Wavelength Example Question

Calculate the de Broglie wavelength of a 0.20 kg ball moving with a speed of 15 m/s.

$$\lambda =\frac{h}{p}=\frac{h}{mv}=\frac{6.6\ast 10^{-34}Js}{(0.2kg)(15m/s)}=2.2\ast 10^{-34}$$

Determine the wavelength of an electron that has been accelerated through a potential difference of 100 V.

$$100eV=0.5m{v}^2$$ $$v=5.9\ast 10^{6}m/s$$ $$\frac{(6.6\ast 10^{-34}Js)}{(0.91\ast 10^{-31}kg)(5.9\ast 10^{6}m/s)}$$

The wave nature of electrons is manifested in experiments where an electron beam interacts with the atoms on the surface of a solid. By studying the angular distribution of the diffracted electrons, one can indirectly measure the geometrical arrangement of atoms.

Assume that the electrons strike perpendicular to the surface of a solid, and that their energy is low, $EK=100eV$, so that they interact only with the surface layer of atoms. If the smallest angle at which a diffraction maximum occurs is at 24°, what is the separation d between the atoms on the surface?

The smallest angle will occur when d sin θ = λ. The electrons are not relativistic, so the wavelength can be found from the kinetic energy: $\lambda =0.123nm$. Then the spacing is $\lambda /\sin \theta =0.30nm$.