Bloch's theorem

In a periodic potential, energy eigenstates have the form $e^{ik\cdot x}{u}_k(x)$ with lattice-periodic $u_k$.

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.

Invariance under spatial inversion about a specified origin.

The Quantum Hamiltonian acting on the cell-periodic part of a Bloch state at fixed wave vector.

For a real scalar lattice-periodic potential, let $|n⟩$ be a nondegenerate zone-centre periodic eigenstate with energy $E_n$. Conjugation by $e^{i\mathbf{k}\cdot \mathbf{r}}$ gives $H(\mathbf{k})=H(0)+\hslash \mathbf{k}\cdot \mathbf{p}/m+{\hslash }^2{k}^2/(2m)$. The zone-centre state can be chosen real, so $⟨n|\mathbf{p}|n⟩=0$. Second-order nondegenerate perturbation theory then givesTime-reversal symmetry makes this isolated spinless band even in $\mathbf{k}$. In a direction $\mathbf{e}$, the curvature givesA degeneracy requires a matrix treatment before extracting curvatures; a nonlocal potential or spin-orbit terms can also alter the simple momentum vertex.

A quantum particle in a one-dimensional lattice experiences a periodic potential. Bloch states organize its spectrum into bands and gaps; the Kronig-Penney model realizes this behavior with an idealized array of barriers or narrow delta barriers.

The delta-comb Kronig-Penney potential is

For $V(x)=\alpha {\sum }_j\delta (x-ja)$ with $\alpha >0$, put $P=m\alpha a/{\hslash }^2$ and $x=ka$. The Floquet discriminant of the delta-comb Kronig-Penney model is $f(x)=\cos x+P\sin x/x$. Extended Bloch states require $|f(x)|\le 1$. Since $f(0)=1+P>1$, there is a positive lower band edge; a repulsive potential does not support negative-energy states in this energy convention.

The $j$th positive-energy band ends at $x=j\pi$. Its lower edge lies in $((j-1)\pi ,j\pi )$ and solves $f(x)=(-1{)}^{j-1}$. A finite periodic ring selects only the discrete allowed values of crystal momentum within these bands. Nonzero band gaps do not by themselves imply an insulator: the band filling must also place the Fermi energy in a gap.

For $E={\hslash }^2{k}^2/(2m)$ and $\gamma =m{V}_0/({\hslash }^{2}k)$,Thus the allowed bands satisfy $|\cos (ka)+\gamma \sin (ka)|\le 1$.

Forthe wavefunction is continuous across every lattice point and obeysAt negative energy $E=-{\hslash }^2{\mu }^2/(2m)$, an allowed energy band satisfiesAt positive energy $E={\hslash }^2{k}^2/(2m)$, it satisfies

With well strength $\lambda$ and spacing $a$, negative-energy Bloch states obey $\tanh y\le 2y/(\lambda a)\le \coth y$, where $y=\kappa a/2$. There is one negative band: it reaches zero when $\lambda a\le 2$ and is separated from zero when $\lambda a>2$.

If one symmetric cell has reflection and transmission amplitudes $r,t$, thenAt a periodic or antiperiodic band edge, with $\mathrm{tr}M=2\sigma$ and $\sigma =\pm 1$, setting $z=e^{-ika}$ gives

Electronic band structure is the collection of allowed electron energy bands and gaps in a periodic medium, usually represented as energy versus crystal momentum. An energy band is one branch of that structure.

For a local periodic scalar potential, writing a Bloch wave as $e^{i\mathbf{k}\cdot \mathbf{r}}{u}_{n\mathbf{k}}$ gives the displayed cell-periodic Hamiltonian before additional spin-orbit terms. Momentum matrix elements mix zone-centre bands when k is varied. Perturbatively eliminating remote bands changes the effective mass; a multiband retained subspace describes degeneracy, nonparabolicity and coupled hole bands. Spin-orbit coupling and the chosen reference wavevector must be included consistently.

An approximation $E(k)=E_0+{\hslash }^2{k}^2/(2m^{\ast })$ defining an effective mass near a band extremum. A fitted positive curvature gives positive effective mass.

An interband transition moves an electron between distinct energy bands. A direct optical transition approximately conserves crystal momentum because optical photon momentum is small compared with a reciprocal lattice vector. A phonon-assisted transition can supply the momentum difference needed near an indirect band gap. Transition energies alone do not fix their optical intensities, which also depend on matrix elements and occupations.

