Phonon

A phonon is a bosonic quantum of a crystal's normal modes. Because phonon number is not conserved in thermal equilibrium, its chemical potential is zero.

An internal harmonic vibration spectrum with lowest positive frequency ${\omega }_{\min }$ has the displayed low-temperature thermal phonon population, where $g_{\min }$ is its degeneracy. The continuum bulk law fails when thermally occupied wavelengths reach the sample size. Mechanical boundary conditions determine ${\omega }_{\min }$; rigid translation of the entire solid is not an internal phonon mode.

The allowed phonon energies, including their wavevector, branch and phonon polarization labels when resolved. A phonon dispersion relation records the branch-resolved spectrum, whereas the phonon density of states sums over those labels.

The displacement eigenvector of a lattice normal mode, including relative amplitudes and phases for the atoms in a primitive cell. Longitudinal and transverse describe its orientation relative to the phonon wavevector in simple isotropic cases.

An optical phonon is a quantum of vibration on an optical phonon branch, involving relative motion of atoms within a primitive cell. Its zone-center frequency can remain nonzero. Some such modes couple strongly to light when the relative displacement changes electric polarization; the name does not imply that every optical phonon is optically active. Their finite energies produce thresholds for electron emission and thermally suppressed absorption at low temperature.

In a polar insulating crystal, a band with negative real permittivity can occur between the transverse and longitudinal optical resonances. A lossless bulk transverse wave is evanescent there, producing high normal-incidence reflectance. Damping rounds the edges and permits absorption. The reflection minimum beyond the band can occur where the refractive index approaches one; it is not exactly the zero of permittivity.

A rattling mode is a relatively soft, often localized vibration of a guest atom within a more rigid surrounding cage. An approximately harmonic narrow-frequency contribution can be modeled as an Einstein solid term with frequency $\Omega$ and degeneracy $g$, giving the heat capacity in the title. It freezes out exponentially at low temperature and approaches $gR$ per mole at high temperature, so it can bend a plot of $C/T$ versus $T^2$ away from its pure electronic-plus-acoustic straight line.

Calorimetry of Ba4Na2Ge25 shows an additional low-energy contribution fitted by an Einstein energy $\hslash \Omega /k_B\simeq 32\mathrm{K}$, assigned to a barium guest site.^[ref] This is a measured-material example; the narrow-mode description remains an approximation to a crystal's full vibrational spectrum.

An acoustic phonon is a quantum of vibration on an acoustic phonon branch. At long wavelengths the atoms in each primitive cell move together and translational invariance makes the frequency approach zero. A three-dimensional crystal has three acoustic branches, corresponding to longitudinal and transverse polarizations. A linear small-wavevector dispersion has slope equal to the appropriate sound speed.

The electron-phonon interaction couples electronic states to the atomic displacements represented by phonons. A lattice vibration changes the potential seen by an electron; its quantized displacement permits phonon absorption and emission. This interaction affects electrical resistance, electronic thermal conductivity, lattice dynamics and pairing in conventional superconductors.

For electron energy transfer $\Delta$ and phonon energy ${\Omega }_q=\hslash {\omega }_q$, the symmetrized pair amplitude is $V_{\mathrm{eff}}=2|M_q{|}^2{\Omega }_q/({\Delta }^2-{\Omega }_q^2)$. It is attractive for $|\Delta |<{\Omega }_q$. This interaction is retarded, and screened Coulomb repulsion must also be considered.

A second-quantized electron-phonon Hamiltonian with free electrons, harmonic phonons and terms linear in phonon displacement, $M_q{c}_{k+q,\sigma }^{†}{c}_{k,\sigma }(b_q+b_{-q}^{†})$. The name is used for both general linear electron-phonon models and the more specific long-range coupling to polar optical modes; the intended dispersion and matrix element must be stated.

An anti-Hermitian generator $S$ is chosen so $[S,H_0]=-H_{e\text{-}ph}$. Expanding $e^{S}H{e}^{-S}$ leaves $H_0+[S,H_{e\text{-}ph}]/2$ to second order. Eliminating virtual phonons produces an effective electron-electron interaction, plus energy shifts. Resonant denominators and higher orders limit this perturbative elimination.

