Charge distributions of fission fragments of low- and high-energy fission of Fm, No, and Rf isotopes

Abstract

The charge (mass) distributions of fission fragments resulting from low- and high-energy fission of the even-even nuclei 254−260,264Fm, 258−264No, and 262−266Rf are studied with the statistical scission-point model. The calculated results are compared with the available experimental data. In contrast to the experimental data, the calculated mass distribution for 258Fm(s.f.) is strikingly similar to the experimental one for 257Fm(s.f.). The transformation of the shape of charge distribution with increasing isospin and excitation energy occurs gradually and in a similar fashion like that of the mass distribution, but slower. For 254Fm(i.f.), 257Fm(nth,f), and 260Fm(s.f.), the unexpected difference (symmetric or asymmetric) between the shapes of charge and mass distributions is predicted for the first time. At some critical excitation energy, the saturation of the symmetric component of charge (mass) yields is demonstrated.

Full text

PHYSICAL REVIEW C 97, 034621 (2018)
Charge distributions of fission fragments of low- and high-energy fission of Fm, No, and Rf isotopes
H. Paşca,1,2A. V. Andreev,1G. G. Adamian,1and N. V. Antonenko1,3
1Joint Institute for Nuclear Research, 141980 Dubna, Russia
2“Babeş-Bolyai” University, Faculty of Physics, 400084 Cluj-Napoca, Romania
3Mathematical Physics Department Tomsk Polytechnic University, 634050 Tomsk, Russia
(Received 12 February 2018; published 26 March 2018)
The charge (mass) distributions of fission fragments resulting from low-and high-energy fission of the even-even
nuclei 254−260,264Fm, 258−264 No, and 262−266Rf are studied with the statistical scission-point model. The calculated
results are compared with the available experimental data. In contrast to the experimental data, the calculated
mass distribution for 258Fm(s.f.) is strikingly similar to the experimental one for 257Fm(s.f.). The transformation
of the shape of charge distribution with increasing isospin and excitation energy occurs gradually and in a similar
fashion like that of the mass distribution, but slower. For 254Fm(i.f.), 257 Fm(nth,f), and 260Fm(s.f.), the unexpected
difference (symmetric or asymmetric) between the shapes of charge and mass distributions is predicted for the
first time. At some critical excitation energy, the saturation of the symmetric component of charge (mass) yields
is demonstrated.
DOI: 10.1103/PhysRevC.97.034621
I. INTRODUCTION
The mass distributions in low-energy fission of nuclei U–Cf
are known to be mostly asymmetric [1]. The first observation
of the onset of symmetric distribution was made in the sponta-
neous fission of 257Fm [1,2]. For 258 Fm, the spontaneous fission
results in the unexpected narrow symmetric mass distribution
[3]. The symmetric distributions have been also observed in
the spontaneous fission of 259Fm, 259,260Md, 258,262 No, and
259Lr [1]. The thermal-neutron-induced fission of 256,257 Fm [1]
leads to the symmetric mass distribution, but with larger width
than that in spontaneous fission of 258Fm. For spontaneous
fissioning Fm isotopes, an evolution of mass-yield shape with
increasing isospin occurs suddenly as demonstrated in Ref. [3].
However, a survey of literature reveals a lack of data on the
charge distributions of fission products in the spontaneous and
induced fission of Fm, No, and Rf, particularly as a function
of isospin and excitation energy. It is interesting to answer
the question if there is the difference between the shapes
(symmetric or asymmetric) of charge and mass distributions
in fission of nuclei in this transeinsteinium region. Thus, one
can focus the studies on the isospin and excitation energy
dependencies of the charge yields.
In this paper, we employ the improved version of the
statistical scission-point model [4–6], to study the evolution
of the charge (mass) distribution of fission fragments with
increasing mass number and excitation energy of even-even
isotopes of 254–260,264Fm, 258–264No, and 262–266 Rf. Our aim is
to predict the transformation of the shape of charge distribution
with increasing neutron number and excitation energy.
