The lattice shows 24 sample phonon modes (standing waves) summed. Their amplitude follows the Bose-Einstein occupation: u² ∝ (n+½)/ω with n = 1/(e^{ħω/kT}−1). At low T only long-wavelength (low-ω) modes are excited; near T ≈ Θ_D every mode up to the cut-off moves. Drawn amplitudes are illustrative (exaggerated, normalised); the numbers in the readouts are exact Debye-model values.
A crystal of N atoms has 3N modes: g(ω)=9Nω²/ωD³. CV = 9R (T/Θ)³ ∫₀^{Θ/T} x⁴eˣ/(eˣ−1)² dx. Low T: CV → (12π⁴/5) R (T/Θ)³; high T: 3R ≈ 24.94 J/(mol·K).
Simplifications: one sound speed, isotropic crystal, no electronic term, no anharmonicity. Θ values are textbook approximations. Drag to orbit, scroll or pinch to zoom.