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Mon Dec 24, 2018, 04:33 PM

Theoretical Analysis of the Observed Asymmetry in the Fission of Actinides. [View all]

The paper I'll discuss in this post is this one: Impact of pear-shaped fission fragments on mass-asymmetric fission in actinides (Guillaume Scamps & Cédric Simenel, Nature 564, 382–385 (2018))

Used nuclear fuel is very interesting stuff purely from a chemist's perspective because of the wide distribution of elements in it. This, in my view, is a good thing, because it allows the recovery of lots of elements with potential for utilization in the solution of extremely important situations. For example, it has been argued - I've referred to this often in places I've written on the internet - that the world supply of the valuable and technologically important element rhodium available from used nuclear fuel will shortly exceed that available from ores. (cf. Electrochimica Acta 53, 6, 2008, Pages 2794-2801) The exactly symmetric fission of plutonium would give an isotope of silver with 118 or 119 neutrons which would rapidly decay - owing to a large neutron excess - to a stable isotope of tin, 118Sn. Tin is a useful element, but nowhere near as valuable as rhodium.

The fission product distribution (with mass numbers for the abscissa) for plutonium-239 is shown here:

?w=444&h=389

I've been thinking about used nuclear fuels - "nuclear waste" in the parlance of people with far too limited imaginations - for a very long time, but I never paused to think about why the actinide elements - at least up to element 100, Fermium which does almost perfectly undergo symmetric fission - fission in an asymmetric fashion.

The paper cited at the outset addresses this issue.

From the abstract:

Nuclear fission of heavy (actinide) nuclei results predominantly in asymmetric mass splits1. Without quantum shell effects, which can give extra binding energy to their mass-asymmetric shapes, these nuclei would fission symmetrically. The strongest shell effects appear in spherical nuclei, such as the spherical ‘doubly magic’ (that is, both its atomic and neutron numbers are ‘magic’ numbers) nucleus 132Sn, which contains 50 protons and 82 neutrons. However, a systematic study of fission2 has shown that heavy fission fragments have atomic numbers distributed around Z = 52 to Z = 56, indicating that the strong shell effects in 132Sn are not the only factor affecting actinide fission. Reconciling the strong spherical shell effects at Z = 50 with the different Z values of fission fragments observed in nature has been a longstanding puzzle3.


From the paper's introduction:

Atomic nuclei are usually found at a minimum of energy, the ground state, which may be deformed because of quantum correlations. Elongation beyond the ground state costs potential energy, until a maximum is reached at the fission barrier. Increasing the elongation beyond the fission barrier decreases the potential energy, and the system follows a ‘fission valley’ in the potential energy surface until it breaks into two fragments (scission). In the absence of quantum shell effects, all heavy nuclei preferentially fission into two fragments of similar mass (mass-symmetric fission). However, quantum shell effects in the fissioning nucleus can result in several valleys on the way to scission. These may be mass-symmetric or mass-asymmetric.

Although progress has been made recently in describing fission-fragment mass distributions with stochastic approaches9,10, theoretical description of the first stage of fission, from the ground-state deformation to the fission barrier, remains a challenge11. However, the study of the dynamics along the fission valleys is now possible with the time-dependent energy-density functional approach12,13 including nuclear superfluidity7,14,15


Heavy nuclei have long been treated as fluids, beginning with the work of Neils Bohr and George Gamow who described heavy nuclei using the "liquid drop" model. This model was used by Lise Meitner and her nephew Otto Frisch to discover nuclear fission using the laboratory results of Otto Hahn. In this model, these drops deform, and sometimes they deform so much that the electrostatic repulsion of protons in their nuclei overcomes the very locally acting strong nuclear force.

