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
Mar 26, 2015
70 YEARS OF FISSION
Kazimierz 2008
Comment on a frequent error in calculationsof the n/f ratio
W.J. Świątecki, J. Wilczyński and K. S-W
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E = Egs + E* - total energy
En= (Mn+MA-1) c2 = Egs + Bn
Ef – the saddle-point energy
To calculate the ratio Γn/Γf
we need the level density ofthe daugther nucleus (A-1) ρ(E-En) and the level density at thesaddle-point of the nucleus A ρ(E-Ef)
E-En= Egs +E*-Egs-Bn= E*- Bn
E-Ef= Egs+E*-Ef = E*- (Ef-Egs)
dKKEE
)dεεE(EAsr2m
ff
nnnnnnnn
)(
12
f
n
EE
02
EE
03/22
0
f
))((2)(2exp)1))((2(
)(2
3/220
f
ngsf
*fn
*n
gsf*
fn
n*
fn EEEaBEaEEEaa
BEaAr4m
)E2(exp)E( a
mn, sn, εn - mass, spin and kinetic energy of the emitted neutronf - level density of the fissioning nucleus (at saddle) n - level density of the daughter nucleus (A-1)
E – total energyEf – saddle-point energyEn - energy of the system n + (A -1) nucleus
E-En= Egs +E*-Egs-Bn= E*- Bn
E-Ef= Egs+E*-Ef = E*- (Ef-Egs)
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)
)(2)(2exp)1)(2(
)(2
3/220
f
nffnn
ffn
nfn EEaEEaEEaa
EEaAr4m
independent of theexcitation energy
))((2)(2exp)1))((2(
)(2
3/220
f
ngsf
*fn
*n
gsf*
fn
n*
fn EEEaBEaEEEaa
BEaAr4m
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, Ed – damping parameter Eshell – shell correction energy, aLDM- 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)
Ef – Egs = BLDM + Bmicr exp(-E*/Ed)dependent on theexcitation energy
d
* EE*
e1E
E1 shell
LDM* aEa
for super-heavy nuclei BLDM= 0Bmicr= - Eshell(gs)
an, af = constno shell effects in (A-1) nucleusshell effects in fission channel, only via Eshell(gs)
G.G. Adamian et al. PRC 69, 014607 (2004) PRC 62, 064303 (2000) PRC 69, 011601 (2004) PRC 69, 014607 (2004) PRC 69, 044601 (2004)
W. Loveland PRC 76, 014612 (2007)W. Loveland et al. PRC 74, 044607 (2006)R.S. Naik et al. PRC 76, 054604 (2007)
ACN = 266 Bn = 8.22 MeV
BLDM = 0
Bmicro= -Eshell = 5.27 MeV Bf = Bmicroexp(-E*/Ed)
an, af = const
numerical integration, with energy dependent level density parameter, Bf = 5.27 MeV
W. Loveland PRC 76, 014612 (2007)W. Loveland et al. PRC 74, 044607 (2006)R.S. Naik et al. PRC 76, 054604 (2007)
G.G. Adamian et al. PRC 69, 014607 (2004)
PRC 62, 064303 (2000)
PRC 69, 011601 (2004)
PRC 69, 014607 (2004)
PRC 69, 044601 (2004)
ACN = 297 Bn = 6.21 MeV
BLDM = 0
Bmicro= -Eshell = 8.27 MeV Bf = Bmicroexp(-E*/Ed) an, af = const
numerical integration, with energy dependent level density parameter, Bf = 8.27 MeV
In case of the saddle-point energy (erroneously) replaced by the energy dependent fission barrier Bf(E*) = - Eshell(gs) exp(-E*/Ed), the classical fission threshold shifts from Ethr = Bf = - Eshell(gs) to a value satisfying Ethr - Eshell(gs)exp(-Ethr /Ed) = 0
For Z=118
Ethr = Bf = 8.27 MeV → Ethr = 6.40 MeV (for Ed=25 MeV)
Conclusions:The scheme of calculating the Γn/Γf 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 subthreshold excitation energies.
Excitation energy dependence of Γn/Γf for different values of (Ef-Bn). The level density parameters af and an were assumed to be equal (25 MeV-1) and Bn= 6 MeV.
Figure taken from „Nuclear Fission”R. Vandenbosh & J.R. Huizenga Academic Press 1973
Edef (ε)
δshellg.s.
EdefTOT Edef
LDM
εδshell
sd
Bf