CHAPTER VIII
Spontaneous fission
Theoretically, alpha decay and spontaneous fission share the same underlying
mechanism in Physics, i.e. the quantum tunneling effect. The alpha decay process is
considered as an alpha cluster penetrating the Coulomb barrier after its formation in the
parent nucleus. As compared to alpha decay, the situation of spontaneous fission is much
more complex and there are large uncertainties existing in the fission process, such as mass
and charge numbers of the two fragments, the number of emitted neutrons, and released
energy etc. In the case of super heavy elements spontaneous fission and alpha decay are the
main decay modes [249-251]. Super heavy nuclei which have relatively small alpha decay
half life times compared to spontaneous fission half lives will survive fission and thus can be
detected in the laboratory through alpha decay.
The existence of long lived super heavy nuclei is controlled mainly by spontaneous
fission and alpha decay process. In the field of super heavy elements, all elements with
Z > 106 can be identified via alpha decay chain. Alpha decay was first explained [252, 253]
in 1928 within a quantum tunneling through the potential barrier leading from the parent
nucleus to the two emitted fragments; the alpha particle and the daughter nucleus.
Spontaneous fission of heavy nuclei was first observed by Flerov et al [145] in 1940 from
238
U nucleus. The super asymmetric fission, cluster radioactivity was first predicted by
Sandulescu et al [16] in 1980 on the basis of quantum mechanical fragmentation theory
(QMFT).
On the basis of liquid drop model Bohr and Wheeler [45] described the mechanism of
nuclear fission and established a limit Z 2 A ≈ 48 for spontaneous fission. Beyond this limit,
nuclei are unstable against spontaneous fission. The alpha decay of super heavy
185
nuclei [146-151] is possible if the shell effect supplies the extra binding energy and increases
the barrier height of fission. Beta decay could be another possible decay mode for the super
heavy nuclei lying beyond the beta stability line. However, since the beta decay proceeds via
weak interaction, the process is slow and less favored and energy involved (released) is also
less compared to spontaneous fission and alpha decay, which proceed quickly via strong
interaction making these processes more probable. Beta stable nuclei having relatively longer
half life for spontaneous fission than that for alpha decay indicate that the dominant decay
mode for such super heavy nuclei might be alpha decay. The spontaneous fission of a heavy
nucleus and its half life depends not only on the height of the fission barrier but also on its
shape. Fission barrier depends on the way in which the intrinsic energy of the nucleus
changes as its shape varies. The intrinsic energy of the nucleus undergoes a significant
change as its spherically symmetric configuration turns into a strongly deformed
configuration of two nuclei in contact at the scission point. Spontaneous fission can occur
much more rapidly when the nucleus of an atom undergoes super deformation.
The present chapter aims to explore the possibility of finding long lived super heavy
elements [254] by comparing the calculated alpha decay half-lives with the spontaneous
fission half-lives. The alpha decay half-lives of even-even elements with Z = 100-122 have
been studied using the Coulomb and proximity potential model and the spontaneous fission
half-lives of even-even isotopes with Z=100-138 is computed using the phenomenological
formula proposed by Ren et al [163]. An attempt is made to develop a semi-empirical
formula [255] for determining the spontaneous fission half life time with minimum
parameters, the predictions are compared with experimental data and other empirical
formulas.
186
8.1 Review of earlier proposed scaling laws
Spontaneous fission is one of the most prominent decay types, energetically feasible for
heavy and super heavy nuclei with proton number Z ≥ 90. It is also an important limiting
factor that determines the stability of newly synthesized super heavy nuclei. This was first
predicted by Bohr and Wheeler [45] in 1939 on the basis of liquid drop model, after one year
Flerov et al [145] observed this phenomenon from 238U nucleus. At present the spontaneous
fission half lives of several super heavy nuclei have been measured in different
laboratories [10, 83, 85, 86, 256, 257].
Several authors proposed empirical formulas for determining the half lives of
spontaneous fission. The first semi-empirical formula for spontaneous fission was proposed
by Swiatecki [258] in 1955 and this formula is applied to the half-lives of even-even, odd A
and odd-odd nuclei in the ground state and by using this formula the author successfully
reproduced the experimental data. Using the macroscopic- microscopic approach,
Smolanczuk et al [259] analyzed the spontaneous fission properties of deformed super heavy
nuclei in a multidimensional deformation space and proposed a formula for spontaneous
fission half lives given as
TSF = T0 P −1
(8.1)
where P is the probability of the barrier penetration by a nucleus and is obtained in the semi
classical (WKB) approximation as Ρ = [1 + exp 2 S ( L)]−1 , T0 is the half life time when the
probability is equal to unity and its value is determined by the number of assaults of nucleus
on the fission barrier in unit time. The equation for finding T0 is given as T0 = 2π ln 2 / ω0 .
By using this formula the authors reproduced the half lives of super heavy elements with
proton number Z=104-120. Later Muntain et al [260] computed the height of spontaneous
fission barriers of heaviest nuclei with proton number Z= 96-120 using macroscopic and
187
microscopic model. Later Ren et al [163, 164] proposed a phenomenological formula for
spontaneous fission half lives and the detailes of the formula are given in section 4.2
Recently Xu et al [261] proposed an empirical formula for determining the
spontaneous fission half lives and is given as
⎧
⎫
Z2
T1 / 2 = exp⎨2π [C0 + C1 A + C 2 Z 2 + C3 Z 4 + C 4 ( N − Z ) 2 − (0.13323 1 − 11.64)]⎬
A 3
⎩
⎭
(8.2)
where Co = -195.09227, C1=3.10156, C2=-0.04386, C3=1.4030 x10-6 and C4=-0.03199. By
using this formula the authors successfully reproduced the spontaneous fission half lives of
nuclei in the mass region from 232Th to 286114.
