Dr Paul Stevenson MA (Oxon) MSc DPhil MInstP

We present a code in Python3 which takes a square real symmetric matrix, of arbitrary size, and decomposes it as a tensor product of Pauli spin matrices. The application to the decomposition of a Hamiltonian of relevance to nuclear physics for implementation on quantum computer is given.

Hindrance factors for K-forbidden E2 decays from multiquasiparticle isomers are analyzed in relation to mixing matrix elements that are often associated with chance near degeneracies, and a functional relationship is established. Focusing on effective matrix elements for ΔK=6 decays from three-quasiparticle isomers, significant configuration dependence is demonstrated

Time-dependent techniques in nuclear theory often rely on mean-field or Hartree-Fock descriptions. Beyond-mean-field dynamical calculations within the time-dependent density matrix (TDDM) theory have often invoked symmetry restrictions and ignored the connection between the mean field and the induced interaction. We study the ground states obtained in a TDDM approach for nuclei from A=12 to A=24, including examples of even-even and odd-even nuclei with and without intrinsic deformation. We overcome previous limitations using three-dimensional simulations and employ density-independent Skyrme interactions self-consistently. The correlated ground states are found starting from the Hartree-Fock solution, by adiabatically including the beyond-mean-field terms in real time. We find that, within this approach, correlations are responsible for ≈4–5% of the total energy. Radii are generally unaffected by the introduction of beyond-mean-field correlations. Large nuclear correlation entropies are associated with large correlation energies. By all measures, 12C is the most correlated isotope in the mass region considered. Our work is the starting point of a consistent implementation of the TDDM technique for applications into nuclear reactions. Our results indicate that correlation effects in structure are small, but beyond-mean-field dynamical simulations could provide insight into several issues of interest.

We revisit the studies of the isotopic shift in the charge radii of {\it even-even} isotopes of Sn and Pb nuclei at = 82, and 126, respectively, within the relativistic mean-field and Relativistic-Hartree-Bogoliubov approach. The shell model is also used to estimate isotopic shift in these nuclei, for the first time, to the best of our knowledge. The ground state single-particle energies () are calculated for non-linear NL3 \& NL3∗ and density-dependent DD-ME2 parameter sets compared with the experimental data, wherever available. We establish a correlation between the filling of single-particle levels and the isotopic shift in occupation probabilities. The obtained from the relativistic mean-field and Relativistic-Hartree-Bogoliubov approaches are in line with those used in the shell model and experimental data for both the Sn and Pb isotopic chains. The shell model calculated isotopic shift agrees with relativistic mean-field and Relativistic-Hartree-Bogoliubov approaches that explain the experimental data quite well.

Energy dependence of fission observables is a key issue for wide nuclear applications. We studied real-time fission dynamics from low-energy to high excitations in the compound nucleus 240Pu with the time-dependent Hartree-Fock+BCS approach. It is shown that the evolution time of the later phase of fission towards scission is considerably lengthened at finite temperature. As the role of dynamical pairing is vanishing at high excitations, the random transition between single-particle levels around the Fermi surface to mimic thermal fluctuations is indispensable to drive fission. The obtained fission yields and total kinetic energies with fluctuations can be divided into two asymmetric scission channels, namely S1 and S2, which explain well experimental results, and give microscopic support to the Brosa model. With increasing fluctuations, S2 channel takes over S1 channel and the spreading fission observables are obtained.

We studied the nuclear shape evolutions in fission process of 240Pu by the time-dependent Hartree-Fock approach with various Skyrme forces. Calculations are performed for the later phase of the fission with large initial deformations toward the scission. We show that calculations with Skyrme forces with large surface energies and large symmetry energies can have extremely long fission evolution time. The symmetry energy plays a role in the evolution of neutron-rich necks. In addition, we also demonstrated the shape oscillations of fission fragments after the fission. We see that particularly the heavy near-spherical fragments have remarkable octupole oscillations.

This note presents a minor alteration to, and subsequent use of, the nuclear time-dependent Hartree–Fock code to calculate ion-ion potentials in the Frozen Hartree–Fock approximation. An example of 16O + 16O is presented.

