9780030839931, Chapter 24, Problem 1P

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Chapter 24, Problem 1P Problem (a) Draw diagrams showing some possible two-phonon processes in which a neutron enters with momentum p and leaves with momentum p'. In labeling the diagrams take due account of the conservation laws. (b) Repeat (a) for three-phonon processes. Step-by-step solution Step 1 of 4 (a) Physically, a lattice phonon does not carry any momentum, but it interacts with other particles and fields as if it has a momentum where K represents the wave vector of the phonon. Also from the de Broglie relation, p h = It is apparent that a phonon of wavelength carries a momentum hK The quantity hK is sometimes called the crystal momentum. The physical significance of hK is provided by the momentum conservation laws in crystals. Step 2 of 4 The wave vector conservation law for elastic scattering or the Bragg's diffraction of X-ray photons from crystals is given by, k' = k + G Here k' and k represent the wave vectors for the scattered and incident photons respectively and G is the reciprocal lattice vector. The conservation of momentum and energy gives, hk' = + hG And = Step 3 of 4 The below figure shows some possible emission and absorption two-phonon process in which a neutron enters with momentum P and leaves with momentum p'. p'(k') Neutron Neutron (w') p (k) p (k) Neutron Neutron K K Phonon (Ω) Phonon (Ω) Step 4 of 4 (b) In realistic, it is very difficult to draw the three-phonon process, because we have no idea about the conservation principle for three phonon system.