How this instrument works
Arthur Compton spent 1922 firing molybdenum Kα X-rays of 70.9 picometres at a block of graphite in his laboratory at Washington University in St. Louis, catching what bounced off with a calcite Bragg spectrometer. Classical wave theory promised scattered radiation identical in wavelength to incident radiation. He found two lines rather than one: an unchanged peak, and beside it a second, longer-wavelength peak that slid further out as he swung his detector to wider angles. Treating each X-ray as a particle carrying momentum h/λ, then conserving energy and momentum through a two-body collision with one loose electron, reproduced that slide exactly. He published in May 1923; Peter Debye reached an identical result independently within months. A Nobel Prize followed in 1927, and physicists who had spent two decades resisting light quanta ran out of room to argue.
What makes this relation strange is that its shift is absolute, not fractional. Δλ = λ_C(1 − cos θ) hands back a length fixed by electron mass alone — zero straight ahead, one Compton wavelength of 2.4263 pm at right angles, two of them straight back. Nothing about your incoming photon enters anywhere. What varies is whether anyone can notice. Take 2.43 pm off green light at 550 nm and you have altered it by four parts per million, which is why no optics bench ever stumbled onto this. Take it off Compton's 70.9 pm X-rays and a line moves 3.4 per cent, plain on a spectrometer and hard to explain away. Take it off a 511 keV annihilation photon, whose wavelength is 2.43 pm to begin with, and a single sideways bounce costs half its energy.
Two assumptions prop up this formula: electrons are free, and they start at rest. In matter, neither quite holds. Inner-shell electrons bound tightly enough recoil together with their entire atom, and swapping m_e for a carbon nucleus shrinks predicted shift by a factor near 22 000 — that is Compton's unmodified line, sitting where classical theory insisted everything should sit. Outer electrons genuinely move, and their momentum smears each shifted peak into a Compton profile hundreds of electronvolts wide: a nuisance for spectroscopists, a gift for anyone mapping momentum distributions inside solids. Nor does this expression say how often scattering happens at a given angle; that belongs to Klein and Nishina, 1929. Energy can also travel in reverse. Let a photon meet an electron already moving at relativistic speed and it departs richer than it arrived — inverse Compton scattering, responsible for that Sunyaev–Zel'dovich dent hot cluster gas presses into the cosmic microwave background.
- Enter Scattering angle — how far a photon turned from its original path, not the direction a recoiling electron takes. Degrees by default; switch to radians if your working already runs in them.
- Read Wavelength shift in metres, then switch its unit to nanometres for comparison against atomic spacing: at 90 degrees it reads 0.0024263102 nm.
- Add that shift to your incident wavelength to get the scattered one. Only Δλ comes back, because Δλ truly does not care what you sent in.
- Bracket your result by sweeping the angle: 0 shifts nothing at all, 90 degrees gives one Compton wavelength, 180 degrees gives exactly two, and nothing gives more.
Worked example — Compton's 90-degree anchor point
Set Scattering angle to 90 degrees. Internally that becomes 1.5707963267949 radians, cos θ collapses to exactly zero, and the bracket (1 − cos θ) is precisely 1. Wavelength shift therefore returns λ_C bare: 2.4263102387 × 10⁻¹² m, or 0.0024263102 nm once you change units.
Drop that figure back into Compton's own apparatus. His molybdenum Kα line arrived at 70.9 pm and left sideways at 73.3 pm — a rise of 3.4 per cent, wide enough for a calcite crystal to resolve without argument. In energy terms 17.48 keV went in, 16.91 keV came out, and 0.57 keV walked off with a recoil electron aimed forward and to one side.
Ninety degrees anchors this whole relation because no trigonometry survives it. Whatever appears there is h/(m_e·c) and nothing else, so a discrepancy indicts your constants rather than your arithmetic. Swing round to 180 degrees and the answer doubles to 4.8526204774 × 10⁻¹² m, which is the largest shift one electron can impose on any photon whatsoever.
Questions
Why is the shift independent of the incoming wavelength?
Because momentum conservation against a stationary free electron fixes recoil in absolute terms. Whatever arrives, an electron bites out a length set purely by its own mass — up to 4.85 pm, never more. What changes is what that bite is worth. Against 550 nm green light it amounts to four parts per million and no instrument has ever resolved it; against a 2.43 pm gamma ray it doubles the wavelength and halves the energy in one collision. So this effect looks invisible in optics and dominant in nuclear work, though its underlying number is identical.
Should I enter degrees or radians?
Either — the field carries a unit selector and defaults to degrees. Internally the cosine takes radians, so 90 degrees becomes 1.5707963267949. Watch for the classic slip: typing 90 with radians selected. Then cos(90 rad) = −0.448, the bracket becomes 1.448, and out comes 3.51 pm. That answer is wrong yet sits innocently inside the legal band between zero and 4.85 pm, so nothing looks obviously broken. Check the unit before trusting anything near the middle of the range.
How do I convert a wavelength shift into energy lost?
Compute scattered wavelength λ′ = λ + Δλ, then take E = hc/λ for both and subtract. For gamma-ray work a direct energy form is quicker: E′ = E / [1 + (E/511 keV)(1 − cos θ)], where 511 keV is the electron rest energy. Caesium-137 emits 662 keV photons; backscattered at 180 degrees they emerge at 184 keV, handing 478 keV to the electron. That 478 keV limit is the Compton edge, a sharp cliff visible in every sodium iodide spectrum, and 184 keV marks a backscatter bump beneath it.
Why does an unshifted line appear alongside the shifted one?
Some photons scatter off electrons bound so tightly that the whole atom recoils as one. Replace m_e in that denominator with an atomic mass — carbon runs about 22 000 electron masses — and predicted shift falls to roughly 0.0001 pm, far below what any spectrometer resolves. Compton's graphite plates show both peaks side by side for exactly this reason. Their ratio tracks atomic number: light elements, loosely held electrons, strong modified line; heavy elements favour the unmodified one.
When does this formula stop being accurate?
Once a target electron is neither free nor still. Orbital motion Doppler-broadens the shifted peak into a Compton profile hundreds of electronvolts wide, which spectroscopists must deconvolve and solid-state physicists deliberately measure. Binding energy also intrudes below roughly 10 keV, where photoelectric absorption dominates anyway and clean Compton peaks grow scarce. At the opposite extreme, a fast-moving electron reverses that trade and hands energy to its photon instead — inverse Compton scattering, which is how relativistic plasma manufactures X-rays from starlight.
Is the Compton wavelength the size of an electron?
No — it is a scale, not a radius, and three nearby lengths get muddled constantly. The Compton wavelength is 2.426 pm. Divide by 2π and you get the reduced Compton wavelength, 386.16 fm, which is what appears in Dirac's equation and marks where pair creation starts to matter. Different again is the classical electron radius, 2.818 fm. Meanwhile Bohr's radius is 52.9 pm, some twenty-two times larger than λ_C. Scattering experiments still show no measurable electron size down to about 10⁻¹⁸ m.