How to compute the photoelectric effect: discovered experimentally by Heinrich Hertz (Germany 1887) and Philipp Lenard (Nobel 1905), explained theoretically by Albert Einstein (Germany-USA 1879-1955) in 1905 "Annus Mirabilis" Annalen der Physik 17:132 with quantum theory of light, earning Einstein the Nobel Prize in Physics 1921. Fundamental formula: KE_max = h*f - W, where KE_max maximum kinetic energy emitted photoelectrons (eV or Joule), h Planck constant (6.626e-34 J*s), f incident photon frequency (Hz), W work function of material (eV). Threshold frequency: f_0 = W/h, below threshold NO emission regardless of light intensity. Threshold wavelength: lambda_0 = h*c/W. Stopping voltage: V_s = KE_max/e (volts). Essential tool for quantum physics teaching high school-university, photomultiplier tube design (Hamamatsu Japan, ETL Russia), photovoltaic solar cells silicon Si bandgap 1.1 eV (Tesla solar roof, SunPower USA, First Solar Tempe AZ), X-ray photoelectron spectroscopy XPS-ESCA chemical surface analysis (Thermo Fisher USA, Kratos UK), EUV lithography 13.5 nm semiconductor chips (ASML Netherlands supplier TSMC Taiwan, Intel USA, Samsung South Korea). Einstein 1905 validity confirmed experimentally by Robert Millikan USA 1916 Nobel 1923 precise h/e ratio measurement.

Photoelectric effect formulas

Einstein's equation 1905: KE_max = h*f - W = h*(f - f_0), where f photon frequency, f_0=W/h threshold frequency. Constants: h=6.62607015e-34 J*s (Planck constant, exact SI 2019), c=2.99792458e8 m/s, e=1.602176634e-19 C (electron charge exact SI 2019). 1 eV = 1.602e-19 J. Photon energy: E_photon = h*f = h*c/lambda. For lambda in nm: E (eV) = 1240/lambda(nm) practical rule.

Threshold frequency and wavelength: f_0 = W/h, lambda_0 = h*c/W = 1240/W(eV) nm. Examples: cesium Cs W=2.14 eV -> lambda_0=579 nm (yellow-green); copper Cu W=4.65 eV -> lambda_0=267 nm (UV-C); platinum Pt W=5.65 eV -> lambda_0=219 nm (UV-C deep). Below threshold (lambda > lambda_0) NO electron emitted.

Stopping voltage: V_s = KE_max/e. Retrograde voltage applied to stop most energetic photoelectrons. Millikan experimental measurement 1916: slope V_s vs f = h/e = 4.136e-15 V*s, intercept = -W/e. Independent confirmation Planck constant precision 0.1%. Pasco Scientific USA, 3B Scientific Germany teaching apparatus.

Photoelectron velocity: v_e = sqrt(2*KE_max/m_e), m_e=9.109e-31 kg electron mass. For KE>1000 eV relativistic correction: v_e = c*sqrt(1 - 1/(1 + KE/511000)^2), 511 keV electron rest energy. Example Cs+UV 254 nm: KE=2.74 eV, v_e=9.8e5 m/s = 0.0033c (non-relativistic). Example XPS Mg-K-alpha 1253.6 eV: v_e=2.1e7 m/s = 0.07c (modest relativistic correction).

Light intensity vs frequency: classical Maxwell PARADOX prediction: KE proportional to intensity. Lenard 1902 experiment: KE INDEPENDENT intensity, depends ONLY on frequency! Einstein 1905 resolves with light quantization: each photon E=h*f transfers energy entirely to ONE electron. Intensity increases ONLY photoelectrons number (current), NOT individual energy. Confirms light corpuscular nature, wave-particle duality.

