Thermal neutrons can be used to probe the structure of crystals because they undergo diffraction similar to X-rays.
(i) Starting from the definitions of kinetic energy and momentum for a non-relativistic particle of mass m, derive an expression that relates its de Broglie wavelength λ to its kinetic energy Ek.
(ii) A crystal has parallel atomic planes spaced d=0.15nm apart. For first-order Bragg reflection (n=1) at a glancing angle of 30∘, obtain an expression for the kinetic energy of neutrons that will satisfy the diffraction condition in terms of the given quantities and fundamental constants.
(iii) Using mn=1.675×10−27kg, calculate the speed of such neutrons.
(iv) Explain briefly how the observation of diffraction of neutrons supports the concept of wave–particle duality.