Mallard's law recast as a Diophantine system: fast and complete enumeration of possible twin laws by [reticular] [pseudo] merohedry
Mallard's law recast as a Diophantine system: fast and complete enumeration of possible twin laws by [reticular] [pseudo] merohedry
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DOI：
10.1107/s0021889801021574
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发表时间：
2002-04
影响因子：
6.1
通讯作者：
Y. Page
中科院分区：
文献类型：
--
作者：
Y. Page
马拉德观察定律指出，已知的网状伪异体孪晶在晶格行和晶格平面之间具有低孪晶指数m和低倾角δ。晶体在孪晶中是通过晶格平面上的精确反射，沿垂直于晶格平面的精确旋转π，或沿晶格行精确旋转2kπ/q，其中q = 2,3,4或6，并且1≤k < q。在已知的情况下，m高达5或6度，δ高达5或6度。Mallard的逆问题是关于找到所有行和面的成对指标，导致双指标不大于m，倾斜度不大于δ。马拉德定律被改写为由等式h·u = n和不等式|h × u| < n tan δ组成的丢番图对。如果使用原始参考系，则整数n为m、2m或3m。给出了该系统在给定n， h， δ和晶格数据下的直接通解。这个简单的解决方案涉及到u的一个很小的二维网格，大大减少了需要考虑的排列数量。如果原始参照系是buerger约简的，则解h的指标模不能超过3m，从而建立了一种生成完全解的简单方法。一个叫做“斜向”的项目就是基于这样的原则设计的。可以在http://ylp.icpet.nrc.ca/oblique/上免费执行一个实现。斜向也是Toth信息系统材料工具包（http://www.tothcanada.com/）中的一个交互式工具，该工具包是一个开发框架，目前在CRYSTMET和ICSD晶体结构数据上运行。以石英晶胞数据为例，mmax = 3, δmax = 6°。
Mallard's law of observation states that known twins by reticular pseudomerohedry have low twin index m and low obliquity δ between a lattice row and a lattice plane. Crystals in the twin are related by exact reflection in the lattice plane, exact rotation by π about the perpendicular to the lattice plane, or by exact rotation about the lattice row by 2kπ/q, with q = 2, 3, 4 or 6, and 1 ≤ k < q. In known cases, m is up to five or six, and δ is up to five or six degrees. Mallard's converse problem is then about finding all pairs of indices for rows and planes leading to twin indices not larger than m and to obliquities that are at most δ. Mallard's law is recast as the Diophantine pair constituted by the equality h · u = n and the inequality |h × u| < n tan δ. If a primitive reference system is used, the integer n is either m, 2m or 3m. A direct general solution of this system for u given n, h, δ and lattice data is detailed. That straightforward solution involves a tiny two-dimensional grid for u, considerably reducing the number of permutations to be considered. If the primitive reference system is Buerger-reduced, then moduli for indices of solutions h cannot exceed 3m, thus establishing a simple way to produce complete solutions. A program called OBLIQUE was designed on such principles. An implementation is available for free execution at http://ylp.icpet.nrc.ca/oblique/. OBLIQUE is also an interactive tool in Toth Information Systems' Materials Toolkit (http://www.tothcanada.com/), an exploitation framework currently operating on CRYSTMET and ICSD crystal structure data. The example of quartz cell data is described with mmax = 3 and δmax = 6°.