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The effective isotropic acoustic speed used to replace separate branches in the Debye model. With atomic number density $n$, counting three branches gives $k_D=(6{\pi }^{2}n{)}^{1/3}$ and ${\omega }_D=v_D{k}_D=k_B{\theta }_D/\hslash$. Matching the low-frequency density of states for one longitudinal and two transverse branches gives the displayed inverse-cube average. A Debye temperature alone does not determine both individual sound speeds.

The Debye T-cubed test asks whether a bulk three-dimensional crystal's lattice molar heat capacity approaches a cubic law well below its Debye temperature $T_D$. It tests the low-frequency acoustic phonon branches in the Debye model, rather than asserting that every contribution to a real sample's heat capacity is cubic. Peter Debye's continuum description makes a striking connection: a thermal measurement can reveal the low-frequency vibrational density of states without resolving individual vibrations.^[ref]

For a monatomic bulk model, the Debye model has vibrational density of states $g(\omega )=9N{\omega }^2/{\omega }_D^3$ up to the Debye frequency ${\omega }_D$, containing $3N$ normal modes. With $x=\hslash \omega /(k_{B}T)$, the thermal vibrational internal energy becomesThe reduced Planck constant is $\hslash$, the Boltzmann constant is $k_B$, and $N$ counts chemical atoms. For $T\ll T_D$, replace the upper limit by infinity. The integral is ${\pi }^4/15$, as in the Stefan–Boltzmann law derivation. Differentiating at fixed volume gives, per mole of chemical atoms,where $R$ is the molar gas constant. The cubic exponent comes from three-dimensional acoustic phonon branches with linear low-wavevector dispersion relations. The coefficient encodes their propagation speeds. Formula-unit normalization in a compound requires its number of chemical atoms per formula unit; do not silently substitute moles of formula units for moles of chemical atoms.

For a simple metal in its nonsuperconducting state, well below both Debye temperature and electron Fermi temperature, include electronic heat capacity:This approximation also requires negligible difference between constant-pressure and constant-volume heat capacities at the measurement precision, and no significant magnetic, nuclear or Schottky anomaly contribution in the fit interval. Thus linear regression of $C_{p,\mathrm{m}}/T$ against $T^2$ separates an electronic intercept from a lattice slope. For an insulating crystal with no relevant low-energy magnetic excitations, the electronic term can instead be negligible; a superconductivity transition changes the electronic response.^[ref]

After subtracting calorimeter addenda, Brando's second gold sample, $72.6\mathrm{mg}$, gave a linear $C_{p,\mathrm{m}}/T$ versus $T^2$ plot in the nonsuperconducting state. The reported electronic intercept was $7.0\times 10^{-4}\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-2}$; deviations from reference heat capacities were below $2\%$.^[ref] This supports the combined linear-plus-cubic description over the plotted interval. Because the platform subtraction used reference gold, it is also an instrument validation with a shared material standard, rather than a wholly independent determination of the Debye model from an uncalibrated instrument.

For a worked original inference, stipulate values of $y=C_{p,\mathrm{m}}/T$ equal to $(0.001100,0.002300,0.004300)\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-2}$ at $T=(1,2,3)\mathrm{K}$. Assign independent $u(y)=0.000020\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-2}$ and negligible temperature error. Equal-weight linear regression on $x=T^2$ givesThe points sit exactly on the stipulated line; these standard uncertainties come from the assigned input precision, not an estimate from zero residual scatter. With $R=8.314462618\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, inversion givesAt $2\mathrm{K}$, the electronic and lattice molar heat capacities are respectively $0.0014$ and $0.0032\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. Their equality occurs at $\sqrt{\gamma /\beta }=1.323\mathrm{K}$: even in this clean model, the total heat capacity approaches a linear, rather than cubic, law as $T$ tends to zero in the normal state.

A straight line over a narrow interval is a useful diagnostic, not a universal proof. Inspect residuals and repeat the fit over several upper and lower temperature cutoffs; curvature can signal higher lattice terms, an unremoved Schottky anomaly, or a faulty calorimeter addenda subtraction. Correlated thermometer calibration and sample-amount errors affect both axes and can dominate the illustrative linear regression uncertainty. Low-dimensional samples, additional low-energy excitations and proximity to a phase transition need other models. The test is valuable precisely because its failures can reveal physics beyond the bulk acoustic approximation.

