---
title: The Photoelectric Effect and the Photon
module: Origins of the Quantum
moduleNumber: 1
lessonNumber: 2
order: 102
summary: >
  Light shone on a clean metal ejects electrons, but the details defied the wave
  theory: the electrons' maximum energy depends on the light's frequency, not its
  brightness, and there is a sharp threshold frequency below which nothing
  happens. Einstein resolved every anomaly by treating light as a stream of energy
  quanta hf, each absorbed whole by one electron, and Millikan's measurement of
  the stopping-potential slope confirmed h to a decade before anyone expected.
topics: [Origins of the Quantum]
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 3 — Quantization of Charge, Light, and Energy; §3-3 The Photoelectric Effect"
  - book: Tipler & Mosca
    ref: "Ch. 34 §34-1 Wave-Particle Duality"
draft: false
---

Planck quantized the oscillators in a cavity wall but held that light itself was
a continuous wave. Einstein removed the qualification: the quantization is a
property of the radiation field. His test case was a laboratory curiosity Hertz
had stumbled on in 1887 while confirming Maxwell's wave theory — light striking a
metal surface knocks electrons loose. The wave theory that predicted the light
also failed to predict what the ejected electrons do. Reconciling the two
required treating the beam as a stream of particles.

## The measurement

Illuminate a clean metal cathode and the electrons it emits — **photoelectrons** —
can be collected at an anode, driving a current through an external circuit. A
variable voltage $V$ between cathode and anode controls how many electrons
arrive: a positive anode attracts them, a negative anode repels them.

$$
% caption: The photoelectric tube. Light ejects electrons from the cathode; a
% bias voltage V between cathode and anode either collects them or turns them
% back, and the ammeter reads the resulting photocurrent.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % evacuated tube outline
  \draw[black, thick] (0,0) ellipse (3.1 and 1.7);
  % cathode plate (left)
  \draw[acc, very thick] (-2.2,-0.9) -- (-2.2,0.9);
  \node[anchor=east, font=\scriptsize, text=acc] at (-2.25,-0.9) {cathode};
  % anode plate (right) with slit
  \draw[acc, very thick] (1.9,-0.9) -- (1.9,-0.15);
  \draw[acc, very thick] (1.9,0.15) -- (1.9,0.9);
  \node[anchor=west, font=\scriptsize, text=acc] at (1.95,0.9) {anode};
  % incoming light
  \draw[->, black] (-3.6,1.3) -- (-2.3,0.4);
  \node[anchor=east, font=\scriptsize] at (-3.6,1.3) {light $hf$};
  % ejected electrons
  \draw[->, black] (-2.1,0.3) -- (-0.4,0.35);
  \draw[->, black] (-2.1,-0.1) -- (-0.4,0.0);
  \draw[->, black] (-2.1,0.6) -- (-0.4,0.7);
  \fill[black] (-0.3,0.35) circle (1.3pt);
  \fill[black] (-0.3,0.0) circle (1.3pt);
  % external circuit
  \draw[black] (-2.2,-0.9) -- (-2.2,-2.3) -- (-0.6,-2.3);
  \draw[black] (1.9,-0.9) -- (1.9,-2.3) -- (0.9,-2.3);
  % ammeter
  \draw[black] (0.15,-2.3) circle (0.28);
  \node[font=\scriptsize] at (0.15,-2.3) {A};
  \draw[black] (-0.6,-2.3) -- (-0.13,-2.3);
  \draw[black] (0.43,-2.3) -- (0.9,-2.3);
  % variable voltage source
  \draw[black] (0.9,-2.3) -- (0.9,-3.15) -- (-0.6,-3.15) -- (-0.6,-2.3);
  \draw[black] (0.02,-3.4) circle (0.25);
  \draw[black] (0.9,-3.15) -- (0.27,-3.15);
  \draw[black] (-0.6,-3.15) -- (-0.23,-3.15);
  \node[font=\scriptsize] at (0.02,-3.4) {V};
  \node[anchor=west, font=\scriptsize] at (1.15,-2.75) {bias voltage};
\end{tikzpicture}
$$

Sweeping $V$ produces the current-voltage curve. At large positive $V$ every
emitted electron is collected and the current saturates. As $V$ turns negative,
only electrons with kinetic energy above $e|V|$ can climb the retarding potential
and reach the anode. Below a certain negative value the current stops entirely.