A feature in phonon dispersion caused by a nonanalytic wavevector dependence of electronic screening. In a quasi-one-dimensional conductor, enhanced response near $2k_F$ can soften a lattice mode and signal an approaching Peierls transition.

A Peierls transition, named for Rudolf Peierls, is a metal-insulator transition caused by a Peierls distortion of an initially metallic chain.^[ref] A modulation at $2k_F$ mixes states near opposite Fermi points and can open a band gap at the Fermi energy. At half filling with one atomic orbital per chemical atom, this modulation doubles the original lattice period and alternates strong and weak bonds. In a weak-coupling zero-temperature description, the Peierls energy gain is proportional to $-({\delta }^2/t)\ln (t/|\delta |)$ whereas the elastic cost is proportional to ${\delta }^2$, where $t>0$ is the undistorted hopping strength and $\delta$ its small alternating change. The logarithm permits a distortion at arbitrarily weak coupling in the ideal model. Finite temperature, finite size, interchain coupling and thermal fluctuations matter for a real transition; strictly one-dimensional reasoning alone does not establish finite-temperature long-range order.

The experiment follows phonon peaks as both momentum and energy transfer are varied. In inelastic neutron scattering, $\mathbf{q}={\mathbf{k}}_i-{\mathbf{k}}_f$ locates the lattice modulation and $\hslash \omega =E_i-E_f$ measures the mode energy, so a softened mode can be distinguished from static diffraction alone. Measurements on deuterated TTF-TCNQ between $60$ and $300\mathrm{K}$ found a growing Kohn anomaly near $2k_F$ below about $150\mathrm{K}$ as the $54\mathrm{K}$ transition was approached.^[ref]

A Peierls distortion modulates the atomic positions and chemical bonds of a one-dimensional chain with a partly filled energy band. In the half-filled example, alternating short and long bonds double the primitive unit cell. The distortion is the structural change; the associated Peierls transition is the change in electronic conduction.^[ref]

For spin-degenerate electrons at atomic spacing $a$, alternate the hopping amplitudes $-(t+\delta )$ and $-(t-\delta )$, with $t>0$ and $|\delta |<t$. The doubled primitive unit cell gives an off-diagonal Bloch Hamiltonian entry $-(t+\delta )-(t-\delta )e^{-2ika}$. Its two eigenvalues are the displayed energy bands, for $|k|\le \pi /(2a)$. The band gap at the edge of the Brillouin zone is $4|\delta |$. At half filling there are two electrons per doubled primitive unit cell, so the lower band is full and the upper empty. This explicit model realizes the period-doubling form of a Peierls distortion.

Fill the lower energy band of the dimerized tight-binding chain with both spin states. Take zero on-site energy and measure ground-state energy per doubled primitive unit cell. State counting in a Bloch band and $q=ka$ giveEquivalently, for $|\delta |\le t$, $\mathcal{E}(\delta )=-(8t/\pi )E(\sqrt{1-{\delta }^2/t^2})$, where $E(k)$ is the complete elliptic integral of the second kind in the modulus convention.

For fixed $t>0$ and $|\delta |/t\to 0$, the change isThe logarithm comes from states near the edge of the Brillouin zone. Put $u=\pi /2-q$. To leading order the square root there is $\sqrt{t^2{u}^2+{\delta }^2}$. For a fixed small cutoff $\eta >0$, direct integration givesThe rest of the band, and the higher-order corrections near its edge, contribute only $O({\delta }^2/t)$; multiplying by $-8/\pi$ gives the displayed asymptotic expansion. Here the remainder uses Big O notation. The negative logarithmic term exceeds any fixed quadratic elastic cost $K{\delta }^2$, $K>0$, for sufficiently small nonzero $\delta$. This explains the zero-temperature instability to a Peierls distortion in the ideal chain model.

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.

For free-electron recoil and a linear acoustic branch, $E_e(q)={\hslash }^2{q}^2/(2m_e)$ and $E_{\mathrm{ph}}(q)=\hslash c_{s}q$. Their ratio is $q\lambda$, so equality occurs at $q=1/\lambda$. This is a momentum/localization crossover scale, formally half a sound-speed analogue of the Reduced Compton wavelength. It does not replace the speed of light in electron relativity or determine a mean free path. A Fermi-surface electron emitting a phonon has fractional radial loss $\delta \simeq q/(2\lambda k_F^2)$ when the transferred energy is small.

