Pre-Scission Model Predictions of Fission Fragment Mass Distributions for Super-Heavy Elements Nicolae Carjan Joint Institute for Nuclear Research, Dubna, Russia National Institute for Physics and Nuclear Engineering, Bucharest-Magurele, Romania Fedor Ivanyuk Institut for Nuclear Research, Kiev, Ukraine Yuri Oganessian Fission of SHE, 9–13 April 2018, Trento, Italy

Plan Introduction Model: pre-scission + generalized Cassini ovals Mass and kinetic energy distributions of the fission fragments Heavy actinides (Fm to Sg) Fission of SHE: asymmetric vs symmetric; binary vs ternary; excitation energy dependence. Synopsis

Introduction SHE have zero macroscopic barriers but fission rarely; instead they undergo α decay: at first sight an oxymoron. However, their ground states are so strongly stabilized by microscopic corrections that even a zero barrier becomes hard to surpass → most studies of fission of SHE are theoretical (macroscopic-microscopic or selfconsistent ). In previous studies the accent was put on barriers and half-lives, the potential energy surfaces (PES) being calculated in the vicinity of ground states and saddle points. Here the accent is put on fission fragment properties and the PES are calculated around the scission point. To provide credible predictions for SHE, one has to test the model on the heaviest nuclei for which such properties have been measured in spontaneous fission (i.e., Fm, No, Rf and Sg isotopes).

It belongs to the class of „scission-point“ models. Formalism N. Carjan, F. A. Ivanyuk, Yu. Oganessian, G. Ter-Akopian, Nucl. Phys. A 942 (2015) 97 It belongs to the class of „scission-point“ models. Two improvements: 1) We use a shape parametrization that is most convenient to describe the fissioning nucleus around the scission point. The Cassinian ovals are generalized by expanding them in series of Legendre polynomials: R0 is the radius of the spherical nucleus, αn the shape parameters and c = (V/V0)1/3 assures the volume conservation. (R,x) is an orthogonal coordinate system introduced in 1971 by Vitaly Pashkevich to describe asymmetric fission with only one parameter α1 . Why is this parametrization the most convenient one around scission?

Obviously it is easier to describe a scission shape starting from curve no (5) than from no (1)

When we generalize the Cassinian ovals we replace ‘ε’ with a new parameter ‘α’ with the property that a shape with rneck =0 has always α=1 , irrespective of the values of αn Where zL (zR ) is the coordinate of the left (right) tip of the nuclear shape and ρ = r2neck . Values of α slightly lower than 1 correspond to two fragments connected by a thin neck. 2) We use a “just-before scission” model. We consider that scission occurs at finite neck radius (rneck ~ 2 fm) and approximate the scission line by α = 0.98. Justification: the scission process ( from the beginning of the neck rupture till the absorption of the neck stubs by the fission fragments) is extremely fast → the mass distribution is frozen and Dcm stays practically unchanged. It is therefore just-before scission that the fragment mass and kinetic energy distributions have to be evaluated (and not when the fragments are already separated).

Main ingredient: potential energy of deformation calculated with the microscopic-macroscopic approach (i.e., Liquid-Drop model + Strutinsky shell correction ): Edef (shape) = ELD (shape) + Eshell (shape) Potential energy surfaces for α = 0.98 are calculated as a function of the three most relevant deformation parameters: α1 (mass asymmetry) , α3 (octupole deformation) and α4 [acting on the quadrupole elongation of each fragment makes the scission shape more compact (α4 <0) or more elongated (α4 >0)]. In addition we minimize with respect to α6 .

Mass distribution of the fission fragments Supposing statistical equilibrium for the collective degrees of freedom normal to the fission direction, the distribution of each point (α1,α4) on the potential energy surfaces is due to thermal fluctuations: with Tcoll =1.5 MeV. Projecting on the α1 –axis one obtains the total yield: The shape parameter α1 determines the mass asymmetry η = (AFH –AFL)/A . The mass distributions are calculated using the formulae from above.

Decomposition of the total mass distribution (black) into two fission modes: one compact (red) (α4 <0) and one elongated (blue) (α4 >0) . An interesting double inversion of these two modes occurs with increasing mass A: The red curve gradually replaces the blue curve as dominant mode and at the same time it (the compact mode) evolves from asymmetric to symmetric mass division.

