Richardson-Dushman Thermionic Emission Current Density Formula (SEM, X-Ray & Klystron Cathodes)

See how work function phi_eV governs electron gun brightness exponentially!

In X-ray tubes, scanning electron microscopes (SEMs), traveling-wave tubes, and hollow-cathode ion thrusters, heating a metal or hexaboride cathode to absolute temperature T (Kelvin) gives conduction electrons enough thermal energy to surmount the surface work function barrier phi (in eV)! Owen Willans Richardson (1928 Nobel Prize) and Saul Dushman derived the saturated thermionic current density: J = A_G * (T^2) * EXP(-phi / (kB * T)), where kB = 8.617333e-5 eV/K is Boltzmann's constant in electron-volts per Kelvin and A_G is the material-specific Richardson constant in A/(cm^2*K^2) (theoretical universal A0 = 120.17 A/(cm^2*K^2))!

Compare pure polycrystalline tungsten (phi = 4.54 eV, A_G = 60 A/(cm^2*K^2)) against lanthanum hexaboride LaB6 (phi = 2.66 eV, A_G = 29 A/(cm^2*K^2)): tungsten at 2,500 K emits J = 0.2641 A/cm^2 (jumping 5.5x to 1.4675 A/cm^2 at 2,700 K), whereas low-work-function LaB6 achieves 3.3517 A/cm^2 at a cool 1,800 K—delivering 10x longer filament life and far brighter SEM beams!

Richardson-Dushman Thermionic Current Density J (A/cm^2)

Evaluates A_G * (T^2) * EXP(-phi_eV / (0.00008617333 * T)) in Amperes per square centimeter.

=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Cathode Inputs", ROUND(B2 * (B3 ^ 2) * EXP(-B4 / (0.00008617333 * B3)), 4))

Total Emitted Cathode Current I_total = J * Area_cm2 (Milliamperes, mA)

Multiplies current density J by active emitter tip area B5 (cm^2) and converts Amperes to mA (* 1,000).

=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Cathode Inputs", ROUND(1000 * B5 * B2 * (B3 ^ 2) * EXP(-B4 / (0.00008617333 * B3)), 2))

Further reading

• Microsoft Support — EXP function