From the paper:

Figure 1 shows the dynamical evolution of isodensity surfaces of 240Pu, starting from a configuration in the asymmetric-fission valley. The final state corresponds to a quantum superposition of different repartitions of the number of nucleons between the fragments, with ⟨Z⟩≈53.8 protons and ⟨N⟩≈85.2 neutrons in average in the heavy (left) fragment. Similar calculations were performed for the actinides 230Th, 234,236U, 246Cm, 250Cf and 258Fm (Extended Data Table 1 and Extended Data Fig. 1). For each system, a range of initial configurations in the asymmetric fission valley has been considered to investigate the influence of the initial elongation (and consequently the initial potential energy) on the final properties of the fragments. In the example of Fig. 1, scission occurs after about 20 zs (1 zs = 10−21 s), but can reach up to 90 zs depending on the initial configuration.


Figure 1:



The caption:

a–d, Isodensity surfaces at 0.08 fm−3 (half the saturation density), computed from the full microscopic evolution, are shown at different times (see Supplementary Information video). The localization function Cn of the neutrons (see Methods) is shown in the projections. Here, scission occurs at t ≈ 20 zs, that is, between c and d. The quadrupole and octupole deformation parameters (see Methods) at scission are β2 ≈ 0.16 and β3 ≈ 0.22 for the heavy fragment (left) and β2 ≈ 0.64 and β3 ≈ 0.4 for the light fragment (right), respectively.


The authors use certain computational quantum mechanical calculation programs (Hartree-Fock and DFT type calculations often used in the study of fermion - electron - interactions, here applied to the alternate quantum mechanical system, Bosons, dictated by the Bose Einstein statistics applied to at least some nuclei.

To wit:

Constrained and time-dependent Hartree-Fock calculations with dynamical Bardeen–Cooper–Schrieffer pairing correlations (CHF+BCS and TDBCS, respectively) were done in a three-dimensional Cartesian geometry with one plane (y = 0) of symmetry using the code of ref. 7 and the Skyrme SLy4d energy density functional31 with a surface pairing interaction of strength Vnn0=1,256MeVfm3 and Vpp0=1,462MeVfm3 in the neutron- and proton-pairing channels, respectively33.


The calculated distribution of fission products in the heavy elements:




The caption:

a, Expectation values of the number of protons (Z) and neutrons (N) in the heavy fragments for various asymmetric fissions. The solid line shows the expected positions of fragments with the N/Z values of 240Pu. The background grey scale quantifies the resistance to octupole (pear-shaped) deformations in the nuclei, as predicted by constrained Hartree–Fock calculations with dynamical Bardeen–Cooper–Schrieffer pairing correlations. It is obtained from the curvature of the octupole deformation energy α=limQ30→0(E/Q230) of the nuclei, where Q30 is the octupole moment (see Methods), near their energy minimum at Q30 = 0 (which corresponds to the curvature at the origin in Fig. 3b). Negative values indicate nuclei likely to exhibit octupole deformations in their ground state. b, Our microscopic predictions of the expectation values ⟨Z⟩ of the number of protons in the fission fragments (vertical lines) are compared with fragment proton number distributions (solid lines) extracted from experimental results of thermal-neutron-induced fission (top six panels)30 and the spontaneous fission of 258Fm (bottom panel)26. Dashed and dotted lines in the bottom panel are Gaussian fits of the asymmetric and symmetric components, respectively.


The authors write about the interplay of the two fundamental forces at play in atomic nuclei:

We also observe in Fig. 1 and Extended Data Fig. 1 that the fragments are formed with a strong deformation at scission. This deformation results from the competition between the long-range Coulomb interaction, which repels the fragments, the short range nuclear attraction in the neck between the fragments, and the deformation energy of the fragments. The latter quantifies the energy cost to deform a fragment, which can be particularly large for spherical doubly magic nuclei, such as 132Sn, or small for non-magic nuclei, which are often deformed in their ground state.

The strong attractive nuclear force between the fragments is responsible for the neck (see Fig. 1b, c), inducing quadrupole (cigar-shaped) and octupole (pear-shaped) deformations of the fragments. Although quadrupole deformation is often taken into account in modelling fission17, octupole deformation is also important for describing scission configurations properly18. The neutron localization function19 Cn (see Methods), shown as projections in Fig. 1, also exhibits strong octupole shapes. The localization function is often used to characterize shell structures in quantum many-body systems such as nuclei20,21 and atoms21.


