Calculation of Nuclear Reaction Cross Sections on Excited Nuclei with the CoupledChannels Method
ABSTRACT We calculate nuclear cross sections on excited nuclei in the fast neutron energy range. We partition the whole process into two contributions: the direct reaction part and the compound nuclear reactions. A coupledchannels method is used for calculating the direct transition of the nucleus from the initial excited state, which is a member of the groundstate rotational band, to the final ground and excited lowlying levels. This process is strongly affected by the channel coupling. The compound nuclear reactions on the excited state are calculated with the statistical HauserFeshbach model, with the transmission coefficients obtained from the coupledchannels calculation. The calculations are performed for a strongly deformed nucleus 169Tm, and selected cross sections for the ground and first excited states are compared. The calculation is also made for actinides to investigate possible modification to the fission cross section when the target is excited. It is shown that both the level coupling for the entrance channel, and the different target spin, change the fission cross section.
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 01/2006; Elsevier., ISBN: 044452035X

Article: Comprehensive nuclear model calculations: Introduction to the theory and use of the GNASH code
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ABSTRACT: A user's manual describing the theory and operation of the GNASH nuclear reaction computer code is presented. This work is based on a series of lectures describing the statistical HauserFeshbach plus preequilibrium version of the code with full angular momentum conservation. This version is expected to be most applicable for incident particle energies between 1 key and 50 MeV. General features of the code, the nuclear models that are utilized, input parameters needed to perform calculations, and the output quantities from typical problems are described in detail. The computational structure of the code and the subroutines and functions that are called are summarized as well. Two detailed examples are considered: 14MeV neutrons incident on â¹Â³Nb and 12MeV neutrons incident on Â²Â³â¸U. The former example illustrates a typical calculation aimed at determining neutron, proton, and alpha emission spectra from 14MeV reactions, and the latter example demonstrates use of the fission model in GNASH. 
Book: Direct Nuclear Reactions
01/1983; Clarendon Press.
Page 1
PHYSICAL REVIEW C 80, 024611 (2009)
Calculation of nuclear reaction cross sections on excited nuclei with the coupledchannels method
T. Kawano,*P. Talou, J. E. Lynn, M. B. Chadwick, and D. G. Madland
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
(Received 21 July 2009; published 31 June 2009)
We calculate nuclear cross sections on excited nuclei in the fast neutron energy range. We partition the whole
process into two contributions: the direct reaction part and the compound nuclear reactions. A coupledchannels
method is used for calculating the direct transition of the nucleus from the initial excited state, which is a member
of the groundstate rotational band, to the final ground and excited lowlying levels. This process is strongly
affected by the channel coupling. The compound nuclear reactions on the excited state are calculated with
the statistical HauserFeshbach model, with the transmission coefficients obtained from the coupledchannels
calculation. The calculations are performed for a strongly deformed nucleus169Tm, and selected cross sections
for the ground and first excited states are compared. The calculation is also made for actinides to investigate
possiblemodificationtothefissioncrosssectionwhenthetargetisexcited.Itisshownthatboththelevelcoupling
for the entrance channel, and the different target spin, change the fission cross section.
DOI: 10.1103/PhysRevC.80.024611 PACS number(s): 24.10.Eq, 24.50.+g, 24.60.Dr
I. INTRODUCTION
Understanding nuclear reactions on excited nuclei is of
physicalinterestforstudyingnucleosynthesis innuclear astro
physics. Nuclear reaction rates for astrophysical applications
are corrected by the stellar enhancement factor (SEF) [1],
taking account of the thermal excitation of the target [2]. In
a natural environment nuclear reactions normally take place
on stable nuclei, with the target nucleus in its ground state or
isomeric state. In the case of shape and spin isomers, because
the excited state hardly couples to the ground state, they can
be treated as if the target was in a ground state, but with
a straightforward modification to the reaction Q value. The
excitation energy of the isomers is released by impinging
neutrons, for example, via a compound nucleus reaction. It is
sometimes referred to as a superelastic process [3,4], because
the outgoing neutron is accelerated.
In a highdensity hightemperature neutron and γray
environment, such as neutron stars or supernovae, it is likely
that neutroninduced nuclear reactions occur on the excited
states, even though the lifetime of the excited state is often in
the order of a nanosecond or shorter. For example, the halflife
of the first excited state of169Tm is 4.08 ns, and that for239Pu
is 36 ps [5]. The nuclear reaction rates on the excited state
could be different from those on the ground state, primarily
because of the difference in phase space of levels accessible:
the spin and parity of the ground and excited states usually
differ, and the target excitation energy shifts the excitation
energyofthecompoundsystem.Thisprocesscanbecalculated
with a standard HauserFeshbach model [6] with an additional
excitation energy on the target nucleus.
