Unified description of fission in fusion and spallation reactions

Abstract

We present a statistical-model description of fission, in the framework of compound-nucleus decay, which is found to simultaneously reproduce data from both heavy-ion-induced fusion reactions and proton-induced spallation reactions at around 1 GeV. For the spallation reactions, the initial compound-nucleus population is predicted by the Li\`{e}ge Intranuclear Cascade Model. We are able to reproduce experimental fission probabilities and fission-fragment mass distributions in both reactions types with the same parameter sets. However, no unique parameter set was obtained for the fission probability. The introduction of fission transients can be offset by an increase of the ratio of level-density parameters for the saddle-point and ground-state configurations. Changes to the finite-range fission barriers could be offset by a scaling of the Bohr-Wheeler decay width as predicted by Kramers. The parameter sets presented allow accurate prediction of fission probabilities for excitation energies up to 300 MeV and spins up to 60 \hbar. Comment: 16 pages, 20 figures. Submitted to Phys. Rev. C

Full text

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arXiv:1007.0963v2 [nucl-th] 16 Jul 2010
Unified description of fission in fusion and spallation reactions
Davide Mancusi
University of Liège, AGO Department, allée du 6 août 17, bât. B5, B-4000 Liège 1, Belgium
Robert J. Charity
Department of Chemistry, Washington University, St. Louis, Missouri 63130, USA
Joseph Cugnon
University of Liège, AGO Department, allée du 6 août 17, bât. B5, B-4000 Liège 1, Belgium
We present a statistical-model description of fission, in the framework of compound-nucleus decay, which
is found to simultaneously reproduce data from both heavy-ion-induced fusion reactions and proton-induced
spallation reactions at around 1 GeV. For the spallation reactions, the initial compound-nucleus population is
predicted by the Liège Intranuclear Cascade Model. We are able to reproduce experimental fission probabili-
ties and fission-fragment mass distributions in both reactions types with the same parameter sets. However, no
unique parameter set was obtained for the fission probability. The introduction of fission transients can be off-
set by an increase of the ratio of level-density parameters for the saddle-point and ground-state configurations.
Changes to the finite-range fission barriers could be offset by a scaling of the Bohr-Wheeler decay width as pre-
dicted by Kramers. The parameter sets presented allow accurate prediction of fission probabilities for excitation
energies up to 300 MeV and spins up to 60 ?.
PACS numbers: 21.10.Ma,24.60.Dr,25.70.Jj
I. INTRODUCTION
Although seventy years have passed since the seminal
works of Bohr and Wheeler [1] and Weisskopf and Ewing
[2] and the establishment of a qualitative understandingof the
de-excitation mechanism of excited nuclei, quantitatively ac-
curateanduniversallyapplicablemodelsdonotyetexist. This
is partly due to the vastness of the amount of nuclear data that
must be fed into the models, and partly to the uncertainties
in the fundamental ingredients of de-excitation, such as level
densities and emission barriers. Even the choice of the math-
ematical formalism, however, is not devoid of confusion, as it
was already pointed out by Moretto [3] and Swiatecki [4].
One way to lift the degeneracy of the ingredients of the
model is to explore diverse regions of the compound-nucleus
parameter space. A systematic study of nuclei with different
masses, excitation energies, spins and isospins would be sen-
sitive to most of the assumptions of the de-excitation model.
The long-term goal of such an investigationwould be to iden-
tify a minimal set of physical ingredients necessary for a uni-
fied quantitative description of nuclear de-excitation chains.
The production of excited compound nuclei can proceed
from several entrancereactions. There has been a long history
of compound-nucleusstudies using heavy-ion-inducedfusion
reactions. Thesereactionsallowonetospecifythecompound-
nucleus mass, charge and excitation energy; however, a dis-
tribution of compound-nucleus spins is obtained. Statistical-
model parameters such as fission barriers are quite sensitive
to spin.Heavy-ion-induced fission probabilities, evapora-
tion spectra, residue masses can generally be reproduced in
statistical-model calculations. However, some fine tuning of
the statistical-model parameters to the mass regionor reaction
is often needed. A unified description over all mass regions
is still lacking even for these reactions. Some work towards
this, concentrating of the parameters describing the shape of
the evaporation spectra, is presented in Ref. [5].
Another typical entrance channel for the production of ex-
cited compound nuclei is spallation. The present interest in
spallationderivesmainlyfromtheapplicationstoAccelerator-
Driven Systems (ADS), namely accelerator-basedreactors for
the transmutation of nuclear waste. At incident energies rel-
evant for transmutation (a few hundred MeV), an appropri-
ate theoretical tool for the description of proton-nucleus reac-
tions is the coupling of an intranuclear-cascade model (INC)
with a nuclear de-excitation model. It is assumed in the INC
framework that the incoming particle starts an avalanche of
binary collisions with and between the target nucleons. When
the cascade stage ends, an excited and thermalized remnant is
formed,with a basically unchangeddensity. Inthe subsequent
de-excitation stage, the remnant gets rid of the excess energy
by particle evaporation and/or fission. For these reactions,
the need for a model to predict the initial compound-nucleus
mass, charge, excitation, and spin distributions adds some un-
certainty in our ability to constrain the statistical-model pa-
rameters by fitting data. However, spallation reactions allow
us to explore different regions of compound-nucleusspin and
excitation energy than can be probed with fusion reactions
alone and thus can be important in parameter fitting.
The role of a transient fission width is currently of some
controversy. Fission transients are where the fission decay
widthis notconstant,but increasesfromzerotowardsits equi-
librium value [6]. Fission transients were first introduced to
help explain the large number of neutrons emitted from a fis-
sioning system before the scission point was reached [7]. The
statistical model assumes there is an equilibriumis all degrees
of freedom including the deformation degrees of freedom as-
sociated with fission. If all compound nuclei have spherical
shapes initially, then they cannot instantaneously fission as it
takes as finite time to diffuse towards the saddle and subse-

