The heat capacity at constant volume C_V of a crystalline solid measures how much thermal energy is needed to raise the temperature. Classical equipartition predicts C_V = 3R per mole (Dulong–Petit law) at all temperatures, but experiments show C_V → 0 as T → 0 — a key triumph of quantum statistical mechanics. Einstein (1907) modeled the solid as 3N independent quantum harmonic oscillators, all with the same frequency ω_E, giving molar C_V = 3R(Θ_E/T)² e^{Θ_E/T}/(e^{Θ_E/T}−1)² with Θ_E = ℏω_E/k_B. This captures the exponential freezing-out of modes at low T but rises to 3R too quickly because a single frequency cannot represent the spread of phonon energies. Debye (1912) treated the crystal as a continuous elastic medium of acoustic phonons up to a cutoff frequency ω_D related to the Debye temperature Θ_D = ℏω_D/k_B. The exact molar heat capacity is C_V = 9R(T/Θ_D)³ ∫₀^{Θ_D/T} u⁴ e^u/(e^u−1)² du. At low temperature C_V ≈ (12π⁴/5) R (T/Θ_D)³ (Debye T³ law), explaining the observed cubic dependence. At high temperature C_V → 3R. This simulator plots Debye and Einstein curves versus T, overlays the low-T T³ asymptote and the 3R plateau, marks a probe temperature, and reports C_V values — the standard bridge from phonon dispersion to thermodynamic properties of solids.
Who it's for: Undergraduate solid-state physics or physical chemistry students after phonon/lattice concepts and before transport or specific-heat experiments.
Key terms
Debye model
Einstein model
Heat capacity C_V
Debye temperature
Dulong–Petit law
T³ law
Phonon density of states
How it works
Compare Einstein and Debye models of the molar heat capacity C_V(T) of a solid: Debye’s T³ law at low temperature, approach to the Dulong–Petit limit 3R at high T, and Einstein’s single-frequency caricature.
Frequently asked questions
Why does C_V go to zero at absolute zero?
Quantum oscillators have spaced energy levels; at low T thermal energy k_B T is insufficient to excite phonons, so adding heat barely raises the internal energy. Both Einstein and Debye models enforce this; the classical 3R law fails at low T.
What is the difference between Θ_D and Θ_E?
Θ_E sets the one frequency of all Einstein oscillators. Θ_D is a cutoff scale for the Debye continuum of acoustic modes — it is related to the maximum phonon frequency and thus to elastic constants and atomic spacing.
When is the T³ law valid?
Only when T ≪ Θ_D and only the long-wavelength acoustic modes contribute appreciably. Roughly for T < Θ_D/10 the Debye curve follows (12π⁴/5)R(T/Θ_D)³ closely; the simulator shades this region.
Why is Einstein worse than Debye for real solids?
Real crystals have a distribution of phonon frequencies (optical and acoustic branches). Einstein’s single frequency cannot reproduce both the T³ low-T slope and the smooth approach to 3R; Debye’s continuum is a much better first approximation for monatomic lattices.