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.

Reading speed

PhononsDebye T³

Governing equation

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

$T \in \left[0.5, 80\right]$

where

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

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

00:0 / 00:08

①

Governing law — red

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

$T \in \left[0.5, 80\right]$

$T = 0.5$ $C_V = 9\times 10^{-6}$

Along the curve

②

Domain of the variable — blue

T_{\min} = 0.5,\quad T_{\max} = 80

$T \in \left[0.5, 80\right]$

$T = 0.5$ $T = 0.5$

Along the curve

③

Response curve — green

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

$T \in \left[0.5, 80\right]$

$T = 0.5$ $C_V = 9\times 10^{-6}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}C_V}{\mathrm{d}T}

$T \in \left[0.5, 80\right]$

$T = 1$ $C_V^{\prime} = 2.34\times 10^{-4}$

Along the curve

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

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

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