Fission Fragments Mass Yields of Actinide Nuclei Institute of Physics Maria Curie Sk lodowska University 20-031 Lublin Poland Krzysztof Pomorskiy Bo_zena Pomorska

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Fission Fragments Mass Yields of Actinide Nuclei ∗

Institute of Physics, Maria Curie Sk lodowska University, 20-031 Lublin, Poland

Krzysztof Pomorski†, Bo˙

zena Pomorska

A new, rapidly convergent Fourier over spheroid parametrization is de-

veloped to describe the shape of a ﬁssioning nucleus: its elongation, non-

axiality and left-right asymmetry and neck formation. The 4D Potential

Energy Surfaces (PES) of even-even actinide nuclei are evaluated within

the macro-micro model. The Langevin trajectories generated on such PESs

allow for obtaining the ﬁssion fragments’ mass and kinetic energy yields.

The charge equilibration at the scission conﬁguration and the post-scission

neutron emission are also discussed.

PACS numbers: 24.75.+i, 25.85.-w,28.41.Ak

1. Introduction

The nuclear ﬁssion phenomenon, discovered in 1938, is still very attrac-

tive to experimentalists and theoreticians. Accurate reproduction of such

observables as the mass and the total kinetic energy yields of ﬁssion frag-

ments or the multiplicities of emitted neutrons is a good test of any modern

theoretical model. A review of the existing ﬁssion models can be found,

e.g., in Refs. [1, 2, 3, 4, 5, 6]. Readers interested in the theory of nuclear

ﬁssion can ﬁnd more details in the textbook [7].

The present research is a continuation of our previous works [9, 10, 11,

12, 13] in which the ﬁssion fragment mass-yields of nuclei from diﬀerent

mass regions were obtained using the Fourier shape parametrization [14].

In the present paper, we extend our 3D Langevin model by adding a mode

responsible for the charge equilibration of the fragments and a Master type

equation describing the multiplicity of emitted neutrons. In addition, an

innovatory, better adopted to make the ﬁssion calculation on a grid, the

Fourier-over-Spheroid (FoS) shape parametrization (see also [15]) is used.

∗Presented at the Zakopane Conference on Nuclear Physics, Zakopane 2022.

†Email: Krzysztof.Pomorski@umcs.pl, ORCID ID: 0000-0002-5557-6037

(1)

arXiv:2210.09637v1 [nucl-th] 18 Oct 2022

2zakopane-pomorski printed on October 19, 2022

One has to stress here that the nuclear shapes generated by FoS are equiva-

lent to those given by the Fourier parameterization [14]. The main features

of our model are described in Section 2, while the neutron emission from

the fragments is discussed in Section 3, followed by the Summary.

2. Nuclear shape parametrization

The potential energy surfaces of ﬁssioning nuclei are obtained using the

macro-micro model. The macroscopic part of the energy is evaluated ac-

cording to the Lublin-Strasbourg-Drop (LSD) formula [16], while the mi-

croscopic energy corrections are calculated using the Yukawa-folded single-

particle potential [17].

The surface of the ﬁssioning nucleus is described in the cylindrical coor-

dinates (ρ, ϕ, z) by the following formula:

ρ2(z, ϕ) = R2

cfz−zsh

z01−η2

1 + η2+ 2ηcos(2ϕ),(1)

where ρ(z, ϕ) is the distance from the z-axis to the surface. Function f(u)

deﬁnes the shape of the nucleus having half-length c= 1:

f(u) = 1 −u2−a4

3−a6

5+. . .cos π

2u−a3sin(π u)

−a4cos 3π

2u−a5sin(2π u)−a6cos 5π

2u−. . . ,

(2)

where −1≤u≤1. The ﬁrst two terms in f(u) describe a sphere, the

third ensures volume conservation for arbitrary deformation parameters

{a3, a4, . . .}. The parameter cdetermines the elongation of the nucleus

keeping its volume ﬁxed, while a3and a4describe the reﬂectional asymme-

try and the neck size, respectively. The half-length is z0=cR0, where R0is

the radius of a sphere with the same volume. The z-coordinate varies in the

range −z0+zsh ≤z≤z0+zsh. The shift zsh =−3/(4π)z0(a3−a5/2 + . . .)

places the nuclear center of mass at the origin of the coordinate system.

The parameter ηdescribes a possible elliptical, non-axial deformation of a

nucleus.

The formula (1) is completely equivalent to those based on the Fourier

expansion and described in Refs. [14]. Here, the deviation from a sphere

with radius ρ= 1 is ﬁrstly expanded in the Fourier series, and subsequently

this deformed object of the length 2R0is scaled to the elongation equal to

2cR0. The formula (1) is more adapted to calculation of the PES made

on a mesh in the multidimensional deformation parameter (c, a3, a4, ..., an)

space, since the range of variability of the akcoeﬃcients does not depend

on the elongation c. In addition, the mass ratio of the fragments, their

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FissionFragmentsMassYieldsofActinideNucleiInstituteofPhysics,MariaCurieSklodowskaUniversity,20-031Lublin,PolandKrzysztofPomorskiy,Bo_zenaPomorskaAnew,rapidlyconvergentFourieroverspheroidparametrizationisde-velopedtodescribetheshapeofassioningnucleus:itselongation,non-axialityandleft-rightasymmetry...

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Fission Fragments Mass Yields of Actinide Nuclei Institute of Physics Maria Curie Sk lodowska University 20-031 Lublin Poland Krzysztof Pomorskiy Bo_zena Pomorska

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