Eur. Phys. J. A (2016) 52: 342
DOI 10.1140/epja/i2016-16342-4
THE EUROPEAN
PHYSICAL JOURNAL A
Regular Article – Experimental Physics
Entrance channel systematics of pre-scission neutron
multiplicities
M. Shareef1,a , A. Chatterjee2 , and E. Prasad1
1
2
Department of Physics, School of Mathematical and Physical Sciences, Central University of Kerala, Kasaragod 671314, India
Inter-University Accelerator Centre, Aruna Asaf Ali Marg, New Delhi 110067, India
Received: 6 April 2016 / Revised: 18 July 2016
c Società Italiana di Fisica / Springer-Verlag 2016
Published online: 24 November 2016 – Communicated by T. Motobayashi
Abstract. Statistical model analysis has been performed for the available neutron multiplicity (νpre ) data
in the literature. Larger νpre values for more symmetric reactions have been observed in comparison with
asymmetric reactions forming the same compound nucleus, in most cases. A reverse trend has also been
noticed in a few cases. A systematic entrance channel dependence of fission timescale is brought out in
this work. Fission timescales calculated using the experimental νpre values fall into two distinct groups
according to the entrance channel mass asymmetry of the reaction with respect to the Businaro-Gallone
critical mass asymmetry. The difference in the delay between these two groups ranges between 20 and
100 zs, which is larger than that reported in some cases.
1 Introduction
Nuclear fission [1, 2] is a clear example for the large-scale
collective re-arrangement of nuclear matter. A fundamental feature of the fission process is the dissipation of collective energy. Even though the role of dissipation in fission
was proposed in the early days of fission discovery [3],
the success of Bohr-Wheeler theory [2] shadowed its acceptance. Clear experimental signature of dissipation in
fission was observed in the early 1980s [4], when measured pre-scission neutron multiplicities were observed to
be much larger than the Bohr-Wheeler predictions. Dissipation increases the fission timescales and thereby suppresses fission with respect to particle evaporation during
the transition of the compound nucleus (CN) from the
ground state deformation to the scission point.
Different experimental probes have been used to study
the role of dissipation in fission and fission timescales,
such as pre-scission particles (charged particles and neutrons) [5–10], giant dipole resonance (GDR) [11, 12]
gamma multiplicities and evaporation residues (ER) [13–
16]. In all these cases, the experimental observables could
not be explained by a model assuming the Bohr-Wheeler
fission width, indicating the dynamic nature of the fission
process [3, 5, 13, 17–31].
Among these probes, pre-scission neutron multiplicity
(νpre ) is one of the most efficient probes to understand
the fusion-fission timescales [5, 6, 32, 33]. Neutrons may be
evaporated from the compound system itself before fisa
e-mail: shareef.m.cuk@gmail.com
sion or from the fission fragments after scission. However
the neutrons from these different origins may be clearly
separated using their respective kinematics. Larger yields
of νpre compared with other probes such as pre-scission
charged particles or GDR gammas ensure better statistics
and thus lower uncertainities in the derived quantities.
Further, the analysis of the charged-particle multiplicities
requires the deformation dependence of the Coulomb barrier and particle binding energies, which are often modeldependent.
Several νpre measurements have been reported in the
literature [29, 30] populating the same CN using different
entrance channels. The fission timescale or viscosity associated with fission have been extracted from these measurements. The analysis of neutron multipicity results has
been performed using a combined dynamical-statistical
model [34] in some cases, where the time evolution of the
system is followed in terms of the Langevin equations. A
rather simple approach is to treat the problem in a statistical model [5, 31].
