XRD Peak Indexing Calculator
Assign hkl indices to a cubic powder pattern, refine the lattice parameter from every peak, and see which reflections are missing.
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What it does
Indexing a powder pattern means deciding which set of crystal planes produced each peak. For a cubic material it reduces to one observation: the squares of the sines of the Bragg angles are proportional to whole numbers, and the constant of proportionality is identical for every peak. Get that constant right and the lattice parameter falls out of every single line, which is also a self-check, because all those independently derived values have to agree.
How it works
For a cubic lattice the interplanar spacing is d = a / √(h² + k² + l²). Substituting that into the Bragg law gives sin²θ proportional to N = h² + k² + l², scaled by (λ / 2a)². Not every N is allowed: a body-centred lattice needs h + k + l to be even, a face-centred one needs all three indices odd or all three even, and the diamond structure adds a further condition that an all-even set must also sum to a multiple of four. That last rule is why silicon shows no 200 or 222 reflection at all.
Worked example
Aluminium metal with copper Kα radiation gives seven peaks between 38.47 and 111.95 degrees. The calculator assigns them in order to 111, 200, 220, 311, 222, 400 and 331, refines a to 0.4050 nm, and reports a spread of 0.045 percent, which is the residual left by rounding the input angles to two decimals. The expected-reflection table flags 420 at 116.56 degrees as missing, simply because the peak list stops at 111.95.
When to use it
Use it to identify an unknown cubic phase, to refine a lattice parameter from a short peak list, or to test whether a pattern really belongs to the structure a paper claims. The other-lattice table is the one to keep in mind, because peak positions alone rarely settle an ambiguity. Pair this with intensities and with what the sample is actually supposed to contain.
FAQ
- What does it mean if the lattice parameter differs from peak to peak?
- That the indexing is probably wrong, or that the peaks do not all belong to the same phase. Every reflection independently determines a from a = λ √N / (2 sin θ). For a correct assignment those values must agree, so the spread of that column is the honest error bar on the answer. Under about 0.2 percent is normal for a well aligned diffractometer and peak positions rounded to two decimals; a spread of several percent means at least one peak has been given the wrong indices.
- Why is the first peak 110 for some metals and 111 for others?
- Selection rules depend on the lattice. In a body-centred cubic crystal h + k + l must be even, so 100 is forbidden and the first allowed reflection is 110. In a face-centred cubic crystal the indices must be all odd or all even, so both 100 and 110 are forbidden and the first allowed reflection is 111. In a diamond-type structure such as silicon an extra condition removes 200 and 222, leaving 111 followed by 220.
- My peaks fit a face-centred lattice, but they also fit a body-centred one. Which is right?
- Both fits are self-consistent, which is exactly the ambiguity to watch for: a face-centred pattern can be indexed as a body-centred lattice with a parameter 1.41 times larger. Peak positions alone cannot settle it, because the reflections that would distinguish the two either fall outside the measured range or overlap. The deciding evidence is intensity, since the strongest line is 111 for face-centred and 110 for body-centred, together with the chemistry: aluminium is face-centred and iron is body-centred.
- A reflection that should be there is missing. What does that mean?
- Usually it does not mean the structure is wrong. Preferred orientation from sample preparation can suppress a line, a weak reflection can disappear into the background, and two reflections at similar angles can merge into a single peak. The useful question is whether lines are missing systematically in a way that suggests a different lattice, or randomly in a way that suggests sample preparation. Lines appearing at angles where the selection rules forbid them are a genuine problem worth investigating.