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2
quently the scission point. The transient time, the time scale
necessary for the system to explore large fluctuations in the
deformation degrees of freedom, is a function of the viscosity
of the nucleus. The predicted fission probability is also very
sensitive to the assumed initial deformation [8] which may
depend on the entrance channel.
The transient time is oftencalled a fission delay as fission is
suppressed during this period. If the excitation energy of the
compound nucleus is large enough, then there will be a prob-
ability of emitting a light or possibility even an intermediate-
mass fragment during the fission delay. Neutron emission
lowers the excitation energy and charged-particle emission
also lowers the fissility of the nucleus by increasing its fis-
sion barrier. These effects will lead to a reduced fission prob-
ability after the fission delay is over. An experimental test of
this idea wouldbe the observationof reducedfission probabil-
ity orenhancedevaporation-residuesurvivalat highexcitation
energies which cannot be explained in terms of the statistical
model. At present there is some controversyover the need for
fission transients.
A number of theoretical studies [9, 10] reproduce exper-
imental fission probabilities and pre-scission neutron multi-
plicities with transient fission widths when the viscosity in-
creases with the mass of the compound nucleus. Transient
fission has also been invoked to explain the relatively large
number of evaporation residues measured for the very fis-
sile216Th compound nuclei formed in32S+184W reactions,
as compared to a statistical-model prediction [11]. Alterna-
tively other studies have reproduced fission probabilities [12]
and both pre-scission neutron multiplicities and fission proba-
bility with no transient effects [13]. Similarly, in very high
excitation-energy data obtained in 2.5-GeV proton-induced
reactions, no transients were needed in reproducing the mea-
sured fission yields [14].
This paperdiscusses the applicationof the GEMINI++ de-
excitation model [15] to the description of fission in fusion
and spallation reactions. In the latter case, the description of
the entrance channel is provided by a coupling to the Liège
Intranuclear Cascade model (INCL) [16]. Both INCL and
GEMINI++ are amongthemostsophisticatedmodelsintheir
ownfields. Thepresentworkalso representsthefirst thorough
discussion of their coupling.
We compare the predictions of the models with experimen-
tal residue yields in spallation studies and with fission and
evaporation-residue excitation functions measured in heavy-
ion induced fusion reactions. The choice of the observables
was motivated by considerations about their sensitivity to fis-
sion and by the availability of experimental data. An ex-
haustive discussion should of course take into account other
observables (e.g. double-differential particle spectra) and the
competition of fission with the other de-excitation channels,
but this is outside the scope of the present paper. Therefore,
we describe how the parameters of the statistical-decay model
have been consistently adjusted to reproduce the data and dis-
cusstowhatextenta successfulunifieddescriptionofthesere-
actions has been achieved. Finally, we explore whether all the
data can be described within the statistical model or whether
transient fission decay widths are needed.
II. THE MODELS
We shall now turn to the description of the most important
features of the models we have considered. The codes will
not be analyzed in detail, but only the most important features
will be outlined.
A. GEMINI++
GEMINI++ is an improved version of the GEMINI sta-
tistical decay model, developed by R. J. Charity [17] with the
goal of describing complex-fragment formation in heavy-ion
fusion experiments. The de-excitation of the compound nu-
cleus proceeds through a sequence of binary decays until par-
ticle emission becomes energeticallyforbiddenor improbable
due to competition with gamma-ray emission.
Since compound nuclei created in fusion reactions are typ-
ically characterized by large intrinsic angular momenta, the
GEMINI and GEMINI++ models explicitly consider the in-
fluence of spin and orbital angular momentum on particle
emission. Moreover, GEMINI/GEMINI++ do not restrict
binary-decaymodesto nucleonandlight-nucleusevaporation,
which are the dominant decay channels, but allow the decay-
ing nucleus to emit a fragment of any mass. The introduction
of a generic binary-decay mode is necessary for the descrip-
tion of complex-fragmentformation and is one of the features
that set GEMINI/GEMINI++ apart from most of the other
de-excitation models.
Emission of nucleons and light nuclei (Z ≤ 2, 3 or 4,
depending on the user’s choice) is described by the Hauser-
Feshbach evaporation formalism [18], which explicitly treats
and conserves angular momentum. The production of heavier
fragments is described by Moretto’s binary-decay formalism
[3]. However for symmetric divisions of heavy compoundnu-
clei, the Moretto formalism employing Sierk’s Finite-Range
calculations [19, 20] fails to reproduce the mass distribution
of decay products (Sec. IIIC). However for light systems,
the Moretto formalism works quite well [17, 19, 21] and is
still used in GEMINI++. Also for the heavier systems, but
for mass-asymmetries outside of the symmetric fission peak,
the Moretto formalism is still used. Otherwise, the total fis-
sion yield is obtained from the Bohr-Wheeler formalism [1]
and the width of the fission-fragment mass distribution is
taken from systematics compiled by Rusanov et al. [22] (see
Sec. IIIC).
Table I summarizes the de-excitation mechanisms featured
by GEMINI++.
The parameters of the model associated with evaporation
have been adjusted to reproducedata from heavy-ion-induced
fusion reactions. This is described in Ref. [5] in more detail,
but we briefly list the important adjustments for that work . In
order to fit experimental light-particle kinetic-energy spectra,
the transmission coefficients in the Hauser-Feshbach formal-
ism were calculated for a distribution of Coulomb barriers as-
sociated with thermal fluctuations. The nature of fluctuations
is not entirely clear, they may be fluctuations in compound-
nucleus shape and/or its density and/or its surface diffuseness.

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Table I. List of de-excitation processes featured by the GEMINI++
model. The symbol Z represents the charge number of the emitted
particle and Zswitchcan be chosen to be 2, 3 or 4.
Process
light-particle
evaporation
binary decay
ModelNotes
Hauser-Feshbach [18]
Z ≤ Zswitch
Moretto [3]
Z > Zswitch
only in heavy
systems
fission
Bohr-Wheeler [1]
partition in fission
Rusanov et al. [22]
Level densities were calculated with the Fermi-gas form:
ρ(E∗,J) ∼ exp
?
2
?
a(U)U
?
(1)
where E∗is the total excitation energy, J is the spin and U
is the thermal excitation energy after the pairing, rotational,
and deformation energies are removed. The level-density pa-
rameter used should be considered an effective value as no
collective-enhancement factors are used in the level density
formula of Eq. (1).
The level-density parameter a(U) is excitation-energy de-
pendentwith an initial fast dependencedue to the washing out
of shell effects following Ref. [23] and a slower dependence
needed to fit the evaporation spectra. The shell-smoothed
level-density parameter was assumed to have the form
? a(U) =
A
k∞− (k∞− k0)exp
?
−
κ
k∞−k0
U
A
?
(2)
which varies from A/k0at low excitation energies to A/k∞
at large values. Here k0=7.3 MeV, consistent with neutron-
resonance counting data at excitation-energies near the neu-
tron separation energy,and k∞=12 MeV. The parameter κ de-
fines the rate of change of ? a with energy and it is essentially
we expect a decrease in the level-density parameter with U
due to decreasing importance of long-range correlations with
increasingexcitationenergy(dueto washingout ofcollective-
enhancement factors and also the reduction of the intrinsic
level-density parameter), the strong mass dependence cannot
be explained at present.
Thestrongexcitation-energydependenceof? aforheavynu-
weak decay channels include n, p, and α evaporation and
thus GEMINI++ calculations predict enhanced evaporation-
residue production consistent with some experimental data.
These enhanced evaporation-residue yields had previously
been interpreted as a consequence of transient fission [11].
Clearly the excitation-energy dependence of ? a is very impor-
nuclei, the yield is decreased relatively little by the increased
temperature.
zero for light nuclei (i.e. a constant ? a value) and increases
roughly exponentially with A for heavier nuclei. Although
clei leads to increased nuclear temperatures which enhance
very weak decay channels. For very fissile systems, these
tant in understanding the role of transient fission. However
we note that for fission, the dominant decay mode in fissile
B.INCL4.5
The INCL model [16] can be applied to collisions between
nucleiand pions,nucleonsor light nucleiof energylowerthan
a few GeV. The particle-nucleus collision is modelled as a se-
quence of binary collisions among the particles present in the
system; particles that are unstable over the time scale of the
collision, notably ∆ resonances, are allowed to decay. The
nucleus is represented by a square potential well whose ra-
dius depends on the nucleonmomentum;thus, nucleonsmove
on straight lines until they undergo a collision with another
nucleon or until they reach the surface, where they escape if
their total energy is positive and they manage to penetrate the
Coulomb barrier.
The latest versionof the INCL model (INCL4.5) includes,
among other things, isospin- and energy-dependent nucleon
potentials, an isospin-dependent pion potential and a new
dynamical coalescence algorithm for the production of light
clusters (up to A = 8 with the default program options). A
comprehensive description will be published in the near fu-
ture [24].
The INCL model simulates a complete cascade event, its
output being the velocities of all the emitted particles. The
characteristics of the remnant (its mass, charge, momentum,
excitation energy and intrinsic angular momentum) are de-
rivedfromthe applicationof conservationlaws and are passed
to the chosen de-excitation code; the latter simulates the de-
cay of the remnant into a nuclear-stable residue plus a number
of nucleons, nuclei and/or gamma rays.
The INCL4.5 model is not to be considered as an ad-
justable model. It does contain parameters, but they are either
takenfromknownphenomenology(such as the matterdensity
radiusofthenuclei)orhavebeenadjustedonceforall (suchas
the parameters of the Pauli blocking or those who determine
thecoalescencemodulefortheproductionofthe lightcharged
clusters). Therefore adjusting INCL4.5/GEMINI++ on the
experimental data basically amounts to the adjustment of
the GEMINI++ parameters. One should keep in mind that
INCL4.5 brings in its own physics features and limitations.
For our purpose here, they essentially determine the distribu-
tions of the remnant properties. These quantities cannot be
compared directly with experimental data, but the predictions
of INCL4.5 concerning those observables that can be con-
fronted directly to experiment, namely the high energy parts
of particle spectra, are of rather good quality, as it was shown
recently [25].
III. ADJUSTMENT OF FISSION YIELDS
The assumption of thermal equilibrium implied by the
statistical-decay hypothesis implies that the excited nucleus
cannot keep any memory of the entrance channel. One of
the main aspirations of the GEMINI++ development is to
provide a unified and coherent description of nuclear de-
excitation in spallation and fusion reactions at the same time.
The degrees of freedom in the model induce different char-
acteristic dependencies of the fission width on the remnant