An optical interband transition that requires another momentum source, commonly a phonon, to connect initial and final electronic wavevectors. Near an indirect band gap, phonon emission gives a threshold photon energy $E_g+\hslash \Omega$, while phonon absorption gives $E_g-\hslash \Omega$ and requires an occupied phonon mode. The temperature dependence of the absorption therefore contains phonon-population information in addition to electronic occupations.

A photon-induced interband transition that approximately preserves the electron's crystal wavevector because the photon momentum is small compared with reciprocal-lattice momenta. Its strength depends on the dipole matrix element and the joint density of states. A direct band gap permits such a transition between the band-edge states, but optical selection rules can still suppress it.

A vertical direct transition at a common wavevector puts an electron above the conduction edge and leaves a hole below the valence edge. Their parabolic kinetic energies are inversely proportional to their effective masses. The excess photon energy is therefore split as $K_e/K_h=m_h^{\ast }/m_e^{\ast }$, rather than equally unless the masses match. The initial valence electron energy is $-K_h$ relative to its band maximum; the final conduction energy is $E_g+K_e$.

Absorption in which a photon excites an electron between different energy bands, requiring an occupied initial state, an available final state and a nonzero transition matrix element. Indirect band-edge transitions also need momentum transfer, commonly through a phonon. Such absorption can prevent optical transmission even above a metal's free-carrier plasma edge.

A superlattice is a spatial periodic structure with a repeat length larger than an underlying atomic lattice period. Its additional reciprocal lattice vectors fold electronic bands into a smaller Brillouin zone, and allowed couplings can open small gaps at the folded crossings. The gap sizes depend on potential Fourier components and symmetry, not on the geometric period alone.

A flat band has energy independent of wavevector, and hence zero group velocity. A nearly flat band has small dispersion and, near a parabolic extremum, large band effective mass. Coupling a flat band to a dispersive band can produce an avoided crossing and a hybridization gap.

Band hybridization mixes two orbital-derived bands through an off-diagonal Hamiltonian matrix element. For a real coupling $\alpha$,An allowed bare crossing becomes an avoided crossing with direct separation $2|\alpha |$. Orbital character transfers continuously between branches through the crossing. A direct separation alone does not guarantee a global band gap; the extrema at different wavevectors must also be compared.

Mix a parabolic energy band ${ϵ}_k={\hslash }^2{k}^2/(2m)$ with a perfectly flat band at $E_d>0$ through momentum-independent $V$. At zero coupling suppose the two orbital sectors have equal populations, with the dispersive sector filled to $k_0=\sqrt{2m{E}_d}/\hslash$. At fixed total density and sufficiently low temperature, nonzero coupling fills only the monotonic lower hybridized branch, whose Fermi wavevector is $k_F=2^{1/3}{k}_0$, provided the available momentum-space states accommodate it. Its small slope at that radius gives the displayed ratio to $v_F^{(1)}=\hslash k_0/m$ for $|V|\ll E_d$. The result uses the singular perfectly flat limit and an empty upper band; it is not implied by Fermi-volume conservation in two hybridized electron bands for arbitrary pairs of dispersive bands. The zero-temperature limit must resolve the small hybridization energy scale.

For two isotropic electron-like bands with equal spin degeneracy, conserving total carrier density conserves the sum of their occupied momentum-space volumes. If both uncoupled radii equal $k_0$,Only when the upper pocket is empty does this imply a single Fermi wavevector $k_F=2^{1/3}{k}_0$; a nonzero mixing matrix element alone does not imply that emptying.

For a counterexample at arbitrarily weak coupling, take $E_a=A{k}^2$, $E_b=C+B{k}^2$, $A>B>0$, and $C=(A-B)k_0^2$. At the uncoupled chemical potential ${\mu }_0=A{k}_0^2$, both pockets have radius $k_0$. Let $\mu ={\mu }_0+\delta \mu$ and $y=k_F^2-k_0^2$. The characteristic equation at the Fermi level becomesFixed density cancels the first-order chemical-potential shift. Thus $\delta \mu =O({\alpha }^2)$ and $k_{F,\mp }=k_0\pm \alpha /(2k_0\sqrt{AB})+O({\alpha }^2)$: both pockets remain occupied at small coupling. This expansion assumes fixed $A,B>0$ before taking $\alpha \to 0$.

