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The three primitive lattice translation vectors join a cube corner to the three adjacent face centres. Thus the Bravais lattice is a face-centered cubic lattice, with conventional cubic side $a$.

Write the given primitive lattice translation vectors as ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3$. Their primitive unit cell volume isThe reciprocal lattice convention ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$ then givesEach reciprocal lattice vector has units ${\mathrm{m}}^{-1}$ when $a$ is in metres. Integer combinations generate a body-centered cubic lattice of conventional cubic side $4\pi /a$; the point $(2\pi /a)(1,1,1)$ is its body centre. Directly taking the nine dot products checks the signs and the $2\pi$ normalization.

Let $N=L/a$ be the integer number of primitive unit cells. A Bloch state can be written as ${\psi }_k(x)=e^{ikx}{u}_k(x)$, with $u_k(x+a)=u_k(x)$. The periodic boundary condition ${\psi }_k(x+L)={\psi }_k(x)$ therefore requiresThe spacing of allowed wavevectors is $\Delta k=2\pi /L$. The first Brillouin zone has width $2\pi /a$, soFor an exact endpoint convention take a half-open Brillouin zone, such as $[-\pi /a,\pi /a)$. Its two endpoints differ by a reciprocal lattice vector and must not both be counted as distinct values of crystal momentum. There are exactly $N$ inequivalent integers $j$ modulo $N$, irrespective of whether $N$ is even or odd. This is state counting in a Bloch band: one allowed wavevector per primitive unit cell per energy band. A separate twofold spin multiplicity gives $2N$ electronic states per energy band, without doubling the number of wavevectors.

The crystal has periodic potential $V(x+a)=V(x)$. By Bloch's theorem, an energy eigenstate may be chosen asExpand the periodic factor in a Fourier series:Multiplication by $e^{ikx}$ gives the stated plane-wave expansion. Here $x$ is position, $a$ is the primitive unit cell length, $G_1$ is the primitive reciprocal lattice wavenumber, $n$ runs over every integer, and $C_{k,n}$ are complex numbers specifying the component wave amplitudes. The wavevector $k$ labels the Bloch state modulo $G_1$; $\hslash k$ is its crystal momentum, rather than generally its measured free-space momentum.

The component $e^{i(k+n{G}_1)x}$ is a plane wave of wavevector $k+n{G}_1$. All these components acquire the same phase factor $e^{ika}$ under $x\mapsto x+a$, since $e^{in{G}_{1}a}=1$. Their mixing therefore respects the crystal's translation symmetry. The coefficients and energy eigenvalue follow from the Schrödinger equation; the expression is not a claim that the coefficients are equal or that only finitely many are nonzero. The energy band index is suppressed in the notation, so different energy bands at a common $k$ generally have different coefficient sets.

Let the nearest-neighbour spacing be $d$, so the primitive unit cell has length $\ell =2d$. Denote the two longitudinal displacements in cell $n$ by $u_n,v_n$, with spring $K$ within each cell and spring $K^{\prime }$ between cells. Hooke's law and Newton's second law giveFor normal modes $u_n=u{e}^{i(qn\ell -\omega t)}$ and $v_n=v{e}^{i(qn\ell -\omega t)}$, the two amplitudes satisfyIts vanishing determinant gives the alternating-spring chain dispersion relationThe minus sign is the acoustic phonon branch, with ${\omega }_{-}(0)=0$ and $u=v$ at the origin. The plus sign is the optical phonon branch, with $u=-v$ at the origin. For positive springs the acoustic phonon branch rises from zero to $\sqrt{2\min (K,K^{\prime })/m}$ at the Brillouin zone edge; the optical phonon branch falls from $\sqrt{2(K+K^{\prime })/m}$ to $\sqrt{2\max (K,K^{\prime })/m}$. Their long-wavelength limits areThus the leading frequency in the strong-bond limit of the alternating-spring chain is $\sqrt{2K/m}$, the internal stretching frequency of a pair with reduced mass $m/2$.

An acoustic phonon branch approaches zero frequency linearly, ${\omega }_{\mathrm{ac}}\simeq v_s|q|$, because a long-wavelength translation has no restoring force. An optical phonon branch has nonzero frequency at $q=0$ and represents relative motion of atoms within a primitive unit cell. Such an optical branch requires more than one independent atom per primitive cell; a truly monatomic simple-cubic model has only the acoustic branches.