II. MODEL
The statistical scission-point model [4–6] relies on the
assumption that the statistical equilibrium is established at
scission of fissioning nucleus where the observable character-
istics of fission are formed. The reliability of this conclusion
is supported by a good description of various experimental
data (mass, charge, kinetic energy distributions, and neutron
multiplicity) with the scission-point models [4–16]. The din-
uclear system (DNS) [13,17–28] is shown to be well suited
for describing the scission configuration. So, the fissioning
nucleus at scission-point is modeled by two nearly touching
coaxial ellipsoids—fragments of the DNS with mass (charge)
numbers AL(ZL) and AH(ZH) for the light (L) and heavy (H)
fragments, respectively. Here, A=AL+AH(Z=ZL+ZH)
is the mass (charge) number of fissioning nucleus. By taking
into consideration the volume conservation, the shape of the
system is defined by the mass and charge numbers of the
fragments, the deformation parameters βi(i=L,H), and
interfragment distance R. The index i=Lor H. The potential
energy [4,5]
U(Ai,Zi,βi,R)
=ULD
L(AL,ZL,βL)+δUshell
L(AL,ZL,βL,E∗
H)
+ULD
H(AH,ZH,βH)+δUshell
H(AH,ZH,βH,E∗
H)
+VC(Ai,Zi,βi,R)+VN(Ai,Zi,βi,R)(1)
of the DNS consists of the energies of the fragments (the liquid-
drop energy ULD
iplus deformation dependent shell-correction
term δUshell
i) and energy V=VC+VNof the fragment-
fragment interaction. The interaction potential consists of the
Coulomb interaction potential VCof two uniformly charged el-
lipsoids and nuclear interaction potential in the double-folding
form [4–6,29]. In the region of fission fragments considered,
the interaction potential has a potential pocket and external bar-
rier located at the distances of about (0.7–1.1) fm and (1.5–2)
fm, respectively, between the tips of the fragments depending
on deformations of the fragments. The internuclear distance
Rin Eq. (1) corresponds to the position R=Rm(Ai,Zi,βi)of
2469-9985/2018/97(3)/034621(12) 034621-1 ©2018 American Physical Society

PAŞCA, ANDREEV, ADAMIAN, AND ANTONENKO PHYSICAL REVIEW C 97, 034621 (2018)
the minimum of this pocket. The decay barrier, Bqf (Ai,Zi,βi),
calculated as the difference of the potential energies at
the bottom of the potential pocket [R=Rm(Ai,Zi,βi)]
and the top of the external barrier [R=Rb(Ai,Zi,βi)], pre-
vents the decay of the DNS in R. Note that the height of the
barrier Bqf decreases with the charge asymmetry.
Because the thermal equilibrium is assumed at scis-
sion point, the excitation energy E∗(Ai,Zi,βi,Rm)=E∗
CN +
[UCN (A,Z,β)−U(Ai,Zi,βi,Rm)] at scission is calculated as
the initial excitation energy of the fissioning nucleus E∗
CN
plus the difference between the potential energy UCN(A,Z,β)
of the fissioning nucleus and one U(Ai,Zi,βi,Rm)ofthe
system at the scission point [4,5]. The excitation energies
E∗
iin the DNS nuclei are assumed to be shared between the
fragments proportionally to their masses.
The shell correction terms are calculated with the Struti-
nsky method and the two-center shell model [30]. The
damping of these terms with excitation energy E∗
iis in-
troduced as δUshell
i(Ai,Zi,βi,E∗
i)=δUshell
i(Ai,Zi,βi,E∗
i=
0) exp[−E∗
i/ED], where ED=18.5 MeV is the damping
constant. The relative formation-decay probability of the DNS
with particular masses, charges, and deformations of the
fragments is statistically calculated as follows [4,5]:
w(Ai,Zi,βi,E∗)
=N0exp −U(Ai,Zi,βi,Rm)+Bqf (Ai,Zi,βi)
T,(2)
where N0is the normalization factor. In Eq. (2), the temperature
is calculated as T=√E∗/a, where a=A/12 MeV−1is
the level density parameter in the Fermi-gas model. In our
calculations, a single value is used for the temperature at the
global potential minimum of Ubefore the shell damping.
