Coincidence site modules in 3-space

The coincidence site lattice (CSL) problem and its generalization to Z-modules in Euclidean 3-space is revisited, and various results and conjectures are proved in a unified way, by using maximal orders in quaternion algebras of class number 1 over real algebraic number fields.Comment: 25 page

Void Growth in BCC Metals Simulated with Molecular Dynamics using the Finnis-Sinclair Potential

The process of fracture in ductile metals involves the nucleation, growth, and linking of voids. This process takes place both at the low rates involved in typical engineering applications and at the high rates associated with dynamic fracture processes such as spallation. Here we study the growth of a void in a single crystal at high rates using molecular dynamics (MD) based on Finnis-Sinclair interatomic potentials for the body-centred cubic (bcc) metals V, Nb, Mo, Ta, and W. The use of the Finnis-Sinclair potential enables the study of plasticity associated with void growth at the atomic level at room temperature and strain rates from 10^9/s down to 10^6/s and systems as large as 128 million atoms. The atomistic systems are observed to undergo a transition from twinning at the higher end of this range to dislocation flow at the lower end. We analyze the simulations for the specific mechanisms of plasticity associated with void growth as dislocation loops are punched out to accommodate the growing void. We also analyse the process of nucleation and growth of voids in simulations of nanocrystalline Ta expanding at different strain rates. We comment on differences in the plasticity associated with void growth in the bcc metals compared to earlier studies in face-centred cubic (fcc) metals.Comment: 24 pages, 12 figure

Magnetic hopfions in solids

Hopfions are an intriguing class of string-like solitons, named according to a classical topological concept classifying three-dimensional direction fields. The search of hopfions in real physical systems is going on for nearly half a century, starting with the seminal work of Faddeev. But so far realizations in solids are missing. Here, we present a theory that identifies magnetic materials featuring hopfions as stable states without the assistance of confinement or external fields. Our results are based on an advanced micromagnetic energy functional derived from a spin-lattice Hamiltonian. Hopfions appear as emergent particles of the classical Heisenberg model. Magnetic hopfions represent three-dimensional particle-like objects of nanometre-size dimensions opening the gate to a new generation of spintronic devices in the framework of a truly three-dimensional architecture. Our approach goes beyond the conventional phenomenological models. We derive material-realistic parameters that serve as concrete guidance in the search of magnetic hopfions bridging computational physics with materials science

The influence of solute diffusion and anisotropic elasticity on the deformation of iron: a study using planar discrete dislocation plasticity

Plastic deformation of crystalline metals is facilitated by crystallographic defects called dislocations. In certain alloys at certain temperatures and loading rates, the motion of these dislocations can become chaotic and unstable. This is a phenomenon related to dynamic strain ageing called the Portevin–Le Châtelier effect. The effect occurs when impurities, such as carbon and nitrogen, segregate to dislocations and obstruct their motion, leading to reduced ductility, hardening, and in the worst case, mechanical failure. In this thesis, I model plasticity with planar discrete dislocation plasticity. I extend planar discrete dislocation plasticity using anisotropic elasticity and introduce discrete diffusing solutes. This is used to understand the deformation of iron; and the onset and progression of the Portevin– Le Châtelier effect. Results from the model show that the nature of the diffusing solute field significantly affects the manifestation of the Portevin–Le Châtelier effect. As the solute concentration increases, the iron specimen’s proof stress, flow stress and solute strengthening increases; and the serrations recorded in a stress-strain curve (symptomatic of the Portevin–Le Châtelier effect) become smaller and occur at a slower frequency. Conversely, as the solute diffusivity increases, the serrations become larger and occur at a faster frequency. Furthermore, I show that anisotropic elasticity alters the mechanisms of plastic deformation of α-iron at elevated temperatures; specifically, that a type edge dislocations, which are energetically unfavourable at moderate temperatures, are nucleated in increasing numbers. This agrees with experimental observations found in the literature but cannot be predicted by isotropic elastic models. The increase in a edge dislocations is symptomatic of the precipitous decrease in the yield strength of iron. My work has important consequences for the application of steels in elevated temperature environments such as generation IV very high temperature nuclear fission reactors.Open Acces

Periodic minimal surfaces of cubic symmetry

A survey of cubic minimal surfaces is presented, based on the concept of fundamental surface patches and their relation to the asymmetric units of the space groups. The software Surface Evolver has been used to test for stability and to produce graphic displays. Particular emphasis is given to those surfaces that can be generated by a finite piece bounded by straight lines. Some new varieties have been found and a systematic nomenclature is introduced, which provides a symbol (a ‘gene’) for each triply-periodic minimal surface that specifies the surface unambiguously

Similar sublattices of the root lattice A4A_4

Similar sublattices of the root lattice $A_4$ are possible, according to a result of Conway, Rains and Sloane, for each index that is the square of a non-zero integer of the form $m^2 + mn - n^2$. Here, we add a constructive approach, based on the arithmetic of the quaternion algebra $\mathbb{H} (\mathbb{Q} (\sqrt{5}))$ and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.Comment: 17 pages, 1 figure; revised version with several additions and improvement

Structural trends in clusters of quadrupolar spheres

The influence of quadrupolar interactions on the structure of small clusters is investigated by adding a point quadrupole of variable strength to the Lennard-Jones potential. Competition arises between sheet-like arrangements of the particles, favoured by the quadrupoles, and compact structures, favoured by the isotropic Lennard-Jones attraction. Putative global potential energy minima are obtained for clusters of up to 25 particles using the basin-hopping algorithm. A number of structural motifs and growth sequences emerge, including star-like structures, tubes, shells and sheets. The results are discussed in the context of colloidal self-assembly.Comment: 8 pages, 6 figure

Stochastic Geometry, Spatial Statistics and Statistical Physics

[no abstract available