Thermionic emission Richardson-Dushman calculator

The thermionic emission calculator determines the current density J of electrons emitted from a heated metallic surface in vacuum, per the Richardson-Dushman equation derived by Owen Willans Richardson 1901 (Nobel Prize Physics 1928 "for his work on the thermionic phenomenon and especially for the discovery of the law named after him") + Saul Dushman 1923 General Electric Research Lab Schenectady, who derived the universal constant from Sommerfeld free-electron gas statistics. The classic formula: J = A · T² · exp(-W/(k_B·T)) [A/m²], with A = 4π·m_e·e·k_B²/h³ = 1.2017×10⁶ A/(m²·K²) universal Richardson constant (ideal emitting surface no quantum reflection), T absolute temperature K, W work function eV, k_B = 1.381×10⁻²³ J/K = 8.617×10⁻⁵ eV/K Boltzmann constant, m_e = 9.109×10⁻³¹ kg electron mass, e = 1.602×10⁻¹⁹ C elementary charge, h = 6.626×10⁻³⁴ J·s Planck constant. Physics: only electrons with kinetic energy perpendicular to surface above potential barrier W (work function) escape; Fermi-Dirac distribution at T determines the available fraction, exponentially sensitive to W/k_BT.

The effective A factor for real metals is 30-120×10⁴ A/(m²·K²) (i.e. 0.25-1.0× A_R universal) due to:

• quantum reflection electrons bounce back at surface (transmission coefficient 0.5-1)
• Schottky correction applied electric field E lowers W: W_eff = W - √(eE/(4π·ε₀))
• patch effect crystallographic surface inhomogeneity. Typical work functions: Tungsten W 4.55 eV (standard filaments); Molybdenum Mo 4.36 eV; Tantalum Ta 4.12 eV; Rhenium Re 4.72 eV; LaB₆ thoriated lanthanum hexaboride 2.6 eV (modern TEM/SEM electron microscopy high brightness); BaO/SrO oxide cathodes 1.4 eV (thermionic valves, hot cathode tubes); Cesium Cs 2.14 eV (lowest pure metals); Thoriated tungsten W-Th 2.6 eV; Tantalum Carbide TaC 3.1 eV. Industrial applications:

• Electron microscopy cathodes (TEM Transmission Electron Microscope, SEM Scanning Electron Microscope, Auger spectroscopy)
• X-ray tubes (medical radiology, industrial, XRD crystallography)
• Thermionic valves (vintage Hi-Fi audio, RF power amplifiers)
• LINAC particle accelerators (electron sources radiotherapy, fundamental physics)
• TWT Travelling Wave Tubes (military radar, Cosmo-SkyMed SAR satellite telecom)
• Klystron and Magnetron (radar, 2.45 GHz domestic microwaves)
• Free-Electron Laser FEL (FERMI@ELETTRA Trieste, ELI-NP)
• Hot cathode fluorescent lamps
• Thermionic Energy Converter TIC (NASA SAFE-30 space reactors). Italian applications:

• ELETTRA Synchrotron Trieste 1993 2-2.4 GeV electrons LINAC injector + booster + storage ring (28 beamlines)
• FERMI@ELETTRA 2010 seeded Free Electron Laser INFN-AREA Trieste photoinjection thermionic cathode
• INFN Frascati LNF 1957 DAFNE 1999 electron-positron 1.02 GeV KLOE-2 thermionic source
• INFN-LNS Catania Cyclotron K800
• ENEA Frascati FTU/DTT plasma fusion electron sources
• ASI Rome 1988 + Leonardo SpA Rome 1948 TWT amplifiers Cosmo-SkyMed satellite missions 2007-2010 SAR (former Galileo Avionica)
• STMicroelectronics Catania 1987 + Agrate Brianza vacuum semiconductor tech
• Galileo Vacuum Lonigo Vicenza 1936 historic vacuum tubes
• Polytechnic Milan 1863 + Polytechnic Turin 1859 applied physics cathodes
• University Bologna 1088 + Pisa Scuola Normale 1810 fundamental research

Italian thermionic emission tables

(Source: Richardson 1901 Phil. Mag. + Dushman 1923 Phys. Rev. + materials data Polytechnic Milan 1863 + ELETTRA Synchrotron Trieste 1993 + Leonardo SpA Rome 1948 TWT handbook).

