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Use the quantum free electron model: conduction electrons move as independent particles in a three-dimensional box of volume $V$, with a uniform positive background replacing the ionic lattice potential. Neglect the detailed periodic potential and explicit electron-electron interactions; the interior one-electron dispersion relation is $ϵ={\hslash }^2{k}^2/(2m_e)$. The boundary confines the particles, and its detailed shape does not affect the bulk state count in a sufficiently large sample.

The electrons are spin-$1/2$ fermions: each spatial quantum state accommodates two opposite spin states, with occupations given by the Fermi-Dirac distribution and the Pauli exclusion principle. A fixed number $N$ of conduction electrons and thermal equilibrium are assumed. Collisions can be represented by a phenomenological momentum relaxation time for transport, but are unnecessary for the equilibrium state counting here. These assumptions distinguish the quantum model from a classical Maxwell-Boltzmann electron gas.

Assume sodium supplies one conduction electron per atom. The supplied Fermi energy givesThe gas is strongly degenerate at room temperature. For one mole, $N=N_A$ and $N_A{k}_B=R=8.31446\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. Using the requested estimate,The supplied mass density determines the answer per unit volume: $N/V=(970/0.023)N_A=2.54\times 10^{28}{\mathrm{m}}^{-3}$. ThereforeThe density is not needed for the molar heat capacity once $E_F$ is supplied. Using the precise Sommerfeld expansion coefficient instead gives $0.327\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$.

Either electronic estimate is only about $1\%$ of the stated measured total. Most of the total heat capacity comes from lattice phonons, whose high-temperature Dulong-Petit law limit is $3R=24.94\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. The remaining difference from $28\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$ need not be electronic: anharmonic lattice effects and the difference between heat capacity at constant pressure and heat capacity at constant volume also contribute. The paper does not specify the measurement constraint, so the two totals should not be equated exactly.

A Bloch state in a crystal experiences a periodic lattice potential. Its band dispersion relation $ϵ(k)$ encodes the lattice response, so an effective mass allows a wavepacket's acceleration under a weak applied force to be described without explicitly following the microscopic periodic forces.

In one dimension the wavepacket group velocity and semiclassical equation for crystal momentum areDifferentiating the first equation with respect to time givesDefining $F=m^{\ast }a$ yieldsThis band effective mass is controlled by band curvature, not the slope. A band minimum has positive $m^{\ast }$; a band maximum has negative electron $m^{\ast }$ and is often more conveniently described with positive-mass electron holes. In three dimensions the inverse effective mass is generally a tensor, $(m^{\ast -1}{)}_{ij}={\hslash }^{-2}{\partial }_{k_i}{\partial }_{k_j}ϵ$. The scalar formula assumes motion along a specified direction in an isolated band, weak fields and negligible interband transitions; the bare electron mass has not physically changed.

Let $e>0$ denote the elementary charge, so an electron's charge is $-e$. In the relaxation-time Drude model with positive transport effective mass,The steady drift velocity is ${\mathbf{v}}_d=-e\tau \mathbf{E}/m^{\ast }=-\mu \mathbf{E}$, defining positive electron mobility $\mu =e\tau /m^{\ast }$. The current density is opposite to the electron drift:By the definition of electrical conductivity, $\mathbf{J}=\sigma \mathbf{E}$, soIts units are $({\mathrm{m}}^{-3})(\mathrm{C})({\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1})=\mathrm{S}{\mathrm{m}}^{-1}$. The mobility definition alone establishes the result in the scalar linear-response regime; the relaxation-time equation supplies a microscopic approximation for $\mu$.

Interpret “3 mm wire” as a circular wire of diameter $3\mathrm{mm}$, and take the current density as uniform. One conduction electron per atom givesUsing $I=neA|v_d|$, the drift velocity corresponding to a current magnitude of $13\mathrm{A}$ isIt points opposite to conventional electric current. A simple distance scale over the $1/50=0.020\mathrm{s}$ cycle is $\boxed{|v_d|/f=2.71\mu \mathrm{m}}$, but this treats the magnitude as constant. For sinusoidal mains, alternating-current drift excursion gives zero net displacement after a full cycle.

