(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

De Broglie wavelength `=2.327xx10^(-12)m`, neutron is not suitable for the diffraction exeriment
Kinetic energy of the neutron, `K=150eV =150xx1.6xx10^(-19)`
`2.4xx10^(-17)J`
Mass of a neutron, `M_(n)=1.675xx10^(-27)Kg`
The kinetic energy of the neutron is given by the relatin : `K=(1)/(2)m_(n)v^(2)`
`m_(n)v=sqrt(2Km_(n))`
Where,
v=Velocity of the neutron
`m_(n)v`=Momentum of the neutron
De-Broglie wavelength of the neutron is given as :
`lambda =(h)/(m_(n)v)=(h)/sqrt(2Km_(n))`
It is clear that wavelength is inversely proportional to the square root of mass. Hence, wavelength decreases with increase is mass and vice versa.
`therefore lambda =(6.6xx10^(-34))/sqrt(2xx2.4xx10^(-17)xx1.675xx10^(-27))`
=`2.327xx10^(-12)m`
It is given in the previous problem that the inter-atomic spacing of a crystal is about `1Å`, 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) De Broglie wavelegth =`1.447xx10^(-10)m` Room temperature ,`T=27^(@)C=27+273=300K` The average kinetic energy of the neutron is given as :
`E=(3)/(2)kT`
Where,
K= Boltzmann constant `=1.38xx10^(-23)JMol^(-1)K^(-1)` The wavelength of the neutron is given as :
`lambda=(h)/sqrt(2M_(n)E)=(h)/sqrt(3M_(n)kT)`
`=(6.6xx10^(-34))/sqrt(3xx1.675xx10^(-27)xx1.38xx10^(-23)xx300)`
`=1.447xx10^(-10)m`
This wavelength is comparable to the inter-atmomic spacing of a crystal. Hence, the high-energy neutron beam should first be thermalised, before using it for diffraction.