For a spherical Fermi surface of radius $k_F$ lying inside a zone with shortest reciprocal lattice vector magnitude $G>2k_F$, a translated neighbouring Fermi sphere is separated from it by $q_U=G-2k_F$. An electron-phonon Umklapp scattering process satisfying $\mathbf{k}+\mathbf{q}={\mathbf{k}}^{\prime }+\mathbf{G}$ therefore needs a phonon wavevector of at least $q_U$. For an approximately linear acoustic dispersion its activation scale isAt $T\ll T_U$ these phonons are sparsely occupied; once they become thermally available, momentum-relaxing scattering can grow rapidly. Normal small-angle scattering and residual disorder need not vanish below this scale, so it is a crossover estimate rather than a sharp universal conductivity threshold.

For acoustic modes ${\omega }_q\simeq c_{s}q$, thermally active wavevectors have $q_T\sim k_{B}T/(\hslash c_s)$. When $q_T\ll k_F$, a nearly elastic event on a spherical Fermi surface has $1-\cos \theta =q^2/(2k_F^2)$, suppressing momentum relaxation. In the clean three-dimensional acoustic model, the thermal relaxation rate scales as $T^3$ while the electrical transport rate gains two extra powers and scales as $T^5$. With electronic heat capacity proportional to $T$, this gives $K_e\propto T^{-2}$ and electrical resistivity proportional to $T^5$ in the phonon-limited regime. Residual elastic scattering cuts off these behaviours as temperature falls further.^[ref]

Phonon scattering changes the occupation or propagation of lattice-vibration normal modes through interactions with boundaries, defects or other phonons. Anharmonic terms permit phonon-phonon scattering. Different processes need not relax heat flow equally: normal processes conserve total crystal momentum within the first Brillouin zone, whereas Umklapp scattering changes its representative by a nonzero reciprocal lattice vector.

Normal phonon collisions conserve the crystal momentum of the distribution, whereas Umklapp scattering transfers momentum by a reciprocal vector. A normal collision can redistribute heat-carrying modes without relaxing a drifting distribution in the same manner as a momentum-relaxing event. Boundaries, impurities and Umklapp processes can then set the resistance. Counting all collision rates as interchangeable inverse transport times can miss collective phonon flow and nontrivial current weighting.

The phonon mean free path is the typical distance traveled by a phonon between the relevant scattering events. For heat transport, the relevant relaxation time must describe relaxation of heat current, not merely count all collisions. An isotropic kinetic estimate is $\kappa \simeq c_V{v}_s\ell /3$, where $c_V$ is heat capacity per unit volume. Boundary-limited $\ell$ and the low-temperature Debye model give $\kappa \propto T^3$.

An Umklapp process conserves reduced crystal momentum modulo a nonzero reciprocal lattice vector. It can occur in electron-phonon scattering as well as phonon-phonon scattering; the lattice takes up the reciprocal-lattice momentum.

In a three-phonon Umklapp scattering process, the first-zone wavevectors obey the displayed equation for a nonzero reciprocal lattice vector $\mathbf{G}$, together with conservation of energy. The crystal momentum transferred to the lattice allows these processes to relax a phonon heat current. They are suppressed at sufficiently low temperature and often determine the approximate $1/T$ high-temperature thermal conductivity of a crystalline electrical insulator.

The Debye–Waller factor suppresses coherent scattering because atoms fluctuate about their equilibrium positions. For Gaussian displacement $u$, the amplitude factor is $⟨e^{iqu}⟩=\exp [-q^2⟨u^2⟩/2]$ and the intensity factor is its squared modulus. It multiplies the lattice interference factor, so exact extrema of the full finite-lattice intensity need not coincide with the constructive-interference centers when this envelope varies appreciably.