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4
02040 6080 100
0
100
200
300
400
500
600
J ???
E??MeV?
020 40 6080
0
100
200
300
400
500
Figure1. (Color online) Comparison of thedistributionsof excitation
energy and spin populated in the19F+181Ta→200Pb fusion reaction
for E∗= 90, 150 MeV (horizontal lines) with the INCL4.5 predic-
tionfor the1-GeV p+208Pbspallation reaction (contours, logarithmi-
cally spaced). The inset shows the distributionof fissioning remnants
for the spallation reaction (same contour levels). The dashed line is
the macroscopic yrast line from Sierk [26].
spin and excitation energy, because fission is at the very least
sensitive to spin, the fade-out of shell and collective effects,
level densities and fission barriers. Since variations in some
of the free parameters can produce similar effects, it is diffi-
cult to disentangle the various contributions and to put strin-
gent constraints on the de-excitation model just by looking
at experiments of a single type. However, fusion and spal-
lation reactions populate different regions of the compound-
nucleus spin/excitation-energy plane. A comparison of the
populations in the E∗-J plane is shown in Fig. 1 for the 1-
GeV p+208Pb spallation reaction and the19F+181Ta→200Pb
fusion reaction. The spallation population is represented by
the contours and two examples of the fusion distributions for
E∗=90, 150 MeV are shown by the thick horizontal shapes,
the thickness of which is proportional to the population. The
inset shows the distributions of remnants leading to fission in
the case of spallation; the contourlevels are the same as in the
main plot. As a guide for200Pb, the macroscopic yrast line
from Sierk [26] is indicated.
For the spallation reaction, the INCL4.5 model predicts
average values of about 167 MeV and 16.5 ?, but both dis-
tributions are quite broad and extend up to ∼ 650 MeV and
∼ 50 ?, respectively. On the other hand, the fusion reactions
we consideredare characterizedby higherspins and lower ex-
citation energies. We concentrate only on complete-fusionre-
actions, where the excitation energy of the compoundnucleus
is defined entirely from energy conservation. The require-
ment of complete fusion restricts us to projectile bombard-
ing energies of less then 10 MeV/A, where incomplete fusion
Table II. Experimental fission and evaporation-residue data used in
this work.
CNreaction
E∗range
[MeV]
25-82
68-94
95-249
63-126
54-95
50-91
65-249
49-153
48-84
125-203
39-86
39-83
26-83
σER
σfus
δJ
[?]
10
2.3
10
4
4.2
4.4
10
4.7
10
10
3
10
4
156Er
158Dy
160Yb
168Yb
178W
188Pt
193Tl
200Pb
200Pb
216Th
216Ra
216Ra
224Th
64Ni+90Zr
19F +139La
60Ni +100Mo
18O +150Sm
19F+159W
19F +169Tm
28Si +165Ho
19F +181Ta
30Si +170Er
32S +184W
19F +197Au
30Si +186W
16O +208Pb
[30]
[27]
[32]
[32, 33]
[32]
[32]
[34]
[35, 36]
[35]
[11]
[38]
[38]
[39]
[31]
[27]
[32, 33]
[32]
[32]
[35, 36]
[35]
[37]
[38]
[38]
[40, 41]
and pre-equilibrium process are small. We can explore some-
whathigherexcitationenergieswithmoresymmetricentrance
channels,but highspins will still bepopulated. Thus it is clear
that the comparison between spallation and fusion data repre-
sents a promising tool to extend the predictive power of the
model over a wide region of mass, energy and spin.
The fusion reactions used in this study are listed in Table II
with the range of excitation energies probed and the appro-
priate references for the data. In most cases we have selected
data where both the evaporation-residueand fission cross sec-
tions have been determined. The sum of these two quantities
givesthe total fusioncross section andthis is used to constrain
the compound-nucleus spin distribution. We assume the spin
distribution has a roughly triangular shape characterized by a
maximum value J0with a smooth cutoff characterized by the
parameter δJ, i.e.
σfus(J) = πλ2(2J + 1)
1
1 + exp?J−J0
δJ
?.
The parameter J0is determined from the total fusion cross
section
σfus=
∞
?
J=0
σfus(J)
and δJ is set to values from 3 to 10 ?, with the larger val-
ues associated with the heavier projectiles. These values are
similar to estimates obtained in Refs. [27–29]
Generally the fission cross section is only sensitive to the
value of δJ at excitation energies where the fission probabil-
ity is small and rises rapidly with J [27]. The values of δJ
assumed in the following calculations are also listed in Ta-
ble II.
For the28Si+165Ho and60Ni+100Mo reactions, only evap-
oration residue data has been measured. However for these
reaction, the J0values associated with fusion-like reactions