If $13\mathrm{A}$ is the usual root mean square current, $v_{\mathrm{pk}}=\sqrt{2}|v_d|$ and the displacement amplitude and total absolute drift distance per cycle areIf $13\mathrm{A}$ instead denotes peak current, these become $0.431\mu \mathrm{m}$ and $1.72\mu \mathrm{m}$. The paper does not state peak versus RMS; the characteristic drift is the boxed $1.35\times 10^{-4}\mathrm{m}{\mathrm{s}}^{-1}$ on whichever current convention is chosen. It also does not explicitly say diameter; taking $3\mathrm{mm}$ as the radius would reduce all these velocities and drift distances by four. None of these distances is the microscopic path travelled at roughly the Fermi velocity between scattering events; that path cannot be obtained from the supplied current alone.

In a normal metal, the electrical resistivity can often be approximated using Matthiessen's rule,Static impurities, defects and grain boundaries scatter conduction electrons even as $T\to 0$, giving the residual resistivity. Thermal lattice phonons add a temperature-dependent contribution through electron-phonon scattering. In a clean three-dimensional acoustic model, small-angle electron-phonon transport at low temperature gives ${\rho }_{\mathrm{phonon}}\propto T^5$ at low $T$; once thermal phonon populations are approximately classical, the resistivity is approximately linear in $T$. An electron-electron contribution proportional to $T^2$ can be relevant in some low-temperature ranges.

Carbon in mild steel, together with associated disorder, strain and microstructural changes, adds scattering and generally raises the residual resistivity relative to purer iron. Thus a dilute alloy may have a similar broad phonon-dependent rise but a larger baseline. The sketch is a generic phonon-plus-disorder model, not measured iron data. Iron's magnetic scattering and phase changes, and real steel's phases and processing history, can add structure; the simple sum is approximate rather than an exact material law. The expected explanation is greater electron scattering from the added carbon and resulting disorder.

Doping changes the mobile carrier population by introducing controlled impurity levels. In an intrinsic semiconductor, the filled valence band is separated from the nearly empty conduction band by a band gap $E_g$. Thermal excitation produces an electron-hole pair, so the electron and hole number densities obey $n=p=n_i$. A electron hole represents a missing valence-band electron and behaves as a mobile positive charge carrier. For nondegenerate bands in thermal equilibrium,Here $N_C,N_V$ are the effective density of states, $E_C,E_V$ the band edges, and $E_F$ denotes the chemical potential, conventionally called the Fermi level. The semiconductor mass-action law holds in this nondegenerate equilibrium regime.

In a tetrahedrally bonded elemental semiconductor such as silicon, a substitutional group-V atom has an extra outer electron relative to the host. It is a donor impurity: a shallow level just below $E_C$ can ionise, releasing an electron to the conduction band and leaving a fixed positively charged donor. The resulting n-type semiconductor has electrons as majority carriers and holes as minority carriers. Conversely a group-III substitution supplies one fewer bonding electron. An acceptor impurity can take an electron from the valence band, leaving a mobile electron hole and a fixed negative acceptor; the resulting p-type semiconductor has holes as majority carriers.

The impurity state is shallow because the electron or hole responds to a screened Coulomb potential and an effective mass, rather than the bare atomic potential. A simple hydrogenic donor estimate iswhere ${ϵ}_r$ is the relative permittivity. This is a useful scaling model, with real band anisotropy and central-cell effects omitted. Dopants are fixed in the lattice; their ionisation changes mobile carrier numbers without making a uniformly doped bulk sample carry a large net electric charge.

The charge neutrality condition isWhen shallow dopants are almost fully ionised and $N_D-N_A\gg n_i$, an n-type region has $n\simeq N_D-N_A$ and $p\simeq n_i^2/n$. A p-type region similarly has $p\simeq N_A-N_D$ and $n\simeq n_i^2/p$. Thus compensation by the opposite impurity type reduces the majority-carrier density; the total dopant concentration is not generally the mobile carrier density. At sufficiently high impurity density the degenerate semiconductor regime requires the full Fermi-Dirac distribution and these exponential approximations fail.

There are three useful temperature regimes. At low temperature, dopant freeze-out leaves many shallow levels un-ionised. In the extrinsic regime, the dopants are largely ionised and set the majority-carrier density. At high enough temperature, intrinsic pair creation dominates and $n,p$ approach $n_i$. The Fermi level moves towards the conduction band with donor doping and towards the valence band with acceptor doping. The electrical conductivity isDoping therefore controls conductivity and its carrier type, but ionised impurities also scatter carriers and can reduce charge carrier mobility. Carrier density and mobility must both be considered.