> **Definition (Stopping potential).** The retarding voltage $V_0$ at which the
> photocurrent falls to zero. The most energetic electron just fails to reach the
> anode, so $V_0$ measures the maximum photoelectron kinetic energy:
> $$
> \left(\tfrac{1}{2} m v^2\right)_{\max} = e V_0.
> $$

Two features of these curves broke the wave theory. First, raising the intensity
raises the saturation current — twice the light, twice the electrons — but leaves
$V_0$ unchanged. Brighter light does not make the electrons faster.

$$
% caption: Photocurrent versus anode voltage at two intensities of the same
% frequency. Brighter light lifts the saturation current but the stopping
% potential is identical, so the maximum electron energy is fixed.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % axes
  \draw[->, black] (-2.6,0) -- (4.2,0) node[anchor=north east] {voltage $V$};
  \draw[->, black] (0,-0.4) -- (0,3.6) node[anchor=south east, align=left] {photocurrent};
  % bright curve (higher saturation)
  \draw[black, thick, densely dashed, smooth] plot coordinates
    {(-1.6,0)(-1.0,0.15)(-0.3,0.9)(0.5,2.2)(1.4,2.85)(2.6,3.0)(3.9,3.0)};
  \node[black, anchor=south west, font=\scriptsize] at (2.4,3.0) {bright};
  % dim curve (lower saturation, same V0)
  \draw[acc, thick, smooth] plot coordinates
    {(-1.6,0)(-1.0,0.1)(-0.3,0.55)(0.5,1.2)(1.4,1.55)(2.6,1.65)(3.9,1.65)};
  \node[acc, anchor=south west, font=\scriptsize] at (2.6,1.65) {dim};
  % stopping potential marker
  \draw[black, dashed] (-1.6,0) -- (-1.6,1.0);
  \node[anchor=south, font=\scriptsize] at (-1.6,1.0) {stopping voltage};
\end{tikzpicture}
$$

Second, changing the frequency shifts $V_0$: bluer light gives a larger stopping
potential, and below a **threshold frequency** $f_t$ no electrons are emitted at
all, no matter how intense the light or how long it shines.

$$
% caption: Photocurrent at two frequencies, same intensity. The higher frequency
% produces the larger stopping potential; below a threshold frequency the current
% vanishes entirely.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-3.0,0) -- (4.0,0) node[anchor=north east] {voltage $V$};
  \draw[->, black] (0,-0.4) -- (0,3.2) node[anchor=south east, align=left] {photocurrent};
  % high freq (larger stopping potential, more negative)
  \draw[black, thick, densely dashed, smooth] plot coordinates
    {(-2.3,0)(-1.7,0.2)(-0.9,0.85)(0.2,1.9)(1.4,2.35)(2.8,2.45)(3.7,2.45)};
  \node[black, anchor=south, font=\scriptsize] at (-2.3,0.15) {high $f$};
  % low freq (smaller stopping potential)
  \draw[acc, thick, smooth] plot coordinates
    {(-0.9,0)(-0.5,0.25)(0.2,0.95)(1.1,1.9)(2.2,2.35)(3.2,2.45)(3.7,2.45)};
  \node[acc, anchor=south, font=\scriptsize] at (-0.9,0.15) {low $f$};
  % markers
  \fill[black] (-2.3,0) circle (1.5pt);
  \fill[acc] (-0.9,0) circle (1.5pt);
\end{tikzpicture}
$$

Classically neither should happen. A wave deposits energy at a rate set by
intensity, so the electron's energy should grow with brightness, and any
frequency of light, given enough time, should eventually shake an electron loose.
There should also be a measurable **time lag** while a low-intensity wave slowly
accumulates the escape energy in one atom. None of these classical expectations
survived.

## Einstein's photon

Einstein applied Planck's quantum to the light itself. A beam of frequency $f$ is
not a smooth wave but a stream of discrete quanta, later called **photons**, each
carrying energy

$$
E = hf = \frac{hc}{\lambda}.
$$

A photon is absorbed by a single electron all at once, delivering its whole
energy $hf$. Freeing the electron from the metal costs a minimum energy $\phi$,
the **work function**, characteristic of the surface. Whatever remains appears as
kinetic energy.