Lattice vibrations contribute a phonon heat capacity. At temperatures well below the Debye temperature, its leading contribution is proportional to $T^3$, whereas the electronic part is proportional to $T$:The lattice term can therefore become important rapidly as temperature rises, even if the electronic term dominates sufficiently close to zero. At higher temperature the lattice contribution approaches the Dulong-Petit law value $3R$ per mole of atoms within the harmonic approximation. This is not a claim that the low-temperature $T^3$ law holds at all temperatures.

The FCC conventional unit cell has four atoms, so the cube containsatoms, neglecting surface corrections. A phonon is one excitation quantum of a vibrational normal mode; there can be many quanta per mode, and zero-point motion does not add half a phonon to the thermal population.

For the low-temperature thermal phonon number, integrate the Bose-Einstein distribution against the Debye model phonon density of states:At $10\mathrm{K}$, ${\Theta }_D/T=21.5$, so extending the upper limit to infinity has negligible effect at the precision required. The supplied integral yieldsThis is the thermally excited phonon number, not the number $3N$ of available modes.

The continuum approximation fails when the thermally occupied wavelengths become comparable with the cube size $L$. From the Debye temperature,A lowest nontrivial acoustic frequency is of order $v/L$; a standing-wave estimate ${\omega }_{\min }\sim \pi v/L$ corresponds to $\hslash {\omega }_{\min }/k_B\sim 0.25\mathrm{K}$. Periodic boundary conditions instead give $2\pi v/L$ and $0.50\mathrm{K}$. The precise spectrum depends on mechanical boundary conditions and branch speeds, neither of which is specified.

This finite-crystal phonon freeze-out requires summing discrete normal modes. For a finite positive lowest frequency, $N_{\mathrm{ph}}\sim g_{\min }{e}^{-\hslash {\omega }_{\min }/(k_{B}T)}$, rather than extrapolating the bulk $T^3$ law indefinitely. Rigid translations and rotations of an isolated crystallite are not internal thermal phonons. Surface acoustic modes can modify the intermediate crossover, but do not justify inventing a unique boundary spectrum.

The Debye model fixes its cutoff by requiring the continuum to contain the $2n$ vibrational degrees of freedom:Therefore the Debye frequency and Debye temperature are

The Debye model treats a solid's low-frequency lattice vibrations as acoustic phonons in an elastic continuum. There are three polarizations per wavevector, one longitudinal and two transverse, with approximately linear dispersion relations. A cutoff is chosen to preserve the total $3N$ vibrational modes of $N$ atoms. The phonon number itself is not fixed; mode occupations follow the Bose-Einstein distribution with zero chemical potential.

For one polarization of speed $v_s$, counting wavevector states in a spherical shell gives $g_s(\omega )=V{\omega }^2/(2{\pi }^2{v}_s^3)$. Define an average Debye speed by $3/v_D^3=1/v_L^3+2/v_T^3$. The total density of states can then be writtenThe Debye wavevector is $k_D=(6{\pi }^{2}N/V{)}^{1/3}$, the Debye frequency is ${\omega }_D=v_D{k}_D$, and the Debye temperature is ${\Theta }_D=\hslash {\omega }_D/k_B$.

Omitting the temperature-independent zero-point energy, the vibrational energy and resulting Debye heat capacity areAt low temperature the upper limit can be extended to infinity. The integral is $4{\pi }^4/15$, givingOnly the long-wavelength acoustic modes are appreciably populated, explaining the Debye T-cubed test. At high temperature every mode contributes $k_B$ by equipartition theorem, so $\boxed{C_V\to 3N{k}_B}$, the Dulong-Petit law.

The model succeeds especially at low temperature, where long-wavelength elasticity is reliable. It replaces real anisotropic branches by a simple continuum and cannot reproduce detailed optical phonon branches or every intermediate-temperature feature. Metals also have an electronic heat capacity approximately proportional to $T$; defects, magnetic excitations or a phase transition can add other terms. Experiments often measure heat capacity at constant pressure, which need not equal the harmonic model's $C_V$ exactly.

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.

In an ordinary three-dimensional crystalline electrical insulator, heat is carried mainly by phonons. Kinetic theory gives the estimatewhere $c_V$ is heat capacity per unit volume, $v_s$ is a typical sound speed, and $\ell$ is the phonon mean free path. The thermal conductivity $\kappa$ has units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$.

