Nuclear fission

The fission decay channel (only for nuclei with \(A > 65\)) is taken into account as a competitor for fragment and photon evaporation channels.

The fission total probability

The fission probability (per unit time) \(W_{fis}\) in the Bohr and Wheeler theory of fission [BW39] is proportional to the level density \(\rho_{fis}(T)\) (approximation Eq. (255)) is used) at the saddle point, i.e.

\[\begin{split}W_{fis} &= \frac{1}{2\pi \hbar \rho_{fis} (E^{*})} \int_0^{E^{*}-B_{fis}} \rho_{fis} (E^{*}-B_{fis}-T)dT \\ &= \frac{1 + (C_f - 1)\exp{(C_f)}}{4\pi a_{fis} \exp{(2\sqrt{aE^{*}})}},\end{split}\]

where \(B_{fis}\) is the fission barrier height. The value of \(C_f = 2\sqrt{a_{fis}(E^{*} - B_{fis})}\) and \(a\), \(a_{fis}\) are the level density parameters of the compound and of the fission saddle point nuclei, respectively.

The value of the level density parameter is large at the saddle point, when excitation energy is given by initial excitation energy minus the fission barrier height, than in the ground state, i. e. \(a_{fis} > a\). \(a_{fis} = 1.08 a\) for \(Z < 85\), \(a_{fis} = 1.04 a\) for \(Z \geq 89\) and \(a_f=a[1.04+0.01(89.-Z)]\) for \(85 \leq Z < 89\) is used.

The fission barrier

The fission barrier is determined as difference between the saddle-point and ground state masses.

We use simple semiphenomenological approach was suggested by Barashenkov and Gereghi [BITG73]. In their approach fission barrier \(B_{fis}(A,Z)\) is approximated by

\[B_{fis} = B^{0}_{fis} + \Delta_g + \Delta_p.\]

The fission barrier height \(B^{0}_{fis}(x)\) varies with the fissility parameter \(x = Z^2/A\). \(B^{0}_{fis}(x)\) is given by

\[B^{0}_{fis}(x) = 12.5 + 4.7 (33.5 -x )^{0.75}\]

for \(x \leq 33.5\) and

\[B^{0}_{fis}(x) = 12.5 - 2.7 (x - 33.5)^{2/3}\]

for \(x > 33.5\). The \(\Delta_g = \Delta M(N) + \Delta M(Z)\), where \(\Delta M(N)\) and \(\Delta M(Z)\) are shell corrections for Cameron’s liquid drop mass formula [Cam57, Cam58] and the pairing energy corrections: \(\Delta_p = 1\) for odd-odd nuclei, \(\Delta_p = 0\) for odd-even nuclei, \(\Delta_p = 0.5\) for even-odd nuclei and \(\Delta_p = -0.5\) for even-even nuclei.

Bibliography

BITG73

V.S. Barashenkov, A.S. Iljinov, V.D. Toneev, and F.G. Gereghi. Fission and decay of excited nuclei. Nuclear Physics A, 206(1):131–144, May 1973. URL: https://doi.org/10.1016/0375-9474(73)90611-8, doi:10.1016/0375-9474(73)90611-8.

BW39

Niels Bohr and John Archibald Wheeler. The mechanism of nuclear fission. Physical Review, 56(5):426–450, Sep 1939. URL: https://doi.org/10.1103/PhysRev.56.426, doi:10.1103/physrev.56.426.

Cam57

A. G. W. Cameron. A REVISED SEMIEMPIRICAL ATOMIC MASS FORMULA. Canadian Journal of Physics, 35(9):1021–1032, Sep 1957. URL: https://doi.org/10.1139/p57-114, doi:10.1139/p57-114.

Cam58

A. G. W. Cameron. NUCLEAR LEVEL SPACINGS. Canadian Journal of Physics, 36(8):1040–1057, Aug 1958. URL: https://doi.org/10.1139/p58-112, doi:10.1139/p58-112.