The visualization of objects moving at relativistic speeds has been a popular topic of study since Special Relativity’s inception. While the standard exposition of the theory describes certain shape-changing effects, such as the Lorentz-contraction, it makes no mention of how an extended object would appear in a snapshot or how apparent distortions could be used for measurement. Previous work on the subject has derived the apparent form of an object, often making mention of George Gamow’s relativistic cyclist thought experiment. Here, a rigorous re-analysis of the cyclist, this time in three dimensions, is undertaken for a binocular observer, accounting for both the distortion in apparent position and the relativistic colour and intensity shifts undergone by a fast-moving object. A methodology for analysing binocular relativistic data is then introduced, allowing the fitting of experimental readings of an object’s apparent position to determine the distance to the object and its velocity. This method is then applied to the simulation of Gamow’s cyclist, producing self-consistent results.

Time-dependent Hartree-Fock calculations have been performed for fusion reactions of4He+4He→8Be∗, followed by 4He+8Be∗. Depending on the orientation of the initial state, a linear chain vibrational state or a triangular vibration is found in12C, with transitions between these states observed. The vibrations of the linear chain state and the triangular state occur at ~9 and 4 MeV respectively

The surface energy is one of the fundamental properties of nuclei, appearing in the simplest form of the semi-empirical mass formula. The surface energy has an influence on e.g. the shape of a nucleus and its ability to deform. This in turn could be expected to have an effect in fusion reactions around the Coulomb barrier where dynamical effects such as the formation of a neck is part of the fusion process. Frozen Hartree-Fock and Time-Dependent Hartree-Fock calculations are performed for a series of effective interactions in which the surface energy is systematically varied, using 40Ca + 48Ca as a test case. The dynamical lowering of the barrier is greatest for the largest surface energy, contrary to naive expectations, and we speculate that this may be due to the variation in other nuclear matter properties for these effective interactions.

The propagation of non-topological solitons in many-nucleus systems is studied based on time-dependent density functional calculations, focusing on mass and energy dependence. The dispersive property and the nonlinearity of the system, which are inherently included in the nuclear density functional, are essential factors to form a non-topological soliton. On the other hand, soliton propagation is prevented by charge equilibration, and competition can appear between soliton formation and disruption. In this article, based on the energy-dependence of the two competitive factors, the concept of conditional recovery of time-reversal symmetry is proposed in many-nucleus systems. It clarifies the possibility of preserving the nuclear medium inside natural or artificial nuclear reactors, at a suitable temperature. From an astrophysical point of view, the existence of the low-temperature solitonic core of compact stars is suggested.

The Skyrme effective interaction, with its multitude of parameterisations, along with its implementation using the static and time-dependent density functional (TDHF) formalism has allowed for a range of microscopic calculations of low-energy heavy-ion collisions. These calculations allow variation of the effective interaction along with an interpretation of the results of this variation informed by a comparison to experimental data. Initial progress in implementing TDHF for heavy-ion collisions necessarily used many approximations in the geometry or the interaction. Over the last decade or so, the implementations have overcome all restrictions, and studies have begun to be made where details of the effective interaction are being probed. This review surveys these studies in low-energy heavy-ion reactions, finding significant effects on observables from the form of the spin–orbit interaction, the use of the tensor force, and the inclusion of time-odd terms in the density functional

Friction coefficients for the fusion reaction 16O+16O→32S are extracted based on both the time-dependent Hartree-Fock and the time-dependent density matrix methods. The latter goes beyond the mean-field approximation by taking into account the effect of two-body correlations, but in practical simulations of fusion reactions we find that the total energy is not conserved. We analyze this problem and propose a solution that allows for a clear quantification of dissipative effects in the dynamics. Compared to mean-field simulations, friction coefficients in the density-matrix approach are enhanced by about 20%. An energy dependence of the dissipative mechanism is also demonstrated, indicating that two-body collisions are more efficient at generating friction at low incident energies.

The level scheme of the neutron-deficient nuclide 168Os has been extended and mean lifetimes of excited states have been measured by the recoil distance Doppler-shift method using the JUROGAM γ-ray spectrometer in conjunction with the IKP Köln plunger device. The 168Osγ rays were measured in delayed coincidence with recoiling fusion-evaporation residues detected at the focal plane of the RITU gas-filled separator. The ratio of reduced transition probabilities B(E2;4+1→2+1)/B(E2;2+1→0+1) is measured to be 0.34(18), which is very unusual for collective band structures and cannot be reproduced by interacting boson model (IBM-2) calculations based on the SkM* energy-density functional.