A hybridization gap is a separation in electronic band structure caused by mixing orbital-derived states. For $E_a=A{k}^2$, $E_b=E_0>0$, $A>0$ and constant coupling $\alpha >0$, the lower eigenvalue is everywhere below $E_0$ and tends to it as $k\to \infty$. The upper eigenvalue has minimum $(E_0+\sqrt{E_0^2+4{\alpha }^2})/2$. The unbounded model therefore has an indirect gap of width $(\sqrt{E_0^2+4{\alpha }^2}-E_0)/2$, approximately ${\alpha }^2/E_0$ at weak coupling. This differs from the direct separation $2\alpha$ at the bare crossing. A finite Brillouin zone bounds the lower band farther below $E_0$.

A Mott insulator is insulating because of electron interactions even where a noninteracting electronic band structure would predict conduction. This mechanism differs from disorder-driven Anderson localization.

An energy band is a continuous branch $E_j(k)$ of the dispersion relation of a periodic quantum system.

The range of energy within one electronic energy band, obtained by varying its wavevector. This differs from an interband band gap.

The electronic bandwidth is the difference between the maximum and minimum of one energy band over the Brillouin zone. It has units of energy, unlike a frequency bandwidth in signal processing. A nearest-neighbour tight-binding model on a primitive cubic lattice with equal real hopping integrals $t$ has $E(\mathbf{k})=E_0+2t{\sum }_{i=1}^3\cos (k_{i}a)$ and $W=12|t|$.

A conical touching of two energy bands, with locally linear dispersion relation. The nearest-neighbour model of graphene supplies an explicit example.

A region of momentum space surrounding a Dirac cone. The two inequivalent Dirac valleys in graphene provide a familiar example; a valley labels a region and is not the electron spin.

A region of wave vector space around a low-energy band feature, such as a band minimum or a Dirac point.

The number of symmetry-related equivalent electronic valleys contributing to a specified low-energy energy band. It multiplies the single-valley density of states, independently of spin degeneracy. Equal electron and hole effective masses alone do not ensure equal total density of states values unless their valley degeneracies and spin degeneracies also agree.

A spatially localized electronic level produced by a defect or foreign atom in a crystalline potential. Such levels can lie in a host band gap, where no extended host state has the same energy, while impurity resonances can also occur inside an energy band. Dilute localized levels are nearly independent; increasing overlap can broaden them into an impurity band. The location and existence of a bound level depend on the impurity potential.

A localized orbital with two electron spin states has energies $E_0=0$, $E_1=ϵ$ and $E_2=2ϵ+U$. Its zero-temperature equilibrium occupation minimizes ${\mathcal{E}}_N=E_N-\mu N$, where $\mu$ is the chemical potential. For repulsion $U>0$, the occupation is successively $0,1,2$ as $\mu$ crosses $ϵ$ and $ϵ+U$. For attraction $U<0$, the singly occupied state is never the minimum: the occupation jumps from $0$ to $2$ at $\mu =ϵ+U/2$. Exactly at a crossing the ground-state occupation is not unique.

At positive temperature, its grand canonical partition function is $\Xi =1+2e^{-\beta (ϵ-\mu )}+e^{-\beta (2ϵ+U-2\mu )}$, where $\beta =1/(k_{B}T)$. The expected value of the occupation follows by differentiating: $⟨N⟩=k_{B}T{\partial }_{\mu }\ln \Xi$. The factor two counts the two singly occupied microstates, not an additional pair state.

In one dimension the band effective mass is defined by $1/m^{\ast }(k)=E^{\prime \prime }(k)/{\hslash }^2$. Together with $v=E^{\prime }(k)/\hslash$ and $\hslash \dot{k}=F$, this gives $\dot{v}=F/m^{\ast }$. Negative band curvature produces negative effective mass for the electron wave packet; near a filled-band maximum the equivalent missing-electron description uses positively responding holes. In several dimensions the inverse mass is the Hessian ${\partial }_i{\partial }_{j}E/{\hslash }^2$.