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5
are very large and the higher J values all go into fission. The
evaporation residue yield is therefore not sensitive to J0and
is entirely determined by fission competition at the lower J
values. Blann et al. [42] termed this a saturation analysis as
the higher J values are saturated by fission.
For spallation reactions, we focused our efforts on proton-
induced fission reactions on197Au [43],208Pb [44] and238U
[45] at 1 GeV, measured in inverse kinematics with the FRag-
ment Separator (FRS) at SIS, GSI, Darmstadt, Germany. An
additional experimental data-set for p+208Pb at 500 MeV ex-
ists [46], but new measurements seem to indicate that the fis-
sion cross section was overestimated by about a factor of 2
[47]. We decided to normalize Fernandez et al.’s total fission
cross section to the cross section measured by the CHARMS
collaboration, assuming that the fission distribution had been
correctly measured. The reader should nevertheless keep in
mind the normalization uncertainties associated with this data
set.
A. Modifications of the Fission Width
The Bohr-Wheeler fission width,
ΓBW=
1
2πρn(E∗,J)
?
dǫρf(E∗− B(J) − ǫ,J),
is sensitive to the choice of the fission barrier B and to the
level-density parameters afand anassociated with the saddle-
point and ground-state configurations. The U dependence of
the level-density parameter was initially assumed to be identi-
calforthegroundstateandthesaddlepoint,andit isdescribed
by Eq. (2). However, the saddle-point level-density parameter
afwas scaled by a constant factor with respect to the corre-
sponding ground-state level-density parameter an, to account
for the increased surface area of the saddle-point configura-
tion [48]. In what follows, we will refer to the scaling factor
as “the af/anratio”, for simplicity.
A numberof modificationsto the Bohr-Wheelerwidth have
been proposed. In a one-dimension derivation of the escape
rate over a parabolic barrier for high viscosity, Kramers [49]
obtained
ΓK=
??
1 +
?γ
2ω
?2
−γ
ω
?
ΓBW
(3)
whereγ is the magnitudeofthe viscosity, ω is frequencyasso-
ciated with the inverted parabolic barrier, and the factor scal-
ing the Bohr-Wheelerdecaywidthis less thanunity. Now ω is
notexpectedto be a strongfunctionof mass orspin, andif γ is
also constant, then the Kramers and the Bohr-Wheeler values
differ by approximatelya constant scaling factor. For this rea-
son we have allowed a constant scaling to the Bohr-Wheeler
width.
Lestone [50] developeda treatment of fission which explic-
itly included the tilting collective degree of freedom at saddle
point. Tilting is where the compound nucleus’s spin is not
perpendicular to the symmetry axis. For strongly-deformed
objects like the saddle-point configuration, this costs energy
and thus decreases the fission probability. The decay width
becomes
?J
where the summation is over K, the projection of the spin on
the symmetry axis and
ΓLestone= ΓBW
K=−Jexp
?
−K2
2Ieff
?
2J + 1
1
Ieff
=
1
I?
−
1
I⊥
and I?and I⊥are the saddle-pointmoments of inertia parallel
and perpendicular to the symmetry axis, respectively. In this
work, the moments of inertia as well as the spin-dependent
saddle-point energies were taken from the finite-range calcu-
lations ofSierk [26]. Deviationsfromthe Bohr-Wheelervalue
are largest for the highest spins and thus the Lestone modifi-
cation will be more important in fusion reactions.
We have tried to reproduce simultaneously fission cross
sections from fusion and spallation experiments by:
• Adding a constant to the Sierk fission barriers for all
spins.
• Scaling the decay width by a constant factor.
• Adjusting the af/anratio.
• Using either the Bohr-Wheeler or the Lestone formal-
ism.
• Introducing a constant fission delay.
B.Fission Probability
Examples of fits to the
evaporation-residue excitation functions are shown in Fig. 2.
As the sum of these quantities (the fusion cross section)
is fixed in the calculations, the degree to which the fission
probability is reproduced is best gauged by the fit to the
smaller quantity, i.e. σfis at low bombarding energies and
σER at the higher values.Good fits were obtained with
ΓBW× 2.46,af/an = 1.00 (long-dashed curves), ΓBW×
1.00,af/an = 1.036 (solid curves), ΓLestone× 7.38,af/an =
1.00 (dotted curves), and ΓLestone× 1.00,af/an = 1.057
(short-dashed curves). The ΓBW× 2.46,af/an= 1.00 calcu-
lation is also almost identical to a ΓBW× 1.00,af/an= 1.00
calculation(notplotted)obtainedwith theSierk fission barrier
reducedby1.0MeV.Withanevenlargerbarrierreductionfac-
tor, one could arrive at a solution where the decay-width scal-
ing factor is less than unity and consistent with the Kramers’
scaling factor in Eq. (3).
As it is impossible to distinguish these different ways of
modifying the fission probability from the fusion data alone,
we now consider the constraint of adding the spallation data
to the analysis. In Fig. 3, we show the equivalent calcu-
lations for the mass distributions of the products of the 1-
GeV p+208Pb spallation reaction.
ties, the ΓBW× 1.00,af/an = 1.036 calculation reproduces
19F+181Ta→200Pb fission and
Of all these possibili-

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6
E* [MeV]
50 100150
[mb]
σ
2
10
3
10
=1.00
n
/a
f
*2.46,a
BW
Γ
Γ
Γ
Γ
=1.036
n
=1.00
n
/a
f
=1.057
n
/a
f
/a
f
*1.00, a
BW
*7.38,a
Lestone
*1.00, a
Lestone
fis
σ
ER
σ
Figure 2. (Color online) Comparison of GEMINI++ predictions to
the experimental evaporation-residue and fission excitation functions
for the19F+181Ta reaction.
1
10
102
4060 80100120140160180200A
σ(A) [mb]
ΓBW*2.46, af/an=1.00
ΓBW*1.00, af/an=1.036
ΓLestone*7.38, af/an=1.00
ΓLestone*1.00, af/an=1.057
Figure 3. (Color online) Comparison of experimental and calculated
residual mass distributions for the 1-GeV p+208Pb reaction. Pre-
dictions of the INCL4.5-GEMINI++ code are shown for different
adjustments of the fission width. Experimental data from Ref. [44].
the yield of the fission peak best.
ΓBW× 2.46,af/an = 1.00 calculation and the reduced fis-
sion barrier calculation with ΓBW × 1.00, af/an = 1.00
(not shown) were again almost identical. Thus the ambigu-
ity between the effect of magnitude of the fission barrier and
Kramers scaling factor was not lifted with the inclusion of
spallation. However the other ambiguities associated with our
fitting parameters were removed. We will continue using the
ΓBW× 1.00,af/an= 1.036 calculation as our best fit to both
sets of experimental data, but it should be noted that with re-
duced fission barriers, an equivalent solution with a Kramers
scaling factor (< 1) can also be obtained. We are just not
able to constrain the magnitudeof the Kramers factor from all
these data.
Comparison of GEMINI++ predictions to the experimen-
tal fission and evaporation-residueexcitations functions listed
for the heavy-ion-induced fusion reactions in Table II are
shown in Figs. 4 to 7. The solid curves show the predic-
We also note that the
E* [MeV]
50100150
[mb]
σ
1
10
2
10
3
10
ER
σ
fis
σ
Pt
188
→
Tm
169
F+
19
(c)
[mb]
σ
1
10
2
10
3
10
Pb
200
→
Er
170
Si+
30
(b)
ER
σ
fis
σ
[mb]
σ
10
2
10
3
10
ER
σ
fis
σ
Pb
200
→
Ta
181
F+
19
(a)
Figure 4. (Color online) Comparison of experimental and calculated
fission and evaporation-residue excitation functions for the indicated
reactions. Solid lines: Bohr-Wheeler fission width, af/an = 1.036,
no fission delay. Dashed lines: Lestone fission width, af/an =
1.057, 1-zs fission delay. Dotted lines: Bohr-Wheeler fission width,
energy-dependent effective af/anratio with r=1.0747.
tions with af/an=1.036, Sierk fission barriers, and no scal-
ing of the Bohr-Wheeler decay width. For spallation, Figs. 8
to 10 show the comparison between measured and calculated
residue mass distributions. Finally, Fig. 11 shows the compar-
ison between measured [51] and calculated excitation curves
for the fission cross section in proton collisions with181Ta, a
low-fissility target.
The central result is that it is possible to reproduce the to-
tal fission cross section for all the studied spallation reactions
by adjusting only one free parameter, namely the af/anratio,
which was set equal to 1.036 in our calculations, whilst the
global scaling of the fission width and of the fission barrier
were kept equal to 1; no Lestone correction was introduced.
A global scaling of the fission width is roughly equivalent
to a reduction of the barrier height, but in both cases, these
adjustment alone do not fit the data. The adjustment of the
af/anratio, on the other hand, is characterized by a different
excitation-energy dependence, which is better suited for the
description of fission from spallation remnants. The Lestone
correction, which suppresses the fission width at high spin,
does not have a large effect on spallation data, some 80% at
most. This is due to the small angular momenta generated in
the intranuclear cascade.
For the heavy-ion-induced fusion data, the GEMINI++

Page 7

7
E* [MeV]
50100150200
[mb]
σ
1
10
2
10
3
10
Ra
216
→
Au
197
F+
19
(c)
ER
σ
fis
σ
[mb]
σ
−1
10
1
10
2
10
3
10
ER
σ
fis
σ
Th
216
→
W
184
S+
32
(b)
Ra
216
→
W
186
Si+
30

