12.1.5: Indexing Patterns and Determining Cell Parameters

Last updated
Save as PDF

Page ID: 18386

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

We have labeled the peaks on the X-ray pattern in Figure 12.30 with hkl indices corresponding to the planes causing diffraction. For identifying unknown minerals, we do not need to know which hkl indices correspond to which d-values, but for other purposes, such as determining unit cell dimensions, we must.

The process of matching d-values to hkl indices is called indexing. The easiest way to index a pattern – unless a computer does it – is to compare it with one for a similar mineral already indexed. For instance, if we measure a pattern of an unknown garnet, it will have the same general appearance as the one in the figure above, so we can easily assign hkl indices to the peaks. This is quick and simple. Without a pattern to compare, the process becomes more complex and, except perhaps for cubic minerals, computer programs are desirable or necessary. In the past it was different, but today analyzing a diffraction pattern is automated. With d-values for many peaks (the more the better), computer programs use the equations in Box 12.1 and a least-squares approach to derive the six cell parameters (a, b, c, α, β, γ). The programs then assign hkl indices to each peak.