Atoms - Rutherford's Alpha Scattering Experiment

  • The chapter on atoms is divided into three parts: experiments (Millikan oil drop and Rutherford's alpha scattering), Bohr's model, and X-rays.
  • Newer versions of the textbook do not include the X-ray and Millikan oil drop experiments.

Rutherford's Alpha Particle Scattering Experiment

  • Purpose: To understand the structure of an atom.
  • Alpha particles (positively charged) were bombarded onto an atom, and their paths were observed.
  • Observations:
    • Most alpha particles went straight, indicating that most of the atom is empty space.
    • Some alpha particles suffered minor deviations.
    • Very few (1 in 1000) took a U-turn (deviated by 180 degrees), indicating a positive charge concentrated in a small space at the center of the atom.
  • Conclusions:
    • Most of the atom is hollow.
    • At the center, there is a small, positively charged region called the nucleus.
  • Comparison of Sizes:
    • The atom is much larger than the nucleus (by a factor of 10,000 to 100,000).
    • If the atom were the size of a football ground, the nucleus would be the size of a football.

Geiger-Marsden Experiment & Impact Parameter

  • Impact Parameter (b): The perpendicular distance from the nucleus to the line of velocity of the alpha particle.
    • It determines how close the alpha particle will pass to the nucleus.
    • Smaller impact parameter means the alpha particle passes closer to the nucleus, experiences more force, and deviates more.
    • If the alpha particle goes straight towards the nucleus, the impact parameter is zero.
  • Formula for Impact Parameter:
    • b = {1
      umberofalphaparticles{4 \pi \epsilon0} * \frac{Ze^2}{E} * \cot(\frac{\theta}{2})}
    • b is directly proportional to cot(\frac{\theta}{2}) ($\theta$ is the deviation angle).
  • Formula for Number of Alpha Particles (N) at an Angle ($\theta$):
    • N(\theta) \propto \frac{1}{\sin^4(\frac{\theta}{2})}

Deflection, Scattering, and Distance of Closest Approach

  • Alpha particles are deflected due to the Coulomb repulsive force between them and the positive charges in the atom's nucleus.
  • Distance of Closest Approach:
    • The distance at which the alpha particle is closest to the nucleus, and its velocity is zero at this point.
    • Kinetic energy of the alpha particle is converted into potential energy.
    • KE = PE; \frac{1}{2}mv^2 = k * \frac{q1q2}{r}, where q1 is the charge on the alpha particle, q2 is the charge inside the nucleus, and r is the distance of closest approach.

Experimental Setup & Observations

  • Alpha particles are generated from radioactive substances (e.g., uranium, thorium) placed inside a lead cavity.
  • These particles are directed towards a thin gold foil (target atom).
  • The alpha particle detector measures deviation angles.
  • Observations:
    • Most alpha particles pass through the gold foil without deviation (atom is mostly hollow).
    • Some are deflected at small angles.
    • A very small number are deflected at large angles (or even retraced their path, 180-degree deflection).
    • The nucleus contains a positive charge concentrated in a small space.

Formulas and Relationships

  • N \propto \frac{Q * n * t * Z^2 * e^4}{\sin^4(\frac{\theta}{2})}
    • N is the number of alpha particles scattered at angle \theta.
    • N is directly proportional to Z^2 (atomic number squared) and inversely proportional to \sin^4(\frac{\theta}{2}).

Impact Parameter

  • The perpendicular distance from the nucleus's center to the initial velocity line of the alpha particle.

Numerical Application

  • If 55 alpha particles are scattered at 90 degrees, calculate the number of alpha particles scattered at 60 degrees.
  • Use the relationship N \propto \frac{1}{\sin^4(\frac{\theta}{2})}.
  • N{60} = 4 * N{90}, If N{90} = 55, then N{60} = 220. This is derived from proportionalities between the different angles.