[mb]
σ
10
2
10
3
10
ER
(a)
σ
fis
σ
Th
224
→
Pb
208
O+
16
Figure 5. (Color online) Same as Fig. 4.
E* [MeV]
50100150
[mb]
σ
−2
10
−1
10
1
10
2
10
3
10
ER
σ
fis
σ
Dy
158
→
La
139
F+
19
(c)
[mb]
σ
1
10
2
10
3
10
ER
σ
fis
σ
Yb
168
→
Sm
150
O+
18
(b)
[mb]
σ
−1
10
1
10
2
10
3
10
W
178
→
Tb
159
F+
19
(a)
ER
σ
fis
σ
Figure 6. (Color online) Same as Fig. 4.
E* [MeV]
50100150200
[mb]
σ
10
2
10
3
10
Er
156
→
Zr
90
Ni+
64
(c)
ER
σ
fis
σ
[mb]
σ
10
2
10
3
10
Yb
160
→
Mo
100
Ni+
60
(b)
[mb]
σ
10
2
10
3
10
Tl
193
→
Ho
165
Si+
28
(a)
Figure 7. (Color online) Same as Fig. 4.
1
10
102
406080100120140160180200
A
σ(A) [mb]
ΓBW, af/an=1.036
ΓLestone, af/an=1.057, tdelay=1 zs
ΓBW, r=1.0747
ΓLestone, tdelay=1 zs, r=1.1
Figure 8. (Color online) Residue-mass distribution for the p+197Au
reaction at 1GeV. Predictionsof the INCL4.5-GEMINI++code are
shown for different adjustments of the fission width. Experimental
data from Ref. [43].
predictions are also generally quite good, however, there are
a couple of reactions where significant deviations are found.
Firstly, for the18O+150Sm →168Yb and19F+139La→158Dy
reactions shown in Figs. 6(b) and 6(c), the fission cross sec-
tion is overestimate by almost an order of magnitude. In com-
parison, the Ni-induced reactions in Figs. 7(b) and 7(c) mak-
ing similar mass compound nuclei are reproduced much bet-

Page 8

8
1
10
102
406080100120140160 180 200A
σ(A) [mb]
ΓBW, af/an=1.036
ΓLestone, af/an=1.057, tdelay=1 zs
ΓBW, r=1.0747
ΓLestone, tdelay=1 zs, r=1.1
Figure 9. Same as Fig. 8 for p+208Pb at 500 MeV. Experimental data
from Refs. [46, 52]. The experimental fission cross sections have
been multiplied by a factor of 146/232 (see text for details).
1
10
102
50 75100125150175200225A
σ(A) [mb]
ΓBW, af/an=1.036
ΓLestone, af/an=1.057, tdelay=1 zs
ΓBW, r=1.0747
ΓLestone, tdelay=1 zs, r=1.1
Figure 10. (Color online) Same as Fig. 8 for p+238U at 1 GeV. Ex-
perimental data from Refs. [45, 53, 54].
ter. It is difficult to understand these18O- and19F-induced
reactions and previous attempts also failed to reproduce the
data [27]. For instance at the highest bombarding energies,
the excitation energies probed in these reactions overlapthose
in the Ni-induced reactions. Similarly the predicted angular-
momentum region over which the fission yield is determined
in the O- and F-induced reactions is similar to that in which
the residue yield is determined in the Ni-induced reactions.
Unless there are significant non-fusionprocesses, such as pre-
equilibriumorincompletefusionoccurring,thesedatasuggest
an entrance-channel dependence of the fission decay proba-
bility which would violate the compound-nucleushypothesis.
However we see no evidence of such an effect for the O- and
F-induced induced reactions with heavier targets. Clearly our
understanding of fission for A < 170 is lacking and more
studies are needed.
The other case where the GEMINI++ predictions fail is
for the32S+184W→216Th reaction in Fig. 5(a). Here the
1
10
020040060080010001200
Ep [MeV]
1400
σfission [mb]
ΓBW, af/an=1.036
ΓLestone, af/an=1.057, tdelay=1 zs
ΓBW, r=1.0747
ΓLestone, tdelay=1 zs, r=1.1
Figure 11. (Color online) Excitation curve for the fission cross sec-
tion in p+181Ta. Predictions of the INCL4.5-GEMINI++ code are
shown for different adjustments of the fission width. Experimental
data from Ref. [51].
evaporation-residue cross section is exceedingly small (∼
0.1mb) andis overpredictedby almost an orderof magnitude.
However, the calculations gets the excitation-energy depen-
denceofthecrosssectionscorrectwhichpreviouscalculations
could not do without invoking an excitation-energy depen-
dence of the dissipation strength [11]. In our calculations the
predicted excitation-energy dependence of the residue cross
section is a consequenceof the assumed excitation-energyde-
pendence of the level-density parameter [5]. Low probability
events in the statistical model are generally quite sensitive to
the statistical-model parameters. In this case, it was demon-
strated that the residue yield is very sensitive the absolute
value of the level-density parameter and its excitation-energy
dependence [5]. For example, the residue yield is increased
by 2-3 orders of magnitude when the level-density parameter
is changed from a = A/7.3 to A/11 MeV−1. Further re-
finement of the value of this parameter at the larger excitation
energies probed in this more symmetric fusion reaction may
be needed in the future.
Alternatively, there is evidence that for this mass region,
quasi-fission completes with fusion reactions even at the
lower ℓ waves associated with evaporation residue produc-
tion [38, 55]. In the case of216Ra compound nucleus in
Figs. 5(b) and 5(c), Berriman et al. have indicated that both
the19F+197Au and30Si+184W reactions have reduced evap-
oration residue cross sections due to quasi-fission competi-
tion [38]. Therefore this higher mass region for the heavy-
ion reactions is subject to more uncertainty in constraining
the statistical-model parameters.
C.Fission-Fragment Mass Distributions
Previous treatments of the fission-fragment mass distribu-
tion have assumed thermal models where the mass division
is determined at either the saddle-point (Moretto’s formal-