When a target nucleus is strongly deformed, we observe a
rotational spectrum on each K band, and this modifies the
accessible phase space depending on which target state is
excited. In the case where the excited target is a member of
the groundstate rotational band, a direct reaction caused by
*kawano@lanl.gov
an incident neutron can deexcite the target without forming
a compound nucleus. In this direct inelastic scattering, the
incoming neutron gains energy during the process. The direct
reaction also may change the compound nuclear reaction
process, because the total compound formation cross section
and corresponding transmission coefficients differ if the target
nucleus is in an excited state.
We study the nuclear reactions on the excited nuclei in
the framework of the coupledchannels (CC) formalism [7,8].
The direct cross sections for the ground or excited states are
calculated by solving the coupled Schr¨ odinger equations, and
transmission coefficients for both ground and excited states
are obtained from the same scattering (S) matrix elements,
using a symmetry property of the S matrix. We calculate
nuclearreactionson169Tm,becauseofpropertiesthatsimplify
our calculations: there are precedent studies by Madland [9],
and Madland and Doolen [10], the target is well deformed
(β2= 0.3) [11], it does not fission by a fast energy neutron,
and charged particle emission channels can be ignored. The
calculations for169Tm are performed at low energies [below
the (n,2n) reaction threshold energy], where we expect a
constraint effect of target spin and partial wave angular
momentum couplings. We also perform studies of the fission
cross sections of239Pu using the HauserFeshbach model that
includes multiplechance fission in the higherenergy range.
The excited nuclei at lowlying states are produced in
various way, e.g., by neutron inelastic scattering, γ ray
absorption, neutron radiative capture, inverse internal conver
sion, and so on. For example, the nuclear reaction process
during ICF (inertial confinement fusion) may involve neutron
interactions on the excited nuclei. It should be emphasized
that measurements of nuclear reaction cross sections on
very shortlived targets are extremely difficult, or impossible,
within an acceptable uncertainty, so that prediction of these
cross sections by model calculations is essential. The method
described in this study is general, and it can be applied to
any medium/heavy nucleus whenever knowledge of nuclear
reactions on the excited target is crucial in applications. This
may provide some insight on possible modifications to the
05562813/2009/80(2)/024611(9)0246111©2009 The American Physical Society
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KAWANO, TALOU, LYNN, CHADWICK, AND MADLANDPHYSICAL REVIEW C 80, 024611 (2009)
nuclear reaction rates on important waiting point nuclei in
nucleosynthesis.
II. THEORY
A. Direct nuclear reaction
The formulation of the CC method is given in Refs. [7,8].
Solving a set of coupled equations gives a scattering matrix
element SJ?
numbers of the incoming/outgoing wave (l,s,j), J? the total
spin and parity, and the target state n that designates the spin
Inand parity πn. n = 0 stands for the ground state, n = 1 is
for the first excited state, the quantities with a prime are for
the exit channel, and so on.
We limit ourselves to consider neutroninduced reactions.
Because the scattering matrix element SJ?
direct cross sections from any nth level can be calculated by
cc? , where the channel c is specified by quantum
cc? is symmetric, the
σ(n)
R=π
k2
n
?
?
J?
?
?
c
δnc,ngJ
?
?
1 −
?
c?
??SJ?
cc?
??2
??2δnc?n?,
?
,
(1)
σ(nn?)
D
=π
k2
n
J?
c
δnc,ngJ
c?
??δcc? − SJ?
cc?
(2)
where gJis the spin factor given by
gJ=
2J + 1
(2s + 1)(2In+ 1),
(3)
where s is the intrinsic spin of neutron, knis the incoming
wave number, ncis the index of the excited level to which the
channel c belongs, σ(n)
cross section when the target is at the nth level, and σ(nn?)
the direct inelasticscattering cross section from the nth to the
n?th level. For a special case, n = n?is for the shape elastic
scattering σ(n)
in terms of the Smatrix elements, or more conveniently,
Ris the reaction (compound formation)
D
is
E. The total cross section σ(n)
T
can be expressed
σ(n)
T
= σ(n)
R+
?
n?
σ(nn?)
D
.
(4)
Figure 1 shows a scheme of CC calculation for the
groundstate rotational band [(1/2)+(3/2)+(5/2)+(7/2)+],
FIG. 1. An example of coupledchannels calculation for the
excited state. The target is at the first (3/2)+state, which is a member
of groundstate rotational band. The coupled equations give a set of
direct cross sections and the compound formation cross section. In
this case the direct transition to the (3/2)+level is elastic scattering
and to other levels is inelastic scattering.
when the target is at the (3/2)+level. In this case the elastic
scattering is a direct transition to the first excited (3/2)+level.