8.2 Competition between alpha decay and spontaneous fission
half lives
The alpha decay half life times of isotopes with Z = 100-122 are computed by taking
potential energy barrier as the sum of Coulomb potential, proximity potential and centrifugal
potential for the touching configuration and for the separated fragments. Using the Coulomb
and proximity potential model [11-13, 262] we have already reported the alpha and cluster
decay half lives of many isotopes in heavy and super heavy region and they are in close
agreement with the experimental values. The Q values are computed using experimental
binding energies of Audi and Wapstra [143] and some masses are taken from the table of
KTUY [144].
Figure 8.1 represents the plot connecting the calculated alpha decay and spontaneous
fission half lives against mass number of even- even isotopes with Z = 100-106. The
computed alpha decay half life times for all the isotopes are compared with the values
calculated by Viola Seaborg empirical formula. In the case of Fm isotopes the highest value
of spontaneous fission half life time (T1/2 = 5.2x107s) is obtained for
252
Fm isotope which is
about 106 times larger than that of alpha decay half life time (T1/2 = 38.02s) value. It is clear
188
from the figure that the points which correspond to alpha decay half lives of isotopes
246-256
Fm are within the spontaneous fission half life inverted parabola. The isotopes situated
within the inverted parabola have spontaneous fission half lives greater than that of alpha
decay half lives, so these isotopes will survive fission and can be identified via alpha decay.
We would like to point out that the alpha decay channel is the dominant decay mode for
246-256
Fm isotopes and these nuclei are already synthesized in the laboratory [263, 264]. The
computed alpha decay half lives of all these fermium isotopes are in agreement with the
experimental value (for e.g. in 250Fm, Present log10 T1 / 2 ( s ) = 3.99, Expt. = 3.25 and for 254Fm,
Present log10 T1 / 2 ( s ) = 4.99, Expt. = 4.14 ). For the case of Nobelium isotopes the maximum
value of spontaneous fission half life time (T1/2 = 1.04x105s) is obtained for
256
No which is
about 104 times larger than that of alpha decay half life time (T1/2 = 2.58s) value. It is clear
from the plot that the
252-260
No isotopes are situated within the inverted parabola, so these
isotopes will survive fission and can be synthesized and identified via alpha decay. Some of
these isotopes are already synthesized and our computed alpha decay half lives are in close
agreement with the experimental values (for e.g. in
Expt. = 0.12; for
252
No, Present log10 T1 / 2 ( s ) = 0.94,
254
No, Present log10 T1 / 2 ( s ) = 1.42, Expt.= 1.86 [265]). Similarly for Rf
isotopes the maximum value for spontaneous fission half life time is for
260
Rf
(T1/2 = 1.21x103s) which is about 103 times greater than the alpha decay half life time
(T1/2 = 0.52s). It is obvious from the figure that the isotopes
256-262
Rf which are situated
within the inverted parabola will survive fission and can be identified via alpha decay. The
isotope 256Rf is already synthesized and our prediction agrees with experimental data (Present
log10 T1 / 2 ( s ) = -0.436; Expt = -0.44). In the case of Sg
isotopes the highest value of
spontaneous fission half life (T1/2 = 1.06x102s) is obtained for
264
Sg isotope which is about
103 times larger than that of alpha decay half life (T1/2 = 0.04s) value. The isotopes
260-266
Sg
situated within the inverted parabola can be synthesized and our computed alpha decay
189
predictions are in close agreement with the experimental data (for e.g. in
260
Sg, Present
log10 T1 / 2 ( s ) = -1.73, Expt. = -2.07; for 266Sg, Present log10 T1 / 2 ( s ) = 1.53, Expt. = 0.39 [136]).
Figures 8.2-8.3 represent the plots showing the competition between the alpha decay
and spontaneous fission half life times for even-even parents with Z=108-122. It is clear from
these figures that the number of isotopes within the inverted parabola increases with
increasing atomic number. It is obvious from the plots that the isotopes
272-280
264-270
108,
268-276
110,
112, 274-284114, 278-290116, 280-294118, 284-300120 and 288-304122 situated within the inverted
parabola have alpha decay channel as the dominant mode of decay and is possible to
synthesis. These isotopes can be identified in the laboratory via alpha decay. Most of these
isotopes (for e.g.
266
108,
270
108,
270
110,
290
116,
294
118 etc.) are already synthesized and
identified via alpha decay [86, 108, 110, 265, 266]. Therefore we presume that our study will
give motivation to future experiments involving the synthesis of super heavy elements. We
would like to point out that our computed alpha decay half life times for all the isotopes are in
close agreement with the experimental data (for e.g. in
264
108, Present = -3.04,
Expt. = -3.60 [267]; for 284114, Present = -0.064, Expt. = -0.10; for 290116, Present = -1.443,
Expt. = -1.82 [86]; for 292116, Present = -0.68, Expt. = -1.74 [83]). Our computed alpha decay
half life time values for all isotopes with Z = 100 - 122 are in close agreement with the values
calculated using Viola Seaborg empirical formula.
Figure 8.4 represents the comparison of present spontaneous fission half lives with the
values taken from Hoffman et al [250] and Smolanczuk et al [259]. It is evident from the plot
that the computed values are in close agreement with the values of Hoffman et al [250] than
Smolanczuk et al [259].
