## Atoms

# Alpha-particle Scattering and Rutherfordâ€™s Nuclear Model of Atom

- According to Ruther fords model, an atom, consists of small and massive central core in which the entire positive charge and almost whole mass concentrated.
- The size of the nucleus is very small (≈ 10
^{−15}m) as compared to the size of the atom (≈ 10^{−10}m). - Total negative charge is equal to the total positive charge and atom is electrically neutral.
- The electrons revolve around nucleus in circular orbits.
- An α-particle is a helium atom from which 2 electrons have been removed.
- Most of α-particles pass through the gold foil or suffer only small deflections.
- A very few α-particles get deflected through 90° or more
- Occasionally an α-particles gets rebounded from gold foil suffering a deflection of nearly 180°.
- The nucleus is surrounded by a cloud of electrons.
- The scattering is proportional to the square of the atomic number of both the incident particle and the target scattering due to the fact that increasing atomic number.
- The distance of closest approach of α-particle \tt r_{0} = \frac{ze^{2}}{\pi \varepsilon_{0} m v^{2}}
- The number of α-particles scattered (N) is directly proportional to thickness of gold foil N ∝ t
- The number of α-particle scattered (N) is directly proportional to square of atomic number (Z) of the foil atoms N ∝ Z
^{2}. - The number of α-particles scattered (N) is inversely proportional to sin
^{4}(θ/2) \tt N \propto \frac{1}{\sin^{4} \theta/2} (θ = scattering angle) - Number of α-particles scattered (N) \tt N = \frac{\theta n t z^{2} e^{4}}{\left(8 \pi \varepsilon_{0}\right)^{2} r^{2} E^{2} \sin^{4} \left(\frac{\theta}{2}\right)} {E = KE, n = number of atoms per unit volume}
- The number of α-particles scattered is inversely proportional to square of KE of α-particles.
- The impact parameter (b) is defined as the perpendicular distance of the velocity vector of the α-particle from the centre of the nucleus.
- Impact parameter \tt b = \frac{1}{4 \pi \varepsilon_{0}} \frac{ze^{2} \cot \theta/2}{\frac{1}{2} mv^{2}}
- For large impact parameter the repulsive force is weak.
- For small impact parameter the repulsive force is large.
- Ruther fords model cannot explain the stability of atom.
- Ruther fords model fail to explain why hydrogen like atoms emit discrete line spectrum.

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1. The trajectory traced by an alpha particle depends on its impact parameter b. Rutherford had analytically calculated the relation between the impact parameter b and the scattering angle θ, given by b=\frac{Ze^{2}\cot\theta/2}{4\pi\varepsilon_{0}K_{\alpha}}

2. At the distance r_{0}, K_{α} = U or \tt \frac{1}{2}\ mv^{2}=\frac{2e.Ze}{r_{0}}\ \frac{1}{4\pi\varepsilon_{0}}