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FROM CURRENT LITERATURE
DATA ON SPONTANEOUS FISSION OF HEAVY NUCLEI
After the discovery by the Soviet physicists G. N. Flerov and K. A. Petrzhak¹ of the spontaneous fission of uranium, it was found that spontaneous fission is also observed for other heavy nuclei, including nuclei of artificially synthesized transuranium elements.
In a recently published note, G. Seaborg gives a summary of data on the half-lives for spontaneous fission and proposes several hypotheses that generalize these data. The summary of the data is given in the table and illustrated graphically (see Fig.).
Summary of data on the rate of spontaneous fission of various nuclei
| Nucleus | Number of fissions in 1 gram per 1 hour | Half-life for spontaneous fission (years) |
|---|---|---|
| $_{90}\mathrm{Th}^{230}_{140}$ | $\leq 1,4$ | $\geq 1,5\cdot 10^{17}$ |
| $_{90}\mathrm{Th}^{232}_{142}$ | $0,15$ | $1,4\cdot 10^{18}$ |
| $_{90}\mathrm{Th}^{232}_{142}$ | $1,2$ | $1,7\cdot 10^{17}$ |
| $_{91}\mathrm{Pa}^{231}_{140}$ | $\leq 20$ | $\geq 10^{16}$ |
| $_{92}\mathrm{U}^{232}_{140}$ | $\leq 25$ | $\geq 8\cdot 10^{12}$ |
| $_{92}\mathrm{U}^{233}_{141}$ | $< 0,7$ | $> 3\cdot 10^{17}$ |
| $_{92}\mathrm{U}^{234}_{142}$ | $< 30$ | $> 7\cdot 10^{15}$ |
| $_{92}\mathrm{U}^{235}_{143}$ | $1,2$ | $1,9\cdot 10^{17}$ |
| $_{92}\mathrm{U}^{238}_{146}$ | $24,8 \pm 0,9$ | $8,0\cdot 10^{15}$ |
| $_{93}\mathrm{Np}^{237}_{144}$ | $\leq 5$ | $\geq 4\cdot 10^{16}$ |
| $_{93}\mathrm{Np}^{239}_{146}$ | $\leq 40$ | $\geq 5\cdot 10^{12}$ |
| $_{94}\mathrm{Pu}^{238}_{144}$ | $5,1\cdot 10^{6}$ | $5,4\cdot 10^{10}$ |
| $_{94}\mathrm{Pu}^{239}_{145}$ | $36$ | $5,5\cdot 10^{15}$ |
| $_{95}\mathrm{Am}^{241}_{146}$ | $\leq 14$ | $\geq 1,4\cdot 10^{13}$ |
| $_{96}\mathrm{Cm}^{242}_{146}$ | $2,7\cdot 10^{10}$ | $7,2\cdot 10^{6}$ |
The indices on the symbols of the nuclei denote the atomic numbers (on the left), mass numbers (upper right), and numbers of neutrons (lower right).
On the graph of the dependence of the half-periods of spontaneous fission on the value of the fission parameter \(\frac{Z^2}{A}\), it is found that, for even-even nuclei, the logarithm of the half-periods is a linearly decreasing function of \(\frac{Z^2}{A}\).
Thus, the dependence \(T_{1/2}=f\left(\frac{Z^2}{A}\right)\) can be represented for even-even nuclei in a form analogous to the Boltzmann distribution law:
\[ T_{1/2}=Me^{-B\frac{Z^2}{A}} \]
or
\[ T_{1/2}=Me^{+\frac{CA}{Z^2}}, \]
where \(M\), \(B\), and \(C\) are constants. Developing further the analogy with the Boltzmann distribution in the kinetics of chemical reactions, one may say that the “activation energy” of spontaneous fission is proportional to \(\frac{Z^2}{A}\). It is interesting that extrapolation of the above relation to the region of immediate spontaneous fission \(\left(T_{1/2}\sim 10^{-20}\ \text{sec.}\right)\) gives the value \(\frac{Z^2}{A}\cong 47\), i.e., close to the critical fission parameter cited in the theoretical work of Bohr and Wheeler.
Fig. 1.
Spontaneous fission of neighboring even-even nuclei and even-odd and odd-even nuclei occurs much more slowly (for example, \(U^{233}\), \(U^{234}\), and \(U^{235}\), or \(Np^{239}\), \(Pu^{238}\), \(Pu^{239}\)). Since the parameter \(\frac{Z^2}{A}\) is, in essence, the ratio \(\frac{Z^2}{r^3}\), rewritten on the assumption \(r\sim A^{1/3}\) (where \(r\) is the nuclear radius), the decrease in the spontaneous-fission rate of nuclei with an odd number of nucleons may be associated with higher values of the radii of such nuclei in comparison with the radii of even-even nuclei.
Since the dependence of \(T_{1/2}\) for spontaneous fission on nuclear radii is very strong,
\[ \left(T_{1/2}\sim e^{-\frac{BZ^2}{r^3}}\right), \]
the maximum deviation on the graph from the linear dependence obtained for even-even nuclei corresponds to an increase in the radius for odd nuclei of only 1%. In the author’s opinion\(^2\), an additional factor acting toward a reduction of the spontaneous-fission rate may be the difference in the zero-point vibrational levels leading to fission of nuclei with different radii.
G. Seaborg points out that the difference between even-even and odd nuclei is also manifested in fission by thermal neutrons. Thus, the fission cross section of $_{96}\mathrm{Cm}^{242}_{146}$ by thermal neutrons is less than $5\cdot10^{-24}\ \text{cm}^2$, although the critical fission energy for $_{96}\mathrm{Cm}^{243}_{147}$ should be of the order of 4 MeV, while the binding energy of the neutron released in the formation of the latter nucleus from $\mathrm{Cm}^{242}$ is of the order of 6 MeV. Apparently, in those cases when the intermediate nucleus in fission is not even-even (as $_{92}\mathrm{U}^{236}_{144}$ or $_{94}\mathrm{Pu}^{240}_{146}$), $\gamma$-processes compete successfully with fission. This may also be connected with the higher photofission thresholds for odd nuclei $\mathrm{U}^{235}$, $\mathrm{U}^{233}$, and $\mathrm{Pu}^{239}$ in comparison with the even-even nucleus $_{92}\mathrm{U}^{238}_{146}$.
In conclusion, G. Seaborg draws attention to the possibility of an especially short half-life for spontaneous fission of nuclei such as $100^{248}$ with filled subshells of 100 protons and 148 neutrons, since the filling of subshells is associated with some decrease in the nuclear radius. This, in particular, may perhaps explain the appreciable fission cross sections by thermal neutrons of the two isomers of americium—$\mathrm{Am}^{242}$ ($6\cdot10^{-21}\ \text{cm}^2$) and $\mathrm{Am}^{242m}$ ($2\cdot10^{-21}\ \text{cm}^2$)3—despite the small fission cross section of the isotope $\mathrm{Am}^{241}$ ($3\cdot10^{-24}\ \text{cm}^2$); in fact, the intermediate nucleus $_{95}\mathrm{Am}^{243}_{148}$ contains 148 neutrons, i.e., a filled subshell.
G. I.
Cited Literature
- K. A. Petrzhak and G. N. Flerov, DAN 28, 500 (1940).
- G. T. Seaborg, Phys. Rev. 85, 157 (1952).
- K. Street, A. Ghiorso, S. Thompson, Phys. Rev. 85, 135 (1952).