The curvature tensor of a nondegenerate energy band, determining the linear relation between force and acceleration in semiclassical band dynamics. From group velocity $v_i={\hslash }^{-1}{\partial }_{k_i}E$ and $\hslash {\dot{k}}_j=F_j$, one obtains ${\dot{v}}_i=(M^{-1}{)}_{ij}{F}_j$. Near a parabolic minimum, $E-E_0={\hslash }^2\delta {\mathbf{k}}^T{M}^{-1}\delta \mathbf{k}/2$. The principal values are reciprocal curvature effective masses; a cyclotron experiment measures a direction-dependent combination rather than every tensor entry separately.

For an isotropic valence-band maximum, the electron curvature is negative. The missing-electron excitation has positive charge and positive mass $m_h^{\ast }={\hslash }^2/|E\prime \prime |$. Its cyclotron angular frequency is $eB/m_h^{\ast }$. An anisotropic orbit requires the corresponding cyclotron mass rather than an arbitrary single directional curvature.

Take the nearest-neighbour hopping $t(a)=-2E_1(1+a/a_1)e^{-a/a_1}$, with $E_1,a_1>0$, in a simple-cubic nearest-neighbour tight-binding band. Substituting into $m^{\ast }=-{\hslash }^2/(2t{a}^2)$ gives the displayed result. With $s=a/a_1$, its logarithmic derivative is $1-2/s-1/(1+s)$, which vanishes only at $s=1+\sqrt{3}>0$. The mass diverges at both positive-domain ends in this formula, so that point is its unique minimum. Its bandwidth has derivative $d\ln W/d\ln a=-s^2/(1+s)$: a small compression increases $W$. The small-$a$ continuation lies beyond reliable localized-orbital tight binding.

For a semiclassical wave packet in an isolated energy band, $v_i={\hslash }^{-1}{\partial }_{k_i}\epsilon$ and $\hslash {\dot{k}}_j=F_j$ imply ${\dot{v}}_i={\sum }_j(m^{\ast -1}{)}_{ij}{F}_j$. The inverse effective-mass tensor is the Hessian matrix of the dispersion divided by ${\hslash }^2$. Along a unit vector $\mathbf{e}$, $1/m_{\mathbf{e}}^{\ast }={\sum }_{ij}{e}_i(m^{\ast -1}{)}_{ij}{e}_j$. The Laplacian is its trace: ${\hslash }^{-2}{\nabla }_k^2\epsilon =\mathrm{tr}(m^{\ast -1})$. Thus in $d$ isotropic dimensions the scalar mass is $d{\hslash }^2/({\nabla }_k^2\epsilon )$, rather than ${\hslash }^2/({\nabla }_k^2\epsilon )$.

In a one-dimensional two-wave nearly-free electron model, set ${\epsilon }_B={\hslash }^2{G}^2/(8m)$ and let the coupling magnitude be $U>0$. Near the Bragg plane,Expansion at $q=0$ gives $E_{\pm }={\epsilon }_B\pm U+[{\hslash }^2/(2m)](1\pm 2{\epsilon }_B/U)q^2+O(q^4)$, henceThe lower edge has negative electronic curvature, and the upper edge positive curvature. The last expression is a leading weak-potential approximation, not the exact curvature of the two-wave dispersion.

Band curvature is the Hessian matrix of an energy band as a function of wavevector. Dividing it by ${\hslash }^2$ gives the inverse band effective mass tensor. At a one-dimensional band minimum the curvature is positive, whereas at a maximum it is negative. Near a weak two-wave avoided crossing with reciprocal wavenumber $G$ and coupling magnitude $U$, the two edge curvatures are $E_{\pm }^{\prime \prime }={\hslash }^2/m\pm {\hslash }^4{G}^2/(4m^{2}U)$. This is a crystal response coefficient, not a change in the bare electron mass.

For $t(a)=-2E_1{e}^{-a/a_1}$ with $E_1,a_1>0$, a nearest-neighbour cubic tight-binding model has zone-centre band effective massIts logarithmic derivative is $1/a_1-2/a$. The unique minimum occurs at $a=2a_1$ and has value ${\hslash }^2\exp (2)/(16E_1{a}_1^2)$. The explicit $a^2$ in band curvature competes with the exponentially weakening hopping integral. Extrapolation to vanishing atomic separation lies outside the localized-orbital approximation.

For a closed semiclassical orbit with area $S(E)$ in a plane of wave vector space, the cyclotron effective mass is defined by the displayed derivative. It need not equal the local curvature-based band effective mass.

