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Consider a chosen primitive cubic lattice $a{\mathbb{Z}}^3$ with $a=0.400\mathrm{nm}$ and one identical point scatterer per primitive unit cell. Retain the incident wavelength $\lambda =0.154\mathrm{nm}$. This is an ideal point model. Its reciprocal lattice vectors are $q=(2\pi /a)(h,k,l)$ for integer triples other than $(0,0,0)$. Write $S=h^2+k^2+l^2$. The associated interplanar spacing and first-order Bragg angle satisfyThe reflection labels $(hkl)$ use the Miller index convention, keeping unreduced triples to encode diffraction order. With that convention, set $n=1$ in Bragg's law for each label. The $(200)$ reflection is equivalently the second diffraction order of $(100)$: $d_{200}=a/2$ does not imply extra scatterers halfway between the occupied $(100)$ planes. Counting both descriptions as distinct reflections would count the same direction twice.

For $(100)$, $(110)$, $(111)$ and $(200)$, respectively, $S=1,2,3,4$ and the calculated interplanar spacings are $0.400000$, $0.282843$, $0.230940$ and $0.200000\mathrm{nm}$. Their calculated deflections $2\theta$ are $22.197435^{\circ }$, $31.594744^{\circ }$, $38.952948^{\circ }$ and $45.287481^{\circ }$. Equal steps in $S$ give equal steps in ${\sin }^2\theta$, not equal steps in $2\theta$. This regularity makes an angular pattern a test of a proposed crystal structure.

A real crystal structure has a Bravais lattice together with an atomic basis of a crystal. The crystal lattice structure factor selects the periodic directions; the crystal basis structure factor combines the scattering amplitudes of atoms within a unit cell and can cancel a reflection. Consequently Bragg's law alone predicts neither allowed reflections nor their wave intensities. ^[ref] Systematic absences further restrict observed families; for example, face-centered cubic lattice centering permits only indices all even or all odd in conventional cubic indexing. The four strips above apply to the stated primitive cubic lattice, not to sodium chloride or a face-centered cubic lattice. ^[ref]

The primitive cubic lattice consists of all integer combinations of three perpendicular, equal-length primitive translations:Its conventional cubic unit cell is already primitive, with volume $a^3$ and one lattice point after shared-boundary counting. Each point has six nearest neighbors at distance $a$.

The reciprocal lattice is also simple cubic, with primitive vectors of length $2\pi /a$ in the usual diffraction convention. A primitive cubic Bravais lattice specifies translations rather than a chemical motif: placing several atoms at each lattice point changes the atomic basis of a crystal without automatically changing the translation lattice. This distinction is essential when interpreting crystal basis structure factors.

Consider an original ideal one-dimensional monatomic harmonic oscillator chain with neighbor separation $a$, mass $M$ and nearest-neighbor restoring coefficient $K>0$. Its displacement equation isSubstitute the normal mode $u_r=u_0{e}^{i(qra-\omega t)}$. Since $e^{iqa}+e^{-iqa}-2=-4{\sin }^2(qa/2)$,Choose $a=4$ ångströms and ${\epsilon }_{\ast }=10\mathrm{meV}$. This is one acoustic phonon branch, distinct from the two-atom elastic example. It has $\epsilon (\pi /(2a))=5\sqrt{2}\mathrm{meV}$ and $\epsilon (\pi /a)=10\mathrm{meV}$.