Despite a considerable amount of data available and
the number of efforts [31, 35–37] made over the years, a
common systematics could not be achieved for the neutron multiplicity data, and hence for the derived quantities. Saxena et al. [37] reported a formation delay depending on the entrance channel mass asymmetry (α) relative
to the Businaro-Gallone (BG) critical mass asymmetry
(αBG ) [38]. However the systematics of Baba et al. [39]
contradicted the above observation as they could not notice any discernible relationship between BG point and
the magnitude of the dissipation coefficient. The empirical
υpre
Page 2 of 8
Eur. Phys. J. A (2016) 52: 342
4
4
3
3
2
2
16
1
υpre
0
70
72
74
76
181
F+
Ta [30]
Hf [30]
78
80
19
O+
178
18
0
82
4
4
3
3
2
2
12
1
19
12
1
50
60
198
C+
O+
70
40
50
60
80
204
C+
197
F+
Pb [29]
Au [29]
1
11
16
0
Pt [40]
Os [41]
192
70
0
80
50
52
54
*
56
58
232
B+
O+
Np [37]
232
Th [37]
60
62
*
E (MeV)
E (MeV)
∗
Fig. 1. Experimental neutron multiplicity νpre as a function of CN excitation energy E for reactions populating 197 Tl [30],
210
Po [40, 41], 216 Ra [29] and 243,248 Cf [37] compound systems. Solid lines are a guide to the eye. Red lines and symbols are
used for more symmetric systems while less symmetric systems are represented by black lines and symbols.
formulation in ref. [36] also did not consider any entrance
channel dependence such as mass asymmetry or angular
momentum in νpre systematics.
Pre-scission neutron multiplicity measurements in
refs. [29, 30] populating the same CN through different
entrance channels showed that the experimental νpre values are larger for more symmetric systems in comparison
to the asymmetric systems at the same CN excitation energy (E ∗ ). The difference observed in the νpre values was
attributed to the spin distribution and a larger delay associated with the formation of the CN in these works. Examples of the cases where νpre values are observed to be larger
for more symmetric systems compared with less symmetric systems at the same excitation energies are shown in
fig. 1 for different systems reported in the literature.
However, a reversed trend has also been observed for a
few systems [26, 41, 42], where the measured νpre for more
symmetric entrance channel is smaller than that for less
symmetric entrance channel for the same CN. Examples
from the literature are shown in fig. 2.
The role of the nuclear shell structure has also been
speculated to be [26–28] a possible reason for the different
behaviour of νpre values observed in the neighbouring
isotopes of nuclei around the neutron shell closure at
N = 126. Recently, the N/Z dependence of νpre has also
been probed [28] by populating compound nuclei with the
same Z but different neutron numbers.
In this context, we present the statistical model [43]
analysis of the neutron multiplicity data for 26 reactions
reported in the literature in this article. Different reactions
considered in this work populate the same or similar CN
through different entrance channels and cover a fissility
range of 0.64 to 0.83, where the fissility (X) of the CN is
given by
2 2
Z
Z
(1)
X=
A
A crit
with
Z /A crit = 50.883 ∗
2
1 − 1.7826 ∗
A − 2Z
A
2 . (2)
Here, Z and A are the atomic and mass number of the
nucleus, respectively.
υpre
Eur. Phys. J. A (2016) 52: 342
Page 3 of 8
4
4
3
3
2
16
2
182
W [42]
28
170
Si+ Er [41]
O+
1
40
50
60
70
16
197
19
194
Au [41]
Pt [26]
O+
F+
80
1
40
50
*
60
70
80
*
E (MeV)
E (MeV)
∗
198
Fig. 2. Experimental νpre as a function of E for the reactions populating
Pb [41, 42] and 213 Fr [26, 41] compound systems.
Solid lines are a guide to the eye. Red lines are for the more symmetric system and black lines for the less symmetric system.
2 Statistical model analysis
where α is given by
In the analysis presented in this work we used the BohrWheeler fission width with separate delays for the presaddle and post-saddle phases of fission. In the pre-saddle
phase, diffusion over the barrier is modelled by introducing
a transient delay (td ). Fission is suppressed if the Monte
Carlo estimate of the elapsed time is less than td . Neutrons
emitted during this phase are accounted as pre-fission neutrons, provided that fission still takes place in a subsequent
step of the cascade. Using td this way amounts to a stepfunction approach, that models a more realistic exponential growth of the fission width. An increase in td would
result in an increase of νpre as the system gets more time
to spend in the pre-saddle stage. This may lead to the enhancement of ER production probablilities as some of the
exit channels may be diverted from the fission to the ER
formation.
