What a powder pattern loses

A single crystal held in a beam scatters into a pattern of spots, each in its own direction, each identified by its indices. Grind the same crystal into a powder and every orientation is present at once, so every reflection appears in every direction — and what is measured is a set of rings, or, plotted against angle, a set of peaks on a line.

Two dimensions of information collapse into one. Most of what is lost was redundant, some of it was not, and the part that was not is the subject of this page.

What survives the collapse #

A powder pattern retains the spacings and the total intensity at each spacing, and discards everything about direction.

That is enough for a great deal. The list of spacings is a fingerprint of the phase present, and matching a measured list against a database identifies a compound in seconds — which is why powder diffraction is the standard tool for phase identification in mineralogy, in metallurgy, in cement chemistry and in pharmaceutical manufacture. It is also enough to refine a known structure, since the intensities are still a function of the atom positions.

What is lost is the ability to say which reflection contributed what. A peak at a given spacing holds every reflection with that spacing, and the measurement gives their sum.

Multiplicity, which is mostly symmetry #

Most of the reflections sharing a peak are related by symmetry, and that part of the collapse costs nothing.

On a square lattice the reflections $(3\,1)$, $(1\,3)$, $(-3\,1)$ and their relatives all have the same spacing, because the lattice’s point group carries each onto the others. They have the same amplitude as well, necessarily, since a symmetry operation of the structure maps one to another. So a single crystal measuring all eight is measuring the same number eight times.

The count of reflections in such a family is its multiplicity, and it is counted here rather than looked up: the figure bins by spacing, counts what lands in each bin, and prints the count. A structure’s contribution to a powder peak is its structure factor times the multiplicity, and getting the multiplicity wrong is a straightforward way to get a refined structure wrong.

The loss from symmetry-related overlap is therefore nil. Nothing distinguishable was thrown away.

Accidental overlap, which is not symmetry #

The other kind of overlap is the interesting one, and the square lattice gives the sharpest example available.

The reflections $(5\,0)$ and $(4\,3)$ both satisfy $h^2 + k^2 = 25$, so their reciprocal vectors have exactly the same length and they land in exactly the same place in a powder pattern. They are not related by any operation of the square lattice’s point group — the eight signed axis swaps carry $(5\,0)$ to $(0\,5)$, $(-5\,0)$ and so on, and never to $(4\,3)$ — so they are genuinely independent reflections with genuinely independent amplitudes.

The coincidence is arithmetic. It is the fact that $25 = 5^2 = 4^2 + 3^2$, which is the smallest Pythagorean triple, and it has no symmetry content whatever.

The figure finds this case rather than being told about it: it computes the orbit of each reflection under the point group, groups the reflections in each peak by orbit, and reports the first peak whose members span more than one. The assertion is that such a peak exists in the window searched, which would fail if the orbit computation were wrong or if the binning tolerance were too tight to detect a genuine coincidence.

No measurement of a powder pattern can separate those two reflections. They arrive at the same angle with no distinguishing feature, and their sum is what is recorded. A single crystal measures them independently; a powder cannot, ever, at any resolution.

Indexing, which is the first thing that has to work #

Before any structure can be considered, the peaks have to be assigned indices, and doing that from spacings alone is a genuinely hard problem with a long history of failure.

For a single crystal it is easy: the spots have directions, the directions determine the reciprocal lattice, and the indices follow. For a powder there are only lengths, and the task is to find a lattice whose reciprocal vector lengths reproduce the measured list. That is a search, and it has three characteristic ways of going wrong.

A sublattice. A cell twice as large as the true one indexes every observed peak and predicts many that are not there. The extra predictions look like centring absences, so the result is self-consistent and wrong.

An impurity peak. One peak from a second phase, included in the list, makes the true cell fail to fit and sends the search off to a larger and less symmetric one. Powder indexing routines are notoriously sensitive to this, and the standard advice is to try the search repeatedly with each peak omitted in turn.

