One-component order parameter

The previous treatment deals with a one-component order parameter (such as for a commensurate Peierls distortion) but does not apply to situations where the order parameter is complex with an amplitude and a phase (superconductivity, incommensurate Peierls, or spin density wave transitions). The latter situation is analogous to classical moments which can rotate freely in an XY plane. The coherence length of the XY model is less strongly divergent at low temperature than for the Ising model,... 

Another example of phase coexistence that is described by a single scalar one-component order parameter is provided by incompressible binary mixtures. One considers a dense liquid of two components, A and B the composition of... 

Let us consider a ferroic nanorod of radius R, height h and the axially-symmetric one-component order parameter qs (z, p) directed along the rod axis z (hereinafter... 

Fig. 10. Adsorbate superstructures on 100) surfaces of cubic crystals. Atoms in the top-layer of substrate are shown as white circles, while adsorbate atoms are shown as full black circles. Upper part shows the two possible domains of the c(2x2) structure, ohtained by dividing the square lattice of preferred adsorption sites into two sublattices following a checkerboard pattern either the white sublattice or the black sublattice is occupied with adatoms. The (2x1) structure also is a 2-sublattice structure, where full and empty rows alternate. These rows can be interchanged and they also can run either in -t-direction (middle part) or y-direction (lower part), so four passible domains result and one has a two-component order parameter.

In order to make contact with the Landau expansion, however, we consider now the special case q = 3 and expand F in terms of the two order parameter components 0] = ni — 1/3 and 02 = 2 — 1/3 (note that all rij — l/q in the disordered phase). One recognizes that the model for q = 3 has a two-component order parameter and there is no symmetry between 0,-and —0<. So cubic terms in the expansion of F arc expected and do occur, whereas for a properly defined order parameter, there cannot be any linear term in the expansion ... 

We consider mixtures of a flexible polymer and a nematogen described by one conserved order parameter (volume fraction of a nematogen) and one non-conserved order parameter (orientational order parameter of a nematogen). Since the orientational order parameter is a traceless symmetric tensor, its components can be expressed as [14] ... 

In the copper-containing oxides with perovskite structure of the R i Ba ifZuO, type, the arrangement of the planes occupied by rare-earth R and Ba ions is of the 1 2 ty Q.. ..RBaRBaRBa. These planes are perpendicular to the four-fold axis of the parent phase. The ordered R u Ba ifOuO state is described by one component of the six-component order parameter [[32] ... 

For a one-component fluid, the vapour-liquid transition is characterized by density fluctuations here the order parameter, mass density p, is also conserved. The equilibrium structure factor S(k) of a one component fluid is... 

The construction of the phase diagram of a heteropolymer liquid in the framework of the WSL theory is based on the procedure of minimization of the Landau free energy T presented as a truncated functional series in powers of the order parameter with components i a(r) proportional to Apa(r). The coefficients of this series, known as vertex functions, are governed by the chemical structure of heteropolymer molecules. More precisely, the values of these coefficients are entirely specified by the generating functions of the chemical correlators. Hence, before constructing the phase diagram of the specimen of a heteropolymer liquid, one is supposed to preliminarily find these statistical characteristics of the chemical structure of this specimen. Here a pronounced interplay of the statistical physics and statistical chemistry of polymers is explicitly manifested. 

Let us try to understand deeper the nature of the order parameter. As usually, we start with a gas as a simplest one-component system. An important role in theoretical physics belongs to a model of classical ideal gas in which molecules (particles) obey the laws of Newtonian mechanics and do not interact with each other. 

The property of the spin-triplet components fi,2,z un) = — /i,2,3(—w ) means that their presence is not easy to observe. For example, the order parameter A is related to the sum X n=-oo /(wn) in which all contributions of the odd functions f 1,2,3 cancel. However, there are phenomena where the presence of the spin-triplet pairing plays a crucial role. One of them is the effect of the Tc dependence on the mutual orientation of magnetizations in the F/S/F structure. Another one is the predicted long-range proximity effect based on the spin-triplet component, which should lead to a Josephson current in F/S/F/S structures with anomalously thick F-layers.[10] The latter is relevant for experimental results of Ref. [5]. 

The method of Fominov et al. [25, 29] was formulated for the case of an F/S bilayer. Following it, Eq. (4) is first solved analytically in the F layers and for the fs component in the S layer. This is possible because in all these cases the equation is uniform. The order parameter A is zero in the F layers and has only singlet component in the S layer. After that, the boundary condition (6) - there is only one F/S (i = 1) boundary in this case - can be re-expressed through the fs part of the anomalous function on the S layer side only... 

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