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A Wigner-Seitz cell is the region of real space at least as close to a chosen Bravais lattice point as to any other. It is bounded by perpendicular bisectors of vectors to neighboring lattice points and is a primitive unit cell.

A reciprocal lattice vector $\mathbf{G}$ satisfies $e^{i\mathbf{G}\cdot \mathbf{R}}=1$ for every lattice translation $\mathbf{R}$. With primitive vectors ${\mathbf{a}}_i$, reciprocal primitive vectors satisfy ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$, and $\mathbf{G}={\sum }_j{n}_j{\mathbf{b}}_j$ for integers $n_j$.

The first Brillouin zone is the Wigner-Seitz cell of that reciprocal lattice, centered at $\mathbf{k}=0$. Its boundaries are Bragg planes, $\mathbf{k}\cdot \mathbf{G}=G^2/2$. Higher Brillouin zones are the regions separated from the origin by successively more such planes: an interior point of zone $j$ has crossed exactly $j-1$ planes. A second Brillouin zone is generally disconnected in this extended-zone representation; it is not just a translated first-zone rectangle.

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.

The two sites give phases one and $\exp [i\pi (h+k+l)]=(-1{)}^{h+k+l}$. ThusFor real nonresonant form factors, even $h+k+l$ gives the square of the sum and odd $h+k+l$ the square of the difference. The factor $K_{hkl}$ represents the remaining experimental corrections, not another basis-interference term. Since the two ions differ, a half-body-diagonal translation does not preserve the decorated crystal. Its Bravais lattice is therefore simple cubic, with a two-ion basis; CsCl is not a monatomic body-centered cubic Bravais lattice.

Replacing both sites by identical Fe atoms changes the amplitude toNow the body-centering translation does preserve the structure. It is a body-centered cubic lattice with one Fe atom per primitive cell, or two per conventional cubic cell.

For 111 the parity is odd: CsCl generally has a nonzero amplitude $f_{\mathrm{C}{\mathrm{s}}^{+}}-f_{\mathrm{C}{\mathrm{l}}^{-}}$, whereas the identical-Fe 111 reflection is systematically absent. A numerical absolute CsCl intensity requires the angle-dependent form factors and experimental normalization; no such values are supplied. Forward electron counts, 54 for Cs+ and 18 for Cl-, explain why their amplitudes are substantially unequal, but are not substituted for the actual finite-angle factors without an additional approximation.

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 $a^3/2$.

A phonon is a quantum of a normal mode of Bravais lattice vibration. A harmonic mode of angular frequency $\omega$ has energy $\hslash \omega (n+1/2)$, and adding one phonon increases its energy by $\hslash \omega$. The phonon carries crystal momentum $\hslash \mathbf{q}$, defined modulo a reciprocal lattice vector.

The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice about the origin: the set of wavevectors at least as close to the origin as to any other reciprocal lattice point. Its boundaries are reciprocal-space perpendicular bisectors. It gives one representative of each wavevector modulo reciprocal lattice translations.

A Bravais lattice is the set of pointswhere the primitive lattice translation vectors ${\mathbf{a}}_i$ are linearly independent. Every lattice point has the same translational environment.

An atomic basis is the motif of atoms attached to each point of the Bravais lattice, specified by positions ${\mathbf{r}}_p$ and atomic species. The full crystal consists of atoms at $\mathbf{R}+{\mathbf{r}}_p$, with $\mathbf{R}\in \Lambda$. The Bravais lattice describes translations; the atomic basis describes what is repeated. A primitive cell contains one lattice point but can contain several atoms.

For the cubic crystal system, the Miller indices satisfyThe inverse squared interplanar spacings have ratios approximately $1:2.004:3.012:4.010$. With the lattice constant close to $0.4\mathrm{nm}$, the appropriate squared-index sums are $1,2,3,4$. Thus the four powder diffraction rings areThe braces collect all symmetry-related lattice planes with the same interplanar spacing. In particular, $(100)$ is forbidden for both a body-centered cubic lattice and a face-centered cubic lattice, but occurs for a primitive cubic lattice. These observations identify the Bravais lattice as primitive cubic lattice, with a multi-atom atomic basis, consistent with a perovskite structure.

Each ring independently gives $a_j=d_j\sqrt{j}$:Their average is $0.42413\mathrm{nm}$. Allowing for the rounded measured angles, a suitable final value isA larger centred unit cell could relabel reflections, but would conflict with the stated approximate lattice constant.

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.

Given independent primitive lattice translation vectors ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3$, defineThen ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$. The reciprocal lattice consists ofequivalently all wavevectors for which $e^{i\mathbf{G}\cdot \mathbf{R}}=1$ for every direct lattice vector $\mathbf{R}$.

For the face-centered cubic lattice, chooseTheir scalar triple product is $a^3/4$, and direct evaluation of the cross products givesEvery integer combination has coordinates $(2\pi /a)(H,K,L)$ with $H,K,L$ all even or all odd. Conversely, for such coordinates the coefficients $(K+L)/2,(H+L)/2,(H+K)/2$ are integers, so every such point occurs. The even-coordinate points are the corners of cubes of edge $4\pi /a$, and the odd-coordinate points are their body centres. ThereforeThis is a property of the Bravais lattice; the two-atom atomic basis changes structure factors rather than these reciprocal lattice translation vectors.

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.

To obtain a crystal's point group, discard the translation parts of its space-group operations. A screw becomes a rotation and a glide becomes a mirror. Thus the tetragonal group's $4/n$, $2_1/m$ and $2/m$ operations giveFor the orthorhombic structure the three perpendicular twofold/glide or screw/mirror directions giveThe primitive Bravais lattice symbol does not itself add a point operation; the fractional translations matter for the space group but disappear from this point-group reduction.

