Question

(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 × 10⁻²⁷ kg)

(b) Obtain the de Broglie wavelength associated with thermal neutrons at room temperature (27 ºC). Hence explain why a fast neutron beam needs to be thermalised with the environment before it can be used for neutron diffraction experiments.

Answer 1

(a) De Broglie wavelength =; 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, m_n = 1.675 × 10⁻²⁷ kg

The kinetic energy of the neutron is given by the relation:

Where,

v = Velocity of the neutron

m_nv = Momentum of the neutron

De-Broglie wavelength of the neutron is given as:

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.

(b) De Broglie wavelength =

Room temperature, T = 27°C = 27 + 273 = 300 K

The average kinetic energy of the neutron is given as:

Where,

k = Boltzmann constant = 1.38 × 10⁻²³ J mol⁻¹ K⁻¹

The wavelength of the neutron is given as:

This wavelength is comparable to the inter-atomic spacing of a crystal. Hence, the high-energy neutron beam should first be thermalised, before using it for diffraction.