Thermal Properties — Einstein and Debye Models

The heat capacity of an insulating solid is the clearest test of lattice dynamics. Classical statistical mechanics predicts a constant value independent of temperature; experiment finds that value only at high temperature and sees the heat capacity fall to zero as $T \to 0$ , proportional to $T^{3}$ . The resolution is the quantization of the phonon modes: modes with $ℏ ω ≫ k_{B} T$ are frozen out and stop absorbing heat. This lesson works through two models of increasing fidelity — Einstein's single frequency and Debye's linear spectrum — and extracts the universal low- temperature law.

The Dulong–Petit law and its failure

The lattice energy is a sum over modes of the mean phonon energy. Using the Bose occupation and including the density of states $g (ω)$ ,

U = 0 \int \infty d ω g (ω) ℏ ω (\frac{1}{e ^{ℏ ω / k_{B} T} - 1} + \frac{1}{2}) .

The zero-point term is temperature-independent and drops out of the heat capacity $C = (\partial U / \partial T)_{V}$ . Classically, equipartition assigns $k_{B} T$ to each of the $3 N$ vibrational modes (kinetic plus potential), giving $U = 3 N k_{B} T$ and the constant heat capacity known as the Dulong–Petit law:

C_{DP} = 3 N k_{B} = 3 n R,

with $n$ the number of moles and $R = N_{A} k_{B} = 8.314 J mol^{- 1} K^{- 1}$ . The molar value $3 R \approx 24.9 J mol^{- 1} K^{- 1}$ matches most solids at room temperature. Below a material-specific temperature it fails: $C$ drops steadily toward zero. Equipartition assumes every mode carries $k_{B} T$ , but a mode of frequency $ω$ holds that energy only when $k_{B} T ≳ ℏ ω$ . As $T$ falls, the high-frequency modes freeze first, then progressively lower ones, and the heat capacity follows them down.

Molar heat capacity versus temperature. At high temperature C approaches the classical Dulong-Petit plateau 3R; below a characteristic temperature the modes freeze out and C falls smoothly to zero, vanishing as T cubed at the lowest temperatures.

The Einstein model

Einstein modelled every atom as an independent three-dimensional oscillator of a single frequency $ω_{E}$ , so $g (ω) = 3 N δ (ω - ω_{E})$ . The energy and heat capacity follow from one oscillator times $3 N$ :

U = 3 N ℏ ω_{E} (\frac{1}{e ^{ℏ ω_{E} / k_{B} T} - 1} + \frac{1}{2}), C = 3 N k_{B} (\frac{Θ _{E}}{T})^{2} \frac{e ^{Θ_{E} / T}}{( e ^{Θ_{E} / T} - 1 ) ^{2}},

where the Einstein temperature $Θ_{E} = ℏ ω_{E} / k_{B}$ sets the scale. Two limits check the result. For $T ≫ Θ_{E}$ , expanding the exponentials gives $C \to 3 N k_{B}$ , recovering Dulong–Petit. For $T ≪ Θ_{E}$ ,

C \approx 3 N k_{B} (\frac{Θ _{E}}{T})^{2} e^{- Θ_{E} / T},

which does fall to zero, capturing the qualitative freeze-out. The exponential is too abrupt, though: real solids follow $T^{3}$ , not $e^{- Θ_{E} / T}$ . The Einstein model fails at low $T$ because it gives every mode the same high frequency, so all modes freeze together. In a real crystal the acoustic branches extend to arbitrarily low frequency, and those low-frequency modes stay thermally active down to much lower temperature. Einstein's picture works well for the nearly dispersionless optical branches, which do cluster near one frequency.

The Debye model

Debye kept the low-frequency acoustic modes that Einstein discarded. He replaced the true dispersion with a linear one, $ω = v_{s} k$ , valid for the long- wavelength acoustic modes that dominate at low $T$ , and used it for all modes. The density of states of a linear, isotropic spectrum in three dimensions is

g (ω) = \frac{V}{2 π ^{2}} \frac{3 ω ^{2}}{v _{s}^{3}} = \frac{9 N}{ω _{D}^{3}} ω^{2}, ω \leq ω_{D},

quadratic in $ω$ . Because a crystal of $N$ atoms has only $3 N$ modes, the spectrum must be cut off at a maximum frequency, the Debye frequency $ω_{D}$ , fixed by $\int_{0}^{ω_{D}} g (ω) d ω = 3 N$ . The corresponding Debye temperature is $Θ_{D} = ℏ ω_{D} / k_{B}$ .

