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A potential of the form ULJ(r) = 4€ *e [C%)*- (-)) describes the energy of interaction between a pair of like particles separated by distance r, where o and € are constants. At sufficiently low temperature, these particles condense into a cubic crystalline solid, having a molar energy of U solid = 3NAUL counting only interactions between the particles and their nearest neighbours. Showing your work, determine the number of nearest neighbours in lattice. State what type of unit cell describes the lattice, giving your reasoning, The internal energy of a harmonic oscillator can be written as U(a) = 0; + 3w+ma? where w is the frequency, m is the oscillator mass, and x is the displacement. a Estimate w for a particle in the lattice. Recall that a Taylor series expansion is given by f'(6) f"(6) f'"() f(y) = f(b) + 1! (y-b) + 2! (y - 6)2 + 3! (y-b) +. where f is the function of interest, b is the central value of variable y, and primes denote derivatives with respect to y.