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Some remarks on the fissile isotopes
Annals of Nuclear Energy 37 (2010) 1783–1784
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Technical Note
Some remarks on the fissile isotopes Yigal Ronen * Ben-Gurion University of the Negev, Department of Nuclear Engineering, Beer-Sheva 84105, Israel
a r t i c l e
i n f o
Article history: Received 7 March 2010 Received in revised form 15 July 2010 Accepted 19 July 2010 Available online 13 August 2010
a b s t r a c t Data on the fission cross-section of actinides, obtained from the cross-section libraries, support the Fissile Rule. Some observations on the fissile properties of odd and even isotopes are also presented. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Fissile isotopes 2Z N correlation Fission cross-sections
In a previous work (Ronen, 2006), we suggested a rule for determining fissile isotopes. This rule (Fissile Rule) states: heavy isotopes with 90 6 Z 6 100 and 2Z N = 43 ± 2, with few exceptions, are fissile (N – number of neutrons and Z – number of protons). In determining this rule, only nuclei with measured thermal fission cross-sections were considered. However, if calculated (JANIS 3.2, 2010) thermal fission cross-sections are considered as well, the number of exceptions is reduced to two, for which there is no available data. The fissile isotopes are presented in Table 1. If we consider the rf of even-Z compared to odd-Z fissile isotopes we see that, in general, the rf of the odd-Z isotopes is higher than that of the even-Z isotopes. A comparison is made for 2Z N = 41, 43 and 45, presented in Figs. 1–3. From these figures we can see that, except for some nuclides (in particular 94Pu 237 ), we have rf (odd-Z) > rf (even-Z). We would like to find out if simple fission barrier penetrabilities can explain the difference in rf for odd-Z and even-Z nuclei. Barrier penetrability is given (Wageman, 1991) by the approximate expression:
Pi ðEÞ ¼
1 h i 1 þ exp 2phðExi EÞ i
ð1Þ
where i = A or B for the two barriers. We have calculated PA and PB using the values EA, hxA and EB, hxB for the relevant nuclei given in Bjornholm and Lynn (1980). We took the value E = Sn, which is the binding energy of the last neutron (the neutron separation energy). * Tel.: +972 8 6461327; fax: +972 8 6472955. E-mail address: [email protected] 0306-4549/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2010.07.006
In all the cases we studied, we found that PB(Sn) is very close to 1.0, so only the penetrability of the first barrier (PA) is of relevance to our study. The results of the penetrabilities PA for the relevant nuclei for which we have experimental barrier data, are given in Table 2. In Table 2, we can see that the penetrability probabilities PA(Sn) of even-Z isotopes, except for the target nuclei 96Cm 247 , are high. When considering the penetrability probabilities of the availAm;m Bk able odd-Z isotopes (95Am;g 242 , 95242 , 97248 ), we found that the values of the PA(Sn) are lower for these odd-Z isotopes, when compared to the even-Z isotopes, as can be seen in Table 2, yet their fission cross-sections are high. We do not have experimental values for the barrier parameters of the other odd-Z fissile isotopes; however, the binding energy of the last neutron (Sn) for even-Z compound nuclei is higher than that of the neighboring compound nuclei with odd-Z (see Table 3). This is due to the fact that, for even-Z, this compound nucleus is an even-Z even-N one, and, as a result, it is more stable than that of the odd-Z even-N nucleus. Given that the Sn of even-Z fissile isotopes is higher than it is in odd-Z fissile isotopes, this indicates that, unless the odd-Z fissile isotopes have very low fission barriers, we should have Pn(Sn – even-Z) > Pn(Sn – odd-Z). Yet, in most of the cases, we have rf (odd-Z) > rf (even-Z). Therefore, simple considerations based on barrier penetrability cannot explain the fact that rf (odd-Z) > rf (even-Z) in the majority of the cases. More rigorous analysis is needed. In light of the above remarks, we reached three conclusions. First of all, the number of exceptions to the Fissile Rule is substantially reduced with the availability of more data. Only two cases of fissile isotopes of 90 6 Z 6 100 remain for which we do not yet have data, after factoring in the values of rf. Secondly, in most of
1784
Y. Ronen / Annals of Nuclear Energy 37 (2010) 1783–1784
Table 1 Fissile isotopes with 2Z N = 43 ± 2. 2Z N = 41 Nuclei
rf (2200)
2Z N = 43
2Z N = 45
Nuclei
rf (2200) (b)
Nuclei
rf (2200) (b) n.a.
