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