CND-06

Debye T³ heat capacity

C_V = (12 π⁴ / 5) N k_B (T/θ_D)³ at T ≪ θ_D. The phonon T³ law.

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PhononsDebye T³

Governing equation

C_V=\dfrac{12\pi^4}{5}Nk_B(T/\theta_D)^3

where

T: Temperature (K)
\theta_D: Debye temperature (K)
n: Amount (mol)
C_V: Heat capacity (J/K)

Lecture brief

Historical brief

Drude electrons, Bloch waves, BCS pairing (1957) and Wiedemann–Franz heat are the first solids-and-metals laws. The lab is conductivity, gap and phonon heat in closed form. This sheet (CND-06 — Debye T³ heat capacity) is the form associated with Debye T³. Working symbols: $T$ , $\theta_D$ , $n$ $\rightarrow$ $C_V$ . Three acoustic branches, linear ω = v k, density of states ∝ ω². Dulong–Petit recovered at high T.

Purpose

Purpose: compute

C_V

from

T

\theta_D

n

in Condensed matter via

C_V=\dfrac{12\pi^4}{5}Nk_B(T/\theta_D)^3

C_V = (12 π⁴ / 5) N k_B (T/θ_D)³ at T ≪ θ_D. The phonon T³ law. Use it when a real condensed matter question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

T = 10.000\,\mathrm{K}

\theta_D = 300.000\,\mathrm{K}

n = 1.000\,\mathrm{mol}

, the governing relation

C_V=\dfrac{12\pi^4}{5}Nk_B(T/\theta_D)^3

yields

C_V = 0.0720\,\mathrm{J/K}

. A C(T) that starts as T³. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Temperature T10 KDebye temperature \theta_D300 KAmount n1 mol

Outputs

Heat capacity C_V0.0720 J/K

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Narration of this film

A C(T) that starts as T³.

Three acoustic branches, linear ω = v k, density of states ∝ ω². Dulong–Petit recovered at high T.

Reading speed

Watch on YouTube