The intrinsic curvature of solids

I igiire 2.h (right) Tlie C sCl structure with its zero potential surfaces. The velkuv C ions lie at the centre of the body-centred cubic unit ceil. 

These surfaces of zero potential formed in different salts are very close to periodic minimal surfaces [9], whose mean curvature, defined as the arithmetic mean of the main curvatures, is everywhere zero (see Chapter 1) . On these minimal surfaces the Gaussian curvature is everywhere negative or 

The reason that this structure forms is as follows units of octahedral W3C crystallise on one side of an interface, that separates solid W3C from Fe. The Fe atoms are drawn into the interstices of the W3C elements, with a driving force dependent on the Gaussian curvature of the interface. This is one way of looking at intrinsic curvature in solids. As will become apparent, there are other ways that link the interactions between atoms, molecules and larger aggregates to local curvature. 

Many more examples of interpenetration in inorganic chemistry lead to a recognition of the ubiquity of hyperbolic surfaces of infinite genus -exemplified by three-periodic minimal surfaces - that demands consideration. In the giant structure of Cu4Cd3 the Cu atoms are separated from the Cd atoms by a surface that resembles a minimal surface. In diamond, cubic ice and cristobalite, all the atoms are located on one side of the surface and the space on the other side is empty. If ice is subjected to very high pressure, the same structure appears on both sides of a minimal surface (double ice or ice IX), with almost double the density of ordinary ice (Fig. 2.8). Similarly, diamond is expected to transform to a double-diamond structure with metallic properties at sufficiently high pressure. 

The concept of intrinsic curvature is particularly useful when dealing with the intricate and beautiful structures formed by zeolites. Zeolites are commonly used as technical materials. They exhibit many special properties, due to their extraordinary ability to selectively absorb a large range of molecules. The forces that act are weak, physical not chemical, and we characterise them by invoking the idea of intrinsic curvature. 

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