For a positive definite parabolic effective mass tensor $M$, the Lorentz force equation is $M\dot{\mathbf{v}}=e\mathbf{B}\times \mathbf{v}$. Its nonzero eigenfrequencies are $\pm eB/m_c$, giving the displayed mass. With principal values $(m_t,m_t,m_l)$, a field along the long valley axis gives $m_c=m_t$; a perpendicular field gives $\sqrt{m_t{m}_l}$. For equally oriented cubic valleys viewed along $[111]$, all valleys give $m_c=\sqrt{3m_t^2{m}_l/(2m_t+m_l)}$.

Oscillations of electrical resistance as a magnetic field moves Landau levels through the Fermi energy.

A finite-temperature measurement averages the oscillatory response over the derivative of the Fermi-Dirac distribution. For the fundamental harmonic and a locally constant cyclotron effective mass, this gives $R_T=X/\sinh X$ with $X=2{\pi }^2{k}_{B}T/(\hslash {\omega }_c)$. The oscillations become small when thermal smearing substantially exceeds their Landau-level spacing.

The Dingle factor describes the damping of quantum oscillations by Landau-level broadening. For a Lorentzian half-width $\hslash /(2{\tau }_q)$, the fundamental harmonic has $R_D=\exp [-\pi /({\omega }_c{\tau }_q)]$. Since ${\omega }_c$ decreases with field, the amplitude decays at large inverse field. The lifetime here is the quantum lifetime, not automatically the mobility-derived transport time.

An allowed energy band is an interval of energies supporting extended Bloch states. Adjacent bands may be separated by a band gap.

A band gap is an energy interval between adjacent energy bands containing no allowed bulk state.

A direct band gap compares two energy bands at the same crystal momentum. A global gap instead requires the lowest energy of the upper band to exceed the highest energy of the lower band, allowing the two extrema to occur at different momenta. For example, the periodic toy bands $E_1(k)=A\cos k$ and $E_2(k)=A\cos k+\delta$, with $A>0$ and $0<\delta <2A$, have positive direct separation $\delta$ everywhere but global overlap $2A-\delta$. A positive direct splitting at each Brillouin zone boundary therefore does not alone establish an insulating filled band. At a suitable electron count the overlap can produce both electron and hole pockets.

An indirect band gap separates a valence band maximum and a conduction band minimum at different values of crystal momentum. A near-edge radiative transition then usually needs a phonon or another momentum-transfer process in addition to the photon, whose momentum is small on the Brillouin zone scale. A direct band gap has the extrema at the same crystal momentum and more readily supports efficient band-edge light emission. A material can have direct transitions at higher energy while its smallest gap is indirect.

For $-V_0(\cos Qx+\cos Qy)$, two-dimensional bands are sums of one-dimensional bands. At first order, the first-band maximum is $E_Q/2-V_0$ at the corner, while the second-band minimum is $E_Q/4+V_0/2$ at an edge centre. Hence ${\Delta }_{\mathrm{ind}}\simeq 3V_0/2-E_Q/4$, giving a rough onset $V_0\sim E_Q/6$. Higher plane waves shift this estimate; insulating filling must be checked separately.

A band gap whose conduction band minimum and valence band maximum occur at the same crystal momentum, allowing an optical transition without a momentum-balancing phonon.

An avoided crossing occurs when two parameter-dependent eigenvalues approach one another but an interaction prevents equality. Near a two-level crossing, a nonzero off-diagonal matrix element opens a positive minimum gap.

For spin-one-half electrons, one nondegenerate orbital band contains two one-particle states per primitive cell, one for each spin. A partially filled band has nearby empty states into which an electric field can move electrons and is ordinarily conducting. A completely filled band carries no net current under a small field because all crystal momenta are occupied symmetrically.

With spin degeneracy two, one full square-zone band has density $Q^2/(2{\pi }^2)$. Thus $n_e-n_h=n-Q^2/(2{\pi }^2)$. If $Q=\alpha \sqrt{2\pi n}$, compensation requires $\alpha =\sqrt{\pi }$ and the real-space cell contains two electrons. A global gap is also necessary for an insulator.

A band insulator has an integer number of completely filled energy bands, with the Fermi energy inside a positive band gap. At zero temperature, no arbitrarily low-energy charged excitation is then available.