The term exp [−Bqf /T ] describes the decay probability of
the system. With increasing elongation and decreasing charge
(mass) asymmetry the value of Bqf decreases, the system
becomes more unstable and decays.
In order to obtain the mass-charge distribution of fission
fragments, one should integrate Eq. (2) over βLand βH:
Y(Ai,Zi,E∗)=N0dβLdβHw(Ai,Zi,βi,E∗).(3)
The ratio of the yields of fragments with different charge/mass
numbers is governed by the difference in energy and in width
between the corresponding potential minima in the plane (βL,
βH), as seen in Eq. (2). For two potential energy surfaces with
the minima close in energy, a higher yield stems from the DNS
with a wider and more shallow minimum, and a lower yield
emerges from an abrupt and narrow minimum. This is a direct
result of Eq. (3)[4,5]. Note that in the case of fast (slow) growth
of the liquid-drop surface energy with increasing deformations,
the minimum in the (βL,βH) plane is positioned at smaller
(larger) deformations and is deep and narrow (shallow and
wide).
For the calculations of mass and charge distributions, one
should sum Y(Ai,Zi,E∗) over Ziand Ai, respectively. As
follows, the yield with mass AL(charge ZL) depends on the
number of fragmentations with the same AL(ZL) but different
ZL(AL).
The calculations were restricted only to even-even fission
fragments which mainly define the shapes of the charge and
mass distributions. The inclusion of the odd-even and odd-
odd fission fragments can elucidate even-odd effects but can
not appreciably change the smooth part of charge (mass)
distributions in which we are interested in. The even-odd
effects would add some oscillations to this smooth part. In
order to obtain a smoother curve for the mass distribution
and to simulate the minimal experimental uncertainty in the
measurement of the mass number of the fission fragment, each
calculated yield is smeared by the Gaussian with the width 1.5
amu [4–6,15,16]. At energies considered, we assume that the
neutron emission prior to fission will not cause major change
in the fission yield of isotopes of Fm and No.
The explanation of the asymmetric fission mode at high
excitations by the multichance fission or fission after consec-
utive neutron evaporations is strongly model dependent and
under discussion. As known from the experimental data [31],
the number of pre-scission neutrons does not exceed 1–2 at
excitation energies 50–60 MeV of heavy actinides. In nuclei
considered the estimated fission barrier is smaller than neutron
separation energy. Thus, the role of multichance fission is
expected to be suppressed in these nuclei.
III. CALCULATED RESULTS
In Figs. 1–8, the calculated charge and mass distribu-
tions resulting from the spontaneous and induced fission of
254,256,258,260,264Fm, 258,260,262,264No, and 262,264,266 Rf isotopes
are presented. The measured mass distributions [32–37]are
well reproduced for the spontaneous fission of 254,256Fm,
262No, 262 Rf, and for the thermal-neutron-induced fission of
255,257Fm. In the cases of fission of 254,256Fm, the measured
mass distributions present two asymmetric peaks as in our
theoretical results. The peak-to-valley ratio decreases with
increasing neutron number of Fm. In the fission of 262No and
262Rf , the measured [36,37] and calculated mass distributions
are symmetric with the large widths. In contrast to the exper-
imental data [3], the calculated mass distribution for 258Fm
is rather wide and has asymmetric bumps, even though the
symmetric yields are extremely enhanced, and is strikingly
similar to the experimental one for 257Fm (Fig. 1)[2]. For
258No (Fig. 5), the symmetric mass yields are larger than the
asymmetric mass-yields that is in a good agreement with the
experimental data [3].