A p-n junction forms when p-type and n-type regions are joined within a semiconductor. Initially the concentration gradients drive diffusion current: electrons diffuse from the n region to the p region and holes in the opposite direction. Recombination removes mobile majority carriers near the join, exposing fixed positive donors on the n side and fixed negative acceptors on the p side. This forms a depletion region and an electric field directed from n to p. Its drift current opposes diffusion. At equilibrium the net current vanishes and the Fermi level is uniform, although the band edges vary spatially through band bending.

For an abrupt nondegenerate junction with fully ionised dopants, its built-in potential isUnder the depletion approximation, the space-charge densities are $-e{N}_A$ on a p-side width $w_p$ and $+e{N}_D$ on an n-side width $w_n$. Integrating Poisson equation and imposing zero field outside gives $N_A{w}_p=N_D{w}_n$ andMost of the depletion width therefore extends into the more lightly doped side. The schematic shows the impurity levels before contact and the resulting space charge and potential after contact; there is no externally extractable equilibrium battery voltage from the built-in field alone.

A forward bias, with p positive relative to n, reduces the barrier from $V_{\mathrm{bi}}$ towards $V_{\mathrm{bi}}-V$ and injects minority carriers across the junction. In the ideal low-injection, diffusion-controlled regime this produces the Shockley diode equation, $I=I_s[e^{eV/(k_{B}T)}-1]$. Reverse bias increases the barrier and the depletion region; only a small thermally generated minority-carrier current flows before breakdown. Reverse-biased depletion regions also give a voltage-dependent junction capacitance. Real devices have recombination, leakage, series resistance and breakdown mechanisms beyond the ideal law. Controlled doping creates majority and minority carriers, and their junction produces a depletion barrier that makes rectification possible.

Take an abrupt equilibrium p-n junction with the p side at $x<0$ and the n side at $x>0$. In the quasi-neutral bulk, electron holes are the majority carriers on the p side and electrons are the majority carriers on the n side. With ionized dopant densities $N_A,N_D$ and nondegenerate statistics,where $n_i$ is the intrinsic carrier concentration. The semiconductor mass-action law $np=n_i^2$ applies at equilibrium. Both mobile populations are small in the depletion region compared with the dopant densities.

The charge and carrier density of an abrupt p-n junction must distinguish mobile populations from net charge. Diffusion leaves negatively charged ionized acceptor impurities on the p side and positively charged ionized donor impurities on the n side. Under the depletion approximation,The neutral bulks have large carrier densities but nearly zero net charge density. The electric field points from the positive donor region toward the negative acceptor region, balancing diffusion at equilibrium.

The figure uses equal dopant densities only to show a simple symmetric sketch. Generally the more lightly doped side has the wider depleted interval. The carrier profiles are smooth, whereas the step-shaped net charge density is the ideal depletion approximation.

Treat sodium as monovalent, with one mobile electron per atom. Its mass density is ${\rho }_m=0.97\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}=970\mathrm{kg}{\mathrm{m}}^{-3}$, and its molar mass is $M=0.023\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$. Using the Avogadro constant,The electron mobility $\mu =e\tau /m_e$ relates the magnitude of drift velocity to electric field. The Drude model then givesUsing $m_e=9.109\times 10^{-31}\mathrm{kg}$,The inferred time is the momentum relaxation time. Identifying it with the mean time between collisions is the intended simple isotropic Drude model assumption; small-angle scattering need not relax directed momentum in a single event.

The quantum free electron model treats conduction electrons as independent particles moving in an approximately constant potential, with parabolic dispersion relation $E={\hslash }^2{k}^2/(2m)$. Fermi-Dirac statistics successfully explains important features such as a Fermi energy and a small electronic heat capacity, but a constant potential gives no band gaps and cannot by itself explain why different band fillings produce metals, semiconductors and band insulators. It also misses lattice-dependent effective masses, detailed directional responses and the topology of real Fermi surfaces. Scattering and finite electrical resistivity require additional physics.

The nearly-free electron model introduces a weak periodic lattice potential. Bloch's theorem replaces individual free plane waves by Bloch states, and Fourier components of the potential mix plane waves whose wavevectors differ by a reciprocal lattice vector. Near degeneracy at a Brillouin zone boundary, this coupling opens a band gap; for a two-wave crossing its size is $2|V_G|$. Away from the crossing, energies remain close to free-particle values. Band curvature produces band effective masses, and band filling determines whether the Fermi energy lies inside an allowed band or a gap.

