TABLE OF CONTENTS In Part I we discussed the structures of crystals, quasicrystals, and liquid crystals. They are all the result of broken spatial translational and orientational symmetries. In this chapter we would like to go beyond geometry, in order to clarity the physical reason for their formation. Landau theory may be the first step in this direction, but more microscopic theories are needed for further elucidation.
Starting from a homogeneous and isotropic liquid and lowering the temperature gradually, mass or compositional density waves will arise in the liquid. Below a certain temperature, some density wave modes are locked in and an ordered solid is formed. One would like to explain the structure from first-principle calculations, taking into account the actual electronic properties of the constituent atoms. However, such a calculation of crystal stability is very elaborate, and some understanding of the relative stability of solids and liquids can be obtained by using postulated periodic density waves within the framework of the phenomenological Landau theory of phase transitions.
Consider a two-or three-dimensional liquid which has full translational and rotational symmetries corresponding to the Euclidean group. The liquid phase may condense into the solid phase, and we would ask what are the possible ordered structures that arise at low temperatures. To simplify this problem, we may ignore the difference in mean density between liquid and solid, so the Gibbs free energy may be replaced by the Helmholtz free energy F.
In the...
