A photon with a momentum, and a collision that proves it

Send a beam of X-rays of one wavelength at a block of graphite and look at what comes off to the side. Some of it has the wavelength it went in with. Some of it is longer, and how much longer depends only on which direction it is being looked at from.

The absence is the whole result. A wave shaken loose from a bound electron re-radiates at the frequency it was shaken at, so a wave picture predicts no change at all; and any picture in which the shift depends on the strength of the binding, or the density of the target, or the intensity of the beam, predicts one but the wrong one. What is measured depends on a single angle, and its scale is set by two constants and the mass of the electron.

Treating it as a collision #

Compton’s move in 1923 was to stop thinking about a wave driving an oscillator and start thinking about two particles hitting each other.

Give the incoming quantum an energy hf — that much was already established by the photoelectric effect — and also a momentum h/λ. Then let it strike a free electron initially at rest and require the two conservation laws to hold, with the electron treated relativistically because it recoils fast.

Everything follows from writing down the books and refusing to look at the middle, which is the standard move for a collision. Momentum is a vector and gives two equations; energy gives a third; and eliminating the electron’s unknown speed and direction between them leaves one relation between the photon’s wavelengths before and after:

$$\Delta\lambda = \frac{h}{m_e c}\,(1-\cos\theta).$$

The constant in front is h/m_ec = 2.426 picometres, the Compton wavelength of the electron. It is not a size and it is not a wavelength the electron has; it is the combination of constants with the dimensions of a length that this collision produces, and it is the scale at which a photon’s momentum becomes comparable to m_ec.

Why nothing else appears in the answer #

The independence is worth taking apart, because each thing that is absent is absent for its own reason.

The incident wavelength does not appear because the shift is additive rather than proportional. A collision transfers momentum; the change in the photon’s momentum for a given deflection is set by the electron’s mass, and converting a momentum change into a wavelength change gives a difference of wavelengths, not a ratio. So a 71-picometre X-ray gains 2.43 picometres at ninety degrees and so does a 7-picometre gamma ray.

That is also why the effect was not seen for two centuries of optics. Visible light has a wavelength around 500 nanometres, so the same 2.43-picometre shift is a relative change of five parts in a million — undetectable with anything available before X-ray spectrometers, and swamped by every other broadening in the apparatus.

The material does not appear because the electron is being treated as free. In graphite the outer electrons are bound by a few electronvolts, and the incident X-ray carries around 17,000 — so on the scale of the collision the binding is negligible and the electron might as well be sitting in space. That approximation is doing real work and it is visible in the data: the unshifted peak that also appears is scattering from electrons that stayed bound, where the recoiling mass is the whole atom rather than one electron, and the shift is smaller by the mass ratio, which for carbon is a factor of 22,000.

The intensity does not appear because the process is one photon meeting one electron. Doubling the beam doubles the count and changes nothing about each event — the same division of labour the photoelectric effect established, showing up again in a completely different measurement.

What the triangle asserts #

The momentum diagram is not decoration. It is the claim, and it is checkable in a way the formula is not.

Three arrows drawn to one scale from the two photons’ wavelengths and the electron’s recoil have to form a closed figure. If they do not, momentum is not conserved and the model is finished. The generator behind the figure computes the electron’s momentum as the vector difference and draws it; the check that runs over the site’s figures then measures the three drawn arrows and requires them to close to under half a pixel, and separately recovers the electron’s kinetic energy from the length of its drawn arrow through the relativistic energy–momentum relation and requires it to match the energy the photon lost.

That second check is the one worth pointing at, because the drawing never used energy conservation. The triangle was built from momentum alone. Energy conservation then holds anyway, to a per cent, which is either a coincidence or evidence that the model is right about both.

The second peak, and what it is for #

A Compton measurement does not produce one line. It produces two: a shifted one and an unshifted one, at every angle, with the ratio between them depending on the target.

