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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.

A lattice translation moves every lattice point of a Bravais lattice onto another lattice point. Its vector is an integer combination of primitive lattice translation vectors. The Laue equations require the wave phase factor of every such translation to be one.

For incident and outgoing unit propagation vectors ${\mathbf{s}}_{\mathrm{in}}$ and ${\mathbf{s}}_{\mathrm{out}}$, wavelength $\lambda >0$, and primitive lattice translation vectors $a_1,a_2,a_3$, the Laue equations areAll three must hold for the same wavelength and outgoing direction. The integers label a reciprocal lattice vector; they need not be coprime. For wavevectors ${\mathbf{k}}_{\mathrm{in},\mathrm{out}}=(2\pi /\lambda ){\mathbf{s}}_{\mathrm{in},\mathrm{out}}$, equivalentlyThis uses angular wavenumbers; the crystallographic convention without $2\pi$ instead uses $\mathbf{s}/\lambda$.^[ref]

For a short proof, translate a scatterer through $a_i$. Its incident wave phase changes by ${\mathbf{k}}_{\mathrm{in}}\cdot a_i$ and propagation to the same distant observation direction contributes $-{\mathbf{k}}_{\mathrm{out}}\cdot a_i$. The relative wave phase factor is therefore $\exp (-i\mathbf{G}\cdot a_i)$. It equals one for all lattice translations exactly when $\mathbf{G}$ belongs to the reciprocal lattice. A finite crystal lattice structure factor broadens the ideal maxima; the crystal basis structure factor can cancel them. Geometric phase matching is necessary for a sharp reflection, but does not guarantee a visible spot.

For elastic scattering, the additional constraint isIt is an Ewald sphere centered at $-{\mathbf{k}}_{\mathrm{in}}$ in the space of transfer vectors $\mathbf{G}$. The parent elastic Bragg scattering condition uses $q={\mathbf{k}}_{\mathrm{in}}-{\mathbf{k}}_{\mathrm{out}}=-\mathbf{G}$, so its displayed condition has the opposite sign. Both give Bragg's law for the same lattice planes.

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.

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.