Eon Jean-Guillaume
Instituto de Química, Universidade Federal do Rio de Janeiro, Avenida Athos da Silveira Ramos, 149 Bloco A, Cidade Universitária, Rio de Janeiro 21941-909, Brazil.
Acta Crystallogr A. 2011 Jan;67(Pt 1):68-86. doi: 10.1107/S0108767310042832. Epub 2010 Dec 15.
Crystal-structure topologies, represented by periodic nets, are described by labelled quotient graphs (or voltage graphs). Because the edge space of a finite graph is the direct sum of its cycle and co-cycle spaces, a Euclidian representation of the derived periodic net is provided by mapping a basis of the cycle and co-cycle spaces to a set of real vectors. The mapping is consistent if every cycle of the basis is mapped on its own net voltage. The sum of all outgoing edges at every vertex may be chosen as a generating set of the co-cycle space. The embedding maps the cycle space onto the lattice L. By analogy, the concept of the co-lattice L* is defined as the image of the generators of the co-cycle space; a co-lattice vector is proportional to the distance vector between an atom and the centre of gravity of its neighbours. The pair (L, L*) forms a complete geometric descriptor of the embedding, generalizing the concept of barycentric embedding. An algebraic expression permits the direct calculation of fractional coordinates. Non-zero co-lattice vectors allow nets with collisions, displacive transitions etc. to be dealt with. The method applies to nets of any periodicity and dimension, be they crystallographic nets or not. Examples are analyzed: α-cristobalite, the seven unstable 3-periodic minimal nets etc.
由周期性网络表示的晶体结构拓扑，通过带标签的商图（或电压图）来描述。由于有限图的边空间是其圈空间和余圈空间的直和，通过将圈空间和余圈空间的一个基映射到一组实向量，可得到导出周期性网络的欧几里得表示。如果基的每个圈都映射到其自身的网络电压上，则该映射是一致的。每个顶点处所有出边的和可被选作余圈空间的一个生成集。嵌入将圈空间映射到晶格(L)上。类似地，余晶格(L^)的概念被定义为余圈空间生成元的像；一个余晶格向量与一个原子与其相邻原子重心之间的距离向量成比例。对((L, L^))形成了嵌入的一个完整几何描述符，推广了重心嵌入的概念。一个代数表达式允许直接计算分数坐标。非零余晶格向量允许处理具有碰撞、位移转变等情况的网络。该方法适用于任何周期性和维度的网络，无论它们是否为晶体学网络。分析了一些例子：α-方石英、七个不稳定的三维周期性最小网络等。
