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A goniometer measures angles by comparing the orientations of a sample, beam or viewing arm. An optical implementation can rotate a specimen and detector about a common center, measuring the directions of reflected, refracted or diffracted light.

For a prism experiment, a collimated input and a viewing telescope determine the angle through which the beam is deviated as the prism rotates. For a reflection experiment, the incident and viewing directions define the scattering geometry. Accurate centering matters: an off-center sample can move out of the measured beam as the arm rotates, changing signal strength for reasons unrelated to angular response.

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.

A reference material is established as suitable for a specified measurement use because the relevant properties are sufficiently homogeneous and stable. The requirement concerns particular properties and an intended use, rather than every possible attribute of the material.^[ref] Reference material homogeneity concerns variation within or between the portions used; reference material stability concerns maintaining the relevant properties over time and under stated conditions.

Reference materials can support measurement precision control even without an assigned quantity value. An assigned value is needed when the material serves in measurement calibration or a check of measurement trueness.^[ref] A certified reference material adds authoritative documentation specifying property values with associated uncertainties and traceabilities obtained through valid procedures.^[ref]

A concrete physical example is the archived 2009 certificate for National Institute of Standards and Technology SRM 1922, a mineral oil characterized for refractive index. Its characterization used a precision spectrometer and goniometer arrangement with the minimum deviation method, oil in a hollow optical prism, and water-bath temperature control. Measurements covered several wavelengths and temperatures; multiple portions were examined to assess homogeneity.^[ref]

The certificate's Table 1 reports, at wavelength 589.3 nm and temperature $20{}^{\circ }\mathrm{C}$, refractive index 1.46945 with expanded uncertainty $6\times 10^{-5}$ and coverage factor 2. This wavelength's value was obtained through a dispersion fit. The listed temperature coefficient is $-3.68\times 10^{-4}{\mathrm{K}}^{-1}$.^[ref]

An original calculation using that linear coefficient predictsThe predicted change, $0.000736$ in magnitude, is larger than the quoted expanded uncertainty at the reference conditions. Temperature therefore belongs with the assigned optical value; it is not an incidental label. This calculation illustrates the certificate's model, rather than supplying a new certified value or uncertainty.

Record the material identity, certificate version, property, wavelength, temperature and handling history with a measurement result. The archived certificate supplies a documented example of those links, rather than a claim about a currently available product or the validity of a particular bottle today.