MP Board · Class 12 · Physics · Dual Nature of Radiation and MatterWhat is de Broglie wavelength? Derive the expression for the de Broglie wavelength of an electron accelerated through a potential difference of $V$ volts. Describe Davisson and Germer experiment with a neat labeled diagram to verify the wave nature of electrons.

Concept of de Broglie Wavelength\nIn 1924, the French physicist Louis de Broglie proposed that moving particles of matter (like electrons, protons, neutrons, etc.) should exhibit wave-like properties under certain conditions. These matter waves are called de Broglie waves. \nThe wavelength associated with a particle of mass $m$ moving with a velocity $v$ is given by:

$$\lambda = \frac{h}{p} = \frac{h}{mv}$$\nwhere $h$ is Planck's constant and $p$ is the linear momentum of the particle.

Derivation for de Broglie Wavelength of an Accelerated Electron\nLet an electron of mass $m$ and charge $e$ be accelerated from rest through a potential difference of $V$ volts. \nThe work done on the electron by the electric field appears as its kinetic energy ($K$):

$$K = eV$$\nWe know that the kinetic energy and linear momentum ($p$) are related by: $$K = \frac{p^2}{2m} \implies p^2 = 2mK = 2meV$$ $$p = \sqrt{2meV}$|\nSubstituting the value of momentum in de Broglie's formula, the wavelength $\lambda$ becomes: $$\lambda = \frac{h}{\sqrt{2meV}}$$\nSubstituting the standard values of Planck's constant ($h = 6.63 \times 10^{-34}\text{ J s}$), mass of electron ($m = 9.1 \times 10^{-31}\text{ kg}$), and charge of electron ($e = 1.6 \times 10^{-19}\text{ C}$): $$\lambda = \frac{6.63 \times 10^{-34}}{\sqrt{2 \times 9.1 \times 10^{-31} \times 1.6 \times 10^{-19} \times V}}$$ $$\lambda = \frac{1.227}{\sqrt{V}}\text{ nm} = \frac{12.27}{\sqrt{V}},\mathring{\text{A}}$$

Davisson and Germer Experiment

• Objective: To experimentally verify the wave nature of electrons (de Broglie hypothesis) through electron diffraction.
• Experimental Arrangement:
  • Electron Gun: Consists of a tungsten filament $F$ heated by a low-tension battery to emit electrons by thermionic emission. The electrons are accelerated to a desired velocity by applying a high positive potential $V$ using a high-tension battery.
  • Collimator: A fine cylindrical anode cylinder focuses the emitted electrons into a fine parallel beam.
  • Target (Nickel Crystal): The fine electron beam is allowed to fall normally on the surface of a single crystal of Nickel (face cut along a specific crystallographic plane). The crystal acts as a three-dimensional diffraction grating.
  • Faraday Cylinder (Detector): The scattered electrons from the crystal are collected by a movable Faraday cylinder connected to a sensitive galvanometer. The detector can be rotated on a circular scale to measure the scattering intensity at different angles $\theta$.
• Observations and Conclusion:
  • It was observed that the intensity of the scattered electron beam varies with the angle of scattering $\theta$ and the accelerating potential $V$.
  • At an accelerating potential of $V = 54\text{ V}$, a sharp peak in scattering intensity was observed at a scattering angle of $\theta = 50^\circ$.
  • Using Bragg's law for diffraction ($2d \sin \theta = n\lambda$), the wavelength of electrons calculated from the crystal plane spacing matched precisely with the theoretical de Broglie wavelength ($\lambda = \frac{12.27}{\sqrt{54}} \approx 1.66,\mathring{\text{A}}$).
  • This conclusively proved that electrons possess wave nature and undergo diffraction, confirming de Broglie's hypothesis.