In an independent-electron crystal with two spin states and no spin splitting, each completely filled energy band contains two electrons per primitive cell. An odd electron count per primitive cell therefore cannot fill an integer number of such bands. Enlarging the actual primitive cell, breaking the relevant degeneracy or including correlated-electron physics changes the premises. Even filling is necessary in this model but not sufficient: bands can overlap.

A Kondo insulator has a bulk insulating state associated with mixing between relatively localized correlated electronic states and itinerant conduction states. A renormalized two-band picture captures a narrow hybridization gap, although an uncorrelated orbital-mixing model does not explain the formation and temperature dependence of the coherent low-energy state. Samarium hexaboride is an example.

If the minimum of the next energy band lies at or below the maximum of the nominally filled band, electrons can lower their energy by occupying the next band and leaving holes in the first. The resulting partially filled bands prevent a band-insulating state.

Suppose a model replaces a set of free-electron boundary energies $E^{(0)}$ by lower and upper edges $E^{(0)}-V$ and $E^{(0)}+V$, with $V>0$. If these are the relevant global band edges, the indirect band gap is $(E_{\min }^{(0)}+V)-(E_{\max }^{(0)}-V)$, giving the displayed expression. A positive direct gap $2V$ at each crossing need not be enough: a band insulator needs $2V>E_{\max }^{(0)}-E_{\min }^{(0)}$. This is a band-edge estimate, not a general three-dimensional nearly-free-electron theorem; multiple Bragg planes at a corner can require a larger coupling matrix.

In a single isolated energy band, a wave packet subject to a constant force satisfies $\hslash \dot{k}=F$. Its group velocity is periodic in $k$ with reciprocal-lattice period $2\pi /a$, so its position oscillates with period $2\pi \hslash /(|F|a)$ when its mean velocity over a band is zero. For $E(k)=E_0-2J\cos ka$, integrating the sinusoidal velocity gives the oscillation explicitly. This ideal description neglects scattering and interband tunnelling.

For $ϵ(k)={ϵ}_0-2J\cos ka$ and electron force $-eE$, $k(t)=k_0-eEt/\hslash$. Integrating the group velocity gives $x(t)-x(0)=2J[\cos (k_{0}a-{\omega }_{B}t)-\cos (k_{0}a)]/(eE)$, with ${\omega }_B=eEa/\hslash$ for positive $E$. The amplitude about the mean is $2|J|/(e|E|)$ and the peak-to-peak extent is twice that.

In a 1996 experiment, ultracold cesium atoms were loaded into the lowest band of an optical standing wave made with $852\mathrm{nm}$ light. Chirping the frequency difference of its counterpropagating beams accelerated the lattice, producing a constant inertial force in its frame. Switching off the lattice after different evolution times allowed momentum-distribution measurements. The observed oscillations tested the periodic band velocity underlying a Bloch oscillation; the lattice period and imposed acceleration controlled the oscillation frequency.^[ref]

The lattice translations commute with one another and with a lattice-periodic Hamiltonian. They can therefore be diagonalized simultaneously within each energy eigenspace. Every unitary character of a Bravais lattice has the form $r\mapsto e^{-ik\cdot r}$, with $k$ defined modulo the reciprocal lattice. Hence $T_r{\psi }_k=e^{-ik\cdot r}{\psi }_k$, so $u_k(x)=e^{-ik\cdot x}{\psi }_k(x)$ is lattice-periodic.

A Bloch state is an energy eigenstate of a lattice-periodic Hamiltonian that is simultaneously an eigenstate of every lattice translation. It can be written as ${\psi }_k(x)=e^{ik\cdot x}{u}_k(x)$ with lattice-periodic $u_k$.

Crystal momentum is the translation quantum number $\hslash k$ of a Bloch state. Since $k$ and $k+G$ give the same translation character whenever $G$ belongs to the reciprocal lattice, crystal momentum is defined modulo reciprocal-lattice vectors.

A charge $-e$ starting at a band minimum when the field $E_0(\sin \omega t,\cos \omega t)$ is applied acquires the displayed crystal momentum. Its orbit is a shifted circle. For masses $m_x,m_y$, the real-space coordinates are proportional to $(\sin \omega t-\omega t)/m_x$ and $(\cos \omega t-1)/m_y$: an elliptic oscillation plus a uniform drift. Omitting the integration constant changes the physical initial condition and reduces the maximum light-axis kinetic energy by a factor of four.