A second contribution to the fission delay, tssc appears
during the transition of the system from the saddle point
to the scission point. Unlike in the pre-saddle phase, tssc
does not divert any flux from the fission channel to the
evaporation channel, as the system has already passed the
saddle point and is committed to scission. However, the
neutron emission in the post-saddle phase is increased as
a result of this delay. The overall fission timescale is thus
given by
tf iss = tstat + tdelay = tstat + td + tssc ,
(3)
where tstat is the Bohr-Wheeler timescale. While tstat has
a probability distribution (due to the input angular momentum distribution), td and tssc are considered to be
fixed numbers independent of E ∗ and angular momentum,
l in this work.
For convenience, we labelled the data according to the
reduced mass asymmetry r defined as the ratio of the entrance channel mass asymmetry (α) to αBG ,
r = α/αBG
(4)
α = (AT − AP )/(AT + AP )
(5)
and αBG is given by
⎧
⎪
⎪0,
⎨
αBG =
⎪
⎪
⎩p
when X < XBG ,
(X − XBG )
,
(X − XBG ) + q
(6)
when X > XBG ,
where p = 1.12, q = 0.240 and XBG = 0.396 [38].
The emission of neutrons during the evolution of the
compound system from the saddle to scission has been
incorporated by considering a deformation approximately
midway between the saddle and scission configurations.
We used the deformation-dependent level densities from
the prescriptions of Tōke and Swiatecki [44] in this work.
For most of the systems considered, fission and/or ER
cross section data are available [10, 15, 45–57]. The rotating finite-range model (RFRM) fission barrier [58] has
been considered without any scaling of the barrier height
and varied the ratio (af /an ) of level density parameters at
the saddle and ground state deformations for reproducing
the fission and ER data. Calculated fission and ER cross
sections depend on the pre-saddle delay td and the same
value was used consistently in our analysis. The sensitivity
of this procedure is shown in fig. 3, as an example, where
experimental fission cross section for the 28 Si + 170 Er reaction [5] is compared with calculations assuming RFRM
barrier and different values of af /an .
Fusion Q-values were calculated using the experimental masses. Shell-corrected masses of the target and the
projectile nuclei were taken from ref. [59] while the mass
of the CN is taken from ref. [60] without shell correction.
Particle transmission coefficients were obtained from the
global optical model potentials [43]. Fusion cross sections
Page 4 of 8
Eur. Phys. J. A (2016) 52: 342
200
3
10
fission(mb)
tdelay (zs)
150
Ref. [5]
af/an=1.08
af/an=1.04
af/an=1.00
2
10
55
60
*
65
70
203
210
Po, 0.98
Fr, 0.95
At, 0.85
Th, 0.73
224
Th, 0.98
229
248
Cf, 0.96
251
Es, 0.94
Np, 0.93
252
201
Bi, 0.95
192
Pb, 0.85
197
50
Pb, 0.83
243
Tl, 0.96 203Bi, 0.96
Am, 1.06
216
189
214
Ra,
1.02
Re, 0.95 198
Rn,
0.99
179
Pb, 1.0
Re, 0.97
213
210
Fr, 0.99
188
197
Pt, 0.96 Tl, 1.0 Po, 1.04
0
75
Fig. 3. Comparison of statistical model calculations using different af /an values with the experimental fission cross section
for the 28 Si + 170 Er reaction.
were calculated using the expression [61]
π (2l + 1)
,
k2
1 + exp[(l − lc )/δ]
213
100
Ra, 0.95
Fm, 0.93
198
E (MeV)
σf us =
220
216
(7)
l
where the values of the critical angular momentum (lc ) for
fusion and diffuseness (δ) were determined from empirical
expressions [61, 62].
The transition of the system from the equilibrium configuration to the saddle point requires a finite transient
time prior to the diffusion over the barrier. It is reasonable to assume a fixed value for the pre-saddle delay, td for
all the reactions considered in this work. This assumption
follows from the systematics presented in ref. [37] which
is in consensus with the theoretical predictions reported
in [63]. On the basis of a quantitative argument, it was
also reported that the chosen combination of td and tssc ,
in order to reproduce the observed neutron, proton and alpha multiplicities together for the same system, limits the
value of td ≤ 10 zs [8], where 1 zs = 10−21 s. Even though
the composite system is committed to fission at the saddle point, particles may be emitted during the dynamical
evolution from saddle to scission. This excess emission of
particles results in a larger delay in the post-saddle region.
