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, u2, ..., U2n-1, Uzn, Uzn+1, .... The harmonic forces between nearest-neighbour atoms are characterised by the alternating interatomic force constants B, and B2. (a) Develop: (i) The equation of motion for the 2nth 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 u, = Aei(wt-kna) and u,n41 = Beilwt-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 = only has a non-zero solution for x and y when С11 С12 = 0, C21 C22 obtain an expression for w?. (d) Making use of the approximation 1 q Vp2 – qx2 : - -- 2 p 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. b а U2n-1 U2n U2n+1 U2n+2

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, u2, ..., U2n-1, Uzn, Uzn+1, .... The harmonic forces between nearest-neighbour atoms are characterised by the alternating interatomic force constants B, and B2. (a) Develop: (i) The equation of motion for the 2nth 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 u, = Aei(wt-kna) and u,n41 = Beilwt-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 = only has a non-zero solution for x and y when С11 С12 = 0, C21 C22 obtain an expression for w?. (d) Making use of the approximation 1 q Vp2 – qx2 : - -- 2 p 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. b а U2n-1 U2n U2n+1 U2n+2