1. Assume that a one-dimensional chain of Argon atoms with lattice spacing a are each tied to their nearest neighbour by a Lennard-Jones potential of form where a = 3.80100 m, e=1.6710-2 J. 12 12 X (a) Use calculus to verify that this potential has a minimum value - achieved when x = a. (iii) (1) (b) Assume that you can neglect all interactions except between nearest neighbours, and that the atoms can only vibrate along the chain, not perpendicular to it. In terms of e, a and the mass m of the atoms, find (i) (ii) a formula for the dispersion relation co(a) of the phonons by calculating the second derivative of Eqn (1), (i) a formula for the speed of sound and (ii) a numerical value for the speed of sound on this Ar crystal by calculatibng the third derivative of Eqn (1), (i) a formula for the thermal expansion coefficient and (ii) a numerical value for the thermal expansion coefficient of this Ar crystal, in the limit of high temperature