Actinide Symmetric/Asymmetric Nucleon-Induced Fission up to 200 MeV

The fission cross sections of the symmetric SL-mode and the asymmetric lumped (S1+S2)-mode of the 238U(n,F), 235U(n,F), 232Th(n,F), and 238U(p,F) reactions are calculated up to En=200 MeV within a statistical model. To reproduce the measured branching ratio of symmetric and asymmetric fission events in the 238U(n,F) reaction, a preferential contribution of fission from neutron-deficient nuclei is assumed. This assumption is of critical importance to reproduce the observed 232Th(n,F) cross section. The branching ratio of symmetric and asymmetric fission events is shown to increase, with the increase of the incident neutron energy, faster for the higher fissility of the target nuclide.


Introduction
In neutron-induced fission of actinides, the contribution of the symmetric or superlong (SL) fission mode [1,2] increases with the excitation energy [3][4][5][6][7].At incident neutron energies E n 6 MeV, pre-fission neutrons might be emitted, before the fissioning nuclide reaches the outer saddle point.We will call them pre-saddle.This peculiarity considerably complicates the interpretation of fission observables.A number of nuclides might contribute to the fission observables, about ∼ 20 for the 238 U(n, xnf) fission reaction at E n ∼ 200 MeV [8].In other words, an ensemble of nuclides, which emerge after emission of x pre-saddle neutrons, contributes to the symmet-E-mail address: maslov@sosny.bas-net.by(V.M. Maslov).
ric and asymmetric fission.The branching ratio of symmetric fission events to the total observed fission events r sym = σ nFSL /(σ nFSL + σ nF(S1+S2) ) was obtained for 238 U(n, F) by Zoller et al. [9] at E n up to ∼ 500 MeV.The values of r sym were obtained from the deduced yield distributions of primary fission fragments and their average total kinetic energies TKE as a function of mass.Description of the branching ratios r sym by Zoller et al. [9] up to E n ∼ 200 MeV favors the major contribution to the observed fission cross section of fission chances with a larger number of pre-saddle neutrons [8].With increasing fissility of the target nuclide this number might be somewhat lower.We will apply the same approach, which was previously used for the description of the cross section and the branching ratio of symmetric/asymmetric fission of the observed 238 U(n, F) reaction, to which either first chance 238 U(n, f) and emis-sive fission 238 U(n, xnf) reactions contribute, to describe measured fission cross sections in the reactions 235 U(n, F), 237 Np(n, F) and 238 U(p, F).In particular, we will predict the 235 U(n, F) sym , 237 Np(n, F) sym and 235 U(n, F) asym , 237 Np(n, F) asym fission cross sections.Simultaneous analysis of 238 U(n, F) and 238 U(p, F) reaction cross sections would help to investigate the dependence of observed fission cross sections on the projectile.

Statistical model
We assume that the fission fragments are emitted from a chain of U(Np) nuclei after pre-equilibrium (PE) emission and evaporation of neutrons [10].We do not take into account charged-particle emission, for justification see the discussion in [8].A coupledchannels model, fitting the 238 U total fission cross sections [11] up to E n ∼ 200 MeV is employed.The contribution of the SL-mode σ nFSL to the observed fission cross section, originating from (n, xnf) fission reactions, was calculated using the fission probability P J π fSLx (U ) for symmetric fission of the fissioning xth nucleus, is the population of the (x + 1)th nucleus at excitation energy U after emission of x neutrons.The excitation energy U max is defined by incident neutron energy E n and energy removed by pre-saddle evaporated neutrons.The modeling of the first-chance fission cross section σ nfSL is described elsewhere [7].The lumped contribution of the asymmetric fission standard 1 (S1) and standard 2 (S2) modes is defined in an analogous way.
The nuclear level density ρ(U, J, π) is represented as factorized contribution of quasiparticle and collective states [12].The quasiparticle level densities ρ qp (U, J, π) were calculated with the phenomenological model of Ignatyuk et al. [13] as where K rot (U, J ) and K vib (U ) are the factors of rotational and vibrational enhancement.At saddle and ground-state deformations K rot (U ) is defined by the symmetry class, adopted from shell-model calculations [2,14,15].At inner saddle we assume axial symmetry for neutron numbers N 144 and triaxial shape for N > 144.At outer saddle triaxiality is assumed for mass symmetric mode, while axial shape is assumed for the mass asymmetric mode.For more extensive discussions see [2,7,8] and references therein.At excitations U U r , damping of rotational modes was anticipated [16].Damping of rotational modes contribution to the nuclear level density ρ(U, J, π) might be different for axially symmetric and triaxial nuclei [17], i.e., (3) Here, σ 2 and σ 2 ⊥ are the spin-distribution parameters.The mass asymmetry for the S1(S2)-modes at the outer saddles doubles the rotational enhancement factors as defined by Eqs.(3), (4).Shell effects in the level density are modeled with the dependence of the aparameter on the shell correction δW as recommended by Ignatyuk et al. [13]: a(U ) = ã(1 + δWf ( Ũ)/( Ũ)).The value of the asymptotic a n -parameter ãn is defined by fitting to the neutron-resonance spacing D obs or by systematics.We assume ãn = ãf , then the a f /a n ratio depends on values of δW f(n) , taken from [18] (δW n ) and [19] (δW f ).

