(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 differaction 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 differaction experiments.

Explanation

(a) Here, K.E. of neutron, `E=150 eV =150 xx1.6xx10^(-19) J`, mass of neutron, `m=1.675xx10^(-27)kg`.
We know, K.E. of neutron, `E=1/2mv^(2) or mv=sqrt(2Em)`
`:. lambda=h/(mv)=h/(sqrt(2Em))= (6.63xx10^(-34))/(sqrt(2xx150xx1.6xx10^(-19)xx1.675xx10^(-27)))=2.33xx10^(-12)m`
The interatomic spacing `~1Å(=10^(-10)m)` is about a hundred times greater than this wave length. Therefore, a neutron beam of energy 150 eV is not suitable for differaction experiment.
(b) here, `T=27+273=300K`, Boltzmann's constant, `k=1.38xx10^(-23)Jmol^(-1)K^(-1)`
We know, average K.E. of neutron at absolute temperature T is given by, `E=3/2kT`. Where k is the Boltzmann's constant.
Now, `lambda=h/(sqrt(2Em))=h/(sqrt(3mkT))=(6.63xx10^(-34))/(sqrt(3xx1.675xx10^(-27)xx1.38xx10^(-23)xx300)) =1.45xx10^(-10)m`
Since this wavelength is comparable to interatomic spacing `(~1Å)` in a crystal, therefore, thermal neutrons are suitable probe for different experiments : so a high energy neutrons beam should be first thermalized befor using it for diffraction.