Jπ, and γ-ray branching rationcan be tabulated in a supplemental fileIAEA maintains RIPL

quantities calculated with nuclear mean-field theoriesground state deformation β2, β4 (also in RIPL)single-particle levels and occupation probabilitiesfission barriers

A natural extension of our Hauser-Feshbach codes is to include somesimple nuclear mean-field models.



Mean-Field Models Added to CoH3

197Au

FRDM Finite-Range Droplet ModelHF-BCS Skyrme Hartree-Fock BCS Model

clone of Bonneau’s code, rewritten in C++

0

5

10

15

20

R [fm]

-10

-5

0

5

10

Z [fm]

-80

-70

-60

-50

-40

-30

-20

-10

0

0

5

10

15

20

R [fm]

-10

-5

0

5

10

Z [fm]

-80

-70

-60

-50

-40

-30

-20

-10

0

FRDM HF-BCS



Combining FRDM and HF-BCS

Use FRDM potential as an initial potential in the Hartree-Fock iteration

Pt-194

macro energy micro energy total

iteration converges quicklycan avoid being trapped by local minimaif spherical WS is used, HF iteration can go either prolate / oblateshape depending on the initial conditionT. Kawano (T-2, LANL) Challenges beyond Hauser-Feshbach for Nuclear Reaction ModelingOct. 14–17, 2014 14 / 23


Direct/SemiDirect (DSD) Capture with HF-BCS

DSD Amplitudes

Td ∝√

S l jK〈Rl jK(r)|r|RLJ(r)〉 Ts ∝√

S l jK〈Rl jK(r)|h(r)|RLJ(r)〉 (8)

DSD with Hartree-Fock BCS Theoryspectroscopic factor S l jK

single-particle occupation probabilities,v2 = 1 − u2

single-particle wave-function, Rl jK(r)

HF-BCS calculation and decompositioninto spherical HO basis

consistent treatment for all nuclei fromspherical to deformed nuclei

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16 18 20

Cro

ss S

ectio

n (m

b)

Incident Neutron Energy (MeV)

238U(n,γ)Drake (1971)

McDaniels (1981)DSD(SLy4)

Bonneau, TK, et al. PRC 75, 054618 (2007)



Coupled-Channels Calculation with FRDM

Expansion of FRDM shape by Spherical Harmonics

r(θ, φ) = R0 + R0

∑l=1

l∑m=−l

βlmYml (θ, φ) (9)

165Ho (ε2 = 0.267, ε4 = 0.02, ε6 = 0.017)

' (β2 = 0.291, β4 = 0.009, β6 = −0.0197, β8 = −0.006, . . .) (10)

We cannot use FRDM potential directly as a real part of opticalpotential (R0 tends to be smaller)

Use the deformation for the global CC potential by Kunieda et al.



FRDM + Kunieda Potential Calculation, Ho-165

4000

6000

8000

10000

12000

14000

0.01 0.1 1 10

Tot

al C

ross

Sec

tion

[mb]

Neutron Incident Energy [MeV]

Meadows 1971Foster 1971

Marshak 1970Pineo 1974Kellie 1974Fasoli 1973Fasoli 1969Islam 1973

KuniedaKoning-Delaroche

10

100

1000

10000

0.001 0.01 0.1 1 10

Cap

ture

Cro

ss S

ectio

n [m

b]


Czirr 1973Poenitz 1975

Gibbons 1961Block 1961

Bensch 1971Lepine 1972

Bokhovko 1991Kunieda

Koning-Delaroche

The calculated capture cross section also depends on other quantities,such as level density, photon-strength function. This is just an example!