We used a fixed value of td equal to 10 zs in this analysis
and varied tssc to fit the νpre data for the reactions given
in table 1.
3 Results
The fission delay extracted for different reactions studied
in this work is plotted against CN fissility (X) in fig. 4.
The dependence of fission delay on the entrance channel mass asymmetry is a remarkable effect noticed in this
work. The data systematically fall into two distinct groups
corresponding to the entrance channel mass asymmetry.
Reactions with reduced mass asymmetry r between 0.73
0.65
0.7
0.75
fissility
243
Cf, 1.01
0.8
Fig. 4. The fission delay as a function of CN fissility for the
systems given in table 1. Data points are connected to guide
the eye. The CN formed in reactions and the corresponding
reduced mass asymmetry are indicated near each data point.
and 0.98 are grouped together in the upper (blue) curve,
while reactions with r greater than 0.99 are grouped in
the lower (red) curve. The two curves in fig. 4 show a
systematic increase with fissility.
At low fissility, the curves tend to merge with tssc ≈
0. In such systems the path to fission may be short and
the fission delay is entirely in the pre-saddle phase. As
the fissility increases, the path to scission becomes longer
and the difference between the two groups appears to be
significant. Slight variations observed in each group may
be due to the experimental uncertainties and statistical
fluctuations.
An important observation is the absence of reverse
trend in the fission delay obtained, unlike the νpre values
noticed in some of the reactions illustrated in fig. 1 and
fig. 2. For example, the 198 Pb when populated through
16
O + 182 W and 28 Si + 170 Er yielded fission delay of 10 zs
and 50 zs respectively, in this analysis, though the experimental νpre values were larger for the 16 O + 182 W reaction
at matching E ∗ . Similar observations may also be made
for the 213 Fr nucleus, where the extracted fission delays
are 100 zs for the 19 F + 194 Pt reaction and 20 zs for the
16
O + 197 Au reaction, though the νpre values showed an
opposite trend.
4 Discussion
Statistical model analysis of νpre excitation functions has
been performed for a number of reactions available in
the literature for a range of CN excitation energies in
this work. In this analysis, we have considered the postsaddle emitter to be located midway between saddle and
scission, unlike a previous work [64] where this has been
fixed by considering the mean neutron emission energies.
We have also varied the excitation energy of the saddleto-scission emitter and the results are observed to be
qualitatively similar for the entrance channel dependence.
Eur. Phys. J. A (2016) 52: 342
Page 5 of 8
Table 1. Reactions considered for the analysis in this work. αBG is calculated using eq. (6). Fission time delay (tdelay = td +tssc )
has been deduced from the statistical model analysis. References for the measurements are given in the last column.
System
CN
Fissility
α
αBG
r
tdelay (zs)
Ref.
16
O+
Ta
197
Tl
0.693
0.837
0.833
1.00
10 ± 5
[30]
19
F + 178 Hf
197
Tl
0.693
0.807
0.833
0.96
40 ± 6
[30]
181
16
O+
W
198
Pb
0.704
0.838
0.839
1.00
10 ± 8
[42]
28
Si + 170 Er
198
Pb
0.704
0.717
0.839
0.85
50 ± 15
[41]
12
C + 198 Pt
210
182
Po
0.711
0.885
0.843
1.04
10 ± 7
[40]
18
O+
Os
210
Po
0.711
0.828
0.843
0.98
70 ± 20
[41]
16
O + 197 Au
213
Fr
0.742
0.849
0.860
0.99
20 ± 9
[41]
19
F+
194
Pt
213
Fr
0.742
0.821
0.860
0.95
100 ± 12
[26]
12
C + 204 Pb
216
Ra
0.750
0.888
0.864
1.02
25 ± 6
[29]
19
F + 197 Au
216
Ra
0.750
0.824
0.864
0.95
120 ± 12
[29]
11
B+
232
Np
243
Cf
0.832
0.909
0.899
1.01
40 ± 9
[37]
16
