The Davisson-Germer experiment is a landmark in physics that provided the first direct experimental confirmation of the wave nature of electrons. By demonstrating that particles of matter can exhibit diffraction—a property previously thought exclusive to waves—it solidified the concept of wave-particle duality and the de Broglie hypothesis.
Figure 1: Schematic arrangement of the Davisson-Germer Experiment showing the electron gun, Nickel target, and detector.
The Davisson-Germer Experiment: Proof of Matter Waves
1. Historical Context and Objective
In 1924, Louis de Broglie proposed that matter possesses a dual nature, acting as both a particle and a wave. In 1927, C.J. Davisson and L.H. Germer at Bell Telephone Laboratories verified this hypothesis by observing the diffraction of electrons from a Nickel crystal.
2. Experimental Setup
The apparatus consisted of the following key components:
- Electron Gun: A tungsten filament coated with barium oxide is heated to emit electrons via thermionic emission.
- Accelerator: An electrostatic field setup using a low-tension battery (for the filament) and a high-tension battery (to accelerate electrons). The voltage \( V \) could be varied.
- Nickel Target: A single crystal of Nickel cut effectively to present the (111) lattice planes to the incident beam.
- Detector (Faraday Cup): A collector connected to a sensitive galvanometer. It could rotate on a circular scale to measure the intensity of the scattered electron beam at various angles.
3. Observations and The "Bump"
Davisson and Germer plotted the intensity of the scattered electrons against the scattering angle \( \phi \) (the angle between the incident beam and the scattered beam) for various accelerating voltages.
The Findings:
At lower voltages, the scattering was fairly uniform. However, as the voltage approached 54 Volts, a distinct maximum (a "bump") appeared in the intensity curve.
Figure 2: Polar graphs showing the development of the intensity peak at 54V.
4. Mathematical Analysis and Proof
To confirm the wave nature, we compare the theoretical wavelength (from de Broglie's hypothesis) with the experimental wavelength (calculated using diffraction laws).
Step A: Theoretical Calculation (de Broglie Hypothesis)
According to de Broglie, the wavelength \( \lambda \) of an electron accelerated through potential \( V \) is given by:
\[ \lambda = \frac{h}{p} = \frac{h}{\sqrt{2mK}} \]
Since Kinetic Energy \( K = eV \), we have:
\[ \lambda = \frac{h}{\sqrt{2meV}} \]
Substituting the standard constants:
Planck's constant \( h = 6.626 \times 10^{-34} \text{ Js} \)
Mass of electron \( m = 9.1 \times 10^{-31} \text{ kg} \)
Charge of electron \( e = 1.6 \times 10^{-19} \text{ C} \)
We get the simplified formula:
\[ \lambda = \frac{12.27}{\sqrt{V}} \: \mathring{A} \]
For the peak voltage \( V = 54 \) Volts:
\[ \lambda_{theoretical} = \frac{12.27}{\sqrt{54}} \approx \mathbf{1.67 \: \mathring{A}} \]
Step B: Experimental Calculation (Using Bragg's Law)
The electrons were diffracted by the atomic planes of the Nickel crystal. We treat the atomic planes as a diffraction grating.
1. Determining the Glancing Angle (\( \theta \)):
The scattering angle is \( \phi = 50^\circ \). The glancing angle \( \theta \) is the angle between the incident beam and the crystal plane.
\[ \theta + \phi + \theta = 180^\circ \] \[ 2\theta = 180^\circ - 50^\circ = 130^\circ \] \[ \theta = 65^\circ \]
2. Applying Bragg's Law:
For constructive interference (maxima) in the first order (\( n=1 \)):
\[ 2d \sin \theta = n\lambda \]
For Nickel, the interplanar spacing (distance between atomic planes) determined by X-rays is \( d = 0.91 \: \mathring{A} \).
\[ \lambda = 2(0.91) \sin(65^\circ) \] \[ \lambda = 1.82 \times 0.9063 \]
\[ \lambda_{experimental} \approx \mathbf{1.65 \: \mathring{A}} \]
5. Conclusion and Interpretation
The experimental value obtained from the diffraction pattern (\( 1.65 \: \mathring{A} \)) is in excellent agreement with the theoretical value predicted by de Broglie (\( 1.67 \: \mathring{A} \)).
Physical Significance: This experiment confirms that moving electrons behave like waves. If electrons were purely particles, they would scatter roughly equally in all directions or bounce off like billiard balls. The existence of a diffraction pattern (the peak at a specific angle) is a characteristic exclusively associated with wave motion.
6. Applications
- Electron Microscopy: The wave nature of electrons allows for the construction of microscopes with much higher resolution than optical microscopes.
- LEED (Low-Energy Electron Diffraction): A technique used to study the surface structure of crystalline materials.