The Photoelectric Effect: Light Comes in Packets

The photoelectric effect is the emission of electrons from a material, usually a metal, when light falls on it; the electrons released are called photoelectrons. Its discovery was accidental. In 1887 Heinrich Hertz, while testing Maxwell's electromagnetic waves, noticed that the sparks jumping across his receiver's gap grew shorter when he shut the receiver in a dark box. A sheet of glass between transmitter and receiver had the same damping effect; a sheet of quartz did not. Glass absorbs ultraviolet light and quartz transmits it, so ultraviolet light from the transmitter's spark had been helping charge escape across the gap.

In 1888 Wilhelm Hallwachs showed that ultraviolet light discharges a negatively charged zinc plate but not a positively charged one: light was driving negative charge out of the metal. In 1899, J. J. Thomson measured the charge-to-mass ratio of these carriers and found it matched that of the cathode-ray particles he had identified two years earlier. They were electrons, and the metal does not surrender them for free: they are held in by the attraction of its positive ions, and freeing even the most loosely held one costs a minimum energy called the work function, written with the Greek letter φ: about 2.1 eV for cesium, 2.3 eV for sodium, 4.3 eV for zinc and 4.7 eV for copper. These are typical values; a real surface can differ by a few tenths of an electron-volt with its crystal face and cleanliness. The electron-volt, eV, is a tiny energy unit, 1.602×10⁻¹⁹ joules, that Part 6 explains.

The decisive measurements began with Philipp Lenard's detailed study of 1902. His apparatus is also the one in your lab. Inside an evacuated glass tube, light falls on a metal plate, the emitter. Escaping electrons fly to a second electrode, the collector, completing a circuit, so a sensitive meter registers a photocurrent: a current carried by photoelectrons. Make the collector slightly negative and it pushes back on arriving electrons; only those with enough kinetic energy, the energy of motion, get through. Raise this retarding voltage until the current stops, and you have measured the energy of the fastest electrons. Lenard did this with a powerful arc lamp whose intensity, the power delivered per square metre, he could vary widely.

Lenard's work, and sharper experiments over the following decade, established three facts. First, a threshold: for each metal there is a threshold frequency, f₀, below which no electrons emerge at all, however intense the light. Red light at any brightness ejects nothing from zinc; ultraviolet does. Second, emission is instantaneous, even in dim light. In 1928 Ernest Lawrence and Jesse Beams chopped light into flashes with a fast electro-optical shutter and found that emission from a potassium hydride surface begins less than 3×10⁻⁹ seconds after illumination starts. Third, the maximum kinetic energy of the photoelectrons does not depend on intensity. Brighter light produces more electrons, and the photocurrent grows in proportion, but the fastest electrons are no faster. Only the frequency changes their energy: the bluer the light, the faster they fly.

Consider the classical picture of 1900. Light is a continuous electromagnetic wave whose energy is spread smoothly across each wavefront, with an intensity proportional to the square of its amplitude, as chapter 2 showed. An electron in the metal is shaken by the wave's oscillating electric field and soaks up energy steadily until it has enough to escape. The picture is reasonable, and every one of its predictions here is wrong.

First, intensity. Brighter light means a larger field amplitude, which should shake electrons harder and fling them out faster. Instead, the maximum speed stays fixed and only the number of electrons rises. Next, the threshold. A wave delivers energy at any frequency; low frequency merely means slower shaking, so with enough intensity or patience, red light should eventually free electrons. Instead, below f₀ nothing happens, ever. Nor could classical theory say why the electrons' energy should depend on frequency at all, let alone rise in a straight line with it.

Finally, the delay, where a back-of-the-envelope estimate is devastating. A wave spreads its energy over the whole surface, so each electron can harvest only the light falling on roughly one atom's cross-section. Be generous: assume no light is reflected, and let one electron in a potassium surface collect everything that falls on a disc of radius 0.1 nm, roughly atomic size, about 3.1×10⁻²⁰ square metres, under dim light of 1.0×10⁻³ W/m². That electron receives about 3.1×10⁻²³ watts, and gathering a work function of 2.3 eV, or 3.7×10⁻¹⁹ joules, takes about 1.2×10⁴ seconds: more than three hours. Real electrons emerge within nanoseconds. Three facts, three failures. Back to the blackboard!

