The Photon: Photoelectric and Compton Effects
Black-Body Radiation and Planck's Hypothesis ended with a constant \(h\) that governs how a cavity wall trades energy with the field, and with Planck's own insistence that this says nothing about the field itself. This chapter is about the evidence that it does. Einstein's 1905 proposal [Einstein:1905c] was that light of frequency \(\nu\) behaves in its emission, propagation and absorption as though it consisted of independent quanta of energy \(h\nu\) — a statement about light, not about oscillators, and one that the thermodynamics of the previous chapter does not entail. He offered the photoelectric effect [Hertz:1887] [Hallwachs:1888] [Lenard:1902] as a test, because its observed dependence on frequency and its independence of intensity are inexplicable if the energy of a wave arrives spread over a wavefront.
The chapter sets out the evidence in the order in which it accumulated and grades it honestly. Millikan's decade of measurement [Millikan:1916] verified Einstein's linear relation to better than a percent while its author continued to reject the hypothesis behind it; and he was not simply being obstinate, for the photoelectric equation alone can be obtained from a quantized atom absorbing from a classical field [Lamb:1969]. What settled the matter was momentum: Compton's shift in the wavelength of scattered X-rays [Compton:1923] [Debye:1923], and then the coincidence experiments [Bothe:1925] [Compton:1925] that destroyed the last classical alternative [Bohr:1924] by showing that energy and momentum balance in every individual scattering event and not merely on average. The quantum thereby acquires the two attributes of a particle, and the field theory that makes this consistent is Part XI — Quantum Field Theory and the Standard Model.
The Photon: Photoelectric and Compton Effects: all derivations of this chapter are pending.
The photoelectric effect
Discovery: Hertz and Hallwachs
[Reserved: Hertz's observation, made while producing the first electromagnetic waves [Hertz:1887], that ultraviolet light on the spark gap lowers the potential at which the spark passes — an unwanted effect in an experiment about something else, and reported as such. Hallwachs' isolation of the phenomenon the following year [Hallwachs:1888]: a freshly polished zinc plate, negatively charged, is discharged by ultraviolet light and a positively charged one is not, so the carriers are negative. Identification of the carriers with Thomson's corpuscles (Atomic Models and Spectra) by their charge-to-mass ratio.]
Lenard's measurements and the classical impasse
[Reserved: Lenard's systematic study with a monochromated source and a retarding potential [Lenard:1902], which established the three facts that no wave theory accommodates: the maximum kinetic energy of the emitted electrons is independent of the intensity; it increases with the frequency; and below a threshold frequency no electrons are emitted at any intensity. To be added, the fourth: the emission begins without measurable delay, whereas a classical wavefront needs minutes to deliver an electron's worth of energy to an atomic cross-section — the estimate should be made explicitly here, because it is the sharpest form of the contradiction. Only the photocurrent scales with intensity.]
Ultraviolet light falling on a clean metal surface ejects electrons, and the emission has four properties that no theory treating light as a continuous wave accommodates [Hertz:1887] [Hallwachs:1888] [Lenard:1902]:
-
below a threshold frequency \(\nu_{0}\), characteristic of the emitting metal, no electrons appear at any intensity and after any exposure;
-
above the threshold the maximum kinetic energy of the emitted electrons is independent of the intensity of the light;
-
that maximum increases with the frequency of the light;
-
the emission begins without measurable delay, even at intensities so feeble that a wave spread uniformly over the surface would need minutes to deliver one electron's binding energy to a region of atomic size.
Only the number of electrons emitted per second — the photocurrent — is proportional to the intensity.
The classical accumulation time: the energy an electromagnetic wave of a stated irradiance delivers per second to a target of atomic cross-section, compared with the work function of the metal, giving the delay a wave theory requires before the first electron can be emitted; and the demonstration that no redistribution of energy within the classical field shortens it, which is what makes item 4 above a contradiction rather than an estimate.
The light-quantum hypothesis
Einstein's heuristic viewpoint of 1905
[Reserved: the structure of Einstein's argument [Einstein:1905c], which is thermodynamic and not electrodynamic: in the Wien regime of Black-Body Radiation and Planck's Hypothesis the entropy of radiation in a volume \(V\) varies with \(V\) exactly as that of an ideal gas of \(E/h\nu\) independent particles, so monochromatic radiation of low density behaves as if composed of that many localized quanta. Three consequences were offered as tests — Stokes' rule for fluorescence, the photoelectric effect, and photoionization. The chapter should stress that Einstein derived the quantum from the empirical Wien law, not from Planck's counting, and regarded the two arguments as independent.]
