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Choose a primitive cubic lattice with $a=0.400\mathrm{nm}$, one identical point scatterer per primitive unit cell, incident direction ${\mathbf{s}}_{\mathrm{in}}=(0,0,1)$, and the continuous wavelength band $0.080\le \lambda \le 0.160\mathrm{nm}$. This is a hypothetical model, not the zinc blende specimen of the Laue X-ray diffraction experiment of 1912. Write a candidate reciprocal lattice vector as $\mathbf{G}=(2\pi /a)(h,k,-m)$ with $m>0$, and set $S=h^2+k^2+m^2$. The Laue equations and elastic scattering giveA transmission photographic plate at $z=L>0$ intercepts only $s_{\mathrm{out},z}>0$. Its coordinates relative to the direct beam areFor $(h,k,-m)=(2,1,-1)$ and $L=50\mathrm{mm}$, $S=6$, $\lambda =0.13333\dots \mathrm{nm}$ and ${\mathbf{s}}_{\mathrm{out}}=(2/3,1/3,2/3)$, so $(X,Y)=(50,25)\mathrm{mm}$. Its length is one, and each component difference satisfies the corresponding Laue equations. This verifies a direction and a selected wavelength, without predicting an exposure or measured wave intensity.

The first panel has fourfold symmetry because its chosen Bravais lattice, beam axis and square detector share that symmetry. For the tilted panel, apply a rotation matrix $R$ to each index vector, set $v=R(h,k,l)$, and use $\lambda =-2a{v}_z/|v{|}^2$ and ${\mathbf{s}}_{\mathrm{out}}=(0,0,1)+(\lambda /a)v$. All indices with $|(h,k,l)|\le 2a/{\lambda }_{\min }=10$ suffice: elastic scattering bounds $|\mathbf{G}|\le 2|{\mathbf{k}}_{\mathrm{in}}|$. Thus the plotted spectral and angular selection is complete within this ideal point model and detector window. Multiples of an index vector select inversely scaled wavelengths and can land on the same spot; a white-beam spot need not identify a unique diffraction order.

In the Laue method, a fixed single crystal structure receives a collimated beam containing many wavelengths. Each reciprocal lattice vector can select its own wavelength; waves with different wavelengths need not interfere with one another. For $\mathbf{G}\ne 0$, the elastic scattering constraint gives the particularly useful testA candidate reflection requires ${\lambda }_{\mathbf{G}}>0$ within the source spectrum and an outgoing ray that reaches the detector. Varying wavelength changes the radius and center of the Ewald sphere while leaving the reciprocal lattice fixed. A monochromatic wave instead fixes the radius: rotating the crystal structure or varying the incident wavelength in a scan finds reflections sequentially. A white-beam photograph integrates many spectral selections in one exposure; it is not an angular scan at one known wavelength.

The Laue method is useful for orienting a single crystal structure: tilting it changes which reciprocal lattice vectors meet the spectral and detector conditions. Absolute scale has an ambiguity if the wavelengths are unknown. Scaling all direct lattice translations by $c>0$ and all selected wavelengths by $c$ leaves every outgoing direction unchanged. Independently known spectral information or length calibration is needed to remove this ambiguity.

Max von Laue proposed crystal X-ray diffraction and wrote the theoretical section; Walther Friedrich and Paul Knipping performed the experiments.^[ref]

An X-ray tube, lead apertures and shielding supplied a narrow beam through a crystal on a goniometer. Photographic plates recorded the direct beam and separate spots. Copper(II) sulfate trials preceded oriented zinc blende. Optical alignment was checked with X-rays. Removing the crystal removed the extra spots; grinding destroyed their ordered arrangement; translation preserved it, while tilt changed it. Rotation about the beam rotated the pattern.^[ref]

Spots and orientation dependence supported wave interference and three-dimensional crystal periodicity. They determined neither all atomic positions nor an absolute wavelength scale. The authors left open whether the selected radiation originated as crystal fluorescence or was already in the incident beam. This differs from the modern Laue method.^[ref]

The 1914 Nobel Prize in Physics went to Max von Laue; the discovery's experimental account remains jointly credited to Walther Friedrich and Paul Knipping.^[ref] The subsequent inference of particular crystal structures is developed under X-ray crystallography.

