LA-UR-22-30081 Collective enhancement in the exciton model M. R. MumpowerD. Neudecker H. Sasaki T. Kawano A. E. Lovell M. W. Herman and I. Stetcu

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LA-UR-22-30081

Collective enhancement in the exciton model

M. R. Mumpower,∗D. Neudecker, H. Sasaki, T. Kawano, A. E. Lovell, M. W. Herman, and I. Stetcu

Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

M. Dupuis

CEA, DAM, DIF, F-91297 Arpajon, France and

Universit´e Paris-Saclay, CEA, Laboratoire Mati`ere sous Conditions Extrˆemes, 91680 Bruy`eres-Le-Chˆatel, France

(Dated: October 24, 2022)

The pre-equilibrium reaction mechanism is considered in the context of the exciton model. A

modiﬁcation to the one-particle one-hole state density is studied which can be interpreted as a

collective enhancement. The magnitude of the collective enhancement is set by simulating the

Lawrence Livermore National Laboratory (LLNL) pulsed-spheres neutron-leakage spectra. The

impact of the collective enhancement is explored in the context of the highly deformed actinide,

239Pu. A consequence of this enhancement is the removal of ﬁctitious levels in the Distorted-Wave

Born Approximation often used in modern nuclear reaction codes.

I. INTRODUCTION

Nuclear reaction modeling for strongly deformed nu-

clei remains an open challenge for contemporary theoret-

ical studies. Modern reaction codes separate the reac-

tion mechanisms into three broad categories. In a direct

reaction, the incident particle interacts on a fast time

scale with a single nucleon that generally resides near

the surface of the target system. The direct reaction

cross section evolves slowly as a function of incident par-

ticle energy [1]. In contrast, compound nucleus formation

occurs when a large number of nucleons participate for a

suﬃciently long enough time that a thermal equilibrium

ensues in the residual system [2]. This mechanism occurs

at low energies inside the volume of the residual system.

The cross section of this mechanism may vary strongly

with small change in the incident-particle energy.

Pre-equilibrium is the third, intermediate reaction

mechanism that embodies both direct- and compound-

like features. Pre-equilibrium reactions occur on a

longer timescale than a direct reaction but on a shorter

timescale than compound nucleus formation [3]. This

mechanism is characterized by an incident particle that

continually enables subsequent scattering. As the scat-

tering proceeds, increasingly more complex states are cre-

ated in the residual system with each successive process

gradually losing information contained in the initial reac-

tion. This reaction mechanism is important to consider

with highly energetic incident particles. If the residual

system has suﬃcient excitation energy, creation of sub-

sequent particles may be possible [4].

There are two distinct approaches to describe the

pre-equilibrium process for nucleon-induced reactions on

medium- to heavy- mass nuclei: purely quantum mechan-

ical models and phenomenological-based models. Quan-

tum mechanical models use the Distorted-Wave Born

Approximation (DWBA) for the multi-step process to

∗mumpower@lanl.gov

couple to the continuum in a residual nucleus. These

models adopt diﬀerent statistical assumptions, mainly

for the two-step process, where 2-particle-2-hole conﬁgu-

rations are created by the NN interaction. Examples

of quantum mechanical models are Feshbach-Kerman-

Koonin (FKK) [5], Tamura-Udagawa-Lenske (TUL) [6],

Nishioka-Weidenm¨uller-Yoshida (NWY) [7], and Luo-

Kawai [8].

Because the angular momentum conservation is prop-

erly included in these quantum mechanical models, they

better reproduce the γ-ray production data that are sen-

sitive to the spins of initial and ﬁnal states [9]. While

these models provide more fundamental insight into nu-

clear reaction mechanisms, the downside of their appli-

cation in nucleon-induced reactions is their high compu-

tational cost for the description of the relatively small

fraction of the total reaction cross section.

The second approach is phenomenological in nature.

An example is the exciton model which treats pre-

equilibrium scattering as a chain of particle-hole exci-

tations [10, 11]. In this context, the particle and hole

degrees of freedom are referred to collectively as excitons

and the exciton number for a single component system

is given by n=p+h. Transitions between particle-hole

conﬁgurations with the same exciton number, n, have

equal probability. The time-dependent master equations

controls the evolution of the scattering process through

transitions to more or less complex conﬁgurations. At

any step in this process an outgoing particle may be

emitted which is referred to as pre-equilibrium emission.

The time integrated solution provides the energy aver-

aged particle spectra. Central to the exciton model is

the set of particle-hole state densities that govern the

magnitude of the excitations. In particular, the relative

magnitude of the state densities are not fully constrained

by diﬀerential data.

A practical step forward is to combine both of these

approaches: feed the quantum mechanical calculations

to the exciton model. For example, the angular mo-

mentum transferred to a 1-particle-1-hole conﬁguration

is calculated by FKK, and the spin distribution of the

arXiv:2210.12105v1 [nucl-th] 21 Oct 2022

populated ﬁnal states are parameterized in the exciton

model [9, 12, 13]. This technique enables a more realis-

tic spin transfer to the residual nucleus, while the whole

pre-equilibrium strength can be determined by the more

established exciton model framework.

