Parametrization of triply periodic minimal surfaces. III. General algorithm and specific examples for the irregular class

Abstract

JORDI RIUS 409 result is obtained for N~arge = 220 and Nsmal I = 200 with 32.5% of the refined sets possessing mean phase errors less than 35°: 7.5, 10.0 and 15.0% in the ranges 0-25, 25-30 and 30-35 ° , respectively.The TVAL structure was selected to show the capability of the tangent formula (12) to refine random phases in the case of a relatively large structure (156 nonhydrogen atoms in the unit cell). The best result was obtained for Nlarg e = 300 and Nsmal ! = 300 (EH)= 1.24). From a total of 200 sets, 12 (6%) show a mean phase error between 30 and 35 ° . Concluding remarksThe viability of solving crystal structures from the direct interpretation of the nonorigin Patterson peaks as a function of the phases has been demonstrated. This result, however, should not be surprising since these peaks contain all the information regarding the atomic arrangement in the structure and, in addition, the atomicity information contained in the removed Patterson origin peak has already been considered in the derivation of the Fourier coefficients GH(q0). AbstractThe construction algorithm of the first two papers of this series [Fogden & Hyde (1992). Acta Cryst. A48, 442-451,575-591] is extended to a general treatment of triply periodic minimal surfaces (containing as a special case the 'regular' class analysed previously). A detailed outline of the parametrization procedure for an arbitrary 'irregular' class surface is provided and verified via a systematic rederivation of the C(P) surface described by Neovius [Bestimmung Zweier Speciellen Periodische Minimalfliichen (1883), Helsinki: Frenckel]. The method is further employed in parametrizing various empirically generated surfaces.

Uniqueness of minimal surfaces, Jacobi fields,and flat structures

“…This recovers, up to associate families, the triply periodic minimal surface Σ a>0 in the rPD family [10,11,27] with the Weierstrass data…”

Self-intersecting three-periodic minimal surfaces forming two-periodic (flat) labyrinths

“…e.g. Anderson, Davis, Scriven & Nitsche, 1990;Cvijovic  & Klinowski, 1992a±c, 1993Fischer & Koch, 1987, 1989a±c, 1990, 1992Fogden 1993Fogden , 1994Fogden & Hyde, 1992a, b;Hyde, 1989;Karcher, 1989;Karcher & Polthier, 1996;Koch & Fischer, 1988, b, 1990, 1993aLidin, 1989;Lidin & Hyde, 1987;Mackay, 1985;Mackay & Klinowski, 1986;Neovius, 1883;Schoen, 1970;Schwarz, 1890Schwarz, , 1894Stessmann, 1934). In comparison, only little is known on self-intersecting minimal surfaces (cf.…”

“…e.g. Fischer & Koch, 1987Fogden 1993Fogden , 1994Fogden & Hyde, 1992b;Karcher, 1989;Karcher & Polthier, 1996;Koch & Fischer, 1988, 1993aNeovius, 1883;Schoen, 1970;Schwarz, 1890Schwarz, , 1894Stessmann, 1934). The resulting infinite surface may be either embedded or self-intersecting.…”

Minimal Twin Surfaces

“…The rPD-family is a 1-parameter family of TPMS with rhombohedral symmetries, generalizing Schwarz' D and P surfaces. A Weierstrass representation for the translational unit of an rPD surface is given in [Fogden, 1993] and [Fogden and Hyde, 1999]…”