The calculated charge distributions of spontaneous fission
of 254,256,258,260,264Fm have the asymmetric shape (Fig. 2). The
tendency to a more symmetric charge splitting of fermium
isotopes with increasing neutron number is clearly seen. In the
case of the spontaneous fission of 258Fm (Fig. 2), the charge
distribution presents two peaks, with a small minimum for
the symmetric Sn + Sn charge split. This is explained by the
fact that the high yield of Cd (ZL=48) or Te (ZH=52)
mainly originates from three mass fragmentations 122Cd +
136Te , 124Cd +134Te, and 126Cd +132Te which have almost
the same potential energies U. The yields of symmetric charge
split mostly originate from the 126Sn +132 Sn configuration,
for which the potential energy Uis larger by about 2 MeV
than those for the Cd + Te configurations. There are two more
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CHARGE DISTRIBUTIONS OF FISSION FRAGMENTS OF … PHYSICAL REVIEW C 97, 034621 (2018)
(nth,f)
254Fm (c)(b)
E* = 15 MeV
254Fm
Y
0.02
0.04
0.06
256Fm (e)(d) 256Fm (f)
256Fm
Y
0
0.04
0.08
0.12
258Fm (h)(g) 258Fm (i)
258Fm
Y
0
0.05
0.1
0.15
0.2 260Fm (l)(k)(j)
(o)(n)(m)
260Fm 260Fm
Y
0
0.06
0.12
0.18
0.24
0.3
A i
80 100 120 140 160
264Fm
A i
80 100 120 140 160
264Fm
A i
80 100 120 140 160
264Fm
Y
0
0.02
0.04
0.06
0.08 (s.f.)
254Fm (a)
FIG. 1. The calculated mass distributions (solid lines) resulting from the spontaneous (E∗=0 MeV, the first column) and induced (E∗=15
MeV, the third column) fission of the indicated nuclei 254,256,258,260,264 Fm, and thermal-neutron-induced (E∗∼6.3 MeV, the second column)
fission of nuclei 253,255,257,259,263Fm. The calculations are performed for even-even mass and charge fragmentations. The distributions are
normalized to unity. The open symbols represent experimental data of Refs. [3,32–35]. The spontaneous fission data for the 258Fm were
multiplied by the factor 0.8. In (h), the dashed line represents the provisional mass deduced in Ref. [35], and the open square represent
radiochemically deduced secondary yield, while the open circles are the derived primary distributions [33].
034621-3

PAŞCA, ANDREEV, ADAMIAN, AND ANTONENKO PHYSICAL REVIEW C 97, 034621 (2018)
(nth, f)
254Fm
E* = 15 MeV
254Fm
Y
0
0.05
0.1
0.15
256Fm 256Fm 256Fm
Y
0
0.05
0.1
0.15
0.2
0.25
258Fm 258Fm 258Fm
Y
0
0.1
0.2
0.3
0.4
260Fm 260Fm 260Fm
Y
0
0.2
0.4
0.6
0.8
Zi
35 40 45 50 55 60
264Fm
Zi
40 45 50 55 60
264Fm
Zi
40 45 50 55 60 65
264Fm
Y
0
0.05
0.1
0.15
0.2 (s.f.)
254Fm (c)(b)
(e)(d)
(g)
(f)
(h) (i)
(l)(k)(j)
(o)(n)(m)
(a)
FIG. 2. The calculated charge distributions (solid lines) resulting from the spontaneous (E∗=0 MeV, the first column) and induced
(E∗=15 MeV, the third column) fission of the indicated nuclei 254,256,258,260,264 Fm, and thermal-neutron-induced (E∗∼6.3 MeV, the second
column) fission of nuclei 253,255,257,259,263Fm. The calculations are performed for even-even mass and charge fragmentations. The distributions
are normalized to unity.
034621-4

CHARGE DISTRIBUTIONS OF FISSION FRAGMENTS OF … PHYSICAL REVIEW C 97, 034621 (2018)
Y
0.01
0.02
0.03
0.04
0.05
E* = 25 MeV
254Fm
E* = 35 MeV
254Fm
E* = 50 MeV
254Fm
Y
0.01
0.02
0.03
0.04
0.05
0.06
256Fm 256Fm 256Fm
Y
0
0.05
0.1
0.15
0.2
0.25
0.3
A i
80 100 120 140 160
264Fm
A i
80 100 120 140 160
264Fm
A i
80 100 120 140 160 180
264Fm
Y
0.04
0.08
0.12
0.16
258Fm 258Fm 258Fm
Y
0.05
0.1
0.15
0.2
260Fm 260Fm
260Fm
(c)(b)
(e)(d) (f)
(h)(g) (i)
(l)(k)(j)
(o)(n)(m)
(a)
FIG. 3. The same as in Fig. 1,butforE∗=25 MeV (the first column), E∗=35 MeV (the second column), and E∗=50 MeV (the third
column) excitation energies.
mass fragmentations 124Sn +134 Sn and 128Sn +130 Sn which
contribute to the symmetric yields. Because their potential
energies are higher by 4 MeV and 3 MeV, respectively,
than those for the Cd + Te configurations, their contribu-
tions to the symmetric yields are rather small. Note that our
previously calculated charge distribution [6] of spontaneous
fission of 258Fm has to be corrected based on the present
results.
