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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.

A Bragg point is a crystal momentum at which two plane waves differing by a reciprocal lattice vector are degenerate. A periodic potential mixes the waves there, producing an avoided crossing and opening a band gap.

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 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 electron hole is an unoccupied electron quantum state in an otherwise nearly full energy band, often the valence band. “Filled” refers to occupation of the allowed quantum states, not to filling all space with a continuous fluid of electrons. Each state, including its spin label, can hold at most one electron by the Pauli exclusion principle. The almost full energy band and its few missing occupants are therefore simpler to describe as a background plus a few electron holes. The electron hole is a quasiparticle, not a Positron inserted into the crystal.

Removing an electron of electric charge $-e$ changes the background's electric charge by $+e$. Thus an electron hole carries positive electric charge relative to that background. Promoting an electron into the conduction band creates an electron hole without changing the whole crystal's electric charge: $(-e)+(+e)=0$. An unbound pair still consists of two separate charge carriers; only a bound state of their relative motion is an exciton.

The local cartoon explains why a vacancy can move: when a neighboring electron fills it, another vacancy appears where that electron was. The vacancy has moved opposite to that particular transfer of an electron. In an extended Bloch state, use crystal momentum instead of imagining a permanently empty atomic site. For a completely filled energy band the summed electric current vanishes. Omitting one electron with group velocity ${\mathbf{v}}_v$ removes its contribution $-e{\mathbf{v}}_v$, leaving a contribution $+e{\mathbf{v}}_v$. This fixes both the positive electric charge and the velocity convention of the electron hole.

More precisely, near a parabolic valence band maximum, set its energy $E_v(0)$ as reference and define the electron hole wave vector by ${\mathbf{k}}_h=-{\mathbf{k}}_v$. ThenThe occupied electron branch has negative curvature there, while the electron hole branch has positive effective mass $m_h^{\ast }$. The velocity equality compares the electron hole with the missing Bloch state's group velocity; it does not contradict the opposite vacancy motion in a local transfer cartoon. The sign conventions are listed in Phys 460 Lecture 17.

In electron-phonon scattering, an electron changes its energy by $\pm \hslash {\omega }_{\mathbf{q}}$ while its crystal momentum changes by a phonon wavevector modulo a reciprocal lattice vector. Charge-transport relaxation weights a scattering angle $\theta$ by $1-\cos \theta$. Small-angle events can therefore relax heat efficiently while relaxing electrical current weakly. At temperatures above the phonon excitation scale, scattering becomes approximately elastic in the electron thermal shell; residual elastic disorder often dominates at very low temperature.

The exciton binding energy is the positive energy needed to separate a bound electron and electron hole into the relevant unbound scattering states. Compare the exciton energy $E_X$ with the continuum threshold $E_{\mathrm{cont}}$ at the same total crystal momentum and for the same energy bands. The negative internal energy eigenvalue $-E_B$ and the positive separation cost $E_B$ are different sign conventions for related quantities. An exciton may move freely through a perfect crystal while remaining internally bound: “free exciton” does not mean an unbound pair.

For the ideal Wannier-Mott exciton in energy bands whose edges occur at the same crystal momentum, the pair excitation energy, measured from the unexcited crystal, isHere $\hslash \mathbf{K}$ is the pair's total crystal momentum; the last term is center of mass kinetic energy. For a permitted photon absorption transition with negligible recoil energy, $\hslash \omega \simeq E_g-E_{B,n}$. An electric-dipole selection rule and a nonzero transition matrix element are still needed: a bound state need not produce an observable spectral line. If the relevant energy band edges have different values of crystal momentum, a phonon can also be needed to conserve crystal momentum; subtracting a binding scale from a band gap alone does not specify the optical threshold.

For example, the lowest internal bound state in this hypothetical model costs $1.489115\mathrm{eV}$ to create optically, yet needs only $0.010885\mathrm{eV}$ more to dissociate. The large creation energy and the small exciton binding energy answer different physical questions. An electron-hole recombination event can instead release excitation energy by photon emission or to other degrees of freedom; an emitted photon need not reproduce the original absorption energy if relaxation intervened.

The crystal momentum of a phonon is defined modulo a reciprocal lattice vector. Extending its dispersion relation periodically beyond the first Brillouin zone labels equivalent excitations by different extended wavevectors. A scattering transfer $Q=G+q$ can be outside that zone while its reduced phonon wavevector $q$ is inside. The measured neutron recoil depends on the actual transfer $Q$, not on the reduced representative alone.