The direct process to the ground state (1/2)+is one of the
inelasticscattering channels, although the outgoing neutron
energybecomeshigherthantheincidentneutronenergy,which
is often called superelastic (or sometimes superinelastic). In
this article we regard this as one of the inelasticscattering
process. Not only the direct reaction but also the compound
process to the ground state, A∗+ n → (A + 1)∗→ A + n?,
releases the target excitation energy too.
As the reaction cross section σRis calculated in Eq. (1), the
generalized transmission coefficient T(n)
the probability of formation of compound nucleus on the nth
state by a neutron having the orbital angular momentum and
spin of l,j, is given by
?
lj, which is defined as
T(n)
lj
=
?
J?
?
c
2s + 1
2jc+ 1gJ
1 −
?
c?
??SJ?
cc?
??2
?
δnc,nδlc,lδjc,j,
(5)
where δlc,lis the Kronecker delta. Equation (5) gives a partial
wave contribution to the total reaction cross section as
?
The transmission coefficient in Eq. (5) is the one we can use
in the HauserFeshbach model. In many previous calculations
Tljin the HauserFeshbach model is calculated for the ground
state, and the decay channel transmission coefficients T(n)
are replaced by the ground state T(0)
energy, T(n)
correct T(n)
involved in our HauserFeshbach calculations.
σ(n)
R=π
k2
n
lj
2j + 1
2s + 1T(n)
lj.
(6)
lj
lj
calculated at a shifted
lj(E) = T(0)
ljfor all excited states, such an approximation is not
lj(E − Ex). Because our method gives a
B. Compound nuclear reaction
The HauserFeshbach theory [6] needs to be modified
when the number of open channels is small [12]. This
width fluctuation correction has been studied for the spherical
nucleus case [13–16]. Two independent computational studies
ofIgarasi[17]andHilaire,Lagrange,andKoning [18]showed
that the integration method with Monte Carlo simulation for
the channel degreeoffreedom ν by Moldauer [14] gives
almost an identical result to the most exact solution by
Verbaarschot et al. of the energyaveraged Smatrix element
using the Gaussian orthogonal ensemble (GOE) [16].
Kawai, Kerman, and McVoy (KKM) [19] formulated the
compound nuclear reaction in terms of the CC method, in
which the nuclear deformation is automatically taken into
account. In KKM, the generalized transmission coefficient in
Eq. (5) is not used, but the compound cross section is defined
in terms of a penetration matrix P, which is calculated from
the S matrix as [20]
Pcc? = δcc? −
?
c??
Scc??S∗
c??c?.
(7)
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CALCULATION OF NUCLEAR REACTION CROSS ...
PHYSICAL REVIEW C 80, 024611 (2009)
Other techniques deal with the direct channels in the com
pound reaction by a unitary transformation (Engelbrecht
Weidenm¨ uller transformation) [21,22] to diagonalize the S
matrix. A method of Nishioka, Weidenm¨ uller, and Yoshida
[23] is a natural extension of the GOE triple integral to
the penetration matrix, which is, however, not practical for
numerical calculations. We have shown [24] that the KKM
theory gives very similar cross sections to that of Moldauer’s
method, and we do not expect a large modification to the
calculated result because this is a correction to the width
fluctuation correction factor (inclusion of direct channel,
which is typically about ∼5–10% contribution to the total
scattering.) Therefore, in this study, the compound cross
section is not calculated from Eq. (7), but the spinaveraged
form in Eq. (5).
In our calculations, therefore, the width fluctuation cor
rection to the HauserFeshbach theory is calculated by the
Moldauer’s method [14], using the generalized transmission
coefficient in Eq. (5). This is a socalled detailed balance
calculation [7]. The systematics of ν are replaced by a recent
study of Ernebjerg and Herman [25], which better reproduces
theGOEcalculations.Whenthetargetisinanexcitedstate,the
width fluctuation elastic enhancement occurs for the excited
state. This method has been applied to calculate neutron
radiative capture cross sections on actinides [26,27], and it
was shown that the calculated capture cross sections on237Np
and241Am well reproduce the experimental data from Los
Alamos.
