Light: wave or particle?
By the late 1800s, interference and diffraction had firmly established that light is a wave, and Maxwell showed it is an electromagnetic wave. Yet experiments on how light ejects electrons from metals could not be explained by waves. The resolution was that light has a dual nature.
Electron emission
Free electrons in a metal are held inside by the attraction of the positive ions. The minimum energy needed to pull an electron out of a metal surface is its work function φ₀, usually measured in electron volts (1 eV = 1.6 × 10⁻¹⁹ J).
Electrons can be emitted by giving them energy through heat (thermionic emission), a strong electric field (field emission), or light (photoelectric emission).
The photoelectric effect
Hertz first noticed (1887) that ultraviolet light helped sparks jump across a gap. Hallwachs and Lenard studied it in detail: light falling on a metal surface ejects electrons, called photoelectrons.
In a photocell, light falls on a metal plate (the emitter), and the ejected electrons are collected by another plate. The current, and the reverse voltage needed to stop it, are measured.
Experimental findings:
Intensity: for a given frequency (above the threshold), the photoelectric current is proportional to the intensity of light.
Stopping potential: a negative potential applied to the collector can stop even the fastest electrons. This stopping potential V₀ gives their maximum kinetic energy: K_max = eV₀. It is independent of intensity.
Frequency: V₀ (and so K_max) increases linearly with the frequency of the light.
Threshold frequency: for each metal, there is a minimum frequency ν₀ below which no electrons are emitted, however intense the light.
No time lag: emission begins instantly (within about 10⁻⁹ s), even with very dim light.
Why the wave theory fails
According to wave theory, energy is spread continuously over the wavefront, so:
brighter light should give electrons more kinetic energy, but K_max does not depend on intensity;
any frequency should eject electrons if the light is intense enough, but there is a threshold frequency;
with dim light, an electron would need time to soak up enough energy, but emission is instant.
All three predictions are wrong.
Einstein's photoelectric equation
Einstein (1905) proposed that light energy comes in packets called quanta or photons, each with energy E = hν. One photon is absorbed by one electron, and gives up all its energy at once.
Part of the energy frees the electron (the work function); the rest becomes kinetic energy:
K_max = hν − φ₀ = h(ν − ν₀), and eV₀ = hν − φ₀
where φ₀ = hν₀. This explains every observation:
below ν₀, a single photon doesn't have enough energy, whatever the intensity;
K_max depends on the photon energy hν, not on the number of photons;
more intense light means more photons, so more electrons: a larger current;
one photon is absorbed in a single event, so there is no delay.
Plotting V₀ against ν gives a straight line with slope h/e. Millikan used this to measure Planck's constant: h = 6.63 × 10⁻³⁴ J s.
Example. Light of frequency 1.5 × 10¹⁵ Hz falls on a metal with work function 4.0 eV. Photon energy = hν = 6.63 × 10⁻³⁴ × 1.5 × 10¹⁵ ≈ 9.95 × 10⁻¹⁹ J ≈ 6.2 eV. K_max = 6.2 − 4.0 = 2.2 eV, and V₀ = 2.2 V.
Properties of photons
Energy E = hν = hc/λ. A handy form: E (in eV) ≈ 1240 ⁄ λ (in nm).
Momentum p = hν/c = h/λ.
Photons travel at the speed of light and have zero rest mass.
They are electrically neutral and aren't deflected by electric or magnetic fields.
In collisions with particles, energy and momentum are conserved, though the number of photons may change.
Matter waves
If light, a wave, can act like particles, can particles act like waves? de Broglie (1924) proposed that every moving particle has a wavelength:
λ = h ⁄ p = h ⁄ mv
For everyday objects the wavelength is far too small to detect. A 0.12 kg ball moving at 20 m/s has λ ≈ 2.8 × 10⁻³⁴ m. But for electrons it is comparable to the spacing of atoms in crystals.
Electron accelerated through a potential V: its kinetic energy is eV, so p = √(2meV) and
λ = h ⁄ √(2meV) ≈ 1.227 ⁄ √V nm
Example. For V = 100 V, λ ≈ 1.227/10 ≈ 0.123 nm, about the size of an atom.
The Davisson-Germer experiment
In 1927, Davisson and Germer fired a beam of electrons accelerated through 54 V at a nickel crystal and measured how many were scattered at each angle. They found a strong peak at 50°, exactly what is expected from diffraction of waves by the regular rows of atoms.
The wavelength they measured, about 0.165 nm, matched de Broglie's formula. This confirmed the wave nature of electrons.
The electron microscope uses the very short wavelength of electron beams to see far finer detail than an optical microscope.
Summary
Photoelectric current ∝ intensity; K_max depends on frequency only; no emission below ν₀; no time lag.
Einstein: K_max = eV₀ = hν − φ₀; light consists of photons of energy hν.
Photon momentum p = h/λ; photons have zero rest mass.
de Broglie: λ = h/mv; for electrons λ ≈ 1.227/√V nm; confirmed by Davisson and Germer.