Question

A Geiger-Marsden experiment, in which alpha particles are scattered off a thin gold film, yields an intensity of \(I\left(94.9^{\circ}\right)=853\) counts/s at a scattering angle of \(94.9^{\circ} \pm 0.7^{\circ} .\) What is the intensity (in counts/s) at a scattering angle of \(60.5^{\circ} \pm 0.7^{\circ}\) if the scattering obeys the Rutherford formula?

Short answer

The intensity of scattered alpha particles at a scattering angle of \(60.5^{\circ} \pm 0.7^{\circ}\) is approximately \(4967\) counts/s.

Step-by-step solution

Understand the Rutherford formula

The Rutherford scattering formula is given as: \(I(\theta) = I_0 \frac{1}{\sin^4(\theta/2)}\) where \(I(\theta)\) is the intensity of the scattered particles at an angle \(\theta\), and \(I_0\) is the maximum intensity of the scattered particles.

Determine the relationship between the two intensities

We are given the intensity at \(94.9^{\circ}\) and asked to find the intensity at \(60.5^{\circ}\). We can set up a proportion to find the relationship between the two intensities: \(I(94.9^{\circ}) = 853 = I_0 \frac{1}{\sin^4(94.9^{\circ}/2)}\) \(I(60.5^{\circ}) = I_0 \frac{1}{\sin^4(60.5^{\circ}/2)}\) Divide the second equation by the first equation: \(\frac{I(60.5^{\circ})}{853} = \frac{\frac{1}{\sin^4(60.5^{\circ}/2)}}{\frac{1}{\sin^4(94.9^{\circ}/2)}}\)

Solve for the intensity at \(60.5^{\circ}\)

Rearrange the equation in step 2 to find \(I(60.5^{\circ})\): \(I(60.5^{\circ}) = 853 \cdot \frac{\sin^4(94.9^{\circ}/2)}{\sin^4(60.5^{\circ}/2)}\) Now, input the angles and calculate the intensity at \(60.5^{\circ}\): \(I(60.5^{\circ}) = 853 \cdot \frac{\sin^4(47.45^{\circ})}{\sin^4(30.25^{\circ})} ≈ 4966.85\) The intensity (in counts/s) at a scattering angle of \(60.5^{\circ} \pm 0.7^{\circ}\) is approximately \(4967\) counts/s.

Key concepts

Geiger-Marsden Experiment

The Geiger-Marsden experiment, also known as the gold foil experiment, was a groundbreaking test led by scientists Hans Geiger and Ernest Marsden under the direction of Ernest Rutherford that led to the development of the Rutherford model of the atom. This experiment involved firing a stream of alpha particles at a thin sheet of gold foil and observing their scattering behavior.

Most of the alpha particles passed straight through the foil, suggesting that the majority of the atom was empty space. However, some particles were deflected at large angles, while a very few even bounced back, indicating the presence of a small, dense, positively charged nucleus at the center. This observation contradicted the then-popular plum pudding model and was pivotal in advancing our understanding of the atomic structure.

Intensity of Scattered Particles

The intensity of scattered particles, in the context of the Rutherford scattering formula, refers to the number of alpha particles per unit time that are detected at a certain angle away from the original direction of the alpha particle beam. The Rutherford formula provides a way to predict this intensity based on the angle of scattering.

It is important to understand that this intensity is an outcome of both the number of alpha particles scattered by the nucleus and the detection method used. In a typical experiment setup, the intensity can be measured by a detector such as a scintillation counter or a Geiger-Müller tube, providing the counts per second at various angles.

Angle of Scattering

The angle of scattering \theta is the angle at which an alpha particle deviates from its original path as a result of its interaction with the nucleus in the gold foil. According to the Rutherford scattering formula, the intensity of scattered particles is inversely proportional to the fourth power of the sine of half the scattering angle, represented as \( I(\theta) = I_0 \frac{1}{\sin^4(\theta/2)} \).

This relationship shows that the smaller the angle of scattering, the higher the intensity of scattered particles detected. A smaller scattering angle means the alpha particle had a glancing interaction with the nucleus, while a large angle indicates a closer, more direct interaction. The actual calculation of intensity based on varying scattering angles requires careful handling of the formula and an understanding of trigonometric functions.