Laue and Bragg diffraction conditions

Choose incident and outgoing probe wavevectors with magnitude $2\pi/\lambda$ for elastic scattering, and define momentum transfer $\mathbf Q=\mathbf k_{\mathrm{out}}-\mathbf k_{\mathrm{in}}$. The lattice amplitude contains a sum of phase factors over translation vectors. Large constructive peaks occur when each translation changes the phase by an integer multiple of two pi, giving Q equal to a reciprocal lattice vector. Reversing the definition of Q changes its sign but leaves the set of candidate reciprocal peaks unchanged because the lattice contains both G and minus G.

Elasticity adds an independent constraint: incoming and outgoing wavevectors have equal magnitude. Combining it with Q equal to G gives the Ewald-sphere geometry. A reciprocal point must be accessible for the chosen incident wavelength and orientation, not merely exist somewhere in reciprocal space. Rotating a single crystal or using many powder orientations can bring different points onto the elastic sphere. The Laue condition is therefore a selection rule within a scattering setup, rather than a promise that one fixed exposure records every reciprocal vector.

For a family of parallel planes separated by d, waves reflected from successive planes have path difference $2d\sin\theta$, where theta is the grazing angle measured from the planes. Constructive interference requires that difference to be an integer number n of wavelengths. The angle between the incident and outgoing beams is twice theta. With our reciprocal convention the fundamental plane-normal vector has magnitude two pi divided by d, and its nth harmonic supplies the nth diffraction order.

In a cubic cell, plane spacing indexed by h, k, l is the conventional edge divided by the square root of the sum of their squares. This simple formula depends on cubic metric geometry and cannot be copied unchanged for an oblique cell. Bragg order and Miller indices also overlap in their description: an nth-order reflection of one plane family can be indexed as the first-order reflection of multiplied indices. Count the physical reciprocal vector once, rather than treating these two descriptions as different scattered beams.

A finite crystal produces peaks of finite width rather than exact reciprocal delta functions. A longer coherent array gives a narrower principal peak because its phase sum tolerates a smaller mismatch before cancellation. Atomic form factors, basis interference, polarization, thermal motion, and experimental geometry affect intensity. The next lesson examines exact basis cancellations. Until then, distinguish a reflection that is geometrically accessible from one with a nonzero scattering amplitude under the chosen probe and crystal structure.

The derivation assumes coherent scattering and a kinematic single-scattering treatment. Strong multiple scattering can redistribute intensity and requires dynamical diffraction theory. Inelastic neutron or x-ray scattering also transfers energy, changing the equal-wavevector-magnitude condition and revealing lattice vibrations. These processes are not failures of reciprocal reasoning: they use additional energy and momentum accounting. For basic elastic indexing, however, the safest procedure is to compute the reciprocal vector, check the wavelength constraint, assign the correct angular geometry, and only then evaluate its structure factor.