To determine a crystal structure, scatter probes whose wavelength is comparable to atomic spacings, typically $0.1\mathrm{nm}$. X-ray diffraction, neutron diffraction and electron diffraction probe periodic order through coherent elastic scattering. With incident and outgoing wavevectors ${\mathbf{k}}_i,{\mathbf{k}}_f$, define $\mathbf{Q}={\mathbf{k}}_i-{\mathbf{k}}_f$. Elastic scattering has $|{\mathbf{k}}_i|=|{\mathbf{k}}_f|$ and constructive interference occurs when $\mathbf{Q}=\mathbf{G}$, a reciprocal lattice vector. The equivalent Bragg's law iswhere the detector scattering angle is $2\theta$, not $\theta$. Peak positions reveal plane spacings and the unit cell; systematic absent reflections constrain centring and the atomic basis.

For X-ray diffraction, a laboratory tube or a synchrotron supplies a collimated beam. In a monochromatic single-crystal experiment a goniometer rotates the specimen and an area detector records reflection positions and integrated wave intensities. A powder experiment uses many randomly oriented crystallites; the reflections form cones recorded as rings or peaks versus $2\theta$, permitting indexing and determination of lattice constants without selecting one crystallite. For a cubic cell, for example, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$. The white-beam Laue method is especially useful for crystal orientation.

The crystal basis structure factor isafter geometric and thermal corrections, including the Debye–Waller factor. For X-rays, $f_j$ describes scattering from the atomic electron density. Thus positions provide lattice geometry while relative wave intensities constrain atomic positions and species. Wave intensities do not directly give the phases of $F_{\mathbf{G}}$: structure refinement or other constraints must address the phase problem.

Neutron diffraction uses a reactor or spallation source, wavelength selection and a detector bank or time-of-flight detection. Neutrons interact with nuclei through nuclear scattering lengths that vary between elements and isotopes rather than simply with atomic number. This makes them useful for light atoms, isotope contrast and distinguishing some neighbours in the periodic table; their magnetic interaction also probes magnetic order. Electron diffraction in a transmission electron microscope gives strong signals from very small thin crystallites, although strong multiple scattering often complicates quantitative wave intensity analysis. These techniques complement one another.

A phonon dispersion relation requires both momentum and energy transfer, so elastic diffraction alone is insufficient. In coherent inelastic neutron scattering, the measured transfers for phonon creation areHere $\mathbf{q}$ is the phonon wavevector reduced into a Brillouin zone, and $s$ labels an acoustic phonon branch or optical phonon branch. Negative energy transfer corresponds to annihilating an already occupied phonon. For neutrons $E={\hslash }^2{k}^2/(2m_n)$, so wavelength and energy selection are linked. Typical thermal-neutron energies of a few to tens of millielectronvolts ($1\mathrm{meV}=10^{-3}$ electronvolts) are well matched to many lattice vibrations.

A triple-axis spectrometer uses a monochromator crystal to select ${\mathbf{k}}_i$, the sample as the second scattering axis, and an analyser crystal to select ${\mathbf{k}}_f$ before detection. Rotate these elements and the specimen to hold $\mathbf{Q}$ fixed while scanning $E_i-E_f$; inelastic peaks locate ${\omega }_s(\mathbf{q})$. Repeat at different $\mathbf{q}$ along symmetry directions to map the phonon dispersion relation. The dynamic structure factor governs peak weights, including the selection of phonon polarization by the scattering geometry. A time-of-flight spectrometer instead infers outgoing neutron energy from travel time over a known flight path and records many directions simultaneously.

Inelastic X-ray scattering can also resolve phonons when very high energy resolution is available. Optical Raman scattering and Brillouin scattering measure vibrations near the zone centre because photon momentum transfers are small compared with a typical reciprocal lattice spacing; they complement, rather than generally map the entire zone like, momentum-resolved neutron measurements. Ultrasound measurements give the long-wavelength sound speed, the initial slope $d\omega /dq$ of an acoustic phonon branch.

X-ray diffraction is the angular redistribution of X-ray wave intensity through wave interference between scattered waves. In a crystal structure, repeated lattice translations select discrete outgoing directions by the Laue equations. A photographic plate can therefore turn invisible atomic periodicity into a visible constellation of spots. Their geometry constrains the Bravais lattice and its orientation; their wave intensities also depend on the crystal basis structure factor. Determining atomic positions belongs to X-ray crystallography and requires more information than recognizing a spot pattern.

The Laue X-ray diffraction experiment of 1912 established this phenomenon experimentally. The Laue method uses a fixed crystal structure with a range of incident wavelengths; the Bragg crystal reflection experiment of 1913 instead measured angularly resolved signals with a moving detector. These are different ways of finding the same coherent elastic scattering directions.