034621-5

PAŞCA, ANDREEV, ADAMIAN, AND ANTONENKO PHYSICAL REVIEW C 97, 034621 (2018)
Y
0.05
0.1
0.15
E* = 25 MeV
254Fm
E* = 35 MeV
254Fm
E* = 50 MeV
254Fm
Y
0.05
0.1
0.15
256Fm 256Fm 256Fm
Y
0.05
0.1
0.15
0.2
0.25 258Fm 258Fm 258Fm
Y
0.1
0.2
0.3
260Fm 260Fm 260Fm
Y
0
0.2
0.4
0.6
0.8
Zi
35 40 45 50 55 60
264Fm
Zi
40 45 50 55 60
264Fm
Zi
40 45 50 55 60 65
264Fm
(c)(b)
(e)(d) (f)
(h)(g) (i)
(l)(k)(j)
(o)(n)(m)
(a)
FIG. 4. The same as in Fig. 2,butforE∗=25 MeV (the first column), E∗=35 MeV (the second column), and E∗=50 MeV (the third
column) excitation energies.
As seen in Figs. 1and 2, in the case of spontaneous fission
of 260Fm with the symmetric mass distribution, the charge
distribution is asymmetric one with charge split Cd + Te. With
increasing excitation energy the charge distribution becomes
symmetric one (Figs. 2and 4). For the thermal-neutron-
induced fission of 257Fm (Figs. 1and 2), the experimental
and calculated mass distributions are symmetric ones whereas
the predicted charge distribution is asymmetric. However, at
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CHARGE DISTRIBUTIONS OF FISSION FRAGMENTS OF … PHYSICAL REVIEW C 97, 034621 (2018)
E* = 25 MeV
258No
E* = 50 MeV
258No
Y
0.05
0.1
0.15
258No
260No 260No 260No
Y
0.05
0.1
0.15
262No 262No 262No
Y
0
0.05
0.1
Ai
80 100 120 140 160
264No
Ai
80 100 120 140 160
264No
Ai
80 100 120 140 160 180
264No
Y
0.02
0.04
0.06
0.08
0.1
0.12 (s. f.)
(c)(b)
(e)(d) (f)
(h)(g) (i)
(l)(k)(j)
(a)
FIG. 5. The calculated mass distributions (solid lines) resulting from the fission of the indicated nuclei 260,262,264 No at excitation energies
E∗=0 MeV (the first column), E∗=25 MeV (the second column), and E∗=50 MeV (the third column). The calculations are performed
for even-even mass and charge fragmentations. The distributions are normalized to unity. In (a) and (g), the histograms represent experimental
data of Refs. [3]and[36], respectively.
034621-7

PAŞCA, ANDREEV, ADAMIAN, AND ANTONENKO PHYSICAL REVIEW C 97, 034621 (2018)
260No 260No
262No
Z i
40 50 60
264No
Z i
40 50 60
264No
E* = 50 MeV
258No
262No
Y
0
0.1
0.2
0.3
Z i
40 50 60
264No
Y
0.05
0.1
0.15
0.2
0.25
0.3 (s. f.)
258No
Y
0.1
0.2
0.3
262No
E* = 25 MeV
258No
Y
0.1
0.2
0.3
260No
(c)(b)
(e)(d) (f)
(h)(g) (i)
(l)(k)(j)
(a)
FIG. 6. The calculated charge distributions (solid lines) resulting from the fission of the indicated nuclei 260,262,264No at excitation energies
E∗=0 MeV (the first column), E∗=25 MeV (the second column), and E∗=50 MeV (the third column). The calculations are performed for
even-even mass and charge fragmentations. The distributions are normalized to unity.
