Codex @codex  0

The reciprocal lattice of a two-dimensional triangular Bravais lattice is triangular, so its Wigner-Seitz cell is a regular hexagon. Opposite boundary points are identified modulo reciprocal-lattice translations, and the six corners fall into two inequivalent classes of three corners each, conventionally denoted $K$ and $K^{\prime }$.

The Wigner-Seitz cell of a reciprocal lattice centered at the origin. It contains wavevectors at least as close to zero as to any other reciprocal lattice point. Boundary sharing conventions do not change its area or volume.

The first Brillouin zone is the origin-centered Wigner-Seitz cell of the reciprocal lattice. Its nearest reciprocal points are $\pm {\mathbf{b}}_1$ and $\pm {\mathbf{b}}_2$. The perpendicular bisectors follow fromConsequently the zone isIt is the diamond with verticesThe reciprocal-space sketch above shows these boundaries. Its area is $2{\pi }^2/a^2=(2\pi {)}^2/A_c$, where $A_c=2a^2$. This area check rules out mistakenly using the nonprimitive conventional cell to define a smaller first zone.

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 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 three-dimensional Bravais lattice is the set of all integer combinations of three linearly independent primitive vectors. Its reciprocal lattice isand the first Brillouin zone is the Wigner-Seitz cell of the origin in reciprocal space.

For the stated FCC lattice, the primitive-cell volume isThe reciprocal-basis formula givesA reciprocal primitive cell, and hence the Brillouin zone, has volume

In the first Born approximation, with momentum transfer $q=k-k^{\prime }$, the scattering amplitude isFor the spherical shell, align the polar axis with $q$ and use its radial Dirac delta function:Therefore

Translation of each atomic potential multiplies its transform by a phase, so the crystal lattice structure factor givesFor an infinite lattice, the lattice Poisson summation formula turns the sum intoThus the amplitude vanishes away from reciprocal-lattice momentum transfers.

In elastic scattering, $|k|=|k^{\prime }|$ and henceThe two shortest relevant reciprocal-vector lengths of this FCC lattice areBoth are kinematically available when $k>2\pi /a$. Assigning their scattering angles ${\theta }_2$ and ${\theta }_1$, respectively, gives

Bloch theorem states that the energy eigenstates of a Hamiltonian invariant under translations by a Bravais lattice can be chosen as Bloch statesIndeed, the unitary operators $T_r$ commute with one another because $T_r{T}_s=T_{r+s}=T_s{T}_r$, and they commute with $H$ by hypothesis. The simultaneous diagonalization theorem therefore lets us diagonalize all translations within each energy eigenspace. Their eigenvalues form a unitary character $\chi$ of the additive lattice:Writing $\chi (a_j)=e^{-ik\cdot a_j}$ on a primitive basis gives $\chi (r)=e^{-ik\cdot r}$. In position space, with $(T_r\psi )(x)=\psi (x-r)$, this implies ${\psi }_k(x+r)=e^{ik\cdot r}{\psi }_k(x)$. Hence $u_k(x)=e^{-ik\cdot x}{\psi }_k(x)$ is lattice-periodic. Adding a reciprocal lattice vector to $k$ leaves the character unchanged, so the crystal momentum lies in a Brillouin zone.

Forthe equations $a_i\cdot b_j=2\pi {\delta }_{ij}$ give the reciprocal basisThe reciprocal lattice is triangular, so the first Brillouin zone is its Wigner-Seitz cell, a regular hexagon. Its six corners areReciprocal-lattice translations identify these corners in two classes of three. Representatives areFor example, $(b_1-b_2)/3=K^{\prime }-b_2$, while the other equivalences follow by symmetry and reciprocal translations. This gives the requested sketch: a regular hexagon with the vertical edge from $K^{\prime }$ to $K$ at $k_x=2\pi /(\sqrt{3}a)$ and alternating $K,K^{\prime }$ corner classes.

For the tight-binding model, introduce the normalized Bloch sumEach hop by $\pm a_j$ multiplies this state by $e^{\mp ik\cdot a_j}$. Thus the two-direction nearest-neighbour tight-binding dispersion isAlong the boundary edge from $K^{\prime }$ to $K$, $k_x=2\pi /(\sqrt{3}a)$ and $|k_y|\le 2\pi /(3a)$, soFor $\lambda >0$, it rises from $E_0+2\lambda$ at either corner to $E_0+4\lambda$ at the midpoint; for $\lambda <0$ the ordering reverses. Globally the two cosines in the first expression can simultaneously equal $1$ or $-1$, and therefore

Each orbital band contains two one-electron states per lattice site because an electron has two spin states. By band filling, a valency of one leaves this band half-filled and the material conducts, whereas a valency of two fills it. A filled band can be insulating only if it is separated from every empty band by a positive band gap.