In March 1905 Albert Einstein, a 26-year-old patent examiner in Bern, submitted a paper whose title translates as On a Heuristic Point of View Concerning the Production and Transformation of Light; a heuristic is a guide to discovery rather than a finished theory. His starting point was your last chapter. For blackbody radiation at high frequencies, where Wien's formula holds, he showed that the entropy of radiation in a box changes with volume exactly as that of a gas of independent particles, as though light consisted of separate packets, each carrying energy hf in today's notation. Planck had quantized the oscillators in the walls; Einstein quantized light itself. Its energy, in one English translation, 'consists of a finite number of energy quanta that are spatially localized at points of space, move without dividing and are absorbed or generated only as a whole.' He told his friend Conrad Habicht the work was very revolutionary.

Today we call such a packet a photon, a name popularized by the American chemist Gilbert Lewis in 1926. A photon of frequency f carries energy E = hf, where h = 6.626×10⁻³⁴ J·s is Planck's constant, and absorption is all or nothing: one photon surrenders its entire energy to one electron. The electron must spend at least the work function φ to escape and keeps the rest as kinetic energy. The least-bound electrons, at the surface, pay exactly φ and emerge with the maximum, KE_max = hf − φ. Electrons that start deeper, or more tightly bound, lose extra energy on the way out, so the emitted electrons range from zero up to KE_max.

Now every fact falls into place. Threshold: if hf is less than φ, no single photon can free an electron, and packets cannot be pooled, so the threshold frequency is f₀ = φ/h. Instant emission: the energy arrives concentrated, so the first photon absorbed by a suitable electron frees it at once; dim light means fewer photons, not weaker ones. Intensity: at fixed frequency, intensity is the number of photons per second per square metre times hf. Double it and you double the photons, the photoelectrons and the current, but each photon is unchanged, so KE_max is unchanged. Frequency sets the energy per photon; intensity sets the number of photons. Einstein also made a bold prediction: KE_max should rise in a straight line with frequency, with slope h for every metal. Testing it took a decade.

To measure KE_max, use the retarding voltage from Part 2. If the collector sits at a voltage V below the emitter, each crossing electron loses kinetic energy e·V, where e = 1.602×10⁻¹⁹ coulombs is the elementary charge, the size of the electron's charge. As V rises, slower electrons are turned back, until at the stopping potential, V₀, even the fastest just fail and the current reaches zero. So e·V₀ = KE_max, and Einstein's equation becomes V₀ = (h/e)·f − φ/e. Plotted against frequency, the stopping potential should form a straight line that meets the frequency axis at the threshold f₀ and has slope h/e for every metal, so different metals give parallel lines. That was Einstein's sharp, falsifiable prediction.

Robert Millikan at the University of Chicago spent about ten years testing it, expecting failure. The alkali metals sodium, potassium and lithium have work functions low enough for visible and near-ultraviolet light, but their surfaces tarnish almost instantly. So Millikan built what he called 'a machine shop in vacuo': electromagnets outside the glass worked a rotating knife inside the vacuum that shaved fresh metal. He lit the surfaces with lines of a quartz mercury arc and measured stopping potentials together with the contact potential, the voltage that appears between two different metals. A contact potential shifts every stopping potential by the same amount, moving the line without tilting it, which is why the slope is the trustworthy quantity.

