3-Periodic Minimal Surfaces Derivable from Laves Nets

Abstract

3-periodic minimal surfaces made up by catenoidlike surface patches may be related to Laves nets. Starting from these nets four new types of such surfaces with selfintersection exclusively along straight lines could be derived.They subdivide R 3 into 2-periodic labyrinths and 1-periodic tubes.Dreifach-periodische Minimalfla È chen, die sich aus Lavesnetzen ableiten lassen Inhaltsu È bersicht. Es besteht eine Beziehung zwischen dreifach-periodischen Minimalfla È chen, die aus katenoidartigen Fla È chenstu È cken aufgebaut sind, und den Laves-Netzen. Ausgehend von diesen Netzen konnten vier neue Typen solcher Fla È chen abgeleitet werden, welche sich nur entlang von Geraden selbst u È berschneiden. Sie unterteilen den R 3 in zweifach-periodische Labyrinthe und einfach-periodische Ro È hren.

Li2PbF6 and SrPbF6 Revisited

“…The Pb-F distances of 2.042(4) Å (four times to F2, forming a square) and 2.060(4) (two times to F1, forming the tips) are in good agreement with reported data of [ [2][3][4]32] Furthermore, the Pb-F distances are now significantly smaller than in the previously reported structure model (Pb-F: 2.24 Å). [2][3][4]32] The previously reported structure model contains trans-vertex-bridged [PbF 6 ] 2octahedra with the Niggli formula 1 ϱ [PbF 4/1 F 2/2 ] with all these octahedra eclipsed. An additional F atom (a "naked" one in this case) had therefore to be present to obtain the correct composition.…”

“…This obsolete structure model by Hoppe and co-workers is shown in Figure S3 In summary, the atom distances of the structure model presented here are in much better agreement with the literature and all reflections in the powder X-ray pattern were indexed. [2][3][4]32] Comparing the [PbF 6 ] 2anions of Li 2 PbF 6 and SrPbF 6 , the Pb-F distance in Li 2 PbF 6 is observed with 2.057(5) Å. So, it is of intermediate length compared to SrPbF 6 with 2.042(4) and 2.060(4) Å.…”

“…Overall they agree within the 3σ criterion as well as to previously reported Pb-F distances of other metal(II) hexafluoridoplumbates(IV). [2][3][4]32]…”

“…We also present the corrected space group and crystal structure for SrPbF 6 Sr, Ba, Mn, Fe, Co, Ni, Pd, Cu, Ag, Zn, Cd, Hg) were reported. [28][29][30][31][32][33][34] Many of these crystallize in the LiSbF 6 structure type.…”

“…Some atom distances were also not in agreement with those of similar compounds from the literature. [2][3][4]32] Also, we observed some additional reflections in the X-ray powder patterns that could not be indexed in space group P4 2 /mmc, however clearly belong to SrPbF 6 . We therefore set out to reinvestigate the crystal structure of SrPbF 6 .…”