Synopsis of fission fragment mass distributions for isotopes of Fm, No, Rf and Sg (chosen around the transition point from asymmetric to symmetric fission). Results with Tcoll =0.75 MeV are also presented (blue curves) for comparison.

Total kinetic energy of the fission fragments For each shape defined by (α1,α4) one calculates Dcm and estimates the Coulomb repulsion of the nascent fragments in the point charge approximation: if one neglects the pre-scission kinetic energy. It is therefore a lower limit for TKE. The TKE distribution can be calculated using It accounts for the finite energy resolution through the parameter Δ that was chosen 6 MeV corresponding to an experimental resolution with a FWHM = 5 MeV.

As a consequence of the fission-mode inversion discussed previously, we observe, with increasing mass A of the fissioning nucleus, a transition from a TKE distribution that deviates from a Gaussian in the high energy part of the spectrum to one that deviates on the low energy part. At the same time the energy difference between the two peaks increases with A reaching 20 MeV for the heaviest isotopes. For 258 Fm the measured value is 25 MeV. So again the agreement is only qualitative.

Remarks so far: The agreement with data (especially for Fm isotopes) is only semi-quantitative in the sense that the transition from symmetric to asymmetric fission is not as sharp as observed. The macroscopic-microscopic model used averages over neighbouring nuclei. Therefore sharp transitions from one nucleus to another are not expected to be explained. An interesting result is the extremely narrow mass distribution of the heaviest isotopes (FWHM ≤ 8 amu). To distinguish it from the regular symmetric fission we call this type of fission “super-symmetric”.

Detailed comparison with existing data on No isotopes; the agreement is surprisingly good. It gives confidence in the simple model used. N.Carjan, F. Ivanyuk, Yu. Oganessian, Journal of Physics: Conf. Series 863 (2017) 012044

The same for Rf isotopes

Comparison with data from the SF of 262 Rf M. R. Lane et al Comparison with data from the SF of 262 Rf M.R. Lane et al. PRC 53 (1996) 2893 FWHM = 22 amu

In Fm isotopes the calculated transition from asymmetric to symmetric fission is smooth (not as sharp as in the experimental data)

Incursion into the region of superheavy elements Super-heavy nuclei that undergo spontaneous fission have already been observed in experiments with 48Ca projectiles in Dubna: (Z,N) = (110,169), (110,171), (111,170), (112,170) and (112,172). Yu. Oganessian, V. Utyonkov, Rep. Prog. Phys. 78 (2015) 036301 F.-P. Hessberger, Eur. Phys. J. A 53 (2017) 75 The production cross sections of these nuclei are only few picobarns but such experiments are considered for the future SHE-Factory under construction at FLNR of JINR-Dubna (Sergey Dmitriev’s talk). Irrespective how difficult is to produce these nuclei the result is worth the effort. Just an example: if 132Sn is the reason for asymmetric fission of actinides (where it is a heavy fragment) then it should also produce asymmetric fission in the super-heavy region (this time being a light fragment). To test this inference is crucial. For the same reason, the SHE may fission into three fragments (two 132Sn and the rest).

Predictions, with the same method as before, for SHE whose spontaneous fission has been detected. The statistics was not sufficient to build experimental distributions Z=110 Z=111 Z=112 Calculated fragment mass (up) and total kinetic energy (down) distributions. Results with and without the inclusion of α3 are presented (red and black curves respectively). Only the inclusion of the octupole term α3P3 leads to asymmetric mass distributions. Parameters: Tcoll = 2.0MeV, ΔE = 20.0MeV.

Potential energy surfaces as a function of octupole deformation (α3) and fission-fragment mass (α1) for 280Ds and 284Cn at a configuration just-before scission. The shapes corresponding to potential minima are also shown. α3 makes the light fragment spherical and the heavy very deformed. The net effect is a slight increase of Dcm

Is, in general, the SHE fission symmetric or asymmetric Is, in general, the SHE fission symmetric or asymmetric? To answer this important question, fragment mass distributions for even-even isotopes of Fl, Lv, Og and 126 were calculated with inclusion of α3. N. Carjan, F. A. Ivanyuk, Yu. Oganessian, Nucl. Phys. A 968 (2017) 453 These isotopes are separated by 4 amu starting with A = 284, 288, 288 and 300 respectively. Parameters: α = 0.98, Tcoll = 2MeV. The constant position of the light fragment peaks (around AL=136) is remarkable.

