Question

A Geiger-Marsden experiment, in which alpha particles are scattered off a thin gold film, is set up with two detectors at \(\theta_{1}=85.1^{\circ} \pm 0.9^{\circ}\) and \(\theta_{2}=62.9^{\circ} \pm 0.9^{\circ} .\) Assuming that the scattering obeys the Rutherford formula, what is the ratio of the measured intensities, \(I_{1} / I_{2} ?\)

Short answer

Answer: The ratio of the measured intensities \(I_1/I_2\) is approximately \(2.39\).

Step-by-step solution

Rutherford Scattering Formula

According to the Rutherford scattering formula, the intensity of scattered particles at a certain angle \(\theta\) is: \[I(\theta) = \frac{K}{\sin^4(\frac{\theta}{2})}\] Where \(K\) is a constant that depends on factors such as the atomic number of the scattering medium and the particle beam intensity, which remains the same for both detectors in our case.

Calculate the Intensity Ratio

Now, let's find the ratio \(I_1/I_2\): \[\frac{I_1}{I_2} = \frac{I(\theta_1)}{I(\theta_2)} = \frac{\frac{K}{\sin^4(\frac{\theta_1}{2})}}{\frac{K}{\sin^4(\frac{\theta_2}{2})}}\] As the \(K\) value is the same for both detectors, it cancels out in our fraction: \[\frac{I_1}{I_2} = \frac{\sin^4(\frac{\theta_2}{2})}{\sin^4(\frac{\theta_1}{2})}\] Now, let's plug in the given angles \(\theta_1 = 85.1^{\circ}\) and \(\theta_2 = 62.9^{\circ}\): \[\frac{I_1}{I_2} = \frac{\sin^4(\frac{62.9^{\circ}}{2})}{\sin^4(\frac{85.1^{\circ}}{2})}\] Calculating the ratio: \[\frac{I_1}{I_2} \approx 2.39\] Thus, the ratio of the measured intensities \(I_1/I_2\) is approximately \(2.39\).

Key concepts

Geiger-Marsden Experiment

The Geiger-Marsden experiment, commonly known as the gold foil experiment, was a groundbreaking test led by Hans Geiger and Ernest Marsden under the supervision of Ernest Rutherford in the early 1900s. This experiment aimed to probe the structure of the atom.

During the experiment, alpha particles, which are helium nuclei, were fired at a thin sheet of gold foil. Most particles passed through without significant deflection, but some were scattered at large angles, and a few even bounced back towards the source. Rutherford explained this surprising observation by proposing that the atom consists of a small, dense nucleus containing most of its mass, and that the nucleus is positively charged. This led to the downfall of the plum pudding model and set the stage for the nuclear model of the atom.

The experiment utilized a circular fluorescent screen to detect the scattered alpha particles. Detectors placed at different angles measured the scattering intensity, which is the number of particles deflected at specific angles. The intensity of scattering was found to decrease sharply with increasing scattering angle, leading to the formulation of the Rutherford scattering formula.

Scattering Intensity Ratio

In the context of the Rutherford Scattering Experiment, the scattering intensity ratio is a comparison between the number of alpha particles scattered at different angles. According to the Rutherford formula, the intensity of scattered alpha particles, detected at a certain angle, is inversely proportional to the fourth power of the sine of half the scattering angle. Mathematically, this relationship can be expressed as:
\[I(\theta) = \frac{K}{\sin^4(\frac{\theta}{2})}\]
where \(I(\theta)\) is the intensity at angle \(\theta\) and \(K\) is a constant that remains the same irrespective of the angle for a particular experiment setup.

This resulting intensity ratio is crucial for understanding atomic structure and confirming the predictions made by Rutherford's model. It is an empirical result that was consistent with the presence of a small, positively charged nucleus at the center of the atom, unlike the fairly uniform distribution of mass and charge in the plum pudding model.

Alpha Particle Scattering

Alpha particle scattering refers to the deflection of alpha particles, which are doubly charged helium nuclei, after they encounter an electric field, such as the one produced by the nucleus of an atom. In Rutherford's experiment, alpha particle scattering revealed the structure of the atom by showing that a small nucleus, not a 'pudding' of distributed positive charge, deflected the particles.

The fact that only a small fraction of the alpha particles were scattered at large angles indicated that the atomic nucleus occupies a very small volume compared to the rest of the atom. The number of particles scattered per unit angle decreases sharply as the angle increases, which is quantified by the Rutherford scattering formula. The greater the deflection, the closer the alpha particle has come to the nucleus, implying a strong central positive charge concentrated in a tiny region of the atom.

Understanding alpha particle scattering and the factors influencing it, such as the scattering medium's atomic number and the energy of the alpha particles, has been instrumental in advancements in both nuclear physics and quantum mechanics.