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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.

Lead zirconate titanate is an oxide solid solution commonly written $\mathrm{Pb}(\mathrm{Z}{\mathrm{r}}_{\mathrm{x}}\mathrm{T}{\mathrm{i}}_{1-\mathrm{x}}){\mathrm{O}}_3$. Its perovskite structure can support switchable electric polarization and a strong piezoelectricity response.

Changing the zirconium-to-titanium ratio changes the competing crystal structures and the response to an applied field or stress. Composition, domain configuration, and processing therefore matter when using the material as a transducer.

Lee J. Todd coauthored research on crystal structure.^[ref]

A crystal structure specifies the chemical species and spatial arrangement of the chemical atoms or ions in a crystal. For an ideal periodic structure, separate two ingredients: a Bravais lattice specifies the translations that repeat the arrangement, and an atomic basis of a crystal specifies the chemical motif repeated at every lattice point. The outward shape, or crystal habit, is a different description: the internal arrangement is what this article describes.

L. S. Dent Glasser, published in research papers as Lesley S. Dent Glasser, worked on crystallography, chemical transformations and science education. A 1968 paper with Ian B. Smith lists the Department of Chemistry at the University of Aberdeen and examines oriented transformations among manganese-containing solids.^[ref]

Their observations connect chemistry with geometry: a reaction can preserve a relationship between the orientation of the starting crystal and that of the product, even while its composition changes. Where their results permitted an interpretation, the authors related the orientations to the oxygen framework and substantial movement of manganese. They also explicitly reported symmetry changes they could not explain. This makes the paper an example of using crystal structure to constrain a reaction mechanism without treating every structural observation as a completed explanation.^[ref]

Her IUCr teaching pamphlet Symmetry by L. S. Dent Glasser develops the relation between finite-object symmetry, periodic arrays and space groups. Its examples show how repeating a motif introduces translations and combined operations beyond the symmetry of an isolated object.^[ref]

A magnetic sublattice groups sites with a common ordered magnetic moment in a description of magnetic order. In the simplest two-sublattice description, assign magnetizations ${\mathbf{M}}_A$ and ${\mathbf{M}}_B$ per total specimen volume, so thatHere $\mathbf{L}$ is one normalization of the staggered magnetization; conventions may include a factor $1/2$. Using the same total volume for both contributions is essential. Equal magnetic moment per site does not imply equal sublattice contributions if the site populations differ.

In this six-site example, three A sites and three B sites give total magnetic moments $6m_0$, $0$, and $3m_0$, respectively, along the chosen axis. The antiferromagnetic case has $\mathbf{M}=0$ but $\mathbf{L}=6m_0\hat{\mathbf{z}}/V$. Cancellation of the sum leaves the alternating information intact.

Materials science relates the preparation and internal structure of materials to their properties and uses. Crystal structure, elasticity and electrical conductivity describe different aspects of that connection.

Max von Laue proposed using a crystal's periodic atomic arrangement to test the wave character of X-rays. Scattering from regularly spaced atoms can interfere constructively only in selected directions, producing a pattern tied to the lattice. The measurements by Walther Friedrich and Paul Knipping connected that idea with photographic evidence. The result simultaneously linked short-wavelength radiation with a wave description and made atomic crystal order accessible to diffraction measurements.^[ref]

A physicist.^[ref]

The Laue X-ray diffraction experiment of 1912 connects a periodic crystal structure to a pattern of diffracted X-rays. Its history links a proposed wave test to the measurements made by Walther Friedrich and Paul Knipping.

MD-2, myeloid differentiation factor 2, is a protein that associates with Toll-like receptor 4 in recognition of bacterial lipopolysaccharide. Its ligand-binding pocket and its contacts with the receptor help connect molecular recognition to assembly of a signaling complex. It is distinct from lipopolysaccharide-binding protein and CD14, other participants in recognition of the same microbial material.^[ref]

A 2009 crystal structure revealed two copies of the receptor–MD-2–ligand unit in a symmetric assembly. Five of the examined ligand's six lipid chains occupied the MD-2 pocket; the remaining chain contributed to contact with the neighboring receptor. The location of a chemical group, rather than just the number of ligand molecules present, helped explain the assembly interface.^[ref]

For an original counting example, two such ligand-bound units contain two MD-2 molecules, two Toll-like receptor 4 molecules and two ligand molecules. Six molecules in total do not imply six independent signaling units: their organization into a particular complex is part of the mechanism. The arithmetic counts the modeled assembly; it is not a dose–response law.

A useful experimental comparison would measure ligand binding and a downstream response separately while perturbing the interface. Preserved binding with impaired response would distinguish recognition from productive assembly. This is an experiment-design illustration, not a reported result from the structural study. A crystal structure supplies spatial evidence, while a cellular response assay tests a different level of function.

A change between a material behaving as an electrical conductor and as an electrical insulator. The distinction concerns the limiting electronic response as temperature tends to zero; finite thermally activated electrical conductivity alone does not establish a metallic state. A Peierls transition opens a band gap through a change in the crystal structure.

Metallurgy studies how to obtain metals from their source materials and control their chemical composition, microstructure and useful properties. Extractive metallurgy deals with separating and refining metals; physical metallurgy connects their internal structure to properties such as mechanical strength, ductility and electrical conductivity.

Steelmaking illustrates the connection between chemistry and materials science: removing oxygen from an iron oxide produces iron, controlling its carbon content produces different steels, and subsequent heat treatment changes the arrangement of crystal structures and hence the mechanical properties. Two samples with similar overall chemical composition can therefore behave differently because their microstructures differ.^[ref][ref]

Mineralogy studies the composition, structure and properties of minerals. A chemical analysis identifies which elements are present, while X-ray diffraction constrains their periodic atomic arrangement. These measurements answer different questions: substances with the same overall composition can have different crystal structures.

