E.2.2 De Broglie wavelength and diffraction (HL only) Notes

De Broglie Hypothesis, Particle Diffraction, and Compton Scattering

The De Broglie Hypothesis: Particles as Waves

1. In 1923, Louis de Broglie proposed a revolutionary idea: particles, such as electrons or protons, can exhibit wave-like properties.
2. He introduced the concept of the de Broglie wavelength, which relates a particle’s momentum to its wavelength: $$\lambda = \frac{h}{p}$$ where:
   1. $\lambda$ is the particle’s wavelength (in meters),
   2. $h$ is Planck’s constant $ 6.63 \times 10^{-34} \, \mathrm{J s} $,
   3. $p$ is the particle’s momentum $ p = mv $, where $ m $ is mass and $ v $ is velocity.

Why Does This Matter?

1. The de Broglie hypothesis bridges the classical and quantum worlds.
2. It shows that particles, which we often think of as discrete points, can behave like waves under certain conditions.
3. This duality particles behaving as waves is a fundamental concept in quantum mechanics.

Particle Diffraction: Evidence for Wave Behavior

1. If particles truly exhibit wave-like properties, we should be able to observe phenomena like diffraction and interference, which are characteristic of waves.
2. This was experimentally confirmed in 1927 by the Davisson-Germer experiment.

The Davisson-Germer Experiment

1. In this experiment, electrons were accelerated through a potential difference and directed at a nickel crystal.
2. The scattered electrons produced a pattern of bright and dark spots, similar to the diffraction patterns seen with light waves passing through a slit.

The key finding: the spacing of the diffraction pattern matched the de Broglie wavelength of the electrons.

It is calculated using:

$$\lambda = \frac{h}{\sqrt{2m_e qV}}$$

where:

1. $m_e$ is the electron’s mass $ 9.11 \times 10^{-31} \, \mathrm{kg} $,
2. $q$ is the electron’s charge $ 1.60 \times 10^{-19} \, \mathrm{C} $,
3. $V$ is the accelerating voltage.

Compton Scattering: Photons as Particles

While the de Broglie hypothesis showed that particles can behave like waves, Compton scattering demonstrated that waves (light) can behave like particles.

The Compton Effect

1. In 1923, Arthur Compton observed that when X-rays scatter off electrons, the wavelength of the scattered X-rays increases.
2. This phenomenon is best explained by treating light as a stream of particles, or photons.
3. When a photon collides with an electron, it transfers energy and momentum to the electron, causing the photon to lose energy (and increase its wavelength).
4. The change in wavelength, $\Delta \lambda $, is described by: $$\Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta)$$ where:
   1. $\Delta \lambda$ is the wavelength shift (in meters),
   2. $m_e$ is the electron’s mass,
   3. $c$ is the speed of light $ 3.0 \times 10^8 \, \mathrm{m \ s}^{-1} $,
   4. $\theta$ is the scattering angle.

Key Observations

1. The wavelength shift depends only on the scattering angle $\theta $, not on the photon’s initial wavelength.
2. The largest shift occurs when $\theta = 180^\circ$ (backscattering).

Why Is This Important?

1. Compton scattering confirmed that photons have momentum, a property traditionally associated with particles.
2. This discovery further solidified the concept of wave-particle duality.