The photoelectric equation and the work function
[Reserved: \(E_{\max}=h\nu-\phi\), with \(\phi\) the work function of the emitting surface [Einstein:1905c]; the stopping potential \(V_{0}=E_{\max}/e\) as the measured quantity, the threshold frequency \(\nu_{0}=\phi/h\), and the slope \(h/e\) of the \(V_{0}\) against \(\nu\) line, which is universal — the same for every metal, which is the content of the prediction. Tabulated work functions in \(\mathrm{eV}\) (caesium about \(2.1\,\mathrm{eV}\), potassium \(2.3\,\mathrm{eV}\), copper \(4.7\,\mathrm{eV}\), platinum \(5.6\,\mathrm{eV}\)), their sensitivity to surface contamination and crystal face, and hence why the experiment is hard. The many-electron theory of \(\phi\) belongs to Electrons in Solids: Band Theory.]
Millikan's verification
[Reserved: the ten-year programme [Millikan:1916] that cut fresh alkali-metal surfaces in vacuum with a magnetically operated knife to defeat oxidation, measured the stopping potential across the visible and near ultraviolet, confirmed the strict linearity, and extracted \(h=6.57\times 10^{-34}\,\mathrm{J}\,\mathrm{s}\) to about \(0.5\,\mathrm{\%}\) — then the best value of Planck's constant by any method. Honest history: Millikan undertook the work to disprove what he called a reckless hypothesis and wrote, on confirming the equation, that the light-quantum theory nonetheless seemed untenable; the section records this because it is a clean case of a prediction surviving a hostile test.]
The stopping potential \(V_{0}\) that just prevents the fastest photoelectrons from reaching the collector is a strictly linear function of the frequency of the incident light,
and the slope of that line is \(h/e\) for every emitting metal, the material entering only through the intercept \(\phi\). Millikan established the linearity across the visible and near ultraviolet on alkali surfaces cut fresh in vacuum, and read from the slope a value of Planck's constant of \(6.57\times 10^{-34}\,\mathrm{J}\,\mathrm{s}\), to about \(0.5\,\mathrm{\%}\) [Millikan:1916] — then the best determination by any method, and in agreement with the value obtained from cavity radiation [Planck:1901], which is a measurement of an entirely different kind.
Derivation. Assume with Einstein [Einstein:1905c] that light of frequency \(\nu\) is emitted and absorbed only in indivisible amounts \(h\nu\), and that one such amount is taken up by a single electron. An electron at the top of the occupied band must pay the work function \(\phi\) to cross the surface, and any electron starting deeper pays more, so the fastest electron leaves with
Three of the four observations of Phenomenon 69.1 follow at once. \(E_{\max}\) depends on the size of the quantum and not on how many quanta arrive per second, which is item 2; it rises linearly with \(\nu\), which is item 3; and it would be negative — so no electron is emitted — for every \(\nu<\nu_{0}:=\phi/h\) whatever the intensity, which is item 1. Raising the intensity multiplies the number of quanta and therefore the photocurrent, and nothing else. Retarding the emitted electrons through a potential difference \(V\) stops all those with \(E_{\max}<eV\), so the measured stopping potential satisfies \(eV_{0}=E_{\max}\), which is Equation (69.1). Its slope \(h/e\) contains no property of the metal at all, and that universality, rather than the linearity, is the content of the prediction.
∎What the photoelectric effect does not prove
[Reserved: the necessary caveat. A quantized atom coupled to a classical radiation field reproduces the threshold, the linearity in frequency and the promptness of the emission, as shown by Lamb and Scully [Lamb:1969]; the discreteness lies in the atomic energy levels of Atomic Models and Spectra, which are independently established. The photoelectric effect is therefore strong evidence for quantized absorbers and weak evidence for quantized light. Naming what an experiment does not decide is editorial rule 1 applied to a case where the textbook claim is routinely overstated; the experiments that do decide it are Section 69.5 and Section 69.6.1.]
Momentum and angular momentum of light
Radiation pressure
[Reserved: the classical prediction \(p=u/c\) from the Maxwell stress tensor (The Maxwell Equations) and its measurement by Lebedev with a torsion balance in vacuum [Lebedew:1901] and, independently and more accurately, by Nichols and Hull [Nichols:1903]. In quantum language each quantum carries \(p=h\nu/c=h/\lambda\), which Einstein obtained from detailed balance in 1917 [Einstein:1917b]; the two descriptions agree for the pressure and differ in the fluctuations. Applications: comet tails, radiation pressure in stellar interiors (Stellar Structure and Nucleosynthesis), optical tweezers and laser cooling (Quantum Optics and the Photon).]