It is generally acknowledged that the time-dependent Hartree–Fock (TDHF) method provides a useful foundation for a fully microscopic many-body theory of low-energy heavy ion reactions. The TDHF method is also known in nuclear physics in the small-amplitude domain, where it provides a useful description of collective states, and is based on the mean-field formalism, which has been a relatively successful approximation to the nuclear many-body problem. Currently, the TDHF theory is being widely used in the study of fusion excitation functions, fission, and deep-inelastic scattering of heavy mass systems, while providing a natural foundation for many other studies. With the advancement of computational power it is now possible to undertake TDHF calculations without any symmetry assumptions and incorporate the major strides made by the nuclear structure community in improving the energy density functionals used in these calculations. In particular, time-odd and tensor terms in these functionals are naturally present during the dynamical evolution, while being absent or minimally important for most static calculations. The parameters of these terms are determined by the requirement of Galilean invariance or local gauge invariance but their significance for the reaction dynamics have not been fully studied. This work addresses this question with emphasis on the tensor force. The full version of the Skyrme force, including terms arising only from the Skyrme tensor force, is applied to the study of collisions within a completely symmetry-unrestricted TDHF implementation. We examine the effect on upper fusion thresholds with and without the tensor force terms and find an effect on the fusion threshold energy of the order several MeV. Details of the distribution of the energy within terms in the energy density functional are also discussed. Terms in the energy density functional linked to the tensor force can play a non-negligible role in dynamic processes in nuclei.

The study of the nuclear equation of state (EOS) and the behavior of nuclear matter under extreme conditions is crucial to our understanding of many nuclear and astrophysical phenomena. Nuclear reactions serve as one of the means for studying the EOS. It is the aim of this paper to discuss the impact of nuclear fusion on the EOS. This is a timely subject given the expected availability of increasingly exotic beams at rare isotope facilities [A. B. Balantekin , ]. In practice, we focus on 48Ca+48Ca fusion. We employ three different approaches to calculate fusion cross sections for a set of energy density functionals with systematically varying nuclear matter properties. Fusion calculations are performed using frozen densities, using a dynamic microscopic method based on density-constrained time-dependent Hartree-Fock (DC-TDHF) approach, as well as direct TDHF study of above barrier cross sections. For these studies, we employ a family of Skyrme parametrizations with systematically varied nuclear matter properties. The folding-potential model provides a reasonable first estimate of cross sections. DC-TDHF, which includes dynamical polarization, reduces the fusion barriers and delivers much better cross sections. Full TDHF near the barrier agrees nicely with DC-TDHF. Most of the Skyrme forces which we used deliver, on the average, fusion cross sections in good agreement with the data. Trying to read off a trend in the results, we find a slight preference for forces which deliver a slope of symmetry energy of L≈50 MeV that corresponds to a neutron-skin thickness of 48Ca of Rskin=(0.180–0.210) fm. Fusion reactions in the barrier and sub-barrier region can be a tool to study the EOS and the neutron skin of nuclei. The success of the approach will depend on reduced experimental uncertainties of fusion data as well as the development of fusion theories that closely couple to the microscopic structure and dynamics.

Present work aims to explicate the effect of entrance channel mass asymmetry on fusion dynamics for the Compound Nucleus 80Sr populated through two different channels, 16O+64Zn and 32S+48Ti, using cross-section and spin distribution measurements as probes. The evaporation spectra studies for these systems, reported earlier indicate the presence of dynamical effects for mass symmetric 32S+48Ti system.The CCDEF and TDHF calculations have been performed for both the systems and an attempt has been made to explain the reported deviations in the α-particle spectrum for the mass symmetric system.  

Nuclear fission is a complex large-amplitude collective decay mode in heavy nuclei. Microscopic density functional studies of fission have previously concentrated on adiabatic approaches based on constrained static calculations ignoring dynamical excitations of the fissioning nucleus and the daughter products. We explore the ability of dynamic mean-field methods to describe induced fission processes, using quadrupole boosts in the nuclide 240Pu as an example. Following upon the work presented in Goddard [], quadrupole-constrained Hartree-Fock calculations are used to create a potential energy surface. An isomeric state and a state beyond the second barrier peak are excited by means of instantaneous as well as temporally extended gauge boosts with quadrupole shapes. The subsequent deexcitation is studied in a time-dependent Hartree-Fock simulation, with emphasis on fissioned final states. The corresponding fission fragment mass numbers are studied. In general, the energy deposited by the quadrupole boost is quickly absorbed by the nucleus. In instantaneous boosts, this leads to fast shape rearrangements and violent dynamics that can ultimately lead to fission. This is a qualitatively different process than the deformation-induced fission. Boosts induced within a finite time window excite the system in a relatively gentler way and do induce fission but with a smaller energy deposition. The fission products obtained using boost-induced fission in time-dependent Hartree-Fock are more asymmetric than the fragments obtained in deformation-induced fission or the corresponding adiabatic approaches.

