An interactive demo created by Claude Opus 5.5, based on the lecture notes by Sang Hoon Lee (Chapter 7, Section 7.5, Debye Theory of Solids)
The atoms of a crystal don’t vibrate independently: they move together in sound waves, and the quanta of those waves, phonons, are bosons just like photons. Treating them that way explains why the heat capacity of every solid falls as \(T^3\) at low temperature.
The Einstein model treats a solid as a collection of independent quantum harmonic oscillators, each storing energy in units of \(\varepsilon = hf\) (with \(E_n = (n+\tfrac12)\hbar\omega\)). One oscillator has \(Z = 1/(1 - e^{-\beta hf})\) and \(\bar{E} = hf/(e^{\beta hf}-1)\). For \(N\) atoms vibrating in three dimensions, \(N \to 3N\), and
At high temperature this gives \(C_V \to 3Nk_{\mathrm{B}}\), the equipartition result. But as \(T \to 0\) it predicts \(C_V \propto e^{-\varepsilon/k_{\mathrm{B}}T}/T^2\), falling exponentially, whereas experiments find \(C_V \propto T^3\). The problem is that atoms in a crystal do not vibrate independently. Low-frequency modes, in which large groups of atoms move together, carry small energy units and stay active at very low temperatures, long after the high-frequency modes are frozen out.
The modes of a crystal are much like those of the electromagnetic field, with three differences. Sound waves travel at the speed of sound \(c_s\) (taken as constant), they have three polarizations (two transverse, one longitudinal: S-waves and P-waves in seismology), and they cannot have wavelengths shorter than twice the atomic spacing. Each bump must contain at least one atom. Try it on a row of 12 atoms with fixed ends.
Each mode has equally spaced energy levels with unit \(\varepsilon = hf = hc_s/\lambda = hc_s n/2L\), where \(L\) is the length of the crystal; in three dimensions \(n\) becomes a vector \(\vec{n}\). In equilibrium the average number of units is the Planck distribution, \(\bar{n}_{\mathrm{Pl}} = 1/(e^{\varepsilon/k_{\mathrm{B}}T} - 1)\): just as for photons with \(c \to c_s\). These quanta are called phonons, bosons with \(\mu = 0\). The total thermal energy is
with the 3 counting polarizations. Unlike photons, though, not every \(\vec{n}\) is allowed.
Since \(n\) can’t exceed the number of atoms in a row, \(N^{1/3}\), the real sum runs over a cube in \(n\)-space, \([1, N^{1/3}]^3\). Peter Debye’s clever idea was to pretend instead that the relevant region is an eighth of a sphere with the same total volume \(N\):
This is exact in both limits. At high \(T\), all that matters is the total number of modes, which the sphere preserves. At low \(T\), modes with large \(\vec{n}\) are frozen out anyway, so it doesn’t matter how we count them. The picture shows a cube of \(8^3\) modes; brightness shows how active each mode is (its contribution to \(C_V\)).
Drag to rotate, or focus the picture and use the arrow keys. Faded points are frozen out.
Both regions are treated as continua, as for a macroscopic crystal, so the difference comes only from the shape. It vanishes at low \(T\), is small at high \(T\), and peaks at a few percent near \(T \approx 0.25\,T_D\).
Integrating over the eighth-sphere with \(x = hc_s n/2Lk_{\mathrm{B}}T\), the upper limit becomes \(x_{\max} = T_D/T\), where
is the Debye temperature. Then
For \(T \gg T_D\), \(x\) is always small, \(\int x^3/(e^x-1)\,dx \approx \int x^2\,dx\), and \(U \approx 3Nk_{\mathrm{B}}T\), in agreement with equipartition and with the Einstein model. For \(T \ll T_D\), the upper limit can go to infinity, \(\int_0^\infty x^3/(e^x-1)\,dx = \pi^4/15\) as for the photon gas, and
The \(T^3\) law agrees beautifully with low-temperature experiments on almost any solid. In between, the integral is done numerically.