(b) 90Th 229
31
90Th 227
202
90Th 225
91Pa 232
1517 (a) 700 (b) 584
91Pa 230
1500
91Pa 228
1200 (f)A
92U 233
530
92U 231
260
93Np 236
2770 (a,b) 3011 (c) 747
93Np 234
904
94Pu 241
2027 (a) 2070 (b) 1012
94Pu 237
95Am 244
2300
95Am;m 242
6400
95Am 240
2100 (a,c) 2447 (b) 2397 (d) 2295 (e) 2830 (d) A1500 (e)A
96Cm 247
82 (a) 111 (b) 959
96Cm 245
2141
96Cm 243
613 3085 (d)A 1800 (e)A
98Cf 251
98Cf 249
1665 (a,b,d) 1633 (c)
99Es 256
1136 (c,d) 1300 (e) n.a.
715 (d) 2000 (e)A 2862
97Bk 246
99Es 254
1966
99Es 252
4800 (d)A 2000 (e)A
100Fm 259
s.f.
100Fm 257
3012
100Fm 255
3360
92U 235 93Np 238
97Bk 250 98Cf 253
94Pu 239
97Bk 248
A
Fig. 3. Fission cross-sections for isotopes with 2Z N = 45.
Table 2 Fission-barrier parameters of some fissile isotopes.
n.a.: Not available. s.f.: Spontaneous fission. A Calculated value: (a) JEFF – 3.1.1; (b) JENDL – 3.3; (c) ENDF/B – VII.0; (d) JEFF – 3.0/A; (e) JENDL/AC – 2008; (f) JEFF – 3.0/A.
a
rf
Target nuclei
Compound nuclei
Sn (MeV)
EA (MeV)
xA h (MeV)
PA(Sn)
92U 233 92U 235 94Pu 237 94Pu 239 95Am;g 242 95Am;m 242 96Cm 243 96Cm 244 96Cm 247 97Bk 248
92U 234 92U 236 94Pu 238 94Pu 240 95Am 243 95Am 243 96Cm 244 96Cm 246 96Cm 248 97Bk 249
6.844
5.6
1.04
0.9995
530
6.545
5.6
1.04
0.9967
584
7.000
5.5
1.04
0.9999
2100
6.534
5.6
1.04
0.9965
747
6.365
5.9
0.8
0.9747
2100
6.412
5.9
0.8
0.9824
6400
6.801
5.8
1.04
0.9976
613
6.457
5.7
1.04
0.9898
2141
6.213
5.7
1.04
0.9569
82
6.253
6.1
0.8
0.7688
715a
(b)
Calculated value. The uncertainty in the barrier heights is about 0.5 MeV.
Table 3 Separation energy Sn of fissile isotopes. 2Z N = 41 Fig. 1. Fission cross-sections for isotopes with 2Z N = 41.
2Z N = 43
2Z N = 45
Compound nuclei
Sn (MeV)
Compound nuclei
Sn (MeV)
Compound nuclei
Sn (MeV)
90Th 230
6.794
90Th 228
7.105
90Th 226
7.184
91Pa 233
6.529
91Pa 231
6.820
91Pa 229
7.097
92U 236
6.545
92U 234
6.845
92U 232
7.268
93Np 239
6.215
93Np 237
6.577
93Np 235
6.983
94Pu 242
6.310
94Pu 240
6.534
94Pu 238
7.000
95Am 245
6.053
95Am 243
6.365
95Am 241
6.647
96Cm 248
6.213
96Cm 246
6.458
96Cm 244
6.801
97Bk 251
5.795
97Bk 249
6.253
97Bk 247
6.549
98Cf 254
6.032
98Cf 252
6.172
98Cf 250
6.625
99Es 257
5.886
99Es 255
5.974
99Es 253
6.352
100Fm 260
6.367
100Fm 258
6.340
100Fm 256
6.384
Reference Fig. 2. Fission cross-sections for isotopes with 2Z N = 43.
the cases rf (odd-Z nuclei) > rf (even-Z nuclei). Thirdly, simple fission barrier penetrabilities cannot sufficiently explain the second conclusion, so more study is required.
Bjornholm, S., Lynn, J.E., 1980. The double-humped fission barrier. Rev. Mod. Phys. 52, 725. JANIS 3.2 – JAVA-based Nuclear Data Display Program. OECD Nuclear Energy Agency.
