Ginzburg Landau order parameter

The range of coherence follows naturally from the BCS theory, and we see now why it becomes short in alloys. The electron mean free path is much shorter in an alloy than in a pure metal, and electron scattering tends to break up the correlated pairs, so dial for very short mean free paths one would expect die coherence length to become comparable to the mean free path. Then the ratio k i/f (called the Ginzburg-Landau order parameter) becomes greater than unity, and the observed magnetic properties of alloy superconductors can be derived. The two kinds of superconductors, namely those with k < 1/-/(2T and those with k > l/,/(2j (the inequalities follow from the detailed theory) are called respectively type I and type II superconductors. 

This fomi is called a Ginzburg-Landau expansion. The first temi f(m) corresponds to the free energy of a homogeneous (bulk-like) system and detemiines the phase behaviour. For t> 0 the fiinction/exliibits two minima at = 37. This value corresponds to the composition difference of the two coexisting phases. The second contribution specifies the cost of an inhomogeneous order parameter profile. / sets the typical length scale. 

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

Analytic teclmiques often use a time-dependent generalization of Landau-Ginzburg ffee-energy fiinctionals. The different universal dynamic behaviours have been classified by Hohenberg and Halperin [94]. In the simple example of a binary fluid (model B) the concentration difference can be used as an order parameter m.. A gradient in the local chemical potential p(r) = 8F/ m(r) gives rise to a current j... 

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

Random interface models for ternary systems share the feature with the Widom model and the one-order-parameter Ginzburg-Landau theory (19) that the density of amphiphiles is not allowed to fluctuate independently, but is entirely determined by the distribution of oil and water. However, in contrast to the Ginzburg-Landau approach, they concentrate on the amphiphilic sheets. Self-assembly of amphiphiles into monolayers of given optimal density is premised, and the free energy of the system is reduced to effective free energies of its internal interfaces. In the same spirit, random interface models for binary systems postulate self-assembly into bilayers and intro-... 

We start from the time-dependent Ginzburg-Landau equation for a non-conserved order parameter 0... 

Near the ODT, the composition profile of ordered microstructures is approximately sinusoidal (Fig. 2.1).The phase behaviour in this regime, where the blocks are weakly segregated, can then be modelled using Landau-Ginzburg theory, where the mean field free energy is expanded with reference to the average composition profile. The order parameter for A/B block copolymers may be defined as (Leibler 1980)... 

Within Landau-Ginzburg theory, the free energy functional near a second-order or weakly first-order phase transition is expanded in terms of an order parameter rj>(q) ... 

The phase behaviour of blends of homopolymers containing block copolymers is governed by a competition between macrophase separation of the homopolymer and microphase separation of the block copolymers. The former occurs at a wavenumber q = 0, whereas the latter is characterized by q + 0. The locus of critical transitions at q, the so-called X line, is divided into q = 0 and q + 0 branches by the (isotropic) Lifshitz point. The Lifshitz point can be described using a simple Landau-Ginzburg free-energy functional for a scalar order parameter rp(r), which for ternary blends containing block copolymers is the total volume fraction of, say, A monomers. The free energy density can be written (Selke 1992)... 

Summary. On the basis of phenomenological Ginzburg-Landau approach we investigate the problem of order parameter nucleation in a ferromagnetic superconductor and hybrid superconductor - ferromagnetic (S/F) systems with a domain structure in an applied external magnetic field H. We study the interplay between the superconductivity localized at the domain walls and between the domain walls and show that such interplay determines a peculiar nonlinear temperature dependence of the upper critical field. For hybrid S/F systems we also study the possible oscillatory behavior of the critical temperature TC(H) similar to the Little-Parks effect. 

An alternative method is to start directly from the Ginzburg-Landau approach [12]. In this method, one introduces (in general complex) order parameters as in Table 8, and expands the free energy in powers of the order parameters. For real order parameters and up to quartic terms one obtains... 

Table IV 1 8. Order parameters in the Ginzburg-Landau approach...

Using the potential V(h)°ch2 in Eq. (110), one recognizes that Eq. (110) is formally identical to a Ginzburg-Landau theory of a second-order transition for T>TC(D), with h(x,y) the order parameter field [186,216]. Therefore, it is straightforward to read off the correlation length , associated with this transition at TC(D), namely... 

The RPA can be improved on by the Landau-Ginzburg (LG) formalism [47] appropriate in a quasistatic regime. One introduces a complex order parameter i[f( ) (dimensions of energy) associated with Apld(jc), which can also be related to the amplitude of the lattice distortion [Eqs. (4 and 5)] qt oc e,2fc, vjf(jtj) + e 2kF i (xi). It is complex because the phase of the CDW or BOW at +2kF is independent of the one at -2kF. TTie partition function is expressed as a functional integral weighing all fluctuations in the order parameter Z = J3)i ie-p/w, where the free-energy functional is... 

The report of the Meissner effect stimulated the London brothers to develop the London equations, which explained this effect, and which also predicted how far a static external magnetic field can penetrate into a superconductor. The next theoretical advance came in 1950 with the theory of Ginzburg and Landau, which described superconductivity in terms of an order parameter and provided a derivation for the London equations. Both of these theories are macroscopic or phenomenological in nature. In the same year, 1950, the... 

Schmahl WW, Swainson IP, Dove MT, Graeme-Barber A (1992) Landau free energy and order parameter behavior of the a-p phase transition in ciistobalite. Z Kristallogr 201 125-145 Sollich P, Heine V, Dove MT (1994) The Ginzburg interval in soft mode phase transitions Consequences of the Rigid Unit Mode picture. JPhys Condensed Matter6 3171-3196 Strauch D, Domer B (1993) Lattice dynamics of a-quartz. 1. Experiment. J Phys Condensed Matter 5 6149-6154... 

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