Although this combined approach compensates deﬁ-

cient information of angular momentum transfer in the

exciton model, it is insuﬃcient to provide individual

contributions from diﬀerent particle-hole conﬁgurations

to the total pre-equilibrium energy spectrum. It is

known that deformed nuclei at relatively low excitation

energies show collective behavior, which can be evalu-

ated by the Quasi-particle Random Phase Approxima-

tion (QRPA) [14, 15], as shown by Kerveno et al. [9]. This

collective excitation can be interpreted as an eﬀective

enhancement in the partial state density for 1-particle-

1-hole conﬁgurations. Ergo, incorporating a collective

enhancement for the 1-particle-1-hole state density into

the exciton model may oﬀer better modeling of the en-

tire nuclear reaction occurring in highly deformed nuclei

such as the actinides. Crucially, this procedure can be in-

tegrated into the Hauser-Feshbach theory which follows

the statistical decay of the residual nucleus.

In this paper we study this combined practical tech-

nique. We propose an increase to the 1-particle-1-hole

state density used in the exciton model and include it

in the Los Alamos statistical model framework, CoH3

[16, 17]. We study the impact of this enhancement in the

context of neutron-induced reactions on 239Pu. We use

feedback from Lawrence Livermore National Laboratory

(LLNL) pulsed-sphere neutron-leakage spectra to set the

magnitude of the enhancement factor and ﬁnd that this

scale factor is signiﬁcantly above unity. We present the

changes to the cross sections in the results section and

summarize our ﬁndings in the ﬁnal section.

II. THEORY

A. Exciton model

We employ the two-component exciton model [18, 19],

which distinguishes neutron and proton in the particle-

hole conﬁgurations. This is denoted schematically in

Fig. 1. Since this model has been well established and ex-

tensively applied to particle emission data analysis, only

a brief description of some of the relevant parts of the

model is given below.

We denote the particle-hole conﬁguration by c, which

abbreviates the number of particles and holes in the

neutron and proton shells as c≡(pν, hν, pπ, hπ). We

also deﬁne the total number of excitons, nν=pν+hν,

nπ=pπ+hπ, and nt=nν+nπ. For a particle hav-

ing z-protons and n-neutrons emitted in output channel

b, the residual conﬁguration will be designated by cb,

that stands for pπ−zand pν−n. In the case of an

incident neutron on a target system with Z-protons and

N-neutrons, the composite system would be the nucleus

… To equilibrium …

3 excitons 5 excitons

Fermi Energy

FIG. 1. (Color Online) A schematic depiction of the ﬁrst

few stages of the 2-component exciton model from an initial

excitation with a neutron. The particles, in this case nucleons

(neutrons and protons), are shown as ﬁlled circles with holes

indicated by open circles. The solid lines represent equally

spaced single-particle states.

— before compound nucleus formation — (Z,N+1), and

the residual system might be (Z,N) after emission of the

neutron, e.g. in the case of inelastic scattering.

For the pre-equilibrium nuclear reaction, (a, b), with

input channel, a, and output channel, b, the emission

rate of the outgoing particle bis written as

Wb(c, E, b) = 2sb+ 1

π2~3µbσCN

b(b)b

ω(cb, U)

ω(c, E)fFW ,(1)

where Eis the total energy of the composite system,

Uis the excitation energy in the residual nucleus, and

ω(c, E) is the composite state density at the excitation

energy E. A commonly used step function, fFW , is em-

ployed to limit the hole state conﬁguration within the

potential depth [20]. The values of ,sb, and µb, denote

the emission energy, the intrinsic spin of particle b, and

the reduced mass respectively. The compound formation

cross section for the inverse reaction calculated by the

particle transmission coeﬃcient is σCN

b(b).

The pre-equilibrium emission takes place at diﬀerent

particle-hole conﬁgurations, which is characterized by the

occupation probability P(c) and its lifetime τ(c). The

observed energy-diﬀerential cross section is a convolution

of all the conﬁgurations

dσ

db

=σCN

a(a)X

P(c)τ(c)Wb(c, E, b),(2)

where σCN

ais the compound nucleus formation cross sec-

tion for channel a.

We employ the τ(c) calculation proposed by

Kalbach [21] and adopt the closed-form expression for

P(c). The most important ingredients of this model are

the single-particle state densities, g, and the eﬀective av-

erage squared matrix element M2for the two-body inter-

action. The eﬀective average squared matrix element is

considered as an adjustable model parameter in the exci-

ton model, and often phenomenologically parameterized

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LA-UR-22-30081CollectiveenhancementintheexcitonmodelM.R.Mumpower,D.Neudecker,H.Sasaki,T.Kawano,A.E.Lovell,M.W.Herman,andI.StetcuLosAlamosNationalLaboratory,LosAlamos,NM,87545,USAM.DupuisCEA,DAM,DIF,F-91297Arpajon,FranceandUniversiteParis-Saclay,CEA,LaboratoireMatieresousConditionsExtr^emes,91680B...

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LA-UR-22-30081 Collective enhancement in the exciton model M. R. MumpowerD. Neudecker H. Sasaki T. Kawano A. E. Lovell M. W. Herman and I. Stetcu

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