The Bose occupation is $N_q=[e^{\hslash {\omega }_q/(k_{B}T)}-1{]}^{-1}$. At low temperature, appreciably populated acoustic modes haveAs temperature falls, the available phonon wavevectors concentrate near the zone centre and optical modes freeze out when $k_{B}T\ll \hslash {\omega }_{\mathrm{op}}(0)$. These are thermal-occupation restrictions, not a change in the allowed geometric Brillouin zone.

For state counting in a Bloch band, take a primitive cubic lattice of spacing $a$ and impose periodic boundary conditions on lengths $L_i=N_{i}a$. The allowed wavevectors are $k_i=2\pi l_i/L_i$. The first Brillouin zone has side $2\pi /a$, so it contains $N_i$ distinct values in each direction and hence $N_x{N}_y{N}_z=N$ wavevectors. Each orbital-derived energy band has one state per wavevector per spin orientation. ThereforeThe phrase “per atom” attached to $2N$ in the PDF is a counting error: $2N$ is the total, not the per-atom count.

By the Pauli exclusion principle, a monovalent crystal supplies $N$ valence electrons and half-fills a spin-degenerate band; its Fermi level lies within allowed states, giving a metal in the independent-electron picture. Two electrons per atom can fill one band. If a band gap separates it from all empty bands, the crystal is a band insulator. An even valence count only permits insulation: overlapping bands can still produce a metal. Conversely, electron interactions can produce a Mott insulator, and symmetry breaking can permit insulation outside this simple spin-degenerate electronic band structure criterion. The relevant count in a crystal with a basis is per primitive unit cell.

Let adjacent equal masses $m$ be separated by $a$, with alternating strong and weak spring constants $K$ and $\kappa$, $K\gg \kappa >0$. The pattern repeats after $2a$, even though the masses are identical. Thus the first Brillouin zone isChoose the strong bond inside each two-atom primitive unit cell. With displacement amplitudes $(u,v)e^{i(2naq-\omega t)}$, the normal mode equations areThe alternating-spring chain dispersion relation is consequentlyThe lower acoustic phonon branch has in-phase motion within the strongly bound pair and ${\omega }_{-}(0)=0$. Its small-$q$ slope is the speed of sound,The upper optical phonon branch has opposite motion within each pair, with ${\omega }_{+}(0)=\sqrt{2(K+\kappa )/m}$. Here “optical” names the gapped branch; equal neutral masses need not couple to light through an electric dipole.

In the strong-bond limit of the alternating-spring chain, the pairs act almost as rigid dimers in the low-frequency branch, while their internal stretch gives a high, nearly flat branch:At the zone edge the two frequencies are $\sqrt{2\kappa /m}$ and $\sqrt{2K/m}$, separated by a large forbidden frequency interval. Neighbouring cells are out of phase there. Both branches have horizontal tangents at the edge.

A Peierls distortion modulates the atomic positions and chemical bonds of a one-dimensional chain with a partly filled energy band. In the half-filled example, alternating short and long bonds double the primitive unit cell. The distortion is the structural change; the associated Peierls transition is the change in electronic conduction.^[ref]

A continuous family of normal modes labelled by the wavevector and a branch index $s$. For $r$ chemical atoms per three-dimensional primitive unit cell, the dynamical matrix has $3r$ eigenvalues, giving $3r$ branches: three acoustic phonon branches and $3r-3$ optical phonon branches for a mechanically stable ordinary crystal structure.

Consider a chosen primitive cubic lattice $a{\mathbb{Z}}^3$ with $a=0.400\mathrm{nm}$ and one identical point scatterer per primitive unit cell. Retain the incident wavelength $\lambda =0.154\mathrm{nm}$. This is an ideal point model. Its reciprocal lattice vectors are $q=(2\pi /a)(h,k,l)$ for integer triples other than $(0,0,0)$. Write $S=h^2+k^2+l^2$. The associated interplanar spacing and first-order Bragg angle satisfyThe reflection labels $(hkl)$ use the Miller index convention, keeping unreduced triples to encode diffraction order. With that convention, set $n=1$ in Bragg's law for each label. The $(200)$ reflection is equivalently the second diffraction order of $(100)$: $d_{200}=a/2$ does not imply extra scatterers halfway between the occupied $(100)$ planes. Counting both descriptions as distinct reflections would count the same direction twice.

