A new method in the probabilistic theory of the structure invariants
A new method in the probabilistic theory of the structure invariants
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结构不变量概率论的一种新方法
DOI：
10.1107/s0567739475001428
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发表时间：
1975
期刊：
影响因子：
--
通讯作者：
H. Hauptman
中科院分区：
文献类型：
--
作者：
H. Hauptman
假设P1中的晶体结构是固定的，并且还指定了七个非负数R1、R2、R3、R4、R12、R23、R31。假设随机变量（向量）h、k、l、m 均匀且独立地分布在由 |Eh| 定义的倒易空间区域中= R1，|Ek| =R2，|El| = R3，|Em| = R4, (1) |Eh + k| = R12, |Ek + l| =R23, |El + h| = R31, (2) 且 h + k + l + m = 0。 (3) 然后，结构不变式 (phi = phih + phik + phil + phim，作为原始随机变量 h、k、l、m 的函数，本身就是一个随机变量，并且在给定 (1) 和 (2) 的情况下推导其条件概率分布，并与仅给定 (1) 时的条件分布进行比较。该分布导致对 cos phi 的估计得到改进此处获得的结果表明用“邻域概念”进行了概括；并且该术语又允许与插值公式进行类比。
It is assumed that a crystal structure in P1 is fixed and that the seven non-negative numbers R1, R2, R3, R4, R12, R23, R31 are also specified. The random variables (vectors) h, k, l, m are assumed to be uniformly and independently distributed in the regions of reciprocal space defined by |Eh| = R1, |Ek| =R2, |El| = R3, |Em| = R4, (1) |Eh + k| = R12, |Ek + l| =R23, |El + h| = R31, (2) and h + k + l + m = 0. (3) Then the structure invariant, (ϕ = ϕh + ϕk + ϕl + ϕm, as a function of the primitive random variables h, k, l, m, is itself a random variable, and its conditional probability distribution, given (1) and (2), is derived and compared with the conditional distribution when only (1) is given. The distribution leads to improved estimates for cos ϕ in terms of the seven magnitudes (1) and (2). The results secured here suggest a generalization which is described in terms of the 'neighborhood concept'; and a 'principle of nested neighborhoods' is formulated. This terminology permits, in turn, an analogy with interpolation formulas.