The unshifted line is the same collision with a different recoiling mass. An electron tightly bound into an atom cannot be knocked out by a glancing 17-keV photon; what recoils is the whole atom, and putting the atom’s mass into the formula in place of the electron’s shrinks the shift by the mass ratio. For carbon that is a factor of 22,000, which puts the shift at 0.0001 picometres — unmeasurable, and indistinguishable from no shift at all.

Why the heavier target barely moves is ordinary mechanics. A light body striking a much heavier one at rest bounces off with nearly its original speed and hands over almost nothing, and the fraction of momentum transferred falls as the mass ratio grows. That arithmetic is what decides which of the two Compton lines a given electron contributes to: an electron loosely enough bound to recoil alone shifts the wavelength, while one bound tightly enough to drag the whole atom with it is scattering off a mass thousands of times larger and shifts it immeasurably.

So the relative strength of the two lines reports how many of a material’s electrons are loosely enough bound to count as free at that photon energy. In graphite, with four valence electrons out of six, the shifted line dominates. In a heavy element with most of its electrons deeply bound, the unshifted line does. Compton chose carbon deliberately, and the choice was the experiment’s design rather than convenience.

That is also the honest answer to a question the formula invites: at what binding energy does an electron stop counting as free? There is no sharp answer, because both processes always happen — what changes is their relative probability. The free-electron treatment is an approximation whose domain is E_photon ≫ E_binding, and the second peak is the part of the data that lives outside it.

The pressure light exerts #

A momentum of h/λ per photon is small, and the number of photons is not, so light pushes on things. Sunlight at the Earth’s orbit delivers about 4.6 micropascals of pressure to an absorbing surface and twice that to a mirror, which is roughly the weight of a grain of sand spread over a square metre.

Small as it is, it is the dominant force in three settings. It is what holds a star up: radiation pressure in a massive star’s interior contributes a large fraction of the support against gravity, and above a certain mass it wins outright and blows the outer layers away. It is what makes a comet’s ion tail point away from the Sun regardless of which way the comet is travelling. And it is the operating principle of a solar sail, which converts a micropascal into a delta-vee by having no fuel to carry.

Reversed, it is a refrigerator. An atom moving toward a laser tuned slightly below its absorption frequency sees the light Doppler-shifted up into resonance, absorbs a photon and takes h/λ of momentum against its motion — then re-emits in a random direction, which averages to nothing. The net effect is a viscous drag proportional to velocity, and it cools a gas of atoms to microkelvin. Laser cooling is the momentum on this page applied 10⁴ times per second per atom, and it is what makes ultracold atomic physics possible.

The cost: a photon has momentum but no mass #

Giving light a momentum raises an immediate objection. Momentum in mechanics is mv, and light has no mass, so how can it have momentum?

The answer is that p = mv is the low-speed limit of something more general, and the general relation is

$$E^2 = (pc)^2 + (mc^2)^2,$$

which is where mass and energy meet. Setting m = 0 gives E = pc — a massless object whose energy is entirely momentum, moving at exactly c, with no rest frame to be at rest in. Combining that with E = hf and f = c/λ gives p = h/λ directly, so the photon’s momentum is not an extra postulate. It is forced by the two things already known about it.

A photon sits at the end of a relation rather than outside it. A massive particle’s energy rises from its rest energy as its speed climbs, with the momentum term growing without limit while the mass term stays fixed at $mc^2$. The photon is the case where the mass term is zero from the outset: all of its energy is momentum, there is no rest energy for it to sit at, and there is no speed other than $c$ at which it can be found. That is not an exception bolted onto the relation — it is the relation read at $m = 0$, and it is why a thing with no mass can still hit something.

The relation also explains why the shift saturates. Backscattering costs the photon the most energy, but even at 180° it cannot lose all of it: Δλ is bounded by 2h/m_ec, so a long-wavelength photon comes back nearly intact and a very short one loses almost everything. The dividing line is at λ ≈ λ_C, which is a photon energy of 511 keV — the electron’s rest energy, appearing as the scale at which light and matter trade on equal terms.