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.

Inelastic X-ray scattering measures energy and momentum transferred from X-rays to matter. With sufficiently high energy resolution, it can map a phonon dispersion relation by measuring inelastic peaks at different momentum transfers. Crystal momentum is determined modulo a reciprocal lattice vector.

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.

For filled valence and empty conduction energy bands, the joint density of states counts pairs of states at a prescribed interband energy difference. In a direct optical transition the photon momentum is usually negligible compared with a Brillouin zone dimension, so initial and final states have nearly the same crystal momentum. The absorption spectrum weights this count by transition matrix elements; its onset and structure can constrain the band gap and electronic band structure. Indirect transitions additionally involve phonons to supply crystal momentum.

Start from electrons moving in the periodic equilibrium ionic potential. Displace ion $l$ by ${\mathbf{u}}_l$ and expand its contribution as $V(\mathbf{r}-{\mathbf{R}}_l-{\mathbf{u}}_l)\simeq V(\mathbf{r}-{\mathbf{R}}_l)-{\mathbf{u}}_l\cdot \nabla V(\mathbf{r}-{\mathbf{R}}_l)$. Quantizing the harmonic normal modes expresses ${\mathbf{u}}_l$ as a sum of creation and annihilation operators with displacement scale $\sqrt{\hslash /(2{\rho }_{\mathrm{ion}}V{\omega }_{\mathbf{q}})}$ for a single simple branch. Expanding the electronic field in bloch states then gives a band-projected Fröhlich HamiltonianHere $c,c^{†}$ obey canonical anticommutation relations and remove/create an electron with crystal momentum $\mathbf{k}$ and spin $\sigma$; $b,b^{†}$ are boson operators for lattice modes. The matrix element $M_{\mathbf{q}}$ contains the derivative of the ionic potential, phonon polarization, displacement normalization and electronic wavefunction overlaps. With phases chosen appropriately, $M_{-\mathbf{q}}=M_{\mathbf{q}}^{\ast }$ ensures a Hermitian Hamiltonian. The displayed one-band, one-branch notation suppresses additional branch and band labels. “Fröhlich Hamiltonian” here means the linear electron-phonon model used in the question; it is not a requirement to use specifically the polar-optical-phonon coupling of a polar semiconductor. A harmonic lattice, small displacements and a weak enough coupling to organize perturbation theory are the underlying approximations.

The Fröhlich canonical transformation removes real linear phonon operators from the electron-only effective problem to first order. Choose an anti-Hermitian $S$ with $[S,H_0]=-V_{e\text{-}ph}$. For $A_{\mathbf{kq}\sigma }=c_{\mathbf{k}+\mathbf{q},\sigma }^{†}{c}_{\mathbf{k}\sigma }$ and ${\Delta }_{\mathbf{kq}}={ϵ}_{\mathbf{k}+\mathbf{q}}-{ϵ}_{\mathbf{k}}$, one possible generator isIts Hermitian conjugate is its negative after relabelling momenta. The Baker--Campbell--Hausdorff formula givesThe commutator describes virtual emission and absorption of a phonon. normal ordering yields an electron-electron term as well as one-electron energy renormalizations and phonon terms. Retain the former, absorb single-particle shifts into ${ϵ}_{\mathbf{k}}$ and eliminate the explicit phonon sector. For elastic pair scattering the two virtual time orderings have the combined amplitudeUsing the usual factor one-half to count unordered pairs gives the coefficient in the printed Hamiltonian, with $\Delta ={ϵ}_{\mathbf{k}}-{ϵ}_{\mathbf{k}-\mathbf{q}}$. The two outgoing momenta are $\mathbf{k}-\mathbf{q}$ and ${\mathbf{k}}^{\prime }+\mathbf{q}$, so total momentum and the spins are conserved. A fully off-shell effective operator can instead use symmetrized denominators for both external legs; the displayed expression is the usual elastic-scattering reduction. Eliminating nearly resonant real phonons requires more care than this second-order construction.