History and context

Heinrich Hertz (Germany 1857-1894) discovers 1887 photoelectric effect accidentally during electromagnetic waves experiments Karlsruhe University: sparks easier under UV light. Wilhelm Hallwachs (Germany 1859-1922) Dresden 1888 systematic confirmation. Philipp Lenard (Germany 1862-1947) Nobel Physics 1905 detailed experimental measurement: 1) electrons number proportional intensity; 2) KE independent intensity; 3) KE proportional frequency; 4) frequency threshold f_0. Classical Maxwell-Lorentz PARADOX fails. Albert Einstein (Germany-Switzerland-USA 1879-1955) Annus Mirabilis 1905 ETH Bern: five revolutionary physics articles - special relativity, mass-energy equivalence E=mc^2, Brownian motion, photoelectric effect (light quantization hf), black body fluctuations. Nobel Physics 1921 specifically photoelectric effect (NOT relativity judged too speculative at time). Robert Millikan (USA Caltech 1868-1953) Nobel 1923: 1) oil drop experiment 1909-1913 measure electron charge e=1.602e-19 C; 2) detailed experimental verification 1914-1916 Einstein's KE=hf-W equation for alkali metals, confirms h with 0.5% precision, initial skepticism toward Einstein 1905 reconsidered 1916 experimental results. Niels Bohr (Denmark Copenhagen 1885-1962) Nobel Physics 1922: quantized atom model 1913, electron energy levels, discrete photon transitions hf=E_n-E_m. Louis de Broglie (France 1892-1987) Nobel Physics 1929: matter wave-particle duality 1924, lambda=h/p electron.

Erwin Schrodinger (Austria 1887-1961) Nobel Physics 1933 quantum wave equation. Werner Heisenberg (Germany 1901-1976) Nobel 1932 matrix mechanics and uncertainty principle. Arthur Compton (USA 1892-1962) Nobel 1927 Compton scattering effect photons complementary photoelectric. Global photodetector industry 30+ billion USD/year: Hamamatsu Photonics Hamamatsu Japan PMT photomultiplier leader, Sony Tokyo CMOS image sensors, Vishay Pennsylvania USA photodiodes, OSRAM Munich Germany LED-laser, Cree Wolfspeed Durham NC USA SiC GaN. USA: National Institute of Standards Technology NIST Gaithersburg MD precision Planck constant measurements, Lawrence Berkeley National Laboratory LBNL Berkeley CA Advanced Light Source ALS synchrotron, SLAC National Accelerator Laboratory Stanford CA Linac Coherent Light Source LCLS X-ray free electron laser. EUV lithography ASML Veldhoven Netherlands monopolist 13.5 nm scanner 200 million USD/machine, Cymer San Diego CA EUV source supplier. TSMC Hsinchu Taiwan, Intel Santa Clara CA, Samsung Suwon South Korea 3nm-2nm chip manufacturing. Recent history: 2016 photoemission angle-resolved ARPES topological insulators Princeton MIT Stanford, 2018 LIGO neutron star merger scintillation photomultipliers.

Application sectors

• medicine PET positron emission tomography Siemens Healthineers Erlangen Germany, GE Healthcare Boston MA
• airport security X-ray baggage scanners Smiths Detection UK, OSI Systems Hawthorne CA USA
• high-energy research CERN Geneva ATLAS-CMS detectors ECAL HCAL 30000+ photomultiplier channels. Confirmed history: quantum mechanics birth year 1900 Planck E=hf black body radiation Berlin Germany -> 1905 Einstein light quantization -> 1913 Bohr atom -> 1925-1927 quantum formalism Heisenberg-Schrodinger-Born-Dirac -> 1948 QED Feynman-Schwinger-Tomonaga