The time-dependent version of nuclear density functional theory, using functionals derived from Skyrme interactions, is able to approximately describe nuclear dynamics. We present time-dependent results of calculations of dipole resonances, concentrating on excitations of valence neutrons against a proton plus neutron core in the neutron-rich doubly-magic 132Sn nucleus, and results of collision dynamics, highlighting potential routes to ternary fusion, with the example of a collision of 48Ca+48Ca+208Pb resulting in a compound nucleus of element 120 stable against immediate fission.

The nucleosynthesis of elements beyond iron is dominated by neutron captures in the and processes. However, 32 stable, proton-rich isotopes cannot be formed during those processes, because they are shielded from the s-process flow and -process, -decay chains. These nuclei are attributed to the and process. For all those processes, current research in nuclear astrophysics addresses the need for more precise reaction data involving radioactive isotopes. Depending on the particular reaction, direct or inverse kinematics, forward or time-reversed direction are investigated to determine or at least to constrain the desired reaction cross sections. The Facility for Antiproton and Ion Research (FAIR) will offer unique, unprecedented opportunities to investigate many of the important reactions. The high yield of radioactive isotopes, even far away from the valley of stability, allows the investigation of isotopes involved in processes as exotic as the or processes.

Explaining observed properties in terms of underlying shape degrees of freedom is a well-established prism with which to understand atomic nuclei. Self-consistent mean-field models provide one tool to understand nuclear shapes, and their link to other nuclear properties and observables. We present examples of how the time-dependent extension of the mean-field approach can be used in particular to shed light on nuclear shape properties, particularly looking at the giant resonances built on deformed nuclear ground states, and at dynamics in highly-deformed fission isomers. Example calculations are shown of 28Si in the first case, and 240Pu in the latter case.

Nuclear fission is a complex large-amplitude collective decay mode in heavy nuclei. Microscopic density functional studies of fission have previously concentrated on adiabatic approaches based on constrained static calculations ignoring dynamical excitations of the fissioning nucleus and the daughter products. We explore the ability of dynamic mean-field methods to describe fast fission processes beyond the fission barrier, using the nuclide 240Pu as an example. Time-dependent Hartree-Fock calculations based on the Skyrme interaction are used to calculate nonadiabatic fission paths, beginning from static constrained Hartree-Fock calculations. The properties of the dynamic states are interpreted in terms of the nature of their collective motion. Fission product properties are compared to data. Parent nuclei constrained to begin dynamic evolution with a deformation less than the fission barrier exhibit giant-resonance-type behavior. Those beginning just beyond the barrier explore large-amplitude motion but do not fission, whereas those beginning beyond the two-fragment pathway crossing fission to final states which differ according to the exact initial deformation. Time-dependent Hartree-Fock is able to give a good qualitative and quantitative description of fast fission, provided one begins from a sufficiently deformed state.

The energies of the canonical (standard, amino-keto) and tautomeric (non-standard, imino-enol) charge-neutral forms of the adenine–thymine base pair (A–T and A*–T*, respectively) are calculated using density functional theory. The reaction pathway is then computed using a transition state search to provide the asymmetric double-well potential minima along with the barrier height and shape, which are combined to create the potential energy surface using a polynomial fit. The influence of quantum tunnelling on proton transfer within a base pair H-bond (modelled as the DFT deduced double-well potential) is then investigated by solving the time-dependent master equation for the density matrix. The effect on a quantum system by its surrounding water molecules is explored the inclusion of a dissipative Lindblad term in the master equation, in which the environment is modelled as a heat bath of harmonic oscillators. It is found that quantum tunnelling, due to transitions to higher energy eigenstates with significant amplitudes in the shallow (tautomeric) side of the potential, is unlikely to be a significant mechanism for the creation of adenine–thymine tautomers within DNA, with thermally assisted coupling of the environment only able to boost the tunnelling probability to a maximum of 2 × 10−9. This is barely increased for different choices of the starting wave function or when the geometry of the potential energy surface is varied.