Phase changes of second order

Differential scanning calorimetry (DSC) Since lc s form phases in a thermodynamic sense, a transition from one phase to another is accompanied by a phase-transition enthalpy. Nevertheless, there are phase transitions of second-order character which can hardly be detected by DSC since there is no phase-transition enthalpy but just a change in heat capacity. A typical example is the transition from orthogonal phases to tilted phases. 

The recoil-free fraction depends on the oxidation state, the spin state, and the elastic bonds of the Mossbauer atom. Therefore, a temperature-dependent transition of the valence state, a spin transition, or a phase change of a particular compound or material may be easily detected as a change in the slope, a kink, or a step in the temperature dependence of In f T). However, in fits of experimental Mossbauer intensities, the values of 0 and Meff are often strongly covariant, as one may expect from a comparison of the traces shown in Fig. 2.5b. In this situation, valuable constraints can be obtained from corresponding fits of the temperature dependence of the second-order-Doppler shift of the Mossbauer spectra, which can be described by using a similar approach. The formalism is given in Sect. 4.2.3 on the temperature dependence of the isomer shift. 

For turbulent flow in single-phase systems, the predicted temperature profile is not changed significantly if the Peclet number is assumed to be infinite. Therefore, in turbulent two-phase systems the second-order terms in Eqs. (9) probably do not have a significant effect on the resulting temperature profiles. In view of the uncertainties in the present state of the art for determining the holdups and the heat-transfer coefficients, the inclusion of these second-order terms is probably not justified, and the resulting first-order equations should adequately model the process. 

Figure 9.3 Schematic illustration of second-order nonlinear optical effects, (a) Second-harmonic generation. Two light fields at frequency go are incident on medium with nonvanishing / 2. Nonlinear interaction with medium creates new field at frequency 2 go. (b) Frequency mixing. One light field at frequency GO and one at frequency go2 is incident on nonlinear medium. Nonlinear interaction with medium creates new field at frequency goi + go2. (c) electro-optic effect. Static electric field E (0) applied over nonlinear medium changes phase of an incoming light field.

The Landau theory predicts the symmetry conditions necessary for a transition to be thermodynamically of second order. The order parameter must in this case vary continuously from 0 to 1. The presence of odd-order coefficients in the expansion gives rise to two values of the transitional Gibbs energy that satisfy the equilibrium conditions. This is not consistent with a continuous change in r and thus corresponds to first-order phase transitions. For this reason all odd-order coefficients must be zero. Furthermore, the sign of b must change from positive to negative at the transition temperature. It is customary to express the temperature dependence of b as a linear function of temperature ... 

The compensating effect becomes quite poor when the order of the two PIPs is changed (Fig. 22d). The distorted excitation prohle can be corrected if an inherited phase

phase coherence in PIPs. 

Another important contribution by Landau is related to symmetry changes accompanying phase transitions. In second-order or structural transitions, the symmetry of the crystal changes discontinuously, causing the appearance (or disappearance) of certain symmetry elements, unlike first-order transitions, where there is no relation between the symmetries of the high- and low-temperature phases. If p(x, y, z) describes the probability distribution of atom positions in a crystal, then p would reflect the symmetry group of the crystal. This means that for T> T p must be consistent with... 

The third type of phase change of the second order is fortunately easy to treat theoretically, at least to an approximation, and it, is the one which will be discussed in the present chapter. This is what is known as an order-disorder transition in an alloy, and can be better understood in terms of specific examples, which we shall mention in the next section. 

Graphs like Fig. XVIII-5 show particularly plainly the difference between phase changes of the first and second order. We can readily imagine that, by slightly altering the mathematical details, the curves could be changed to the form of Fig. XVIII-6, in which, though we have a... 

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