The properties and computation of T surfaces

The geometry of a tD minimal surface depends on the ratio cla of tetragonal axes, and can be fully describd in terms of a single free parameter. We offer a choice of three such parameters, all related to surface geometry, and derive analytical expressions for their relationships to the axes ratio and the normalization factor. The latter is crucial for the matching of specific surfaces to real structures. Parametric equations for normalized tD surfaces make it possible, for the first time, to find the surface corresponding to any given value of the cla ratio, and to compare it with actual structural data. Straightforward physical applications of tD surfaces are therefore possible. We discuss the geometric consequences of this result and show that the z coordinate of any tD surface can be approximated using elementary functions. A rational approximation for the relationship between the free parameter and the cla ratio is also found. We list exact coordinates of tD surfaces corresponding to several prescribed values of the cla ratio. Introduction. Approximately 40 triply periodic embedded minimal surfaces (TPEMS) derived by various methods (now mainly by group theory), have been described [1-3]. Over the last 20 years TPEMS have been applied in many areas of the physical and biological sciences [4]. They are a useful crystallographic concept for the description of condensed matter, as advocated by Scriven [5], Mackay [6-7], Hyde and Andersson [8], Mackay and Klinowski [9] and Sadoc and Charvolin [10]. In principle, the translation symmetry of TPEMS makes it possible to match them to actual structures, I,e, to compare surface coordinates with spatial pattems of atoms, and to establish relationships between surface properties, such as curvature and the volume-to-surface ratio, and relationships between structural properties, especially those provided by X-ray diffraction and solid-state NMR. Modelling of structures also suggests a method of quantifying structural changes by relating them to transformations (such as the Bonnet and Goursat transformations) of minimal surfaces. Finally, studies of interpenetrating crystalline structures with large unit cells and complicated networks of cages and channels exposed the limitations of classical crystallography, showing the need for more appropriate concepts. Since TPEMS have labyrinthine structures, they are likely to lead to simple structural descriptions of such systems. A rigorous classification of all known and hypothetical TPEMS will be of great importance to materials science in general. 2192 JOURNAL DE PHYSIQUE I N° 12 Unfortunately, mathematical arguments linking TPEMS and physical structures are often far from rigorous. The main obstacle to a wider application of minimal surfaces in experimental science is that most of them have been described empirically, without the precise mathematical specification. It is therefore essential to quantify all known TPEMS, to establish which geometric properties are related to the properties of physical systems, and to find a reliable method of computing their coordinates. In particular, it is imperative that approximate computation of TPEMS to a prescribed degree of accuracy be systematically investigated. A TPEMS is described by giving its parametric representation or the Enneper-Weierstrass representation, or by specifying the boundary conditions of a partial differential equation [I1l3]. In general, any surface in three-dimensional space is completely described by the parametric representation of the form (X, y, Z) " IX (U, V), Y(U, V), Z(U, V)j where the coordinates are functions of two parameters, u and v. However, in most cases analytical expressions for the coordinates of TPEMS are unknown. Sometimes a parametric representation of such surfaces cannot be expressed in terms of elementary functions alone. For example, parametric representation of surfaces of the T and CLP family involves special functions [14]. Any minimal surface is described by the following three complex integrals iw x = Re R(r) (I r~) dr