For a chain with lattice spacing $b$ and $N$ atoms, write $x_s=sb+u_s$. The periodic normal-mode condition is $u_N=u_0$, equivalently $x_N=x_0+Nb$ for unwrapped positions. The literal position condition $x_N=x_0$ is incompatible with the equilibrium $x_s=sb$ in that coordinate system. Fourier modes $k_r=2\pi r/(Nb)$ give ${\omega }_r=2\lambda |\sin (k_{r}b/2)|$ for spring coefficient $m{\lambda }^2$. The zero mode describes free center-of-mass translation and must be fixed or handled separately when defining an oscillator vacuum.

A phonon dispersion relation gives its angular frequency $\omega$ as a function of its wavevector $k$. For an isotropic power law, $\omega =C|k{|}^{\alpha }$.

For a translation-invariant monatomic chain with symmetric pair force constants, $M{\ddot{u}}_p={\sum }_{n>0}{C}_n(u_{p+n}+u_{p-n}-2u_p)$. Substituting a traveling normal mode gives the displayed even, reciprocal-periodic phonon dispersion relation. If $\sum n^2{C}_n$ converges and the chain is stable, its long-wavelength speed is $a\sqrt{\sum n^2{C}_n/M}$. A truly long-range interaction need not have a finite second moment or a finite linear acoustic slope.

Let $\alpha =C_2/C_1>0$ and $x={\sin }^2(ka/2)$. The squared frequency in units $4C_1/M$ is $(1+4\alpha )x-4\alpha x^2$. For $\alpha \le 1/4$ its maximum on $0\le x\le 1$ is at the zone edge and equals one. For $\alpha >1/4$ it moves inside the zone to $x=(1+4\alpha )/(8\alpha )$, with the displayed frequency ratio. The small-wavevector speed instead increases by $\sqrt{1+4\alpha }$; maximizing the frequency and finding the sound speed are different operations.

A continuous family of normal modes labelled by the wavevector and a branch index $s$. For $r$ chemical atoms per three-dimensional primitive unit cell, the dynamical matrix has $3r$ eigenvalues, giving $3r$ branches: three acoustic phonon branches and $3r-3$ optical phonon branches for a mechanically stable ordinary crystal structure.

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.

For alternating masses $m,M$ at spacing $d$, equal nearest-neighbour spring constants $K$, and cell length $2d$, a two-component normal mode givesThe minus sign is the acoustic phonon branch and the plus sign is the optical phonon branch. For $M>m$ the edge frequencies are $\sqrt{2K/M}$ and $\sqrt{2K/m}$, leaving a gap. At $k=0$ they are zero and $\sqrt{2K(1/m+1/M)}$.

In an alternating-mass chain with nearest-neighbour spacing $a$ and cell length $2a$, the inter-sublattice Bloch coupling is proportional to $\cos qa$. At the first-zone edge $q=\pi /(2a)$ it vanishes. For $M>m$, the acoustic edge mode moves only the heavy sublattice and the optical edge mode only the light one. Their frequency difference is $\sqrt{2\alpha /m}-\sqrt{2\alpha /M}$. At long wavelength the acoustic displacements are in phase, whereas the optical mode has zero centre-of-mass displacement.

For alternating masses $m_A,m_B$, nearest-neighbour spacing $a$ and equal spring constants $\alpha$, the zone-centre optical angular frequency and acoustic group velocity obeyTherefore $\alpha =m_A{m}_B{\omega }_{\mathrm{op}}^2(0)/[2(m_A+m_B)]$ and $a=v_s\sqrt{(m_A+m_B)/(2\alpha )}$. These infer effective model parameters from a measured intercept and initial slope of a phonon dispersion relation. Fitting a real polar crystal to a nearest-neighbour scalar model does not guarantee that every measured point or its structural spacing is reproduced exactly.

For equal masses $m$ at spacing $d$ joined by nearest-neighbour springs of spring constant $K$, Newton's second law gives $m{\ddot{u}}_j=K(u_{j+1}+u_{j-1}-2u_j)$. A normal mode $u_j=u_0{e}^{i(kjd-\omega t)}$ givesThis single acoustic phonon branch is linear at small wavevector and has zero group velocity at the first Brillouin zone edge.