Common materials work function table

• Cesium Cs (alkali): W=2.14 eV, lambda_0=579 nm (yellow-green)
• Potassium K: W=2.30 eV, lambda_0=539 nm (green)
• Sodium Na: W=2.36 eV, lambda_0=525 nm (green)
• Calcium Ca: W=2.87 eV, lambda_0=432 nm (blue-violet)
• Lithium Li: W=2.93 eV, lambda_0=423 nm (blue-violet)
• Magnesium Mg: W=3.66 eV, lambda_0=339 nm (UV-A)
• Aluminum Al: W=4.20 eV, lambda_0=295 nm (UV-B)
• Silver Ag: W=4.26 eV, lambda_0=291 nm (UV-B)
• Zinc Zn: W=4.33 eV, lambda_0=286 nm (UV-B)
• Iron Fe: W=4.50 eV, lambda_0=275 nm (UV-C)
• Tungsten W: W=4.55 eV, lambda_0=272 nm (UV-C)
• Copper Cu: W=4.65 eV, lambda_0=267 nm (UV-C)
• Graphite C: W=4.81 eV, lambda_0=258 nm (UV-C)
• Gold Au: W=5.10 eV, lambda_0=243 nm (UV-C)
• Selenium Se: W=5.11 eV, lambda_0=243 nm (UV-C)
• Nickel Ni: W=5.15 eV, lambda_0=241 nm (UV-C)
• Platinum Pt: W=5.65 eV, lambda_0=219 nm (UV-C deep)
• Planck constant h: 6.62607015e-34 J*s (exact SI 2019)
• Electron charge e: 1.602176634e-19 C (exact SI 2019)
• Electron mass m_e: 9.1093837e-31 kg = 511 keV/c^2

Detailed numerical examples

Example 1 - Cesium Cs UV-C 254 nm. Mercury Hg lamp 254 nm UV-C line (germicidal). Cs W=2.14 eV. Photon energy E=1240/254=4.88 eV. KE_max=4.88-2.14=2.74 eV. Stopping voltage V_s=2.74 V. Photoelectron velocity v_e=sqrt(2*2.74*1.602e-19/9.11e-31)=9.8e5 m/s=0.0033c (non-relativistic). Cs threshold frequency f_0=W/h=2.14*1.602e-19/6.626e-34=5.17e14 Hz=517 THz=580 nm (yellow-green). Application photocathodes UV-VIS photomultiplier Hamamatsu R6095 sensitivity 200-650 nm.

Example 2 - Sodium Na visible green light 500 nm. Na W=2.36 eV. E_photon=1240/500=2.48 eV. KE_max=2.48-2.36=0.12 eV. Stopping voltage V_s=0.12 V. Na threshold lambda_0=525 nm. Almost at limit, slow photoelectrons v_e=2.1e5 m/s. Application amorphous solar cells a-Si:H bandgap 1.7 eV, ZnO transparent electrode AZO smart windows.

Example 3 - Copper Cu below threshold visible light. Cu W=4.65 eV. E_photon=1240/500=2.48 eV<W=4.65 eV. NO EMISSION whatever light intensity! Cu threshold lambda_0=267 nm UV-C required. Confirms quantum nature Einstein 1905: increasing visible intensity does NOT produce Cu photoelectrons. Photomask UV-DUV 248 nm KrF 193 nm ArF excimer Lasers GigaPhoton Cymer industry.

Example 4 - Zinc Zn UV-C 250 nm classic experiment. Historical Lenard-Hertz zinc plate electroscope experiment. Zn W=4.33 eV, lambda 250 nm. E_photon=1240/250=4.96 eV. KE_max=4.96-4.33=0.63 eV. Stopping voltage 0.63 V. v_e=4.7e5 m/s. Teaching experiment universities MIT 8.04 Quantum Mechanics, Caltech Ph 12, Princeton, Stanford. Replica experiment Pasco Scientific Roseville CA USA Photoelectric Effect Apparatus.

Example 5 - Platinum Pt X-ray Mg-K-alpha XPS spectroscopy. XPS spectroscopy source Mg-K-alpha 1253.6 eV (frequency 3.03e17 Hz=303000 THz). Pt W=5.65 eV. KE_max=1253.6-5.65=1247.95 eV. v_e=2.1e7 m/s=0.07c (relativistic correction). Measure binding energy core electrons 1s 2p 3d chemical surface analysis. Instruments Thermo Fisher Scientific K-Alpha+ Waltham MA USA, Kratos AXIS Manchester UK, ULVAC-PHI Tokyo Japan. Applications industrial catalysis, biomedical biomaterials, lithium Li-ion battery analysis SEI Solid Electrolyte Interphase.

