A theory of infinite spanning sets and bases is developed for the first-order flex space of an infinite bar-joint framework, together with space group symmetric versions for a crystallographic bar-joint framework ${{\mathcal {C}}}$. The existence of a crystal flex basis for ${{\mathcal {C}}}$ is shown to be closely related to the spectral analysis of the rigid unit mode (RUM) spectrum of ${{\mathcal {C}}}$ and an associated geometric flex spectrum. Additionally, infinite spanning sets and bases are computed for a range of fundamental crystallographic bar-joint frameworks, including the honeycomb (graphene) framework, the octahedron (perovskite) framework and the 2D and 3D kagome frameworks.

We describe a model for framework silicates in which the SiO4 (and AlO4) tetrahedra are treated as perfectly rigid and freely jointed. From this model we are able to identify low-energy modes of distortion of the structure, which we call Rigid unit modes. These modes can act as soft modes to allow easy distortions at phase transition. We discuss three forces that will operate at a phase transition in conjunction with the candidate soft modes to determine which of the rigid unit modes will actually precipitate a phase transition, and illustrate these ideas by detailed discussions of the phase transitions in quartz, leucite and cristobalite. The model can also be used to estimate the transition temperature, and the theory highlights an important role for the stiffness of the tetrahedra.