The fissility of a nucleus is defined as the ratio of Coulomb energy of a spherical
sharp surface drop to twice the nuclear surface energy. It scales with the square of nuclear
190
charge divided by mass number Z2/A, the fissionability parameter. The fissility parameter is
normalized to critical fissility beyond which nuclei promptly disintegrate and is given as
X =
( Z 2 / A)
50 .883[1 − 1 .7826 (( N − Z ) / A ) 2 ]
(8.3)
Figures 8.5- 8.6 represent the Seaborg plots connecting the spontaneous fission half
life times versus fissionability parameter (Z2/A) for isotopes with Z = 100-138. For a given Z
value spontaneous fission half lives increases with increasing fissionability parameter,
reaches a maximum and then decreases with increasing fissionability parameter. Several
authors [268, 269] have studied the Seaborg plots connecting the spontaneous fission half life
times versus fissionability parameter (Z2/A) for isotopes with Z = 90-100 which shows a
general trend of decreasing spontaneous fission half lives (exponential decrease) with
increasing proton number Z. But from the present plots we can see that spontaneous fission
half lives increases with the proton number Z, which is a slight increase for Z=100 to Z=118
and a sharp increase for Z=120 to Z=138. This trend is different from the earlier observation
of Studier et al [268]. For a fixed neutron number, the neutron separation energy increases
with increasing proton number. Increase in proton number Z, increases the fissionability
parameter which pulls in maximum of liquid drop energy to make the peak of first hump
higher than second. So when protons are added beyond shell closure Z=120 (see Ref [9]) the
barrier height increases gradually which result in reduced fission probability (increased half
life) from Z=120 to 138.
Figure 8.7 represents the plot connecting the ratio of spontaneous fission and alpha
decay half lives against fissility parameter for isotopes with Z = 100-122. It is evident from
the plot that for a given Z value the ratio of spontaneous fission and alpha decay half lives
increases with increasing fissility parameter reaches a maximum and then decreases with
increasing fissility parameter. It is also clear that the ratio of spontaneous fission and alpha
191
decay half lives increases with increasing proton number. This indicates that as proton
number increases alpha decay is the most dominant mode of decay. Figure 8.8 represents the
plot for the competition of spontaneous fission with alpha decay half lives similar to the one
proposed by Studier et al [268]. The isotopes which differ in Z by two units and A by six
units can be connected by a straight line. At a particular value on this line difference between
spontaneous fission and alpha decay half lives starts decreasing with increasing fissility
parameter. Studier and Bonetti [268, 269] have shown the competition between spontaneous
fission and alpha decay against fissility parameter for Z = 90-102 and the same trend is seen
in the linear line connecting even-even nuclides differing in atomic number by ‘two’ units
and in mass number by ‘six’ units.
Figures 8.9 - 8.10 represent the plots connecting spontaneous fission half lives against
relative neutron excess, I =
(N − Z )
for Z = 100-138. It is evident from these plots that the
(N + Z )
curves converge at a single point (at 0.2) and afterwards it diverges. This point corresponds to
the value Z/A = 0.4 which is the mechanism of nuclear fission established by the limit for
spontaneous fission. i.e. the isotopes with neutron excess less than or equal to 0.2 are the
probable candidates which can survive spontaneous fission. Figure 8.11 represents the plot
for ratio of spontaneous fission with alpha decay half lives against relative neutron excess for
nuclei with Z = 100-122. It is also obvious from this plot that the curves converge to a single
point (at 0.2) and then diverge. It is clear that the isotopes in the first quadrant of this plot
have alpha decay as the dominant mode of decay. So these isotopes (246-256Fm,
252-260
256-262
284-300
Rf,
260-266
Sg,
264-270
108,
268-276
110,
272-280
112,
274-284
114,
278-290
116,
280-294
118,
No,
120
and 288-304122) will survive fission and can be synthesized and identified via alpha decay. We
presume that the present work on spontaneous fission versus alpha decay of super heavy
elements will be a guide to future experiments.
192
Z = 100
10
Z = 102
10
0
0
-10
log10 (T1/2)
-10
TSF
Tα CPPM
Tα VSS
-20
-20
-30
-30
-40
240
244
248
252
256
260
264
268
245
250
255
260
265
270
10
Z = 104
Z = 106
0
0
-10
-10
-20
-20
-30
-30
-40
-40
248
252
256
260
264
268
272
252
256
260
264
268
272
276
Mass Number
Figure 8.1 The plot for the comparison of alpha and spontaneous fission half life time against
mass number of even-even parents with Z = 100-106.
10
Z = 108
Z = 110
0
0
-10
-10
-30
log10 (T1/2)
-20
TSF
Tα CPPM
Tα VSS
-20
-30
-40
-40
256
260
264
268
272
276
280
Z=112
0
260
-10
-20
-20
-30
-30
-40
-40
264
268
272
276
280
284
288
268
272
276
280
284
276
280
284
288
292
Z = 114
0
-10
264
268
272
Mass Number
Figure 8.2 The plot for the comparison of alpha and spontaneous fission half life time against
mass number of even-even parents with Z = 108-114.
193
20
10
Z = 116
Z = 118
10
0
0
-10
-10
TSF
Tα CPPM
Tα VSS
-20
log10 (T1/2)
-30
-20
-30
-40
-40
272
20
276
280
284
288
292
296
276
Z = 120
280
284
288
292
296
300
292
296
300
304
308
Z = 122
30
20
10
10
0
0
-10
-10
-20
-20
-30
-30
-40
280
284
288
292
296
300
304
284
288
Mass Number
Figure 8.3 The plot for the comparison of alpha and spontaneous fission half life time against
mass number of even-even parents with Z = 116-122.
Ref [250]
Present
Ref [259]
24
18
Cf
Cf
Cf
250
248
246
Cm
Cm
250
Cm
248
246
Cm
Cm
244
242
Pu
Cm
240
244
242
Pu
Pu
Pu
240
238
U
U
U
U
236
Pu
238
236
234
232
Th
232
230
260
Rf
Rf
258
256
Rf
Rf
No
254
260
No
256
254
No
No
252
Fm
No
250
Fm
260
256
254
Fm
Fm
Fm
252
Fm
250
Fm
248
246
Cf
Fm
244
256
Cf
254
Cf
8
4
0
-4
252
log10 (TSF)
Th
12
4
0
112
114
276
274
272
Ds
112
270
276
Ds
Ds
274
272
Ds
270
Ds
268
Hs
270
Hs
268
Hs
Sg
Hs
266
264
266
Sg
264
262
Rf
260
Sg
262
Sg
-8
112
274
114
-4
Figure 8.4 The plot for the comparison of present spontaneous fission half lives (in seconds)
with the values taken from Hoffman et al [250] and Smolanczuk et al [259]
194
0
Z = 100
Z = 102
Z = 104
Z = 106
Z = 108
Z = 110
Z = 112
Z = 114
Z = 116
Z = 118
log10 (TSF)
-50
-100
-150
-200
-250
35
40
45
50
55
2
Z /A
Figure 8.5 The plot for spontaneous fission half life time against fissionability parameter of
even-even parents with Z = 100-118.