For $|\Delta |<{\Omega }_{\mathbf{q}}$ the denominator is negative: virtual phonon exchange is attractive in this low-energy channel. Physically, an electron changes the ionic density and a second electron can gain energy by responding to the delayed distortion. This is a retarded interaction, not an unscreened instantaneous attraction at every energy transfer. At large transfer the sign changes, and screened Coulomb repulsion must also be included when deciding the net low-energy pairing strength. The connection between lattice dynamics and superconductivity is supported by the isotope effect, but the mathematical attraction alone does not determine a measured transition temperature.^[ref]

For the pair calculation, make the model restrictions explicit. The filled Fermi sea is inert, and $|0⟩$ denotes that sea with no added particles, not an empty electronic vacuum. Project the interaction onto the previously empty shell $0\le {\xi }_{\mathbf{k}}\le D$, where ${\xi }_{\mathbf{k}}={\hslash }^2{k}^2/(2m)-E_F$ and $D=\hslash {\omega }_D$. Measure the added-pair energy relative to $2E_F$; this is the negative $E$ in the final formula. If absolute one-electron energies are retained in the preceding eigenvalue equation, its eigenvalue must instead be $E_{\mathrm{abs}}=2E_F+E$. A shell-restricted state would not in general be an exact eigenstate of the completely unprojected, all-momentum contact Hamiltonian.

Write $g={\gamma }^2>0$ and $|\mathbf{k}⟩=c_{-\mathbf{k}\downarrow }^{†}{c}_{\mathbf{k}\uparrow }^{†}|0⟩$. Within the zero-total-momentum pair subspace the two opposite-spin terms cancel the interaction's factor one-half, while same-spin s-wave terms cancel by fermionic antisymmetry. Thus the projected Hamiltonian acts asFor $|\Psi ⟩={\sum }_{⟨\mathbf{k}⟩}{\phi }_{\mathbf{k}}|\mathbf{k}⟩$ and $C={\sum }_{⟨\mathbf{k}⟩}{\phi }_{\mathbf{k}}$, matching each basis coefficient givesThis is the stated coefficient formula in the consistent Fermi-level energy convention. Its even dependence on $\mathbf{k}$ produces an even orbital state with a spin singlet. The constant $C$ is fixed up to an overall phase by normalization; for a bound state, $|C{|}^{-2}=g^2{\sum }_{⟨\mathbf{k}⟩}(E-2{\xi }_{\mathbf{k}}{)}^{-2}$. It cannot be set to zero for the nontrivial bound solution.

Put $E=-B$ with $B>0$ and impose the sum defining $C$:Here $\rho (E_F)$ counts orbital states for one spin species, as appropriate to the unrestricted $\mathbf{k}$ sum in this pair basis, and is approximately constant over the thin shell. If a two-spin total density of states is used, it must be divided by two in this integral. State-counting and the volume convention in $g$ and $\rho$ must be consistent. With $\xi =g\rho (E_F)$ as the paper's dimensionless coupling, inversion yieldsThe first approximation is the constant-density-of-states, restricted-shell model; the final exponential additionally assumes weak attraction. The binding energy is nonanalytic at zero coupling, so a finite-order power expansion in $g$ cannot produce it.

Cooper's problem asks whether two electrons added above an inert Fermi sea bind despite Pauli exclusion of lower states. An arbitrarily weak net attraction binds a zero-momentum singlet below the free-pair threshold, because the near-Fermi-surface integral diverges logarithmically as $B\downarrow 0$. Its wavefunction draws on many time-reversed momentum states and becomes dominated by a narrow region near the Fermi surface at weak coupling; the corresponding pair is spatially extended. This establishes a Cooper instability of the unpaired reference state.^[ref] The full superconducting ground state needs a coherent many-electron treatment in which overlapping pairs and the reorganized sea are included. BCS theory supplies that step; the two-electron binding scale alone is neither a calculation of the many-body gap nor a derivation of zero resistance.^[ref]

A direct band gap has its conduction band minimum and valence band maximum at the same crystal momentum. Near-edge absorption or emission can therefore conserve momentum with the electron and photon alone, since optical photon momentum is small on the Brillouin-zone scale. An indirect band gap has its extrema at different wavevectors and normally needs a phonon or another momentum-transfer process in addition to the photon.

Direct-gap materials are usually preferred for efficient light emission, because the radiative transition avoids the extra phonon-assisted process. Their stronger near-edge absorption also permits thinner photovoltaic absorbers. Indirect-gap materials can nevertheless make excellent solar cells: sufficiently thick optical paths, light trapping, long carrier lifetimes and suitable material quality can compensate for weaker absorption. The direct/indirect distinction alone does not rank every device.