His 1916 paper in Physical Review reported sodium data on a straight line with slope 4.128×10⁻¹⁵ volt-seconds, giving h = 6.57×10⁻³⁴ J·s, with a precision he estimated at about 0.5 percent. Einstein's equation, he wrote, 'appears in every case to predict exactly the observed results.' Yet the same paper called the light corpuscle 'the bold, not to say the reckless, hypothesis', because it 'flies in the face of the thoroughly established facts of interference.' Millikan accepted the equation long before he accepted the photon. Since 2019, h has been fixed exactly at 6.62607015×10⁻³⁴ J·s.

For single photons, physicists measure energy in electron-volts. One electron-volt is the energy an electron gains when accelerated through one volt, exactly 1.602176634×10⁻¹⁹ joules since 2019. It is natural here because a stopping potential in volts reads off KE_max directly in electron-volts: a stopping potential of 0.78 V means the fastest electrons left with 0.78 eV. On this scale, work functions run from about 2 to 6 eV, and visible photons carry from about 1.65 eV at the red end, 750 nm, to 3.26 eV at the violet end, 380 nm. The two are comparable, which is why some metals respond to visible light and others demand ultraviolet.

Combine E = hf with c = fλ to get E = hc/λ. The product hc equals 1239.84 eV·nm, so a photon's energy in electron-volts is about 1240 divided by its wavelength in nanometres: E ≈ 1240/λ. Red light at 650 nm carries 1.91 eV, a green laser at 532 nm 2.33 eV, a violet laser at 405 nm 3.06 eV, and a germicidal lamp at 254 nm 4.88 eV. Cesium, at 2.1 eV, has a threshold wavelength of 1240/2.1 ≈ 590 nm, so green, blue and violet light all work, which is why cesium, usually in compounds, coats many light detectors. Zinc, at 4.3 eV, has a threshold near 288 nm, well into the ultraviolet: that is why my red laser failed and the feeble germicidal lamp succeeded.

The same arithmetic counts photons. A beam of power P and frequency f delivers R = P/(hf) photons per second. A 1-milliwatt green laser pointer emits about 2.7×10¹⁵ photons every second. Dim it to a nanowatt, a millionth as bright, and it still sends about 2.7 billion photons per second, every one a full-sized packet of 2.33 eV. Dimming light reduces the number of packets, never their size.

In November 1922 the Royal Swedish Academy of Sciences awarded Einstein the 1921 Nobel Prize in Physics, held over for a year, 'for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.' Relativity, still controversial, goes unmentioned; the Nobel Committee preferred a law that experiment had confirmed beyond dispute. Einstein, travelling in Japan, missed the ceremony, and when he finally delivered his Nobel lecture, in Gothenburg in July 1923, its subject was relativity. Mwahaha, measurably!

Einstein's rule now runs a vast technology, in two forms. In the external photoelectric effect, the one Einstein explained, electrons leave the material entirely. In the internal effect, in semiconductors, a photon lifts an electron across the band gap, the forbidden range of energies separating bound electrons from those free to move through the crystal. The electron stays inside, but it and the vacancy it leaves, a hole, can now carry current; the band gap plays the role of the threshold. In a solar cell, the built-in electric field of a p-n junction, the boundary between two regions of silicon laced with different trace impurities, sweeps these charges apart and drives a current: the photovoltaic effect. Silicon's band gap of 1.12 eV lets it use photons with wavelengths shorter than about 1,100 nm. Photodiodes, built the same way, read fibre-optic signals and remote-control flashes.

A camera sensor is millions of photodiodes in a grid, each pixel collecting charge in proportion to the photons that land on it. The charge-coupled device, or CCD, invented in 1969 by Willard Boyle and George Smith, earned them a share of the 2009 Nobel Prize in Physics; most phone cameras now use CMOS sensors on the same principle. In a photomultiplier tube, a photocathode releases an electron per detected photon and a chain of electrodes called dynodes multiplies it a million times or more, enough to count single photons. A night-vision image intensifier does this for a whole picture: faint light strikes a photocathode, of gallium arsenide in third-generation tubes; a microchannel plate, a disc of millions of tiny glass channels, multiplies the photoelectrons thousands of times; and they light a green phosphor screen.