For coherent creation of one phonon in a crystal, the excitation's wave vector $q$ and the measured transfer obey $Q=G+q$, with $G$ a reciprocal lattice vector. To connect the dispersion relation to an illustrative intensity, take zero temperature, longitudinal displacements and the first interval $Q=q\in (0,\pi /a]$, choosing $G=0$. Expand the scattering phase to first order, $e^{iQ(ra+u_r)}\simeq e^{iQra}(1+iQ{u}_r)$. The constant term supplies elastic scattering; the linear term creates one phonon. Its displacement matrix element is proportional to $1/\sqrt{\omega }$, so the squared amplitude has weight $Q^2/\epsilon (Q)$, apart from common factors. Normalize this weight to one at $Q=\pi /a$:The limit $w(0)=0$ follows from $\sin (\pi x/2)\sim \pi x/2$. Set $\sigma =0.4\mathrm{meV}$ and replace the ideal sharp line by a normalized Gaussian function:This chosen broadening illustrates readout width; it is not a measured resolution or a full spectrometer resolution function. Integrating over all $\Delta E$ gives $w(x)$. The small negative-energy tail is a broadening artifact, not a zero-temperature phonon-annihilation signal.

These panels separate three kinds of information: allowed reciprocal lattice positions, cancellation within a crystal basis structure factor, and an excitation's energy as a function of wave vector. The chain has no magnetic degrees of freedom, optical branches, damping or multi-phonon processes. It illustrates how one excitation could be read; it proves nothing about the completeness of a real material's excitation spectrum.

The coherent scattering amplitude associated with a specified spatial arrangement, obtained by summing the site amplitudes with their relative wave phases. Distinguish the crystal lattice structure factor, which sums over lattice sites, from the crystal basis structure factor, which sums the chemical atoms inside a unit cell. In a periodic crystal the two factors multiply. A basis factor can vanish even at an allowed reciprocal lattice vector, causing an extinct reflection.

A systematic absence is a class of diffraction reflections forced to vanish by symmetry or a repeated structural motif, rather than merely being weak by accident. It follows from exact cancellation in a crystal lattice structure factor or crystal basis structure factor within the stated scattering model.

For identical body-centering-related scatterers, $1+(-1{)}^{h+k+l}$ vanishes whenever $h+k+l$ is odd. Changing the motif can lift the cancellation, as in chemical ordering and body-centering extinction. Additional effects such as unequal species amplitudes, disorder or multiple scattering can also produce intensity where a simplified model predicts zero. Reflection conditions therefore help infer structure, but an observed weak or missing spot needs a measurement and model qualification before it is treated as an exact symmetry statement.

Choose a one-dimensional crystal with cell translation $a$ and two atoms per cell at $0$ and $d=a/2$. Here $d$ is an intracell displacement, distinct from the interplanar spacing $d_{\mathrm{p}}$ above. In a single-scattering approximation, a row of $N$ cells has scattering amplitudeThe sum selects reciprocal lattice positions $Q_h=2\pi h/a$ as $N$ becomes large. At these positions, the two-site crystal basis structure factor and its squared magnitude areConsequently $b_1=b_2=b$ gives even reflections of strength $4b^2$ and extinguishes the odd reflections. Choosing $b_1=b$, $b_2=b/2$ instead gives $9b^2/4$ and $b^2/4$: the odd/even ratio is $1/9$. Equal atomic numbers would not establish equal $b_j$ if the isotopes differ. The figure below plots these chosen coefficients, without assigning them to actual elements. Missing reflections can encode destructive wave interference within a cell, not the absence of matter at a site.

X-ray crystallographic structure inference is an inverse problem: infer a crystal structure from measured X-ray diffraction. A proposed unit cell and atomic basis of a crystal predict a pattern; structure determination asks which proposals survive comparison with that pattern. The useful separation is between directions and strengths. Peak positions constrain the reciprocal lattice and hence interplanar spacings; relative wave intensities constrain the motif through the crystal basis structure factor. Knowing a repeat distance does not locate all the chemical atoms within that repeat.

The lattice constant from two indexed reflections example isolates the geometric inference. Basis intensities and phase ambiguity then holds the translations fixed while changing the motif. Even complete, noiseless wave intensity data need not determine all wave phases: this is the phase problem. Chemical composition, symmetry, and additional measurements can supply constraints beyond the original pattern.^[ref]

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.