150
100
Z = 120
Z = 122
Z = 124
Z = 126
Z = 128
Z = 130
Z = 132
Z = 134
Z = 136
Z = 138
log10 (TSF)
50
0
-50
-100
-150
-200
-250
-300
44
48
52
56
60
64
2
Z /A
Figure 8.6 The plot for spontaneous fission half life time against fissionability parameter of
even-even parents with Z = 120-138 parents.
195
40
30
Z = 100
Z = 102
Z = 104
Z = 106
Z = 108
Z = 110
Z = 112
Z = 114
Z = 116
Z = 118
Z = 120
Z = 122
log10 (TSF/ Tα )
20
10
0
-10
-20
-30
-40
-50
0.84
0.88
0.92
0.96
1.00
1.04
1.08
1.12
Fissility Parameter
Figure 8.7 The plot connecting the ratio of spontaneous fission and alpha decay half lives
against fissility parameter of even-even parents with Z = 100-122.
40
log10 (TSF / Tα )
30
20
298
120
10
114
112
0
116
284
280
118
114
120
308
298
116
290
114
122
118
294
304
296
-40
122
302
116
288
-30
306
118
292
112
120
296
-10
284
300
116
114
112
122
294
290
286
282
-20
304
288
300
120
118
116
0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05
Fissility Parameter
Figure 8.8 The plot connecting the ratio of spontaneous fission and alpha decay half lives
against fissility parameter of even-even parents with Z = 112-122.
196
0
Z = 100
Z = 102
Z = 104
Z = 106
Z = 108
Z = 110
Z = 112
Z = 114
Z = 116
Z = 118
log10 (TSF)
-50
-100
-150
-200
-250
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
Relative Neutron Excess
Figure 8.9 The plot for spontaneous fission half life time against relative neutron excess of
even-even parents with Z = 100-118.
150
100
log10 (TSF)
50
0
Z = 120
Z = 122
Z = 124
Z = 126
Z = 128
Z = 130
Z = 132
Z = 134
Z = 136
Z = 138
-50
-100
-150
-200
-250
-300
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
Relative Neutron Excess
Figure 8.10 The plot for spontaneous fission half life time against relative neutron excess of
even-even parents with Z = 120-138 parents.
197
Z = 100
Z = 102
Z = 104
Z = 106
Z = 108
Z = 110
Z = 112
Z = 114
Z = 116
Z = 118
Z = 120
Z = 122
50
40
log10 (TSF/ Tα )
30
20
10
0
-10
-20
-30
-40
-50
0.14
0.16
0.18
0.20
0.22
0.24
0.26
Relative Neutron Excess
Figure 8.11 The plot connecting the ratio of spontaneous fission and alpha decay half lives
against relative neutron excess of even-even parents with Z = 100-122.Figure taken
from [270]
198
8.3 Semi empirical formula for spontaneous fission half lives
The ratio of Coulomb energy of a spherical sharp surface drop to twice the nuclear
EC( 0 )
surface energy is called the fissility parameter of a nucleus and is given as X =
.
2 E S( 0)
It scales with the square of nuclear charge divided by mass number
Z2
, the fissionability
A
parameter. In spontaneous fission the probability for tunneling through a potential barrier
depends exponentially on the square root of the barrier height and inversely related to
fissionability parameter. Figure 8.12 represents the plot connecting the experimentally
⎛ Z2 ⎞
⎟⎟ for even-even
observed spontaneous fission half lives versus fissionability parameter ⎜⎜
⎝ A⎠
isotopes with Z = 94-100. It is clear from the plot that for a given Z value spontaneous fission
half lives increases with increasing fissionability parameter reaches a maximum and then
decreases with increasing fissionability parameter. The same trend is obtained when we
drawn the graph connecting spontaneous fission half lives versus square of the fissionability
2
⎛ Z2 ⎞
parameter ⎜⎜ ⎟⎟ .
⎝ A⎠
It is already a known fact that the relative neutron excess, I =
N −Z
have a major
N +Z
role in spontaneous fission half lives. Figure 8.13 represents the plot connecting the
experimentally observed spontaneous fission half lives against the relative neutron excess for
even-even isotopes with Z = 94-100. This plots also showed a trend similar to that in the
Figure 8.12. i.e. for a particular Z value the spontaneous fission half lives increases with
increasing relative neutron excess reaches a maximum and then decreases with increasing
relative neutron excess. The same behavior is obtained when we draw a graph connecting
spontaneous fission half lives versus square of the relative neutron excess. We have
199
computed the spontaneous fission half lives of even-even isotopes with Z ranging from
100-138 using the phenomenological formula of Ren et al [163], studied its variation against
relative neutron excess is given in Figures 8.9-8.10. It is obvious from the plot that the curves
converge at a single point (at 0.2) and afterwards it diverges. In view of all these facts we
2
2
N −Z
Z2 ⎛ Z2 ⎞
⎛N −Z ⎞
, ⎜⎜ ⎟⎟ ,
and ⎜
have taken the variables
⎟ for developing a new semi
A ⎝ A ⎠ N +Z
⎝N +Z ⎠
empirical formula for explaining spontaneous fission by making least squares fit to the
available experimental data. The new formula obtained for logarithmic half-life time for
spontaneous fission is given by
2
2
⎛Z2 ⎞
Z2
⎛ N −Z ⎞
⎛ N −Z ⎞
⎟⎟ + c⎜
log10 (T1 / 2 / yr ) = a
+ b⎜⎜
⎟ +e
⎟ + d⎜
A
⎝N +Z⎠
⎝N +Z⎠
⎝ A⎠
(8.4)
Where, the constants are a = -43.25203, b = 0.49192, c = 3674.3927, d = -9360.6 and
e = 580.75058.