As schematically shown in Fig. 2 all the transmission
coefficients for the compound nucleus decay channels are
easily obtained by the CC calculation with the detailed
balance technique. For example, the transmission coefficient
from the compound nucleus to the excited (3/2)+state (exit
channel)isidenticaltothatofcompoundformationprobability
(entrance channel). A transmission coefficient to uncoupled
states (shown by the dotted arrow in Fig. 2) is given by solving
a spherical optical model on this excited state, which forms
the same compound state as the entrance channel. The optical
FIG. 2. Transmission coefficients used in the HauserFeshbach
calculation. The transmission coefficients for the rotational band
membersareobtainedbytheCCcalculation.Fortheuncoupledstate,
we perform a spherical optical model calculation.
potential for these spherical calculations would be different
from that for the CC calculation. Indeed, it is possible to
increase the imaginary potential phenomenologically or apply
a global spherical optical potential for those uncoupled states
to account for eliminating the direct channels in the CC
calculation. In this study we simply adopt the same optical
potential parameters of the CC model for the uncoupled states
and perform spherical optical model calculations. The same
procedure is used both for calculations of targets in their
ground state and in their excited states.
III. RESULTS AND DISCUSSION
A. Coupledchannels calculation for169Tm
The CC optical potential parameter for169Tm was taken
from Ref. [9]: they are
V = 46.87 − 0.25E MeV,
Ws= 3.6 + 0.6E MeV,
Vso= 6.0 MeV,
(8)
(9)
(10)
where V is the WoodSaxon central potential depth, Ws is
the derivative WoodsSaxon imaginary potential, Vso is the
Thomastypespinorbitpotential,andE istheincidentneutron
energy in MeV. The radius r and diffuseness a for each
potential are rv= rw= rso= 1.27 fm, av= aso= 0.63 fm,
and aw= 0.48 fm, in common notation. The deformation
parameters of
coupled five levels of the groundstate rotational band,
(1/2)+,(3/2)+,(5/2)+,(7/2)+,and(9/2)+,andcalculatedthe
direct cross sections for two cases: (1) the target is in the
ground state and (2) the target is in the first excited state,
(3/2)+8.41 keV. The neutron incident energies considered
are from 1 keV up to 20 MeV, but we are mostly interested
in the lowenergy region. We expect noticeable differences
in the calculated cross sections at low energies, because the
number of incoming partial waves is not so large, which limits
the accessible spinspace in the reaction and amplifies the
calculated differences for the target in its ground state versus
excited states.
The calculated total, shape elastic, and reaction (compound
formation) cross sections are shown in Fig. 3. The thick lines
are the calculated results when the target is in the ground
state, and the thin lines are for the first excited state case.
Differences are clearly observed in the lowenergy region, as
weexpected.At1keV,σ(1)
0.77. The increase in σRat low energies is due to the swave
transmission coefficient, and therefore can be related to the
swave strength function. The calculated S0for the ground
state and first excited cases are 1.81 × 10−4and 2.44 × 10−4,
respectively. However, the pwave strength function S1 for
both cases are not so different (less than 5% difference).
Above 5 MeV or so, differences in these cross sections
become very small, and this is expected because the excitation
energy of the first level is only 8.41 keV, which is only 0.1% of
the incident energies. Inaddition, there aremany partialwaves
that couple to the total J?, which washes out the difference
in the target spin (which is only 1¯ h anyway).
169Tm are β2= 0.31 and β4= −0.01. We
R/σ(0)
Ris1.33andσ(1)
E/σ(0)
Eareabout
0246113
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KAWANO, TALOU, LYNN, CHADWICK, AND MADLANDPHYSICAL REVIEW C 80, 024611 (2009)
0
0.001
5
10
15
20
25
30
0.01
Incident Neutron Energy [MeV]
0.1 1 10
Cross Section [b]
Ground State, Total
Shape Elastic
Compound Formation
1st Excited State, Total
Shape Elastic
Compound Formation
FIG. 3. Calculated total (solid line), shape elastic scattering
(dashed line), and reaction cross sections (dotted line) for169Tm.
The thick lines are for the groundstate case, and thin lines are for the
first excited state.
ThetargetspineffectappearsonlyintheCCmodel,because
spherical optical model (SOM) calculations are target spin
independent. In the SOM case, the transmission coefficients
on the excited states are calculated regardless of the Iπof the
state. Note that this implies that the same spherical optical
potential obtained for the groundstate target can be applied
to the excited states, although there might be a nonoverlapped
phase space; compound states that satisfy the spin selection
rule I − j ? J ? I + j are different.
