Compton Scattering Calculator

Background.

When X-rays bounce off the electrons in a light element such as carbon, part of the scattered beam comes back with a longer wavelength than it went in with. Classical wave theory has no room for that: a wave that shakes a free charge should be re-radiated at the frequency it arrived with. Arthur Compton measured the shift in 1923 and explained it by treating the X-ray as a particle carrying momentum h/λ, colliding with an electron the way one billiard ball strikes another. The electron recoils and takes some of the photon's energy, so the photon leaves with less energy and a longer wavelength. The result was among the strongest early evidence that light comes in quanta, and it won Compton a share of the 1927 Nobel Prize in Physics.

The striking feature is that the size of the shift depends on nothing but the angle. Conserving energy and momentum gives Δλ = (h/mₑc)(1 − cos θ), where θ is the angle between the photon's old and new directions. The constant h/mₑc, about 2.426 picometres, is the Compton wavelength of the electron. A photon that carries straight on at 0° keeps its wavelength; one turned through 90° gains exactly one Compton wavelength; one sent straight back gains two, 4.853 pm.

Whether a fixed handful of picometres matters depends on the wavelength you start with. Visible light at 500 nm changes by less than one part in 100,000 even at the largest angle, which is why the effect never shows up in everyday optics. A 100 pm X-ray scattered through 90° lengthens by about 2.4%. For the gamma rays used in medicine and radiation detection, whose wavelengths are only a picometre or two, the shift can exceed the original wavelength.

The page returns Δλ in metres for any angle from 0° to 180°, assuming the electron is free and at rest. No incident wavelength is needed.

What is compton scattering calculator?

Compton scattering is the scattering of a photon by a charged particle, usually an electron, in which the photon hands over part of its energy and momentum to the particle. The Compton shift Δλ is the resulting increase in the photon's wavelength. For an electron that is free and initially at rest it equals the electron's Compton wavelength, h/mₑc, multiplied by (1 − cos θ), so it runs from zero at θ = 0° to twice the Compton wavelength at θ = 180°. The result is given in metres; one picometre is 10⁻¹² m.

How to use this calculator.

1. 01Enter the Scattering angle in degrees, measured between the direction the photon travelled before the collision and the direction it travels after: 0° means undeflected, 180° means sent straight back.
2. 02Keep the angle between 0° and 180°; larger values are refused. A deflection described as 200° one way is the same geometry as 160° the other, so enter 160.
3. 03Read the Compton wavelength shift in metres, and multiply by 10¹² to express it in picometres, the unit most textbooks use.
4. 04To get the scattered wavelength, add the shift to the wavelength of the photon you started with.
5. 05Check the Model scope note before comparing with a measured spectrum: electrons bound in atoms, their motion and repeated scattering all change what a detector records.

The formula.

Δλ = (h / mₑc) × (1 − cos θ). Here h is the Planck constant, mₑ the rest mass of the electron and c the speed of light; together they form the electron's Compton wavelength, which the code takes as the CODATA value 2.42631023538 × 10⁻¹² m. Since the 2019 SI revision h and c have exact defined values, so the only uncertainty in that constant comes from the measured electron mass, and it is far too small to matter here. All the variation sits in the angular factor (1 − cos θ): it is 0 at 0°, 0.5 at 60°, 1 at 90° and 2 at 180°, so the shift grows slowly at small angles and fastest around 90°. The equation follows from conserving relativistic energy and momentum in a two-body collision, and the mass in the denominator is why scattering from anything heavier than an electron produces a far smaller shift. Angles are converted to radians before the cosine is taken.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Compton AH. A Quantum Theory of the Scattering of X-rays by Light Elements. Physical Review. 1923.
2. [2]OpenStax, University Physics Volume 3, section 6.3, The Compton Effect.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.