A common approximation to the interatomic forces in a material is the Lennard-Jones potential between neighboring...

A common approximation to the interatomic forces in a material is the Lennard-Jones potential between neighboring atoms in a solid:

U(r) = A/r¹² - B/r⁶

This can be used to find various information about a solid material.

Assume that for the lattice of a particular alloy of copper, the Lennard-Jones constants are:

A = 9.08 ⨯ 10^-133 J·m¹²
B = 3.96 ⨯ 10^-76 J·m⁶

Find the following:

(a) The equilibrium distance between neighboring atoms in the copper lattice (this is called the lattice constant a).

(b) Model this as an approximate potential around the equilibrium position

U_eff(r) = (1/2)k(r-a)² + C

(i) What is the effective spring stiffness k?

(ii) What is the energy minimum C?

(c) Find the resonance frequency of a copper atom in a linear chain of atoms? [Assume only nearest neighbor interactions]

(d) Find the vibration amplitude at the temperature T = 5⁰C. Assume that the oscillator has one degree of freedom.

[ Use k_B = 1.38⨯10^-23 J/K/atom as Boltzmann's constant. ]

Expert Solution

genius_generous answered 5 months ago

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