A near-degeneracy. Two quite different cells can reproduce the same twenty spacings to within measurement error, and choosing between them requires information the pattern does not contain.

None of these is a failure of the instrument. All three follow from having lengths without directions, which is exactly the information the collapse destroyed.

Why it gets worse, not better #

The natural expectation is that a longer measurement or a better instrument recovers what a powder pattern loses. The opposite happens, and the reason is a counting argument.

The number of reflections with reciprocal length below $g$ grows like $g^2$ in two dimensions and $g^3$ in three, because they fill a disc or a ball. The number of peaks — distinct spacings — grows like $g$, because the spacings are spread along a line. So the number of reflections per peak grows as the resolution improves, and at high angle a powder pattern is a continuous overlapping band rather than a set of resolved peaks.

That is the real limit on powder methods, and it is structural rather than instrumental. Sharper peaks help by separating spacings that were merely close; they do nothing at all about spacings that are exactly equal.

The consequence for structure determination is direct. Solving a structure from powder data means extracting individual intensities from overlapping peaks, and the extraction is underdetermined wherever the overlap is exact. Every method for doing it imports an assumption — that the overlapping reflections have intensities in the ratio their multiplicities suggest, or that the structure is chemically reasonable — which is the same move the phase problem requires, applied to a different missing quantity.

What is checked here #

The figure computes rather than illustrates, and three claims are asserted.

The collapse is real. The count of reflections in the window is compared against the count of distinct peaks, and the second must be smaller. That is a weak claim and it would catch a binning routine that had failed to bin.

An accidental overlap exists. The orbit computation is run, the peaks are examined, and the first one holding reflections from two different orbits is found and reported. On a square lattice this always succeeds inside a reasonable window; the assertion states the requirement rather than assuming it.

The absences are where they should be. When a group is supplied, the atoms are its orbit of a general position, so a structure with a glide has reflections that vanish, and they vanish in the powder pattern too — appearing as a peak with a lower multiplicity than its neighbours rather than as a missing spot.

What is not checked is the intensity of a real powder pattern, which involves several factors this site does not model: the Lorentz–polarisation correction, absorption, preferred orientation, and the temperature factor. The heights here are $|F|^2$ times multiplicity and nothing else, which is the geometric content and not a simulation of an experiment.

Where the exactness stops #

The overlaps here are exact and the ones in a laboratory are not, and the difference matters in both directions.

An exact coincidence — $(5\,0)$ against $(4\,3)$ — stays exact at any resolution, and is a fact about integers. A near coincidence, where two spacings differ by less than the peak width, is far commoner and depends on the instrument: a better diffractometer separates it and a worse one does not. So a real analysis has a resolution-dependent list of overlaps, of which the exact ones are a small and permanent subset.

The binning here uses a tolerance of one part in a billion, which for computed spacings is effectively exact and is not comparable to any experimental peak width. That is deliberate — the point of the figure is the arithmetic coincidences, not the instrumental ones — and it means the peak counts on this page are lower bounds on what an experiment would resolve.

And the standing limit: this is a two-dimensional calculation. Three-dimensional powder patterns have the same structure with $h^2 + k^2 + l^2$ in place of $h^2 + k^2$, and accidental overlaps are correspondingly commoner, since a number has more representations as a sum of three squares than as a sum of two.

The same collapse on a hexagonal lattice #

The square lattice is the extreme case for accidental overlap, and running the same computation on another lattice shows what changes and what does not.

Two effects pull against each other, and which wins depends on the lattice.

Higher symmetry means larger families. A twelve-operation point group puts up to twelve reflections in each symmetry family, against eight for the square lattice, so more of the collapse is the harmless kind. That is a gain.

Higher symmetry also means more coincidences. The spacings on a hexagonal lattice are governed by $h^2 + hk + k^2$, which — like $h^2 + k^2$ — takes some values in more than one essentially different way, and the same arithmetic accidents occur. That is a loss.

