Richardson Thermionic Emission

Worked Example: Thermionic Current

The Richardson–Dushman equation gives the idealized current density
J = A_R * T² * exp(-W / (kB * T))
where A_R = 1.2e6 A/(m²·K²), T is the absolute temperature, W is the work function, and kB = 8.617333262145e-5 eV/K is the Boltzmann constant.

1. Suppose T = 1500 K and W = 2 eV.
2. Thermal energy: kB * T = 8.617333262145e-5 * 1500 = 0.129259998932175 eV.
3. Boltzmann factor: exp(-2 / 0.129259998932175) = 1.9067587713797286e-07.
4. Pre-exponential factor: A_R * 1500² = 1.2e6 * 2250000 = 2.7e12.
5. Current density expected result: J = 2.7e12 * 1.9067587713797286e-07 = 514824.8682725267 A/m².

Experiment with the Module

You can interact with these variables in the Richardson Thermionic Emission module.

1. Set the Temperature K to 2000.
2. Set the Work function eV to 2.
3. Observe the new thermal energy and resulting exponentially larger current density due to the higher temperature.

Assumptions and Limitations

The ideal Richardson–Dushman equation assumes an ideal metal with a perfectly clean surface emitting into a vacuum. In real devices, the effective Richardson constant depends heavily on the material and surface conditions, and is typically much lower than the theoretical 1.2e6 A/(m²·K²). Furthermore, space-charge effects (where a cloud of emitted electrons repels further emission) and the Schottky effect (field-enhanced emission) are omitted from this simple model. Finally, this describes thermally activated electron emission, which is entirely distinct from photoelectric photon emission.

References

The concepts presented in this module are based on established physical models of thermionic emission described in classical and modern physics literature, which systematically review the historical and theoretical background.

• Thermionic Emission (Reviews of Modern Physics, 1930) - Comprehensive historical review of thermionic emission theories and experiments.