A beam of light exerts a mechanical force on the surface it falls upon. At normal incidence the pressure is
with \(I\) the incident irradiance. Lebedev detected the effect on light and dark vanes hung from a torsion fibre in an evacuated vessel, the whole difficulty of the measurement being to separate it from the far larger radiometric and convective forces [Lebedew:1901]; Nichols and Hull, independently, confirmed the predicted magnitude to a few percent [Nichols:1903].
Derivation. In the quantum description the beam is a flux of \(N\) quanta per unit area per unit time, each of energy \(h\nu\) and — by the detailed-balance argument Einstein gave in 1917 [Einstein:1917b] — of momentum \(h\nu/c\) directed along the propagation direction. The irradiance is the energy flux, \(I=Nh\nu\), so the momentum arriving per unit area per unit time is \(Nh\nu/c=I/c\). Force per unit area is momentum delivered per unit area per unit time, so an absorber feels \(P=I/c\); a reflector returns each quantum with its momentum reversed and therefore receives twice as much, \(P=2I/c\). This is Equation (69.3).
The same two numbers follow classically from the Maxwell stress tensor of The Maxwell Equations, and it must be said plainly that Equation (69.3) is therefore not evidence for quanta. The two descriptions agree on the mean pressure and differ in its fluctuations; what the quantum reading adds is that the momentum comes in the same indivisible packets as the energy, in the fixed ratio \(1/c\), and it is that ratio which Phenomenon 69.5 tests directly.
∎Angular momentum of light
[Reserved: circularly polarized light carries angular momentum \(\hbar\) per quantum, measured mechanically by Beth [Beth:1936], who suspended a birefringent half-wave plate on a quartz fibre and detected the torque exerted as the handedness of the transmitted beam was reversed. This is the direct evidence that the photon has spin one, and with the absence of a longitudinal polarization state (Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism) it fixes the field's representation content under the Poincaré group (Particles as Poincaré Representations).]
A circularly polarized beam of angular frequency \(\omega\) and power \(P\) transports angular momentum about its own axis at the rate \(P/\omega\) — one unit \(\hbar\) for every \(\hbar\omega\) of energy it carries, and of the sign set by the handedness. Beth suspended a birefringent half-wave plate from a quartz fibre inside an evacuated enclosure and measured the steady torque exerted on it when the plate reversed the handedness of the transmitted beam, finding the predicted magnitude and the predicted reversal of sign with the handedness of the incident light [Beth:1936]. This is a mechanical measurement of a property of light that has no analogue in the mechanics of a scalar wave.
The angular momentum of a circularly polarized wave: the flux of angular momentum computed from the Maxwell stress tensor for a beam of finite transverse extent, its ratio to the energy flux, and the torque this predicts on a half-wave plate that reverses the handedness; and the identification of that ratio with one unit of the reduced Planck constant per quantum, which is what makes the result a statement about the spin of the photon rather than about the field alone.
Compton scattering
The 1923 experiment
[Reserved: Compton's measurement of molybdenum K\(\alpha\) X-rays (\(17.5\,\mathrm{keV}\), \(71\,\mathrm{pm}\)) scattered from graphite and analysed with a Bragg spectrometer [Compton:1923]; the scattered radiation contains, besides the unshifted line, a component displaced to longer wavelength by an amount depending on the scattering angle alone and not on the wavelength, the material, or the intensity. Classical Thomson scattering (Radiation and Scattering of Electromagnetic Waves, [Jackson:1999]) predicts no shift whatever, since a driven electron re-radiates at the driving frequency. Apparatus, angles, and the observed shift with uncertainties to be tabulated.]
X-rays scattered from a light element contain, besides a component at the incident wavelength, a component displaced to longer wavelength. The displacement depends on the scattering angle \(\theta\) and on nothing else — not on the incident wavelength, not on the scattering material, not on the intensity —
where the Compton wavelength \(\lambda_{\text{C}}\) is \(2.42631\,\mathrm{pm}\) [Mohr:2025]. Compton measured the shift on the molybdenum K\(\alpha\) line scattered from graphite, analysing the scattered radiation with a Bragg crystal spectrometer [Compton:1923]. Classical scattering from a charge driven by the incident wave predicts no shift at all, at any angle, because a driven oscillator re-radiates at the frequency that drives it [Jackson:1999].
