Second-kind phase transition

We will investigate the conditions for the stable spontaneous polarization state forg4 > 0 or g4 < 0 in the following paragraph. Forg4 > 0, the phase transition is called second-kind phase transition. The positive value for is possible in Eq. (5.35) only wheng2 < 0, i.e. for the temperatures < o. If the coefficient ge is small (it has qualitatively not important contribution to G potential with respect to the value of Ps in the vicinity of the temperature o) it could be expressed... 

Fig. 5.9 Temperature dependence of the reciprocal dielectric permittivity of Triglycine Sulfate at the second-kind phase transition at c = o=49.92 = C (Gonzalo 1966)...

The relation between diamond and zinc blende shown above is a formal view. The substitution of carbon atoms by zinc and sulfur atoms cannot be performed in reality. The distortion of the NiAs structure according to Fig. 18.4, however, can actually be performed. This happens during phase transitions (Section 18.4). For example, MnAs exhibits this kind of phase transition at 125 °C (NiAs type above 125 °C, second-order phase transition another transition takes place at 45 °C, cf. p. 238). 

Fig. 29. Phase diagram of the model Eq. (22) for coadsorption of two kinds of atoms in the temperature-coverage space. Circles indicate a second-order phase transition, while crosses indicate first-order transitions. Point A is believed to be a tricritical point and point B a bicritical point. The dashed curve shows the boundary from the Blume-Capel model on a square lattice with a nearest-neighbor coupling equal to 7 in the present model (for - 0 Eq. (22) reduces to this model), only the ordered phase I then occurs. From Lee and Landau. )...

Phase transitions of the system such as chain ordering transitions of lipids, appear in the isotherm as regions of constant pressure in the case of first order phase transitions involving the coexistence of two phases, or as a kink in the isotherm corresponding to a second order phase transition. These kinds of surface measurements are highly sensitive to impurities and must be carried out using very pure water and sample materials. 

A nonmagnetic anomaly had been discovered previously around 17.5 K in URu2Si2, but not characterised in the low Tphase, and therefore called the hidden order .No anomaly was apparent in ° Ru NQR measurements around this T, with the results indicating that the four-fold axis of the Ru site survived even at lower T. Space group analysis has shown that one type of second-order phase transition did not require any kind of lattice distortion in the system, and allowed the Ru frequency to remain unchanged, thus providing a possible explanation. The characteristics of the hidden order were discussed, based on a local 5/ electron picture. 

Percolation transition is one kind of phase transitions (or critical phenomena). Unlike the melting or evaporation phase transition phenomena, which are second-order phase transitions, the percolation transition is a first-order phase transition without involving the temperature and volume changes in the system. It can be universally expressed as a power law or scaling law as shown below ... 

For the exact solution of A -electron atoms at the large dimension limit, the symmetry breaking is shown to be a first-order phase transition. For the special case of two-electron atoms, the first-order transition shows a triple point where three phases with different symmetry exist. Treatment of the Hartree-Fock solution reveals a different kind of symmetry breaking where a second-order phase transition exists for N — 2. The Hartree-Fock two-electron atoms in weak external electric field exhibit a critical point with mean-field critical exponents ( = j, a = Odis, 5 = 3, and y — 1). ... 

The initial classification of phase transitions made by Ehrenfest (1933) was extended and clarified by Pippard [1], who illustrated the distmctions with schematic heat capacity curves. Pippard distinguished different kinds of second- and third-order transitions and examples of some of his second-order transitions will appear in subsequent sections some of his types are unknown experimentally. Theoretical models exist for third-order transitions, but whether tiiese have ever been found is unclear. 

A second, less common kind of pressure-induced phase transition (but in one instance at least, of very great importance) is that in which there is no change in primary coordination number and, to a good approximation, no change in nearest-neighbour bond lengths but, nevertheless, a substantial decrease in volume. We consider next, two examples of this latter type of phase transition. 

Therefore the model avoids two main difficulties the large amount of computer time which is normally needed for simulations and the loss of structural information which occurs in simple theoretical models (mean-field models) which do not take into account the structural aspects of the adsorbate layer. Mean-field-kind models fail in the prediction of phase transitions of the second order because at these points the long-range correlations appear. They also fail in describing the system s behaviour in the neighbourhood of the point of first-order kinetic phase transition. 

The course of the process at a later stage, where the second assumption is not satisfied, was studied by I. M. Lifshitz and V. V. Slezov.1 The kinetics of phase transitions of the first kind near absolute zero, where fluctuations have a quantum character, were described by I. M. Lifshitz and Yu. M. Kagan2 and by S. V. Iordanskii and A. M. Finkelshtein.3 In these works the ideas of Ya.B. s paper also play an important role. 

Horn et al. simulated phase transitions of the first and second kind, and a three-critical point, via a Wien-bridge oscillator. Such behavior is fairly well accounted for by the Landau theory except for very close to the critical point. Furthermore, Horn et al. observed that the phase of their oscillator had a... 

In the proposed two-step process, it is important to attain high efficiencies for conversion of coal to COx (CO -I- CO2) in the first-step reaction and then for conversion of CO2 to CO in the second-step reaction by an external heat input. From the thermodynamic conditions and the low cost, the redox pair of Fe304/a-Fe was one of the promising redox systems for the two-step process, but it still required the operating temperature above 1200°C[2]. It is well known that many kinds of metal ions can be incorporated into the spinel lattice structure of magnetite by replacing ferrous or ferric ions. There is the possibility that metal-substitution for Fe or Fe " in magnetite causes a phase transition to the metallic phase, which proceeds readily even at low temperatures and improves the conversion efficiencies of coal and CO2 to CO in the two-step process. 

Simulations of octahedral molecular clusters at constant temperature show two kinds of structural phase changes, a high-temperature discontinuous transformation analogous to a first-order bulk phase transition, and a lower-temperature continuous transformation, analogous to a second-order bulk phase transition. The former shows a band of temperatures within which the two phases coexist and hysteresis is likely to appear in cooling and heating cycles Fig. 10 the latter shows no evidence of coexistence of two phases. The width of the coexistence band depends on cluster size an empirical relation for that dependence has been inferred from the simulations. 

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