The improvement is the inclusion of lattice periodicity, rather than a switch from classical to quantum statistics: the quantum free electron model already uses Fermi-Dirac statistics. The strong delta barriers in the later numerical example are treated exactly by the Kronig-Penney model, rather than by assuming that their strength is perturbatively small.

With length measured in units of $a=1$, plot the real functionOnly the portions with $|f(k)|\le 1$ can equal $\cos \varphi$, so the strip between the horizontal lines $+1$ and $-1$ identifies allowed energy bands. The band edges of a repulsive delta-comb potential have the following approximate first intervals:Each interval corresponds to energy $E={\hslash }^2{k}^2/(2m)$; the intervening intervals are forbidden. The exact upper endpoints follow from $f(j\pi )=(-1{)}^j$. The lower endpoints solve $f(k)=(-1{)}^{j-1}$ inside $((j-1)\pi ,j\pi )$. The divergence suggested by $\sin k/k$ at the origin is removable, and $f(0)=11>1$, so there is no zero-energy state in the repulsive model.

The periodic barriers produce separated energy bands and band gaps, unlike the continuous free-particle spectrum. A finite $N$-atom loop has discrete states within these bands. Whether the material conducts still depends on band filling: one electron per primitive cell half-fills a spin-degenerate band, while two electrons per cell can fill the first band and place the Fermi energy in a gap. Band gaps alone do not establish that every filling is insulating. At high $k$, the perturbing term $10\sin k/k$ becomes relatively small and the spectrum approaches free-electron behaviour in relative terms.

The Hall effect is a transverse electric response to a magnetic force on moving charge carriers. Take conventional current density $\mathbf{J}=J_x\hat{\mathbf{x}}$ and magnetic field $\mathbf{B}=B_z\hat{\mathbf{z}}$ in a conducting slab. For one carrier type of number density $n$ and signed electric charge $q$, the drift velocity is ${\mathbf{v}}_d=\mathbf{J}/(nq)$. Carriers initially deflect sideways until a Hall electric field balances the magnetic Lorentz force:The Hall coefficient is thereforeFor electrons it is $-1/(ne)$, whereas a simple hole-dominated conductor gives $+1/(pe)$. The sign reveals the effective carrier sign; the magnitude estimates their density in the one-carrier approximation.

For width $w$ along $y$ and thickness $t$ along $z$, $I=J_{x}wt$. Define the signed Hall voltage by $V_H=E_{y}w$, namely the electric potential at the lower-$y$ edge minus that at the higher-$y$ edge. ThenThis explicit contact convention fixes the voltage sign. A Hall bar with longitudinal and transverse contacts measures both electrical resistivity and the Hall response. Reverse the field and take the part of the transverse voltage odd in $B_z$ to suppress longitudinal-voltage pickup from misaligned contacts; reversing current provides another useful check.

A Hall measurement is useful for carrier identification, semiconductor doping measurements and magnetic-field sensing. Combining a one-carrier $R_H$ with electrical conductivity gives a mobility magnitude $\mu =\sigma |R_H|$. However, mixed electron and hole conduction is not described by a single density: in weak fields,The response weights mobilities differently from ordinary conductivity. Complicated energy bands, anisotropic Fermi surfaces and multiple bands can likewise change the simple interpretation in metals; a positive Hall coefficient does not mean that the material contains positively charged fundamental electrons.

Semiconductor doping introduces controlled impurities to alter equilibrium carrier populations and electrical conductivity. In an intrinsic semiconductor, thermally excited electrons in the conduction band and electron holes in the valence band have equal densities $n=p=n_i$. A shallow donor impurity, such as phosphorus in silicon, supplies an electron from an impurity level just below the conduction-band edge. Once ionized it is positively charged, and the material is n type. An acceptor impurity, such as boron in silicon, accepts a valence electron into a level just above the valence-band edge, leaving a mobile hole and a negatively charged ionized impurity; the material is p type.

With dopant densities $N_D,N_A$, charge neutrality requiresFor fully ionized, nondegenerate dopants at equilibrium, the semiconductor mass-action law gives $np=n_i^2$. In a substantially n-type sample, $N_D-N_A\gg n_i$, soThe corresponding p-type result is $p\simeq N_A-N_D$ and $n\simeq n_i^2/(N_A-N_D)$. The majority population grows while the minority population decreases; adding donors does not simply add equal numbers of free electrons and holes.