A nearest-neighbour chain with the Lennard-Jones potential $V(x)=4ϵ[(\sigma /x{)}^{12}-(\sigma /x{)}^6]$ has equilibrium spacing $a=2^{1/6}\sigma$ and spring constant $K=V^{\prime \prime }(a)=72ϵ/a^2$. The monatomic-chain dispersion relation gives speed of sound $v=a\sqrt{K/m}=\sqrt{72ϵ/m}$ and maximum angular frequency $2v/a$. The cancellation of $\sigma$ from the sound speed holds in this chain model; it does not remove the spacing from the cutoff frequency.

With real wave number and time convention $e^{-i\omega t}$, an on-site friction force $-\Gamma {\dot{u}}_j$ shifts the frequencies as displayed. Underdamped oscillations have amplitude envelope $e^{-\Gamma t/(2m)}$, while overdamped modes have two nonoscillatory rates. At small wave number a stable acoustic chain has slow decay $\omega \simeq -im{c}_s^2{k}^2/\Gamma$. Exactly uniform displacement remains a zero-restoring-force mode, so a constant displacement is not forced to vanish.

A periodic chain of $N$ sites and length $Na$ has allowed signed wave numbers separated by $2\pi /(Na)$. The continuum state density on the signed axis is $Na/(2\pi )$. Combining opposite propagation directions onto $0<|q|<\pi /a$ doubles it to $Na/\pi$. This counts the same $N$ modes, not two independent acoustic phonon branches.

An acoustic phonon branch has $\omega (k)\to 0$ as $k\to 0$. Its long-wavelength group velocity is the speed of sound.

An optical phonon branch has a nonzero frequency as $k\to 0$, corresponding to relative motion of degrees of freedom within one primitive cell.

For equal masses $m$ separated by $a$ and alternating spring constants $\lambda$ and $\alpha \lambda$, the primitive-cell length is $2a$ and the two branches areThe minus and plus signs give the acoustic phonon branch and optical phonon branch, respectively.

Take equal masses $m$, atomic spacing $a$, and alternating springs $K\gg \kappa >0$. The primitive unit cell contains a strong-bonded pair, so its length is $2a$ and the Brillouin zone ends at $\pm \pi /(2a)$. Expanding the exact alternating-spring chain dispersion relation givesThe low-frequency acoustic phonon branch moves almost rigid dimers of mass $2m$ coupled by weak springs; its speed of sound tends to $a\sqrt{2\kappa /m}$. The high-frequency optical phonon branch describes internal dimer stretch. At the zone edge the exact frequencies are $\sqrt{2\kappa /m}$ and $\sqrt{2K/m}$.

For two polarizations in area $A$ and $\omega =C|k{|}^{\alpha }$, counting wavevector states in a disk gives

The Debye model replaces the crystal's vibrational spectrum by a continuum of phonon modes up to a cutoff chosen to preserve the total number of vibrational degrees of freedom.

Integrating the Bose-Einstein distribution against the three-dimensional Debye model phonon density of states gives $N_{\mathrm{ph}}=9N(T/{\Theta }_D{)}^3{\int }_0^{{\Theta }_D/T}{x}^2/(e^x-1)dx$. For $T\ll {\Theta }_D$, expand $(e^x-1{)}^{-1}={\sum }_{j\ge 1}{e}^{-jx}$ and integrate termwise: the limiting integral is $2{\sum }_{j\ge 1}{j}^{-3}=2\zeta (3)$. This counts thermal quanta, not available modes or half quanta of zero-point energy.

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.

Suppose a three-dimensional Debye model is normalized to $3N_m$ vibrational modes in volume $V$. Its low-temperature heat-capacity coefficient is the one displayed, and its spherical wavevector cutoff obeys $k_D^3=6{\pi }^2{N}_m/V$. The Debye temperature is ${\Theta }_D=\hslash v_D{k}_D/k_B$. At fixed measured $\beta$ and volume, changing the counting convention by $N_m\mapsto c{N}_m$ raises both ${\Theta }_D$ and $k_D$ by $c^{1/3}$; the inferred average sound speed is unchanged.

For a mole of formula units containing $r$ atoms each, a conventional full atomic-mode count uses $N_m=r{N}_A$ and tends to lattice heat capacity $3rR$ at high temperature. A reduced convention using $N_m=N_A$ instead has three effective modes per formula unit. These are different continuum normalizations, not a claim that the real crystal has only three modes per formula unit. A real multi-atom primitive cell has three acoustic branches and additional optical branches; a Debye sphere preserving the full atomic count is an effective representation of that spectrum, rather than its literal first Brillouin zone.