Page 9

9
primary
A
6080100120140
(A) [mb]
σ
1
10
2
10
E*=67 MeV
E*=125 MeV
Pb
198

→
W
182
O+
16
Figure 12. (Color online) Comparison of experimental (data points)
and simulated (curves) distributions of the primary fission-fragment
masses for the198Pb compound nucleus formed in the16O+182W
reaction at the two indicated excitation energies. The total fission
cross section has been calculated using the Bohr-Wheeler formal-
ism, af/an = 1.036, without any fission delay. The dotted curves
was obtained using the Moretto formalism with Sierk’s conditional
barriers to define the mass distributions. The solid and dashed curves
were obtained using the Rusanov systematics using the saddle and
scission-point temperatures, respectively.
ism) or at the scission configuration [56]. In reality, a com-
pletedescriptionprobablyrequiresonetofollowthetrajectory
from saddle to scission including fluctuations, for example by
Langevin simulations [57, 58]. However for large-scale sim-
ulations, this is too time consuming, so a simpler and faster
procedure for determining mass division is required.
Experimental mass distributions for heavy-ion-induced
fusion-fission reactions are shown in Figs. 12 and 13. In these
figures, the experimental fission-fragment masses was not di-
rectly measured, but rather the ratio of primary masses (be-
fore post-scission particle evaporation) was inferred from ei-
ther the ratio of the measured fission-fragment velocities or
kinetic energies. The absolute primary mass was assumed to
be equal to the compound-nucleus mass, which of course ig-
nores the pre-scission evaporationof light particles. However,
the distributions simulated by GEMINI++ (the curves in the
figures) were analyzed in the same manner as the experimen-
tal data and thus contain the same deficiencies.
The data in Fig. 12 is for the16O+182W→198Pb reactions.
The relative mass distribution were obtained from Ref. [59]
and absolute normalization was achieved using the fission
cross sections measured in Ref. [60]. The data in Fig. 13 are
for the216Ra compound nucleus at E∗∼ 60 MeV with three
entrance channels:
The fission excitation function for the latter two are shown in
Figs. 5(b) and 5(c).
In Fig. 12, the dotted curves show the mass distribution de-
termined from Moretto’s formalism using interpolated values
of Sierk’s finite-range calculations for the conditional barriers
[15]. The total fission width (the total width for all mass di-
visions associated with the peak in the mass distribution) was
normalized to the Bohr-Wheeler value for these calculations.
Therefore in this figure, only the shape of the mass distribu-
12C+204Pb,19F+197Au, and28Si+186W.
primary
A
6080 100120140
(A) [mb]
σ
1
10
2
10
W
186
Si+
28
2)
×
Au(
197
F+
19
2)
×
Pb(
204
C+
12
60 MeV
≈
E*
Figure 13. (Color online) Comparison of experimental (data points)
and simulated (curves) distributions of the primary fission-fragment
masses for the216Ra compound nucleus at E∗∼ 60 MeV formed
in the12C+204Pb,19F+197Au and30Si+186W reactions. The total
fission cross section has been calculated using the Bohr-Wheeler for-
malism, af/an = 1.036, without any fission delay. To aid in viewing,
the data and curves have been scaled by the indicated amounts.
tion is determined from the Moretto formalism. Clearly these
distributions are much wider than the experimentalquantities.
This is quite typical of other cases where a peak exist in the
mass distribution at symmetry. See for example the study of
151Eu compound nuclei in Ref. [61] and the light, but high
spin110Sn compound nuclei studied in Ref. [62]. For less
fissile nuclei where the mass distribution has a minimum at
symmetric division, the Moretto formalism (with Sierk barri-
ers) give a much better description of the experimental data.
See for example the studies of111In [19],102Rh and105Ag
[17] and75Br [21] compound nuclei.
The cause of this inadequacy for the heavier systems could
either be an incorrect asymmetry dependence of Sierk’s con-
ditionalbarriersora failureofMoretto’sformalism. Thelatter
predicts the asymmetry distribution at the ridge line of condi-
tional saddle points and assumes that the mass asymmetry is
unchanged during the descent from saddle to scission.
As an alternative to using Moretto’s formalism, we have
used the systematics of fission-fragment mass distributions
complied by Rusanov et al. [22]. The mass distribution is as-
sumed to be Gaussian and its variance is parameterized as
σ2
A=
A2
CNT
16d2V
dη2(Z2/A,J)
where
ergy surface with respect to the mass-asymmetry deformation
parameter(η = 2A1−A2
sion). This quantity is parameterized as a function of the fis-
sility Z2/A and spin J. The quantity T is the nuclear temper-
ature where
d2V
dη2 is the second derivative of the potential en-
A1+A2where A1and A2is the mass divi-
1
T=dlnρ
dU
.
Rusanov et al. considered three parameterizations ofd2V
with three different temperatures. Either 1) the temperature
dη2,

Page 10

10
of the fission nucleus at the saddle-point is used, but no pre-
saddlelight-particleevaporationsareallowed, 2)as above,but
pre-saddle evaporations are allowed, or 3) the temperature at
the scission point is used. The first of these is not realistic
and was not considered and the second is basically consistent
with the ideas of the Moretto formalism. The latter two can
be called saddle-point and scission-point models where the
mass distributionsare bothdeterminedthermally. Inthese two
cases, the quantityd2V
try dependence of the potential-energy surface at the saddle
and scission points, respectively.
In GEMINI++, oncefission is decidedfor an event, evapo-
ration during the saddle-to-scission transition is allowed. This
is important for the scission model, as we need to determine
the temperature at the scission point. The saddle-to-scission
evaporation is treated in a simplified manner using spherical
level densities and transmission coefficients in the Weisskopf-
Ewing evaporation formalism with the deformation-plus-
rotational energy removed from the total excitation energy.
The deformation-plus-rotational energy of the scission con-
figurationis determinedas the sum of fission-fragmentkinetic
energy from Viola’s systematics [63] and the fission Q-value.
Evaporationduringthesaddle-to-scissiontransitionoccursfor
a period proportional to the difference in energy between the
saddle and scission points, i.e.
dη2 shouldbe identified with the asymme-
tss= kss(Esaddle− Escission)
consistent with large viscosity. The parameter kssis related to
the magnitude of this viscosity was fixed to kss=1 zs/MeV by
fitting pre-scission neutron multiplicities from Ref. [7].
The solid and dashedcurves in Fig. 12 show the predictions
with the Rusanov saddle and scission-point systematics, re-
spectively. These predictions are almost identical, and for the
lowest excitation energy, the curves completely overlap and
cannot be distinguished. This is not surprising as both Ru-
sanov systematics are fits to3He induced fission and fusion-
fission data including the data set of Fig. 12. However, at
higher excitation energies such as those sampled in spalla-
tion reactions, the two systematics give quite different pre-
dictions as the thermal excitation at scission increases much
more slowly with compound-nucleusexcitation than does the
saddle-point value [7]. Figure 14 compares, the two system-
atics for the 1-GeV p+Pb spallation reaction. In this case, the
predictedmass distribution obtainedwith the scission system-
atics (dashed curve) is too narrow, while the saddle systemat-
ics (solid curve) gives good agreement.
The success of the Rusanov’s saddle-point systematics thus
suggestthat thefission mass divisionis determinedquiteclose
to the saddle-point configuration. It addition it indicates that
the Moretto formalism is still applicable for near symmetric
divisions of heavy nuclei. However it should not be used with
Sierk’s conditional barriers in this region.
The differences between the mass-asymmetry dependence
of Sierk’s conditional barriers and the Rusanov systematics
are shown directly in Fig. 15 for149Tb and194Hg compound
nuclei at J=0. The dashed curves are parabolic functions with
curvatures from the Rusanov systematics and with the sym-
metric fission barriersfromSierk’s calculations. The Rusanov
10
-1
1
10
40 50 6070 80 90100 110 120 130A
σ(A) [mb]
Figure 14. (Color online) Comparison of experimental (data points)
and simulated (curves) distributions of the fission-fragment masses
for the 1-GeV p+Pb spallation reaction. The blue dotted curve was
obtained using the Moretto formalism with Sierk’s conditional barri-
ers to define the mass distributions. The black solid and red dashed
curves were obtained using the Rusanov systematics using the saddle
and scission-point temperatures, respectively.
results have larger curvatures at symmetry than Sierk’s pre-
dictions and thus give narrower fission-fragment mass distri-
butions. The differences between Sierk’s predictions and the
Rusanov systematics is much larger for the heavier194Hg nu-
cleus. For even heavier nuclei, the asymmetry coordinate in
the finite-range calculations becomes undefined as the saddle-
pointconfigurationhas no well definedneck [64]. If this is the
case, then the Moretto formalism is no longer applicable for
these systems and the mass asymmetry is determined during
the descent from saddle to scission. In such cases the inter-
pretation of the Rusanov systematics in terms of a Moretto-
type may be suspect. We note that the Z2/A dependence of
d2V/dη2in Rusanov systematics has an abrupt slope change
at Z2/A=24 possibly related to this effect. However, even in
the p+U spallation reaction we produce Z2/A ratios that are
below this value.
The Rusanov saddle systematics was used for the other
spallation predictions in Figs. 8 to 10 and gives quite good
agreement. However, for the p+238U reaction in Fig. 10, the
simulation fails to reproduce the small shoulder in the fission
mass distribution for higher mass. The Rusanov systematics
only gives the width of the distribution and will not predict
finer structures linked to shell effects, such as this.
In Fig. 13, the simulated mass distributions (from the sad-
dle systematics) for the216Ra compoundnuclei reproducethe
data reasonablywell with the exceptionof the12C+204Pb data
where experimental distribution is somewhat narrower. Ber-
riman et al. [38] suggest that12C+204Pb data is all fusion-
fission while the19F+197Au and30Si+184W data both con-
tain quasi-fission contributions making the mass distributions
wider. This would imply that for the more massive compound
nucleus, the Rusanov systematics overestimate the width of
the statistical fission mass distributions as many of the heavy-
ion data used in these systematics have contributions from

