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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

Ueff(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 kB = 1.38⨯10-23 J/K/atom as Boltzmann's constant. ]

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