Classification and Effects of Symmetry of Mechanical Structure and its Application in Design

Symmetry widely exists in natural objects and man-made objects. Mechanical structures, as man-made objects, have the property of symmetry without exception. The existence of symmetry affects the function and performance of mechanical products. Therefore, on the basis of analyzing a large number of examples and referring to the Schoenflies symbol of crystal, the symmetry of mechanical structures is divided into point group symmetry and space group symmetry, and these two types are further subdivided according to the types and spatial positions of the symmetry elements. Then, the general effects of symmetry are summarized according to symmetry types and functions, and several symmetry rules for design are further refined. Finally, after defining the requirements of speed change and technology background, a multispeed device for bicycle shaft drive is proposed by applying symmetry knowledge comprehensively.

Synthesis, Crystal Structures and Thermal Properties of Ammine Barium Borohydrides

Ammine metal borohydrides show large compositional and structural diversity, and have been proposed as candidates for solid-state ammonia and hydrogen storage as well as fast cationic conductors. Here, we report the synthesis method of ammine barium borohydrides, Ba(BH4)2·xNH3 (x = 1, 2). The two new compounds were investigated with time-resolved temperature-varied in situ synchrotron radiation powder X-ray diffraction, thermal analysis, infrared spectroscopy and photographic analysis. The compound Ba(BH4)2·2NH3 crystallizes in an orthorhombic unit cell with space group symmetry Pnc2, and is isostructural to Sr(BH4)2·2NH3, forming octahedral [Ba(NH3)2(BH4)4] complexes, which are connected into a two-dimensional layered structure, where the layers are interconnected by dihydrogen bonds, N–Hδ+⋯−δH–B. A new structure type is observed for Ba(BH4)2·NH3, which crystallizes in an orthorhombic unit cell with space group symmetry P212121, forming a three-dimensional framework structure of [Ba(NH3)(BH4)6] complexes. The structure is built from distorted hexagonal chains, where NH3 groups form dihydrogen bonds to the nearby BH4−-groups within the chain. Ba(BH4)2·2NH3 is unstable at room temperature and releases NH3 in two subsequent endothermic reactions with maxima at 49 and 117 °C, eventually reforming Ba(BH4)2. We demonstrate that the thermal stability and composition of the gas release for the ammine alkaline earth metal borohydrides can be correlated to the charge density of the metal cation, but are also influenced by other effects.

Multiplicity-weighted Euler's formula for symmetrically arranged space-filling polyhedra

The famous Euler's rule for three-dimensional polyhedra, F − E + V = 2 (F, E and V are the numbers of faces, edges and vertices, respectively), when extended to many tested cases of space-filling polyhedra such as the asymmetric unit (ASU), takes the form Fn − En + Vn = 1, where Fn, En and Vn enumerate the corresponding elements, normalized by their multiplicity, i.e. by the number of times they are repeated by the space-group symmetry. This modified formula holds for the ASUs of all 230 space groups and 17 two-dimensional planar groups as specified in the International Tables for Crystallography, and for a number of tested Dirichlet domains, suggesting that it may have a general character. The modification of the formula stems from the fact that in a symmetrical space-filling arrangement the polyhedra (such as the ASU) have incomplete bounding elements (faces, edges, vertices), since they are shared (in various degrees) with the space-filling neighbors.

Crystal Structure of Kristiansenite from Szklarska Poręba, Southwestern Poland

Kristiansenite, ideally Ca2ScSn(Si2O7)(Si2O6OH), a rare late-stage hydrothermal Sc-bearing sorosilicate mineral, was found in a gadolinite-fergusonite-type pegmatite of the MI-REE subclass related to the Karkonosze granite, exposed in a quarry at Szklarska Poręba, Lower Silesia, Poland. Kristiansenite occurs in an association with andradite, epidote, allanite-(Ce), titanite, fersmite, scheelite, Sc-bearing columbite-(Fe), a YNbO4 mineral as fergusonite-(Y) or fergusonite-(Y)-beta, silesiaite and wolframite. Single-crystal study of the mineral (R1 of 4.96%), with composition Ca2.00(Sn0.97Sc0.69Fe3+0.17Mn0.05Ti0.04Zr0.03Nb0.02Al0.02Ta0.01)Σ2(Si2O7)[(Si1.98Al0.02)Σ2O6.03(OH)0.97], corroborates its triclinic structure with space group-symmetry C1, Z = 2, and unit-cell parameters a = 10.0304(5), b = 8.4056(4), c = 13.3228(6) Å, α = 90.001(3), β = 109.105(3), γ = 89.997(3)° and V = 1061.40(9) Å3. In the structure of the mineral, the Ca and Si sites are dominantly occupied with Ca and Si, whereas the M1–M4 sites are disordered. The M3 and M4 sites are occupied dominantly by Sn and subordinately Sc, whereas the M1 and M2 sites are occupied dominantly by Sc and subordinately by remaining occupants, including Sn.

Validation of missed space-group symmetry in X-ray powder diffraction structures with dispersion-corrected density functional theory

More than 600 molecular crystal structures with correct, incorrect and uncertain space-group symmetry were energy-minimized with dispersion-corrected density functional theory (DFT-D, PBE-D3). For the purpose of determining the correct space-group symmetry the required tolerance on the atomic coordinates of all non-H atoms is established to be 0.2 Å. For 98.5% of 200 molecular crystal structures published with missed symmetry, the correct space group is identified; there are no false positives. Very small, very symmetrical molecules can end up in artificially high space groups upon energy minimization, although this is easily detected through visual inspection. If the space group of a crystal structure determined from powder diffraction data is ambiguous, energy minimization with DFT-D provides a fast and reliable method to select the correct space group.

Crystal structure of Cr-bearing Mg3BeAl8O16, a new polytype of magnesiotaaffeite-2N′2S

The crystal structure of a new polytype of magnesiotaaffeite-2N′2S, ideally Mg3BeAl8O16(trimagnesium beryllium octaaluminium hexadecaoxide), is described in space-group symmetryP-3m1. It has been identified in a fragment of a mineral sample from Burma (Myanmar). The new polytype is composed of two Mg2Al4O8(S)- and two BeMgAl4O8(N′)-modules in a stacking sequenceN′SSN′′ which differs from theN′SN′S-stacking sequence of the known magnesiotaaffeite-2N′2Spolytype. The crystal structure can be derived from a close-packed arrangement of O atoms and is discussed with regard to its polytypism and its Cr3+chromophore content.