The Fermi energy shifts toward the conduction band for n-type material and toward the valence band for p-type material. In the nondegenerate approximation,where $N_C,N_V$ are effective band-edge state densities. At very low temperature, dopants may not be ionized and carrier freeze-out occurs. At intermediate temperatures, ionized dopants dominate the carrier density; at sufficiently high temperature, intrinsic excitation can dominate. Heavy doping can produce degenerate carrier statistics, invalidating these Boltzmann expressions.

The electrical conductivity is $\boxed{\sigma =e(n{\mu }_e+p{\mu }_h)}$. Doping usually raises the majority carrier density, but ionized impurity scattering can lower mobility, so conductivity need not scale indefinitely with dopant concentration. Joining p-type and n-type regions creates a p-n junction, with a depletion region and built-in electric field established by carrier diffusion and compensating drift. The fixed ionized dopants, not an absence of all charge, account for the depleted region's net charge density.

In a metal, mobile electrons transport heat as well as electric charge. A temperature gradient creates an imbalance between carriers arriving from warmer and colder regions. The resulting heat current obeys Fourier's law ${\mathbf{j}}_Q=-\kappa \nabla T$ for an isotropic material, where thermal conductivity $\kappa$ has units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$. For a bar of area $A$ carrying steady heat flow $P$, $P=-\kappa AdT/dx$ when parasitic losses are negligible.

A kinetic estimate for the electronic contribution iswhere $C_e$ is the electronic heat capacity per volume, $v_F$ the Fermi velocity, and ${\ell }_{\kappa }=v_F{\tau }_{\kappa }$ the heat-transport mean free path. Fermi-Dirac statistics and the Pauli exclusion principle restrict thermal rearrangements to states near the Fermi surface, rather than allowing every electron to contribute a classical heat capacity. A degenerate parabolic band givesUsing the Drude model form $\sigma =n{e}^2{\tau }_{\sigma }/m$ for electrical transport, the relation between thermal and electrical relaxation times in metals becomesFor scattering that relaxes heat and charge currents similarly, ${\tau }_{\kappa }\simeq {\tau }_{\sigma }$ and the Wiedemann-Franz law follows:Thus good electrical conductors are usually good thermal conductors. The correct quantum Lorenz number differs from a naive classical electron-gas estimate.

Scattering from impurities, boundaries and phonons limits the heat-transport path. In an impurity-dominated low-temperature regime with approximately constant relaxation time, ${\kappa }_e$ is approximately proportional to $T$. Where phonon-limited electrical resistivity is proportional to $T$ and the Wiedemann-Franz approximation is applicable, ${\kappa }_e$ can be roughly temperature independent. Inelastic processes can relax heat and charge currents differently, so the Lorenz ratio is not universal at all temperatures. Lattice phonons add a separate positive thermal contribution; the measured total is $\kappa ={\kappa }_e+{\kappa }_{\mathrm{ph}}$, and the electronic Wiedemann-Franz expression should not be applied blindly to that total.

The photoelectric effect, black-body radiation and the Davisson–Germer experiment challenge different parts of a fully classical picture. Classical electromagnetic waves successfully describe propagation, interference and diffraction, while classical particles follow trajectories determined by forces. The difficulty is combining that picture with the observed exchange of energy between radiation and matter, the equilibrium distribution of radiation, and electron scattering. The three experiments expose the roles of discrete energy exchange and coherent matter waves.

In a photoelectric effect experiment, monochromatic light illuminates a metal cathode in vacuum and a second electrode collects emitted electrons. Varying the collecting voltage measures a photocurrent; reversing it until collection ceases determines the stopping potential. The observations distinguish two controls: at fixed frequency above threshold, increasing light intensity increases the emission rate, whereas the largest emitted kinetic energy is determined mainly by frequency. There is a material-dependent photoelectric threshold frequency, and emission above threshold does not require the long charging-up delay predicted by a simple picture of gradual classical energy accumulation at low intensity.

