CND-05

Phonon oscillator energy

U = ħω / (e^{ħω/kT} − 1). Mean energy of one Bose mode (zero-point omitted).

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PhononsPhonon

Governing equation

U=\dfrac{\hbar\omega}{e^{\hbar\omega/kT}-1}

$T \in \left[5, 800\right]$

where

f: Frequency (THz)
T: Temperature (K)
U: Mode energy (meV)

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

U=\dfrac{\hbar\omega}{e^{\hbar\omega/kT}-1}

$T \in \left[5, 800\right]$

$T = 5$ $U = 2.97\times 10^{-20}$

Along the curve

②

Domain of the variable — blue

T_{\min} = 5,\quad T_{\max} = 800

$T \in \left[5, 800\right]$

$T = 5$ $T = 5$

Along the curve

③

Response curve — green

U=\dfrac{\hbar\omega}{e^{\hbar\omega/kT}-1}

$T \in \left[5, 800\right]$

$T = 5$ $U = 2.97\times 10^{-20}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}U}{\mathrm{d}T}

$T \in \left[5, 800\right]$

$T = 10$ $U^{\prime} = 2.33\times 10^{-7}$

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-05 — Phonon oscillator energy) is the form associated with Phonon. Working symbols: $f$ , $T$ $\rightarrow$ $U$ . The heat capacity of a solid is the T-derivative of a sum of such modes.

Purpose

Purpose: compute

U

from

f

T

in Condensed matter via

U=\dfrac{\hbar\omega}{e^{\hbar\omega/kT}-1}

U = ħω / (e^{ħω/kT} − 1). Mean energy of one Bose mode (zero-point omitted). 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

f = 5.000\,\mathrm{THz}

T = 300.000\,\mathrm{K}

, the governing relation

U=\dfrac{\hbar\omega}{e^{\hbar\omega/kT}-1}

yields

U = 16.877\,\mathrm{meV}

. A lattice spring, a Bose occupation. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Frequency f5 THzTemperature T300 K

Outputs

Mode energy U16.877 meV

Reading speed

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Narration of this film

A lattice spring, a Bose occupation.

The heat capacity of a solid is the T-derivative of a sum of such modes.

Reading speed

Watch on YouTube