Order parameter macroscopic

The excess thermodynamic properties correlated with phase transitions are conveniently described in terms of a macroscopic order parameter Q. Formal relations between Q and the excess thermodynamic properties associated with a transition are conveniently derived by expanding the Gibbs free energy of transition in terms of a Landau potential ... 

Carpenter, 1988). Because the excess Gibbs free energy of transition must always be at a minimum with respect to the macroscopic order parameter Q, i.e. ... 

Figure 2.8 shows the behavior of the macroscopic order parameter 2 a function of TIT (F/r rans for first-order transitions). 

For a first-order transition, the macroscopic order parameter shows a jump at Ftrans from 2 = 0 to 2 = gtrans—i-e., beloW Ftrans,... 

Obviously, if we know experimentally the behavior of the macroscopic ordering parameter with T, we may determine the corresponding coefficients of the Landau expansion (eq. 2.52). However, things are not so easy when different transitions are superimposed (such as, for instance, the displacive and order-disorder transitions in feldspars). In these cases the Landau potential is a summation of terms corresponding to the different reactions plus a couphng factor associated with the common elastic strain. 

The most important coupling to deformations of the network is the one that is linear in both the strain of the network and the nematic order parameter. As has been discussed earlier in this section this leads to the consequence that the strain tensor can be used as an order parameter for the nematic-isotropic transition in nematic sidechain elastomers, just as the dielectric or the diamagnetic tensor are used as macroscopic order parameters to characterize this phase transition in low molecular weight materials. But it has also been stressed that nonlinear elastic effects as well as nonlinear coupling terms between the nematic order parameter and the strain tensor must be taken into account as soon as effects that are nonlinear in the nematic order parameter are studied [4, 25]. So far, no deviation from classical mean field behavior concerning the critical exponents has been detected in the static properties of this transition and correspondingly there are no reports as yet discussing static critical fluctuations. 

As no symmetry breaking occurs during the process t/ should transform as the totally symmetric irreducible representation of the group. The physical quantity AV/Vo is a good candidate for use as a macroscopic order parameter compatible with the symmetry requirement. Minimization of F leads to two solutions... 

Fig. 4.10 a V/Vq plot for BagSi4 as a function of pressure (points) along with a third-order Birch-Mumaghan equation of state (line). The macroscopic order parameter corresponds to spontaneous i c., the Variation of the volume cmrected from the compressibility, b Aspo ,a eous as a function of pressure fits with the theoretical analysis based on the Landau theory of phase transition null before the transition pressure, a jump at the transition and a square-root evolution after the transition. Such a behaviOT is also observed for Si atomic displacement parameters that can be used as microscopic order parameter [82]. The correlation between these physical quantities underlines the relationships between the isostructural transition and disordering of the Si sub-lattice... 

In conclusion, the Landau theory provides a welcome quantitative description of the thermodynamics and kinetics of phase transitions in minerals, including ferroelectric ceramics, by using macroscopic order parameters and their relationship to physical properhes and symmetry, as shown below. For an excellent review of the apphcation of Landau theory to displacive phase transitions in minerals, see Dove (1997). 

The critical quantity investigated to describe ferroelectric transitions is the macroscopic order parameters P (polarization). 

Using Eqs. (3.9) and (3.10) in Ax = 3x Qzz — Qxx)j it is found that, as before, Qzz = 2 A x/9x This shows that a knowledge of A% and Aij is sufficient to determine the macroscopic order parameter Qzz, which distinguishes the nematic phase from the isotropic phase. However, in Eq. (3.9), it is seen that at least two microscopic parameters are needed to describe the orientational order of a biaxial molecule in a uniaxial medium. Unless the molecule possesses a threefold (C3) symmetry or higher, it is necessary to use S and D to describe its orientational order. These microscopic order parameters cannot be simultaneously determined from a single measurement of a bulk property like Xa/3- If> however, the molecule has a Cs or higher symmetry axis (772 = rjs and D = 0), then Qzz (2n A rj/9x)S ... 

The quantity V X ttax refers to the anisotropy of the tensor for a fully aligned state for which the order parameter is one. For a biaxial phase (i.e. a phase which has different properties along each of the three principal axes), the macroscopic order parameter in principal axes can be written as ... 

In deriving a macroscopic order parameter, we will use the diamagnetic susceptibility as an example. However, any other macroscopic property, e.g., refractive index or dielectric response, could be used as well. 

Relationship Between Microscopic and Macroscopic Order Parameters... 

The order parameters defined previously in terms of the directional averages can be translated into expressions in terms of the anisotropies in the physical parameters such as magnetic, electric, and optical susceptibilities. For example, in terms of the optical dielectric anisotropies Ae = one can define a so-called macroscopic order parameter which characterizes the bulk response... 

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