Thermionic Emission Calculator
Enter cathode temperature T and work function W to compute emission current density J=A·T²·exp(-W/kT). Tungsten W=4.5 eV, T=2500 K → J≈6369 A/m²; Richardson A=1.2×10⁶ A/(m²·K²).
Input Data
Results
At a glance:Thermionic emission: electrons in a metal follow the Fermi–Dirac distribution; those in the high-energy tail with energy above the work function φ can escape the surface. The Richardson–Dushman equation J=A·T²·exp(−W/(kT)) gives the emission current density, where A=4π·m·e·k²/h³=1.2×10⁶ A/(m²·K²) (theoretical Richardson constant; experiments are slightly lower due to a reflection coefficient), k=1.380649×10⁻²³ J/K Boltzmann's constant, W the work function (J) and T absolute temperature (K). The T² term comes from the electron density of states; the exponential is the Boltzmann factor (fraction of high-energy tail electrons). Emission is extremely temperature-sensitive: a 100 K rise can increase J 5–10×. Common cathodes: (1) pure tungsten — W=4.5 eV, works at 2500 K, J≈6369 A/m², melting point 3695 K, ion-bombardment resistant, used in electron guns and X-ray tubes; (2) thoriated tungsten — a monolayer of thorium lowers W to 2.6 eV, works at 2000 K, J≈1e4 A/m², used in broadcast tubes; (3) barium-oxide cathode — W=1.1 eV, works at 1000 K, J≈1e4 A/m², long-lived but ion-sensitive, used in CRT and microwave tubes; (4) lanthanum hexaboride LaB₆ — W=2.7 eV, works at 1800 K, stable emission, used in electron microscopes; (5) tungsten carbide — a compromise. Classic example: tungsten W=4.5 eV=7.21e-19 J, T=2500 K, kT=3.452e-20 J, W/(kT)=20.89, exp(−20.89)=8.85e-10, J=1.2e6×2500²×8.85e-10≈6637 A/m² (literature ~6369, the difference is from the experimental A and reflection). History: Edison discovered the Edison effect in 1883; Richardson derived the law in 1901 (Nobel 1928); Fleming invented the diode in 1904; De Forest the triode in 1906. Applications: electron guns (microscopes, e-beam lithography, CRT); X-ray tubes (medical, industrial); vacuum tubes (audio, broadcast); thermoelectric conversion (space nuclear power, waste-heat recovery); neutron sources.
Formula
Emission: J = A·T²·exp(−W/(k·T))
Richardson constant: A = 4π·m·e·k²/h³ ≈ 1.2×10⁶ A/(m²·K²)
Boltzmann factor: exp(−W/(k·T))
Boltzmann constant: k = 1.380649×10⁻²³ J/K
W = eV-value × 1.602×10⁻¹⁹ J/eV
$$J = A T^2 \exp\left(-\frac{W}{kT}\right), \quad A = \frac{4\pi m_e e k^2}{h^3} \approx 1.2 \times 10^6\ \mathrm{A/(m^2 \cdot K^2)}$$How to Use
- Enter cathode temperature T (K) and work function W (eV).
- The tool computes J=A·T²·exp(−W/kT), the Boltzmann factor and A.
- Typical: W-tungsten W=4.5 eV, T=2500 K → J≈6369 A/m²; thoriated W=2.6 eV, T=2000 K → J≈1e4 A/m².
Case Studies
X-ray tube tungsten filament and life
Medical X-ray tubes use a tungsten filament φ=4.5 eV heated to 2500 K, giving J≈6369 A/m².
At 100 kV and 100 mA, tube power is 10 kW, needing larger area or higher T (3000 K) at the cost of life (down to 1000 h).
Tungsten evaporation limits life; Hong Kong hospital CT tubes last ~300k exposures at 60 kW per scan.
Space RTG thermoelectric power
Spacecraft RTGs use Pu-238 decay heat (600°C) → electricity; a thermionic converter with W-cathode at 2500 K and anode at 1000 K reaches 15–20% efficiency.
Voyager 1 (1977) carried 3 RTGs totalling 470 W, still working in 2024 at ~220 W after 47 years.
Modern space nuclear power prefers static thermocouples (no moving parts, >50-year life).
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.