Finally, the different initial conditions to set up the calculation space for each type of HF or TDHF calculation are detailed.
Box sizes are in fm and the mesh grid size is ∆x = 0.8 fm. Box sizes are listed in
x−y−z direction with the collision axis (if relevant) being in thex direction. All time dependent calculations have a plane of symmetry atz= 0 to reduce calculation time.
1. For all HF ground state nuclei, which are calculated as an eighth of the nucleus (as perev8), there are 14 points from the centre of the nucleus to the edge of the box, corresponding to a size of11.23 fm3.
2. For vibrations calculations in Chapter 3 that are fully time dependent calculations, the box size must be at least as large as holding half of a nucleus, corresponding to a box size of22.4×22.4×11.2 fm3.
3. For both static and dynamic reactions appearing in Chapter 4 and central collisions in Chapter 5, the box size is67.4×22.4×11.2fm3. This setup gives two nuclei an
initial separation distance between the centres of mass of 44.8 fm. This separation distance is chosen at such a large distance so that Coulomb excitation between the colliding partners are negligible at initial time.
4. For non-central collisions appearing in Chapter 5, the box size is increased in they direction by a factor of two making a final box size of 67.4×44.8×11.2 fm3. The
initial separation distance between collision partners is again 44.8 fm.
When discretising both sides of the time-dependent Schrödinger equation (2.1.2) for the HF Hamiltonian Hˆ, the spatial mesh size is related to time step size by
1 (∆x)2 ∝
1 ∆t,
considering just one spatial dimension. In order to optimise the number of points on which to solve the wavefunctions versus computational time, the values∆x= 0.8 fm and ∆t= 1.5×10−24s are a good compromise [25, 101, 176]. Because of the square relation
from the spatial dimension, if the spatial mesh is decreased by for example 0.2 fm in order to have more points in the wavefunction solution, then the time mesh must be
decreased from 1.5×10−24s to 1.0×10−24 s to compensate. Reduced time and spatial
meshes increase computational time significantly.
For dynamic reaction calculations the initial centre of mass energy for the system can be set at a particular value. From this input, initial momentum is applied to each nucleon via a Galilean boost on the wavefunction
ϕj(t0) = exp (ika·ˆr)ϕgj.s., j = 1, ..., A,
acting on the ground state HF single particle wavefunctionsϕgj.s. in nucleus awith mass numberA.
3
Vibrations
3.1 Overview
Studying collective vibrations of nuclei is interesting because it sheds light on the structure of the individual nucleus itself and also because collective vibrations play important roles in reaction studies, which will be explored in the following chapter. Many vibrational modes of nuclei across the chart have been observed experimentally. Those modes that are low in excitation energy (low-lying) are of particular interest as they play a more important role in reactions [74] compared to modes with high excitation energy or giant resonances.
To start discussions of vibrations, consider first the basic nuclear property of ground state spin, J, and parity, π. This property cannot be predicted by macroscopic models such as the liquid drop model (LDM). Models such as LDM only consider the bulk nature of a nucleus and consequently do not account for any physical phenomena arising from quantum shell effects. Microscopic approaches provide more insight into properties that arise from nuclear shell structure. Fully independent particle models, for example the shell model, can indeed explain experimentally observed ground statesJπ both for odd mass nuclei and also even-even nuclei for which the ground state is always 0+. For some
excited states observed experimentally, pure independent particle states are not sufficient to describe them. A collective model, built upon the idea of coherent superposition of independent particle states, extends independent particle models and can account for collective motion in a nucleus.
A large portion of collective motion in nuclei can be generalised into two main categories of vibrations and rotations. Rotational motion refers to a nucleus that is statically deformed in its ground state, for example any of the actinides, and will naturally have rotational degrees of freedom when excited. This type of collective motion is not considered in this chapter. Instead, vibrations of nuclei are explored and are based on the assumption that the nucleus is spherical or near spherical in its ground state shape and that vibrational motion is not coupled to any degrees of rotational freedom. This applies to nuclides with magic shell closure. Collective vibrational motion of nuclei result in oscillations of electromagnetic multipole moment.
A standard way to study collective vibrations with a microscopic approach is with the
random phase approximation (RPA) method, first appearing in application to oscillations in electron gases [22] and then applied to nuclear physics [57]. It introduces a time- dependent external field or potential to the Hamiltonian of the many-body system, and the response of the particles to this is analysed. The response includes single particle excitations and also two-body correlations within the ground state. The time-dependent Hartree–Fock (TDHF) method with linear response theory [210] is also used to study collective vibrations. In the small amplitude limit, the TDHF equations are the same as the RPA equations [154]. Collective vibrations within this approach come from the attractive residual interaction in the mean field potential. For linear responses, the residual interaction of one-particle one-hole type is taken into account by TDHF and can be observed with one-body operators. Residual interactions describing higher order collisions are not included in the mean field approximation.
In this chapter, TDHF with linear response theory is used to calculate vibrational states of a chain of calcium nuclides. The systematics of one phonon octupole and quadrupole vibrations are studied focussing on the low-lying modes. This application of TDHF is a precursor to the application of TDHF to reaction studies as it involves just one nucleus at a time. Ground states of nuclei are calculated from the static HF method (outlined in section 2.5) which are then used to start a TDHF calculation that introduces a time dependent external potential that includes a vibrational one body operator on top of the usual HF Hamiltonian. The response of the nucleus to this external potential is used to obtain information such as excitation energies and transition strengths of particular multipole excitations.
Expressions of multipole moments on spherical nuclei arising from the electric field induced by a nucleus is outlined first, followed by details of linear response theory. Physical quantities from the calculated multipole moments are then presented. Finally, the single particle excitations contributing to the vibrations, using the single particle levels from HF ground state, are discussed.