The same as before but without inclusion of α3 : the mass division is symmetric in all cases → the octupole deformation at scission plays a crucial role in determining the main feature of the mass distribution: symmetric or asymmetric The isotopes are separated by 4 amu starting with A = 284, 288, 288 and 296 respectively The widths of the distributions vary from one isotope to the other, the narrowest having NL=NH=87 (a neutron deformed shell known already in the 50’s).

The deformation energy just-before scission (α=0 The deformation energy just-before scission (α=0.98) shows a deep minimum at α3 ≠ 0; hence from energetic point of view octupole shapes are clearly favored. But: will the dynamical paths reach these minima ?

Neutron and proton microscopic corrections to the total deformation energy at α=0.98 (just-before scission) for Fl (Z=114) isotopes. For A<280 the neutrons prefer a symmetric division while the protons an asymmetric one. Since the mass distribution is symmetric we conclude that the neutrons impose their pattern. The protons follow since Z/N ratios in the fragments have to be the same (unchanged charge distribution). For A>290 both neutrons and protons prefer asymmetric divisions but only in the neutron case the asymmetry increases with A ( as in the mass distribution) → in the SHE region neutrons determine the mass split.

Is the fission of SHE binary or sequential-ternary Is the fission of SHE binary or sequential-ternary? (preliminary) Minimization with respect to α3 leads to “ternary” scission shapes one almost spherical fragment with AL ≈ 135 and one extremely deformed (neck shaped) complementary fragment. What consequences may this have? The first neck rupture will separate the spherical light fragment from the rest. The question is: will the very-deformed heavy-fragment undergo a second scission or will it recover a more or less compact shape?

Study of the potential energy surfaces of the heavy fragment before scission in order to locate the saddles and the valleys as well as the Businaro-Gallone mountain. In the fission of 284Cn it is 148Ce. It has a well defined valley that is slightly mass-asymmetric. A very-asymmetric valley corresponding to two double-magic fragments is also expected (16O ,132Sn).

Heavy fragments from the fission of 296Fl and 312Og 161Sm: well defined symmetric and very asymmetric (AL=27, AH=134) valleys 177Ho: slightly symmetric and very asymmetric (AL=43, AH=134) valleys

To find the injection point of the heavy fragment (HF) at scission into the corresponding PES, we fitted its shape with generalized Cassinian ovals and extracted the values of (α, AL) According to the results that we have so far, the most probable post-scission evolutions of the HF are: 1) 148Ce will move into the direction of increased mass asymmetry reaching a compact shape. 2) 161Sm lying in the very asymmetric valley (few MeV under the barrier top) may pass over the barrier due to its kinetic energy 3) From the ridge of the BG mountain, 177Ho will descend into the very asymmetric valley and may pass the saddle. In conclusion the fission of 284Cn is mainly binary while that of 296Fl and 312Og may be ternary.

Mass distributions for neutron deficient isotopes of Fl calculated with α3. These isotopes are separated by 2 amu starting from A = 274. A transition asymmetric (282Fl) to symmetric (278Fl) fission takes place. In the latter nucleus the neutron numbers of the fragments (NL and NH) are equal to 82 (278=114+2x82) showing again that the magicity of the neutrons is the decisive factor. For 276Fl and 274Fl, the distributions are the narrowest. In this way the fission systematics of SHE and of heavy-actinides (where the neutron rich isotopes fission symmetrically) join smoothly.

Synthesis of the results obtained showing consistency between the two regions where calculations have been performed. The transition from asymmetric to symmetric fission (arrows) and the narrowest symmetric fission (dashed verticals) are indicated for each series of isotopes.

