A Geiger-Marsden experiment, where \(\alpha\) particles are scattered off of a thin gold film, yields an intensity of particles of \(I\left(90^{\circ}\right)=100 .\) counts/s at a scattering angle of \(90^{\circ} \pm 1^{\circ} .\) What will be the intensity of particles at a scattering angle of \(60^{\circ} \pm 1^{\circ}\) if the scattering obeys the Rutherford formula?

The Geiger-Marsden experiment, often referred to as the gold foil experiment, was a pivotal scientific undertaking that led to the overthrow of the plum pudding model of the atom and the establishment of the nuclear model. Conducted by Hans Geiger and Ernest Marsden under the supervision of Ernest Rutherford in the early 1900s, the experiment involved firing alpha particles—helium nuclei—at a thin sheet of gold foil.

Most of the alpha particles passed through the foil, supporting the idea that atoms are mostly empty space. However, some particles were deflected at various angles, and a very few even bounced back toward the source. This suggested the presence of a dense, positively charged nucleus at the center of the atom, around which the negatively charged electrons orbited. This experiment fundamentally changed our understanding of the atomic structure.

Understanding Alpha Particles

Alpha particle scattering is a process that involves alpha particles colliding with a target material and deflecting from their original path. Alpha particles are charged, having two protons and two neutrons, which makes them relatively heavy and positively charged compared to other forms of radiation such as beta particles or gamma rays.

Scattering Phenomena

The interaction of alpha particles with the atoms of the target material is governed by the Coulomb force due to their charge. As they approach the dense nucleus of an atom, the positive charges repel, resulting in the deflection of the alpha particles. The extent of this deflection can vary greatly, from very small angles to nearly 180 degrees, indicating a direct hit to the nucleus. This scattering forms the basis of our understanding of the nuclear structure of atoms.

The angular intensity distribution in the context of particle scattering describes how the scattering intensity varies with respect to the angle of deflection. The intensity, denoted as 'I(θ)', measures how many particles are detected at a given angle per unit time.

According to the Rutherford scattering formula, the intensity of alpha particles scattered by a nucleus follows an inverse proportionality with the fourth power of the sine of half the scattering angle, expressed as ewline ewline \(I(\theta) = \frac{k}{\sin^4{\frac{\theta}{2}}}\) ewline ewline where 'k' is a proportionality constant that is characteristic of the scattering conditions and the specific setup, including the type of target material and the energy of the alpha particles.

The crucial takeaway from this formula is the steep decline in intensity with a decrease in the scattering angle, showing that most alpha particles pass through with little deflection and few are scattered at large angles, thus validating the nuclear model of the atom.