The Maier-Saupe model of the nematic phase with an orientational order parameter is extended to the smectic $A$ phase by introducing a new order parameter, the amplitude of a density wave in the direction of the nematic preferred axis. Self-consistent equations for the two order parameters are derived from an anisotropic model interaction and are solved numerically. We calculate the order parameters, the entropy, and the specific heat as a function of temperature for several values of dimensionless interaction strength $\ensuremath{\alpha}$ for the smectic $A$ phase. The transition temperatures plotted versus $\ensuremath{\alpha}$ provide a theoretical phase diagram which resembles experimental plots of transition temperature versus alkyl chain length for homologous series of compounds. The model qualitatively reproduces chemical trends in transition entropies. Experiments are suggested to measure the order parameters in the smectic $A$ phase.

Simple Molecular Model for the Smectic A Phase of Liquid Crystals

This paper reviews the experimental studies of displacive phase transitions in solids. Primary emphasis is upon inelastic light scattering and neutron scattering; related infrared reflectivity measurements, as well as x-ray and EPR analyses are also summarized. Several prototype structures are considered in detail: (1) the rocksalt IV-VI semiconductors PbTe, SnTe, and GeTe; (2) the ferroelectric perovskites exemplified by PbTi${\mathrm{O}}_{3}$ and BaTi${\mathrm{O}}_{3}$; (3) perovskites which exhibit cell-doubling transitions, such as LaAl${\mathrm{O}}_{3}$, SrTi${\mathrm{O}}_{3}$, and KMn${\mathrm{F}}_{3}$; (4) crystals having the $\ensuremath{\alpha}$-quartz structure, including Ge${\mathrm{O}}_{2}$, Si${\mathrm{O}}_{2}$, and AlP${\mathrm{O}}_{4}$; (5) the "improper ferroelectrics" ${\mathrm{Gd}}_{2}$${(\mathrm{M}\mathrm{o}{\mathrm{O}}_{4})}_{3}$ and ${\mathrm{Tb}}_{2}$${(\mathrm{M}\mathrm{o}{\mathrm{O}}_{4})}_{3}$; (6) the V-VI-VII semiconductors typified by SbSI; (7) the hydrogen-bonded ferroelectrics of the K${\mathrm{H}}_{2}$P${\mathrm{O}}_{4}$ family; (8) Jahn-Teller systems such as DyV${\mathrm{O}}_{4}$ and RbCo${\mathrm{F}}_{3}$, in which structural distortions occur as secondary effects; (9) order-disorder systems such as NaN${\mathrm{O}}_{2}$ and the ammonium halides (N${\mathrm{H}}_{4}$Cl, N${\mathrm{H}}_{4}$Br), in which no "soft mode" occurs in the spectral region ($\ensuremath{\omega}g{10}^{11}$ Hz) probed by ir, Raman, and neutron spectroscopy; (10) $\ensuremath{\beta}$-tungsten ($A\ensuremath{-}15$) structures such as ${\mathrm{V}}_{3}$Si and ${\mathrm{Nb}}_{3}$Sn, which exhibit high-temperature superconductivity. These crystal categories are used to illustrate several phenomena of current physical interest: Specifically, we discuss harmonic and anharmonic mode coupling; "critical exponents" $\ensuremath{\beta}$ differing from one-half in the temperature dependences of the order parameter $\ensuremath{\phi}(T)={\ensuremath{\phi}}_{0}{(T\ensuremath{-}{T}_{0})}^{\ensuremath{\beta}}$ and of the soft-mode frequency $\ensuremath{\omega}(T)={\ensuremath{\omega}}_{0}{(T\ensuremath{-}{T}_{0})}^{\ensuremath{\beta}}$; and the recently discovered "central" modes centered at zero frequency, which grow in intensity as the transition temperature ${T}_{0}$ is approached from above or below. The review covers the period 1940-1972. A few 1973 works are mentioned for which the author had preprints in 1972 or very early 1973. This review is in no sense a comprehensive survey of ferroelectricity. Readers are referred to the following earlier reviews on that subject: Silverman (1966, 1969), Cochran and Cowley (1967), Blinc (1968), Murzin et al. (1968), Nettleton (1970), and Blinc and Zaks (1972).

