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An unoccupied oxygen site in a reference crystal structure. Removing an oxide ion gives an effective charge $+2e$ relative to the perfect lattice; the charge of the actual missing ion is $-2e$. A physical sample must satisfy overall charge neutrality through other defects, electronic carriers or changed cation valences.

A Bravais lattice is the translation set $\Lambda =\{n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2+n_3{\mathbf{a}}_3:n_i\in \mathbb{Z}\}$ generated by primitive lattice translation vectors. Translating by any member of $\Lambda$ leaves its surroundings equivalent. A crystal structure requires both this lattice and an atomic basis attached to every lattice point. The basis can lower the symmetry of the bare point lattice; a familiar-looking geometrical arrangement of atoms is not necessarily itself a Bravais lattice. For example, the diamond cubic structure is a face-centred cubic lattice with an identical-atom basis at fractional positions $(0,0,0)$ and $(1/4,1/4,1/4)$ in conventional cubic coordinates. The displacement between those two basis atoms is not itself a lattice translation.

The seven three-dimensional crystal systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic; the distinct translation types give 14 Bravais lattices. A conventional cell can contain several lattice points, while a primitive unit cell contains one. Simple cubic primitive lattice translation vectors can be chosen along the cube edges; for face-centred cubic they can be $(a/2)(0,1,1)$, $(a/2)(1,0,1)$ and $(a/2)(1,1,0)$. Their primitive volume is $a^3/4$, explaining the four lattice points per conventional face-centred cell.

A crystallographic point group comprises the orthogonal parts of the crystal's spatial symmetry operations. It constrains possible rotations, reflections, inversion and their combinations, while forgetting translations. Lattice preservation restricts proper rotational orders in three dimensions to $1,2,3,4,6$; a fivefold axis cannot be a symmetry of a periodic three-dimensional lattice. One quick argument restricts a rotation's integral lattice representation through $1+2\cos \theta$, an integer trace, producing these orders. There are 32 crystallographic point-group types. Examples are the cubic full octahedral group $O_h$ with Hermann–Mauguin symbol $m\bar{3}m$, the tetrahedral class $T_d$ with symbol $\bar{4}3m$, and the polar tetragonal class $C_{4v}$ with symbol $4mm$.

The space group retains the full isometry. In Seitz notation $\{R\mid \mathbf{t}\}$ acts by $\mathbf{x}\mapsto R\mathbf{x}+\mathbf{t}$, and multiplication isIts translation subgroup is normal and the quotient is the crystallographic point group. A screw axis combines rotation with translation along the axis; for example, applying a $2_1$ screw twice gives one whole lattice translation. A glide plane combines reflection with a translation parallel to the plane; applying the glide twice likewise gives a translation. These operations can preserve the complete crystal without their rotation or reflection alone preserving it.

A symmorphic space group admits an origin and representatives for which the point-group operations have no translation component. A nonsymmorphic space group has essential fractional translations that cannot all be removed by moving the origin. There are 230 three-dimensional space-group types, distinguished more finely than their point groups. These counts refer to ordinary, nonmagnetic periodic space groups; magnetic and aperiodic generalizations are different classifications.^[ref]

For concrete comparisons, a monatomic simple-cubic crystal has space group $Pm\bar{3}m$ and point group $m\bar{3}m$. Rock salt has a face-centred cubic lattice, a two-species basis and symmorphic space group $Fm\bar{3}m$. Diamond has the same cubic point group and translation-lattice type, but its basis leads to nonsymmorphic $Fd\bar{3}m$. Replacing the two diamond basis sites by distinct species gives the zinc blende structure, exemplified by GaAs, with $F\bar{4}3m$ and point group $\bar{4}3m$; inversion that would interchange the species is no longer a symmetry. The lattice specifies translations, the point group specifies the linear symmetry classes, and the space group specifies their complete spatial implementation. These examples show why none alone uniquely describes an atomic structure.