From experimental spontaneous fission half life time values of 45 nuclei, the standard
deviation is estimated using equation 7.13 and is found to be 1.427353. We have also
computed the standard deviations for the emperical formulas of Ren et al [163] and
Xu et al [261] are found to be 2.853711 and 1.27461 respectively. It is clear from the
standard deviation
values that the present formula is better than the formula of
Ren et al [163]. The formula of Xu et al [261] is better than the present formula but it
employs seven parameters and five variables A, Z2, Z4, (N-Z) 2 and
Z2
while the present
A1/ 3
formula employs only five parameters and four variables. The deviations of the present
spontaneous fission half lives, the predictions of Ren et al [163] and Xu et al [261] from the
experimental values against mass number of parent nuclei are presented in Figure 8.14.
200
8.3.1 Results and discussion
The spontaneous fission half lives of nuclei in all mass regions from 232Th to 286114 are
computed by using the present semi-empirical formula and these values are compared with
experimental data [250, 271] and with the formula predictions of Ren et al [163] and
Xu et al [261] shown in Table 8.1. We would like to point out that our semi empirical
formula predictions are in good agreement with the experimental values. From Table 8.1, we
can see that the maximum deviation in the present spontaneous fission half life time with
experimental values are found for 250Cf; log(Ti / Texp ) = 2.14 , 252Fm; log(Ti / Texp ) = 3.439 and
for
258
Sg, log(Ti / Texp ) = 2.229 . Such a large deviation is due to the sub magic neutron shell
closure at N = 152. Similarly in the case of 284112 isotope the deviation, log(Ti / Texp ) = 2.98
it is an interesting fact that this isotope has a neutron shell closure at N=172. It is obvious
from these facts that the shell effects have a greater role in spontaneous fission. Xu et al [261]
also showed the importance of the neutron shell closures at N=152, 172 in spontaneous
fission. Figure 8.15 represents the comparison of our spontaneous fission half life time for
232
Th to
250
Cm with the formulas of Ren et al [163] and Xu et al [261] and with the
experimental values. From this figure we can see that our half lives are in good agreement
with the experimental data and with the formulas of Ren et al [163] and Xu et al [261]. From
the figure it can also be seen that the spontaneous fission half life time values decrease with
increasing proton number.
Tables 8.2- 8.4 represent the comparison between computed alpha decay half lives
with the present spontaneous fission half lives of Z = 90-122 even-even isotopes. We would
like to point out that the elements which have relatively small alpha decay half life time
compared to spontaneous fission half lives will survive fission and thus can be synthesized
and detected in the laboratory through alpha decay. From the table it can be seen that the
201
parents
232
Th,
228-238
U,
232-244
Pu,
240-246
Cm,
240-248
Cf,
246-254
Fm,
254-258
No and
254
Rf in the
heavy region and the super heavy parents 270-274Ds, 272-278112, 272-282114, 274-292116, 276-298118,
276-308
120 and
278-314
122 have alpha decay half lives less than that of the spontaneous fission
half lives. Thus all these isotopes can be synthesized and identified via alpha decay and it
should be mentioned that most of these isotopes (for e.g. 270110, 290116, 292116, 294118 etc.) are
already synthesized and identified via alpha decay [86, 110]. Figure 8.16 represents the
comparison of measured spontaneous fission and alpha decay half lives with our predictions.
The experimental data for spontaneous fission are taken from Ref [250, 271] and those of
alpha decay are taken from [136, 272]. Our predicted half life time values are in agreement
with the experimental data, for e.g. in the case of
log10 TSFPr esent ( s ) = −2.69 ; log10 TαExpt . ( s ) = −4.0
Ds, log10 TSFExpt. ( s) = −1.10 ,
270
log10 TαPr esent ( s) = −3.49 . Therefore we
presume that our study will be a guide to future experiments involving the synthesis of super
heavy elements.
Figure 8.17 represents the plot connecting computed alpha decay half lives and barrier
penetrability (log10(P)) against neutron number of the even-even parent isotopes with Z
ranging from 96-108. In the half lives part of the plot, two peaks appear at N = 152 and 162
and correspondingly two dips are present in the barrier penetrability values. In alpha and
cluster radioactivity, the decay which has maximum half life time value shows the presence
of a doubly magic parent. Therefore the peaks at N=152 and 162 are indicating the
presence of shell closures at these values. We would like to point out that many
authors [7, 147, 273, 274] have predicted sub magic neutron shell closures at
N= 152 and 162.
202
6
11.5
5
Pu
Cm
log10 (TSF /yr)
4
11.0
3
36.6
36.8
37.0
37.2
37.4
37.0
37.5
38.0
38.5
2
-2
1
-3
Cf
0
Fm
-4
-1
-5
-2
-6
38.0
38.5
39.0
39.5
39.0
39.5
40.0
40.5
2
Z /A
Figure 8.12 The plot connecting experimentally observed spontaneous fission half life times
versus fissionability parameter for Pu, Cm, Cf and Fm isotopes.
11.8
6
11.6
log10 (TSF /yr)
11.4
5
Pu
11.2
Cm
4
11.0
3
10.8
0.205
0.210
0.215
0.220
0.225
0.20
0.21
0.22
0.23
2
-2
1
-3
Cf
0
Fm
-4
-1
-5
-2
-6
0.192
0.200
0.208
0.216
0.224
0.232
0.19
0.20
0.21
0.22
Relative Neutron Excess
Figure 8.13 The plot connecting experimentally observed spontaneous fission half life times
versus relative neutron excess for Pu, Cm, Cf and Fm isotopes.
