X-ray and Neutron Diffraction

The spacing of atoms in a crystal is a few tenths of a nanometer, matching the wavelength of X-rays, thermal neutrons, and fast electrons. A wave of that wavelength scattering off the periodic array interferes constructively only in sharply defined directions, and the pattern of those directions reveals the lattice while their intensities reveal the basis. The geometry of the allowed directions is fixed entirely by the reciprocal lattice: diffraction occurs when the scattering vector equals a reciprocal-lattice vector.

The Bragg condition

Bragg treated the crystal as a stack of parallel atomic planes, spacing $d$ , each reflecting a small fraction of the incident wave specularly. Rays reflected from adjacent planes travel path lengths differing by $2 d sin θ$ , where $θ$ is the glancing angle measured from the plane. Constructive interference requires this to be a whole number of wavelengths.

nλ = 2 d sin θ .

Bragg reflection. Waves reflecting from adjacent planes a distance d apart differ in path by 2 d sin theta; when this equals an integer number of wavelengths, the reflections add in phase.

The Bragg picture is heuristic — atoms are points, not mirror planes — but the condition it gives is exact. Its limitation is that it says nothing about intensity; that requires the structure factor below.

The Laue condition and its equivalence

Von Laue treated every atom as a scatterer and summed the outgoing amplitudes. An incident plane wave with wavevector $k$ scatters into $k^{'}$ ; the path difference between waves scattered from lattice points separated by $R$ is $R \cdot (\hat{k} - \hat{k}^{'})$ , so all scatterers add in phase when

(k^{'} - k) \cdot R = 2 π m for all R .

By the definition of the reciprocal lattice this holds exactly when the scattering vector $Δ k = k^{'} - k$ is a reciprocal-lattice vector $G$ . For elastic scattering $∣ k^{'} ∣ = ∣ k ∣$ , so writing $k^{'} = k + G$ and squaring gives the standard Laue form,

2 k \cdot G = G^{2} .

This is the Bragg condition in disguise. Take $G$ to be the shortest reciprocal vector normal to a plane family, $∣ G ∣ = 2 π / d$ . The component of $k$ along $G$ is $k sin θ$ , so $2 k sin θ = G = 2 π / d$ , and with $k = 2 π / λ$ this is $2 d sin θ = λ$ . Higher orders $n$ correspond to reciprocal vectors $n$ times as long.

The Ewald sphere

The Laue condition has a geometric solution. Place the tail of $k$ so its head lands on a reciprocal-lattice point (the origin). Sweep out the sphere of radius $k$ centered at the tail. A reciprocal point $G$ lies on this Ewald sphere exactly when $k^{'} = k + G$ has length $k$ , i.e. when the Laue condition holds. Diffraction occurs for every reciprocal point the sphere passes through.

The Ewald sphere. The incident wavevector k_i ends on the origin of the reciprocal lattice; the sphere of radius k centered on its tail intersects a reciprocal point G, and the vector to that point is the diffracted wavevector k_f. Only points on the sphere diffract.

The construction shows why a stationary single crystal in monochromatic radiation usually gives no reflections: the sphere passes through the origin but rarely through a second point. Rotating the crystal, using a range of wavelengths (Laue method), or using a powder (all orientations) brings reciprocal points onto the sphere.

The geometric structure factor

The Bragg and Laue conditions locate the allowed reflections but not their strength. When the basis has more than one atom, the waves scattered by the atoms within a cell can interfere and cancel. Summing over the basis positions $d_{j}$ with atomic scattering amplitudes $f_{j}$ , the amplitude of the reflection $G = (hk l)$ is the structure factor

S_{G} = j \sum f_{j} e^{- i G \cdot d_{j}} = j \sum f_{j} e^{- 2 π i (h u_{j} + k v_{j} + l w_{j})},

where $d_{j} = u_{j} a_{1} + v_{j} a_{2} + w_{j} a_{3}$ . The measured intensity is $I \propto ∣ S_{G} ∣^{2}$ . When $S_{G} = 0$ the reflection is systematically absent, even though the Bragg condition is satisfied.

Systematic absences in reciprocal space. For bcc (left) only points with h plus k plus l even survive (filled); for fcc (right) only all-even or all-odd indices survive. Open circles are reflections extinguished by the structure factor.

The atomic form factor

The amplitude $f_{j}$ is not a constant. Each atom scatters from its whole electron cloud $n (r)$ , and the contributions from different parts of the cloud interfere. The atomic form factor is the Fourier transform of the electron density,

f (G) = \int n (r) e^{- i G \cdot r} d^{3} r .

At $G = 0$ (forward scattering) the phases align and $f (0) = Z$ , the number of electrons. As $∣ G ∣$ grows — equivalently as $sin θ / λ$ grows — the cloud's finite size dephases the sum and $f$ falls off. Light atoms with few electrons scatter X-rays weakly, and hydrogen is nearly invisible, which motivates neutron diffraction below.

The atomic form factor falls from f equals Z in the forward direction as the scattering angle grows, because the finite electron cloud dephases the scattered waves at large G. Heavier atoms (more electrons) start higher.

Methods and probes

Two experimental geometries dominate. In the single-crystal method one oriented crystal is rotated to bring successive reciprocal points onto the Ewald sphere, mapping the full three-dimensional reciprocal lattice. In the powder (Debye-Scherrer) method a mass of randomly oriented crystallites presents every plane orientation at once, so each family $(hk l)$ with $S_{G} \neq = 0$ diffracts into a cone of half-angle $2 θ$ , recorded as a ring. The ring radii give the $d$ -spacings and identify the lattice without a single crystal.

A powder pattern. Randomly oriented crystallites turn each allowed reflection into a cone, recorded as a ring; the ring radii encode the d-spacings d = lambda over 2 sin theta of the surviving reflections.

The scattering amplitude depends on the probe. X-rays scatter from electrons, so $f \propto Z$ and light atoms are hard to locate. Neutrons scatter from nuclei with amplitudes that vary irregularly with isotope rather than with $Z$ , so they see hydrogen and distinguish neighboring elements; and because the neutron carries a magnetic moment, it also scatters from ordered electron spins, making neutron diffraction the primary probe of magnetic structure. Electrons interact strongly through the Coulomb potential, giving intense scattering from thin samples and surfaces, at the cost of multiple-scattering complications. The three probes are complementary: X-rays for the heavy-atom skeleton, neutrons for light atoms and magnetism, electrons for thin films and surfaces.

The diffraction pattern is the experimental face of the reciprocal lattice. Its peak positions fix the Bravais lattice and cell dimensions, and its intensities, through $∣ S_{G} ∣^{2}$ and the form factor, fix the basis — the two ingredients that define the crystal structure.