Limits and considerations

1. KE_max = h*f - W only for photons above threshold (f>f_0). Below threshold ZERO emission.
2. W (work function) varies with: crystallographic orientation (Cu 100 vs 111), surface contamination (oxides, adsorbates), modest temperature.
3. Light intensity affects only photoelectron number (current), NOT individual KE.
4. Photoelectron velocity requires relativistic correction for KE>1000 eV (X-ray XPS).
5. Photoelectric vs Compton scattering: photoelectric dominates E<100 keV on heavy elements (Z^4-Z^5).
6. Single-photon photoemission linear: 1 photon absorption -> 1 electron. Multi-photon non-linear intense pulsed lasers.
7. Angle-dependent photoemission: ARPES Angle-Resolved studies electronic band structure.
8. Quantum yield (efficiency): typically 0.001-0.1 photons/electron metals, up to 0.2 cesium alkalis.
9. Threshold effect Millikan 1916 experiment direct experimental confirmation Einstein 1905.
10. Teaching experiment requires vacuum 10^-5 mbar minimum to avoid photoelectron air scattering.

Common errors

1. Confusing intensity with frequency: increasing visible light intensity does NOT extract Cu electrons.
2. Calculating KE in Joule without converting to eV: 1 eV = 1.602e-19 J, common factor 6e18 mistake.
3. Neglecting work function W dependency on oxidized surface contamination.
4. Assuming KE proportional to exposure time: it is instantaneous on nanosecond scale.
5. Confusing photoelectric effect (UV-VIS) with Compton effect (X-gamma).
6. Neglecting relativistic correction velocity for XPS X-ray (KE>1 keV).
7. Assuming stopping voltage equals applied voltage: V_s only stops most energetic photoelectrons.
8. Confusing thermionic emission (Edison-Richardson) with photoelectric (Hertz-Einstein): different mechanism.
9. Calculating lambda from E with wrong rule: correct E(eV)=1240/lambda(nm), NOT 1240*lambda.
10. Neglecting quantum yield efficiency typically 0.001-0.1 metals, complicates photoelectron count.

Numbered applications

1. Photomultiplier PMT: Hamamatsu Hamamatsu Japan, ETL Russia, PerkinElmer Waltham MA USA single-photon scintillation detector.
2. Photovoltaic solar cells: silicon Si bandgap 1.1 eV (Tesla solar roof, SunPower San Jose CA, First Solar Tempe AZ).
3. X-ray photoelectron spectroscopy XPS-ESCA: Thermo Fisher Waltham MA USA, Kratos Manchester UK, ULVAC-PHI Tokyo surface analysis.
4. EUV 13.5 nm lithography: ASML Veldhoven Netherlands 200 mln USD/scanner, TSMC Taiwan Intel Samsung 3 nm chip production.
5. PET medicine tomography: Siemens Healthineers Erlangen Germany, GE Healthcare Boston MA, BGO LYSO scintillator photomultipliers.
6. Airport security X-ray scanners: Smiths Detection Watford UK, OSI Systems Hawthorne CA USA, Rapiscan baggage analysis.
7. CERN high-energy physics: ATLAS-CMS calorimeters ECAL HCAL 30000+ photomultipliers scintillation Cherenkov.
8. USA NIST-LBNL-SLAC research: precision Planck constant, ALS synchrotron Berkeley, LCLS X-ray FEL Stanford.
9. Educational teaching: Quantum Mechanics MIT 8.04, Caltech Ph 12, Princeton, Stanford, Pasco Scientific apparatus.
10. DUV 193 nm lithography: ArF excimer laser GigaPhoton Tokyo, Cymer San Diego, ASML scanner 7 nm chip production.

Support

For technical questions or reports: info@calcoi.com. Educational tool for preliminary quantum physics. For professional research consult Einstein A 1905 Annalen der Physik 17:132 'Uber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt', Millikan RA 1916 Phys Rev 7:355 'A Direct Photoelectric Determination of Planck h', Eisberg-Resnick 'Quantum Physics' 1985 Wiley, Griffiths 'Introduction to Quantum Mechanics' 2018 Cambridge, Hertz H 1887 Annalen der Physik 31:983 first observation.

Updated July 2026