Derivation. Treat the process as an elastic collision between a quantum of energy \(h\nu\) and momentum \(h\nu/c\) — the pair used already in Phenomenon 69.3 — and a free electron of mass \(m\) at rest, and impose the relativistic conservation laws of Relativistic Dynamics. Let the scattered quantum have frequency \(\nu'\) and travel at an angle \(\theta\) to the incident direction. Momentum conservation gives the electron momentum \(\vect{p}_{\text{e}}=\vect{p}-\vect{p}\,'\), whose square is
and energy conservation gives \(E_{\text{e}}=h\nu-h\nu'+mc^{2}\). Substituting both into the mass-shell relation \(E_{\text{e}}^{2}=p_{\text{e}}^{2}c^{2}+m^{2}c^{4}\),
Expanding the square on the left and cancelling \(h^{2}\nu^{2}+h^{2}\nu'^{2}+m^{2}c^{4}\) from both sides leaves
that is \(mc^{2}\left(\nu-\nu'\right)=h\nu\nu'\left(1-\cos\theta\right)\). Dividing through by \(mc\,\nu\nu'\) and using \(\lambda=c/\nu\),
which is Equation (69.4). The result carries no adjustable parameter and no property of the target, exactly as observed; Debye published the same kinematics independently and within weeks [Debye:1923]. The unshifted component comes from electrons bound too tightly to recoil freely, for which \(m\) must be replaced by the mass of the whole atom, some five orders of magnitude larger, and the shift becomes unobservable.
∎Kinematics and the Compton wavelength
[Reserved: treat the process as an elastic collision between a quantum of energy \(h\nu\) and momentum \(h\nu/c\) and a free electron at rest, and impose the relativistic conservation laws of Relativistic Dynamics. The result is \(\Delta\lambda= \lambda_{\text{C}}(1-\cos\theta)\) with the Compton wavelength \(\lambda_{\text{C}}=h/m_{\text{e}}c=2.42631\,\mathrm{pm}\), a pure prediction with no adjustable parameter. Why the unshifted line survives (scattering from tightly bound electrons, for which the recoiling mass is the whole atom); the recoil electrons, seen in a cloud chamber at the predicted energies and angles; and the identification of \(\lambda_{\text{C}}\) as the length scale at which The Dirac Equation takes over from single-particle mechanics.]
Debye's independent derivation
[Reserved: Debye published the same kinematics from the same hypothesis within weeks of Compton and without knowledge of the experiment [Debye:1923], a fact usually omitted; the relation is occasionally called the Compton–Debye formula on the continent. The section records the double attribution because the treatise's citation rule is to name the source that contains the claim, and here there are two.]
The Klein–Nishina cross-section
[Reserved: the kinematics fix the shift but not the rate. The angular distribution follows from the Dirac equation (The Dirac Equation) as the Klein–Nishina formula [Klein:1929a], one of the earliest quantitative successes of relativistic quantum mechanics; its low-energy limit is the Thomson cross-section \(\sigma_{\text{T}}=6.6524587\times 10^{-29}\,\mathrm{m}^{2}\) [Mohr:2025] and its high-energy behaviour falls as \(\log(E)/E\). Compton scattering as the dominant photon interaction in matter between about \(100\,\mathrm{keV}\) and \(10\,\mathrm{MeV}\), hence its role in detector physics and in the inverse-Compton emission of Cosmic Rays and Astroparticle Physics.]
Coincidence: the last classical alternative
The Bohr–Kramers–Slater proposal
[Reserved: the 1924 attempt by Bohr, Kramers and Slater [Bohr:1924] to keep a continuous classical field by surrendering strict conservation: energy and momentum would hold only statistically, the atom being coupled to a “virtual” radiation field that fixes transition probabilities. It is the sharpest counter-theory the light quantum ever faced, and it makes a definite, falsifiable prediction — that a scattered quantum and its recoil electron are not correlated event by event.]
Bothe–Geiger coincidences
[Reserved: Bothe and Geiger placed two needle counters, one sensitive to scattered X-rays and one to recoil electrons, on opposite sides of a hydrogen-filled scattering chamber, and recorded their discharges on a moving photographic tape [Bothe:1925]. Coincident events appeared at a rate far above chance, with a resolving time of about \(10^{-4}\,\mathrm{s}\), so energy and momentum balance in each individual act and the statistical proposal of Section 69.5.1 is excluded. This is the founding use of the coincidence method, for which Bothe was awarded the Nobel Prize, and it is the technique behind every particle detector in Part XI — Quantum Field Theory and the Standard Model.]