For three-dimensional Debye model mode density $D(\omega )=9N{\omega }^2/{\omega }_D^3$, the Bose-Einstein distribution becomes $n_{\omega }\simeq k_{B}T/(\hslash \omega )$ when $T\gg {\Theta }_D$. Integrating the occupations gives $N_{\mathrm{ph}}\simeq (9/2)NT/{\Theta }_D$. The number of phonon quanta is not the number $3N$ of vibrational modes and need not be bounded by it. If a stipulated electron-phonon scattering model makes resistivity proportional to this phonon number with a temperature-independent coefficient, the high-temperature electrical resistivity is proportional to $T$.

The three-dimensional Debye model normalizes its phonon density of states to $3N$ modes: $g(\omega )=9N{\omega }^2/{\omega }_D^3$ below the Debye frequency ${\omega }_D$. The thermal energy is $U={\int }_0^{{\omega }_D}g(\omega )\hslash \omega /[e^{\hslash \omega /(k_{B}T)}-1]d\omega$. Differentiate at fixed volume and substitute $x=\hslash \omega /(k_{B}T)$ to obtain the displayed heat capacity at constant volume.

At $T\ll {\Theta }_D$, integration by parts gives ${\int }_0^{\infty }{x}^4{e}^x/(e^x-1{)}^{2}dx=4{\int }_0^{\infty }{x}^3/(e^x-1)dx=4{\pi }^4/15$. Therefore $C_V\sim (12{\pi }^4/5)N{k}_B(T/{\Theta }_D{)}^3$. At $T\gg {\Theta }_D$, each mode approaches the equipartition theorem value $k_B$ of heat capacity, so $C_V\to 3N{k}_B$, the Dulong-Petit law. Fixed harmonic zero-point energy does not contribute to either limit.

For a monovalent three-dimensional free electron gas, the spin-summed molar density of states is $g(E_F)=3N_A/(2E_F)$. Combining the shell estimate $C_{el}=3k_B^{2}g(E_F)T$ with $C_{la}=(12{\pi }^4/5)R(T/{\theta }_D{)}^3$ gives the displayed nonzero crossover. The controlled Sommerfeld expansion replaces $3$ by ${\pi }^2/3$ and multiplies $T_{\ast }^2$ by ${\pi }^2/9$. A Fermi energy alone does not determine the DOS in an unspecified band structure.

A neutral two-dimensional linear electronic spectrum has heat capacity proportional to $A{T}^2$, while three-dimensional acoustic phonons have Debye heat capacity proportional to $N{T}^3/{\Theta }_D^3$. Thus their ratio scales as $A{k}_B^2{\Theta }_D^3/({\hslash }^2{v}^{2}NT)$. At fixed crystal composition and shape, $A/N\propto 1/L$ for linear size $L$, favoring the electronic surface contribution in smaller or colder specimens. The continuum laws require thermally populated wavelengths small compared with sample size; their formal low-temperature crossover need not lie inside that regime.

The spherical cutoff in the three-dimensional Debye model preserves three vibrational modes per atom. For atom density $n_a$, $q_D=(6{\pi }^2{n}_a{)}^{1/3}$ and ${\omega }_D=c_s{q}_D$ in a single-speed approximation. A monovalent free-electron metal has $k_F=(3{\pi }^2{n}_a{)}^{1/3}$, giving $q_D=2^{1/3}{k}_F$. This cutoff is a mode-counting approximation, not an exact spherical boundary of the crystal's Brillouin zone.

The Debye frequency is the upper frequency cutoff fixed by equating the continuum mode count to the crystal's number of vibrational degrees of freedom.

The Debye temperature is $T_D=\hslash {\omega }_D/k_B$.

For two polarizations, $n$ atoms in area $A$, and $\omega =C|k{|}^{\alpha }$, mode counting givesAt low temperature its energy scales as $T^{1+2/\alpha }$ and its constant-volume heat capacity as $T^{2/\alpha }$.