Compton scattering, discovered by Arthur Compton in 1923, is a crucial phenomenon in quantum physics that provides compelling evidence for the particle nature of light (photons). It describes the inelastic scattering of a high-energy photon (typically X-ray or gamma ray) when it collides with a free charged particle, usually an electron.

The collision is treated as a two-body interaction between the photon (behaving like a particle with momentum and energy) and a stationary electron. During the collision, the photon transfers some of its energy and momentum to the electron, causing the photon to recoil at a different angle ($\theta$) and the electron to be ejected.

Because the photon loses energy, its frequency must decrease, and consequently, its wavelength must increase. This increase in wavelength, known as the **Compton shift** ($\Delta\lambda$), is independent of the initial photon wavelength, depending only on the scattering angle, which validates the conservation of energy and momentum principles applied to quantum particles.

The phenomenon is governed by the formula derived from special relativity and quantum theory: $$ \lambda’ – \lambda = \frac{h}{m_e c} \left( 1 – \cos \theta \right) $$

Where:

• $\lambda$: Initial wavelength of the incident photon.
• $\lambda’$: Wavelength of the scattered photon.
• $h$: Planck’s constant ($6.62607015 \times 10^{-34} \text{ J}\cdot\text{s}$).
• $m_e$: Electron rest mass ($9.1093837 \times 10^{-31} \text{ kg}$).
• $c$: Speed of light ($2.99792458 \times 10^8 \text{ m/s}$).
• $\theta$: Scattering angle (the angle between the incident and scattered photon direction).

How to Use the Calculator

1. Enter the Initial Wavelength ($\lambda$): Input the initial wavelength of the photon in **meters**. For typical X-rays, this value will be very small (e.g., $10^{-11}$ or $10^{-12}$). Use scientific notation (e.g., `1.0e-11`) for accuracy.
2. Enter the Scattering Angle ($\theta$): Input the angle in **degrees** at which the photon is scattered, ranging from $0^\circ$ to $180^\circ$.
3. Click ‘Calculate Compton Shift’: The tool will instantly calculate the Wavelength Shift ($\Delta\lambda$) and the final Scattered Wavelength ($\lambda’$).
4. Review Results: The outputs are displayed in scientific notation (meters) to maintain precision, reflecting the tiny scales involved in this quantum interaction.