Question

The de Broglie wavelength, \(\lambda\), of a 5-MeV alpha particle is \(6.4 \mathrm{fm}\), as shown in this chapter, and the closest distance, \(r_{\text {min }}\), to the gold nucleus this alpha particle can get is \(45.5 \mathrm{fm}\) (calculated in Example 39.1). Based on the fact that \(\lambda \ll r_{\text {min }}\), one can conclude that, for this Rutherford scattering experiment, it is adequate to treat the alpha particle as a a) particle. b) wave.

Short answer

Answer: (a) particle

Step-by-step solution

Compare the given wavelength and closest distance values

We are given the de Broglie wavelength of the alpha particle (λ = 6.4 fm) and the closest distance to the gold nucleus (r_min = 45.5 fm). To understand whether the alpha particle should be treated as a wave or a particle, we have to compare these two values.

Analyze the relationship between wavelength and closest distance

The de Broglie wavelength (λ) of a particle is related to its momentum and is given by the formula: λ = \frac{h}{p} where h is the Planck constant and p is the momentum of the particle (mass times velocity). If λ is much smaller than the closest distance to the gold nucleus (r_min), this indicates that the wave-like behavior of the alpha particle is less significant because the wavelength is too short to interact with the nucleus.

Compare the values and draw conclusions

Now we compare the given values: λ = 6.4 fm r_min = 45.5 fm We were asked to assess if it is adequate to treat the alpha particle as a particle or a wave. It is mentioned that λ << r_min. Based on this relationship, it is adequate to treat the alpha particle as: a) particle b) wave Since \(\lambda \ll r_{\text {min }}\) (6.4 fm << 45.5 fm), we can conclude that treating the alpha particle as a particle is adequate for this Rutherford scattering experiment. This is because the relatively short de Broglie wavelength does not significantly interact with the gold nucleus. So, the answer is (a) particle.

Key concepts

Rutherford Scattering Experiment

The Rutherford scattering experiment, conducted by Ernest Rutherford in 1909, was a groundbreaking procedure that revealed the presence of a small, dense nucleus at the center of the atom.

Prior to this experiment, the prevailing model was J.J. Thomson's 'plum pudding' model, which suggested that electrons were scattered within a positively charged 'soup'. Using alpha particles emitted from a radioactive source, Rutherford directed these particles at a thin sheet of gold foil and observed their deflection through detectors.

Surprisingly, while most alpha particles passed through with little or no deflection, some rebounded at sharp angles. This indicated that the positive charge and most of the atom's mass were concentrated in a tiny central nucleus, with the electrons orbiting around it, much like planets around the sun. This led to the planetary model of the atom and has shaped the foundation of modern atomic physics.

Importance for Quantum Mechanics

Rutherford's experiment was pivotal in quantum theory development as it highlighted the need for a new atomic model, eventually leading to the quantum mechanical model which incorporates wave-particle duality, where particles like electrons have properties of both particles and waves.

Alpha Particle

An alpha particle is a type of ionizing radiation made up of two protons and two neutrons bound together. These particles are relatively massive and positively charged, making them a key component in nuclear and atomic physics experiments, such as the Rutherford scattering experiment.

In the context of this experiment, alpha particles acted as probes into the structure of the atom. When directed towards the gold foil, their massive nature allowed some of them to approach close to the gold nuclei, providing clues to the internal structure of the atom.

The characteristics of alpha particles, such as their mass and charge, are essential in calculating interactions with other particles or potentials, and understanding their behavior requires a blend of classical and quantum mechanical analysis.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It departs from classical mechanics primarily at the quantum realm of atomic and subatomic length scales.

One of the intriguing aspects of quantum mechanics is the concept of wave-particle duality as posited by de Broglie. According to de Broglie, every particle can be described as having wave-like properties. This is characterized by the de Broglie wavelength, which is inversely proportional to the momentum of a particle: \( \lambda = \frac{h}{p} \).

Quantum mechanics uses probability to predict where an electron might be found around the nucleus, which is a stark contrast to the definitive orbits of electrons envisioned in Rutherford's model. This theory has been supported by numerous experiments and has become foundational in understanding and designing technologies such as semiconductors, lasers, and even medical imaging and treatments.