For $(100)$, $(110)$, $(111)$ and $(200)$, respectively, $S=1,2,3,4$ and the calculated interplanar spacings are $0.400000$, $0.282843$, $0.230940$ and $0.200000\mathrm{nm}$. Their calculated deflections $2\theta$ are $22.197435^{\circ }$, $31.594744^{\circ }$, $38.952948^{\circ }$ and $45.287481^{\circ }$. Equal steps in $S$ give equal steps in ${\sin }^2\theta$, not equal steps in $2\theta$. This regularity makes an angular pattern a test of a proposed crystal structure.

A real crystal structure has a Bravais lattice together with an atomic basis of a crystal. The crystal lattice structure factor selects the periodic directions; the crystal basis structure factor combines the scattering amplitudes of atoms within a unit cell and can cancel a reflection. Consequently Bragg's law alone predicts neither allowed reflections nor their wave intensities. ^[ref] Systematic absences further restrict observed families; for example, face-centered cubic lattice centering permits only indices all even or all odd in conventional cubic indexing. The four strips above apply to the stated primitive cubic lattice, not to sodium chloride or a face-centered cubic lattice. ^[ref]

The reciprocal lattice of a body-centered cubic lattice is face-centred cubic, and that of a face-centered cubic lattice is body-centred cubic. In both cases its conventional cubic side is $4\pi /a$ if the direct conventional side is $a$. Thus a direct BCC-to-FCC phase transition reverses the reciprocal lattice types. The reciprocal primitive cell volume is $(2\pi {)}^3/\Omega$, so thermal expansion contracts reciprocal spacings, while a discontinuous change in direct primitive unit cell volume gives an inverse change in reciprocal cell volume.

A first-order structural transition need not continuously deform one perfect reciprocal array into another. Phase coexistence, new orientation variants, domains and loss of single-crystal alignment can produce two sets of diffraction peaks or many differently oriented sets. The two endpoint lattice types alone do not specify the transformation path or orientation relationship.

For a divalent square Bravais lattice, let $E_X={\hslash }^2{\pi }^2/(2m_e{a}^2)$ and real periodic potential coefficients $V_{10}<V_{11}<0$. The leading two-wave edge levels are $E_X\pm V_{10}$. The four-wave corner levels are $2E_X+V_{11}\pm 2V_{10}$ and a doublet at $2E_X-V_{11}$. Comparing the lowest corner level with the upper edge level gives the displayed indirect band gap. The first spin-degenerate energy band can hold the two electrons per primitive unit cell; it becomes isolated when $-E_X-V_{11}-3V_{10}>0$ in this truncation. This follows from resonant degenerate perturbation theory, and higher plane waves can shift the threshold.

A crystal with $N$ primitive unit cells has $N$ distinct wavevectors in one Brillouin zone. For a rectangular periodic sample, $k_i=2\pi l_i/L_i$, and the product of the numbers of allowed values along the three directions equals the number of cells. Each orbital-derived energy band therefore holds $N$ states per spin orientation, or $2N$ with twofold spin degeneracy. This means two states per primitive cell per spin-degenerate band, independently of how many atoms the cell contains.

Take equal masses $m$, atomic spacing $a$, and alternating springs $K\gg \kappa >0$. The primitive unit cell contains a strong-bonded pair, so its length is $2a$ and the Brillouin zone ends at $\pm \pi /(2a)$. Expanding the exact alternating-spring chain dispersion relation givesThe low-frequency acoustic phonon branch moves almost rigid dimers of mass $2m$ coupled by weak springs; its speed of sound tends to $a\sqrt{2\kappa /m}$. The high-frequency optical phonon branch describes internal dimer stretch. At the zone edge the exact frequencies are $\sqrt{2\kappa /m}$ and $\sqrt{2K/m}$.

A region whose translates tile a lattice's ambient space, together with the lattice points represented in it. A primitive unit cell represents one lattice point; a conventional unit cell may represent several to display symmetry.

Putting $M=m$ in the diatomic-chain dispersion relation with equal springs gives ${\omega }_{-}=2\sqrt{K/m}|\sin (kd/2)|$ and ${\omega }_{+}=2\sqrt{K/m}|\cos (kd/2)|$ in the reduced zone $|k|\le \pi /(2d)$. The branches touch at its edge. If the sites are actually identical, the true primitive unit cell has length $d$; the apparent optical phonon branch is the outer part of the monatomic-chain dispersion relation folded by an unnecessarily doubled cell. A gap does not appear merely from redrawing the zone.