Page 11

11
)
2
+A
1
|/(A
2
−A
1
|A
0 0.20.40.6 0.81
B [MeV]
10
15
20
25
30
35
Hg
194
Tb
149
Sierk
Rusanov
Figure 15. (Color online) Comparison of asymmetry dependences of
conditional barriers for149Tb and194Hg nucleus at J=0. The solid
curves are the predictions from Sierk’s finite-range calculations. The
dashed curves are parabolic functions with curvatures taken from the
Rusanov systematics and with the symmetric fission barriers taken
from the finite-range calculations. The mass-asymmetry coordinate
is defined in terms of A1 and A2, the two masses following binary
division.
1
10
102
40 6080 100120 140 160 180200A
σ(A) [mb]
tdelay=0 zs
tdelay=1 zs
tdelay=10 zs
Figure 16. (Color online) Same as Fig. 3 for the Bohr-Wheeler fis-
sion width, af/an = 1.036 and three different values of the fission
delay.
quasi-fission. However, the Rusanov systematics also con-
tains the lower-spin3He-induced fission data in this mass re-
gion and here quasi-fission is expected to be absent. Thus
spallationmass distributionswhichsamplelowerspins arenot
expected to suffer from this problem.
D. Fission Delays
Apart from the lighter compound nuclei, we have demon-
strated that a unified description of fission widths in fusion
and spallation reactions can be obtained.
unique, apart from an ambiguity between the height of the
fission barrier and the Kramers scaling factor. However we
The solution is
1
10
102
40 6080 100120 140160 180200A
σ(A) [mb]
tdelay=0 zs
tdelay=1 zs
tdelay=10 zs
Figure 17. (Color online) Same as Fig. 3 for the Lestone fission
width, af/an = 1.057 and three different values of the fission delay.
willnowshowthatanotherambiguityariseswhenfissiontran-
sients are considered. To show the sensitivity of predictionsto
transients, we have incorporated a simple implementation of
these in GEMINI++; the fission width is set to zero for a
time tdelay, after which it assumes its asymptotic value. Dur-
ing this fission-delayperiod, the compoundnucleuscan decay
by light-particle evaporation and intermediate-mass-fragment
emission. The fission delay is expectedon theoretical grounds
to be logarithmically dependent on nuclear temperature [6],
but this weak dependence (and any mass dependence) has
been neglected in a first approximation. Figure 16 shows the
dependence of the predicted mass distributions for the 1-GeV
p+208Pb reaction with tdelay=0, 1, and 10 zs. Even a small
1 zs delay has a large effect on the yield in the fission peak.
Therefore,the spallation reactions should be quite sensitive to
the fission transients. Tishchenko et al. also expected large re-
ductions is the fission probability in 2.5-GeV p+197Au,209Bi,
238U reactions due to fission transients; however, they were
also able to reproduce the fission yield within the standard
statistical-model framework [14].
Jing et al. [65] find the effect of increasing the fission de-
lay can be largely counteracted by increasing the value of the
af/an parameter. Both parameters have little effect on the
fission probability at low excitation energies. However with
increasing excitation energy, the fission probability becomes
ever more sensitive to both tdelayand af/an. Even with the
large range of excitation energies explored in this work, we
found it is impossible to break the ambiguity between tdelay
and af/an. To illustrate this, Fig. 18 compares the200Pb fu-
sion data to GEMINI++ calculations with fission delay for
both the Bohr-Wheeler [Fig. 18(a)] and Lestone [Fig. 18(b)]
formalisms. The values of tdelayand af/anlisted in these fig-
ures were obtained by reproducing the fission cross section in
the 1-GeV p+Pb spallation reaction. For the Bohr-Wheeler
case in Fig. 18(a), one see that the calculations with tdelay= 1
and 10 zs are almost identical and within 30% of the experi-
mental values. The calculation with tdelay= 0 zs fits the data
somewhat better, but all calculations can be deemed accept-

Page 12

12
E* [MeV]
50100150
[mb]
σ
2
10
3
10
=1.043
n
=1.057
n
=1.077
n
/a
/a
f
=0 zs, a
delay
t
/a
f
=1 zs, a
delay
t
f
=10 zs, a
delay
t
(b) Lestone
[mb]
σ
2
10
3
10
=1.036
n
=1.050
n
=1.069
n
/a
/a
f
=0 zs, a
delay
t
/a
f
=1 zs, a
delay
t
f
=10 zs, a
delay
t
(a) Bohr−Wheeler
fis
σ
ER
σ
Figure 18. (Color online) Comparison of GEMINI++ predictions
using (a) the Bohr-Wheeler and (b) the Lestone fission formalism to
the experimental evaporation-residue and fission excitation functions
for the19F+181Ta reaction. The curves are labeled by the tdelay and
af/an values obtained from fitting the fission cross section for the
1-GeV p+Pb reaction.
able.
For the Lestone formalism in Fig. 18(b), the inclusion of a
delay with tdelay> 1 zs improves the agreement with the data.
As in Fig. 18(a), the calculations with tdelay≥ 1 zs are again
almost identical. The Lestone prescription with fission delay
also allows good agreement with the other data sets we have
considered, see the dotted-curves in Figs. 4 to 7 (fusion) and
Figs. 8 to 10 (spallation) which were obtained with af/an=
1.057and tdelay=1 zs. Calculations with the largertdelayvalues
produce a similar level of agreement. It is thus clear that the
magnitude of the fission transients cannot be deduced from
the fission probability alone.
IV. FISSION AT VERY HIGH EXCITATION ENERGY
Fission cross sections in fusion and spallation reactions
are dominated by the most densely populated regions of the
compound-nucleus E∗-J plane (Fig. 1). The successful re-
production of these data thus indicates that the GEMINI++
model gives an efficient descriptionof fission from compound
nuclei with excitation energy up to ∼ 300 MeV and spin up
to ∼ 60 ?.
It is possible to probe beyond this region if one considers
E* [MeV]
0200400 6008001000
fission
p
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p + U
p + Bi
p + Au
Figure 19. (Color online) Fission probability as a function of the
excitation energy of remnants in 2.5-GeV p+Au, Bi and U reac-
tions. Solid lines: Bohr-Wheeler width, af/an = 1.036. Dashed
lines: Lestone, af/an = 1.057, tdelay = 1 zs. Dotted line: Bohr-
Wheeler width, af/an = 1.02. Dash-dotted line: Bohr-Wheeler
width, af/an = 1.00.
other types of data. Tishchenko et al. [66] studied proton-
induced spallation reactions at 2.5 GeV on gold, bismuth and
uranium targets. They measured the fission probability in co-
incidence with neutron, hydrogen and helium multiplicities,
which can be used to reconstruct the excitation energy af-
ter the intranuclear cascade. They were able to reproduce
the measurements with an old version of the INCL-GEMINI
model by tuning the value of af/anon a system-by-system
basis, ranging from 1.000 for the uranium target to 1.022 for
the gold target. These af/anvalues are smaller than those dis-
cussed in the present work.
Fig. 19 indeed shows that our candidate parameter sets
(Bohr-Wheeler, af/an = 1.036; Lestone, af/an = 1.057,
tdelay = 1 zs) largely overestimate the fission probability at
high excitation energy deduced by Tishchenko et al. for p+U.
Note that the fission probability is well reproduced up to a
few hundred MeV, which is coherent with the results of the
previous section. The shape of the curve is indeed very sensi-
tive to the value of af/an, as Fig. 19 shows. We can interpret
this result an indication of the fact that while a large value of
af/anvalue is appropriate at low excitation energy, the effec-
tive af/anvalue at high excitation energy should be smaller.
An energy-dependent af/an ratio can naturally appear,
among other things, as a consequence of the fade out of long-
range correlations. To obtain a better reproduction of the the
Tishchenko data, we have considered a simple refinement of
the formula for the level-density parameter at saddle point,
Eq. (2), as follows:
? af(U) =
The r variable,which replaces the af/anratio, is a free param-
eter that describes the difference in the effect of long-range
correlations for the saddle point. In the limit of zero excita-
A
k∞− r(k∞− k0)exp
?
−f
κ
k∞−k0
U
A
?.(4)