The theory that gave up conservation to avoid the photon #

The formula fitted the data in 1923 and the interpretation was resisted for another two years, and the resistance produced one of the most interesting wrong theories in physics.

Bohr did not accept light quanta. The wave theory accounted for interference and diffraction, which the quanta did not, and he regarded introducing particles of light as a step backwards. With Kramers and Slater he proposed in 1924 an alternative in which the field remains classical: atoms communicate through a “virtual radiation field”, transitions are induced by it probabilistically, and — this is the remarkable part — energy and momentum are conserved only statistically, over many events, and not in each individual interaction.

That is an enormous thing to give up, and it was given up deliberately, as the price of keeping the field classical. It also makes a prediction that can be tested.

If conservation holds only on average, then in Compton scattering the recoil electron and the scattered photon need not be produced together. The electron would be knocked out at some rate, the radiation scattered at some rate, and the two would be uncorrelated in time and in direction. If conservation holds event by event, they must appear simultaneously and their directions must satisfy the momentum triangle.

Bothe and Geiger tested the timing in 1925 with two point counters — one arranged to detect scattered X-rays, one recoil electrons — and a circuit that recorded whether they fired together. They found genuine coincidences at a rate far above the accidental one: the two particles appear together, within the resolving time of the apparatus.

Compton and Simon tested the directions in the same year, using a cloud chamber so that the recoil electron’s track was visible and, when the scattered photon happened to scatter again, its direction too. The pairs of directions satisfied the Compton relation, event by event.

The theory was abandoned within months, and Bohr wrote that there was nothing to do but give the revolutionary effort as honourable a funeral as possible. Slater, who had never liked the abandonment of conservation, went on to a long career elsewhere.

Two things came out of the episode besides the answer. Coincidence counting, invented to settle it, became the standard technique of nuclear and particle physics — Bothe received a Nobel Prize for the method twenty-nine years later, and every detector that asks whether two things happened together descends from that circuit. And the shape of the argument is worth keeping: an alternative theory was constructed that saved the older picture at the cost of a conservation law, it was taken seriously by the best people in the subject, it made a distinguishable prediction, and it was killed in a year by an experiment it had provoked.

The result nobody could reproduce #

Before any of that, the measurement itself was in dispute for two years, and the dispute is a good illustration of what a difficult experiment looks like from inside.

Compton’s data came from a Bragg spectrometer looking at X-rays scattered from graphite, and the shift he was measuring is a couple of picometres out of seventy — a three per cent change in a wavelength, read from the position of a diffraction peak. It is a demanding measurement with the apparatus of 1922, and it is sensitive to stray scattering from everything in the room.

William Duane’s group at Harvard could not reproduce it. What they found instead was attributed to spurious effects of the apparatus — radiation scattered from the walls of the enclosure and secondary processes in the target — and Duane argued publicly against the quantum interpretation through 1923 and 1924. He was a serious experimentalist with a serious apparatus, and his failure to see the effect was a substantial obstacle to its acceptance.

It was resolved in the ordinary way: Duane’s own group traced the discrepancy to their own arrangement, found the source of the spurious signal, and on repeating the measurement obtained the shift. Duane conceded publicly at the end of 1924.

There is a priority footnote as well. Debye derived the same relation independently, from the same premise, and his paper appeared in April 1923 — a month before Compton’s, though Compton’s had been submitted earlier and was accompanied by the measurements. Debye had no data and never claimed the effect; it is occasionally called the Compton–Debye shift, and the theory was clearly available to anyone willing to grant the photon a momentum.

Compton received the Nobel Prize in 1927, sharing it with C. T. R. Wilson — whose cloud chamber was the instrument that had made the individual recoil electrons visible, and therefore the instrument that settled the argument in the previous section.

