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Rutherford’s scattering formula fails for very small scattering angles because:
This question has multiple correct options
(A) the gold foil is very thin
(B) the kinetic energy of $\alpha$-particles is very high
(C) the full nuclear charge of the target atom is partially screened by its electron.
(D) there is strong repulsive force between the $\alpha$-particles and nucleus of the target

Last updated date: 17th Jul 2024
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Hint: The Rutherford scattering formula indicates that the number of particles which is deflected by an angle $\theta$ because of the scattering from fixed nuclei is inversely proportional to the fourth power of the sine function of one half the angle of deflection. In this question two options are correct.

Complete step by step answer:
After the failure of the plum pudding model by J.J Thomson, another scientist named Ernest Rutherford did an experiment and gave a new outlook to the atomic structure. The experiment was named as the gold foil experiment. He bombardes the $\alpha$ -particles to the thin sheet of gold foil from a radioactive source in order to study the deflection done by the $\alpha$-particles. A fluorescent screen was fixed which surrounded the gold foil.
The following observations were made after the experiment.
(i) A high volume of $\alpha$-particles passed through the gold foil without getting deflected which represents that the huge amount of space available in the atom is vacant.
(ii) Some of the $\alpha$-particles get deflected from a very small angle which shows that the positive charged particles present in the atom are localized in small space.
(iii) Very few $\alpha$-particles get deflected back by the angle of $180^\circ$which shows that the positive charged particles are present in small amounts in comparison to the total volume of the atom.
From the above observation, Rutherford discovered the atomic structure and named it Rutherford atomic model.
According to the model:
(i) The positively charged particles are concentrated in the atom in a small region known as nucleus.
(ii) The negatively charged electrons revolve around the nucleus in a circular manner. This circular path is called an orbit.
(iii) The negatively charged electrons and positively charged protons are joined together by electrostatic force of attraction.
But Rutherford's scattering formula fails for very small scattering angles because due to the strong repulsive force present between the $\alpha$-particles and the nucleus and also the full nuclear charge of the target atom is partially screened by its electrons.
Therefore the correct option is C and D.

The Rutherford scattering was first referred to as the Coulomb scattering as it depends only on the static electric potential and the minimum distance between the particles.