The direct cross sections are depicted in Fig. 4. The thick
curves are the usual CC calculations, in which the target
is at the ground state. Our unique calculation is the direct
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 5
Incident Neutron Energy [MeV]
10 15 20
Inelastic Scattering Cross Section [b]
from G.S. to 3/2
5/2
7/2
Total
from 1st Ex. to 1/2
5/2
7/2
Total
FIG. 4. Calculated direct inelastic scattering cross sections for
two cases; target at the ground state (thick curves), and at the first
excited state (thin curves). Although we coupled the (9/2)+state too,
the cross sections are not plotted here for clearer visibility. For the
groundstatecase,thesolidlineisthedirectcrosssectiontothe(3/2)+
state. In the case of excited target, the solid line is the direct cross
section to the (1/2)+ground state. The dotdashed curves are the sum
of all direct reactions.
TABLEI. Nmerical comparison of thecalculated crosssections
for the target in its ground state (1/2)+and in its first excited state
(3/2)+at the neutron incident energy of 100 keV, with those by
Madland [9]. The cross sections are in barns.
Present Madland [9]
(1/2)+
(3/2)+
(1/2)+
(3/2)+
Total
Shape elastic
Compound formation
Total inelastic
Capture
10.1
5.86
4.26
1.03
0.625
8.89
4.34
4.53
0.569
0.852
10.3
5.91
4.35
1.17
0.625
8.98
4.45
4.52
0.650
0.941
transition from the first excited (3/2)+state to the ground
state,shownbythethinsolidcurve.Thisprocesshasapositive
Q value, which means the scattered neutrons are accelerated.
This cross section is a factor of 2 smaller than the normal
process: from the ground state to the first excited state (thick
solid curve). However, the inelastic scattering to the (7/2)+
state is significantly enhanced. It is interesting to note that the
sum of all direct inelastic scattering cross sections, shown by
the dotdashed curve, are not so different. Although we see
20% difference in the total inelasticscattering cross section at
20 MeV, this is only 1% of the total reaction cross section.
The direct cross section from the (3/2)+state to the ground
state is about a half of the cross section from the ground
state to the (3/2)+excited state, and this is the same ratio as
(2I0+ 1)/(2I1+ 1).ThisisrelatedtothespinfactorofEq.(3).
Because the Smatrix is symmetric whichever the incident
particle channel is in, the difference in the direct cross section
σ(nn?)
D
comes from gJand a channel selection δnc,nand δnc?,n?
in summation in Eq. (2), and the simplest case is just equal to
the ratio of gJ’s. In addition, differences also occur because
we adopted the energydependent optical potential parameters
of Eqs. (8)–(10).
We also compared the present results numerically with
the independent calculations in Ref. [9]. Table I shows the
calculated cross sections at 100 keV. The table shows that the
two independent solutions for the ground and excitedstate
scattering cross sections yield essentially the same results, and
this gives credence to the approach.
B. Statistical model calculation for169Tm
1. Neutron radiative capture
ThecalculatedtransmissioncoefficientsofEq.(5)arefedto
the statistical HauserFeshbachMoldauer model calculations.
The CC calculation for the entrance channel is performed by
coupling the five groundstate rotational band members [up
to 332 keV (9/2)+level]. Discrete levels are included up to
938keV(13/2)−level,andtheirspinandparityaretakenfrom
the reference input parameter library version 2, RIPL2 [28].
Above 938 keV, the GilbertCameron leveldensity formulas
[29] with a parameter systematics in Ref. [30] are employed.
There is a 316keV (7/2)+level, whose excitation energy is
lower than the 332keV (9/2)+level. We included this level as
an uncoupled state in the HauserFeshbach model calculation.
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PHYSICAL REVIEW C 80, 024611 (2009)
0.01
0.1
1
10
0.001 0.01
Neutron Incident Energy [MeV]
0.1 1
Radiative Capture Cross Section [b]
Xia (1988)
Xu (1986)
Jiang (1982)
Joly (1979)
Block (1961)
Gibbons (1961)
Ground
Excited
FIG. 5. Comparison of calculated neutron radiative capture cross
sectionwithexperimentaldata.Thesolidcurveisforthegroundstate
case,andthedashedcurveisfortheexcitedstate.Alltheexperimental
data are for the ground state.
The E1 γray strength function for the radiative capture
channel is calculated with the generalized Lorentzian form
[31], with the giant dipole resonance parameter in Ref. [28].
For higher multipole radiations, we include E2 and M1
transitions. The γray strength function is then renormalized
to the complied ??γ?/D0value taken to be 0.0118 [32]. This
normalization factor for the γray strength function must be
obtained for the ground state, because the ??γ?/D0value is
for the ground state. Then, of course the same normalization
factorisappliedfortheexcitedstatecases,becauseweassume
that a giantdipole state on the excited state is the same as on
the ground state by the BrinkAxel hypothesis.