Two molecules of the carboxylic acid form a cyclic carboxylic-acid dimer. Let Ar denote the meta-methylphenyl group. One molecule donates its OH hydrogen to the other's carbonyl oxygen, and the second donates back:
The two contacts close an eight-membered hydrogen-bonded ring, shown explicitly below.

Both molecules therefore act once as hydrogen-bond donor and once as hydrogen-bond acceptor, giving two O–H···O=C contacts. This is the conventional self-complementary dimer motif; the schematic is not a prediction of every weaker contact in an unprovided crystal structure.

A suitable representative structure is a three-dimensional network of corner-sharing CO${}_4$ tetrahedra, with C–O single bonds. Each carbon is four-coordinate; every bridging oxygen is bonded to two carbons. Sharing four oxygens between two tetrahedra gives $4/2=2$ oxygens per carbon, so the network retains empirical formula CO${}_2$ without containing separate O=C=O molecules.

High pressure favours a compact, highly coordinated network through its smaller molar volume: at fixed temperature, increasing pressure changes relative Gibbs energies according to $\mathrm{d}(\Delta G)=\Delta V\mathrm{d}p$. Heating helps overcome the large reconstructive barrier to breaking and redistributing molecular pi bonding and establishing an extended single-bond network. It should not be argued that high temperature by itself favours the lower-entropy network. Nor is merely freezing molecular CO${}_2$ sufficient: dry ice remains a molecular solid.

The sketch shows connectivity and local tetrahedral coordination, not a unique predicted crystal structure. Real amorphous carbonia need not have perfect periodic order or exclusively one coordination environment. The original carbonia experiments combined high-pressure preparation with infrared, Raman and diffraction evidence for loss of molecular character and formation of an amorphous network.^[ref]

At fixed number density, attempt frequency, nearest-neighbour distance and barrier, the conductivities have ratio ${\sigma }_{\mathrm{sc}}:{\sigma }_{\mathrm{bcc}}:{\sigma }_{\mathrm{fcc}}=1:4/3:2$. Thus the face-centred cubic array is largest. Numerically,The small value is consistent with the low supplied carrier number density and strongly activated hopping. This comparison holds $n$ fixed rather than comparing fully occupied crystal structures, whose site densities at the same neighbour distance differ.

A Bravais lattice is an infinite periodic set of equivalent points $\mathbf{R}=n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2+n_3{\mathbf{a}}_3$, with integer $n_i$ and independent primitive translations ${\mathbf{a}}_i$. It describes translation symmetry, without assigning a chemical species to a point. An atomic basis of a crystal is the motif of atoms, with their species and relative positions, repeated identically at every lattice point. The resulting crystal structure consists of the lattice decorated by that basis; the basis can contain several atoms even when the lattice is primitive.

A primitive unit cell tiles space under lattice translations and contains exactly one lattice point, counting shared boundary points fractionally. Its volume is $|{\mathbf{a}}_1\cdot ({\mathbf{a}}_2\times {\mathbf{a}}_3)|$. A conventional unit cell is chosen to display useful crystal symmetry and may contain several lattice points. Thus a centered conventional cell is not primitive, but the crystal still has a primitive cell. Neither cell size nor the number of dots drawn on its boundary should be confused with the number of atoms in the crystal basis.

A Bravais lattice is a set of points generated by integer translations along three independent primitive lattice translation vectors:Every lattice point has the same environment under lattice translations. A crystal structure consists of this Bravais lattice with an atomic basis of a crystal repeated at every lattice point; the set of atomic positions need not itself be a Bravais lattice.

The two positions in the conventional cubic cell of alpha-tungsten give a body-centred cubic lattice, with one tungsten atom per primitive cell and two per conventional cell. One possible set of primitive lattice translation vectors isTheir primitive-cell volume is $a^3/2$.

