When high-energy photons collide with electrons, the photons can lose energy and change direction in a process known as Compton scattering. This effect, first observed by Arthur Compton in 1923, provided early evidence for the particle-like behavior of light. In classical theory, light waves scattering from a free electron would not change wavelength. However, Compton discovered that X-rays scatter with a longer wavelength than they had initially, and the shift depends on the angle of scattering. The phenomenon supported the idea that photons carry discrete energy $E=h\mathrm{\backslash nu}$, where $h$ is Planck’s constant and $\mathrm{\backslash nu}$ is frequency.

Compton derived a simple expression for the change in wavelength. The shift $\mathrm{\backslash Delta\backslash lambda}$ between the incident and scattered photon equals

$\mathrm{\backslash Delta\backslash lambda}=\mathrm{\backslash lambda\backslash \_C}(1-\backslash cos\mathrm{\backslash theta})$

where $\mathrm{\backslash theta}$ is the scattering angle and $\mathrm{\backslash lambda\backslash \_C}$ is the electron’s Compton wavelength $\mathrm{\backslash lambda\backslash \_C}=\frac{h}{\mathrm{m\backslash \_e}c}$. Numerically, $\mathrm{\backslash lambda\backslash \_C}$ is about 0.002426 nm. The scattered wavelength is then $\mathrm{\backslash lambda\text{'}}=\mathrm{\backslash lambda}+\mathrm{\backslash Delta\backslash lambda}$. This calculator implements that formula so you can estimate how much a photon’s wavelength increases for any scattering angle.

The Compton effect is most pronounced for photons in the X-ray or gamma-ray region, where wavelengths are comparable to the electron Compton wavelength. At optical wavelengths, the shift is extremely small. Nevertheless, understanding Compton scattering is crucial in fields ranging from medical imaging to astrophysics. It underlies the production of secondary photons in detectors and helps scientists interpret X-ray spectra from cosmic sources like pulsars and black holes.

To use this calculator, enter the incident wavelength in nanometers and the scattering angle in degrees. The script computes $\mathrm{\backslash Delta\backslash lambda}$ using the formula above and displays both the shift and the final wavelength. Angles near 0° correspond to forward scattering with minimal change, while angles close to 180° give the largest possible shift—twice the Compton wavelength.

The concept of photon momentum is key to understanding Compton’s formula. Even though photons have no rest mass, they carry momentum $p=\frac{h}{\mathrm{\backslash lambda}}$. When a photon collides with an electron, conservation of energy and momentum dictate how both particles move afterward. Solving these conservation equations yields the wavelength shift equation above. The electron recoils with a kinetic energy that depends on $\mathrm{\backslash theta}$ as well, though this calculator focuses solely on the photon’s new wavelength.

Historically, Compton’s discovery had a major impact on modern physics. It convinced many skeptical scientists that light behaves as a stream of particles under certain circumstances—a critical step toward quantum mechanics. The experiment involved scattering X-rays off graphite and carefully measuring the wavelengths of the scattered photons at various angles. The systematic increase in wavelength, dependent on angle, was unmistakable. Compton received the Nobel Prize in Physics in 1927 for this work.

In laboratory settings, Compton scattering enables determination of electron densities and material composition. In astrophysics, high-energy photons from cosmic sources scatter off electrons in interstellar gas and high-energy plasmas, subtly shifting the observed spectra. Gamma-ray telescopes and X-ray observatories must account for these effects to deduce the physics of distant objects. Even in medical imaging, scattering influences the design of protective shielding and the interpretation of diagnostic scans.

Beyond a single scattering event, multiple Compton scatterings can create a broad distribution of photon energies. This Compton continuum is visible in gamma-ray spectroscopy, where detectors pick up not only the photopeak from photons that deposit all their energy but also a smear of lower energies from those that scatter and exit the detector. Modeling this distribution often involves simulations based on the same underlying equation implemented in this calculator.

Although the basic Compton formula assumes the electron is free and initially at rest, real materials contain bound electrons. For high enough photon energies, the binding energy is negligible, and the free-electron approximation holds. At lower energies, modifications accounting for atomic binding may be necessary. Nevertheless, the simple formula remains remarkably accurate for many practical applications.

By experimenting with different angles and wavelengths in this calculator, you can see firsthand how scattering redistributes photon energy. This intuition helps in designing detectors, interpreting spectra, and appreciating how fundamental principles of quantum physics manifest across the universe.