Notes : Alpha-Particle Trajectory and Impact Parameter

Alpha-Particle Trajectory and Impact Parameter

Class 12 Physics | Chapter 12: Atoms

12.2.1 Alpha-Particle Trajectory

The trajectory (path) of an alpha-particle depends upon the impact parameter (b).

Definition of Impact Parameter

The impact parameter (b) is the perpendicular distance between the centre of the nucleus and the initial direction of motion (velocity vector) of the alpha-particle when it is far away from the nucleus.

A beam of alpha-particles contains particles having different values of impact parameter. Therefore, they are scattered through different angles. However, all alpha-particles possess nearly the same kinetic energy.

Effect of Impact Parameter on Scattering

1. Small Impact Parameter (Small b)

• The alpha-particle passes very close to the nucleus.
• Strong electrostatic repulsion acts between the positively charged nucleus and alpha-particle.
• The scattering angle (θ) becomes large.
• For a head-on collision, the alpha-particle rebounds back.

Result: Large deflection.

2. Large Impact Parameter (Large b)

• The alpha-particle passes far from the nucleus.
• The electrostatic force is comparatively weak.
• The scattering angle (θ) is small.

Result: Small deflection or nearly undeviated path.

Important Conclusion from Rutherford's Experiment

Only a very small fraction of alpha-particles were scattered through large angles or bounced back.

This led Rutherford to conclude that:

• Most of the space inside an atom is empty.
• Almost the entire positive charge and mass of the atom are concentrated in a very small central region called the nucleus.

Rutherford scattering also provides an upper limit to the size of the nucleus.

12.2.2 Alpha-Particle Trajectory and Impact Parameter

Definition of Scattering Angle

The scattering angle (θ) is defined as the angle between:

• the initial direction of the alpha-particle, and
• the final direction after scattering.

Rutherford's Formula for Impact Parameter

The relation between impact parameter and scattering angle is:

$$b=\frac{Ze^2}{4\pi\varepsilon_0(\frac{1}{2}mu^{2})}\cot\left(\frac{\theta}{2}\right)$$

Hence,

b ∝ cot(θ/2)

Symbols Used

Symbol	Meaning
b	Impact parameter
Z	Atomic number of the target nucleus
e	Electronic charge
θ	Scattering angle
ε0	Permittivity of free space
m	Mass of alpha-particle
u	Initial velocity of alpha-particle
½mu²	Kinetic energy of alpha-particle

Special Cases of Impact Parameter

Case 1: Large Value of b

Since:

b ∝ cot(θ/2)

Large b implies large cot(θ/2). Therefore, the scattering angle (θ) is small.

Conclusion: Alpha-particles passing far from the nucleus undergo small deflections.

Case 2: Small Value of b

Small b implies small cot(θ/2). Therefore, the scattering angle (θ) is large.

Conclusion: Alpha-particles passing close to the nucleus undergo large deflections.

Case 3: Zero Impact Parameter (b = 0)

When the alpha-particle moves directly towards the centre of the nucleus:

b = 0

From Rutherford's relation:

cot(θ/2) = 0

Therefore:

θ/2 = 90°

Hence:

θ = 180°

Conclusion: The alpha-particle retraces its path completely and rebounds backward.

General Rule

As the impact parameter increases, the scattering angle decreases.

Larger the impact parameter, smaller is the scattering angle; smaller the impact parameter, larger is the scattering angle.

Frequently Asked Questions (FAQs)

Q1. What is the impact parameter?

The impact parameter is the perpendicular distance between the nucleus and the initial path of the alpha-particle.

Q2. What happens when the impact parameter is very small?

The alpha-particle experiences strong repulsion and is scattered through a large angle.

Q3. What is the scattering angle for a head-on collision?

For a head-on collision (b = 0), the scattering angle is 180°.

Q4. Why do most alpha-particles pass undeflected?

Because most of the atom is empty space, and only a few alpha-particles come sufficiently close to the tiny nucleus.

Q5. What did Rutherford conclude from the scattering experiment?

He concluded that the atom contains a small, dense, positively charged nucleus where almost all the mass of the atom is concentrated.