Photoelectric Effect Equation

No. Increasing the intensity increases the number of photons arriving per second, which increases the number of ejected electrons (the photocurrent), but each electron still carries the same maximum kinetic energy K_max = hf − φ. The energy is set entirely by the photon frequency and the material's work function. This was one of the most surprising early results and is directly explained by the photon model — each electron absorbs exactly one photon, so more photons means more electrons but not faster ones. At very high intensities, two-photon processes can occur, but those require laser intensities far beyond everyday light sources.

The work function is the energy needed to move an electron from the Fermi level (the highest occupied electron energy in the metal) to the vacuum level just outside the surface. It depends on how tightly the metal holds its valence electrons, which in turn depends on the atomic number, the crystal structure, and the exposed crystal face. Alkali metals like cesium have a single loosely bound valence electron, so their work function is low (2.14 eV). Transition metals like platinum have tightly bound d-electrons, so their work function is high (6.35 eV). Surface contamination can change the work function by more than 1 eV, which is why photocathodes must be kept in ultra-high vacuum.

The stopping potential V₀ is the reverse voltage needed to stop even the fastest photoelectrons from reaching the anode. It satisfies eV₀ = K_max, so measuring V₀ in volts gives K_max in electronvolts directly. In a typical experiment, monochromatic light illuminates a photocathode in an evacuated tube, and the anode current is measured as a function of reverse bias. The current drops to zero at V₀. Repeating the measurement at several frequencies and plotting V₀ against f gives a straight line of slope h/e; the x-intercept is the threshold frequency f₀ = φ/h. Millikan's 1916 measurements of this kind were the first precise verification of Einstein's equation and gave one of the best early values of Planck's constant.

Both are photon–electron interactions, but at different energy scales. The photoelectric effect dominates at photon energies of a few eV to a few keV, where the photon is completely absorbed and its energy is transferred to a bound electron, which may escape the material. The Compton effect dominates at higher energies (keV to MeV), where the photon scatters off a loosely bound or free electron, transferring only part of its energy and changing direction. The photoelectric cross-section scales roughly as Z⁵/E³ (strongly favoring heavy elements and low energies), while the Compton cross-section scales roughly as Z/E (weaker dependence). At very high energies, pair production takes over.

In principle, yes — every material has a work function, so every material exhibits photoemission above some threshold frequency. In practice, metals are the most common photocathodes because their electrons are delocalized and can escape from the bulk, whereas in semiconductors and insulators the electrons must first be excited from the valence band to the conduction band and then escape, requiring a higher photon energy (the sum of band gap and electron affinity). The lowest practical threshold is around 1.1 eV for cesium–antimony intermetallic compounds, which respond to near-infrared light. Negative-electron-affinity semiconductors can have effective thresholds below the band gap and are used in high-efficiency photocathodes.

Use λ = c/f where c = 2.998×10⁸ m/s. For visible light, wavelengths are conventionally given in nanometres (nm), so λ(nm) = 3×10⁸ / f(Hz) × 10⁹ = 3×10¹⁷ / f. Equivalently, the photon energy is hf = hc/λ, and using hc = 1239.84 eV·nm, E(eV) = 1239.84 / λ(nm). Memorize the visible range: red 620–700 nm (1.77–2.00 eV), green 500–560 nm (2.21–2.48 eV), blue 450–490 nm (2.53–2.76 eV), violet 380–450 nm (2.76–3.26 eV). UV begins below 380 nm and X-rays below 10 nm.

Yes, weakly. The work function of a metal decreases by roughly 10⁻⁴ eV per kelvin as temperature rises. At 1,000 K this amounts to about 0.1 eV — small but measurable. The temperature dependence comes from thermal expansion of the lattice, changes in the electron density at the surface, and the temperature dependence of the chemical potential. For most practical purposes at room temperature this is negligible compared to the 0.5–1 eV variation between crystal faces of the same metal, or the 1–2 eV shift caused by surface contamination (oxidation, adsorbed gases, or a monolayer of cesium).