CND-21

Richardson–Dushman current

J = A T² exp(−φ / k T). Thermionic emission over a work function φ.

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Electrons in solidsRichardson

Governing equation

J=AT^2 e^{-\varphi/kT}

$T \in \left[600, 2800\right]$

where

\varphi: Work function (eV)
T: Temperature (K)
A: Richardson A (A/m²/K²)
J: Current density (A/m²)

The figure, part by part

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①

Governing law — red

J=AT^2 e^{-\varphi/kT}

$T \in \left[600, 2800\right]$

$T = 600$ $J = 6.87\times 10^{-27}$

Along the curve

②

Domain of the variable — blue

T_{\min} = 600,\quad T_{\max} = 2800

$T \in \left[600, 2800\right]$

$T = 600$ $T = 600$

Along the curve

③

Response curve — green

J=AT^2 e^{-\varphi/kT}

$T \in \left[600, 2800\right]$

$T = 600$ $J = 6.87\times 10^{-27}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}J}{\mathrm{d}T}

$T \in \left[600, 2800\right]$

$T = 613.8$ $J^{\prime} = 1.24\times 10^{-26}$

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-21 — Richardson–Dushman current) is the form associated with Richardson. Working symbols: $\varphi$ , $T$ , $A$ $\rightarrow$ $J$ . A ≈ 1.2×10⁶ A/m²K². The work function is E_vac − E_F.

Purpose

Purpose: compute

J

from

\varphi

T

A

in Condensed matter via

J=AT^2 e^{-\varphi/kT}

J = A T² exp(−φ / k T). Thermionic emission over a work function φ. 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

\varphi = 4.500\,\mathrm{eV}

T = 1800.000\,\mathrm{K}

A = 1.200e+6\,\mathrm{A/m^{2}/K^{2}}

, the governing relation

J=AT^2 e^{-\varphi/kT}

yields

J = 0.978\,\mathrm{A/m^{2}}

. A hot cathode, an escaping electron cloud. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Work function \varphi4.5 eVTemperature T1800 KRichardson A A1200000 A/m²/K²

Outputs

Current density J0.978 A/m²

Reading speed

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Narration of this film

A hot cathode, an escaping electron cloud.

A ≈ 1.2×10⁶ A/m²K². The work function is E_vac − E_F.

Reading speed

Watch on YouTube