Davisson-Germer Experiment: An Experiment that confirms the existence of de Broglie waves.

The Davisson-Germer Experiment is a groundbreaking experiment that provided key evidence supporting the wave-particle duality of matter, specifically confirming the wave nature of electrons as predicted by Louis de Broglie's hypothesis. 

de Broglie's Hypothesis

In 1924, Louis de Broglie proposed that particles, such as electrons, possess wave-like properties. According to his hypothesis, any moving particle has an associated wavelength, called the de Broglie wavelength. The relationship between the wavelength (λ) and the momentum (p) of a particle is given by the equation:

$$λ = \frac{h}{p}$$

Where:

• $λ$ = de Broglie wavelength
• $h$ = Planck’s constant ( $6.626 \times 10^{-34}$ Js)
• $p$ = momentum of the particle ( $p=m\mathrm{v,\; where }$$m$ is the mass and $v$ is the velocity)

This concept introduces matter waves, where particles such as electrons can exhibit behaviors traditionally associated with waves, like diffraction.

Davisson-Germer Experiment (1927)

The experiment, conducted by Clinton Davisson and Lester Germer, sought to study the scattering of electrons off a nickel crystal. Surprisingly, the experiment revealed that electrons exhibit diffraction patterns, which is a property of waves, thereby confirming their wave-like behavior.

Experimental Setup

1. Electron gun: Produces a beam of electrons.
2. Nickel target: A nickel crystal serves as a diffraction grating.
3. Electron detector: Measures the intensity of scattered electrons at various angles.
4. Accelerating voltage: Adjusts the speed (and momentum) of the electrons.

Working of the Experiment

• Electron emission: Electrons are emitted from the electron gun and accelerated by a voltage (V). The kinetic energy ($K.E$) of the electrons is:

$$K.E = eV$$

Where:

• $e$ = charge of the electron ( $1.6\times 10^{-\mathrm{19\; C)}}$

• $V$ = accelerating voltage

• Momentum: The momentum of the electrons is related to their kinetic energy:

$$p = \sqrt{2m_e eV}$$

Where:

• $m_e$ = mass of the electron ($9.11 \times 10^{-31}$ kg)

• Electron diffraction: When the electrons hit the nickel crystal, they are scattered by the atoms in the crystal, causing diffraction. The crystal structure acts like a diffraction grating for the electron waves.

• Measurement of angles: The scattered electrons are detected at different angles, and a sharp intensity peak is seen at specific angles, indicating constructive interference—a characteristic of wave behavior.

Bragg’s Law

The diffraction pattern observed in the experiment can be explained using Bragg’s Law, which relates the diffraction angle (θ) to the wavelength (λ) and the spacing between crystal planes (d):

$$nλ = 2d \sin θ$$

Where:

• $n$ = order of diffraction (typically $n = 1$ for first-order diffraction)
• $d$ = spacing between crystal planes
• $θ$ = angle of incidence

Verifying de Broglie’s Hypothesis

Using the de Broglie equation $λ = \frac{h}{p}$, the wavelength of the electron can be calculated based on its momentum, which is determined by the accelerating voltage. The experiment demonstrated that the observed diffraction pattern of electrons corresponded to the wavelength predicted by de Broglie’s hypothesis.

Results

At an accelerating voltage of about 54V, a sharp diffraction peak was observed at an angle of approximately 50°. Using Bragg’s law, the wavelength of the electrons was calculated and found to match the de Broglie wavelength. This confirmed that electrons, like light, exhibit wave-like behavior.

Key Takeaways for Students:

1. Wave-particle duality: The experiment confirms that electrons can behave both as particles and as waves.
2. Diffraction patterns: The diffraction of electrons through a crystal proves that they exhibit constructive and destructive interference, similar to light waves.
3. Experimental evidence for de Broglie’s hypothesis: The calculated de Broglie wavelength from the experiment matched theoretical predictions, supporting the concept of matter waves.

This experiment played a crucial role in the development of quantum mechanics, illustrating that matter on small scales behaves in ways that challenge classical physics, embodying both particle and wave-like properties.