In Compton scattering the scattered quantum and the recoil electron appear together, in each single act, at the angles and energies that Equation (69.4) and momentum conservation assign to that act — not merely on the average over many acts. Bothe and Geiger placed one needle counter sensitive to scattered X-rays and one to recoil electrons on opposite sides of a hydrogen-filled scattering chamber and recorded coincidences at a rate far above chance with a resolving time of about \(10^{-4}\,\mathrm{s}\) [Bothe:1925]. Compton and Simon photographed both the recoil electron track and the track of a second electron ejected by the scattered quantum in the same cloud chamber, and found the two directions correlated as the kinematics require, within the measurement errors [Compton:1925]. This excludes the one serious counter-theory, in which conservation would hold only statistically [Bohr:1924], and it was withdrawn at once.
The angular correlation implied by the collision kinematics: the recoil-electron direction as a function of the scattering angle of the quantum, obtained from the same conservation laws that give the wavelength shift; the rate of accidental coincidences to be subtracted from the observed rate, given the counting rates and the resolving time; and the contrasting prediction of statistical-only conservation, under which the two directions are independent and the coincidence rate is the accidental rate alone.
Compton–Simon cloud-chamber pairs
[Reserved: the complementary experiment [Compton:1925], in which a cloud chamber records the recoil electron's track and the track of a second electron ejected by the scattered quantum, so that both directions are measured in the same event; the angular correlation follows the kinematics of Section 69.4.2 within the measurement errors. Together with Section 69.5.2 this closed the question in 1925, and Bohr withdrew the proposal at once — a model of how a theory should be abandoned.]
The photon as a countable object
Anticorrelation at a beam splitter
[Reserved: the modern decisive experiment. Grangier, Roger and Aspect prepared single quanta by heralding on the second photon of a calcium radiative cascade, sent them onto a beam splitter and measured the joint detection rate at the two outputs [Grangier:1986]; the observed anticorrelation parameter is well below the value that any classical field, however weak, can produce, while the same source in a Mach–Zehnder interferometer still shows full-visibility interference. One indivisible quantum, taking both paths: the statement of Interpretations (Evidence-Anchored) rests on data of this kind and not on the photoelectric effect.]
Light prepared one quantum at a time and sent onto a beam splitter is detected in one output arm or in the other, and not in both. Writing \(\alpha\) for the ratio of the joint detection rate at the two outputs to the product of the two singles rates, every classical field — however weak, and whatever its statistics — obeys \(\alpha\geq1\), because the joint rate is an average of a squared intensity and the Cauchy–Schwarz inequality bounds it from below. Grangier, Roger and Aspect prepared single quanta by heralding on the partner photon of a calcium radiative cascade and measured \(\alpha=0.18\pm0.06\), far below the classical bound; the same source in a Mach–Zehnder interferometer nevertheless produced interference of full visibility [Grangier:1986].
The two bounds that make the measurement decisive: the classical inequality on the normalised joint detection rate, from the Cauchy–Schwarz inequality applied to the intensity correlations of any positive classical intensity distribution; and the quantum prediction that a one-quantum state incident on a beam splitter gives a vanishing joint rate, together with the correction for the imperfect heralding and the accidental coincidences of a real source.
Counting photons
[Reserved: photomultipliers, avalanche photodiodes and transition-edge sensors as devices whose output is a count; the photon-number distribution of thermal, coherent and squeezed light, distinguished by their second-order correlation and developed in Quantum Optics and the Photon; single-photon sources and detectors as the working hardware of the Bell tests of Experiment: Bell Tests. Historical footnote: the word “photon” is Lewis' [Lewis:1926], coined for an entity quite unlike the one it now names.]
Creation and annihilation of quanta
Pair production and annihilation
[Reserved: above \(2m_{\text{e}}c^{2}=1.022\,\mathrm{MeV}\) a photon in the Coulomb field of a nucleus can convert into an electron and a positron; the discovery of the positron in cosmic rays by Anderson [Anderson:1933] and the first photographs of pairs by Blackett and Occhialini [Blackett:1933], who identified the process as the materialization of a quantum. Annihilation into two back-to-back \(511\,\mathrm{keV}\) quanta, the basis of positron emission tomography and the cleanest demonstration that photon number is not conserved. The experiment chapter is Experiment: The Positron, and the theory is the Dirac equation of The Dirac Equation.]
The photon in quantum electrodynamics
[Reserved: what a photon actually is in the settled theory — an excitation of the quantized electromagnetic field, first written down by Dirac [Dirac:1927], a massless spin-one particle with two helicity states and no rest frame, created and destroyed by field operators rather than moved about. Present experimental limits on the photon mass and charge [Navas:2024], quoted here as the numbers that make the idealization exact for every purpose in this treatise. The full development is Canonical Quantization of Fields and Quantum Electrodynamics and Renormalization; the non-technical account in [Feynman:1985] remains the best statement of what the theory says.]