A common approximation to the interatomic forces in a material is the Lennard-Jones potential between neighboring atoms in a solid: Utr) = Afr\" - Bfr This can be used to nd various information about a solid material. Assume that for the lattice of a particular alloy of copper, the Lennard-Jones constants are: A = 1.5 x 1(1'133J-m12 B = 3.23 x 10'\" J'mf' Find the following: (a) The equilibrium distance between neighboring atoms in the copper lattice (this is called the lattice constant a). E (b) Model this as an approximate potential around the equilibrium position uelr) = {1f2)k{r-a}2 + C 3!- {i} What is the effective spring stiffness k {use the second derivative)? H (ii) what is the energy minimum C {use the original formula for U)? H {c} Find the (angular) resonance frequency of a copper atom in a linear chain of atoms? [Assume only:r the two nearest neighbor interactions with the effective above] H (d) Find the vibration amplitude at the temperature T : lac. Assume that the oscillator has one degree of freedom, so 1 E=l