(a) Obtain the de-Broglie wavelength of a neutron of kinetic energy 150 eV. As you have seen in previous problem 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. Given `m_(n)=1.675xx10^(-27)kg`.
(b) Obtain the de-Broglie wavelength associated with thermal neutrons at room temperature `(27^(@)C)`. Hence explain why a fast neutrons beam needs to be thermalised with the environment before it can be used for neutron diffraction experiments.

Explanation

(a) Given , of neutron , `E_k=150eV`
Mass of neutron , `m_n=1.675xx10^(-27)` kg
Kinetic energy of neutron is given by
`E_k=1/2m_m_nv^2`
`:. "Momentum " , p = mv = sqrt(2E_km_n)`
The de Broglie wavelength of neutron is given by
`lamda = h/p = h / (sqrt(2m_nE_k))`
`=(6.63xx10^(-34))/(sqrt(2xx150xx1.6xx10^(-19)xx1.675xx10^(-27)))`
`=2.34xx10^(-12)m `
The interatomic spacing `(=1Å)` is about a hundred times greater . A neutron beam of 150 eV energy is therefore not suitable for diffraction experiments.
(b) Given , temperature , T = 300 K
The de Broglie wavelength associated with the thermal neutrons is
`lamda=h/(sqrt(2mE))`
`implieslamda=h/(sqrt(3mkT))" "(becauseE=3/2kT)`
`lamda=(6.625xx10^(-34))/(sqrt(3xx1.675xx10^(-27)xx1.38xx10^(-23)xx300))`
`lamda=1.45xx10^(-10)m`
which is comparable to interatomic spacing in a crystal . Clearly form (a) and (b) above, thermal neutrons are a suitable poroble for diffraction experiments , so a high - energy neutron beam should be first thermalised before using it for diffraction.