In 1887, during his experiments to confirm electromagnetic waves, Hertz noticed that a spark jumped more easily across a gap when ultraviolet light fell on it. The next year Hallwachs found that a negatively charged zinc plate gradually loses its charge when ultraviolet light shines on it, while a positively charged plate does not change. With a sheet of glass between the plate and the lamp, the charge no longer decreases, because glass does not let ultraviolet light through.
In 1899 and 1900, Thomson and Lenard measured the ratio of charge to mass of what comes out of a metal under light, and showed that it is the same as for the electron, discovered in 1897. The emission of electrons from the surface of a metal under light is called the photoelectric effect, and the electrons that leave are called photoelectrons. The negative plate lost its charge because electrons were leaving it.
Electrons inside a metal move freely within it but cannot leave through the surface. Getting one out requires giving it at least a certain amount of energy, and that minimum is called the work function, written W. The work function depends on the metal: about 2.3 eV for sodium, 4.3 eV for zinc and 5.6 eV for platinum. One electronvolt, 1 eV, is the energy an electron gains when accelerated through 1 V, 1.6 \times 10^{-19} J.
Experiments use a phototube. Inside an evacuated glass tube, a metal plate that receives the light (the cathode) faces an electrode that collects the electrons (the anode), and the current is measured while a voltage is applied between them. When the anode is at a higher potential than the cathode, almost all the electrons that come out reach the anode.
When the anode is at a lower potential than the cathode, the electrons are pushed back. To reach the anode against a retarding voltage V, an electron needs a kinetic energy of at least eV. As the voltage is raised, at some value even the fastest electrons stop arriving and the current falls to 0. This voltage is called the stopping voltage, written V_0, and the largest kinetic energy of the photoelectrons is K_{\mathrm{max}} = eV_0.
In 1902 Lenard studied photoelectrons while changing the intensity of the light. Stronger light increased the current in proportion to the intensity, but the stopping voltage stayed the same. More electrons come out, but the kinetic energy of the fastest one does not change. That is why the arrows for the fastest electrons have the same length on both sides of the figure.
This does not fit a wave picture of light. The energy a wave carries is proportional to the square of its amplitude, and stronger light has a larger electric field. The force with which the field shakes an electron is then larger too, so with stronger light the electrons should gain more energy and come out faster.
That electrons come out at once even in weak light does not fit a wave picture either. Suppose light of 1 W/m² falls on sodium, and an electron can collect only the energy falling on the cross section of one atom, about 3 \times 10^{-20} m² for a radius of 0.1 nm. That is 3 \times 10^{-20} J per second, and collecting the work function of 2.3 eV (3.7 \times 10^{-19} J) would take about 12 seconds.
In fact, electrons come out as soon as the light arrives. In 1928 Lawrence and Beams measured the delay and found it shorter than 3 \times 10^{-9} seconds, below what they could resolve. Weaker light reduces only the number of electrons; it does not change how soon they begin to come out.
Lower the frequency of the light and, below a certain value, not a single electron comes out however strong the light is. This boundary frequency is called the threshold frequency, written \nu_0. In terms of wavelength, light longer than \lambda_0 = \dfrac{c}{\nu_0} releases no electrons, and \lambda_0 is called the threshold wavelength.
Above the threshold frequency, electrons come out however weak the light is. What decides whether they come out is not the intensity but the frequency. For visible light a difference in frequency is a difference in color, so red light does not release electrons from sodium however strong it is.
The threshold frequency differs from metal to metal, and the larger the work function, the higher it is. For sodium the threshold wavelength is about 540 nm, so light shorter than green releases electrons. For zinc it is about 290 nm, and only ultraviolet light releases electrons. That is why, in Hallwachs's experiment, the zinc plate stopped losing charge when glass was placed between it and the lamp.
In 1905 Einstein proposed that light of frequency \nu is a collection of particles each carrying energy E = h\nu, and that light is absorbed one such particle at a time. h is the constant Planck had introduced for blackbody radiation. Einstein called the particle a light quantum; today it is called a photon.
The energy of a photon of wavelength \lambda is E = \dfrac{hc}{\lambda}, and the value hc = 1240 eV·nm makes it easy to compute. A photon of red light at 700 nm has 1.8 eV, green at 550 nm 2.3 eV, violet at 400 nm 3.1 eV, and ultraviolet at 250 nm 5.0 eV.
The intensity of light is set by the number of photons arriving per second. A green beam with a power of 1 mW, for example, carries about 3 \times 10^{15} photons each second. With numbers this large, the fact that light consists of separate particles is not noticeable in light of ordinary brightness.
An electron in the metal absorbs one photon whole. Of the h\nu it receives, at least W is used to leave the surface, and the rest becomes kinetic energy. Electrons that were just at the surface come out with the most energy, K_{\mathrm{max}} = h\nu - W. Electrons from deeper inside lose part of their energy on the way out and are slower.
This accounts for the two measurements about intensity. Stronger light means more photons, so the current increases, but the energy of each photon is unchanged, so K_{\mathrm{max}} stays the same. And since each photon gives its energy to one electron all at once, there is no time spent collecting energy.
If h\nu < W, an electron that absorbs one photon cannot leave. An electron almost never absorbs two photons together, so no electrons come out however strong the light is. The threshold is set by h\nu_0 = W; from sodium's W = 2.3 eV the threshold wavelength is \dfrac{1240}{2.3} nm, about 540 nm.
Combining K_{\mathrm{max}} = eV_0 with K_{\mathrm{max}} = h\nu - W gives eV_0 = h\nu - W. With frequency on the horizontal axis and eV_0 on the vertical axis this is a straight line with slope h, crossing the horizontal axis at the threshold frequency \nu_0 and the vertical axis at -W. With V_0 on the vertical axis instead, the slope is \dfrac{h}{e}.
Einstein's equation predicted that the slope of this line would be the same h for every metal. In 1905, however, no measurement was precise enough to test it, because a metal surface oxidizes quickly in air and its work function changes.
Millikan measured the stopping voltage of sodium, potassium and lithium at many frequencies, renewing the surfaces by cutting them with a blade inside the vacuum. In the results he published in 1916 the measured points lay on a straight line, and the h obtained from its slope was 6.57 \times 10^{-34} J s, 0.3% from the 6.55 \times 10^{-34} J s Planck had obtained from blackbody radiation.
Changing the metal changes the work function, so the line moves up or down while staying parallel, and the slope does not change. In the figure, the line for the other metal moves while keeping its slope for this reason.
Millikan later wrote that he had doubted the idea of light as particles and had expected his measurements to refute it. Instead they confirmed Einstein's equation. Einstein received the 1921 Nobel Prize in Physics for discovering the law of the photoelectric effect, and Millikan received the same prize in 1923 for his measurements of the elementary charge and of the photoelectric effect.