Analysis
The total fission cross section σ nF = σ nFSL + σ nF(S1+S2) depends on the contributions of both terms.These contributions strongly depend on the asymptotic value ãf (A) of the a f parameter of the fissioning nuclei, while the branching ratio r sym depends both on the contributions of fission chances and the damping of the contribution of the triaxial collective modes to the nuclear level density (see Eqs. ( 4), (5) for the SL fission mode [8].The contributions of fission chances to the observed fission cross section σ nF are affected by decreasing the asymptotic value of the a f parameter with energy as . This expression was obtained by consistent description of observed Th, U and Pu fission cross sections [20] as well as branching ratio r sym for 238 U(n, F) reaction [8].The heights of the outer fission barrier B of the SL mode for 239 U and 236 U fissioning nuclides were derived to be higher than those of the asymmetric modes, i.e., (E fBSL − E fBS1(S2) ) ∼ 3.5 MeV, while hω BSL = 2.25 MeV [7].The contributions of lighter U nuclides via (n, xnf) reactions to the observed symmetric fission might be obtained assuming for each of them the same difference of the outer barriers for the symmetric SL and the asymmetric fission S1(S2) modes, the shell-correction values being defined as (δW fBSL − δW fBS1(S2) ) ∼ 3.5 MeV, assuming δW fBS1(S2) ∼ 0.6 MeV [19].The assumption that the difference of mass-symmetric and mass-asymmetric fission barriers do not vary with the neutron number might seem too crude (see Schmidt et al. [21], however, we believe that weak isotopic dependence of (E fBSL − E fBS1(S2) ) could be compensated by slight variation of asymptotic value of the a f parameter (see Eq. ( 6)).The inner and outer fission-barrier parameters of uranium for the double-humped fission-barrier model [22], relevant for the saddle asymmetries, predicted by SCM calculations [14] for asymmetric fission, were defined in [23,24].This simple systematics of level-density and fission-barrier parameters for U nuclides allowed to reproduce the observed 235 U(n, F) fission cross section [25].The same approach works in the case of neutron-induced fission of 237 Np target nuclides.
It might be anticipated that further sophistication of the model, i.e., inclusion of the temperature and angular momentum dependence of fission barriers and shell corrections, influence of neutron shell N = 126 on collective enhancement in neutron channel [17], or use of calculated with SCM method [2] fission barriers for symmetric and asymmetric fission modes of U nuclei, would not change pattern of emissive fission contributions.Lowering of asymptotic value ãf of a f parameter at saddle deformations might be considered as reproducing lumped effect of all these factors.We anticipate also that because of strong emissive fission nature of 238 U(n, f) reaction for E n 200 MeV, vanishing of the distinction of symmetric and asymmetric valleys at high excitation of fissioning nuclei [26] would not be much pronounced.
Fig. 1 shows the calculated symmetric 235 U(n, F) sym , asymmetric 235 U(n, F) asym and symmetric + asymmetric 235 U(n, F) fission cross sections for the energy-dependent asymptotic ãf (U, A) of a f parameter (Eq.( 6)).The set of solid curves shows the 235 U(n, F) sym and 235 U(n, xnf) sym cross sections, while the dashed lines show those for the asymmetric neutron-induced fission of 235 U target nuclide.
Fig. 2 shows the symmetric 237 Np(n, F) sym , asymmetric 237 Np(n, F) asym and symmetric + asymmetric 237 Np(n, F) fission cross sections.The lines on Fig. 2 have the same meanings as on Fig. 1.The sum of the calculated 237 Np(n, F) sym and 237 Np(n, F) asym reaction cross sections (dash-dotted line) is quite compatible with the observed fission cross section [27] up to E n ∼ 200 MeV.The data by Hambsch et al. [3] for the symmetric fission yield are also reproduced.
In case of the proton-induced fission reaction 238 U(p, F), the observed fission reaction cross section could be calculated using the fission-barrier and leveldensity parameters of Np nuclei, obtained by fitting to the observed fission cross section σ nF could be controlled by comparing the calculated branching ratio r sym with the measured data by Zoller et al. [9] (see Fig. 4) for the 238 U(n, F) reaction and with the measured data below the fission threshold by Hambsch [5] for 238 U(n, f), Vives et al. [4] for 235 U(n, f) and Hambsch et al. [3] for 237 Np(n, f).The relative contributions of fission chances with low and high number of prefission neutrons have a major influence on the energy dependence of r sym at E n 25 MeV.When energy-dependent asymptotic ãf (A) of a f parameter is employed (see Eq. ( 6)), higher chances make a predominant contribution to the observed fission cross section, and damping of triaxial collective modes contribution (see Eqs. ( 4), ( 5)) at outer symmetric fission saddle allows to reproduce the measured data by Zoller et al. [9] for 238 U(n, F) at E n 10 MeV.Below the fission threshold there is a systematic  difference of branching-ratio data by Hambsch [5] for 238 U(n, f) and Vives et al. [4] for 235 U(n, f).
The calculated branching ratios r sym for 235 U(n, F) and 238 U(n, F) reactions are different around the