Italian thermionic emission examples

• Tungsten W cathode X-ray tube hospital radiology Italy: T = 2,800 K, W = 4.55 eV, A_eff = 0.6·A_R = 7.2×10⁵ A/(m²·K²). Exponent -W/(k_BT) = -4.55/(8.617×10⁻⁵·2,800) = -18.86. exp(-18.86) = 6.4×10⁻⁹. J = 7.2×10⁵·(2,800)²·6.4×10⁻⁹ = 3.6×10⁴ A/m² = 3.6 A/cm². Typical X-ray tube radiology (Leonardo Healthcare, former Esaote Genoa)
• LaB₆ lanthanum hexaboride TEM/SEM IIT Genoa 2003: T = 1,700 K (W would melt), W = 2.6 eV, A_eff = 0.3·A_R. Exponent -2.6/(8.617×10⁻⁵·1,700) = -17.75. J = 3.6×10⁵·(1,700)²·exp(-17.75) = 2×10⁴ A/m² = 2 A/cm². Brightness 10× tungsten + lifetime 1,000 h
• BaO/SrO oxide cathode CRT vintage Galileo Vacuum Lonigo: T = 1,100 K (low!), W = 1.4 eV, A_eff = 0.01·A_R (oxide very low). Exponent -1.4/(8.617×10⁻⁵·1,100) = -14.77. J = 1.2×10⁴·(1,100)²·exp(-14.77) = 5×10³ A/m² = 0.5 A/cm². Excellent for RF Klystron but short lifetime evaporation
• TWT Cosmo-SkyMed satellite Leonardo SpA Rome: Cs/W mix cathode T = 1,300 K, W = 2.0 eV. J ≈ 5×10³ A/m². Modulated density anodes → space-charge wave → 60 dB RF gain X-band SAR
• Thoriated tungsten W-Th audio valves Garrard EL34 KT88 Galileo Vacuum: T = 2,000 K, W = 2.6 eV, A_eff = 0.3·A_R. J ≈ 3×10⁴ A/m² = 3 A/cm². Audiophile premium
• FERMI@ELETTRA Trieste 2010 FEL photoinjection cathode: photosensitive Cu cathode + thermal complementary T = 1,500 K; J pulse 200 A/cm² femtoseconds 4-10 nm EUV/soft X
• Klystron military radar Leonardo SpA Rome 1948: W cathode T = 1,300 K oxide mix W = 1.8 eV. J ≈ 2×10⁴ A/m² CW continuous wave. Schottky note: field E = 10⁶ V/m lowers effective W by √(eE/(4πε₀)) = 0.12 eV; J can rise 2-3× (relevant DAFNE Frascati linac injection)

Italian thermionic research

Italian thermionic research:

• ELETTRA Synchrotron Trieste 1993 2-2.4 GeV electrons storage ring + thermionic LINAC injector + booster (28 beamlines X-ray scattering, photoemission, microscopy); FERMI@ELETTRA 2010 seeded Free Electron Laser fs pulses EUV/soft X-ray (Trieste University 1924 + AREA Science Park) Cu photoinjector + thermionic backup
• INFN Italian National Institute of Nuclear Physics 21 sections: LNF Frascati 1957 Edoardo Amaldi DAFNE 1999 Φ-factory 1.02 GeV electron-positron KLOE-2 thermionic + LINAC; LNS Catania National Laboratories of South Cyclotron Superconducting K800; LNGS Gran Sasso 1987 1,400 m underground (XENONnT, DarkSide, Borexino); LNL Legnaro Padua 1968 ALPI superconducting linac Tandem ions; INFN Pisa LFI Planck Low Frequency Instrument 30-70 GHz radiometers
• ENEA Frascati Research Center 1960 FTU Frascati Tokamak Upgrade fusion + DTT Divertor Tokamak Test 2025-2030 (€1.3B) plasma electron sources
• ASI Rome 1988 TWT cathode missions: Cosmo-SkyMed 4 SAR satellites X-band 9.6 GHz (2007-2010 + Cosmo-SkyMed Second Generation 2019-2025) Leonardo Galileo Avionica TWT amplifiers 1-2 kW; Sentinel ESA Copernicus 30% Italian contribution; BeppoSAX 1996-2002 ASI X-ray Bruno Rossi Caraveo
• Leonardo SpA Rome 1948 (former Finmeccanica): Defense Electronics Pomezia + Genoa satellite radar TWT amplifiers; historic seat Pomezia + Genoa 6,000 engineers; €14B revenue 2024
• STMicroelectronics SGS-Thomson 1987 Catania + Agrate Brianza vacuum semiconductor fab + plasma deposition sputtering MOCVD
• Galileo Vacuum Lonigo Vicenza 1936 historic thermionic audio + industrial valves (now Solidor)
• Polytechnic Milan 1863 Electronics Department vacuum electronics; Polytechnic Turin 1859 microelectronics DET; University Bologna 1088 CNR-NANO
• ICTP Trieste 1964 Abdus Salam + SISSA Trieste 1978 condensed matter physics; IIT Genoa 2003 Roberto Cingolani (3,000 researchers) NanoTechnology + cathode nanofabrication; INRIM Turin 1923 electrical metrology vacuum gauges; CINECA Bologna 1969 Marconi100 + Leonardo HPC 250 PFlops 5th TOP500 Monte Carlo cathode simulations
• AGCOM 1997 telecommunications; Magneti Marelli Corbetta Milan automotive electronics thermo sensors. Italian Physics Nobel Prizes: Marconi 1909, Fermi 1938, Segrè 1959, Carlo Rubbia 1984 CERN W/Z bosons, Riccardo Giacconi 2002 X-ray. Italian R&D tax bonuses 2026: R&D Tax Credit art. 1 par. 200 L. 160/2019 up to 25%; Patent Box L. 178/2020 50% deduction; Industry/Transition 5.0 Tax Credit art. 38 DL 39/2024 up to 35%; Hyper-amortization 250%; PNRR Mission 4 Component 2 research €6.9B; PNRR Mission 1 Component 2 hydrogen electrolysis cathode lines

Updated July 2026