A lattice-translation-invariant interaction conserves total crystal momentum modulo a reciprocal lattice vector $G$. On a periodic chain of $M$ sites with unit spacing, the discrete Fourier transform supplies ${\sum }_{j=1}^M{e}^{i(k_1+k_2-k_3-k_4)j}=M{\delta }_{k_1+k_2,k_3+k_4(\mathrm{mod}2\pi )}$. This Fourier orthogonality proves the selection rule directly; it permits transfer by a reciprocal lattice vector rather than requiring equality of unrestricted momenta.

A periodic potential satisfies $V(x+a)=V(x)$ for some lattice period $a$. Its Fourier coefficients couple plane waves whose wavevectors differ by reciprocal-lattice vectors.

The nearly-free electron model treats a weak periodic potential by degenerate perturbation theory. At a Bragg crossing, a Fourier coefficient $V_n$ couples the two degenerate plane waves and opens an energy gap of width $2|V_n|$.

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.

For rectangular real-space periods $a>b$, the free first-zone corner has energy ${\hslash }^2{\pi }^2(a^{-2}+b^{-2})/(2m)$, whereas the free second-zone minimum is ${\hslash }^2{\pi }^2/(2m{a}^2)$. Their difference is the displayed overlap. A divalent spin-degenerate system is a band insulator only when the actual first-band maximum lies below the second-band minimum. A rigid symmetric splitting $\Delta$ removes the overlap if $\Delta >{\Delta }_{\mathrm{overlap}}$, but an arbitrary periodic potential does not shift every corner and face by the same amount. The exact criterion uses the indirect band gap, not a geometry-only local direct-gap width.

For two electrons per conventional monovalent BCC cell, $k_F=(6{\pi }^2{)}^{1/3}/a$ lies inside the first zone. The shortest reciprocal vector is $G=2\sqrt{2}\pi /a$. The free folded-branch difference ${\hslash }^2(G^2-2Gk)/(2m_e)$ is smallest on the occupied Fermi-sphere point toward G, giving the displayed ratio. The destination has magnitude $G-k_F>k_F$ and is empty. A weak periodic potential supplies band mixing and optical matrix elements; its finite corrections and absorption peak shape are not fixed by this leading threshold.

For $-V_0\cos Qx$, the Fourier coupling between the two free states at $k=\pm Q/2$ is $-V_0/2$. The split energies are $E_Q/4\mp V_0/2$, with $E_Q={\hslash }^2{Q}^2/(2m)$. The direct face gap is $V_0$, not $2V_0$.

Consider a spin-degenerate two-dimensional free electron model with rectangular periods $a>b$ and two electrons per unit cell. Write $K=\pi /a$, $L=\pi /b$, and $k_F^2=4\pi /(ab)$. If $K<k_F<L$, the circular Fermi contour crosses only the vertical boundaries of the first Brillouin zone. Folding translates its protruding caps back by a reciprocal lattice vector $(2K,0)$. A weak periodic potential mixes the intersecting plane waves, reconnecting the contours into two open first-band sheets and a closed second-band pocket. The pocket is centred on a zone boundary: its halves on opposite sides of a reduced-zone plot are identified periodically and belong to one closed sheet.

Put $C={\hslash }^2/(2m)$, $v=|V_G|$, $\mathbf{G}=(2K,0)$, and $k_x=K+q$. The upper eigenvalue of the two coupled plane waves isAt fixed Fermi energy $C{k}_F^2$, its closed Fermi contour hasThus $A(v)=4{\int }_0^{q_m}y(q)dq$ can be inverted numerically to estimate the Bragg scattering coupling. The pocket disappears at $v=C(k_F^2-K^2)$. The estimate assumes a fixed chemical potential and neglects coupling to other plane waves; a measured pocket area alone does not determine all Fourier coefficients of an arbitrary lattice potential.

In rectangular-lattice Fermi-surface reconstruction, the zero-coupling closed orbit is the overlap lens of two equal circles of radius $k_F$ with centres separated by $2K$. Two circular sectors minus their triangular portions giveAt fixed chemical potential, the upper hybridized energy band rises when the magnitude of the Fourier coefficient increases. Its occupied pocket therefore shrinks, so this lens is an upper bound in the two-wave model. It is not a universal bound for arbitrary changes of chemical potential or additional energy bands.