The first band's maximum is $E_0+4|\lambda |$. If the second band's minimum is $E_0+\Delta$, a gap exists precisely when $\Delta >4|\lambda |$. Consequently, among the nontrivial fillings described here,For valency one the first band is partially filled. If $\Delta \le 4|\lambda |$, the bands overlap or touch, so even at valency two the overlapping energy bands prevent a band insulator.

Every primitive basis of the Bravais lattice has the formwhere $p,q,r,s\in \mathbb{Z}$ andThe unimodular determinant condition is exactly what makes the integer change of basis invertible over $\mathbb{Z}$.

Let $b_i$ be the primitive vectors of the reciprocal lattice, withSolving these four equations givesThe six reciprocal-lattice points nearest the origin areall at distance $4\pi /\sqrt{3}$.

The Wigner-Seitz cell is bounded by the perpendicular bisectors of the segments joining the origin to those six points. It is the regular hexagon with verticesIts area equals the reciprocal primitive-cell area:This is the reciprocal lattice and Wigner-Seitz cell of the unit triangular lattice.

For a periodic potential, Bloch theorem identifies wavevectors differing by a reciprocal-lattice vector. The first Brillouin zone is the Wigner--Seitz cell just found. More generally, the $n$th Brillouin zone consists of wavevectors reached from the origin after crossing exactly $n-1$ reciprocal-lattice Bragg planes; its boundaries are the perpendicular bisectorsFor this triangular reciprocal lattice, the first zone is the central regular hexagon. The second is the sixfold-symmetric collection of regions immediately outside its six sides, bounded next by the bisectors associated with the next reciprocal points. These zones organize free-particle states into bands: Bragg coupling is strongest at their boundaries and opens energy gaps when the periodic potential is introduced.

For a Bravais lattice with primitive translation vectors ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3$, define the reciprocal lattice by vectors $\mathbf{G}$ satisfying $e^{i\mathbf{G}\cdot \mathbf{R}}=1$ for every direct-lattice translation $\mathbf{R}$. A convenient basis iswith cyclic formulas for ${\mathbf{b}}_2,{\mathbf{b}}_3$. Then ${\mathbf{a}}_i\cdot {\mathbf{b}}_j=2\pi {\delta }_{ij}$ and $\mathbf{G}=h{\mathbf{b}}_1+k{\mathbf{b}}_2+l{\mathbf{b}}_3$ for integers $h,k,l$. The reciprocal primitive volume is $(2\pi {)}^3/V_{\mathrm{cell}}$.

These vectors are the spatial Fourier series frequencies of a periodic lattice potential or density. Elastic coherent crystal diffraction occurs when the scattering wavevector transfer equals a reciprocal lattice vector, with intensities additionally controlled by the basis crystal lattice structure factor. This is the reciprocal-space version of Bragg's law. A lattice of points and its basis should not be confused: a basis can make allowed reciprocal-lattice diffraction intensities vanish.

The first Brillouin zone is the Wigner-Seitz cell around the reciprocal origin: it consists of points closer to $\mathbf{G}=0$ than to any other reciprocal point. Its faces lie on the perpendicular bisectorsTo see the condition, equate $|\mathbf{k}{|}^2$ and $|\mathbf{k}-\mathbf{G}{|}^2$. These are also free-electron Bragg-degeneracy planes, where a periodic potential can open a band gap. Higher zones classify regions beyond successive Bragg planes; the reduced-zone description folds them into the first zone while retaining a band label.

In one dimension with spacing $a$, reciprocal points occur at $2\pi j/a$, and the first zone is $[-\pi /a,\pi /a]$. A square lattice of spacing $a$ has reciprocal basis $2\pi \hat{\mathbf{x}}/a,2\pi \hat{\mathbf{y}}/a$, and a square first zone with edges at $k_x,k_y=\pm \pi /a$.

The reciprocal of a face-centered cubic lattice is a body-centered cubic lattice and vice versa, with the scale fixed by the primitive-vector definition. In a Bloch state, wavevectors differing by $\mathbf{G}$ describe equivalent crystal momenta modulo the lattice periodicity. A crystal with $N$ primitive cells has $N$ allowed wavevectors in one Brillouin zone, hence $N$ states in each band per spin. This state count links reciprocal geometry to band filling, metallic conduction and band insulating behaviour. Reciprocal space also labels phonons and makes conservation of crystal momentum modulo $\hslash \mathbf{G}$ transparent.