#### Question

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

#### Solution

De Broglie wavelength = `1.447 xx 10^(-10) m`

Room temperature, *T* = 27°C = 27 + 273 = 300 K

The average kinetic energy of the neutron is given as:

`E = 3/2 kT`

Where,

*k* = Boltzmann constant = 1.38 × 10^{−23} J mol^{−1} K^{−1}

The wavelength of the neutron is given as:

`lambda = h/sqrt(2m_n E) = h/(sqrt(3m_n KT))`

`= (6.6 xx 10^(-34))/sqrt(3xx1.675 xx 10^(-27) xx 1.38 xx 10^(-23) xx 300)`

`= 1.447 xx 10^(-10) m`

This wavelength is comparable to the inter-atomic spacing of a crystal. Hence, the high-energy neutron beam should first be thermalised, before using it for diffraction.