203
Present
Ref [163]
Ref [261]
12
log10 (TSF/TExpt.)
8
4
0
-4
-8
-12
232
240
248
256
264
272
280
Mass Number
Figure 8.14 Deviations of present and other semi empirical formula predictions from the
experimental half life values against mass number of parent nuclei.
Expt.
Present
Ref. [163]
Ref. [261]
log10(T1/2)
20
15
10
234
232
Th
U
236
U
238
U
236
Pu
238
Pu
240
Pu
242
Pu
244
Pu
240
Cm
242
Cm
244
Cm
246
Cm
248
Cm
250
Cm
5
Figure 8.15 The plot giving the comparison of present semi empirial formula predictions
with other semi empirical formula predictions and with experimental spontaeous fission half
life time values for various parent isotopes.
204
25
Tα Present
Tα Expt.
TSF Present
TSF Expt.
log 10 T (s)
20
15
10
5
0
Ds
270
No
256
Fm
248
Cf
Cf
246
246
242
Cm
Cm
242
Cm
240
Pu
240
Pu
238
238
U
-5
Figure 8.16 The plot giving the comparison of measured spontaneous fission and alpha decay
half lives with the present alpha decay and spontaneous fission half lives.
log10(P)
-20
Cm
Cf
Fm
No
Rf
Sg
Hs
-25
-30
log10(Tα)
-35
144
148
152
156
160
164
168
172
144
148
152
156
160
164
168
172
12
8
4
0
-4
Neutron Number of Parent
Figure 8.17 Computed logarithmic barrier penetrability and half life times against neutron
number for various parent nuclei with Z =96 -108 emitting alpha decay.
205
Table 8.1 The comparison of the present semi empirical formula predictions with
experimental spontaneous fission half lives and other semi empirical formula predictions for
parents with Z ranging from 90-114
==================================================================
Parent
log10(TSF) (year)
Parent
log10(TSF) (year)
Nuclei
Nuclei
Ref.
Ref.
Present Expt.
Present
Present Expt.
Present
[261]
[261]
==================================================================
232
Th
23.61 21.08
21.08 21.88 250Fm
-1.81
-0.1
-0.63
-1.57
234
17.60
16.18
16.18
16.03
252
-1.34
2.1
0.22
-0.92
236
17.44
16.4
16.36
16.56
254
-1.52
-0.2
-0.29
0.98
238
U
16.53
15.91
16.26
16.38
256
-2.32
-3.48
-2.13
-1.76
236
Pu
11.05
9.18
10.46
9.71
252
-6.19
-6.54
-6.12
-6.04
238
11.81
10.68
11.41
10.99
254
-4.96
-3.04
-3.48
-4.65
240
11.82
11.06
11.78
11.55
256
-4.41
-4.77
-2.48
-3.97
242
11.14
10.83
11.57
11.4
254
Rf
-10.62
-12.1
-13.51
-10.62
244
9.80
10.82
10.81
10.54
256
Rf
-8.68
-9.71
-8.55
-8.48
5.02
258
Rf
-7.41
-9.35
-5.54
-7.06
6.33
260
Rf
-6.78
-9.2
-4.41
-6.36
Rf
-6.77
-7.18
-5.13
-6.36
Sg
U
U
Pu
Pu
Pu
Pu
240
242
Cm
Cm
5.69
6.61
6.28
6.85
5.53
6.94
Fm
Fm
Fm
No
No
No
244
Cm
6.80
7.12
7.47
6.92
262
246
Cm
6.32
7.26
7.17
6.8
258
-12.34
-10
-15.65
-12.34
248
Cm
5.19
6.62
6.03
5.96
260
-10.39
-9.65
-10.1
-10.17
250
Cm
3.44
4.05
4.1
4.41
262
-9.10
-9.32
-6.72
-8.72
Sg
Sg
242
Cf
-0.76
-1.33
-1.95
-1.27
264
-8.44
-8.93
-5.47
-7.98
246
Cf
2.06
3.26
2.88
2.12
266
Sg
-8.37
-7.86
-6.3
-7.96
248
Cf
2.41
4.51
3.58
2.74
264
Hs
-11.33
-10.2
-10.64
-11.02
250
Cf
2.09
4.23
3.17
2.65
270
Ds
-10.19
-8.6
-6.05
-9.46
252
Cf
1.14
1.93
1.68
1.84
282
112
-10.09
-10.6
-13.99
-9.39
254
Cf
-0.42
-0.78
-0.86
0.32
284
112
-11.48
-8.5
-23.03
-11.43
286
114
-8.72
-8.08
-11.61
-7.12
246
-4.81
-6.6
-6.56
-5.01
248
-2.95
-2.94
-2.88
-2.93
Fm
Fm
Sg
==================================================================
206
Table 8.2 The competition between computed alpha decay half lives and the present
spontaneous fission half lives of even-even isotopes with Z = 92-122. (*) represents the Q
values computed from the tables of Ref. [143, 144]. Half life is in seconds
=============================================================
log10(Tα) log10(TSF)
log10(Tα) log10(TSF) Parent
Qα
Parent
Qα
Nuclei (MeV)
Nuclei (MeV)
=============================================================
232
Th
4.08
20.07
31.11 252Fm
7.15
6.06
6.16
228
254
U
6.8
4.13
21.01
Fm
7.31
5.33
5.98
230
256
U
5.99
8.01
23.21
Fm
7.03
6.54
5.18
232
258
*
U
5.41
11.3
24.57
Fm
7.10
6.18
3.79
234
260
*
U
4.86
14.95
25.13
Fm
6.18
10.76
1.85
236
250
*
U
4.57
17.12
24.94
No
8.96
0.15
-0.6
238
252
U
4.27
19.6
24.03
No
8.55
1.47
1.31
232
254
Pu
6.72
5.41
14.68
No
8.23
2.58
2.53
234
Pu
6.31
7.34
17.02 256No
8.58
1.3
3.09
236
*
Pu
5.87
9.64
18.55 258No
8.15
2.81
3.02
238
260
*
Pu
5.59
11.23
19.31
No
7.70
4.54
2.34
240
262
*
Pu
5.26
13.28
19.33
No
7.10
7.11
1.1
242
264
*
Pu
4.98
15.17
18.65
No
6.59
9.54
-0.7
244
266
*
Pu
4.66
17.54
17.3
No
6.97
7.65
-3.01
240
254
*
Cm
