Explain the distance of closest approach and impact parameter.

The distance of closest approach is a concept that refers to the minimum distance an α-particle can approach the center of a nucleus when it is directed head-on toward the nucleus. This is a critical concept in nuclear physics and describes the point at which an α-particle comes closest to the nucleus before being repelled due to electrostatic forces. Understanding the distance of closest approach is essential for explaining the behavior of particles in nuclear reactions, such as Rutherford's scattering experiment.

Distance of Closest Approach

Imagine an α-particle being directed straight toward the nucleus of an atom. As the α-particle approaches the nucleus, the positively charged nucleus exerts a repulsive force on the α-particle. This repulsion causes the kinetic energy of the α-particle to be converted into electrostatic potential energy, gradually slowing the particle down. At a certain point, the α-particle's velocity becomes zero, and it stops for an instant. It then reverses its direction and moves away from the nucleus, scattering at an angle of 180°. This point of zero velocity, where the α-particle is closest to the nucleus, is called the distance of closest approach.

At the distance of closest approach, all the initial kinetic energy of the α-particle is completely transformed into electrostatic potential energy. Mathematically, the relationship between the initial kinetic energy and the electrostatic potential energy at this distance can be expressed as:

1/2 mv² = (1 / (4πε₀)) * (Ze² / r₀)

Where:

• m is the mass of the α-particle
• v is the velocity of the α-particle
• Ze is the charge of the nucleus (Z is the atomic number)
• r₀ is the distance of closest approach
• ε₀ is the permittivity of free space

Rearranging this equation gives us an expression for the distance of closest approach as:

r₀ = (1 / (4πε₀)) * (Ze² / Kα)

Where Kα is the initial kinetic energy of the α-particle, represented as:

Kα = 1/2 mv²

Impact Parameter

The impact parameter is another important concept in scattering theory. It refers to the perpendicular distance from the velocity vector of the α-particle to the center of the nucleus when the particle is far away from the nucleus. The impact parameter is denoted by the symbol b.

Rutherford derived a mathematical relation between the impact parameter b and the scattering angle θ, which describes how the angle of deflection changes as a result of the α-particle's interaction with the nucleus. The relationship is given by:

b = (Ze² / (4πε₀)) * (cot(θ / 2)) / (1/2 mv²)

This formula helps in understanding how the impact parameter influences the scattering behavior of the α-particle. The following points summarize the effect of b on the scattering angle:

• If the impact parameter b is large, the scattering angle θ will be small, meaning the α-particle will suffer a small deflection.
• If the impact parameter b is small, the scattering angle θ will be large, meaning the α-particle will suffer a large deflection.
• If b = 0, the scattering angle θ = 180°, indicating that the α-particle is directed straight at the nucleus and retraces its path, which corresponds to the distance of closest approach.

Conclusion

The distance of closest approach is a key concept in understanding the behavior of charged particles like α-particles when they interact with atomic nuclei. It represents the point where the α-particle's kinetic energy is entirely converted into potential energy due to the repulsive forces of the nucleus. The impact parameter, on the other hand, determines the nature of the scattering event, influencing the angle through which the α-particle is deflected. Both of these concepts play a crucial role in nuclear physics and are integral to explaining experimental results like Rutherford's gold foil experiment.