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Technical Note
Some remarks on the fissile isotopes Yigal Ronen * Ben-Gurion University of the Negev, Department of Nuclear Engineering, Beer-Sheva 84105, Israel
a r t i c l e
i n f o
Article history: Received 7 March 2010 Received in revised form 15 July 2010 Accepted 19 July 2010 Available online 13 August 2010
a b s t r a c t Data on the fission cross-section of actinides, obtained from the cross-section libraries, support the Fissile Rule. Some observations on the fissile properties of odd and even isotopes are also presented. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: Fissile isotopes 2Z N correlation Fission cross-sections
In a previous work (Ronen, 2006), we suggested a rule for determining fissile isotopes. This rule (Fissile Rule) states: heavy isotopes with 90 6 Z 6 100 and 2Z N = 43 ± 2, with few exceptions, are fissile (N – number of neutrons and Z – number of protons). In determining this rule, only nuclei with measured thermal fission cross-sections were considered. However, if calculated (JANIS 3.2, 2010) thermal fission cross-sections are considered as well, the number of exceptions is reduced to two, for which there is no available data. The fissile isotopes are presented in Table 1. If we consider the rf of even-Z compared to odd-Z fissile isotopes we see that, in general, the rf of the odd-Z isotopes is higher than that of the even-Z isotopes. A comparison is made for 2Z N = 41, 43 and 45, presented in Figs. 1–3. From these figures we can see that, except for some nuclides (in particular 94Pu 237 ), we have rf (odd-Z) > rf (even-Z). We would like to find out if simple fission barrier penetrabilities can explain the difference in rf for odd-Z and even-Z nuclei. Barrier penetrability is given (Wageman, 1991) by the approximate expression:
Pi ðEÞ ¼
1 h i 1 þ exp 2phðExi EÞ i
ð1Þ
where i = A or B for the two barriers. We have calculated PA and PB using the values EA, hxA and EB, hxB for the relevant nuclei given in Bjornholm and Lynn (1980). We took the value E = Sn, which is the binding energy of the last neutron (the neutron separation energy). * Tel.: +972 8 6461327; fax: +972 8 6472955. E-mail address: [email protected] 0306-4549/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2010.07.006
In all the cases we studied, we found that PB(Sn) is very close to 1.0, so only the penetrability of the first barrier (PA) is of relevance to our study. The results of the penetrabilities PA for the relevant nuclei for which we have experimental barrier data, are given in Table 2. In Table 2, we can see that the penetrability probabilities PA(Sn) of even-Z isotopes, except for the target nuclei 96Cm 247 , are high. When considering the penetrability probabilities of the availAm;m Bk able odd-Z isotopes (95Am;g 242 , 95242 , 97248 ), we found that the values of the PA(Sn) are lower for these odd-Z isotopes, when compared to the even-Z isotopes, as can be seen in Table 2, yet their fission cross-sections are high. We do not have experimental values for the barrier parameters of the other odd-Z fissile isotopes; however, the binding energy of the last neutron (Sn) for even-Z compound nuclei is higher than that of the neighboring compound nuclei with odd-Z (see Table 3). This is due to the fact that, for even-Z, this compound nucleus is an even-Z even-N one, and, as a result, it is more stable than that of the odd-Z even-N nucleus. Given that the Sn of even-Z fissile isotopes is higher than it is in odd-Z fissile isotopes, this indicates that, unless the odd-Z fissile isotopes have very low fission barriers, we should have Pn(Sn – even-Z) > Pn(Sn – odd-Z). Yet, in most of the cases, we have rf (odd-Z) > rf (even-Z). Therefore, simple considerations based on barrier penetrability cannot explain the fact that rf (odd-Z) > rf (even-Z) in the majority of the cases. More rigorous analysis is needed. In light of the above remarks, we reached three conclusions. First of all, the number of exceptions to the Fissile Rule is substantially reduced with the availability of more data. Only two cases of fissile isotopes of 90 6 Z 6 100 remain for which we do not yet have data, after factoring in the values of rf. Secondly, in most of
1784
Y. Ronen / Annals of Nuclear Energy 37 (2010) 1783–1784
Table 1 Fissile isotopes with 2Z N = 43 ± 2. 2Z N = 41 Nuclei
rf (2200)
2Z N = 43
2Z N = 45
Nuclei
rf (2200) (b)
Nuclei
rf (2200) (b) n.a.