At low temperature, $T\ll {\Theta }_D$, the Debye model gives $c_V\propto T^3$. With phonon-phonon collisions frozen out, diffuse scattering at the sample boundaries makes $\ell$ approximately constant, and $v_s$ is nearly constant. HenceAt high temperature, of order or above the Debye temperature, the Dulong-Petit law makes $c_V$ approximately constant. Anharmonic phonon scattering, especially resistive Umklapp scattering, has a collision rate approximately proportional to $T$. Thus $\ell \propto T^{-1}$ andThese two regimes usually give an intermediate maximum of thermal conductivity. The assumptions matter: glassy materials, strong defects or specular boundary reflection can change the low-temperature law; these are not universal laws for every electrical insulator.

The Wiedemann-Franz law applies to a normal degenerate metal when the same itinerant electrons carry heat and charge, scattering is approximately elastic over the thermal shell, and the transport probability varies slowly across the energy interval $k_{B}T$ around the Fermi level. The comparison uses the electronic thermal conductivity $K_e$, with any lattice contribution removed.

It works particularly well as $T\to 0$ when elastic impurity or defect scattering dominates. It also often holds approximately for ${\Theta }_D\ll T\ll T_F$, where thermally populated phonons act as a nearly elastic classical scattering field. Room temperature need not lie above the Debye temperature for every metal.

At intermediate temperatures, inelastic small-angle electron-phonon scattering can relax heat much more efficiently than charge. Then ${\tau }_Q\ne {\tau }_{\sigma }$ and $K_e/(\sigma T)$ can be smaller than $L_0$. Strong energy dependence of scattering, additional neutral heat carriers, or a transition out of the normal metallic state also invalidates the simple derivation. An observed deviation alone does not identify which assumption failed.

The Fermi-Dirac distribution gives the mean occupancy of a single fermion state:Its maximum occupancy of one expresses the Pauli exclusion principle. A spatial orbital with two spin states can hold two electrons because they are different single-particle states. At zero temperature, the occupation is a step at the Fermi energy $E_F$; at low nonzero temperature, occupation changes appreciably only in an energy shell of width of order $k_{B}T$ about the chemical potential.

Consequently, most conduction electrons cannot absorb small thermal energies into already occupied states. Only a fraction of order $T/T_F$ participates, where $T_F=E_F/k_B$. For a normal metal with a smooth density of states $g(E_F)$, including both spin states and using the total-system normalization, the Sommerfeld expansion at fixed particle number givesand thereforeFor a three-dimensional free electron gas, $g(E_F)=3N/(2E_F)$, so $C_{\mathrm{el}}=({\pi }^2/2)N{k}_B(T/T_F)$. This is much smaller than a classical equipartition estimate at $T\ll T_F$. The chemical potential must adjust with temperature to maintain $N$; treating it as fixed while differentiating an energy integral gives the wrong fixed-$N$ calculation in general.

The lattice adds a Debye contribution $C_{\mathrm{ph}}={\beta }_D{T}^3$ at temperatures well below the Debye temperature. HenceA plot of measured $C/T$ against $T^2$ can separate the electronic intercept from the phonon slope. calorimetry must account for the heat capacity of the sample holder, heat leaks, thermal equilibration and thermometer calibration. A magnetic impurity contribution or a superconducting transition can invalidate this two-term fit; the prediction is for the normal metallic state in an appropriate low-temperature interval. Differences from a free-electron coefficient can reveal quasiparticle effective-mass enhancement or nontrivial electronic structure rather than a failure of Fermi statistics.

For a phonon mode of angular frequency $\omega$, the equilibrium occupation isWell above the Debye temperature, the relevant vibrational energies are small compared with $k_{B}T$. Expanding the exponential givesThe spectrum and its cutoff are fixed in the model, so the remaining sum is temperature independent. ThusFor example, the three-dimensional Debye model with $D(\omega )=9N{\omega }^2/{\omega }_D^3$ gives $N_{\mathrm{ph}}\simeq (9/2)NT/{\Theta }_D$. This is the high-temperature phonon number in the Debye model. The number of phonon quanta can exceed $3N$; $3N$ counts modes, not their total occupation. Under the stipulated proportionality to phonon number, only the phonon contribution to resistivity has been calculated. A temperature-independent residual resistivity would add an intercept in a real sample.