Theory Of Second-order Phase Transitions Research Articles

Thermodynamic aspects of the glass transition: The applicability of the Landau theory of second-order transitions

Abstract Gordon has shown that the Landau theory of second-order phase transitions when applied to the glass transition provides a plausible explanation of the apparent coincidence of the temperature To of viscosity divergence and the T2 of the Kauzmann paradox. It can be shown however that two current statistical mechanical theories of the transition, the Gibbs-DiMarzio theory and one proposed by Goldstein which takes into account nonconfigurational contributions to the entropy, do not follow the Landau theory. We are thus presented with a dilemma: if the Landau theory is applicable to the glass transition, neither of the two statistical mechanical theories can be, and vice versa.

Lifshitz's Criterion for Symmetry Changes in Second-Order Phase Transitions

Some points implicit in the Landau theory of second-order phase transitions are clarified. Lifshitz's criterion is then directly derived by requiring the absence of k -linear terms A k ( p , T ), the second-order coefficient in Landau's expansion of free energy. It is shown that the presence of k -linear terms leads to invariants bilinear in the order parameter and its spatial derivative as proposed by Lifshitz. The theory is developed in parallel with the shallow impurity problems in semiconductors and the k · p perturbation theory. Lifshitz's criterion is shown to be equivalent to the absence of k -linear terms in energy bands. The equivalence makes the application of Lifshitz's criterion easier

Application of Landau’s theory of second-order phase transition to systems with two order parameters

Landau’s theory of second-order phase transitions for systems with two order parameters is discussed and applied to a meanfield Hamiltonian for an Ising model with bilinear and biquadratic pair interactions. Second-order phase transition temperatures and tricritical points are determined for several spins.

The magnetic structure of Bi2Fe4O9– analysis of neutron diffraction measurements

The compound Bi2Fe4O9 belongs to the space group Pbam ( D92h), with two formula units per unit cell. Neutron diffraction measurements showed that it is paramagnetic at room temperature and undergoes a transition to an antiferromagnetic state at TN = (264 ± 3) K in agreement with previous susceptibility and Mössbauer measurements. Analysis of the 80 K neutron diffraction pattern yielded a magnetic structure with the following features: (a) The basic translations ao, bo, co of the chemical lattice change into antitranslations in the magnetic lattice. (b) The spins are perpendicular to co. (c) The magnetic structure belongs to the PC2/m space group and is a basis vector to an irreducible space under the Pbam irreducible representations, in accord with Landau's theory of second-order phase transition. The position parameters of the Fe3+ ions in the unit cell were refined. The magnetic moment of the compound was found to be (4.95 ± 0.08) μB, compared with the value of 5 μB for the Fe3+ free ion. The temperature dependence of the { 131 } magnetic reflection peak intensity was measured and found to be in agreement with the sublattice magnetization predicted by the molecular field approximation.

Some problems of the statistical physics of polymer chains with volume interaction

The present state of the theory of volume effects in a polymer chain is reviewed. The theory of coils (i.e., chains with predominately repulsive volume interaction of monomers) and the analogy between the excluded volume problem and the theory of second-order phase transitions are briefly described. The theory of globules, formed by attractive interaction of monomers or by external attractive fields, is considered in greater detail. The coil-globule transition under various conditions is analyzed. A theory is constructed for the simplest model of the polymer chain---the model of interacting beads on a flexible string. The connection of this model with more realistic ones is discussed. This review is based on the approach proposed by I. M. Lifshitz in 1968. While attention is focused on the physical aspects of the problems, some questions concerning biological applications are discussed in the conclusion.

Teaching the renormalization group

The renormalization group theory of second-order phase transitions is described in a form suitable for presentation as part of an undergraduate statistical physics course.

The Landau theory of phase transitions in TTF-TCNQ (tetrathiafulvalene-tetracyanoquinodimethane)

The observed low-temperature charge-density-wave (CDW) transitions in tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) are analyzed in the framework of the Landau theory of second-order phase transitions. A careful treatment of the symmetry requirements is given. It is found that the (commensurate) transition at 54 K involves CDW distortions on both TCNQ and TTF chains. The possibilities for the subsequent (noncommensurate) transition near 49 K are analyzed. A continuous transition involving a phase modulation of the first transition may happen for a range of values of the Landau parameters. The next (commensurate) transition, at 38 K is discontinuous. The formulation requires a thorough group-theoretical analysis: We discuss the validity of the usual practice of using only a simple basis set for an irreducible representation associated with a transition. A discussion of the absence or presence of commensurability energies of the various transitions is also given.