(b) 90Th 229
31
90Th 227
202
90Th 225
91Pa 232
1517 (a) 700 (b) 584
91Pa 230
1500
91Pa 228
1200 (f)A
92U 233
530
92U 231
260
93Np 236
2770 (a,b) 3011 (c) 747
93Np 234
904
94Pu 241
2027 (a) 2070 (b) 1012
94Pu 237
95Am 244
2300
95Am;m 242
6400
95Am 240
2100 (a,c) 2447 (b) 2397 (d) 2295 (e) 2830 (d) A1500 (e)A
96Cm 247
82 (a) 111 (b) 959
96Cm 245
2141
96Cm 243
613 3085 (d)A 1800 (e)A
98Cf 251
98Cf 249
1665 (a,b,d) 1633 (c)
99Es 256
1136 (c,d) 1300 (e) n.a.
715 (d) 2000 (e)A 2862
97Bk 246
99Es 254
1966
99Es 252
4800 (d)A 2000 (e)A
100Fm 259
s.f.
100Fm 257
3012
100Fm 255
3360
92U 235 93Np 238
97Bk 250 98Cf 253
94Pu 239
97Bk 248
A
Fig. 3. Fission cross-sections for isotopes with 2Z N = 45.
Table 2 Fission-barrier parameters of some fissile isotopes.
n.a.: Not available. s.f.: Spontaneous fission. A Calculated value: (a) JEFF – 3.1.1; (b) JENDL – 3.3; (c) ENDF/B – VII.0; (d) JEFF – 3.0/A; (e) JENDL/AC – 2008; (f) JEFF – 3.0/A.
a
rf
Target nuclei
Compound nuclei
Sn (MeV)
EA (MeV)
xA h (MeV)
PA(Sn)
92U 233 92U 235 94Pu 237 94Pu 239 95Am;g 242 95Am;m 242 96Cm 243 96Cm 244 96Cm 247 97Bk 248
92U 234 92U 236 94Pu 238 94Pu 240 95Am 243 95Am 243 96Cm 244 96Cm 246 96Cm 248 97Bk 249
6.844
5.6
1.04
0.9995
530
6.545
5.6
1.04
0.9967
584
7.000
5.5
1.04
0.9999
2100
6.534
5.6
1.04
0.9965
747
6.365
5.9
0.8
0.9747
2100
6.412
5.9
0.8
0.9824
6400
6.801
5.8
1.04
0.9976
613
6.457
5.7
1.04
0.9898
2141
6.213
5.7
1.04
0.9569
82
6.253
6.1
0.8
0.7688
715a
(b)
Calculated value. The uncertainty in the barrier heights is about 0.5 MeV.
Table 3 Separation energy Sn of fissile isotopes. 2Z N = 41 Fig. 1. Fission cross-sections for isotopes with 2Z N = 41.
2Z N = 43
2Z N = 45
Compound nuclei
Sn (MeV)
Compound nuclei
Sn (MeV)
Compound nuclei
Sn (MeV)
90Th 230
6.794
90Th 228
7.105
90Th 226
7.184
91Pa 233
6.529
91Pa 231
6.820
91Pa 229
7.097
92U 236
6.545
92U 234
6.845
92U 232
7.268
93Np 239
6.215
93Np 237
6.577
93Np 235
6.983
94Pu 242
6.310
94Pu 240
6.534
94Pu 238
7.000
95Am 245
6.053
95Am 243
6.365
95Am 241
6.647
96Cm 248
6.213
96Cm 246
6.458
96Cm 244
6.801
97Bk 251
5.795
97Bk 249
6.253
97Bk 247
6.549
98Cf 254
6.032
98Cf 252
6.172
98Cf 250
6.625
99Es 257
5.886
99Es 255
5.974
99Es 253
6.352
100Fm 260
6.367
100Fm 258
6.340
100Fm 256
6.384
Reference Fig. 2. Fission cross-sections for isotopes with 2Z N = 43.
the cases rf (odd-Z nuclei) > rf (even-Z nuclei). Thirdly, simple fission barrier penetrabilities cannot sufficiently explain the second conclusion, so more study is required.
Bjornholm, S., Lynn, J.E., 1980. The double-humped fission barrier. Rev. Mod. Phys. 52, 725. JANIS 3.2 – JAVA-based Nuclear Data Display Program. OECD Nuclear Energy Agency.