The Reciprocal Lattice and Diffraction

Here is a problem that ought to be impossible. An atom is about one ten-billionth of a metre across — thousands of times smaller than the wavelength of visible light. No optical microscope, however perfect, can ever resolve it: you cannot see something much smaller than the ruler you measure it with. And yet we know the arrangement of atoms in salt, in DNA, in a superconductor, to a fraction of an \text{\AA}\text{ngstr\"om}. How?

We do not look at the crystal — we bounce X-rays off it and read the shadow-pattern of interference that comes out. X-rays have wavelengths of about 1\ \text{\AA}, comparable to the spacing between atomic planes, so a crystal acts as a three-dimensional diffraction grating. The pattern of bright spots that results is not a blurry photo of the atoms. It is something stranger and more beautiful: it is a direct image of the crystal's reciprocal lattice — the Fourier-space twin of the real crystal lattice. This page teaches that one idea and the law that makes it work.

The reciprocal lattice: the crystal in Fourier space

Everything periodic has a natural description in terms of the Fourier transform. A crystal repeats with the period of its lattice, so its density n(\mathbf{r}) is built from waves whose wavevectors are special: only those that share the crystal's periodicity survive. That discrete set of allowed wavevectors is itself a lattice — the reciprocal lattice.

Given the real-space primitive vectors \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3, the reciprocal primitive vectors \mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3 are the ones satisfying the defining relation:

Read that last line twice. The direction of a reciprocal-lattice vector encodes the orientation of a family of atomic planes, and its length is inversely proportional to the spacing of those planes. The reciprocal lattice is a filing system for every set of planes in the crystal, indexed by the very same Miller indices (hkl) we met before — which is exactly why diffraction can read the structure off.

Bragg's law: reflections off atomic planes

The most intuitive picture is due to William and Lawrence Bragg, a father-and-son team who won the Nobel Prize in 1915 (Lawrence, at 25, remains the youngest science laureate). Treat each family of parallel atomic planes, spacing d, as a set of half-silvered mirrors. An X-ray beam glancing in at angle \theta partially reflects off each plane. The wave reflected from the second plane travels an extra distance compared with the wave from the first — reveal the figure to see it — and simple geometry shows that extra path is 2d\sin\theta.

The reflected waves reinforce — a bright spot appears — only when that extra path is a whole number of wavelengths. That single condition is Bragg's law:

n\lambda = 2d\sin\theta, \qquad n = 1, 2, 3, \ldots

Everything about X-ray crystallography flows from this. Shine in monochromatic X-rays of known \lambda, measure the angles \theta at which bright spots appear, and solve for the plane spacings d — the geometry of the crystal, decoded.

The Laue condition: Bragg's law in reciprocal space

Bragg's picture of mirrors is handy but a little cartoonish — atoms are not mirrors. Max von Laue's formulation is the deeper one, and it is where the reciprocal lattice earns its keep. When a wave of wavevector \mathbf{k} scatters into a new direction \mathbf{k}' (with the same length, since the scattering is elastic), define the scattering vector \Delta\mathbf{k} = \mathbf{k}' - \mathbf{k}. Constructive interference from the whole crystal happens if and only if:

So the two descriptions are one and the same, viewed from real space (Bragg) or reciprocal space (Laue). The Laue form makes the punchline unmissable: a diffraction pattern is the reciprocal lattice made visible. Each bright spot corresponds not to an atom but to a \mathbf{G}-vector — to a whole family of planes.

d-spacings for cubic crystals

For the cubic crystals that dominate the periodic table, the plane spacing has a wonderfully simple form in terms of the Miller indices and the lattice constant a:

d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}.

Planes with large indices are closely spaced (small d); the widely spaced low-index planes — (100), (110), (111) — are the ones that give the strongest, lowest-angle Bragg peaks. The graph shows how d shrinks as h^2 + k^2 + l^2 grows; drag the slider to change the lattice constant and watch the whole curve scale.

Worked example — solving Bragg's law. Copper is FCC with a = 3.61\ \text{\AA}. The (111) spacing is

d_{111} = \frac{3.61}{\sqrt{1^2+1^2+1^2}} = \frac{3.61}{\sqrt3} \approx 2.08\ \text{\AA}.

With copper-K_\alpha X-rays (\lambda = 1.54\ \text{\AA}) the first-order (n=1) reflection appears where \sin\theta = \dfrac{n\lambda}{2d} = \dfrac{1.54}{2(2.08)} \approx 0.370, i.e. at \theta \approx 21.7^\circ — exactly where the diffractometer finds the copper (111) peak.

Worked example — big cell, small reciprocal cell. The reciprocal spacing scales as 2\pi/a. Double the real-space lattice constant and every reciprocal-lattice vector halves. This inverse relationship is the signature of the Fourier transform, and it is why a large unit cell produces a dense, finely spaced diffraction pattern while a small cell produces a sparse, widely spaced one.

No — this is the deepest and most common misconception, so hold onto it. A diffraction pattern is not a magnified image of the atoms. It is the reciprocal lattice — the Fourier transform of the crystal — projected onto a detector. Each bright spot is a \mathbf{G}-vector, standing for a whole family of planes, not for an individual atom. Bright spots that are far apart on the film correspond to planes that are close together in the crystal, because reciprocal distance is inverse real distance.

A vivid consequence: the reciprocal lattice of an FCC crystal is a BCC lattice, and the reciprocal of BCC is FCC. So a face-centred crystal produces a body-centred pattern of spots — a face swaps for a body-centre when you cross into Fourier space. If you ever catch yourself trying to "read atom positions straight off the film," stop: you must transform back. Turning the pattern into an actual atomic picture requires the intensities and the missing phases — the notorious "phase problem" of crystallography.

Right at the centre of it. The amplitude scattered into direction \Delta\mathbf{k} is literally the Fourier transform of the electron density n(\mathbf{r}): F(\Delta\mathbf{k}) = \int n(\mathbf{r})\,e^{-i\Delta\mathbf{k}\cdot\mathbf{r}}\,d^3r. Because n(\mathbf{r}) is periodic, this integral is zero everywhere except when \Delta\mathbf{k} = \mathbf{G} — the Laue condition drops straight out of the maths — and at those points it equals the structure factor F_{hkl} = \sum_j f_j\, e^{-i\mathbf{G}\cdot\mathbf{r}_j}, a sum over the atoms of the basis.

The structure factor is where the basis re-enters the story. For some structures certain terms cancel: in BCC, reflections with h+k+l odd vanish; in FCC, spots survive only when h, k, l are all even or all odd. These systematic absences are how a crystallographer tells BCC from FCC at a glance — missing spots are as informative as present ones. The whole of X-ray crystallography is one grand application of the Fourier transform.