The calculation is performed up to the neutron incident
energy of 8 MeV. At higher energies, the dominant neutron
capture reaction becomes the direct/semidirect process [33]
that requires a target state wave function. Anyway we do not
expect a large difference in the calculated cross sections for
both cases at high energies, because the total reaction cross
sections σ(0)
Comparison of the calculated neutron capture cross section
with the experimental data are shown in Fig. 5. The solid line
is a calculation when the target is in its ground state, and the
dashed line is for the first excited state case, respectively. The
numericalcomparisonwiththeresultsinRef.[9]isalsoshown
in Table I. The groundstate calculation is compared with the
experimental data available [34–38]. Agreement between the
groundstate calculation and the experimental data is seen and
it is fairly good, though in this figure we are more focused on
a prediction of differences between ground state and excited
statecapture.Theshapeofthecalculatedcapturecrosssections
on both states are very similar, but the absolute magnitudes
differ by 20–30% below 100 keV. One of the reasons of this
difference is the compound formation cross sections, which
are different for both cases as shown in Fig. 3. However,
the differences between σ(0)
100 keV. Another reason could be a difference in the total
spin of the compound nucleus. Assuming the incoming partial
Rand σ(1)
Rare almost the same.
Rand σ(1)
Rare visible only below
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
0.001 0.01
Neutron Incident Energy [MeV]
0.1 1
Capture Cross Section Ratio
CoupledChannels
Spherical OM
FIG. 6. Ratios of the calculated capture cross sections of excited
state to the ground state. The solid line is for the coupledchannels
calculation, and the dashed line is for the spherical optical model
case. The arrows on the x axis are the inelastic channel threshold
energies.
wave is just an swave, the spin of compound state is I ± 1/2,
and this gives a different γray cascading pattern from the
compound state and different competition of neutron emission
processes.
When an SOM calculation to the entrance channels is
employed, the compound formation cross section does not
depend on the target spin (however, the spin distribution of
the compound nucleus is still different). To see the effect
of channel coupling, we also performed the neutron capture
calculation with the spherical optical model (SOM) poten
tial. The global optical potential parameters of Koning and
Delaroche [39] were used. Figure 6 shows a ratio of the
calculated capture cross sections on the excited state to the
ground state. Because the shape of both curves are similar
above 100 keV, the enhancement of the capture cross sections
in the energy range 100 keV to 1 MeV is probably due
to the target spin effect. At low energies CC and SOM
calculations give somewhat different tendencies. The SOM
calculation enhances the capture cross section significantly
from 10 to 100 keV. Although an oldfashioned spherical
optical model and HauserFeshbach model gives the same
compound formation cross section, there would still be an
enhancement to the capture cross section, because of J?
coupling.
2. Neutron elastic and inelastic scattering
The calculated neutron elasticscattering cross section is
shown in Fig. 7. The dashed lines are the shape elastic
scattering, which are identical to those in Fig. 3. The
dotdashed lines are the compound elastic, and the solid line is
thetotalelasticscatteringcrosssections.Fortheexcitedtarget
case, reduction in the shape elastic scattering σ(1)
the similar total cross sections for both cases, results in an
increase in the compound formation σ(1)
compound elasticscattering cross section [σ(1)
the compound elastic and neutron capture below 8.41 keV].
E, having
R, which enhances the
Ris the sum of
0246115
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KAWANO, TALOU, LYNN, CHADWICK, AND MADLANDPHYSICAL REVIEW C 80, 024611 (2009)
0
0.001
5
10
15
20
0.01 0.1 1
Elastic Scattering Cross Section [b]
Neutron Incident Energy [MeV]
Shape Elastic (target g.s.)
Compound Elastic (target g.s.)
Total Elastic (target g.s.)
Shape Elastic (target 1st)
Compound Elastic (target 1st)
Total Elastic (target 1st)
FIG. 7. Comparison of calculated neu
tron elasticscattering cross sections. The
thick lines are for the groundstate target, and
the thin lines are for the excited state. The
solid lines are the total inelasticscattering
crosssections,dashedlinesaretheshapeelas
tic, and dotdashed lines are the compound
elastic scattering.
The total elasticscattering cross sections look identical
below 10 keV. This is, however, probably a coincidence,
because the total elastic scattering at low energies depends on
howlargetheneutroncapturecompetitionis.Above2MeVthe
compoundelasticscatteringcrosssectionbecomesnegligible,
because many neutron inelasticscattering channels open.