Soft-mode spectroscopy: Experimental studies of structural phase transitions

The co-operative Jahn-Teller effect is a phase transition which is driven by the interaction between localized orbital electronic states and the crystal lattice. The possible origins, symmetries and properties of the electron-lattice interactions and how they lead to possible hamiltonians for the coupled system are discussed. The relation of these interactions with quadrupolar interactions and magnetostriction is included. The methods of solution of the hamiltonians lead to an understanding of the electronic states, the phonon spectrum and the mixed normal modes. The wide variety of experimental techniques used on this problem are reviewed in detail and the results are compared with theoretical expressions whenever possible. The application of external stress and magnetic field is of particular significance in the case of the rare earth compounds, because they can product effects which are larger than the low-temperature spontaneous effects.

http://iopscience.iop.org/article/10.1088/0034-4885/38/1/001/pdf

Co-operative Jahn-Teller effects

Thermodynamic and static properties of liquid crystals

An outline is given of the effects of the various processes occurring at a structural phase transformation on the quantities measured in an ultrasonic experiment: sound velocity and attenuation. Various theoretical approaches are discussed as well as experimental results.

The study of structural phase transitions by means of ultrasonic experiments

Dynamical aspects of the ferroelectric phase transition in KH 2 PO 4 -type crystals are elucidated on the basis of the coupling between the proton tunneling mode and the optical mode vibration of the [K-PO 4 ] complexes along the c-axis. Two coupled modes are obtained within the framework of a linear theory. The mode in which both systems oscillate in phase is shown to be responsible for the ferroelectric phase transition. The isotope effect in the Curie point is explained qualitatively and the Curie constant is estimated.

Dynamical Theory of the Phase Transition in KH2PO4-Type Ferroelectric Crystals

A theory of translational and orientational melting with application to liquid crystals is presented in a manner analogous to that of Kirkwood and Monroe by introducing both translational and orientational order parameters.

Theory of Translational and Orientational Melting with Application of Liquid Crystals. I

The dynamical aspects of non-polar classical liquids are investigated by extending the Kirkwood's static treatment of the fusion to the dynamical one in a manner quite analogous to the Landau-Vlasov's formulation of the plasma oscillation. The phonon-like collective modes are obtained in the collisionless regime of which velocity is quite different from that of the ordinary sound waves in classical liquids. The dynamical behaviour of liquids near the freezing point is also considered and a non-propagating mode with a critical wave number k c somewhat similar to the entropy fluctuation mode is obtained.

Dynamical Aspects of Classical Liquids

The dielectric functions due to π-electrons in graphite are calculated on the basis of both Coulson-Taylor's two dimensional model and Wallace's three dimensional model. The dielectric function σ(ω) in the frequency region including the peak at 4.8 eV is fairly well reproduced. The calculated overall behavior of σ(ω) is discussed and compared with the observed one.

Dielectric Function of Graphite Due to Interband Transitions of π-Electrons

The collective proton tunneling mode is derived in hydrogen bonded biopolymers, in particular, in α-helical polypeptides. The coupling of this mode with an acoustical vibration of residues is investigated. Some remarks on the signal transmission in biopolymers are given in terms of these modes.

Kenji K. Kobayashi

Papers

Dynamical Theory of the Phase Transition in KH2PO4-Type Ferroelectric Crystals

Theory of Translational and Orientational Melting with Application of Liquid Crystals. I

Dynamical Aspects of Classical Liquids

Dielectric Function of Graphite Due to Interband Transitions of π-Electrons

Correlated Processes in Hydrogen-Bonded Biopolymers