A classical wave carries continuously distributed energy, with intensity proportional to squared field amplitude. On a fully classical absorption picture, increasing intensity would be expected to increase the energy delivered to an electron, and sufficiently long exposure could accumulate the escape energy even below the observed threshold. It does not naturally produce the measured universal straight-line relation between stopping potential and light frequency. In 1905, Albert Einstein instead treated light-energy transfer as localized quanta. His argument distinguished the fastest escaping electrons from electrons losing energy before escape.^[ref]

In the single-photon regime, conservation of energy gives the Einstein photoelectric equationwhere $\Phi$ is the surface work function. The photon supplies energy $h\nu$; the escape cost is paid first, and the remainder sets the upper electron-energy edge. Increasing intensity at fixed frequency increases the number of available photons, so it raises current without raising this edge. Below ${\nu }_0$, one photon cannot pay the escape cost. In an actual measurement, stable surface preparation, suppression of unwanted higher-frequency light, and the photoelectric contact-potential correction are needed to test the slope $h/e$ reliably. Robert Andrews Millikan's experiment used selected mercury spectral lines, freshly exposed metal surfaces and an electrometer to make such a quantitative test.^[ref] The interpretation is for weak single-photon illumination; multiphoton photoemission at sufficiently intense fields is a different regime.

Black-body radiation tests equilibrium physics rather than the escape of individual electrons. A small aperture in an isothermal absorbing cavity approximates an ideal blackbody: radiation entering the hole undergoes repeated interactions with the walls, and the equilibrium spectrum depends on temperature rather than wall material. A spectrally resolving detector outside the aperture measures how emitted power varies with frequency or wavelength. Experimentally the spectrum has a finite peak, is strongly suppressed at short wavelengths, shifts toward shorter wavelengths when the cavity is heated, and has integrated emission proportional to $T^4$.

For a large cavity, the electromagnetic mode density in a box, including two independent transverse polarizations, is $g(\nu )=8\pi {\nu }^2/c^3$ per volume per unit frequency. The classical equipartition theorem assigns mean energy $k_{B}T$ to each mode, leading to the Rayleigh-Jeans lawThis agrees at low frequencies but diverges as frequency increases; integrating it over all frequencies gives infinite energy density at any nonzero temperature. This is the ultraviolet catastrophe, a failure of classical equipartition applied to the electromagnetic modes.

Max Planck's essential modification was that energy at a frequency is exchanged in units $h\nu$. In a modern oscillator description, the thermal excitation levels are $nh\nu$, $n=0,1,\dots$. Using the Boltzmann distribution and summing a geometric series givesMultiplying by the mode density yields Planck's law for the spectral energy density,When $h\nu \ll k_{B}T$, expansion of the denominator recovers the classical law. When $h\nu \gg k_{B}T$, the large excitation cost suppresses occupancy exponentially, removing the divergence. Its integral gives the Stefan–Boltzmann law, while its peak gives Wien's displacement law. Thus quantization repairs the spectral shape, total emission and temperature dependence together. The displayed thermal energy excludes the oscillator's temperature-independent zero-point energy.

The Davisson–Germer experiment supplies the complementary evidence that matter cannot always be treated as classical particles with definite scattering trajectories. In 1927, Clinton J. Davisson and Lester Halbert Germer sent a narrow, accelerated electron beam normally onto a single nickel crystal in vacuum. A movable collector measured the current of scattered electrons as direction and accelerating voltage were varied; a retarding potential selected electrons retaining approximately the incident energy. Sharp angular beams depended on crystal orientation and beam energy, with a prominent feature near $54\mathrm{V}$ and detector colatitude $50^{\circ }$. These are scattering directions and currents, not photographs of electron paths.^[ref]

The de Broglie wavelength predictsAt $U=54\mathrm{V}$, this gives $\lambda \simeq 0.167\mathrm{nm}$, comparable to atomic spacings. A crystalline arrangement supplies regularly spaced scattering sites, so scattered probability amplitudes can add constructively in selected directions. For a suitable family of crystal planes, Bragg's law expresses the condition as $2d\sin {\theta }_B=n\lambda$, with ${\theta }_B$ measured relative to those planes. The detector colatitude is not automatically this angle. The measured angular structure and its variation with electron momentum are described by electron diffraction with the de Broglie wavelength. A classical point-particle deflection model has no phase wavelength $h/p$ and does not supply that interference condition.

The common quantum account preserves the successes of wave propagation and particle detection while changing what is quantized. Photons exchange energy in frequency-dependent units; a thermal distribution of those excitations gives the blackbody spectrum. Electrons have a wavefunction whose amplitudes interfere on scattering from a crystal, while detection transfers localized charge. The Born rule turns squared amplitudes into event probabilities. These results invalidate a fully classical description of matter and radiation; the photoelectric energy law alone does not establish every property of a quantized electromagnetic field, but the three phenomena together provide complementary evidence for the quantum description.