Where the billiard-ball picture stops #

Three limits, in increasing order of how badly they break it.

Bound electrons. The derivation assumes a free electron at rest. Real electrons are bound and moving, and the momentum they already have smears the scattered line — Compton profile broadening, which is a nuisance in the original experiment and an instrument in modern ones, because the width of the smeared line measures the electron’s momentum distribution inside the material.

Low energies. As the photon’s energy falls below the binding energies, the free-electron treatment fails entirely and the scattering becomes elastic: Thomson scattering at the low limit, in which the wave picture is exactly right and there is no shift at all. Rayleigh scattering off a molecule is the same regime, and it is why the sky’s colour owes nothing to this page.

High energies. Above 1.02 MeV a photon near a nucleus can convert into an electron and a positron, and the “collision” picture — one particle in, one particle out — stops being the right accounting. Compton scattering, photoelectric absorption and pair production are three regimes of one interaction, dominant in different energy bands, and the boundaries move with the atomic number of the material.

Push the other way, down to quanta of a few electronvolts, and the picture fails entirely. The photoelectric effect operates there and destroys the quantum outright rather than deflecting it; Compton scattering operates on quanta of tens of thousands of electronvolts and merely turns them. The same object and the same constant, separated by four orders of magnitude in energy — which is why a single picture of “light hitting a metal” was never going to cover both, and why the two effects were discovered twenty years apart and argued about separately.

What the figures cannot show #

The curve shows a shift against an angle and hides the fact that most photons are not scattered at all. Scattering is a probabilistic process with a cross-section, and the angular distribution — how likely each angle is — is a separate function, the Klein–Nishina formula, which the drawing says nothing about. A figure of what happens given that something happened is silent on how often it happens.

The geometry figure hides something more basic. It draws one event with definite momenta before and after, which is a legitimate description of an outcome and a misleading description of the process: the outgoing photon does not have a direction until it is detected in one, and the “collision” is not a trajectory. The picture is the bookkeeping of an outcome, drawn as though it were a mechanism.

And neither figure shows the recoil electron’s fate. It leaves with kilo-electronvolts of energy, ionises a track of atoms and stops within microns of where it started — which is the process by which X-rays actually deposit their dose in tissue, and the reason Compton scattering is the dominant interaction in radiotherapy at treatment energies.

Made into an instrument #

Compton scattering is now used in three quite different ways, and the independence that makes the effect striking is exactly what makes it useful.

As a spectrometer. Because the shift depends only on the angle, a detector at a known angle converts a measured wavelength into a known incident wavelength, and the profile’s width reports the target electrons’ momentum distribution. Compton profile measurements are one of the few direct probes of the electron momentum density in a solid.

As an imager. A gamma camera can be built with no collimator at all by measuring the scattering angle in one detector layer and the absorption position in a second, and reconstructing the source direction from the cone the angle defines. Compton cameras trade a heavy, absorbing collimator for arithmetic.

As an astronomical thermometer. The inverse process — a low-energy photon scattering off a fast electron and gaining energy — is how the cosmic microwave background is distorted as it passes through the hot gas in a galaxy cluster. The distortion’s size measures the gas pressure along the line of sight, and it is redshift-independent, which makes it a way of finding clusters at any distance.

Where the ladder goes next #

The rungs from here: the Klein–Nishina cross-section, and how the angular distribution changes shape as the photon energy passes the electron’s rest energy; the inverse effect and its role in high-energy astrophysics; pair production, where the conversion of energy into two masses closes the loop E = mc² opened; and the double-slit experiment done with single photons, which is the experiment this rung’s picture is least able to survive.

The claim to carry forward is what the constants say. A photon’s energy is hf and its momentum is h/λ, and the two are not two assumptions — they are one assumption plus the relation between energy and momentum that relativity already required. The Compton shift is the measurement that tests both at once, and it does so by producing a number in which almost everything about the experiment has cancelled out.