For type I, a magnetic unit cell uses the magnetic Bravais translations $(0,a,a)$, $(a,0,a)$ and $(a,a,0)$. They form an FCC lattice with conventional cubic parameter $2a$. A two-spin basis is an up spin at $(0,0,0)$ and a down spin at $(a,0,0)$, or fractional conventional coordinates $(0,0,0)$ and $(1/2,0,0)$. Every translation preserves the parity of $n_x+n_y+n_z$.

At $\mathbf{Q}=(2\pi /a)(h,k,l)$, the FCC translation sum allows $h,k,l$ all integers or all half-odd-integers. The spin basis contributes $S[1-e^{2\pi ih}]$, which removes the integer set and leavesEquivalently the spin field is $S(-1{)}^{n_x+n_y+n_z}$, so magnetic Bragg peaks are nuclear reciprocal vectors shifted by $(1/2,1/2,1/2)$.

For type II with parallel spins along $z$, use $(a,a,0)$, $(-a,a,0)$ and $(0,0,a)$ as magnetic translations: a primitive tetragonal cell with side lengths $\sqrt{2}a,\sqrt{2}a,a$, containing an up spin at zero and a down spin at $(a,0,0)$. The reciprocal translations have $h,k$ both integers or both half-odd-integers, with $l$ integer. The same basis factor removes the integer-$h$ set:This also follows from $S(-1{)}^{n_x+n_y}$. Other domains permute the three coordinate directions. A magnetic neutron cross section further multiplies these lattice conditions by the magnetic form factor and by the component of ordered moment perpendicular to $\mathbf{Q}$; that polarization factor can suppress an otherwise allowed peak.

Use fractional coordinates in a conventional cubic cell with an atom at the origin. The diamond cubic structure consists of an FCC Bravais lattice with the two-atom basisThe FCC translations within the conventional cell are $(0,0,0)$, $(0,\frac{1}{2},\frac{1}{2})$, $(\frac{1}{2},0,\frac{1}{2})$ and $(\frac{1}{2},\frac{1}{2},0)$. Adding the basis gives eight atoms per conventional cell. Equivalently, two identical FCC sublattices are displaced by $a[111]/4$. It is not a monatomic close-packed arrangement: every atom has four tetrahedrally arranged nearest neighbors rather than twelve.

In the projection, label each atom by its fractional height. The $(0,0)$ and $(1/2,1/2)$ columns have height0; $(0,1/2)$ and $(1/2,0)$ have height1/2. The additional four columns are $(1/4,1/4)$ and $(3/4,3/4)$ at height1/4, and $(1/4,3/4)$ and $(3/4,1/4)$ at height3/4. Boundary columns repeat periodically; they must not be counted as full extra atoms.

The nearest-neighbor displacement is $a[111]/4$, of length $a\sqrt{3}/4$. Touching hard spheres therefore obeyThere are eight atoms per conventional cubic cell, givingFor monatomic FCC close packing, four atoms per cell touch along a face diagonal, so $r_{\mathrm{atom}}=a/(2\sqrt{2})$ andThus the diamond packing fraction is much smaller. The tetrahedral network leaves substantial open volume. Although its Bravais lattice is FCC, the two-atom motif and the shorter hard-sphere contact distance prevent it from inheriting the FCC close-packing fraction.

The nickel-rich $\gamma$ matrix is a disordered substitutional solid solution on a face-centered cubic lattice. The ordered ${\gamma }^{\prime }$ phase has the L12 structure, approximately $\mathrm{N}{\mathrm{i}}_3\mathrm{Al}$: aluminium occupies cube corners and nickel occupies face centres. Its occupied-site geometry is often called ordered FCC, but its chemically labelled Bravais lattice is primitive cubic, with a four-atom basis. A face-centre translation exchanges chemical sublattices and is not a symmetry of the ordered crystal.

Each Al atom has twelve nearest-neighbour Ni atoms at Euclidean distance $a/\sqrt{2}$. Each Ni atom has twelve nearest neighbours too, but these are four Al and eight Ni. The chemical distinction matters for slip. In disordered $\gamma$, the shortest perfect Burgers vector isFor the ordered primitive cubic translation lattice, the shortest order-preserving perfect vector isThis does not imply that $⟨100⟩$ slip must dominate. A vector must lie in a favourable glide plane and have an accessible core structure and activation stress. Matrix dislocations commonly glide on $\{111\}⟨110⟩$ systems. A single $a⟨110⟩/2$ passage through ${\gamma }^{\prime }$ leaves an antiphase boundary; a second matching passage restores order, giving total vector $a⟨110⟩$. Such a superdislocation has a longer total vector than $a⟨100⟩$, yet its constituent dislocations can move on the favourable close-packed planes and interact with the matrix. Paired motion balances line and antiphase-boundary energies, so the shortest abstract lattice translation alone does not select the active slip system.^[ref]

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.

Successive planes normal to $[001]$ are chemically ordered Au and Cu layers. The supplied four-site cell is a convenient doubled description of the primitive tetragonal lattice: valid primitive translations areThey give a square basal cell of side $a/\sqrt{2}$, containing an Au basis atom at the origin and Cu at fractional $(1/2,1/2,1/2)$. That last vector exchanges chemical species and is not a Bravais translation. Thus a face-centred-looking atomic arrangement does not make the chemically ordered material an FCC Bravais lattice. This is the L10 crystal structure; the following indices retain the larger cell specified in the question.

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.

A periodic crystal cannot have a fivefold crystallographic rotation axis. The crystallographic restriction theorem permits rotation orders $1,2,3,4,6$ for a periodic Bravais lattice. A $72^{\circ }$ rotation cannot preserve such a lattice. This restriction concerns exact symmetry of the infinite crystal structure; an isolated finite cluster with fivefold symmetry is a different object.