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

Solution

We are given,
Room tempertaure, T = 27 + 273 = 300 K
Boltzmann's constant, k = 1.38 x 10^–23 J mol^–1 K^–1We know, average kinetic energy of neutron at absolute temperature T is given by,

$E = \frac{3}{2} kT .$
where k is the Boltzmann's constant.

Now, wavelength, $λ = \frac{h}{\sqrt{2 mE}} = \frac{h}{\sqrt{3 mkT}}$
$∴$ $λ = \frac{6.63 \times 10^{- 34}}{\sqrt{3 \times 1.675 \times 10^{- 27} \times 1.38 \times 10^{- 23} \times 300}} = 1.45 \times 10^{- 10} m$

Since this wavelength is comparable to interatomic spacing (~ 1 Å) in a crystal, therefore, thermal neutrons are suitable for diffraction experiments. So, a high energy neutron beam should be first thermalised before using it for diffraction.

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Wave Optics

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

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