Twice the quadratic coefficient from the previous part is the band curvature $E_{\pm }^{\prime \prime }(G_1/2)$. Thus the band effective masses areHere $U=|V(G_1)|$ and $E_B={\hslash }^2{G}_1^2/(8m)$. In the weak-potential regime $U\ll E_B$,Both have units of mass; their magnitudes are much smaller than $m$ in this asymptotic regime. The upper energy band has a local minimum and positive effective mass, while the lower energy band has a local maximum and negative effective mass. The exact lower-branch sign changes at $U=2E_B$ in the two-wave formula; the intended weak-coupling limit lies below that threshold.

The physical meaning follows from the semiclassical equations for an electron wave packet in one energy band:An upper-band electron near its minimum accelerates in the direction of the applied force, whereas a lower-band electron near its maximum accelerates opposite to it. At the extremum itself the group velocity is zero. A nearly filled lower energy band is conveniently described by electron holes with positive electric charge $+e$ and positive effective mass $m_h^{\ast }=-m_{-}^{\ast }$. Negative curvature is a response of a crystal energy eigenstate, not a negative bare electron mass. These statements assume slow forcing that does not induce transitions across the band gap.

In a metal, a partially filled energy band or overlapping energy bands supplies electrons with nearby unoccupied states. At thermal equilibrium, oppositely directed group velocities are equally populated and the net electric current vanishes. An applied electric field distorts the occupation distribution, producing a small mean drift velocity. A filled energy band alone has no such low-field intraband response because its velocity contributions cancel over the entire Brillouin zone.

In the Drude model, let $n$ be the mobile electron number density, let $e>0$ be the elementary charge, and let $\tau$ be the momentum relaxation time. The mean motion obeysThe steady state is ${\mathbf{v}}_d=-e\tau \mathbf{E}/m$. The current density is opposite to the electron drift velocity:This gives microscopic Ohm's law, with electrical conductivity $\sigma$ in $\mathrm{S}{\mathrm{m}}^{-1}$ and electrical resistivity ${\rho }_{\mathrm{el}}$ in $\Omega \mathrm{m}$. For a uniform wire of length $\ell$ and cross-sectional area $A$, its electrical resistance is $R={\rho }_{\mathrm{el}}\ell /A$. The dissipated power per volume is $\mathbf{J}\cdot \mathbf{E}=\sigma E^2$, the Joule heating supplied by the external source.

The free electron model with Fermi-Dirac statistics improves the microscopic picture. The Pauli exclusion principle fills states up to the Fermi energy at low temperature. For a spherical, parabolic energy band with two spin states,Only states near the Fermi surface change their occupation in a weak-field response; the large equilibrium Fermi velocity is not the small drift velocity. The same $n{e}^2\tau /m$ formula results for this isotropic parabolic model when $\tau$ is evaluated for transport near the Fermi surface. In a crystal, the energy band group velocity, effective mass and possibly direction-dependent scattering replace the simplest bare-particle picture. The mean free path is approximately ${\ell }_{\mathrm{mfp}}=v_F\tau$.

A perfectly periodic static lattice is already included in Bloch states and does not by itself give a finite electrical resistivity. Momentum relaxation comes from defects, impurities, boundaries, and electron-phonon scattering. Independent scattering mechanisms approximately add their rates, giving Matthiessen's rule,The approximately temperature-independent ${\rho }_0$ is the residual resistivity. In ordinary simple metals, the phonon contribution is approximately proportional to temperature at high temperature; a three-dimensional acoustic-phonon transport model gives a $T^5$ contribution well below the relevant Debye temperature. Other mechanisms can dominate in particular materials or regimes, so neither power law is universal.

The classical Drude model incorrectly treats the equilibrium carrier statistics. Fermi-Dirac statistics explains the small electronic heat capacity, while electrons carrying both charge and heat give the approximate Wiedemann-Franz law ${\kappa }_e/(\sigma T)={\pi }^2{k}_B^2/(3e^2)$ for degenerate carriers with appropriate nearly elastic scattering. Both the simple electrical conductivity formula and this thermal relation depend on the assumptions about energy bands and scattering, rather than being exact for all metals.

The Debye model approximates a three-dimensional crystal as an elastic continuum supporting quantized normal modes, or phonons. At long wavelengths, the acoustic phonon branches have approximately linear dispersion relations ${\omega }_s(k)=v_{s}k$: one longitudinal branch and two transverse branches. Replace their complicated actual spectra by these linear forms, then introduce a cutoff chosen to preserve the total $3N$ vibrational degrees of freedom for $N$ atoms. This is an approximation to the spectrum, not a claim that a real crystal has no non-acoustic branches.