In the Einstein curve, \(\varepsilon\) is chosen so the two models agree at high temperature, which puts \(\varepsilon/k_{\mathrm{B}} = \sqrt{3/5}\,T_D \approx 0.77\,T_D\). Note how much flatter the Einstein curve is at low temperature, especially on logarithmic axes. Since \(C_V\) reaches 95% of its maximum at \(T = T_D\), the Debye temperature tells you roughly when equipartition is good enough. Typical values range from 88 K for lead (soft and dense) to 1860 K for diamond (stiff and light); the others shown are typical handbook values. In practice \(T_D\) is usually chosen to fit measured heat capacities rather than computed from \(c_s\).
In a metal, the conduction electrons add their own heat capacity, linear in \(T\) (Section 7.3): \(C_{\text{el}} = \frac{\pi^2}{2}\frac{Nk_{\mathrm{B}}^2}{\varepsilon_F}T\). At low temperature,
so a plot of \(C/T\) against \(T^2\) is a straight line: the intercept is the electronic coefficient \(\gamma\), and the slope gives \(T_D\).
Lines use typical measured values of \(\gamma\) and \(T_D\), per mole. Intercept: electrons. Slope: phonons.
Below that temperature the electrons dominate, above it the phonons. The free-electron \(\gamma\) uses \(\varepsilon_F\) from the free-electron model; the real value is somewhat larger.
Problem 7.64. In a ferromagnet such as iron, each elementary dipole strongly prefers to align with its neighbors. At \(T = 0\) all are lined up and the magnetization is about \(2\mu_B N\). At somewhat higher temperatures the excitations are spin waves, in which the dipoles precess in cones with a phase that advances along the crystal. Their quanta, magnons, each reduce the magnetization by about \(2\mu_B\).
A long-wavelength spin wave costs little energy, because neighboring dipoles differ only slightly in direction. Unlike sound, the frequency goes as the square of \(1/\lambda\), so (watch the animation) shortening the wavelength speeds the precession up quadratically, and \(\varepsilon = p^2/2m^*\), with \(m^* = 1.24 \times 10^{-29}\ \mathrm{kg}\) for iron, about 14 electron masses. Magnons have only one polarization. At low temperature,
using \(\int_0^\infty x^{3/2}/(e^x-1)\,dx \approx 1.78329\). For iron, \(T_0 \approx 4200\ \mathrm{K}\) and \(T_1 \approx 2700\ \mathrm{K}\), so the magnetization drops only slowly at room temperature. Compare the magnon and phonon heat capacities (iron’s \(T_D = 470\ \mathrm{K}\)):
* The low-\(T\) law only holds well below \(T_D\), roughly under 0.1 \(T_D\) ≈ 47 K, so its value here is not physical.
At 300 K, the low-temperature phonon formula gives \(C_V/Nk_{\mathrm{B}} \approx 61\), as in the lecture, but 300 K is \(0.64\,T_D\), outside that formula’s range; the full Debye formula gives about 2.7. Either way phonons win by a wide margin, while magnons give \(1/27\). Magnons would only take over below about 2 K, and in iron the conduction electrons dominate there anyway.
For a two-dimensional array of dipoles, the magnon count becomes \(N_m \propto \int_0 dx/(e^x - 1)\), which diverges at small \(x\), that is, for the longest wavelengths. The number of magnons, and so the reduction in magnetization, grows without bound as the system gets larger: no spontaneous magnetization at any finite temperature in 2D, at least in this model. (The Ising model in Section 8.2 is a two-dimensional model in which magnetization does occur.) The lower limit \(x_{\min}\) shrinks as the system grows, roughly as \(1/L^2\):
The 3D integral settles at 2.31516; the 2D one equals \(-\ln(1 - e^{-x_{\min}})\) and keeps growing like \(\ln(1/x_{\min})\).