FRDM + Kunieda Potential Calculation, Sm-152

0

5000

10000

15000

20000

0.01 0.1 1 10

Tot

al C

ross

Sec

tion

[mb]


Pineo 1974Kellie 1974

Sherwood 1970Foster 1971

KuniedaKoning-Delaroche

JENDL-4

1

10

100

1000

0.001 0.01 0.1 1 10C

aptu

re C

ross

Sec

tion

[mb]


Macklin 1957Lyon 1959

Macklin 1963Bensch 1971

Peto 1967Chaubey 1966

SiddappaZhou 1984

Luo 1994Tan 2008

Kononov 1977Trofimov 1987Trofimov 1987

CoH3with M1

It turned out that capture is sensitive to a small M1 photon strength at lowenergy — scissors/twist mode(E = 3 MeV, Γ = 3 MeV, and σ0 = 0.5 mb).

J. L. Ullmann et al. PRC 89, 034603 (2014); M. Guttormsen et al. PRC 89, 014302 (2014)



Strutinsky Method for Single-Particle State Density

Level (state) densities in the pre-equilibrium process

ω =gn(E − Aph)n−1

p!h!(n − 1)!

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

0 50 100 150 200

Sin

gle

Par

ticle

Ene

rgy

[MeV

]

Number of Levels

N = 126

208Pb single-particle levels

0

2

4

6

8

0 20 40 60 80 100 120 140

s.p.

sta

te d

ensi

ty [/

MeV

]

Z, N

neutronproton

(Z,N) / 15(Z,N) / 20

g is given by the tangent at EF in the FRDMstrutinsky method



Mean-Field Single-Particle States for MSC/MSD

The quantum mechanical pre-equilibrium calculations benefit from themean-field models — a smooth transition of DWBA from discrete finalstates to a continuum.

They tend to be lengthy calculations, and not so practical in thenuclear data evaluation.Simplification applied

TUL in Empire, or Koning and Akkermans [PRC 47, 724 (1993)]random sample of particle-hole pairs by Kawano and Yoshida (2001).

There is very little progress in this area nowadays, except for at CEA.

Dupuis first time implemented QRPA in MSD.



P-H Excitation for One-Step MSD Reaction

d2σba

dEdΩ=

(2π)4

k2a

∑µ

|〈χ(−)b umµ|V|χ

(+)a u0〉|

2ρmµ(Ex) (11)

Unperturbed State Density from FRDM / HF

ρ(0)m (E) =

∑µ

δ(E − εmµ) (12)

Exciton State Density for fixed Jπ

ρm(E) = −∑µ

1π

Im1

E − εmµ − σm(E)(13)

Saddle Point Equation with Second MomentM of M.E.

σm(E) =∑

n

Mmn

∫ρ(0)

n (ε)1

E − ε − σn(E)dε (14)

TK, S. Yoshida, PRC, 64, 024603 (2001)



Calculated One-Step MSD DDX208Pb(n, n′) reaction at Ein = 14.5, Eout = 7.5 MeV

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0 5 10 15

Mm

n [M

eV2 ]

J

M22

M44

M66

M88

M24

M46

M68

100

101

0 30 60 90 120 150 180

d2 σ/dE

dΩ [m

b/sr

MeV

]

θ [deg]

Takahashi (1983)Gul (1985)

Elfruth (1986)M3Y Paris

1-Yukawa V0=70 MeV

TK, S.Yoshida, unpublished.


Conclusion

Concluding Remarks

Direct reaction channels should be properly included in theHauser-Feshbach codes.

Engelbrecht-Weidenmuller transformation, or KKM

Nuclear structure (mean-field calculation) inputwill reduce the number of phenomenological adjustable parametersThe M1 photon strength will be a key to handle spiky capture channel

Quantum mechanical pre-equilibrium processwe should shed light on this again for better understanding of neutroninelastic scattering from actinidesmaybe need more practical models for nuclear data evaluations


Documents

Challenges beyond Hauser-Feshbach for Nuclear …...FRDM HF-BCS T. Kawano (T-2, LANL) Challenges beyond Hauser-Feshbach for Nuclear Reaction ModelingOct. 14–17, 2014 13 / 23 Combining