bases were introduced in [Smy77] where they are called “R-structures”. Examples of abstract bases are concrete bases of continuous domains, of course, where the relation≺ is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element. Other examples are partially ordered sets, where (INT) is satisfied because of reflexivity. We will shortly identify posets as being exactly the bases of compact elements of algebraic domains. In what follows we will use the terminology developed at the beginning of this chapter, even though the relation ≺ on an abstract basis need neither be reflexive nor antisymmetric. This is convenient but in some instances looks more innocent than it is. An idealA in a basis, for example, has the property (following from directedness) that for everyx ∈ A there is another element y ∈ A with x ≺ y. In posets this doesn’t mean anything but here it becomes an important feature. Sometimes this is stressed by using the expression ‘ A is a round ideal’. Note that a set of the form↓x is always an ideal because of (INT) but that it need not contain x itself. We will refrain from calling ↓x ‘principal’ in these circumstances. Definition 2.2.21. For a basis〈B,≺〉 let Idl(B) be the set of all ideals ordered by inclusion. It is called theideal completionof B. Furthermore, leti : B → Idl(B) denote the function which maps x ∈ B to ↓x. If we want to stress the relation with whichB is equipped then we write Idl(B,≺) for the ideal completion. Proposition 2.2.22.Let 〈B,≺〉 be an abstract basis.

Domain theory

The contribution that the technique of ferromagnetic resonance (FMR) has made to the understanding of the magnetic behaviour of ultrathin single films is reviewed. Experimental methods to measure FMR in situ in ultrahigh vacuum are presented. The temperature dependence of the magnetization, of the magnetic relaxation rate in the vicinity of the Curie temperature, and of the second- and fourth-order magnetic anisotropy energy (MAE) constants can be measured by FMR in situ for magnetic monolayers. Using the cases of Ni/Cu(001) and Gd/W(110) as examples, the role of the MAE for the quantitative description of temperature- and thickness-dependent reorientation transitions of the magnetization is discussed. Initial results for the anisotropy of the g-factor which is related to the anisotropy of the orbital moment (and the MAE) are presented.

https://iopscience.iop.org/article/10.1088/0034-4885/61/7/001/pdf

Ferromagnetic resonance of ultrathin metallic layers

Basic properties of cold Bose atoms in optical lattices are reviewed. The main principles of correct self-consistent description of arbitrary systems with Bose-Einstein condensate are formulated. Theoretical methods for describing regular periodic lattices are presented. A special attention is paid to the discussion of Bose-atom properties in the frame of the boson Hubbard model. Optical lattices with arbitrary strong disorder, induced by random potentials, are treated. Possible applications of cold atoms in optical lattices are discussed, with an emphasis of their usefulness for quantum information processing and quantum computing. An important feature of the present review article, distinguishing it from other review works, is that theoretical fundamentals here are not just mentioned in brief, but are thoroughly explained. This makes it easy for the reader to follow the principal points without the immediate necessity of resorting to numerous publications in the field.

Cold Bosons in Optical Lattices

Electrical currents flowing in ferromagnetic materials are spin-polarised as a result of the spin-dependent band structure. When the spatial direction of the polarisation changes, in a domain structure, the electrons must somehow accommodate the necessary change in direction of their spin angular momentum as they pass through the wall. Reflection, scattering, or a transfer of angular momentum onto the lattice are all possible outcomes, depending on the circumstances. This gives rise to a variety of different physical effects, most importantly a contribution to the electrical resistance caused by the wall, and a motion of the wall driven by the spin-polarised current. Historical and recent research on these topics is reviewed. Contents PAGE 1. Introduction5862. Spin-polarised current587 2.1. Tunnelling current spin polarisation589 2.2. Ballistic current spin polarisation592 2.3. Diffusive current spin polarisation5933. Magnetic Domain Walls598 3.1. Basics of domain walls598 3.1.1. Domain wall thickness and...

/pdf/spin-polarised-currents-and-magnetic-domain-walls-1cmsmewys0.pdf

Spin-polarised currents and magnetic domain walls

Phase transitions and heterophase fluctuations

The present state of understanding of the dynamics of magnetic domain walls and magnetic bubbles is reviewed The theory of domain wall motion for the linear and non-linear regions is outlined Experimental techniques for straight walls and magnetic bubbles are discussed An extensive comparison between theory and experiment is made Topics included are peak and saturation velocities, mobility, inertial effects and overshoot, hard bubbles, wall states and state transformations in magnetic bubbles Origins of wall damping are also discussed

http://iopscience.iop.org/article/10.1088/0034-4885/43/6/001/pdf

Dynamic properties of magnetic domain walls and magnetic bubbles

The dynamical behavior of straight magnetic domain walls and propagating magnetic bubbles in single‐, double‐, and triple‐layer lanthanum gallium garnet films is reported. In the study of straight walls in the single‐layer films, a good agreement is obtained between the observed and the calculated values of the domain‐wall width parameter Δ, the derivative of the peak velocity to in‐plane field at high in‐plane fields, and the derivative of the saturation velocity to in‐plane field at a moderate drive field and high in‐plane fields H1. The saturation velocity (at H1=0) decreases with decreasing film thickness, which is in disagreement with theoretical predictions. A propagating bubble in a single‐layer film exhibits an overshoot. In double‐ and triple‐layer films, the observed and calculated Δ values disagree in most cases, and the straight wall and propagating bubble saturation velocities are significantly higher than in single‐layer films. Overshoots and low‐frequency oscillations are observed in the sa...

R. van den Doel

Papers

Dynamic properties of magnetic domain walls and magnetic bubbles

The dynamical behavior of magnetic domain walls and magnetic bubbles in single‐, double‐, and triple‐layer garnet films