This momentum distinction and the difference between bound exciton lines and the free-pair continuum are treated together in the primary effective-mass absorption theory.^[ref]

A phonon is one quantum of a collective harmonic normal mode of a crystal. A mode labelled by wavevector and branch has energies $E_m=\hslash \omega (m+1/2)$, with bosonic occupation statistics. The crystal momentum $\hslash \mathbf{k}$ is defined modulo a reciprocal lattice vector; it is not generally the mechanical momentum of one atom. The phonon dispersion relation assigns its frequency and polarization to the wavevector and branch.

Inelastic neutron scattering can measure this relation in a single crystal. Determine the incident neutron energy and wavevector, scatter at a chosen angle, and analyze the outgoing energy by time of flight or a spectrometer analyzer. Energy and crystal-momentum conservation giveThe signs distinguish phonon creation and annihilation with consistent momentum convention. Repeating measurements at different momentum transfers maps peaks in the dynamic scattering response into phonon branches. The measured intensity depends on occupation, scattering factors and the projection of the mode polarization along the momentum transfer; a weak or missing peak need not mean that the corresponding mode does not exist.

Inelastic neutron scattering measures both momentum and energy exchanged with the crystal. Comparing incident and outgoing neutron wavevectors gives $\mathbf{Q}={\mathbf{k}}_i-{\mathbf{k}}_f$, while their energy difference gives a created or annihilated phonon energy $\hslash \omega$. Crystal momentum conservation identifies the reduced phonon wavevector modulo a reciprocal-lattice vector. Varying scattering geometry maps dispersion curves and polarization-dependent intensities across a substantial part of the Brillouin zone.

Inelastic X-ray scattering provides a complementary energy- and momentum-resolved probe, particularly when small samples or high-pressure environments favor photon beams. Raman and infrared spectroscopy measure optical-frequency resonances subject to different symmetry selection rules. Photon momentum at these frequencies is usually small on the Brillouin-zone scale, so these methods mostly sample near the zone center rather than an arbitrary phonon wavevector. Brillouin scattering and ultrasonic measurements probe low-frequency acoustic motion and infer sound velocities or elastic constants.

Heat-capacity measurements constrain the thermally weighted vibrational density of states and its low-frequency behavior, but do not by themselves reconstruct a unique wavevector-resolved dispersion. Elastic diffraction and temperature-dependent displacement factors also constrain mean-square motion, while energy-resolved inelastic measurements distinguish individual excitations directly. A monatomic scalar chain has only the acoustic branch derived below; it should not be assigned an optical branch merely because Raman spectroscopy exists for more complicated crystals.

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.

For a periodic single-electron Hamiltonian operator, Bloch's theorem givesTwo bands at the same $\mathbf{k}$ have the same translation eigenvalues but different cell-periodic eigenfunctions, generally different energies and different orbital/nodal character. They may be chosen orthogonal even at a band degeneracy. The basis choices are treated in the two scoped branches below.

For a slowly varying external potential $U(\mathbf{r})$, the semiclassical wavepacket Hamiltonian is ${ϵ}_b(\mathbf{k})+U(\mathbf{r})$. Hamilton's equation for the conjugate variables $\mathbf{r},\hslash \mathbf{k}$ gives $\hslash \dot{\mathbf{k}}=-\nabla U=\mathbf{F}$. Equivalently the electric perturbation changes the translation phase at the rate $q\mathbf{E}\cdot \mathbf{R}/\hslash$. This is the single-band acceleration theorem, requiring a slowly varying field and negligible interband tunnelling/scattering; it refers to crystal momentum, not free-space mechanical momentum.

Let $e>0$ and $K=e{E}_0/(\hslash \omega )$. Integrating from the specified initial condition $\mathbf{k}(0)=0$ gives, with the first field alone,With both components switched on,This is a circle centred at $(-K,0)$, not at the band minimum. Since $v_i=\hslash k_i/m_i^{\ast }$, integration givesThe switch-on drift in a circularly driven parabolic band accompanies unequal transverse scales: it is a stretched cycloid. In a frame moving with that drift, the oscillation is an ellipse with semiaxis ratio $m_y^{\ast }/m_x^{\ast }\gg 1$. A closed real-space ellipse would require different initial momentum. Indeed, after one period $x-x_0=-2\pi e{E}_0/(m_x^{\ast }{\omega }^2)\ne 0$, an explicit check against that interpretation.

