Davisson-Germer Experiment — Revision Notes

⚡ 30-Second Revision

De Broglie Wavelength — $\lambda = h/p = h/(mv)$
For Electron Accelerated by V — $\lambda = \frac{h}{\sqrt{2m_e eV}}$
Simplified for Electron — $\lambda \approx \frac{1.227}{\sqrt{V}}\,\text{nm}$
Bragg's Law — $2d\sin\theta = n\lambda$
Purpose — Experimental proof of wave nature of electrons.
Key Observation — Electron diffraction peak at specific angle ($50^\circ$) for specific voltage ($54\,\text{V}$).

2-Minute Revision

The Davisson-Germer experiment was a landmark in physics, providing the first experimental confirmation of the wave nature of electrons, as hypothesized by Louis de Broglie. They fired a beam of electrons, accelerated by a potential $V$, at a nickel crystal.

The crystal's regular atomic arrangement acted as a diffraction grating. A detector measured the intensity of scattered electrons at various angles. A prominent peak in electron intensity was observed at a scattering angle of $50^\circ$ when the accelerating voltage was $54\,\text{V}$.

This peak indicated constructive interference, a clear wave phenomenon. Using Bragg's Law ($2d\sin\theta = n\lambda$), the wavelength of the electrons was calculated from this diffraction pattern. This experimentally determined wavelength closely matched the de Broglie wavelength ($\lambda = \frac{h}{\sqrt{2m_e eV}}$) predicted for electrons at $54\,\text{V}$.

This agreement conclusively proved that electrons exhibit wave-like properties, establishing wave-particle duality for matter and paving the way for quantum mechanics and technologies like the electron microscope.

5-Minute Revision

The Davisson-Germer experiment is crucial for understanding the dual nature of matter. It experimentally confirmed de Broglie's hypothesis that particles like electrons possess wave characteristics. The setup involved an electron gun to produce and accelerate electrons (via thermionic emission and potential $V$), a nickel crystal target, and a rotatable detector. The nickel crystal's ordered atomic planes served as a diffraction grating.

Key Principle: If electrons are waves, they should diffract from the crystal, producing an interference pattern. This pattern can be analyzed using Bragg's Law, $2d\sin\theta = n\lambda$, where $d$ is interplanar spacing, $\theta$ is the glancing angle, $n$ is the order of diffraction, and $\lambda$ is the electron's wavelength.

Key Observation: For electrons accelerated through $V = 54\,\text{V}$, a strong peak in scattered intensity was observed at a scattering angle $\phi = 50^\circ$. The glancing angle $\theta$ for this peak was $90^\circ - \phi/2 = 65^\circ$. Using $d = 0.091\,\text{nm}$ for nickel, the experimental wavelength was calculated as $\lambda_{exp} = 2d\sin(65^\circ) \approx 0.165\,\text{nm}$.

De Broglie Wavelength Calculation: For an electron accelerated through $V$, its de Broglie wavelength is $\lambda_{dB} = \frac{h}{\sqrt{2m_e eV}}$. Plugging in $V=54\,\text{V}$, $h=6.626 \times 10^{-34}\,\text{J\cdot s}$, $m_e=9.1 \times 10^{-31}\,\text{kg}$, $e=1.6 \times 10^{-19}\,\text{C}$, we get $\lambda_{dB} \approx 0.167\,\text{nm}$.

Conclusion: The close agreement between $\lambda_{exp}$ and $\lambda_{dB}$ provided irrefutable proof of the wave nature of electrons. This experiment, alongside the photoelectric effect, solidified the concept of wave-particle duality. Its applications include the electron microscope, which uses the very small de Broglie wavelength of electrons for high-resolution imaging.

Example: An electron is accelerated through $200\,\text{V}$. Its de Broglie wavelength is $\lambda = \frac{1.227}{\sqrt{200}}\,\text{nm} \approx \frac{1.227}{14.14}\,\text{nm} \approx 0.0868\,\text{nm}$.

Prelims Revision Notes

The Davisson-Germer experiment is a critical topic for NEET, focusing on the wave nature of matter. Remember that it was the first experimental verification of Louis de Broglie's hypothesis, which states that all moving particles have an associated wave.

The de Broglie wavelength is given by $\lambda = h/p$, where $p$ is the momentum. For an electron accelerated from rest through a potential difference $V$, its kinetic energy is $K = eV$. Since $K = p^2/(2m_e)$, the momentum $p = \sqrt{2m_e eV}$.

Substituting this into the de Broglie equation yields $\lambda = \frac{h}{\sqrt{2m_e eV}}$. A highly useful approximation for electrons is $\lambda \approx \frac{1.227}{\sqrt{V}}\,\text{nm}$.

The experimental setup involved an electron gun (producing electrons via thermionic emission and accelerating them), a single crystal of nickel as the target (acting as a diffraction grating), and a rotatable electron detector.

The key observation was a sharp peak in the intensity of scattered electrons at a specific scattering angle (e.g., $50^\circ$) for a particular accelerating voltage (e.g., $54\,\text{V}$). This peak is evidence of constructive interference, a wave phenomenon.

The wavelength of the electrons was then calculated using Bragg's Law, $2d\sin\theta = n\lambda$, where $d$ is the interplanar spacing of the crystal and $\theta$ is the glancing angle (angle between the incident beam and the crystal planes).

For the Davisson-Germer experiment, the glancing angle $\theta$ is often related to the scattering angle $\phi$ by $\theta = 90^\circ - \phi/2$. The remarkable agreement between the experimentally determined wavelength and the de Broglie wavelength provided conclusive proof of the wave nature of electrons.

This experiment is fundamental to understanding wave-particle duality and is the basis for technologies like the electron microscope. Be prepared for numerical problems involving the de Broglie wavelength formula and conceptual questions about the experiment's significance and implications.

Vyyuha Quick Recall

To remember the key aspects of Davisson-Germer: Diffraction Gives Evidence for Waves in Matter.

Davisson-Germer: Electrons Wave-like Matter.

De Broglie's Lambda: Heavy Matter Vibrates Easily (for $\lambda = h/\sqrt{2meV}$). (H for h, M for m, V for V, E for e)