Debye density of states. The quadratic curve g proportional to omega squared is truncated abruptly at the Debye frequency omega_D, the cutoff that keeps the total mode count at 3N. The shaded area equals 3N.

The energy integral becomes, with $x = ℏ ω / k_{B} T$ ,

U = \frac{9 N}{ω _{D}^{3}} 0 \int ω_{D} \frac{ℏ ω ^{3}}{e ^{ℏ ω / k_{B} T} - 1} d ω = 9 N k_{B} T (\frac{T}{Θ _{D}})^{3} 0 \int Θ_{D} / T \frac{x ^{3}}{e ^{x} - 1} d x .

Differentiating gives the Debye heat capacity

C = 9 N k_{B} (\frac{T}{Θ _{D}})^{3} 0 \int Θ_{D} / T \frac{x ^{4} e ^{x}}{( e ^{x} - 1 ) ^{2}} d x,

a universal function of the single ratio $T / Θ_{D}$ . All monatomic solids collapse onto one curve when $C$ is plotted against $T / Θ_{D}$ .

The T³ law

The low-temperature limit is the model's triumph. As $T \to 0$ the upper limit $Θ_{D} / T \to \infty$ and the integral becomes a constant,

0 \int \infty \frac{x ^{3}}{e ^{x} - 1} d x = \frac{π ^{4}}{15} .

The energy is then $U = \frac{3}{5} π^{4} N k_{B} T (T / Θ_{D})^{3}$ , and the heat capacity is the Debye $T^{3}$ law

C = \frac{12 π ^{4}}{5} N k_{B} (\frac{T}{Θ _{D}})^{3} \approx 234 N k_{B} (\frac{T}{Θ _{D}})^{3} .

The physical origin is a mode count. At temperature $T$ only modes with $ℏ ω ≲ k_{B} T$ are thermally active, i.e. those inside a sphere of radius $k_{th} \sim k_{B} T /ℏ v_{s}$ in $k$ -space. The number of such modes scales as the sphere volume, $k_{th}^{3} \propto T^{3}$ , and each contributes about $k_{B}$ , so $C \propto T^{3}$ . The exponent counts the three dimensions of $k$ -space that the acoustic modes fill.

The full interpolation and comparison

Between the two limits the Debye integral must be evaluated numerically, but it interpolates smoothly from $C = 3 N k_{B}$ at high $T$ to $C \propto T^{3}$ at low $T$ . The contrast with Einstein is sharpest at low temperature, where the Einstein curve drops exponentially and the Debye curve as a power law, matching data.

Heat capacity of the Einstein and Debye models against reduced temperature. Both reach the Dulong-Petit plateau at high T. At low T the Einstein curve (dashed) falls exponentially and undershoots the data, while the Debye curve (solid) follows the correct T-cubed power law.

A clean way to separate the lattice from the electronic term in a metal plots $C / T$ against $T^{2}$ . The total low-temperature heat capacity is

C = γ T + β T^{3}, C / T = γ + β T^{2},

with $γ T$ the electronic term (linear in $T$ ) and $β T^{3}$ the Debye lattice term. The plot is a straight line whose intercept is $γ$ and whose slope $β$ gives $Θ_{D}$ through $β = (12 π^{4} /5) N k_{B} / Θ_{D}^{3}$ .

Separating the electronic and lattice terms. Plotting C over T against T squared linearizes C = gamma T + beta T cubed: the intercept is the electronic coefficient gamma, and the slope beta fixes the Debye temperature.

Thermal expansion and the Grüneisen parameter

A strictly harmonic crystal would not expand: the mean position of each atom sits at the symmetric minimum of a parabola regardless of amplitude. Real solids expand because the mode frequencies depend on volume — a mild anharmonic effect captured by keeping harmonic modes but letting each $ω_{s} (k)$ shift as the lattice is compressed or dilated (the quasiharmonic approximation). The sensitivity is the Grüneisen parameter

γ_{G} = - \frac{d ln ω}{d ln V},

the fractional drop in a mode frequency per fractional increase in volume, of order unity for most solids. Thermodynamics then ties the linear thermal- expansion coefficient $α$ to the heat capacity:

α = \frac{γ _{G} C _{V}}{3 B V},

with $B$ the bulk modulus and $C_{V}$ the heat capacity per unit volume; the volume coefficient is $3 α$ . The proportionality to $C_{V}$ predicts that thermal expansion, like heat capacity, falls as $T^{3}$ at low temperature and saturates at high temperature — a link confirmed across many materials. The anharmonic mechanism behind this expansion, and behind the finite thermal conductivity that pure harmonic theory cannot produce, is the subject of the next lesson.