Landau theory continuous transitions

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 ... 

More generally, the occurrence of first- or second-order phase transitions can be rationalized within the framework of Landau theory (1937) that classifies phase transitions by symmetry considerations. In this theory, the symmetry relationship of the descendant phase with the parent phase is of primary importance in the description of the possible phase transitions. Within a group-subgroup filiation, continuous- (second order) or discon-tinuous-(first order)phase transitions are possible. On the other hand, if the two phases are not related by a group-subgroup relationship, then the phase transition cannot be continuous. 

In the equation, the first term of the Taylor expansion is zero because the free energy is a minimum at the average value of the order parameter in the parent phase. According to the Landau theory, near a continuous transition, the coefficient J goes to zero as... 

Although the Landau description was established for continuous phase transitions it is also used for describing discontinuous transitions, though special care should be taken because of a jump in the order parameter. In the case of LCs, the phase transitions are only weakly discontinuous and the description within the Landau theory is useful. 

Certain two-dimensional 5 = quantum antiferromagnets can imdergo a direct continuous quantum phase transition between two ordered phases, an antiferromagnetic Neel phase and the so-called valence-bond ordered phase (where translational invariance is broken). This is in contradiction to Landau theory, which predicts phase coexistence, an intermediate phase, or a first-order transition, if any. The continuous transition is the result of topological defects that become spatially deconfined at the critical point and are not contained in an LGW description. Recently, there has been a great interest in the resulting deconfined quantum critical points. ... 

In Landau s phenomenological theory of the mean field the thermodynamic potential G (or F) is considered as a series in terms of the order parameter Q. I he series coefficients, generally, depend on temperature and pressure. Depending on the presence of the series term with a certain power n Q ), on the coefficients values, and on the relationships between them, the thermodynamic potential assumes a functional dependence being characteristic for the first-order transitions, transitions of the continuous type or for the first-order ones of the near-continuous type. 

Landau developed a theory for second-order phase transition [16], such as from diamagnetic phase to ferromagnetic phase, in which the order parameter increases continuously from zero as the temperature is decreased across the transition temperature Tc from the high temperature disordered phase to the low temperature ordered phase. For a temperature near the order is very small. The free energy of the system can be expanded in terms of the order parameter. 

In the McMillan model, the smectic A-nematic transition can be continuous or discontinuous. If a is less than 0.7, then o decreases to zero continuously and S is continuous at the smectic A- nematic transition. If a is between 0.7 and 0.98, then a jumps to zero discontinuously and S has a small discontinuity at the smectic A-nematic transition. When a is greater than 0.98, the smectic phase transforms directly into the isotropic phase with discontinuities in both order parameters. So just as in the extended Landau-de Geimes theory for the smectic A phase, a tricritical point is predicted at a=0.7, which corresponds to a ratio in the smectic A—nematic transition temperature to the nematic-isotropic transition temperature of 0.87. A great deal of experimental work has been done on the smectic A-nematic transition, and the results seem to indicate that the tricritical point occurs when the ratio of the two transition temperatures is significantly larger than 0.87. 

The lattice gas approach is valid within certain limits for typical metallic hydrides, binaries as well as ternaries. Deviation from this idealized picture indicates that metallic hydrides are not pure host-guest systems, but real chemical compounds. An important difference between the model of hydrogen as a lattice gas, liquid, or solid and real metal hydrides lies in the nature of the phase transitions. Whereas the crystallization of a material is a first-order transition according to Landau s theory, an order-disorder transition in a hydride can be of first or second order. The structural relationships between ordered and disordered phases of metal hydrides have been proven in many cases by crystallographic group-subgroup relationships, which suggests the possibility of second-order (continuous) phase transitions. However, in many cases hints for a transition of first order were found due to a surface contamination of the sample that kinetically hinders the transition to proceed. 

Marcus[195] gave a quantitative interpretation of this idea and above all, the role of solvent rearrangement within the framework of the absolute rate theory. Later, he also extended these concepts to electrochemical processes[196]. Similar concepts were also developed by Hush[197,198]. An important result of this work was the establishment of the relation between the transfer coefficient for adiabatic reactions and the charge distribution in the transient state. Gerischer[93,199] proposed a very useful and lucid treatment of the process of electron transfer in reactions with metallic as well as semiconductor electrodes. While the works mentioned above were mainly based on transition state theory, a systematic quantum-mechanical analysis of the problem was started by Levich, Dogonadze, and Chizmadzhev[200-202] and continued in a series of investigations by the same group. They extensively used the results and methods of solid state physics, and above all the Landau-Pekar polaron theory[203]. 

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