Derive the expression for the de-Broglie wavelength of an electron moving under a potential difference of V volt.
Describe the Davisson and Germer experiment to establish the wave nature of electrons. Draw labelled diagram of the apparatus used.  

Consider an electron of mass 'm' and charge 'e'. Let v be the velocity acquired by electron when accelerated from rest through a potential difference of V volt. 

Gain in K.E of electron = $\frac{1}{2}m{v}^2$ 
Work done = eV

If $\mathrm{\lambda }$ is the De- Broglie wavelegth assosciated with the electron, then 

$\mathrm{\lambda } = \frac{\mathrm{h}}{\mathrm{mv}} = \frac{\mathrm{h}}{\mathrm{m} \sqrt{2\mathrm{eV}/\mathrm{m}}} = \frac{\mathrm{h}}{\sqrt{2\mathrm{meV}}}$. 

In order to establish the wave nature of electrons an experiment was performed by Davison and Germer. 

i) A fine beam of accelerated electrons obtained from electron gun is made to fall normally on the surface of nickel crystal.

ii) Incident electrons are scattered in different directions by the atoms of the crystals.
iii) Intensity of the elctron beam,  is scattered in a different direction. 

iv) By rotating the electron detector on circular scale at different positions, intensity of the scattered beam is measured for different values of scattering angle $\mathrm{\phi }$.

v) The experiment was performed by varying the accelerating voltage 44 V to 68 V. 

vi) In an accelerating voltage at 54 V, a graph with axis of variation of intensity (I) and scattering angle ($\mathrm{\phi }$) was plotted. 

vi) The appearance of peak in a particular direction is due to constructive interference of electrons scattered from different layers of regularly spaced atoms of the crystal, i.e., the diffraction of electrons takes place. 

This established the wave nature of electrons.