6.4
7.82
13.19
Rf
9.38
-0.43
-3.12
242
256
*
Cm
6.22
8.71
14.11
Rf
8.95
0.89
-1.18
244
258
*
Cm
5.9
10.42
14.31
Rf
9.25
-0.11
0.09
246
*
Cm
5.47
12.97
13.82 260Rf
8.90
0.99
0.72
248
262
*
Cm
5.16
14.99
12.69
Rf
8.49
2.39
0.73
250
264
*
Cm
5.17
14.89
10.94
Rf
8.15
3.62
0.16
240
266
*
Cf
7.72
2.96
4.194
Rf
7.51
6.17
-0.98
242
268
*
Cf
7.52
3.72
6.74
Rf
8.10
3.74
-2.64
244
258
*
Cf
7.33
4.47
8.52
Sg
9.68
-0.62
-4.84
246
260
*
Cf
6.86
6.5
9.56
Sg
9.93
-1.4
-2.9
248
262
*
Cf
6.36
8.92
9.91
Sg
9.60
-0.44
-1.6
250
264
*
Cf
6.13
10.11
9.59
Sg
9.21
0.75
-0.94
252
266
Cf
6.22
9.59
8.64
Sg
8.88
1.79
-0.87
254
*
Cf
5.93
11.17
7.08 268Sg
8.40
3.48
-1.36
246
270
*
Fm
8.37
1.34
2.69
Sg
9.10
1.01
-2.41
248
272
*
Fm
8
2.66
4.55
Sg
8.31
3.76
-3.97
250
264
Fm
7.56
4.35
5.69
Hs
10.58
-2.52
-3.83
=============================================================
207
Table 8.3 The competition between computed alpha decay half lives and the present
spontaneous fission half lives of even-even isotopes with Z = 92-122. (*) represents the Q
values computed from the tables of Ref. [143, 144]. Half life is in seconds
=============================================================
log10(Tα) log10(TSF)
log10(Tα) log10(TSF) Parent
Qα
Parent
Qα
Nuclei (MeV)
Nuclei (MeV)
=============================================================
266
*
-2.54 288112
-8.12
Hs *10.35
-1.91
8.48
5.43
268
268
*
*
114
-1.85
Hs
9.91
-0.69
13.26
-6.73
-12.49
270
270
*
114
-1.74
Hs
9.30
1.15
13.11
-6.46
-8.86
272
272
*
*
114
-2.19
Hs
10.11
-1.33
12.94
-6.14
-5.88
274
274
*
*
-3.17
114
Hs
9.51
0.44
12.72
-5.71
-3.52
276
276
*
*
-4.65
114
Hs
8.80
2.77
12.49
-5.24
-1.77
260
Ds *12.21
-5.72
-15.48 278114 *12.31
-4.87
-0.60
262
*
280
*
Ds
11.90
-5.05
-11.58
114
11.99
-4.17
0.02
264
*
282
*
Ds
11.77
-4.79
-8.379
114
11.48
-2.96
0.10
266
*
284
*
Ds
11.58
-4.37
-5.85
114
10.35
0.06
-0.32
268
*
286
Ds
11.92
-5.18
-3.96
114 10.34
0.05
-1.22
270
*
288
Ds
11.20
-3.49
-2.69
114 10.09
0.76
-2.60
272
*
290
*
Ds
10.76
-2.38
-2.01
114
9.06
4.12
-4.42
274
*
292
*
Ds
11.40
-4.05
-1.88
114
8.38
6.65
-6.67
276
*
272
*
Ds
10.60
-2.01
-2.30
116
13.89
-7.41
-9.64
278
*
274
*
Ds
9.99
-0.34
-3.23
116
13.69
-7.06
-6.16
280
*
276
*
Ds
9.31
1.79
-4.66
116
13.49
-6.70
-3.32
282
*
278
*
Ds
8.66
4.33
-6.56
116
13.56
-6.88
-1.08
284
*
280
*
Ds
7.95
6.74
-8.91
116
12.97
-5.69
0.57
264
282
*
112 *12.70
-6.18
-14.44
116
12.44
-4.56
1.67
266
*
284
*
112
12.63
-6.07
-10.66
116
11.82
-3.13
2.22
268
*
286
*
112
12.38
-5.56
-7.56
116
11.18
-1.53
2.27
270
*
288
*
112
12.14
-5.06
-5.11
116
10.84
-0.65
1.82
272
*
290
112
11.92
-4.56
-3.28
116 10.57
-0.85
0.89
274
*
292
112
11.68
-4.02
-2.05
116 10.80
-0.62
-0.48
276
*
*
294
112
10.16
1.20
-2.29
11.34
-3.22
-1.39
116
278
*
*
296
112
10.51
0.16
-4.52
11.13
-2.72
-1.28
116
280
276
*
*
112
118
14.39
-7.83
-5.87
10.99
-2.41
-1.68
282
278
*
*
112
118
14.63
-8.29
-2.57
9.97
0.46
-2.59
284
*
280
*
112
9.30
2.57
-3.98
118
14.13
-7.39
0.13
286
*
282
*
112
8.74
4.48
-5.83
118
12.74
-4.56
2.23
=============================================================
208
Table 8.4 The competition between computed alpha decay half lives and the present
spontaneous fission half lives of even-even isotopes with Z = 92-122. (*) represents the Q
values computed from the tables of Ref. [143, 144]. Half life is in seconds
=============================================================
log10(Tα) log10(TSF)
log10(Tα) log10(TSF) Parent
Qα
Parent
Qα
Nuclei (MeV)
Nuclei (MeV)
=============================================================
284
118 *12.75
-4.62
3.77 310120 *12.46
-3.71
-7.02
286
288
290
292
294
118
*
118
*
118
*
118
*
12.34
11.91
11.65
11.47
118
11.81
118
*
118
*
118
274
120
*
276
120
*
120
*
120
*
296
298
300
278
280
282
284
286
288
290
292
294
296
298
10.95
11.12
11.04
*
16.99
120
*
120
*
120
*
120
*
120
*
120
*
120
*
120
*
16.47
15.14
15.01
14.26
12.68
13.44
13.11
12.84
12.72
12.50
12.23
120
*
300
120
*
11.89
302
120
*
11.79
120
*
120
*
304
306
308
*
11.63
11.52
13.23
-3.72
-2.72
-2.11
-1.68
-2.58
-0.35
-0.85
-0.66
-11.52
-10.28
-8.61
-8.40
-7.05
-3.78
-5.50
-4.81
-4.25
-4.02
-3.55
-2.94
4.77
312
5.25
314
5.23
316
4.73
318
3.77
320
2.37
276
0.55
278
-1.67
280
-14.35
282
-9.30
284
-4.92
286
-1.18
288
1.94
290
4.46
292
6.42
294
7.83
296
8.72
298
9.10
300
8.99
302
8.43
304
120
*
11.81
-2.16
-10.68
120
*
10.13
