The reciprocal lattice

Every symmetry statement made on this site so far has been about a pattern somebody can look at. Real crystals are not like that. Their atoms are a fraction of a nanometre apart, no microscope resolves them directly, and everything known about their arrangement is inferred from where they scatter radiation.

That inference runs through a second lattice, and the relationship between the two is the most useful thing to know about the subject.

The definition #

Given a lattice with basis vectors $\mathbf{a}_1$ and $\mathbf{a}_2$, the reciprocal lattice has basis vectors $\mathbf{b}_1$ and $\mathbf{b}_2$ defined by

$$\mathbf{a}_i \cdot \mathbf{b}_j = \delta_{ij},$$

which reads: each reciprocal vector is perpendicular to all the direct vectors except its own partner, with which it has unit dot product.

In two dimensions that determines them uniquely. $\mathbf{b}_1$ is perpendicular to $\mathbf{a}_2$, with length inversely proportional to the spacing of the rows of lattice points running along $\mathbf{a}_2$; and correspondingly for $\mathbf{b}_2$.

The consequence is the one the figure shows. A long axis in the crystal gives a short reciprocal vector, so the spots along that direction are closely spaced. A short axis gives widely spaced spots. Everything about a diffraction pattern’s proportions is the inverse of the crystal’s.

Where it comes from #

The definition looks arbitrary until it is derived from the physics, which takes one step.

A wave scattered from a point at position $\mathbf{r}$ arrives with a phase $2\pi \mathbf{q} \cdot \mathbf{r}$, where $\mathbf{q}$ is the change in wave vector. Sum over all atoms and the scattered amplitude is a sum of complex exponentials.

For a periodic array, that sum is large only when every lattice point contributes in phase — which requires $\mathbf{q} \cdot \mathbf{a}$ to be a whole number for every lattice vector $\mathbf{a}$. The set of $\mathbf{q}$ satisfying that condition is exactly the reciprocal lattice.

So the reciprocal lattice is not a mathematical convenience laid over the physics. It is the set of directions in which scattering is not cancelled, and the definition above is what that condition looks like written out.

Bragg’s version #

The same result is usually met first in a different form, and the two are worth reconciling because they look unrelated.

W. L. Bragg’s condition, from 1912, treats a crystal as a stack of parallel planes of atoms and asks when reflections from successive planes add in phase. The answer is

$$n\lambda = 2d\sin\theta,$$

with $d$ the plane spacing and $\theta$ the glancing angle. It is the version in every introductory account, and it is entirely correct.

The reciprocal-lattice version is the same statement without choosing planes. Each reciprocal lattice point corresponds to a family of planes in the crystal — the point at $h\mathbf{b}_1 + k\mathbf{b}_2$ corresponds to the family with Miller indices $(hk)$ — and its distance from the origin is $1/d$ for that family. Bragg’s law is what the reciprocal-lattice condition looks like when the geometry is expressed in angles.

Bragg’s form is more intuitive and less general. The reciprocal form extends without modification to structures with several atoms per cell, to quasicrystals where there are no planes to stack, and to the calculation of intensities rather than merely positions.

Indexing, which is the first thing done #

Before any structure is solved, every observed reflection is assigned three integers. That step is called indexing, and it is where the reciprocal lattice does its most practical work.

A reflection at position $\mathbf{q}$ in reciprocal space is labelled $(hk\ell)$ when $\mathbf{q} = h\mathbf{b}_1 + k\mathbf{b}_2 + \ell\mathbf{b}_3$. Finding the labels means finding the reciprocal basis that makes every observed spot land on integer coordinates — which is to say, finding the crystal’s unit cell from its diffraction pattern.

The procedure is a search, and it either succeeds cleanly or fails informatively. A successful indexing gives the cell dimensions directly. A failure means one of several interesting things: the sample is not a single crystal, or it is twinned, or the true cell is larger than the one being tried, or the material is not periodic at all.

The number of integers required is itself a measurement, and it is the one that identifies a quasicrystal. Three suffice for a periodic crystal. Six are needed for an icosahedral quasicrystal, because its reflections are integer combinations of six basis vectors projected into three dimensions. A pattern that stubbornly refuses to index on three integers, with no twinning to explain it, is what Shechtman was looking at.

