Question 11.32:

(a) Obtain the de Broglie wavelength of a neutron of kinetic energy 150 eV. As you have seen in Exercise 11.31, an electron beam of this energy is suitable for crystal diffraction experiments. Would a neutron beam of the same energy be equally suitable? Explain. (m= 1.675 x 10-27 kg)

(b) Obtain the de Broglie wavelength associated with thermal neutrons at room temperature (27 $°$C). Hence explain why a fast neutron beam needs to be thermalized with the environment it can be used for neutron diffraction experiments.

(a)
Hint: $$\lambda=\frac{h}{\sqrt{2mK.E}}$$

Step 1: Find the wavelength of a neutron.
The wavelength of a neutron with kinetic energy K.E is given by:
$$\lambda=\frac{h}{\sqrt{2mK.E}}$$
It is given in the previous problem that the inter-atomic spacing of a crystal is about 1 Å, i.e, 10-10 m. Hence, the inter-atomic spacing is about a hundred times greater. Hence, a neutron beam of energy 150 eV is not suitable for diffraction experiments.
(b)
Hint:
$$\lambda=\frac{h}{\sqrt{2mE}}$$
Step 1: Find the average kinetic energy of the neutron.
The average kinetic energy of the neutron is given as:
$\mathrm{E}=\frac{3}{2}\mathrm{kT}$
Where, k = Boltzmann Constant 1.38 x 10-23 J mol-1 K-1
Step 2: Find the wavelength of the neutron.
The wavelength of the neutron is given as:
This wavelength is comparable to the interatomic spacing of a crystal. Hence, the high-energy neutron beam should first be thermalized, before using it for diffraction.