Order parameter component

Here Qm and Q can be separate order parameters or order parameter components. In combination, Equations (4) and (5) provide a means of predicting the elastic constant variations associated with any phase transition in which the relaxation of Q in response to an applied stress is rapid relative to the time scale of the experimental measurement. A classic example of the success that this approach can have for describing the elastic behaviour of real materials is provided by the work of Errandonea (1980) for the orthorhombic monoclinic transition in LaPsOi4 (Fig. 4). 

Tabje 3. Order parameter components for the subgroups of / mSm associated with special points Mj and R4 (after Howard and Stokes 1998). The system of reference axes for these components is that used in Stokes and Hatch (1988) and the group theory program ISOTROPY. 

Space Group Order parameter components Relationships between order parameter components... 

One easily recognizes that the two types of domains shown on the top of fig. 10 simply correspond to ijr = 1. Since for this structure the sublattices (a,c) and (b,d) each can be combined to a single sublattice, the c(2x2) structure has a single order parameter component. But the situation differs for the (2x1) structure, where two components are needed ... 

Note that due to the constraint pe, + pi, + pc + pa — 1/4, there is no fourth independent component. The order parameter components defined in eq. (11) are not orthogonal with each other, and do not bring out the symmetry properties of the structure in a natural way thus in practice one proceeds differently, by considering the expansion of the ordering in terms of mass density waves, as will be discussed below. 

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 ... 

The first Landau rule states that for a second-order transition to be possible there should occur just a single star of q in the description of the ordered phase. Now the order parameter components [Pg.152]

Fig. 17. Schematic variation of the critical exponents of the order parameter fi (a), the order parameter response function y (b), and the correlation length v (c) with the spatial dimensionality, for Lhe m-veclor model. Upper (du) and lower (rf ) critical dimensionalities are indicated. Here m = 1 corresponds to the Ising model, m = 2 to the XY model, m = 3 to the Heisenberg model of magneLism, while the limit of infinitely many order parameter components (m —> oo) reduces to the exactly solved spherical model (Berlin and Kac 1952, Stanley, 1968).

Fig. 69. (a) Part of the body-centered cubic lattice ordered in the B2 structure (left part) and in the Dtp structure (right part). Left part shows assignment of four sublattices a, b, c and d, In the B2 structure (cf. also fig. 66a), the concentrations of A atoms are the same at the a and c sublatticcs, but differ from the concentrations of the b, d sublattices, while in the DOj structure the concentration of the b sublattice differs from that of the d sublatlice, but both differ from those of the a, c sublattices (which are still the same). In terms of an Ising spin model, these sublattice concentrations translate into sublattice magnetizations mu, mu, mc, m,i, which allow to define three order parameter components / = ma + mL- — mu — m,/, fa = m - mc + mu — m,j, and fa = -ma + m., + mu — nij. 

For conventional ferroics aij T) = ajiT — Tq), where is Curie temperature of the bulk material with order parameter component ti (see Chap. 1). 

It is seen, that renormalized Curie temperatures for different order parameter components t) are different due to t, dependence on index i. The shift of... 

We first calculate the elastic strains My as they make an essential contribution into the free energy (4.11). Then obtained elastic solution should be substituted into the Euler-Lagrange equations for the order parameter components... 

The simplest even-parity state is the isotropic state encountered in ordinary superconductors. This state is often referred to as s-wave state . The isotropic order parameter does not depend on the direction k and reduces to a complex constant cj) = Its only degree of freedom is the Josephson phase. By far the most extensively studied examples of anisotropic pairing are the p-wave states realized in the superfiuid phases of He, the d-wave pair state in high-Tc superconductors and the f-wave states in UPts and SrRu204. The odd parity (p, f) states among these examples are characterised by more than one order parameter component with internal phase degrees of freedom which appear in addition to the overall Josephson phase. 

The fluctuations in the order parameter (component concentrations) in polymer mixtures are expressed so weakly that it is quite correct to consider the thermodynamic... 

We will now investigate in more detail what happens when the restriction placed on h(r) in the previous subsection is relaxed. It was pointed out earlier that in such a situation the fluctuations of S(r) cannot be calculated independently from the fluctuations of ri(r), due to interaction terms between them in Eq. [57]. The problem of coupling between different fluctuation modes can be avoided if one considers the thermal fluctuation amplitudes of the tensor order parameter components, Qiy(r), rather than the fluctuation amplitudes of S(r) and h(f). 

One can also see from the relative heights of the density and order parameter components in Figures 2.3a and 2.3b that the overall response of the nematic film is different for the two forms of excitatioa The absorption of infrared photons (A, = 10.6 pm) corresponds to the excitation of the ground (electronic) state s rovibrational manifold, whereas the visible photoabsorption (k = 0.53 pm) corresponds to the excitation of the molecules to the electronically excited states (see Fig. 2.5). The electronic molecular stractures of these two excited states are different and may therefore account for the different dynamical response behavior of the order parameter, which is dependent on the intermolecular Coulombic dipole-dipole interaction. From this observation one may conclude that the dynamical grating technique would be an interesting technique for probing the different dynamical... 

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