The calculated neutron inelasticscattering cross sections,
including both the direct and compound processes, are shown
in Fig. 8. The thick solid line is the total inelastic scattering,
which is a threshold reaction. The thin solid line is the result
for the excited target with a negative Q value. The dashed
and dotdashed lines are the production cross section of the
8.41 and 118keV levels, respectively. The compound cross
sections to the higher energy levels (above 118keV levels)
become similar for both cases, because a large phase space
(many partial waves and large numbers of spin couplings)
washes out the spin selection rule that is important when only
a few partial waves are involved.
One obvious difference seen in Fig. 8 is, of course, the
transition from the excited state to the ground state, which
has no threshold. The superelastic cross section is, however,
only 15% of the total elastic scattering (see Fig. 7). To observe
the superelastically scattered neutrons, very high resolution
experiments would be needed.
C. Fission calculation for239Pu
In the case of actinides, the lowincident neutron energy
regime is dominated by capture and fission reaction processes.
At relatively low neutron energies, the fission cross section is
0
0.001
1
2
3
4
0.01 0.1 1
Inelastic Scattering Cross Section [b]
Neutron Incident Energy [MeV]
1st (target g.s.)
2nd (target g.s.)
Total Inelastic (target g.s.)
0th (target 1st)
2nd (target 1st)
Total Inelastic (target 1st)
FIG. 8. Comparison of calculated neu
tron inelasticscattering cross sections. The
thick lines are for the ground state target, and
the thin lines are for the excited state. The
solid lines are the total inelasticscattering
cross sections.
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PHYSICAL REVIEW C 80, 024611 (2009)
0
0.01
1
2
3
4
5
0.1 1 10
Pu239 Fission Cross Section [b]
Neutron Energy [MeV]
(a)
GNASH Ground State
1st Excited State
AVXSF Ground State
1st Excited State
0.5
1
1.5
2
0.01 0.1
Neutron Energy [MeV]
1 10
Pu239 Fission Cross Section Ratio
(b)
GNASH 1st/GS
AVXSF 1st/GS
FIG. 9. (a) The neutroninduced fission cross section of239Pu calculated in the two cases where the target nucleus is in its first excited state
versus in its ground state; (b) the ratio of fission cross section for excited239Pu to the groundstate target.
dominated by a relatively few transition states at the barrier
deformations. In particular, the compound nucleus240Pu is
believed, from experimental evidence on shape isomers and
theoreticalcalculationsofenergyasafunctionofdeformation,
to have a higher inner barrier peak than outer. Its fission
probability in a state of given total angular momentum and
parity is therefore strongly affected by the small number of
transition states below the pairing energy gap at the inner
barrier. A change in the spin and parity of the target nucleus,
and hence of the compound nucleus, can have a significant
impact on the fission cross section.
We have investigated this question by calculating the
neutron induced fission cross section on239Pu, either in its
ground state [Iπ= (1/2)+] or in its first excited state [Ex=
7.86 keV; Iπ= (3/2)+]. The coupledchannels calculations
was performed using the optical potential of P. G. Young
[28,40] and considering the first five states of the groundstate
rotational band.
We have performed HauserFeshbach calculations using
the GNASH code [41] and more detailed fission crosssection
calculations using the AVXSF code by J. E. Lynn [42,43].
Becausethesetwocodeshavedifferentspecialities,comparing
two results may reduce codespecific problems in the fission
calculations within the HauserFeshbach formalism. The
modeling of the fission channel in GNASH is not particularly
well suited for calculating the fission cross section below
the barrier, as the coupling between class I and class II
states is neglected. We also neglected the width fluctuation
correction to the GNASH calculation. Better fission physics
has been implemented in the AVXSF code, including correct
treatments of the class I and class II states coupling. However,
the neutron transmission coefficients are calculated from the
experimental neutron strength function, and they are assumed
to be the same for both the ground and excited states.
However, GNASH employs the coupledchannels transmission
coefficients.
BeforeweperformedtheHauserFeshbachcalculations,we
compared the reaction cross sections σRcalculated with the
coupledchannelsmethod,justasinFig.3.Itwasobservedthat
these cross sections for both cases are very similar, except that
we saw an increase in σRfor the first excited state. This might
be due to the optical potential employed. At least we could say
here that the comparison of fission calculation may reveal the
target spin dependence of the fission process. An assumption
madeinthe AVXSFcalculationisthatthesameneutronstrength
function is used for both ground and excited states, and this is
supported by the coupledchannels calculation.