The beta phase has a primitive cubic lattice with an eight-atom atomic basis of a crystal, rather than a body-centred cubic lattice. A lattice translation must map the entire atomic arrangement onto itself. For example, the putative body-centering vector $(1/2,1/2,1/2)$ maps $(1/4,1/2,0)$ to $(3/4,0,1/2)$ modulo integer coordinates, where the given atomic basis of a crystal has no atom. None of the seven nonzero basis positions generates a translation that preserves all eight sites. ThereforeThe presence of atoms at both the origin and the body centre alone is insufficient to establish body-centering of the full crystal structure.

Determining antiferromagnetic magnetic order is a good reason to use neutron scattering. The neutron has a magnetic dipole moment, so its scattering amplitude includes the spatial arrangement of electronic magnetic moments. Alternating magnetic moments can produce magnetic neutron diffraction peaks even when there is no change of the nuclear crystal structure. Ordinary nonresonant X-ray diffraction primarily measures the electron density, and is much less sensitive to this magnetic order.

An applied magnetic field changes the alignment of the manganese magnetic moments. The component of their magnetic structure factor perpendicular to the scattering vector contributes to magnetic neutron diffraction. A field-induced uniform magnetization has the same translational periodicity as the nuclear crystal structure, so its scattering amplitude can occur at the existing reciprocal lattice vectors. The diffraction wave intensities can therefore change without moving the peaks. Their angular positions still measure the same interplanar spacings. This mechanism does not require a measurable change of the lattice constant.

A consistent mechanism is a transition to G-type antiferromagnetism at the Néel temperature. On the manganese primitive cubic lattice, take the magnetic moments to beEvery nearest neighbour has the opposite magnetic moment. The nuclear crystal structure can retain its original lattice constant, while the magnetic order introduces the propagation wavevectorSumming the scattering amplitude from the magnetic moments over sites gives peaks at $\mathbf{q}=\mathbf{G}+\mathbf{Q}$, where $\mathbf{G}$ is a nuclear reciprocal lattice vector. Thus all three components of the new Miller indices, expressed relative to the original unit cell, are half-integers. The two smallest squared magnitudes in units of $(2\pi /a{)}^2$ areThe first predicts $2\theta \simeq 21.29^{\circ }$, agreeing with the new ring. The next squared magnitude is $19/4$, outside the measured angular range. Taking the observed first magnetic ring as the calibration, Bragg's law predictsUsing the independently fitted lattice constant gives $41.42^{\circ }$, within the input rounding. One ring alone does not uniquely determine a magnetic order; this proposed nearest-neighbour antiferromagnetic pattern explains both the observed ring and the count of two new rings in the specified interval.

The Bravais lattice is a face-centered cubic lattice. Four inequivalent representatives of its lattice points in the conventional cubic unit cell, in fractional crystal coordinates, areBoundary copies in the drawing do not constitute additional independent lattice points. Attach the two-atom atomic basisto each lattice point. The resulting aluminium phosphide crystal structure is the zinc blende structure. For example, the four phosphorus positions reduced into the conventional unit cell are $(1/4,1/4,1/4)$, $(1/4,3/4,3/4)$, $(3/4,1/4,3/4)$, and $(3/4,3/4,1/4)$, reproducing the heights in the source drawing. There are four aluminium and four phosphorus chemical atoms per conventional unit cell, but only two chemical atoms per primitive unit cell.

Use the long-wavelength linear dispersion relation $\omega \simeq v_L|\mathbf{k}|$ as an estimate. The phonon energy is thenThis extrapolates the measured low-wavevector speed of sound to the $X$ point, the first Brillouin zone boundary along $[100]$. It is not an exact phonon dispersion relation there: the discrete crystal structure generally bends the branch, and the initial slope alone cannot determine its boundary energy. In particular, $(2\pi /a,0,0)$ is not a reciprocal lattice vector of the face-centered cubic lattice; the smallest such axial vector is $(4\pi /a,0,0)$. There is therefore no reason for this acoustic phonon energy to vanish.

