. Write expression only for impact parameter, distance of closest approach and angle of scattering for Ruther fords a- particle experiment.

In Rutherford's alpha-particle scattering experiment, the impact parameter, distance of closest approach, and angle of scattering are crucial concepts that help us understand the interactions between alpha particles and atomic nuclei. Let's break down each of these expressions clearly.

Impact Parameter

The impact parameter, denoted as b, is defined as the perpendicular distance from the center of the nucleus to the initial trajectory of the alpha particle. It can be expressed in terms of the scattering angle (θ) and the distance of closest approach (r_min) using the following relationship:

b = r_min * sin(θ)

This equation indicates that the impact parameter is directly related to how far the alpha particle's path is from the nucleus at its closest approach, influenced by the scattering angle.

Distance of Closest Approach

The distance of closest approach is the minimum distance between the alpha particle and the nucleus during the scattering process. This distance can be derived from the conservation of energy and is given by:

r_min = \frac{Z_1 Z_2 e^2}{4 \pi \epsilon_0 K}

In this formula:

• Z₁ and Z₂ are the atomic numbers of the alpha particle and the target nucleus, respectively.
• e is the elementary charge.
• ε₀ is the permittivity of free space.
• K is the kinetic energy of the alpha particle.

This expression shows that the distance of closest approach depends on the charges of the interacting particles and the energy of the alpha particle.

Angle of Scattering

The angle of scattering, denoted as θ, is the angle at which the alpha particle is deflected after interacting with the nucleus. The relationship between the impact parameter and the angle of scattering can be derived from the geometry of the scattering process. For small angles, it can be approximated as:

θ = \tan^{-1} \left(\frac{r_min}{b}\right)

This equation indicates that the angle of scattering is influenced by both the distance of closest approach and the impact parameter, illustrating how the trajectory of the alpha particle changes due to the electrostatic forces exerted by the nucleus.

Summary

To summarize, in Rutherford's experiment:

• The impact parameter b is related to the scattering angle and the distance of closest approach.
• The distance of closest approach r_min is determined by the charges of the particles and the kinetic energy of the alpha particle.
• The angle of scattering θ can be expressed in terms of the impact parameter and the distance of closest approach.

These expressions provide a comprehensive understanding of the dynamics involved in alpha particle scattering, revealing insights into atomic structure and interactions.