Combined Analysis

Chapter 3

Automatic Indexing of Powder Diagrams

3.1. Principle

Automatic ab initio indexing consists of retrieving unit-cell parameters of the reciprocal lattice from the peak positions (list of d*). This method relies on resolving the following system of linear equations from the quadratic form obtained by squaring the reciprocal-lattice vector d* = ha* + kb* + ℓc*:

[3.1] 

where h, k, ℓ are Miller indices and i: 1 .. N with N the number of observed lines. Equation [3.1] can be rewritten:

[3.2] 

in which ∆QB_i is the error in Q_i that can be accepted for the system.

In this equation, a*, b* and c* are the unknown vectors of the reciprocal lattice. In this system the number of equations (N) is always smaller than the number of unknowns (i.e. six for a triclinic, four for monoclinic, etc., one for cubic lattices, and N sets of three indices h_i, k_i, ℓ_i). Hence there is no algebraic solution to this problem.

Resolving methods to this equation have been described since the earliest times [RUN 17; ITO 49; WOL 57]. The main indexing method developments are described by Shirley [SHI 78; 80] and Boultif and Louër [BOU 91].

Here we will provide only a quick review of the three main approaches:

– the Runge-Ito-de Wolff method, based on the research of zones, lies on the existence of specific ...