O + 232 Th
248
Cf
0.825
0.870
0.897
0.96
110 ± 16
[37]
189
Au
0.681
0.788
0.825
0.95
30 ± 5
[10]
192
20
169
Ne +
28
Si + 164 Er
192
Pb
0.715
0.708
0.846
0.83
50 ± 13
[41]
16
O+
198
214
Rn
0.729
0.850
0.853
0.99
20 ± 4
[27]
28
Si +
175
Lu
203
At
0.734
0.724
0.856
0.85
80 ± 11
[6]
16
O + 208 Pb
224
Th
0.763
0.857
0.870
0.98
120 ± 8
[65]
Tm
Pt
20
Ne +
Bi
229
Np
0.792
0.825
0.883
0.93
115 ± 15
[31]
11
B + 232 Th
243
Am
0.798
0.909
0.886
1.02
32 ± 5
[37]
20
Ne + 159 Tb
179
209
Re
0.647
0.776
0.801
0.97
10 ± 4
[10]
20
Ne +
Th
252
Fm
0.843
0.841
0.903
0.93
90 ± 12
[31]
20
Ne + 168 Er
188
Pt
0.670
0.787
0.818
0.96
10 ± 8
[31]
19
F+
184
203
Bi
0.708
0.812
0.842
0.96
40 ± 14
[66]
20
Ne + 181 Ta
201
Bi
0.712
0.800
0.844
0.95
60 ± 14
[31]
Th
0.768
0.636
0.873
0.73
125 ± 25
[67]
Es
0.834
0.848
0.900
0.94
110 ± 11
[41]
232
W
40
180
Ar +
Hf
220
19
F + 232 Th
251
Table 2. Comparison of fission delay deduced in this work with other analysis previously reported.
System
tdelay (zs) (this work)
tdelay (zs) (other analysis)
Ref.
28
50 ± 15
40
Hinde et al. [5]
Si + 170 Er
169
30 ± 5
15–45
Cabrera et al. [10]
50 ± 13
40
Hinde et al. [5]
20
Ne +
28
Si + 164 Er
28
Si + 175 Lu
80 ± 11
40–120
Ramachandran et al. [6]
16
O + 208 Pb
120 ± 8
105
Schmitt et al. [68]
20
Ne + 159 Tb
10 ± 4
15–45
Cabrera et al. [10]
Tm
40
Ar +
Hf
125 ± 25
90
Kuznetsov et al. [69]
18
O + 192 Os
70 ± 20
80
Mahata et al. [70]
180
Present results are compared with the data available for
few systems in table 2.
It may be noted that the fission timescales obtained in
this work do not differ significantly from other independent results where a similar treatment has been adopted
for the data analysis. Slight differences observed might be
due to the difference in the level density parameters and
the deformation energies of the saddle-to-scission emitter
used in such analysis. However, larger differences have
been observed for a few cases where completely different treatments have been followed in the analysis —for
example for the 40 Ar + 180 Hf reaction, the present analysis results in a delay of 125 zs, where a delay of 90 zs was
reported in ref. [67].
In a different approach, fission dynamics has been explained using an excitation-energy–dependent dissipation
Page 6 of 8
Eur. Phys. J. A (2016) 52: 342
16
200
181
O+178 Ta
F+ 182Hf
16
O+ 170W
28
Si+198 Er
12
C+ 192Pt
18
O+197Os
16
O+194 Au
19
F+ 204Pt
12
C+197 Pb
19
F+ 232Au
11
B+ Np
16
232
O+ 169Th
20
Ne+164 Tm
28
Si+198 Er
16
O+ 175Pt
28
Si+208 Lu
16
O+ 209Pb
20
Ne+
Bi
11
232
B+ 159Th
20
Ne+232Tb
20
Ne+168Th
20
Ne+181Er
20
Ne+180 Ta
40
Ar+232 Hf
19
F+184Th
19
F+ W
19
1
0.95
0.65
0.7
0.75
fissility
0.8
Fig. 5. The ratio of level densities (af /an ) used in the reactions studied in this work.
coefficient [26, 27] instead of the tdelay . The dissipation strength is often used as a free parameter in these
works [26,27], which represents the damping of the collective motion associated with fission. The dissipation coefficient is assumed to be a bulk property of the nucleus in such treatments. Conceptually dissipative effects are expected to increase the fission timescales and
thereby reduce the fission width. A consistent qualitative
trend —in terms of the dissipation strength and the fission delay— has been observed in systems forming the
same CN through different entrance channels, in both approaches, the one using dissipation and the one in the
present work. However, unlike the energy-dependent dissipation strength used in these works [26, 27], a constant
fission delay has been used in the present work, in the
excitation energy range covered for the various reactions
selected.