The ordered crystal structure consists of a primitive cubic lattice with the four-site atomic basisin fractional crystal coordinates. The drawing shows all boundary representatives of these sites. Eight corner gold chemical atoms, shared among eight unit cells each, contribute one gold chemical atom. Six face-centred copper chemical atoms, shared between two unit cells each, contribute three copper chemical atoms. Thuswhich gives the required stoichiometry. Although the unlabelled positions form a face-centered cubic lattice, a face-centering lattice translation exchanges gold and copper and therefore fails to preserve the ordered crystal structure. Its Bravais lattice is primitive cubic lattice.

Replace every coloured site in the drawing by an identical site carrying $\bar{f}$. The face-centering lattice translations now preserve the average crystal structure, whose Bravais lattice is a face-centered cubic lattice. One choice of primitive lattice translation vectors isA coherent Bragg diffraction peak requires $\mathbf{G}\cdot {\mathbf{a}}_j\in 2\pi \mathbb{Z}$ for all three primitive lattice translation vectors. Thus $k+l$, $h+l$, and $h+k$ must all be even, equivalent to all three Miller indices having the same integer parity. This is the face-centred cubic selection rule: $(110)$ is forbidden and $(111)$ is allowed. The restoration of these lattice translations explains the disappearance of the ordered superlattice reflections.

The stoichiometry fixes the site probabilities to $1/4$ gold and $3/4$ copper. Every site in the average crystal structure therefore has the same atomic scattering factorThe average structure factor isAt $(110)$ the bracket vanishes, while at $(111)$ it is four. HenceThe $(111)$ powder diffraction multiplicity is unchanged, so its coherent Bragg diffraction wave intensity equals the ordered value under the stated equal-lattice constant approximation.

The zero refers to the coherent Bragg diffraction peak of the average crystal structure. A random alloy can still give occupational diffuse scattering at the same scattering vector: the average of the squared amplitude includes fluctuations as well as the square of the average amplitude. Such background is not a surviving $(110)$ Bragg diffraction peak.

A Bravais lattice is a set of equivalent points generated by integer combinations of three independent translation vectors. A crystal motif is the arrangement of atoms attached identically to every lattice point. Together they specify the periodic crystal structure: lattice translations repeat the motif without changing its internal arrangement.

A unit cell is a region whose translations tile the structure. A primitive unit cell contains one lattice point after sharing boundary points with neighboring cells. A conventional cell can contain more than one lattice point to display the crystal's symmetry with simple axes and angles; for example, a conventional face-centered cube has four lattice points but a less visually symmetric primitive cell has one.

Rotational symmetry compatible with a periodic three-dimensional lattice has order $1,2,3,4$ or $6$, as expressed by the crystallographic restriction theorem. A crystallographic fivefold axis is excluded from an ordinary periodic crystal. This statement concerns exact lattice symmetry; it does not exclude local fivefold atomic arrangements or nonperiodic ordered structures.

Pearlite is a microstructural arrangement of two phases, ferrite and cementite, rather than a single phase. Ferrite is an iron-rich solid solution; cementite is the carbide $\mathrm{F}{\mathrm{e}}_3\mathrm{C}$. A pearlite colony contains alternating lamellae of these constituents formed by transformation of austenite.

A thermodynamic phase has a specified crystal structure and composition range with locally homogeneous properties on the appropriate scale. The word pearlite instead describes how distinct phases are arranged. Its lamellar spacing and orientation can change while the identities of its two phases remain the same.

In crystal polymorphism, the same chemical composition has different crystal structures; for an chemical element, this is a form of allotropy. Diamond has a three-dimensional tetrahedral network of covalent bonds, giving high hardness and a wide band gap. Graphite has strongly bonded planar sheets, with weaker interactions between sheets. Sliding between sheets makes it a useful solid lubricant; mobile electronic states give good electrical conduction within the sheets. Their different properties come from different bonding and structures, despite both being carbon.

A motif can have fivefold symmetry. For example, repeat an identically oriented regular pentagonal molecular motif at every point of a square translation lattice. Each isolated motif has fivefold rotational symmetry, while the crystal structure as a whole does not: rotating about one motif fails to map the other motif centres onto lattice points. The symmetry of an atomic basis of a crystal need not be a symmetry of its Bravais lattice.