Writing $\theta =\omega t$, the kinetic energy above the band minimum isFor $m_x^{\ast }<m_y^{\ast }$ it is largest at $\theta =\pi$, with value $2e^2{E}_0^2/(m_x^{\ast }{\omega }^2)$. Equating this to the optical-phonon energy gives the emission threshold estimateThis treats the optical branch as approximately dispersionless and assumes a momentum-allowed final state, a cold initial electron, and negligible phonon absorption. It is the amplitude of each Cartesian field component; the circular field has constant magnitude $E_0$. A steady periodic orbit centred at $\mathbf{k}=0$ would instead give twice this amplitude, but fails the stated switch-on condition.

For $N$ ion pairs per unit volume, a relative displacement $x$ produces ionic polarization $P=Nqx$. Solving the driven Lorentz oscillator model with the source's $e^{-i\omega t}$ convention givesThe constant ${ϵ}_{\infty }$ includes the fast electronic response: infinity here means well above the phonon band but below electronic resonances. Writing $\Delta ϵ={ϵ}_s-{ϵ}_{\infty }$ makes the numerator $\Delta ϵ{\omega }_{\mathrm{TO}}^2$. In an undamped longitudinal mode the bulk displacement field vanishes without an external free charge, so a nonzero electric field requires ${ϵ}_r({\omega }_{\mathrm{LO}})=0$. ThereforeThis is the Lyddane-Sachs-Teller relation for the single polar mode.

At normal incidence from vacuum the Fresnel equations giveBelow the transverse resonance the dielectric response increases, raising reflectivity. Between the transverse and longitudinal frequencies the undamped dielectric function is negative, giving an evanescent transmitted wave and nearly total reflection: the Reststrahlen band. Above the longitudinal edge it becomes small and positive, passing through ${ϵ}_r\simeq 1$ at the reflectivity minimum and eventually approaching its electronic background.

Reading the original graph to its limited precision gives $\hslash {\omega }_{\mathrm{TO}}\simeq 30\mathrm{meV}$ and $\hslash {\omega }_{\mathrm{LO}}\simeq 37\mathrm{meV}$, hence ${\omega }_{\mathrm{TO}}\simeq 4.6\times 10^{13}{\mathrm{s}}^{-1}$ and ${\omega }_{\mathrm{LO}}\simeq 5.6\times 10^{13}{\mathrm{s}}^{-1}$. Extrapolating the low-energy reflectivity to roughly $R(0)\simeq 0.25$ givesValues around $9$–$10$ and $5.5$–$6.5$ are appropriate to this graphical estimate. Neither the dip near $39\mathrm{meV}$ nor the finite-energy right endpoint represents the high-frequency asymptote; treating that endpoint as $R(\infty )$ would underestimate ${ϵ}_{\infty }$.

At $\omega ={\omega }_{\mathrm{TO}}$, ${ϵ}_r={ϵ}_{\infty }+iA$ with $A=\Delta ϵ{\omega }_{\mathrm{TO}}/\gamma$. Weak damping gives $A\gg {ϵ}_{\infty }$, so for the complex refractive index $n+i\kappa$,Squaring and substituting $A$ provesUse the reflectivity at the estimated transverse resonance, not automatically the plateau maximum. A representative reading $R_{\mathrm{TO}}\simeq 0.9$, with $\Delta ϵ\simeq 3.1$, gives $\gamma /{\omega }_{\mathrm{TO}}\simeq 0.0043$, $\gamma \simeq 2.0\times 10^{11}{\mathrm{s}}^{-1}$ andGraphical uncertainty in the transverse edge makes this only a few-picosecond estimate: readings $R_{\mathrm{TO}}\simeq 0.85$–$0.93$ give approximately $2$–$11\mathrm{ps}$. The displacement amplitude decays as $e^{-\gamma t/2}$ and has lifetime $2/\gamma$; phonon occupation/energy decays as $e^{-\gamma t}$ and has lifetime $1/\gamma$. Anharmonic forces allow an optical phonon to decay into two lower-energy phonons when energy and crystal momentum are conserved. Spontaneous decay can persist at low temperature; defects and electron–phonon scattering can add further linewidth.