The present work clearly demonstrates the strong entrance channel dependence of fission timescales. Such a
dependence was previously reported by Saxena et al. [37].
A natural question is the selection of af /an values used
in the present analysis influencing the νpre values and fission timescales. The values of af /an used for the reactions studied in this work are shown in fig. 5. It may be
noticed that the af /an values are scattered around unity,
irrespective of the entrance channel mass asymmerty and
thus ruling out any influence on our observation.
When a CN is populated at the same E ∗ through different entrance channels, the l values may differ substantially. Since the fission barrier decreases with increasing
angular momentum, a larger angular momentum would result in higher fission probability at each step of the neutron
emission cascade. In order to explore the sensitivity of the
deduced fission delay on angular momentum, simulations
have been performed for the CN 213 Fr at E ∗ = 65 MeV,
for different angular momentum distributions by varying
the values of critical angular momentum (lc ), but keeping
the diffuseness parameter (δ) constant. In fig. 6 we show
150
tdelay(zs)
af /an
1.05
100
50
0
28
30
32
_
<l> h
34
36
38
Fig. 6. The fission delay required to reproduce νpre = 3.0 for
the CN 213 Fr at E ∗ = 65 MeV as a function of l.
the delay required to reproduce a value of νpre = 3.0 as a
function of l for the 213 Fr at E ∗ = 65 MeV. It may be
noticed from fig. 6 that in order to reproduce the same
νpre value, a larger delay is required at higher l values.
Considering the two entrance channels 16 O + 197 Au and
19
F + 194 Pt populating the same CN 213 Fr, the l value
is smaller for the former reaction compared with the latter reaction. The higher l value results in a larger fission
delay for the 19 F + 194 Pt reaction, even though the experimental νpre values were smaller for this reaction compared
with the 16 O + 197 Au reaction.
The difference in the tdelay observed between systems
with r < 1 and r ≥ 1, populating the same CN is apparently larger than the previous reports [37] for a few systems. A difference of 30 zs has been noticed previously, between the reactions with different entrance channel mass
asymmetry in ref. [37], which was attributed to the CN formation time. In the present work this difference in tdelay
for systems on either side of the BG point is observed to be
20–100 zs. While a major part of this timescale difference
could be attributed to the CN formation time, the possibility of enhanced dissipation effects at higher angular
momentum [71] could also play a role. This needs further
investigation.
5 Conclusion
Pre-scission neutron multiplicity is one of the powerful
probes to understand the fission timescales. Despite sizeable data available for the neutron multiplicities, a common systematics has not been achieved yet, partly due to
the differences in analysis methods and interpretation of
the data. In this work we analysed the available νpre data
using the statistical treatment.
Although νpre values show the reversal trend with entrance channel mass asymmetry for a few systems, no
such trend has been observed in the calculated fission
timescales. This has been attributed to different angular
momentum values populated in different entrance channels forming the same CN. An interesting observation is
Eur. Phys. J. A (2016) 52: 342
the appearance of the systems in two distinct groups depending on their entrance channel mass asymmetry —a
clear evidence of entrance channel properties influencing
the fission timescales.
The difference in fission delay between systems with
different entrance channel mass asymmetry is found to increase with an increase in fissility. Though the major part
of this difference could be attributed to the CN formation
time, dissipative effects at higher angular momentum may
also be influencing the timescales deduced from the νpre
data.
The present analysis calls for more νpre measurements,
for systems populating the same CN with different entrance channels with a wider excitation energy range, to
investigate its entrance channel dependence and systematics in detail in fusion-fission reactions.
One of the authors (SM) acknowledges the financial support of
Kerala State Council for Science Technology and Environment
(KSCSTE) in the form of research fellowship. We are thankful
to Dr. Sugathan, IUAC, for fruitful discussions.
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