Landau's theory of second-order phase transitions and its application to ferromagnetism

Landau's theory of second-order phase transitions and its application to ferromagnetism

Landau's theory of second-order phase transitions and its application to ferromagnetism

A detailed comparison is made between the theory of second-order phase transitions in terms of Landau's original approach of minimizing the thermodynamical potential and in terms of the established group-theoretical approach, which is derived from the general thermodynamical approach. It is shown that the two approaches are not equivalent and a number of spurious features are identified in the existing accounts of the group-theoretical criteria. Thus, minimization of the thermodynamic potential frequently gives some additional information which could not be obtained from the group-theoretical approach.

Phase transitions into inhomogeneous states

In the Landau theory of second-order phase transitions one usually considers the position that the characteristic parameter of the phase transition transforms according to an irreducible representation, with wave vectors terminating at special points of the Brillouin zone. The period of the structure which arises at such transitions is a multiple of the lattice constant of the initial phase (i.e. 1, 2, 3, or 4 times)1. Such transitions are possible only if a special condition (Lifschitz condition) on the transformation properties of the characteristic parameter is fulfilled. Namely, gradient invariants such as ηδζ/δz - ζδη/δz, where q,E are the components of the characteristic parameter (transforming according to an irreducible representation with wave vector qo) must be absent.1 If the Lifschitz condition is not fulfilled, then a second-order phase transition corresponding to the irreducible representation under consideration is impossible. The transition is realized according to the representation with ...

Evolution of coagulating systems

This paper is devoted to the study of some properties of solutions to the kinetic equation of coagulation. A simple transformation of the equation is proposed. Instead of c o ( t) and t, where c o ( t) is the concentration of coagulating particles consisting of g monomers and t is a dimensionless time, new variables τ = f 0 t c 1( t′) dt′ and ν g(τ) = c g c 1 are introduced. These ν 9 are then expanded in powers of τ. This expansion has the convergence radius τ 1 ⩾ τ 0 = f 0 ∞ c 1( t) dt and therefore determines the behavior of the size distribution in the interval 0 ⩽ t < ∞. Simple recurrence relations are obtained for the determination of the coefficients of the expansion. A scaling hypothesis adopted from the modern theory of second-order phase transitions is applied to establish the asymptotic form of size distributions at large g and t. Three exactly soluble models are used to demonstrate the possibilities of the proposed method.

Landau Model for the Critical Susceptibility of a Ferromagnet

Measurements have been made of the low-field magnetic susceptibility of the ferromagnet Gd${\mathrm{Ni}}_{2}$ in the reduced temperature interval ${10}^{\ensuremath{-}3}&lt;\ensuremath{\epsilon}=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}&lt;4$. The study focuses on the transition from Curie-Weiss behavior for $T\ensuremath{\gg}{T}_{c}$ to critical behavior for $T={T}_{c}$. A phenomenological model based on Landau's theory of second-order phase transitions is presented which fits the experimental data in this interpolation region. The viability of a Landau model for predicting microscopic exchange interactions (from an analysis of the high-temperature susceptibility) is discussed critically.

The spectral linewidth of laser noise and of fluctuations near other second-order phase transitions

A simple, exact analytic expression is derived for the effective spectral linewidth of laser intensity fluctuations as a function of output power. Relevance of the work to the general theory of second-order phase transitions is pointed out.

On the theory of the Kondo effect

The Kondo effect is investigated by methods developed in the theory of fields with strong coupling. Renormalizability relations analogous to the scaling laws in the theory of second-order phase transitions are found. The behavior of the scattering amplitude of an electron by an impurity is analyzed as a function of the electron energy. Formulas are derived for the resistance as a function of the temperature and for the magnetic moment as a function of the magnetic field and temperature. For the antiferromagnetic interaction sign the resistance tends to a constant and the magnetic moment to zero asT → 0 according to power laws with universal exponents that depend only on the spin of the impurity. For sufficiently large spin the susceptibility tends to infinity asT → 0.