Excitation energy dependence (preliminary) At Ex =0 and 30 MeV the mass distributions are the same; it is of importance for future measurements that will not produce these nuclei in their ground states

300126 is an exception

The temperature dependence of the energy shell correction F.A.Ivanyuk et al, arXiv:1803.01036 [nucl-th]

The shell correction to the pairing energy

A. Bohr, B. Mottelson, Nuclear Structure, vol.2 (1975) p. 607 Unusual result: at low excitation energy the shell effect is reinforced A. Bohr, B. Mottelson, Nuclear Structure, vol.2 (1975) p. 607 It depends on whether the density of the levels near the Fermi energy is larger or smaller than the average; in the case of 284Fl it is smaller.

Summary and Conclusions An improved scission point model, tested on existing data (fission fragment mass and TKE for Fm, No and Rf isotopes), is used to predict mass distributions for super-heavy elements. Long series of isotopes of the elements Z = 114, 116, 118 and 126 have been considered. The basic input in our calculations is an ensemble of nuclear shapes along a finite-neck scission line defined by α = 0.98. These shapes are obtained by an expansion of Cassinian ovals in Legendre polynomials. Four deformation parameters have been included: α1, α3, α4 and α6. The underlying four dimensional potential energy surfaces are calculated using Strutinsky’s shell-correction procedure. The mass distribution is obtained by weighting these shapes with Boltzman factors and summing, for each mass ratio, their contributions.

Summary and Conclusions (cont.) For all studied nuclei, we predict asymmetric fission if one does include the octupole deformation and symmetric fission otherwise. For α3 ≠ 0, the position of the light-fragment peaks is remarkably constant (AL=136). It is because α3 allows the light fragment to be spherical creating a highly stable pre-scission configuration with NL =82. Naturally, a transition towards symmetric fission occurs for neutron deficient isotopes when NH = NL = 82 → in the SHE region the magicity of the neutrons determine the mass split. In addition of making the light fragment spherical, α3 also creates a highly deformed heavy fragment at scission rising the question whether a sequential fission of the latter is possible as expected from general considerations (two 123Sn and the rest).

To gain some confidance in the scission shapes included in the calculations we studied their main characteristics: the neck radius and the distance between the centers of mass of the nascent fragments. As seen below rneck covers the range 1.5fm to 2.5fm consistent with rneck = 2 fm which is an accepted values.

Dcm covers the range from 20. 0 fm to 23 Dcm covers the range from 20.0 fm to 23.5 fm in agreement with macroscopic dynamical calculations: N. Carjan, A. J. Sierk, J. R. Nix, Nucl. Phys. A452 (1986) 381. → The range of Dcm and rneck is quite narrow. → α = 0.98 is a good definition of the scission line.

Nuclear scission shapes for 254,264 Fm and 254,264 Rf corresponding to the local minima

Comparison of the most probable shapes obtained by the two methods The minimization in the deformation space (α3, α4, α6) always leads to more asymmetric shapes (black) as compared with the “optimal” shapes (red).

The minima corresponding to elongated shapes (both mass symmetric and asymmetric) do not vanish closer to zero-neck shapes (α = 0.99, i.e. rneck = 1.05 fm)

Estimated total excitation energy 252No 256Rf 268Rf 264Sg Fiss modes: red blue red blue red blue red blue [correspond to the color code used in our figures to distinguish the two main fission modes: red = compact and blue = elongated] A_L @ Ymax: 116 113 125 113 134 134 128 119 Z_L (UCD) 47 46 51 46 52 52 51 48 TKE @ Ymax: 214.0 199.1 222.4 203.0 225.3 209.4 229.8 212.6 (Coulomb interaction only) Q-value: 252.0 251.5 270.0 259.5 280.0 280.0 281.7 271.8 (from www-nds.iaea.org/ripl2/) E*=Q-TKE-15: 23.0 37.4 32.6 41.5 39.7 55.6 36.9 44.2 assumed 15 MeV pre-scission kinetic energy is a guess. In the case of 268Rf there is only one mode (red). B_n in the nucleus (52,134) is 7.5 MeV. Hence one can emit about 4 neutrons (in agreement with the data obtained 10 years ago in Dubna) and leave about 10 MeV for gamma rays. So these estimates are also not far from reality.

Comparison with Hulet et al. data.