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NCERT Class XII Chapter
Dual Nature of Radiation and Matter
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Question : 32
Total: 37
(a) Obtain the de Broglie wavelength of a neutron of kinetic energy 150 eV. 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)
(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.
Solution:  
(a) Let us first calculate the wavelength of matter wave associated with neutron of kinetic energy 150 eV.
K =
p2
2m
So, momentum p = √2mK
Wavelength λ =
h
p
 =
h
√2mK
λ =
6.63×10−34
√2×1.675×10−27×150×1.6×10−19
m = 2.33 × 10−12 m
As the interatomic spacing (1 Å = 10–10 m) is about hundred times greater than this wavelength, so a neutron beam of 150 eV energy is not suitable for diffraction experiments.
(b) Average kinetic energy of a neutron at absolute temperature T is
1
2
mv2 =
3
2
 kT or
p2
2m
 =
3
2
 kT or p = √3mkT
∴ de Broglie wavelength , λ =
h
p
 =
h
√3mkT
Given mn = 1.675 × 10–27 kg, k = 1.38 × 10–23JK–1
T = 27 + 273 = 300 K, h = 6.63 × 10–34 J s
λ =
6.63×10−34
√3×1.675×10−27×1.38×10−23×300

=
6.63×10−10
4.56
 m = 1.45 × 10−10 m = 1.45 Å
As this wavelength is comparable to interatomic spacing (≈ 1 Å) in a crystal, so thermal neutrons can be used for diffraction experiments. So high energy neutron beam should be first thermalised before using it for diffraction.
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