Figure 9 shows the calculated fission cross section in the
two situations, with two different codes. Two thick curves are
forthegroundstatecase;thesolidcurveiswith GNASHandthe
dottedcurveiswith AVXSF.Otherthincurvesarefortheexcited
state case; the dashed curve is for GNASH, and the dotdashed
curve is for AVXSF. Note that the AVXSF calculations go up
to 500 keV. Both codes indicate that the calculated fission
cross sections are enhanced when the target is in its excited
state. The cross section ratios of the excited to the ground
states are shown in the lower panel of Fig. 9. Although the
absolute magnitude of the enhancement is different in GNASH
and AVXSF, their tendency is very similar, and the fact that
both calculations lead to the same qualitative conclusion is
comforting and indicates that the underlying physics of the
transition states is correct.
At 10 keV the calculated first excitedstate cross section
with GNASH is about 30% higher than the groundstate cross
section.Thisenhancementbecomes50%forthecaseof AVXSF.
We understand that the enhancement reflects the different
spin of the target state [(3/2)+for the first excited state and
(1/2)+for the ground state], which modifies possible fission
paths through the discrete transition states lying on top of
the barriers. At low energies a few partial waves contribute
to the compound formation, and decay of the compound
nucleus is strongly constrained by the spin selection rule.
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KAWANO, TALOU, LYNN, CHADWICK, AND MADLANDPHYSICAL REVIEW C 80, 024611 (2009)
For example, the swave coupled with the ground state of
the target nucleus produces the compound nucleus states of
0+(with spin statistical weighting of 1/4) and 1+(with spin
statistical weighting of 3/4), whereas two spin states 1+and
2+(with respective weightings of 3/8 and 5/8) are possible in
thecaseoftheIπ= 3/2+excitedstate.The1+transitionstate
at the inner barrier plays a special role as it is expected to lie
much higher in excitation energy than other spinparity states,
due to its intrinsic complexity (e.g., a combination of bending
and massasymmetry vibrations). Therefore the fission cross
section observed for the239Pu in its ground state is hindered
compared to the one for the first excited state. In the AVXSF
calculation the class II intermediate structure associated with
the 1+state has the effect of further reducing the average
fission cross section, and this could explain the higher ratio of
excited state to ground state cross sections shown in Fig. 9
Above 1 MeV neutron incident energy, the two results
with GNASH are practically identical because the number of
incomingpartialwavesbecomeslarge,thentheavailablephase
spaces for both cases get very similar.
With the AVXSF code, we performed more fission cross
section calculations for the cases in which the target nucleus
is in the higher excited states, not only the ground state
rotational band, but also K = (5/2)+band members whose
bandhead energy is 285 keV. Because the angularmomentum
conservation during the fission process is not well understood
(K mixing [42]), a quantitative argument requires more
detailedinformationofnuclearstructureforstronglydeformed
systems. However, qualitatively the calculated fission cross
section is larger than that for the ground state, and this
enhancement tends to be larger if the target spin is higher.
IV. CONCLUSION
We have applied a CC method to calculate nuclear
reaction cross sections for excited nuclei. The direct reac
tions among the members of groundstate rotational band
are calculated with the CC method, and the generalized
transmission coefficients from both ground and excited states
are calculated. These transmission coefficients are fed to
the statistical HauserFeshbach model calculation to obtain
compound reaction cross sections.
We performed a numerical comparison of cross sections
for169Tm. The statistical model calculation on the excited
nucleus gives different cross sections from the calculation for
the ground state. However, the differences are visible only
below neutron energies of about 1 MeV. The difference of the
cross sections comes from both the level coupling effect and
the target spin effect. It was shown that the target spin effect is
important, when a number of contributing partial waves is not
solarge.Thelevelcouplingeffectisalsoimportant,whichwas
shownbycomparingwiththesphericalmodelcalculation.Our
results have confirmed the original excitedstate calculations
of Ref. [9].
Thesametechniqueisalsoappliedtocalculatefissioncross
sections of239Pu, as the level structure of239Pu is very similar
to169Tm. To reduce ambiguities in fission modeling in the
HauserFeshbach framework, we employed two independent
nuclear reaction model codes, GNASH and AVXSF. These two
codes gave relatively similar tendency for the fission cross
section when the target nucleus is in its excited state. At low
energies (below 1 MeV), the calculated fission cross section
for the first excited state is larger than that for the ground state,
and the difference becomes smaller at higher energies. This
observation is consistent with the phasespace argument; low
energy reactions are strongly constrained by the spin selection
rule.
ACKNOWLEDGMENTS
ThisworkwascarriedoutundertheauspicesoftheNational
Nuclear Security Administration of the US Department of
Energy at Los Alamos National Laboratory under Contract
No. DEAC5206NA25396.
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