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An acoustic phonon is a quantum of vibration on an acoustic phonon branch. At long wavelengths the atoms in each primitive cell move together and translational invariance makes the frequency approach zero. A three-dimensional crystal has three acoustic branches, corresponding to longitudinal and transverse polarizations. A linear small-wavevector dispersion has slope equal to the appropriate sound speed.

An acute episode of crystal-associated arthritis in gout. Rapid local inflammation reflects the response to deposited urate rather than a new infection simply inferred from a hot, swollen joint.^[ref]

Anisotropy means that a physical property depends on direction. A strip cut along the fibres of a long-fibre composite can resist stretching much more strongly than a strip cut across them. The directional difference can come from the crystal structure of a constituent or from the arrangement of otherwise isotropic constituents.

For thermal conduction, a thermal conductivity tensor relates the temperature gradient to the heat flux. Along its principal directions the conductivities can differ, and in a general direction the heat flux need not be parallel to the negative temperature gradient. Anisotropic elasticity similarly requires a tensor-valued constitutive equation rather than a single Young's modulus. Measuring a specimen in several orientations therefore reveals information that one axial measurement cannot; averaging over randomly oriented constituents can reduce macroscopic anisotropy, but the result depends on the actual orientation distribution.

Antiferromagnetism is ordered compensation of magnetic moments. In the collinear two-sublattice model, ${\mathbf{M}}_A=-{\mathbf{M}}_B\ne 0$: net magnetization vanishes but staggered magnetization remains. Louis Néel developed this possibility.^[ref]

For an original local energy check, perturb the antiparallel pair in exchange interaction by writing $\theta =\pi -\delta$. ThenFor $J>0$, a small relative rotation costs positive energy. This verifies a pair's preference; it does not prove a bulk ordering temperature. A whole crystal must satisfy many bonds simultaneously, and thermal fluctuations and quantum effects matter. The Heisenberg antiferromagnet and antiferromagnetic Ising model specify different assumptions about those collective degrees of freedom.

Why is the distinction useful? A measurement of zero magnetization cannot decide between an ordered alternating arrangement, a disordered ensemble, and oppositely magnetized magnetic domains. A spatial probe such as magnetic neutron diffraction tests the pattern directly. The manganese monoxide neutron diffraction experiment is a particularly clear example: the order adds a new periodic scattering signal even though the net magnetic moment cancels.

An experimental condensed matter physicist who, with Gene Dresselhaus and Charles Kittel, studied cyclotron resonance in silicon and germanium. Their 1955 work used single crystals near 4 K, light modulation and circularly polarized microwaves. Changing the orientation revealed direction-dependent band effective masses of electrons and electron holes, allowing measurements to reconstruct the shapes of constant-energy surfaces near the edges of energy bands.^[ref]

Composition-sensitive contrast associated with the generally larger electron backscattering yield of higher-atomic-number regions. Detector geometry, crystal orientation, surface topography and beam energy can also affect intensity, so a bright region is not a unique elemental identification.

A strongly chiral liquid-crystal phase containing local double-twist order and frustrated regions. Ordered blue phases I and II have body-centered cubic and simple-cubic director/defect structures respectively, while blue phase III is disordered. The periodicity is orientational/defect order, not a molecular positional crystal lattice.

Bragg scattering is coherent elastic scattering from a periodic lattice, enhanced when the wavevector transfer is a reciprocal lattice vector. Finite crystals have narrow rather than infinitely sharp reciprocal-space peaks.

Differentiate Bragg's law at fixed correctly assigned diffraction order:Divide the second equality by the first. For small signed changes,The Bragg angle error $\delta \theta$ must be in radians. An error quoted for detector deflection $2\theta$ must first be halved. The expression is a first-order Taylor expansion; an unknown diffraction order is a discrete assignment error, not an infinitesimal perturbation.

For error bounds, add magnitudes. For independent zero-mean random errors with standard deviations ${\sigma }_d,{\sigma }_{\theta }$, variance addition instead givesCorrelated errors require a covariance term. A shared calibration bias does not become independent merely because many line positions are measured.

With a chosen angular error $10$ arcseconds, $\delta \theta =4.84814\times 10^{-5}\mathrm{rad}$, the angular contribution is $8.3972\times 10^{-5}$ at $30^{\circ }$ and $1.7646\times 10^{-5}$ at $70^{\circ }$. Larger Bragg angles reduce this contribution because $\cot \theta$ decreases. For a nearby spectral doublet, the inverse relation is $\Delta \theta \simeq \tan \theta \Delta \lambda /\lambda$: a fixed fractional separation spreads farther apart angularly at large Bragg angle.

This does not permit unlimited improvement. The geometry requires $n\lambda \le 2d$; a desired wavelength might not have an accessible large-angle reflection with the chosen crystal. Higher diffraction orders can reach larger Bragg angles but their strengths depend on the crystal lattice structure factor. Finite crystal response, slit widths and source size remain relevant. An error in $d$, including uncorrected thermal expansion, remains even when the angular term is small. Very near $90^{\circ }$, the leading angular derivative vanishes and the second-order Taylor expansion matters.

