Differential Scattering Cross Section of Alpha Particle

The differential scattering cross section measures the probability of an alpha particle being scattered at a specific angle by a target nucleus. The differential cross section is denoted as dσ/dΩ, where dΩ represents the solid angle into which the particle scatters.

Inversely Proportional to the Square of Energy

The differential scattering cross section of an alpha particle is inversely proportional to the square of the energy of the incident particle. This relationship can be derived using the Rutherford scattering formula and the kinetic energy of the alpha particle.

Rutherford Scattering Formula

The Rutherford scattering formula describes the scattering of alpha particles by a target nucleus. It is given by:

dσ/dΩ = (zZe^2 / 4πε₀mv²)² * (1 / sin⁴(θ/2))

where:
- z and Z are the charges of the alpha particle and target nucleus, respectively.
- e is the elementary charge.
- ε₀ is the permittivity of free space.
- m is the mass of the alpha particle.
- v is the velocity of the alpha particle.
- θ is the scattering angle.

Derivation

1. The kinetic energy of the alpha particle can be expressed as:

KE = ½mv²

2. We can substitute the expression for KE into the Rutherford scattering formula:

dσ/dΩ = (zZe^2 / 4πε₀(½mv²))² * (1 / sin⁴(θ/2))

3. Simplifying the equation:

dσ/dΩ = (4z²Z²e⁴ / 16π²ε₀²m²v⁴) * (1 / sin⁴(θ/2))

4. Rearranging the equation:

dσ/dΩ = (z²Z²e⁴ / 4π²ε₀²m²v⁴) * (1 / sin⁴(θ/2))

5. The term (z²Z²e⁴ / 4π²ε₀²m²v⁴) is constant for a given experiment and is denoted as K.

dσ/dΩ = K * (1 / sin⁴(θ/2))

6. Since sin(θ/2) can be approximated as θ/2 for small angles, we can substitute it into the equation:

dσ/dΩ = K * (1 / (θ/2)⁴)

7. Rearranging the equation to highlight the inverse square relationship:

dσ/dΩ = K * (2⁴ / θ⁴)

8. As the scattering angle θ approaches zero, the differential scattering cross section becomes infinite. This is because the probability of scattering in the forward direction increases as the angle decreases.

Conclusion

In conclusion, the differential scattering cross section of an alpha particle is inversely proportional to the square of the energy of the incident particle. This relationship is derived from the Rutherford scattering formula and the kinetic energy of the alpha particle. The inverse square relationship arises due to the scattering angle in the formula and indicates that as the energy of the incident particle increases, the probability of scattering at large angles decreases.