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Question
Obtain the de Broglie wavelength of a neutron of kinetic energy 150 eV. As you have an electron beam of this energy is suitable for crystal diffraction experiments. Would a neutron beam of the same energy be equally suitable? Explain. (mn= 1.675 × 10−27 kg)
Solution
De Broglie wavelength = 2.327 × 10−12 m; neutron is not suitable for the diffraction experiment.
Kinetic energy of the neutron, K = 150 eV
= 150 × 1.6 × 10−19
= 2.4 × 10−17 J
Mass of a neutron, mn = 1.675 × 10−27 kg
The kinetic energy of the neutron is given by the relation:
`"K" = 1/2 "m"_"n""v"^2`
`"m"_"n""v" = sqrt(2 "KM"_"n")`
Where
v = Velocity of the neutron
mnv = Momentum of the neutron
De-Broglie wavelength of the neutron is given as:
`lambda = "h"/("m"_"n""v") = "h"/(sqrt(2 "Km"_"n"))`
it is clear that wavelength is inversely proportional to the square rot of mass.
Hence wavelength decrease with an increase in mass and vice versa.
∴ `lambda = (6.6 xx 10^(-34))/sqrt(2 xx 2.4 xx 10^(-17) xx 1.675 xx 10^(-27))`
= 2.327 × 10−12 m
It is given in the previous problem that the inter-atomic spacing of a crystal is about 1 Å, 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.
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