The net effect for a real structure determination is that high symmetry helps for identification and hurts for solution. Identifying a phase needs only that its list of spacings be distinctive, and a high-symmetry compound has a short, sharp list. Solving a structure needs individual intensities, and a high-symmetry compound has fewer of them independently measurable.

What powder diffraction is for #

The list of losses above should not obscure how much the method does, and it does three things a single crystal cannot.

It works without a single crystal. Most materials of practical interest never form crystals large enough to mount — cements, pharmaceuticals in tablet form, battery electrodes, corrosion products, geological samples. Powder diffraction is often the only structural measurement available.

It measures a mixture. A sample containing several phases gives the superposed patterns of each, and the proportions can be refined. Quantitative phase analysis is a standard industrial measurement and it has no single-crystal counterpart.

It follows a process. Powder patterns can be collected in seconds, so a reaction, a phase transition or a battery charging cycle can be watched as it happens. The time resolution is what matters, and no single-crystal method comes close.

The technique that made structure refinement from powder data routine is Rietveld refinement, from 1969: rather than extracting individual intensities from the overlapping peaks, fit the whole calculated pattern to the whole measured one, and let the overlaps take care of themselves. Hugo Rietveld’s insight was that the overlap problem disappears if nothing is ever extracted — the model predicts the full profile, and the comparison is made there.

Where the collapse is deliberate #

There is a class of experiment that grinds a crystal up on purpose, and it is worth naming because it inverts the logic of everything above.

Texture and strain measurement. A rolled metal sheet is not a powder: its grains are preferentially oriented, so its diffraction rings are stronger in some directions than others. Measuring the variation round a ring is measuring the orientation distribution, which is a mechanical property of the sheet. Here the loss of orientation is not a loss at all — orientation is the quantity being measured.

Particle size from peak width. A very small crystallite gives broader peaks, because there are not enough repeating planes for destructive interference to be complete. Reading widths rather than positions turns a powder pattern into a measurement of grain size, down to a few nanometres.

Phase quantification in a mixture. A sample containing four compounds gives four superposed patterns, and the ratio of their intensities gives the proportions. A single crystal of one component would say nothing at all about the mixture.

In each case the powder geometry is the right one, and a single-crystal measurement would be answering the wrong question. The collapse discards which reflection contributed what, and these three experiments do not need to know.

The surprising part #

The reflections a powder pattern cannot separate are separated by an integer identity, and the identity is the same one that makes a $3$–$4$–$5$ triangle right-angled.

That connection is worth carrying because it makes the difficulty concrete. Which reflections collapse together in a powder pattern is a question about how many ways a number can be written as a sum of two squares — a classical problem in number theory, answered completely by Fermat and Jacobi, and one whose answer is irregular. The number $25$ has two essentially different representations; $50$ has two; $65$ has two; $325$ has four. There is a formula, and it depends on the prime factorisation, and it means the density of accidental overlaps in a powder pattern is a number-theoretic quantity rather than an experimental one.

So a crystallographer deciding whether a structure is solvable from powder data is, without usually knowing it, asking a question about the arithmetic of the cell parameters. A cubic cell has the most overlaps of any, because it has the most symmetry and because sums of three squares repeat often. A triclinic cell with no relations among its parameters has almost none, and its powder pattern is correspondingly the most informative — the exact opposite of the ordering a crystallographer would prefer.

Where the ladder goes next #

The geometry being collapsed is the reciprocal lattice and the dual construction that builds it.

The other half of what an experiment discards is the phase problem, and the information that survives both losses is the systematic absences.

The absence that comes from the description rather than the structure is centring, and the reduction that decides whether two indexings describe the same lattice is the shortest basis.

What the pictures here cannot show. The peaks on this page are geometric — $|F|^2$ times multiplicity, plotted against spacing — and a measured powder pattern differs from them in every respect that makes measurement difficult: peaks have widths, backgrounds slope, intensities are modified by half a dozen corrections, and preferred orientation can change a peak’s height by a factor of two. What the figure shows is which reflections land where, which is the part that no experimental improvement changes.