(a) Obtain the de Broglie wavelength of a neutron of kinetic energy 150 eV. As you have seen in Exercise 11.31, an electron beam of this energy is suitable for crystal diffraction experiments. Would a neutron beam of the same energy be equally suitable? Explain. `(m_(n) = 1.675 xx 10^(-27) kg)`
(b) 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.

Explanation

(a)Here `m_(n)=1.675xx10^(-27)` kg
`K=150 eV=150xx1.6xx10^(-19)J=2.4xx10^(-17)J`
`h=6.63xx10^(-34)Js,c=3xx10^(8) ms^(-1)`
`impliesK=(1)/(2)m_(n)V^(2)=((m_(n)V)^(2))/(2m_(n))=(p^(2))/(2m_(n))`
`therefore p=sqrt(2m_(n)K)`
`implies` de-Broglie wavelength,
`lambda=(h)/(p)=(6.63xx10^(-34))/(sqrt(2m_(n)K))`
`therefore ,lambda=(6.63xx10^(-34))/(sqrt(2xx1.675xx10^(-27)xx2.4xx10^(-17)))`
`therefore lambda=(6.63xx10^(-34))/(2.835xx10^(-22))`
`=2.3386xx10^(12)m`
`therefore lambda~~2.34xx10^(-12)m`
Interatomic distance is about 1Å =`10^(-10)` m which is about 100 times larger than wavelength of neurton .Hence neutron beam of 150 eV is not suitable for diffraction.
(b)At standard temperature T,average kinetic energy of neutron ,
`(1)/(2)m_(n)v^(2)=(3)/(2)k_(B)T` where `k_(B)=`Boltzmann.s constant.
`(p^(2))/(2m_(n))=(3)/(2)k_(B)T`
`therefore` de-Broglie wavelength `lambda=(h)/(p)`
`therefore lambda=(h)/(sqrt(3m_(n)K_(B)T))`
Here,`m_(n)=1.675x10^(-27)Kg`
`k_(B)=1.38xx10^(-23)J mol^(-1)K^(-1)`
T=23+273=300K,`h=6.63xx10^(-34)Js`
`therefore lambda=(6.63xx10^(-34))/(sqrt(3x1.675xx10^(-27)xx1.38xx10^(-23)xx300))`
`=(6.63xx10^(-34))/(45.61xx10^(-25))`
`lambda=0.14536xx10^(-9)`m
`therefore lambda~~1.45xx10^(-10)m=1.45 Å`
`implies` This distance is equal to ineratomic distance.
`implies`From (a) and (b) it is clear that thermal neutron are more suitable for experiment of diffraction.Hence before using high energy neutron should be coverted to thermal neutron and then it should be used in experiment of diffraction.