What decides the intensities #

The reciprocal lattice says where a crystal scatters. It says nothing about how strongly, and the strength is where the structure is hiding.

The amplitude scattered into reflection $(hk)$ is the structure factor

$$F(h,k) = \sum_j f_j \, e^{2\pi i (h x_j + k y_j)},$$

summed over the atoms in one unit cell, with $x_j, y_j$ their fractional coordinates and $f_j$ their scattering strengths. It is a Fourier component of the cell’s contents, evaluated at the reciprocal lattice point.

Two things follow, and both matter for this site’s purposes.

The intensities encode the positions of the atoms, which is why a diffraction experiment can determine a structure at all.

And the intensities encode the symmetry, because a symmetric arrangement makes certain terms in the sum cancel exactly. That cancellation is systematic absence, and it is how a space group is identified in practice.

Superlattice reflections, and what they announce #

One consequence of long-becomes-short is worth following through, because it turns a structural change into a visible one.

Suppose a material’s atoms sit on a lattice, with two species distributed randomly among the sites. The lattice repeat is one site, because the species are indistinguishable to it, and the diffraction pattern is the corresponding reciprocal lattice.

Now cool the material so the species order — alternating regularly rather than sitting at random. The true repeat has doubled along some direction, so the true lattice is a sublattice of what it was, and the reciprocal lattice, being inverse, has gained points: new reflections appear exactly halfway between the ones that were there.

Those are superlattice reflections, and they are among the most useful signals in materials science. Their appearance is direct evidence that ordering has occurred; their intensity measures how complete it is; and their positions say which sites the species chose. All of that is read from spots that were not there before.

The logic is worth stating in general because it recurs. A structural feature that lengthens a repeat shortens a reciprocal spacing, and shortening a reciprocal spacing means new spots appear between the old ones. Doubling in direct space is halving in reciprocal space, and halving is visible.

The second independent route #

This is where diffraction earns its place on a site about symmetry, and the reason is methodological rather than physical.

A pattern figure on this site asserts its group directly from the point set: generate the orbit, hand it to a detector, compare. A diffraction figure computes what that same point set would scatter, by the sum above, and reads the symmetry off which reflections vanish.

Two calculations sharing nothing but the atom positions. When they agree, the agreement is evidence; a single calculation agreeing with itself is not evidence of anything.

That instinct — prefer two routes to one route checked twice — is the same one behind having both an algebraic and a geometric proof of the crystallographic restriction. An error inside a chain of reasoning tends to be invisible from within that chain and obvious from outside it.

Symmetry survives the transformation #

A useful structural fact: the reciprocal lattice has the same holohedry as the direct lattice.

The reciprocal of a square lattice is square. The reciprocal of a hexagonal lattice is hexagonal, rotated by $30°$. The reciprocal of a rectangular lattice is rectangular with the axis ratio inverted. The reciprocal of an oblique lattice is oblique.

So the classification into five lattice types, and with it the constraint on rotation orders, survives the transformation into reciprocal space, and a diffraction pattern’s point symmetry is the crystal’s point symmetry. That is the first thing a crystallographer reads off a pattern, and it works because of this fact rather than by good fortune.

There is one wrinkle. The observed diffraction symmetry always appears to include an inversion centre, whether the crystal has one or not, because intensities at $\mathbf{q}$ and $-\mathbf{q}$ are equal to a good approximation. This is Friedel’s law, and it means diffraction alone cannot distinguish a structure from its mirror image — which was the obstacle to determining the handedness of chiral molecules until anomalous scattering was exploited to break it.

The phase problem #

Now the difficulty that dominates practical crystallography, and it follows directly from what has been said.

A detector measures intensity, which is $|F|^2$. The structure factor is a complex number, with a magnitude and a phase. Squaring destroys the phase, and the phase is where most of the structural information lives.

So a diffraction experiment measures the magnitudes of the Fourier components of the electron density and throws away their phases. Reconstructing the density requires both. This is the phase problem, it is not a technical limitation to be engineered away, and every method of solving structures is a method of recovering phases by some indirect route — heavy-atom substitution, anomalous scattering, molecular replacement, direct methods exploiting the fact that electron density is real and non-negative.