E∗15 MeV the difference between these distributions van-
ish. For the induced fission of 254Fm at E∗=35 and 50 MeV
(Figs. 3and 4), the shape of mass yields becomes symmetric in
contrast to one of the charge distribution. This unexpected dif-
ference between the shapes of mass and charge yields is worth
to be studied experimentally. In the case of spontaneous fission
of 260No (Figs. 5and 6), the mass distribution displays sym-
metric (AL=AH=130) and asymmetric (AL=122, AH=
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CHARGE DISTRIBUTIONS OF FISSION FRAGMENTS OF … PHYSICAL REVIEW C 97, 034621 (2018)
E* = 50 MeV
262Rf
264Rf
Y
0.05
0.1
0.15
262Rf
264Rf
Y
0.04
0.08
0.12
Ai
100 120 140 160
266Rf
Ai
100 120 140 160
266Rf
Ai
100 120 140 160 180
266Rf
E* = 20 MeV
262Rf
Y
0.05
0.1
0.15
0.2 (s. f.)
264Rf
(c)(b)
(e)(d) (f)
(h)(g) (i)
(a)
FIG. 7. The same as in Fig. 5, but for the indicated nuclei 262,264,266Rf. In (a), the symbols connected by line represent the experimental data
of Ref. [37].
138) peaks. For 260No, the peaks of the charge distribution are
at ZL=50 and ZH=52. The spontaneous fission of 264No
(Figs. 5and 6) has the slightly asymmetric mass distribution
with peaks at AL=128 and AH=136 and charge distribution
with peaks at ZL=50 and ZH=52. For the spontaneous
fission of 266Rf (Figs. 7and 8), the mass yield is symmetric
and the charge-yield is asymmetric (ZL=50 and ZH=54).
Note the in the case of spontaneous fission of 264Fm (Figs. 1
and 2), the peaks of charge and mass distributions are at
ZL=48, ZH=52 and AL=128, AH=136, respectively.
The interplay between the liquid-drop surface energy and
the nucleus-nucleus interaction potential at scission is the main
reason of the appearance or disappearance of the asymmetric
fission mode. This interplay depends on the shell effects, exci-
tation energy, and isospin of the fissioning nucleus. The shell
effects affect directly and indirectly (through the deformations
of nuclei) the appearance of the asymmetric minimum.
As a global trend for the isotopes of Fm and No, the sym-
metric components of the mass and charge yields are enhanced
with increasing excitation energy (Figs. 1–6). The saturation of
the symmetric components is reached at around 15–30 MeV. At
larger E∗the distributions becomes wider. The fission of 260 No
is a good example of that (Figs. 5and 6). The shift to more
symmetric mass and charge distributions and the saturation of
the symmetric components with increasing excitation energy
(Figs. 1–6) can be understood in the following way: for each
mass and charge fragmentation the configurations with the
highest yields correspond to local minima on the potential
energy surfaces (βL,βH). These minima of the potential energy
surface (PES) result from the tricky competition between the
macroscopic interaction and liquid-drop (surface) energies,
and the microscopic shell corrections at scission. The strong
shells also affect the macroscopic parts of the potential energy
by fixing the minimum energy at small deformations (βL,
034621-9

PAŞCA, ANDREEV, ADAMIAN, AND ANTONENKO PHYSICAL REVIEW C 97, 034621 (2018)
E* = 50 MeV
262Rf
264Rf
E* = 20 MeV
262Rf
Y
0.05
0.1
0.15
0.2
0.25
0.3
0.35 (s. f.)
262Rf
Y
0.05
0.1
0.15
0.2
0.25
0.3
0.35 264Rf 264Rf
Y
0.05
0.1
0.15
0.2
0.25
Zi
30 40 50 60
266Rf
Zi
40 50 60
266Rf
Zi
40 50 60 70
266Rf
(c)(b)
(e)(d) (f)
(h)(g) (i)
(a)
FIG. 8. The same as in Fig. 6, but for the indicated nuclei 262,264,266 Rf.
βH). With increasing excitation energy, the shell effects are
washed out, and the stiffness of the nuclear surface decreases.