> **Definition (Work function).** The minimum energy $\phi$ needed to remove the
> least tightly bound electron from a metal surface. Values run a few electron
> volts: $\phi = 2.28\ \text{eV}$ for sodium, $6.35\ \text{eV}$ for platinum.

$$
% caption: Energy bookkeeping for one photon. It arrives with hf, pays the work
% function to lift an electron out of the filled band, and the remainder becomes
% the electron's kinetic energy outside the metal.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % potential-energy step at the surface
  \draw[black, thick] (0,0) -- (3.0,0);           % filled band top (inside)
  \draw[black, thick] (3.0,0) -- (3.0,2.4);        % surface barrier
  \draw[black, thick] (3.0,2.4) -- (6.0,2.4);      % vacuum level (outside)
  \node[anchor=north, font=\scriptsize] at (1.5,0) {inside metal};
  \node[anchor=north, font=\scriptsize] at (4.5,2.4) {outside metal};
  % filled states hatching
  \foreach \x in {0.3,0.7,1.1,1.5,1.9,2.3,2.7}
    \draw[black] (\x,-0.75) -- (\x,0);
  \node[anchor=north, font=\scriptsize] at (1.5,-0.85) {occupied states};
  % work function bracket
  \draw[<->, black] (3.25,0) -- (3.25,2.4);
  \node[anchor=west, font=\scriptsize, text=black] at (3.35,1.2) {work function};
  % incoming photon
  \draw[->, acc, thick] (0.4,3.3) -- (1.2,0.15);
  \node[anchor=south, font=\scriptsize, text=acc] at (0.4,3.3) {photon $hf$};
  % ejected electron with kinetic energy above vacuum level
  \draw[<->, black] (4.5,2.4) -- (4.5,3.4);
  \node[anchor=west, font=\scriptsize] at (4.6,2.95) {kinetic energy};
  \fill[black] (4.5,3.4) circle (1.6pt);
  \draw[->, black] (4.7,3.4) -- (5.6,3.4);
\end{tikzpicture}
$$

Energy conservation gives **Einstein's photoelectric equation**:

> **Theorem (Photoelectric equation).** The maximum kinetic energy of a
> photoelectron is the photon energy less the work function,
> $$
> \left(\tfrac{1}{2} m v^2\right)_{\max} = hf - \phi,
> \qquad
> e V_0 = hf - \phi.
> $$
> Electrons that lose energy traversing the metal come off with less; the fastest
> come off with exactly this.

Every anomaly follows at once.

- **Stopping potential independent of intensity.** Intensity is the number of
  photons per second, not the energy each carries. More photons free more
  electrons — a larger current — but each electron still receives one photon's
  $hf$, so the maximum energy, and $V_0$, is fixed by frequency alone.
- **Threshold frequency.** A photon with $hf < \phi$ cannot pay the escape cost,
  and the electron is not emitted regardless of how many such photons arrive.
  Setting $V_0 = 0$,
  $$
  \phi = h f_t = \frac{hc}{\lambda_t}.
  $$
- **No time lag.** The energy is not accumulated gradually; a single absorption
  event delivers the whole $hf$ instantly, so emission can begin the moment the
  light is switched on, even at intensities where a classical wave would need
  minutes.

## Millikan's confirmation

Einstein's equation is a straight line. Plotting $V_0$ against frequency $f$,

$$
V_0 = \frac{h}{e}\, f - \frac{\phi}{e},
$$

predicts a line of slope $h/e$ — the same for every metal — and intercept
$-\phi/e$ on the voltage axis, or $f_t = \phi/h$ on the frequency axis. Millikan,
who disbelieved the photon and set out to disprove the equation, instead
confirmed it in 1914-1916 and measured $h$ from the slope in agreement with
Planck's blackbody value.

$$
% caption: Stopping potential versus frequency. The data lie on a straight line
% of slope h/e, the same for any metal; the intercept on the frequency axis is
% the threshold, and different metals shift the line without tilting it.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-1.1) -- (7.2,-1.1) node[anchor=north east] {frequency $f$};
  \draw[->, black] (0,-1.4) -- (0,3.2) node[anchor=south east, align=left] {stopping\\potential};
  % the zero of V0 axis
  \draw[black] (0,0) -- (7.0,0);
  \node[anchor=east, font=\scriptsize] at (-0.05,0) {$0$};
  % metal A line (slope h/e), threshold at f=1.8
  \draw[acc, thick] (1.8,0) -- (6.4,2.7);
  \draw[acc, thick, dashed] (1.0,-0.47) -- (1.8,0);
  \fill[acc] (1.8,0) circle (1.7pt);
  \node[anchor=north west, font=\scriptsize, text=acc] at (1.85,0) {threshold $f_t$};
  \node[anchor=west, font=\scriptsize, text=acc] at (5.4,2.2) {metal A};
  % metal B line (same slope, larger threshold)
  \draw[black, thick] (3.3,0) -- (6.4,1.82);
  \draw[black, thick, dashed] (2.5,-0.47) -- (3.3,0);
  \fill[black] (3.3,0) circle (1.7pt);
  \node[anchor=west, font=\scriptsize, text=black] at (5.6,1.3) {metal B};
  % slope label
  \node[font=\scriptsize] at (3.6,1.7) {slope $= \frac{h}{e}$};
\end{tikzpicture}
$$

The measurement was doubly important. It confirmed the photon, and it gave an
independent value of Planck's constant from a phenomenon that, on its face, has
nothing to do with cavity radiation. Two unrelated experiments returning the same
$h$ is strong evidence the constant is real.

