CND-20

Umklapp thermal conductivity

κ_ph ~ T⁻¹ exp(θ_D / α T) with α ~ 2. Phonon heat that dies by Umklapp.

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PhononsUmklapp

Governing equation

\kappa\propto T^{-1}\exp(\theta_D/\alpha T)

$T \in \left[10, 400\right]$

where

\kappa_0: Prefactor (W/m/K)
\theta_D: Debye temperature (K)
T: Temperature (K)
\alpha: α (—)
\kappa: Phonon κ (W/m/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.

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①

Governing law — red

\kappa\propto T^{-1}\exp(\theta_D/\alpha T)

$T \in \left[10, 400\right]$

$T = 10$ $\kappa = 6.54\times 10^{7}$

Along the curve

②

Domain of the variable — blue

T_{\min} = 10,\quad T_{\max} = 400

$T \in \left[10, 400\right]$

$T = 10$ $T = 10$

Along the curve

③

Response curve — green

\kappa\propto T^{-1}\exp(\theta_D/\alpha T)

$T \in \left[10, 400\right]$

$T = 10$ $\kappa = 6.54\times 10^{7}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\kappa}{\mathrm{d}T}

$T \in \left[10, 400\right]$

$T = 12.45$ $\kappa^{\prime} = -1.33\times 10^{7}$

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-20 — Umklapp thermal conductivity) is the form associated with Umklapp. Working symbols: $\kappa_0$ , $\theta_D$ , $T$ , $\alpha$ $\rightarrow$ $\kappa$ . Normal processes conserve crystal momentum; Umklapp needs a reciprocal-lattice kick.

Purpose

Purpose: compute

\kappa

from

\kappa_0

\theta_D

T

\alpha

in Condensed matter via

\kappa\propto T^{-1}\exp(\theta_D/\alpha T)

κ_ph ~ T⁻¹ exp(θ_D / α T) with α ~ 2. Phonon heat that dies by Umklapp. 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

\kappa_0 = 200.000\,\mathrm{W/m/K}

\theta_D = 300.000\,\mathrm{K}

T = 80.000\,\mathrm{K}

\alpha = 2.000\,\mathrm{—}

, the governing relation

\kappa\propto T^{-1}\exp(\theta_D/\alpha T)

yields

\kappa = 16.30\,\mathrm{W/m/K}

. Two phonons, a G vector, a falling κ. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Prefactor \kappa_0200 W/m/KDebye temperature \theta_D300 KTemperature T80 Kα \alpha2 —

Outputs

Phonon κ \kappa16.30 W/m/K

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Narration of this film

Two phonons, a G vector, a falling κ.

Normal processes conserve crystal momentum; Umklapp needs a reciprocal-lattice kick.

Reading speed

Watch on YouTube