70 YEARS OF FISSION Kazimierz 2008 Comment on a frequent error in calculations of the n / f ratio W.J. Świątecki, J. Wilczyński and K. S-W

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E = E gs + E * - total energy E n = (M n +M A-1 ) c 2 = E gs + B n E f – the saddle-point energy To calculate the ratio Γ n /Γ f we need the level density of the daugther nucleus (A-1) ρ(E-E n ) and the level density at the saddle-point of the nucleus A ρ(E-E f ) E-E n = E gs +E * -E gs -B n = E* - B n E-E f = E gs +E*-E f = E*- (E f -E gs )

m n, s n, ε n - mass, spin and kinetic energy of the emitted neutron f - level density of the fissioning nucleus (at saddle) n - level density of the daughter nucleus (A-1) E – total energy E f – saddle-point energy E n - energy of the system n + (A -1) nucleus E-E n = E gs +E*-E gs -B n = E*- B n E-E f = E gs +E*-E f = E*- (E f -E gs ) R. Vandenbosch & J.R. Huizenga, Nuclear Fission - formula (VII-3) R. Vandenbosch & J.R. Huizenga, Nuclear Fission - formula (VII-7) Assuming:, and a =const (1) (2)

independent of the excitation energy Shell effects included using: the energy dependent level density parameter ( A.V. Ignatyuk et al., Sov. J. Nucl. Phys. 29 (1975) 255 ) where: E * - excitation energy, E d – damping parameter E shell – shell correction energy, a LDM - the LDM level density parameter or an exponentially dependent fission barrier replacing the saddle-point energy (erroneously postulated by G. G. Adamian, N. V. Antonenko and W. Scheid, Nucl. Phys. A678, 24 (2000), and their followers) E f – E gs = B LDM + B micr exp(-E*/E d ) dependent on the excitation energy for super-heavy nuclei B LDM = 0 B micr = - E shell (gs) a n, a f = const no shell effects in (A-1) nucleus shell effects in fission channel, only via E shell (gs )

G.G. Adamian et al. PRC 69, (2004) PRC 62, (2000) PRC 69, (2004) PRC 69, (2004) PRC 69, (2004) W. Loveland PRC 76, (2007) W. Loveland et al. PRC 74, (2006) R.S. Naik et al. PRC 76, (2007) A CN = 266 B n = 8.22 MeV B LDM = 0 B micro = -E shell = 5.27 MeV B f = B micro exp(-E*/E d ) a n, a f = const numerical integration, with energy dependent level density parameter, B f = 5.27 MeV

W. Loveland PRC 76, (2007) W. Loveland et al. PRC 74, (2006) R.S. Naik et al. PRC 76, (2007) G.G. Adamian et al. PRC 69, (2004) PRC 62, (2000) PRC 69, (2004) PRC 69, (2004) PRC 69, (2004) A CN = 297 B n = 6.21 MeV B LDM = 0 B micro = -E shell = 8.27 MeV B f = B micro exp(-E*/E d ) a n, a f = const numerical integration, with energy dependent level density parameter, B f = 8.27 MeV

In case of the saddle-point energy (erroneously) replaced by the energy dependent fission barrier B f (E * ) = - E shell (gs) exp(-E * /E d ), the classical fission threshold shifts from E thr = B f = - E shell (gs) to a value satisfying E thr - E shell (gs)exp(-E thr /E d ) = 0 For Z=118 E thr = B f = 8.27 MeV E thr = 6.40 MeV (for E d =25 MeV) Conclusions: The scheme of calculating the Γ n /Γ f ratios using the concept of energy dependent fission barrier of Adamian et al. is erroneous and leads to predictions which at low excitation energies may deviate from correctly evaluated values by a factor of 1000 or more. Moreover, it leads to unphysical predictions for the existence of fission at energetically forbidden subt h reshold excitation energies.

Excitation energy dependence of Γ n /Γ f for different values of (E f -B n ). The level density parameters a f and a n were assumed to be equal (25 MeV -1 ) and B n = 6 MeV. Figure taken from Nuclear Fission R. Vandenbosh & J.R. Huizenga Academic Press 1973

E def (ε) δ shell g.s. E def TOT E def LDM ε δ shell sd BfBf