Page 13

13
tion energy, Eq. (4) leads to
af
an
=
k0
k∞− r(k∞− k0).
while for U → ∞, af= an= A/k∞. The value of r thus de-
termines the af/anratio at low energy. We expect on physical
grounds that r should be slightly larger than one, to reflect the
increase in surface area and an enhanced collective enhance-
ment of the saddle-point configuration. This would also lead
to af/an> 1 at small U. The parameter f, on the other hand,
expressesthedifferentfade-outrate oflong-rangecorrelations
at the saddle point compared to the ground state. This quan-
tity is essentially unconstrained by experimental data. How-
ever, we observe that, from Sec. IIIB, the approximation of
an energy-independent af/anratio is a good one at low ex-
citation energies, since we can successfully reproduce fission
cross sections in fusion and spallation. We impose this condi-
tion by requiring that
∂(af/an)
∂U
????
U=0
= 0.
This introduces a correlation between the parameters f and r:
f =k∞− r(k∞− k0)
rk0
. (5)
There is no a priori reason to expect that the fade-out rate at
the saddle point (described by f) should be correlated with
the af/anratio at low energy. We make this assumption on a
phenomenologicalbasis. Note that Eq. (5) implies that f < 1
for r > 1, i.e., that long-range correlations should fade out
more slowly at the saddle point than in the ground state. One
should also note that κ has a very strong mass dependence [5]
and therefore the modificationof af/anwith excitation energy
is much stronger for the p+U reaction compared to the lighter
systems. One can indeed see from Fig. 19 that this is the sys-
tem that requires the biggest modification from our previous
solution.
We can finally determine the value of the r parameter by
requiring, for example, that the fission cross section for 1-
GeV p+208Pb be correctly reproduced. For a Bohr-Wheeler
width without fission delay, this condition yields r = 1.0747,
which corresponds to af/an = 1.051 for U = 0. For a Le-
stone width with a 1-zs fission delay, we get r = 1.1 and
af/an= 1.069 at U = 0. Fusion-fission and spallation-fission
are not severely affected by this modification, as shown by
the dotted and dashed-dotted curves in Figs. 4–11. The re-
sulting fission probability curves to the 2.5-GeV reactions are
shown in Fig. 20. We have good quantitative agreement up to
∼ 400 MeV and we can qualitatively reproduce the decrease
of fission probability with excitation energy for the uranium
target. This proves that the fission probability at very high
excitation energies is indeed sensitive to the fade-out of col-
lective effects at saddle point.
We have thus shown that one can obtain similar quality
of agreement with or without a fission delay, provided that
one increases the value of r. Therefore, while we agree with
Tishchenko et al. [14] that no transients are needed to explain
E* [MeV]
0 200 400600 8001000
fission
p
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p + U
p + Bi
p + Au
Figure 20. (Color online) Same as Fig. 19, for a Bohr-Wheeler width
(solid lines) or a Lestone width with a 1-zs fission delay (dashed
lines) with an energy-dependent effective af/anratio, Eq. (4).
their data, one cannot rule out the presence of fission tran-
sients as well. However, the above calculations and conclu-
sions shouldbe takenwith caution. Leveldensities andfission
probabilities are dramatically sensitive to all their ingredients
at such high excitation energies. We have indeed observed
that the shape of the curve can also be modified by e.g. in-
troducing surface terms (∝ A2/3) in the level-density param-
eter formula, or by considering different functional forms for
the fade-out of long-range correlations. Even the competition
with evaporationcannot be neglected. Finally, very high exci-
tation energies will eventually give rise to other phenomena,
such as nuclear expansion and multifragmentation, which are
not accounted for in our framework. With all these consider-
ations in mind, we conclude that pursuing perfect agreement
between calculations and measurements of fission probabili-
ties for E∗? 500 MeV is useless for our understandingof the
physics of de-excitation.
V.CONCLUSIONS
We have described the first coupling of the Liège In-
tranuclear Cascade model with the GEMINI++ compound-
nucleus de-excitation model.
calculated using Sierk’s finite-range liquid-drop fission bar-
riers [26] and the excitation-energy-dependent level-density
parameters of Ref. [5] adjusted to reproduce experimental
kinetic-energy spectra of light particles. The latter were very
important to obtain the correct excitation-energy dependence
for heavy systems. It was demonstrated that it is possible to
describe fission cross sections from spallation and heavy-ion
fusion reactions for 160 < A < 230 within the same frame-
work. The simultaneous application of the statistical-decay
model to spallation and fusion actually allows one to lift some
of the degeneracy of the model parameters. However, even
with the large range of spin and excitation energy studied, no
unique parameter set could be obtained and there remained
some ambiguities in the choice of parameters. In particular,
The fission probability was

Page 14

14
the effect of an increasing fission delay associated with fis-
sion transients could be offset by an increase in the parame-
ter af/an, the ratio of level-density parameters at the saddle-
point and ground-state configurations. In addition, modifica-
tions to the height of Sierk’s fission barrier could be offset by
scaling of the fission decay width which could be associated
with the Kramers scaling of the Bohr-Wheeler decay width
due to friction. In spite of these ambiguities, we present two
sets of statistical model parameters suitable for predictions of
fission probabilities for spins up to 60 ? and excitation ener-
gies up to ∼ 300 MeV.
From the study of the width of the fragment-fragmentmass
distributions in both fusion and spallation reactions, we were
able to differentiate between the systematics compiled by Ru-
sanov et al. based on thermal distributions at either the saddle
or scission-point. Only the saddle-point systematics provided
good reproduction of the experimental data in both types of
reactions, thus suggesting the fission mass division is deter-
mined close to the saddle point. The asymmetry dependence
of the saddle-point conditional barriers in the Rusanov sys-
tematics is stronger than Sierk’s prediction, which produces
very wide fission-fragment mass distributions when incorpo-
rated in the Moretto formalism.
We have proven that we can qualitatively describe fission
probabilities at excitation energies higher than 300 MeV by
accommodating different fade-out rates for the ground-state
and the saddle-point configurations. However, we cannot ex-
clude that other solutions are possible, given the uncontrol-
lable sensitivity of the predictions of the model to a large
number of its ingredients. Thus, we conclude that the theoret-
ical uncertainties on fission probabilities at very high excita-
tion energyare too large to permit drawingstrong conclusions
about the physics of highly excited nuclei.
ACKNOWLEDGMENTS
The authors wish to thank S. Leray for commenting on
the manuscript, K.-H. Schmidt for useful discussions and
J. Benlliure for kindly providingthe experimental data for the
p+181Ta system. This work was supported by the U.S. De-
partment of Energy, Division of Nuclear Physics under grant
DE-FG02-87ER-40316.
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