a)
Given that
Kinetic energy of neutron = 150 eV
= 150 * 1.6* 10^{-19} joules
Mass of neutron m_n = 1.675* 10^{-27} kg
Step 1: Determining the de Broglie wavelength of neutron
Recall the formula of momentum and kinetic energy
Momentum p = mv
Kinetic energy K.E = \frac{1}{2} mv^2
Where, p - momentum, m - mass of neutron and v - velocity
K.E = \frac{p^2}{2m} ...........................(1) \because v = \frac{p}{m}
p = \sqrt{2m K.E}
de Broglie wavelength \lambda = \frac{h}{p}
\lambda = \frac{h}{\sqrt{2 m_n K.E}} \because \text{ equation (1)}
Where, h - Planck's constant( 6.63* 10^{-34} ), m_n - mass of neutron
\lambda = \frac{6.638 10^{-34}}{2 * 1.675* 10^{-27} * 150 * 1.6* 10^{-19} }
\lambda = 2.327 * 10^{-12} m
Hence, de Broglie wavelength of neutron \lambda = 2.327 * 10^{-12} m
Step 2: Checking neutron energy is suitable for the crystal diffraction or not
Exercise 11.31:
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Inter atomic spacing \lambda = 1 Angstrom = 10^{-10} m
de Broglie wavelength of neutron \lambda = 2.327 * 10^{-12} m
de Broglie wavelength of neutron < 100 * Inter atomic spacing
Therefore, a 150 eV energy neutron beam does not suit diffraction experiments.
b)
Step 1: de Broglie wavelength neutron at room temperature
Neutron temperature T = 27^\degree C
T = 300 K \because 0\degree C = 273 K
Kinetic energy at temperature T
K.E = \frac{3}{2} kT
Where, k - Boltzmann constant = 1.38 * 10^{-23} J/mol * K
de Broglie wavelength neutron \lambda = \frac{h}{p} ...................................(1)
We know, momentum p = \sqrt{2m K.E}
Substituting p value in equation (1)
\lambda = \frac{h}{\sqrt{2m K.E}}
\lambda = \frac{h}{\sqrt{3 m kT}}
\lambda = \frac{6.63 * 10^{-34}}{3 * 1.675* 10^{-27} * 1.38 * 10^{-23} * 300 }
\lambda = 1.447 * 10^{-10} m
Hence, de Broglie wavelength neutron \lambda = 1.447 * 10^{-10} m
Step 2: Make sure the thermalised neutrons undergoes diffraction.
Broglie wavelength of thermalised neutron \lambda = 1.447 * 10^{-10} m
Inter atomic spacing \lambda = 1 Angstrom = 10^{-10} m
Broglie wavelength of thermalised neutron \approx Inter atomic spacing
Therefore, Before using the high-energy neutron beam for diffraction, it must first be thermalized.