The interaction employed in the Maier-Saupe theory of the nematic state is generalized in a manner consistent with the asymmetry of the molecules that exhibit such a phase. In the molecular field approximation such an interaction then yields a first-order transition from the isotropic to a uniaxial state followed, at lower temperatures, by a secondorder transition to a biaxial state.Organic molecules that exhibit a nematic liquid crystalline state characteristically have molecules which are elongated and flat. 1 Insofar as most observed properties of these liquids are concerned, the nematic state has uniaxial symmetry, that is, while the long axes of the molecules are aligned parallel to each other, the orientational distribution function is independent of the rotational angle of the molecules about their long axes. There does exist one observation of optical activity 2 which may show this to be an inadequate picture.It has recently been shown that there is a smectic phase which is biaxial 3 and one might inquire as to whether there can exist a biaxial nematic phase. The existing molecular field theory of nematic liquids 4 starts from an interaction of the form P 2 (cos9 f J) between two molecules, derived from the lowest order London interaction, where 9 f j is the angle between the long molecular axes. With such an interaction the molecules are, in effect, entities with axial symmetry. The theory predicts a first-order transition from an isotropic to a uniaxial nematic state. We shall show that the simplest generalization of this interaction leads to just such a first-order transition followed, at lower temperature, by a second-order transition to a biaxial state. Such a second transition might account for an anomaly observed in the specific heat of ^-azoxyanisole (PAA) in the nematic state. 5 The energy of interaction between two asymmetrical molecules can be expanded in a series of terms which are functions of the orientations of the molecules and of the line joining their centers of gravity. London forces are probably dominant, but other forces may also play an important role. We can proceed without prejudice on this point because it is only the angular form of the interaction which will be relevant. An effective orientational interaction is obtained if we average over the relative positions of an interacting pair of fixed orientation. Our use of such an averaging process presupposes a two-particle distribution function which is factorable into a spherically symmetric radial and an orientational part, which while not strictly valid, as is transparently the case in the smectic type of order, should give qualitatively reasonable results for nematic and cholesteric liquids.Such a procedure yields an effective pair interaction which is a sum (over I) of contributions of the form m m' m"where the D m , m (I) are the elements of the transformation matrices of the spherical harmonics, Y Im , under rotation, and &/,&/, Y/ are the Euler angles of axes attached to molecule i relative to a fixed set of axes. 6 The Q...

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When the method of Luttinger and Tisza for finding the classical ground state of a system of spins with Heisenberg interactions is applicable, it yields a configuration with the same periodicity as the ordered state existing just below the transition temperature. When the method of Luttinger and Tisza fails to yield the classical ground state then there can occur, with decreasing temperature, either additional transitions or a gradual change in the periodicity of the stable configuration. An example of the latter is the thermal change in pitch of a helical configuration. Such a situation can be described in the internal field approximation when the consistency equations admit as solutions helical states with a continuous range of pitches. The free energy of the stable state with a temperature-dependent pitch can then be obtained as the envelope of the free-energy curves belonging to the family of helical solutions. A one-dimensional diatomic chain whose ground state can be found by a generalization of the method of Luttinger and Tisza illustrates this possibility. It is also pointed out that anisotropic exchange interaction between nearest neighbors can give rise to helically ordered configurations.