Secondary-order parameter

An interesting aspect of many structural phase transitions is the coupling of the primary order parameter to a secondary order parameter. In transitions of molecular crystals, the order parameter is coupled with reorientational or libration modes. In Jahn-Teller as well as ferroelastic transitions, an optical phonon or an electronic excitation is coupled with strain (acoustic phonon). In antiferrodistortive transitions, a zone-boundary phonon (primary order parameter) can induce spontaneous polarization (secondary order parameter). Magnetic resonance and vibrational spectroscopic methods provide valuable information on static as well as dynamic processes occurring during a transition (Owens et ai, 1979 Iqbal Owens, 1984 Rao, 1993). Complementary information is provided by diffraction methods. 

Proper identification of the order parameter of a particular system often needs detailed physical insight, and sometimes is complicated because different degrees of freedom are coupled. For example, there are many reports in the literature that an order-disorder transition of adsorbates on loose-packed substrates causes an adsorbate-induced reconstruction of the substrate surface. In such a situation, the order parameter of the adsorbate order-disorder transition is the primary order parameter whereas the lattice distortion of the substrate surface is a secondary order parameter . However, for pure surface reconstruction transitions (i.e. structural phase transitions of the surface of crystals where no adsorbates are involved) all considered degrees of freedom are atomic displacements relative to positions of higher symmetry. The proper distinction between primary and secondary order parameters is then much more subtle. 

Polarization is a secondary order parameter, which appears only when the molecules are tilted with respect to the smectic layer normal. It is defined as ... 

The two polarizations Pp and Pap may be taken as secondary order parameters coupled with the genuine order parameters. As a result, depending of the model, the theory predicts transitions from the smectic A phase into either the synclinic ferroelectric phase at temperature Tp or into an anticlinic antiferroelectric phase at Tap- One intermediate ferrielectric phase is also predicted that has a tilt plane in the i + 1 layer turned through some angle

tilt plane in the i layer. The models based on the two order parameters are of continuous nature (9 may take any values) and, although conceptually are very important, caimot explain a variety of intermediate phases and their basic properties. 

The discrepancy between small AF moment in URu2Si2 and large thermodynamic anomalies has led to the postulation of a hidden (non-dipolar) order parameter. The cmcial questions about its nature are (i) Is the order primarily involving the localised 5f-CEF split states or the heavy itinerant conduction electrons, (ii) Does the hidden order parameter break time reversal invariance or not. In the former case it may induce AF as secondary order parameter, in the latter the two order parameters are umelated and their appearance at the same temperatnre Tm = To has to be considered as accidental. 

So far it has been assumed that the polarization is the variable that determines the free energy, but it may happen that the transition is connected to, i.e. occurs in a completely different variable, let us call it q, to which a polar order can be coupled. A polarization P appearing this way may affect the transition weakly, but it can only have a secondary influence on the free energy. We will therefore call q and P, respectively, the primary and secondary order parameters, and the simplest Landau expansion in those two parameters can be written... 

Equation (43) shows that this is also true for P as the secondary order parameter. However, we have a new Curie temperature given by Eq. (47), shifted upwards by an amount AT =T -T ... 

The order parameter thus characterizes the transition, and the Landau free energy expansion in this order parameter, and in eventual secondary order parameters coupled to the first, has to be invariant under the symmetry operations of the disordered phase, at the same time as the order parameter itself should describe the order in the condensed phase as closely as possible. In addition to having a magnitude (zero for T> Tc, nonzero for T< T ), it should have the same symmetry as that phase. Further requirements of a good order parameter are that it should correctly predict the order of the transition, and that it should be as simple as possible. As an example, the tensori-al property of the nematic order parameter... 

A group theoretical symmetry analysis by Indenbom and his collaborators in Moscow [197,198] led, around 1977, to the introduction of a very attractive order parameter for the A -C transition which usually is referred to as the Pikin-Indenbom order parameter. It is attractive because it presents, in the simplest form possible, the correct symmetry and a very lucid connection to the secondary order parameter P. It was adopted by the Ljubljana group around Blinc for the description of both static and dynamic properties of the C phase [199-204]. The basic formalism has been described in particular detail by Pikin [40] and by Pikin and Osipov [41]. 

Before going on to this expansion we should, however, make the connection to the secondary order parameter P. Looking onto the smectic layer plane in Figure 59 we have (for a positive material, Po>0) tion between the P direction and the tilt direction such that P advances in phase by 90 degrees in the positive (p direction. If we express P as a complex quantity Px + iPy, we can relate P directly to the c director expressed in complex form, c = Cx + iCy, by... 

This is the relation between the secondary order parameter P= P, Py) and the primary order parameter (V=(n Wy, -n n ), when the latter is expressed in the tilt vector =(( , 2)-... 

As primary and secondary order parameters we have thus used two two-component vectors lying in the smectic plane, perpendicular to the layer normal. The number of degrees of freedom, i.e. the number of components, is evidently more important than subtle differences in symmetry, if the order parameters are used with sound physical arguments. 

To do better, one needs to extend the theory to include secondary order parameters, such as the mass density p. The physical idea is simple when a collection of rod-shaped molecules go from a state of complete orientational... 

In the SmC liquid crystals, the primary order parameter is the director tilt and not the spontaneous polarization. The polarization, therefore, is often called secondary-order parameter, and the SmC materials are called improper ferroelectrics. The main reason for this is the weakness of the dipole-dipole interactions in the molecules. The polar order can be estimated by the ratio of the actual spontaneous polarization and which is the value that would appear when complete polar order is assumed. From its definition, the polarization is the density of the molecular dipoles. Assuming molecular dipoles of 3 Debye ( 10 Cm) and typical molecular weight of 300 g, density of about 1 g/cm = 10 kg/m, we get that in 1 m we have 6 10 molecules, which would give Po 2 W C/m = 2000nC/cm. For a SmC with Pg <100 nC/cnP, this means that less than 5% of the dipoles are ordered in one direction. For this reason, in the first approximation the polarization is proportional to the tilt angle. This relation, indeed, is found to be true for materials with moderate or low polarization. However, for materials with large polarization, like Pg 500 nC/cm, the dipole-dipole interaction becomes considerable, and the proportionality is not true. The deviation is more pronounced at lower temperatures, when the dipole-dipole... 

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