The combined effect is the enlargement of the minima on
the PES and their shift towards much larger deformations.
At large E∗the shell effects are completely damped, the
surface stiffness becomes minimal, and the minima on the
PES reach their maximum widths and final position. At this
point the yields reach the maximal values, and further increase
of excitation energy leads only to the population of more
asymmetric accessible configurations.
In the case of fission of 262,264,266Rf (Figs. 7and 8), the
increase of excitation energy leads to the access of previously
inaccessible asymmetric configurations and to the decrease of
the symmetric component.
IV. CONCLUSIONS
The mass and charge distributions resulting from the
spontaneous and induced fission of even-even nuclei
254,256,258,260,264Fm, 258,260,262,264No, and 262,264,266 Rf were
calculated within the statistical scission-point fission model.
For these fissioning nuclei, the available experimental mass
distributions in the spontaneous and thermal-neutron-induced
fission were well described excepting 258Fm(s.f.). In contrast
to the experimental data, the calculated mass distribution
for the spontaneous fission of 258Fm is rather wide and has
asymmetric bumps, even though the symmetric yields are
extremely enhanced. However, the calculated spontaneous
fission of 258Fm shows slightly asymmetric charge distribution.
In the case of spontaneous fission of 260Fm, we found that the
mass distribution is symmetric but the charge distribution are
asymmetric (ZL=48, ZH=52). For 260No(s.f.), the mass
distribution has asymmetric and symmetric peaks and the
charge distribution has peaks at ZL=50 and ZH=52. For
266Rf (s.f.), the mass yield is symmetric and the charge yield is
asymmetric (ZL=50 and ZH=54).
For the thermal-neutron-induced fission of 257Fm, the asym-
metric charge distribution (ZL=48, ZH=52) was predicted.
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CHARGE DISTRIBUTIONS OF FISSION FRAGMENTS OF … PHYSICAL REVIEW C 97, 034621 (2018)
As well-known, the reaction 257Fm(nth,f ) has the symmetric
mass yield. For the fission of 254Fm at E∗=35 and 50 MeV,
the symmetric shape of mass yields and the asymmetric shape
of charge yields were obtained. For 260,262,264No (266 Rf), the
asymmetric peaks at ZL=50 and ZH=52 (ZL=50 and
ZH=54) is conserved with increasing E∗. The experimental
verifications of these theoretical predictions are desirable.
Our calculations for spontaneously fissioning nuclei Fm,
No, and Rf suggest that an evolution of charge-yield shape
occurs gradually with increasing isospin and excitation energy.
For the isotopes of Fm and No, the symmetric components
of mass and charge distributions are enhanced with increas-
ing E∗. For the first time, we demonstrated that at some
critical E∗the saturation of the symmetric yields occurs.
The transformation of the shape of mass distribution oc-
curs in a similar fashion like that of the charge distribu-
tion, but faster. For 254,256 Fm, the shape of charge yields
evolves slower with increasing E∗than the shape of the mass
yields and remains asymmetric even at high excitation energy
(E∗=50 MeV). In the fission of 258Fm and 260Fm, the sym-
metric component of the charge (mass) distribution is enhanced
much faster with excitation energy due to the rapid transition
from the strong shell-effect-governed to the macroscopic-
governed symmetric fission behavior. Once the shell effects are
damped at high excitation energy, the fission is strictly driven
by the competition between the liquid-drop binding energy and
the interaction energy of the two fragments. Any increase of
excitation energy leads to the access of previously inaccessible
asymmetric configurations. This, together with the smooth
character of macroscopic energies, mainly leads to a saturation
of the symmetric yields with increasing excitation energy and
broadening of the mass and change distributions. Thus, the
evolution of charge and mass distributions with the variation
of excitation energy and isospin is related to the change of the
PES at scission. The saturation effect is worth to be studied
experimentally.
ACKNOWLEDGMENTS
This work was partially supported by the Romania-
JINR(Dubna) Cooperation Programme, the Russian Founda-
tion for Basic Research (Moscow), and DFG (Bonn). The work
of N.V.A. was supported by Tomsk Polytechnic University
Competitiveness Enhancement Program grant.
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