2.56
-14.66
120
*
9.92
3.20
-18.95
120
*
9.68
3.94
-23.53
120
122
*
9.46
4.70
-28.39
*
15.09
-7.87
-13.45
122
*
15.95
-9.38
-7.99
122
*
15.83
-9.22
-3.21
122
*
14.31
-6.51
0.92
122
*
14.32
-6.58
4.44
122
*
13.51
-4.97
7.36
122
*
122
122
14.45
14.03
-6.93
-6.12
9.70
*
11.49
*
13.78
-5.65
12.76
122
*
13.57
-5.25
13.52
122
*
13.40
-4.93
13.79
122
*
13.11
-4.33
13.59
122
*
12.63
-3.28
12.95
122
*
13.06
-4.29
11.87
122
*
12.80
-3.74
10.38
7.43
306
122
*
12.70
-3.54
8.49
-2.18
5.99
308
122
*
14.43
-7.21
6.22
-1.98
4.14
310
122
*
13.83
-6.05
3.58
1.91
312
122
*
13.49
-5.40
0.59
-0.71
314
122
*
12.62
-3.47
-2.74
-1.48
-1.29
-5.37
316
*
120
12.78
-4.41
-3.69
122
12.04
-2.11
-6.39
=============================================================
209
8.4 Conclusion
In this chapter we have studied the competition between alpha decay and spontaneous
fission half-lives of even-even elements with Z = 100-122. The alpha decay study is done
using Coulomb and Proximity Potential Model while spontaneous fission half-lives are
calculated using the phenomenological formula spontaneous fission. The computed alpha
decay half lives for all isotopes are in good agreement with the values calculated by ViolaSeaborg semi empirical relation ship and with the experimental data. We have studied the
Seaborg plots connecting the spontaneous fission half lives versus fissionability parameter
(Z2/A) for isotopes with Z = 100-138. But from the plots we can see that spontaneous fission
half lives increases with the proton number Z, which is a slight increase for Z=100 to Z=118
and a sharp increase for Z=120 to Z=138. This trend is different from the earlier observation
of Studier et al. When protons are added beyond shell closure Z=120 the barrier height
increases gradually which result in reduced fission probability (increased half life) from
Z=120 to 138. We have also studied the ratio of spontaneous fission with alpha decay half
lives against relative neutron excess, I =
(N − Z )
for nuclei with Z = 100-122. It is found
(N + Z )
that the isotopes in the first quadrant of this plot have alpha decay as the dominant mode of
decay. So these isotopes (246-256Fm,
272-280
112,
274-284
114,
278-290
116,
280-294
252-260
118,
No,256-262Rf,
284-300
120 and
260-266
Sg,
288-304
264-270
108,
268-276
110,
122) will survive fission and
can be synthesized and identified via alpha decay. The present work on spontaneous fission
versus alpha decay of super heavy elements will be a guide to future experiments.
A new semi empirical formula is also proposed for determining the spontaneous
fission half lives, which works well for the mass region from
232
Th to
286
114. The computed
spontaneous fission half life times are also compared with other semi empirical formula
predictions. The estimated standard deviation from the present fromula with experimental
210
spontaneous fission half lives is found to be 1.427353. We have also computed the standard
deviations for the emperical formulas of Ren et al, Xu et al and are found to be 2.853711 and
1.27461 respectively. It is clear that the present formula is better than the formula of Ren et
al. The formula of Xu et al is better than the present formula but it employs seven parameters
and five variables A, Z2, Z4, (N-Z) 2 and Z2/A1/3 while the present formula employs only five
parameters and four variables. The alpha decay half lives are systematically computed for
heavy and super heavy region with proton numbers varying from 90 < Z < 122 using
Coulomb and proximity potential model. The comparison between computed alpha decay
half lives and the present spontaneous fission semi empirical formula predictions of eveneven isotopes with Z = 90-122 are also studied. It is found that the parents
232-244
Pu,
240-246
heavy parents
Cm,
240-248
270-274
Ds,
Cf,
246-254
272-278
Fm,
112,
254-258
272-282
No and
114,
274-292
254
232
Th,
228-238
U,
Rf in the heavy region and the super
116,
276-298
118,
276-308
120 and
278-314
122
will survive fission and can be synthesized and identified via alpha decay. Most of these
isotopes are already synthesized and identified via alpha decay. Therefore, we presume that
our study will give motivation to future experiments involving the synthesis of super heavy
elements.
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