Debye model

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 $T^{2/\alpha }$.