The nematic critical point in the molecular field approximation

2014 The classical spin system with biquadratic exchange interaction and single spin anisotropy is analysed in the molecular field approximation. The temperature dependence of important thermodynamic quantities versus anisotropy strength is studied and the crossover from ordinary critical point behaviour to nematic point behaviour is discussed. The power-law parameters describing the temperature dependence are numerically estimated in the experimentally available temperature region of 10-4 ~ |03B5 = (T Tt)/Tt| ~ 10-2 both below and above the transition point Tt. The estimated values of the gap and the correlation exponents at the nematic critical point are : 0394+ = 1.32 ± 0.005, 0394= 1.29 ± 0.01 and 03BD+ = 0.321 ± 0.001, 03BD= 0.357 ± 0.004 respectively. J. Physique LETTRES 42 (1981) L-335 L-338 Classification Physics Abstracts 64. 70E 15 JUILLET 1981, In recent years the problem of the values of critical exponents for the isotropic-nematic phase transition has Become a subject of growing interest [1, 2]. The suggestion [1] that these exponents take values characteristic of a tricritical point now seems to be confirmed experimentally [2]. In this letter a simple model exhibiting a nematic critical point is analysed in the molecular field approximation. It has been found that the estimated values of the power-law exponents for this model agree approximately with those determined experimentally. 1. The model Hamiltonian and the molecular field approximation. A classical spin system with a biquadratic exchange interaction is the simplest theoretical model exhibiting nematic ordering. A unit vector S classical spin is assigned to each nematogenic molecule showing the direction of its main axis. We assume the Hamiltonian of the N particle system is of the form consisting of three terms namely the particle-particle interaction energy, the single-particle anisotropy energy and the energy of interaction with an external field. J (ij ) is the positive particle-particle (spin-spin) interaction constant, and we assume that Jo = ~ J (ij ) j is finite. D = aJo represents the anisotropy field and a > 0 is the anisotropy parameter. H(i) is the external magnetic field at the site of the i-th particle with spin S(i). g is the coupling constant between field and the spin. The behaviour of the system will be discussed in the molecular field approximation obtained by means of the cluster expansion method of Strieb, Callen and Horwitz [3]. In order to separate out that part of the Hamiltonian suitable for the MFA (1) the Ypma, Vertogen and Koster [4] transformations are used here, leading to the Ising type Hamiltonian JC’, after rejecting small terms containing products of the spin x and y components Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019810042014033500 L-336 JOURNAL DE PHYSIQUE LETTRES