A salt of calcium and oxalate that can form crystals in urine and contribute to urinary stones. Crystal formation depends on solution conditions, so detecting crystals does not alone establish an obstructing stone.

Sites at 000 and one half of each cubic coordinate give $F=f_A+(-1{)}^{h+k+l}{f}_B$. Identical sites produce a body-centered cubic lattice with absent odd-parity reflections. Distinct chemical species generally preserve only the simple cubic lattice translations, so those same odd reflections need not vanish. A geometric body center is not by itself a symmetry translation of the decorated crystal.

Gout with persistent tissue deposits called tophi. Repeated crystal deposition can damage joints and nearby structures, linking long-term burden with effects beyond individual acute inflammatory episodes.^[ref]

Clinton Davisson and Lester Germer measured strong angular structure in electron scattering from a nickel crystal. Its dependence on incident electron energy agreed with a wavelength set by the electron momentum through the de Broglie wavelength. The crystal therefore supplied a periodic structure against which electron wave behavior could be tested. The Davisson–Germer experiment linked a tunable electron source and angularly resolved detector with the theoretical idea of matter waves.^[ref]

A physicist whose route into research included a telephone-company job when he could not afford to continue at Chicago. He later worked at Bell Telephone Laboratories, retiring in 1946 after 29 years, and taught physics at Virginia in 1947–1949.^[ref] The Davisson–Germer experiment connects that industrial-laboratory work to electron diffraction and matter waves.

Cold working is permanent shaping by plastic deformation at a temperature low enough that substantial concurrent recrystallization does not erase the deformed microstructure. The relevant comparison is with the material's thermal response, rather than with a universal room-temperature boundary. Drawing wire and rolling sheet are examples of imposing large controlled shape changes.

In a crystalline metal, cold working changes the dislocation population and can create preferred crystal orientations. Dislocation interactions can increase resistance to further deformation through work hardening, while uneven deformation can leave residual stress. A practical comparison measures the stress-strain curve before and after a fixed amount of shaping, then unloads to identify the permanent strain. Subsequent heating can modify the stored microstructure through recovery or recrystallization, so processing history is part of specifying a specimen rather than a detail recoverable from its chemical composition alone.

A colloidal glass is a disordered dense particle assembly with structural relaxation too slow on the observation timescale. Crowding, repulsion or short-range attraction can impede rearrangement. Kinetic arrest must be distinguished from an equilibrium phase transition to a crystal.

Condensed matter comprises physical systems such as solids and liquids in which interactions among many constituents determine the observed properties. The term refers to the material system; condensed matter physics studies its structure, dynamics and phases.

A crystal illustrates the change of viewpoint. Its chemical atoms can have collective vibrational modes described as phonons, while its electronic response can involve energy bands rather than isolated-atom energy levels. Collective excitations and phase transitions show why a many-constituent system can display useful organizing principles that are not apparent from one atom alone.

A conoscopic interference pattern is the direction-dependent pattern obtained when a polarizing microscope examines convergent illumination passing through a birefringent specimen. Different positions in the observed angular pattern correspond to different propagation directions through the specimen, and hence to different accumulated phase retardations.

A thin crystal between crossed polarizers can therefore show bright and dark bands even with nearly uniform thickness. Inspecting the angular pattern, rather than an ordinary image of specimen texture, helps identify optical-axis directions. The pattern depends on crystal orientation, thickness and wavelength; it is not a map of surface height.

A cryogenic direct-detection search using scintillating crystal calorimeters. Calcium-tungstate modules measure deposited energy through phonons and compare a separate light signal to discriminate recoil populations. Oxygen, calcium and tungsten recoil differently for a given WIMP mass; a model fit must include their responses and backgrounds.^[ref]

A crystal is a solid with long-range spatial order in its constituents. In a periodic crystal, translation by a lattice vector reproduces the structure; a unit cell and its constituent arrangement describe the repeating pattern. The atomic structure is distinct from the visible external shape: an irregularly cut specimen can remain a single ordered crystal.

Diffraction probes that order because radiation scattered from many regularly arranged sites can reinforce at selected directions. The resulting pattern reveals structural spacings and orientation, rather than merely showing the specimen's outer faces. A single crystal maintains one coherent orientation across the specimen, while a polycrystalline material contains many crystal grains. Crystal orientation can make properties anisotropic, so a test along one direction need not predict response along another. Surface shape additionally depends on equilibrium surface free energy and growth kinetics.

A macroscopically planar crystal boundary characterized by a crystallographic orientation. Its Miller index specifies that orientation; visible facet area depends on surface thermodynamics and growth kinetics. A low surface energy and a slow growth rate are different properties.