## Worked examples

The convenient combination for photon energies is
$hc = 1240\ \text{eV}\cdot\text{nm}$, so a photon of wavelength $\lambda$ (in nm)
carries $1240/\lambda$ eV.

**Threshold and stopping potential for potassium.** Potassium has threshold
wavelength $\lambda_t = 558\ \text{nm}$. The work function is the threshold photon
energy,

$$
\phi = \frac{hc}{\lambda_t} = \frac{1240\ \text{eV}\cdot\text{nm}}{558\ \text{nm}}
= 2.22\ \text{eV}.
$$

Illuminating with $400\ \text{nm}$ light, each photon carries
$1240/400 = 3.10\ \text{eV}$, so the stopping potential is

$$
V_0 = \frac{hf - \phi}{e}
= 3.10\ \text{eV} - 2.22\ \text{eV} = 0.88\ \text{V}.
$$

**The classical time lag that never appears.** Take $400\ \text{nm}$ light at a
low intensity $I = 10^{-2}\ \text{W/m}^2$, and let a target atom present a disk of
radius $r \approx 10^{-10}\ \text{m}$. Classically the energy accumulates at

$$
\frac{\d E}{\d t} = I\,\pi r^2 = (10^{-2})(\pi)(10^{-20})
= 3.14 \times 10^{-22}\ \text{W}.
$$

Reaching $\phi = 2.22\ \text{eV} = 3.55 \times 10^{-19}\ \text{J}$ would take

$$
t = \frac{3.55 \times 10^{-19}\ \text{J}}{3.14 \times 10^{-22}\ \text{W}}
\approx 1.13 \times 10^{3}\ \text{s} = 18.8\ \text{min}.
$$

Classical physics predicts nearly a twenty-minute wait before the first electron
appears. Experiment shows no delay. In the photon picture the same intensity is
$2 \times 10^{16}$ photons per second per square meter — about one photon per
thousand surface atoms per second — and each carries enough energy to eject an
electron immediately. Emission is prompt but sparse, not delayed and uniform.

## The wave-particle tension

The photoelectric effect established that light exchanges energy with matter in
discrete quanta $hf$, each localized enough to be absorbed by a single electron.
Yet the same light diffracts and interferes, phenomena only a wave explains. Both
descriptions are needed: a particle theory for the energy exchange, a wave theory
for propagation and interference.

| Feature | Wave theory | Photon theory | Observed |
| --- | --- | --- | --- |
| Saturation current vs. intensity | rises with intensity | rises with photon rate | rises with intensity |
| Max electron energy vs. intensity | rises | fixed | fixed |
| Max electron energy vs. frequency | no dependence | linear, slope $h$ | linear, slope $h$ |
| Threshold frequency | none | $f_t = \phi/h$ | sharp threshold |
| Time lag at low intensity | minutes | none | none |

The photon carries energy $hf$; the next lesson shows it also carries momentum
$hf/c$, and that a photon can scatter off an electron like one billiard ball off
another. This
[Compton scattering](/quantum-mechanics/old-quantum-theory/x-rays-and-the-compton-effect)
of X-rays removed the last doubt that light is granular. The full reconciliation —
that particles are also waves — waits for
[de Broglie](/quantum-mechanics/matter-waves/de-broglie-waves-and-electron-diffraction) and the
[uncertainty principle](/quantum-mechanics/matter-waves/the-uncertainty-principle).[^tl-pe][^tl-einstein]

[^tl-pe]: Tipler & Llewellyn, §3-3 — Lenard's apparatus, the stopping-potential and threshold-frequency data, and Einstein's photoelectric equation $eV_0 = hf - \phi$.
[^tl-einstein]: Tipler & Llewellyn, §3-3 — Millikan's 1914-1916 measurement of the $V_0$-versus-$f$ line, its slope $h/e$, and agreement of the resulting $h$ with Planck's blackbody value; work-function table (Table 3-1).