They discuss some properties of one of the prominent fission fragments, Ba-144.

By contrast, fewer nuclei are expected to exhibit octupole deformation in their ground state4,6,23,24,25. Recent experiments have confirmed non-ambiguously that this is the case for 144Ba (Z = 56)4, a possible heavy fragment in asymmetric fission of actinides. Nuclei close to 144Ba in the nuclear chart should also exhibit particularly strong octupole correlations, thereby providing a possible explanation to the favoured production of these nuclei in fission.


144Ba is an important fission product, since it gives rise through rapid beta decays to the interesting and potentially very useful radioactive isotope 144Ce, which has a moderately long half life of 284 days, meaning that it is possible to isolate it for use, before decaying into the quasi-stable naturally occurring isotope found in all samples of neodymium, an element that plays an important role is some types of magnets utilized for the somewhat absurd pretenses of giving a rat's ass about climate change represented by electric cars.

One of the graphics refers to computations involving 144Ba:



The caption:

a, Potential energy surface of 144Ba. The binding energy, obtained using the Skyrme SLy4d functional31, is shown as a function of the quadrupole (Q20) and octupole (Q30) moments (see Methods). The binding energy increases from blue to red, with the isoenergy contour lines separated by an energy of 2 MeV. The red thick line represents the minimum energy at a given quadrupole deformation. b, Microscopic calculations of octupole deformation energy (defined as the binding energy of an octupole deformed nucleus minus the binding energy of the same nucleus without octupole deformation) for several isotopes produced in the fission of actinides. All other multipole moments, including the quadrupole moment, are not constrained. Thus, the curves show the minimum energy for a given octupole moment. The reference energy is shifted up by 1-MeV steps for each curve for clarity. Whereas 144Ba is predicted to have an octupole minimum, these octupole correlations disappear for the magic nucleus Sn (Z = 50).


Here's another graphic touching on the relative stability of fission products, including 144Ba, with respect to fission products.



The caption:

a, b, Neutron (ϵn; a) and proton (ϵp; b) single-particle energies as a function of the quadrupole (Q20; lower scale) and octupole (Q30; upper scale) moments in 144Ba following the path of the red solid line in Fig. 3a. The numbers in the energy gaps correspond to the number of particles that can occupy the single-particle states below the energy gap. 50 and 82 are magic numbers, associated with spherical shells, whereas 52, 56, 84 and 88 are associated with deformed energy gaps. In particular, the opening of the Z = 56 and N = 88 energy gaps with octupole deformation induces collective octupole correlations in the ground state of nuclei around 144Ba. Evidence for such correlations32 includes additional binding, revealed by measured masses in this region, as well as shallow minima in theoretical potential energy surfaces for octupole deformed shapes, as shown in Fig. 3a. However, direct experimental evidence for octupole deformation in this region was only recently found4,5.


Some concluding remarks from the authors:


The fact that both the mass-asymmetric fission of actinides and the TKE measured experimentally can be explained by our time-dependent microscopic calculations gives us confidence in the fission dynamics predicted by these calculations and, in particular, in the major role of the octupole deformation of the heavy fragments. The octupole deformation in Zlight ≈ 34 and Nheavy ≈ 56 nuclei25 could also explain mass-asymmetric fission found experimentally8 in lighter systems. Other properties of fission will also be investigated in the future, such as the excitation energy of the fragments and the number of neutrons emitted during the fission process.


This paper was a wonderful Christmas gift to me resulting from my birthday present from my wife, a subscription to Nature (meaning I can read this journal any time without dragging my fat ass to an open library.)

Whether we figure it out or not in any kind of time to save what still might be saved, our understanding (and use) of nuclear fission is our last, best hope.

I hope your Christmas Eve and Christmas day will prove as wonderful as my holidays have been thus far.

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