Phase Transitions and Soft Librational Modes in Cubic Crystals

Landau's theory of second-order phase transitions and the theory of lattice dynamics are combined in a presentation of an approach to the understanding of the crystallographic phase transitions of complex fcc crystals. The approach is applied to the antifluorite-type crystal ${\mathrm{K}}_{2}$Re${\mathrm{Cl}}_{6}$, which exhibits four phase transitions. The results of a series of crystallographic and spectrographic experiments are described, and are interpreted in light of the developed theory. Finally, a model based on the theory is constructed and is shown to describe the spectral observations. The resultant model leads to the conclusion that two of the phase transitions are driven by a single branch of a "soft" phonon.

Parafluidity in helium near the ?-transition and the generalized time-dependent Landau theory

A phenomenological theory of parafluidity, i.e., an enhancement of fluidity due to order-parameter fluctuations, is presented for helium near the 2 transition. The generalized time-dependent Landau theory of second-order phase transitions is reviewed in general and is applied to the superfluid transition in helium as a particular example. In helium, it is found that parafluidity is manifested in the divergences of the mass diffusivity Dm oc e -~, the thermal conductivity t¢ oc e -~, the first-sound amplitude attenuation ~ ~ lel- ~, and the second-sound damping D 2 ~ (-g)-~, which are all consistent with the dynamic scaling hypothesis. Here a characteristic relaxation time Zo oc ]el- 1 is used, where e = (T - To)/Tc and T o is the transition temperature. Although there are not enough experimental data to confirm our formulas, the present approach is seen to agree in order of magnitude with available experiments. Finally, the sound absorption above a ferromagnetic transition is calculated by adding a diffusion term to the generalized time-dependent Landau equation. The result thus obtained agrees in order of magnitude with experiments in nickel.

Measurement of the magnetic susceptibility of Cu(NH4)2Br4.2H2O near its curie temperature

The zero-field susceptibility of Cu(NH4)2Br4.2H2O has been measured, using an a.c. induction method. In the critical temperature region the susceptibility can be described by the power law:χTC=A(1−TcT)−γ, with γ=1.31 ± 0.02 and A = = 1.22 ∓ 0.15. The transition temperature has been determined independently as Tc=1.773 ± 0.001 K, by heat-capacity measurements on the same sample. Since Cu(NH4)2Br4.2H2O and isomorphous copper salts have been found to be representative of the b.c.c. Heisenberg ferromagnet (S=12) - for which theory predicts γ = = 1.43 ± 0.01 - the obtained γ value is surprisingly low. This suggests that either the presence of a small anisotropy (0.7%),or scaling relations are checked tentatively. influence the exponent considerably. The scaling relations are checked tentatively. A relaxation effect has been observed that shows a narrow maximum just below Tc and vanishes upon application of a constant field of about 50 Oe. Such an effect has been predicted on the basis of the Landau theory of second-order phase transitions.

Landau's Theory of Second‐Order Phase Transitions and the Magnetic Phase Transitions in Some Antiferromagnetic Oxides

AbstractIt is argued that many magnetic phase transitions are not, according to Landau's theory, strictly second‐order phase transitions. A detailed discussion of the application of Landau's theory of second‐order phase transitions to antiferromagnetic UO2, MnO, FeO, CoO, and NiO is given in terms of Jahn‐Teller distortions and of soft phonon modes.

The Landau theory of second-order phase transitions: [$\Gamma$]3 and {$\Gamma$}2 for crystallographic point groups

The symmetrized cube and the antisymmetrized square of the irreducible representations of the point groups are reduced and tabulated; these are the products which are relevant to Landau's theory of second-order phase transitions.

On the theory of magnetic phase transitions. II: “Quasi phase transitions” the magnetic structure of vivianite

The problem of appearance of peaks in the temperature dependence of the specific heat is examined on the basis of Landau's theory of second-order phase transitions. It is pointed out that not only real phase transitions but also called- “quasi phase transitions” cause peaks in the specific heat. The magnetic structure of vivianite is investigated, because it is possible that vivianite undergoes a “quasi phase transition”.