1. A linear lattice with lattice constant a has a basis of two identical atoms of mass m where the equilibrium spacing of the atoms within each unit cell is b (where b <. The displacements of the atoms from their equilibrium positions are given by u1, uz, ...,U2n-1, Uzn, Uzn+1, ... The harmonic forces between nearest-neighbour atoms are characterised by the alternating interatomic force constants B1 and B2. (a) Develop: (i) The equation of motion for the 2nh atom in terms of forces exerted by the (2n – 1)th and (2n + 1)th atoms. (ii) The equation of motion for the (2n + 1)th atom in terms of forces exerted by the 2nth and (2n + 2)th atoms. (b) Using the equations of motion and assuming travelling wave solutions of the form uzn = Aelwt-kna) and uzn+1 = Be(wt-kna-kb). derive two simultaneous equations for A and B. (c) Making use of the fact that a homogeneous system of linear equations C11x + C12y = 0 C21x + c22y = 0 only has a non-zero solution for x and y when C11 C12 = 0, C21 C22 obtain an expression for w?. (d) Making use of the approximation 14 Vp2 – qx² × p 2p for small x, determine the dispersion relation for the acoustic branch in the long-wavelength limit and thus find the group velocity of acoustic waves in the lattice.

1. A linear lattice with lattice constant a has a basis of two identical atoms of mass m where the equilibrium spacing of the atoms within each unit cell is b (where b <. The displacements of the atoms from their equilibrium positions are given by u1, uz, ...,U2n-1, Uzn, Uzn+1, ... The harmonic forces between nearest-neighbour atoms are characterised by the alternating interatomic force constants B1 and B2. (a) Develop: (i) The equation of motion for the 2nh atom in terms of forces exerted by the (2n – 1)th and (2n + 1)th atoms. (ii) The equation of motion for the (2n + 1)th atom in terms of forces exerted by the 2nth and (2n + 2)th atoms. (b) Using the equations of motion and assuming travelling wave solutions of the form uzn = Aelwt-kna) and uzn+1 = Be(wt-kna-kb). derive two simultaneous equations for A and B. (c) Making use of the fact that a homogeneous system of linear equations C11x + C12y = 0 C21x + c22y = 0 only has a non-zero solution for x and y when C11 C12 = 0, C21 C22 obtain an expression for w?. (d) Making use of the approximation 14 Vp2 – qx² × p 2p for small x, determine the dispersion relation for the acoustic branch in the long-wavelength limit and thus find the group velocity of acoustic waves in the lattice.