For periodic boundary conditions in volume $V$, each allowed wavevector occupies $(2\pi {)}^3/V$ in the space of wavevectors. Counting a spherical shell for each branch gives the density of states per unit angular frequencyAn effective speed of sound $v_D$ can be defined by $3/v_D^3=1/v_l^3+2/v_t^3$. In the single-speed Debye model, choose ${\omega }_D=v_D{k}_D$ so thatThus the Debye wavevector, Debye frequency and Debye temperature satisfyHere $g$ has units of seconds when angular frequency is measured in ${\mathrm{s}}^{-1}$; its integral is the dimensionless number of normal modes. The spherical cutoff replaces the real Brillouin zone by an equal-count continuum construction.

Each phonon mode is a quantum harmonic oscillator. Since phonon number is not conserved at thermal equilibrium, the chemical potential is zero and its Bose-Einstein distribution gives mean thermal energy $\hslash \omega /[e^{\hslash \omega /(k_{B}T)}-1]$. The zero-point energy is independent of temperature in the harmonic, fixed-volume approximation and does not contribute to the heat capacity at constant volume. Excluding that constant term,Differentiating the first integral at fixed volume avoids losing the temperature-dependent upper limit in the second one. The resulting Debye heat capacity isFor $T\ll {\Theta }_D$, extend the upper limit to infinity. Expanding $1/(e^x-1)={\sum }_{r\ge 1}{e}^{-rx}$ and integrating term by term gives ${\int }_0^{\infty }{x}^3/(e^x-1)dx=6\zeta (4)={\pi }^4/15$. ThereforeThis $T^3$ law follows from the three-dimensional ${\omega }^2$ density of states and the availability only of low-frequency modes at low temperature. It is a central success of the Debye model.

For $T\gg {\Theta }_D$, the Bose-Einstein distribution reduces to the classical equipartition theorem: every mode contributes thermal energy $k_{B}T$. Consequentlythe Dulong-Petit law, or $3R$ per mole of atoms. The simple model omits detailed phonon dispersion relation, distinct optical phonon branches, and anharmonic effects such as thermal expansion. In a metal, the low-temperature total heat capacity additionally contains a term approximately proportional to $T$ from electrons, so the lattice $T^3$ term should not be mistaken for the whole measurement.

Take a p-n junction made from one semiconductor, with the p-type semiconductor on the left and the n-type semiconductor on the right. At thermal equilibrium the chemical potential $\mu =E_F$ is horizontal throughout. Before contact the chemical potential lies nearer the valence band on the p side and nearer the conduction band on the n side; after contact the band edges bend to permit one common value.

In this orientation both the conduction band edge $E_C$ and the valence band edge $E_V$ decrease from p to n by $e{V}_{\mathrm{bi}}$. For a homojunction their separation $E_g=E_C-E_V$ is constant. The acceptor impurity level $E_A$ lies just above $E_V$ where acceptors are present, and the donor impurity level $E_D$ lies just below $E_C$ where donors are present. These impurity levels follow the local band bending; they do not form one additional horizontal band across the device.

The shaded depletion region has few mobile electrons or electron holes and contains fixed negative ionized acceptor impurities on the p side and positive ionized donor impurities on the n side. Thus its electric field points from n to p. Its neutral outer regions have negligible electric field, so the band edges become flat there. In the depletion approximation, charge neutrality requires $N_A{x}_p=N_D{x}_n$; the symmetric drawing illustrates $N_A=N_D$ rather than assuming that equality is universal. Diffusion and field-driven drift current cancel at equilibrium: no applied voltage does not mean zero internal electric field or zero built-in electric potential difference.

In three dimensions use the $k$-space density of states $(L/2\pi {)}^3$ per spin species and the spherical shell volume $4\pi k^{2}dk$:Thus the three-dimensional free-electron density of states grows as ${ϵ}^{1/2}$. For physical electrons, including both spin states givesAll three total density of states functions have units ${\mathrm{J}}^{-1}$ when energy is in joules; they count states per energy interval. The paper explicitly suppresses spin in (a) but does not repeat that convention in (b), so giving both conventions avoids a hidden factor-of-two ambiguity. Parts (c)--(f) below use the physical two-spin-state electron gas.