On the phase transition in s y m-triazine-mean field theory

Abstract

A Landau mean field description of the nearly second order phase transition in sym-triazine crystals at ∼200 K is presented. A model Hamiltonian is generated which consists of the appropriate symmetry elastic constant terms, molecular rotational energy, and rotation–translation coupling terms (to second order in both strains and rotations). Due to the symmetry of the crystal in the high (R3̄c) and low (C2/c) temperature phases, third order terms in the rotational order parameter are nonvanishing; the transition is thereby a first order one (although only weakly so). This Hamiltonian is then converted to a free energy by addition of an entropy term calculated for an orientation distribution (about the z axis) based on pocket state functions. The Landau mean field model is developed by choosing a set of order parameters Ry (molecular rotation about the y axis) and strains e5 and (e1-e2). The free energy expression is used to calculate relations between order parameters by setting ∂F/∂Ry=∂F/e5=∂F/∂e7=0. Coupling terms including bilinear products of eρ’s and Ry are employed in this development. Renormalized temperature dependent elastic constants are derived. e5(T) is solved for and found to be in good agreement with observed temperature dependences. Librational frequencies are determined from (∂2H/∂RiRj)ep=Iω2iδij. It is found that in the low temperature phase Δω=‖ωy−ωx∝α e5 in lowest order. Observed power laws for frequencies, splittings and strains with respect to ε≡(T−Tc/T) are discussed in light of these new results. The role of third order terms in (Rx, Ry) is considered and found to be an important factor in apparent deviation from mean field exponents.