Microscopy and physical properties provide complementary evidence. Mineral assemblages and textures help reconstruct rock formation, deformation and fluid interaction. Identification must allow for solid solutions, in which composition varies within one structural family, and for alteration that changes only part of a grain.

An activity model in which distinct atomic positions in a crystal structure are occupied independently by specified species. For ideal random mixing, a pure mineral endmember requiring ${\nu }_s$ specified atoms on site type $s$ has thermodynamic activity $a={\prod }_s{X}_s^{{\nu }_s}$ relative to that pure endmember, with other occupations held fixed. Its mixing contribution to the chemical potential is $RT\ln a$. The powers count occupied positions per formula unit; they are not always the mole fraction of whole endmember formulae.

A crystalline particle with nanoscale dimensions. Its shape and growth kinetics depend on exposed interfaces, defects and material supply, not just its bulk crystal structure.

Natalia Mast is the first author of a 2011 study explaining how the cholesterol side-chain cleavage enzyme carries out successive reactions converting cholesterol toward pregnenolone. The investigators studied the enzyme bound to the first oxygenated intermediate, combining a crystal structure with solution measurements.^[ref]

The active site forms an elongated cavity leading toward the haem group. The intermediate's placement and interactions suggested that movement within this cavity permits successive reactions while the sterol remains associated with the enzyme. Solution spectroscopy tested changes in its position.^[ref]

The example illustrates why one catalyst can support a multistep transformation without requiring the intermediate to diffuse freely after every step. A bound structure proposes spatial constraints; measurements in solution test whether the arrangement can change, rather than assuming that a crystal captures the only functional position.

Natallia Strushkevich is the first author of a 2011 structural study of the mitochondrial system that converts cholesterol toward pregnenolone. The team examined the cholesterol side-chain cleavage enzyme together with its electron-transfer partner adrenodoxin.^[ref]

An engineered fusion protein helped hold the partners together for crystal structure determination. The structure related the iron-sulfur cluster of adrenodoxin to the haem group of the catalytic protein, while structures with different bound sterols addressed successive reaction stages.^[ref]

The engineered connection was an experimental aid, not evidence that the native proteins form one continuous chain. Comparing fusion proteins with different linker lengths also tested how the imposed connection affected reduction of the catalytic partner.^[ref] The example connects a useful structure-stabilizing intervention to an explicit check on its functional consequences.

In neutron diffraction, coherent elastic scattering adds scattering amplitudes from regularly positioned scatterers. Since $E_f=E_i$, the neutron scattering kinematics gives $k_f=k_i=2\pi /\lambda$ andFor lattice planes of interplanar spacing $d_{\mathrm{p}}$, constructive addition requires $Q=2\pi n/d_{\mathrm{p}}$ along the plane normal. Equating these expressions proves Bragg's law:For example, stipulate $d_{\mathrm{p}}=2$ ångströms and $\lambda =2$ ångströms. The first diffraction order has $\theta =30^{\circ }$ and detector deflection $60^{\circ }$. A peak position constrains a period; its intensity also depends on the crystal basis structure factor.

Clifford Glenwood Shull describes powdered sodium chloride, a crystal monochromator, and a shielded counter recording angular counts. Inserting cadmium measured background; substantial diffuse scattering complicated peak interpretation.^[ref]

Powdering supplies many crystal orientations. Comparing the angular peaks with Bragg's law constrains crystal structure; a background estimate helps distinguish specimen peaks from unwanted counts. A two-axis diffraction instrument has no separate outgoing energy analysis, so an angular pattern alone does not separate every inelastic contribution. The richer magnetic example, including a warm/cold comparison and its limits, is the manganese monoxide neutron diffraction experiment. Its extra peaks reveal magnetic order beyond the chemical periodicity; the associated magnetic neutron diffraction model is developed there.

Neutron scattering measures how a specimen changes the directions and kinetic energies of incident neutrons. Neutron diffraction uses the elastic scattering pattern to investigate crystal structure and magnetic order; inelastic neutron scattering uses energy transfer to investigate excitations such as phonons and magnons. The two measurements ask complementary questions: what arrangement repeats in space, and what disturbances can propagate through it?

The 1994 Nobel Prize in Physics recognized Clifford Glenwood Shull's diffraction work and Bertram Neville Brockhouse's spectroscopy.^[ref]

Nuclear scattering couples to atomic nuclei; magnetic scattering couples the neutron's magnetic moment to electronic magnetic moments. The latter underlies both magnetic neutron diffraction and magnon spectroscopy. An atomic number is not a nuclear neutron scattering weight: the relevant coefficient in the elementary coherent model is the nuclear neutron scattering length.

Awarded jointly to William Henry Bragg and William Lawrence Bragg for methods of determining crystal structures using X-rays. Bragg's law connects measured angles to lattice-plane spacings.^[ref]

Static fluctuations in which chemical element occupies a site produce scattering beyond coherent Bragg diffraction from the mean crystal structure. For independent binary occupations with probabilities $p$ and $1-p$, the variance of the site's atomic scattering factor is $p(1-p)|f_A-f_B{|}^2$. Writing each site amplitude as its mean plus a zero-mean fluctuation shows that the mean squared total amplitude is the coherent contribution plus a term proportional to the number of sites and this variance. Consequently a forbidden reflection of the average crystal structure can still sit on a diffuse background. Correlated occupations add nontrivial scattering vector dependence.