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Question
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.
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Solution
De Broglie wavelength = 1.447 × 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(2"m"_"n""E") = "h"/(sqrt(3"m"_"n""kT"))`
= `(6.6 xx 10^(-34))/sqrt(3 xx 1.675 xx 10^(-27) xx 1.38 xx 10^(-23) xx 300)`
= 1.447 × 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 for diffraction.
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