Coulomb scattering cross section using Partial wave equation

The Coulomb scattering cross section, often referred to as the Rutherford cross section, describes the likelihood of a charged particle undergoing a scattering event when interacting with the electric field of a nucleus. This concept is fundamental in the study of nuclear and particle physics, particularly in understanding the scattering of charged particles by atomic nuclei.

The formula for the Coulomb scattering cross section is derived from the Rutherford scattering model and depends on several factors:

1. Particle Charge: The charge of the incident particle is represented by "z" (e.g., the charge of an electron is -1, and the charge of a proton is +1).

2. Kinetic Energy of the Incident Particle: Denoted as "E" for the kinetic energy of the particle.

3. Scattering Angle: The angle through which the particle is scattered is represented by "θ."

4. Nuclear Charge: The charge of the nucleus being scattered off is "Z."

5. Impact Parameter: Denoted as "b," it represents the distance between the initial trajectory of the particle and the center of the nucleus.

The Coulomb scattering cross section is given by the formula:

σ(θ) = B²/ 4k² sin⁴(θ/2)

where σ(θ) is the scattering cross section at a specific scattering angle θ.

The scattering cross section σ(θ) tells you how likely it is for a particle to be scattered through a specific angle θ when interacting with the electric field of a nucleus. The term "sin⁴(θ/2)" in the formula represents the angular dependence of the scattering.

Key points regarding the Coulomb scattering cross section:

1. At small scattering angles (θ close to 0), the cross section is inversely proportional to the fourth power of sin(θ/2). This is indicative of the strong forward scattering of particles in the Rutherford model.

2. At larger angles, the cross section decreases rapidly due to the rapidly decreasing sin(θ/2) term.

3. The screening parameter ε takes into account the shielding effect of the atomic electrons on the nucleus and is determined by the specific atomic structure of the target material.

4. The Coulomb scattering cross section is an idealized model and doesn't account for various factors like quantum mechanical effects and the finite size of nuclei. For more accurate predictions, more advanced models, such as the Mott cross section, may be used.

For central potential,

Equation 1 becomes

Let solution be of the form

This is plane wave in positive z direction with amplitude F(𝜉) such that

Using equation 2

This is equation of hypergeometric function so solution are

In experimental and theoretical nuclear physics, the Coulomb scattering cross section is a fundamental tool for understanding the behaviour of charged particles interacting with atomic nuclei and is especially important in the study of nuclear structure and properties.

This partial wave analysis provides a more detailed understanding of the scattering process by considering contributions from different angular momentum states, and it is particularly useful for Coulomb scattering where long-range interactions play a crucial role.

This note is a part of the Physics Repository.