Question B2 Consider a one-dimensional chain of atoms, each joined to its nearest neighbours by a bond of spring constant J. The unit cell, of length a, contains one large atom of mass M, which we label = 1, and two smaller atoms of mass m, which we label l = 2 and l = 3. We denote the displacement of the eth atom in the nth unit cell by ur (note that this does not mean exponentiation): n-1 u +1 mm mmmmm M m m M m m a a) Give an expression for the force on each atom, F1, F2 and F3, in terms of the atomic displacements u. b) Use the standard ansatz u = u exp (i(kna - wt)) to write the equations of motion for this chain in the form (1) 01 = Pū² 03 0 where P is a 3 x 3 matrix. (Do not attempt to diagonalise this matrix.) (2) c) Show that at the Brillouin zone boundary (k = π/a), one solution to your matrix equation (2) is u₁ = 0, u² = u³ = 1. Calculate the angular frequency w of this mode. Explain why this value is independent of M. Is your answer consistent with the solution for a diatomic chain of alternating masses? (You may take the diatomic result from your notes or elsewhere without justification.) d) There is similarly a solution to (2) at the Brillouin zone centre (k = 0) for which the angular frequency is independent of M. Find this solution and determine its angular frequency w.

Question B2 Consider a one-dimensional chain of atoms, each joined to its nearest neighbours by a bond of spring constant J. The unit cell, of length a, contains one large atom of mass M, which we label = 1, and two smaller atoms of mass m, which we label l = 2 and l = 3. We denote the displacement of the eth atom in the nth unit cell by ur (note that this does not mean exponentiation): n-1 u +1 mm mmmmm M m m M m m a a) Give an expression for the force on each atom, F1, F2 and F3, in terms of the atomic displacements u. b) Use the standard ansatz u = u exp (i(kna - wt)) to write the equations of motion for this chain in the form (1) 01 = Pū² 03 0 where P is a 3 x 3 matrix. (Do not attempt to diagonalise this matrix.) (2) c) Show that at the Brillouin zone boundary (k = π/a), one solution to your matrix equation (2) is u₁ = 0, u² = u³ = 1. Calculate the angular frequency w of this mode. Explain why this value is independent of M. Is your answer consistent with the solution for a diatomic chain of alternating masses? (You may take the diatomic result from your notes or elsewhere without justification.) d) There is similarly a solution to (2) at the Brillouin zone centre (k = 0) for which the angular frequency is independent of M. Find this solution and determine its angular frequency w.