Klein-Nishina Formula

Introduction

The Klein-Nishina formula is an essential result in quantum electrodynamics that describes the angular distribution of photons scattered off a single, free electron. Named after Oskar Klein and Yoshio Nishina, who derived the formula in 1928, it takes into account the relativistic and quantum mechanical effects of the scattering process, particularly in the case of high-energy (or hard) photons.

The Compton Effect

Before diving into the Klein-Nishina formula, it’s essential to understand the underlying physics it describes – the Compton effect. This effect is the change in energy (frequency) and direction of a photon when it scatters off a target particle, such as an electron.

In classical terms, the Compton effect is an inelastic collision between a photon and a target particle, and its analysis can be performed using conservation of energy and linear momentum. However, the classical description fails when the energy of the incident photons becomes comparable to the rest energy of the electron, which is where the Klein-Nishina formula comes into play.

Mathematical Formulation

The Klein-Nishina formula for the differential cross-section of Compton scattering is given by:

Here, denotes the differential cross-section per solid angle , is the classical electron radius, is the energy of the incident photon, is the energy of the scattered photon, and is the scattering angle.

The equation demonstrates that the intensity of scattered radiation depends not only on the scattering angle but also on the energy of the incident photons.

Physical Interpretation

The Klein-Nishina formula predicts two crucial effects:

1. Relativistic Effect: As the energy of the incident photon increases (and becomes comparable to the rest mass energy of the electron), the scattering becomes less probable in the forward direction and more probable in the backward direction.
2. Quantum Effect: For photons with very high energy, the scattered photon’s energy does not decrease with the scattering angle, as would be expected from classical physics. Instead, it reaches a minimum at a certain angle (known as the Compton edge) and then increases for larger scattering angles.

Conclusion

The Klein-Nishina formula has had a significant impact on various fields of physics, including nuclear physics, quantum electrodynamics, and astrophysics. Its correctness and predictive power stand as testament to the success of quantum mechanics in describing the nature of the physical world.

Do you prefer video lectures over reading a webpage? Follow us on YouTube to stay updated with the latest video content!

Want to study more? Visit our Index here!