Herbert Hauptman and Jerome Karle received the Nobel Prize in Chemistry in 1985 for the direct methods, which turn the non-negativity of electron density into statistical relationships between phases. That a problem of missing information could be solved at all by assuming so little is among the more surprising results in the field.

What survives the phase problem #

The symmetry does, and that is why this site’s figures are about absences rather than intensities.

A systematically absent reflection has $|F| = 0$, and zero magnitude survives squaring. So the pattern of which reflections are missing is directly observable, phase problem or no, and that pattern is determined by the symmetry.

That is the sense in which symmetry is the easy part of structure determination. The space group can usually be read from the absences and the diffraction symmetry before any phase has been recovered, and knowing the space group constrains everything that follows — how many atoms are in the asymmetric unit, which coordinates are free, and which structures are possible at all.

How these figures are computed #

The diffraction figures on this site are not schematic. Each one evaluates the structure factor over a range of $(h,k)$ and draws what comes out.

The input is a point set generated by applying a wallpaper group to a motif — the same point sets that the pattern figures elsewhere on the site draw directly. For each pair of integers in range, the sum $\sum_j e^{2\pi i (h x_j + k y_j)}$ is evaluated over those points, its magnitude is taken, and a spot is drawn with a radius proportional to it. Where the magnitude is zero, a cross is drawn instead.

The figure then asserts the thing this whole approach exists to check: a group containing a glide must produce absences, and a group without one must produce none. That assertion connects two calculations that share only the atom positions. If the group machinery and the diffraction sum disagreed about which groups have glides, the build would stop.

Two limitations, stated because they matter. The scattering strengths $f_j$ are taken as equal, so these are patterns from identical atoms rather than from a real compound. And the sum is over a finite point set, so the peaks have the width that a finite sample gives — real reflections from a millimetre of crystal are far sharper.

What a diffraction pattern cannot say #

Three limits, stated because diffraction is easy to over-trust.

It cannot see a single unit cell. A diffraction pattern is an average over an enormous number of cells, so it reports the average structure. Local departures — a defect, a substitution, a single molecule in an unusual conformation — contribute to diffuse scattering between the spots and are essentially invisible in the peaks.

It cannot distinguish handedness without help. Friedel’s law makes a structure and its mirror image give identical intensities, and separating them requires anomalous scattering near an absorption edge.

It cannot resolve what the sample does not have. A powder pattern collapses three dimensions onto one, losing the orientation information entirely; a poorly ordered sample gives broadened peaks that limit resolution. Neither is a failure of the method — both are the sample being reported honestly.

Where it came from #

The reciprocal lattice was introduced by Josiah Willard Gibbs in the 1880s, in the context of vector analysis, and reached crystallography through Paul Ewald in 1913.

The experiment came first. Max von Laue directed X-rays at a copper sulfate crystal in 1912, expecting to test whether X-rays were waves, and obtained a pattern of spots that answered that question and simultaneously confirmed the lattice hypothesis that Haüy had proposed in 1784. Laue received the Nobel Prize in Physics in 1914. William Henry Bragg and William Lawrence Bragg — father and son — worked out how to read the patterns as structures and received the prize in 1915, the son at twenty-five, still the youngest laureate in physics.

Ewald’s contribution was the geometrical construction — the Ewald sphere — that makes the diffraction condition visualisable: draw a sphere of radius $1/\lambda$ through the origin of reciprocal space, and reflections occur where the sphere passes through reciprocal lattice points. It converts an algebraic condition into a picture, and it is the reason a crystal must be rotated during a measurement.

Where the ladder goes next #

The immediate sequel is systematic absences — which reflections a glide or a screw axis removes, and why the missing spots are the most informative part of a diffraction pattern.

The wider context is the direct lattice whose reciprocal this is, and the aperiodic structures whose diffraction is sharp without a lattice to be reciprocal to.

What the pictures here cannot show. Every diffraction figure on this page is computed from a finite point set, so its peaks have finite width and its reflections are finite in number. The sharpness and the extent of a real pattern belong to the infinite structure, and no drawing can display an average over $10^{18}$ unit cells.