Harmonicity in phonon transport

[janakiraman]

Well again I'm very new to the field of solid state physics. I understand that phonons are the lattice vibrations which are transferred from one atom to another. In case of harmonic vibration, the phonons are similar to the elastic spring and the atoms are considered like the balls attached to the spring that are vibrating. However i would like to know

1. under what case can we consider the phonons as harmonic vibrations and why?
2. what is the basis of the assumption that phonons are harmonic vibration (because in elastic springs, the force constant varies linearly with respect to the displacement, while the interatomic potential varies non linearly with respect to the displacement).

 

[Gokul43201]

The anharmonicity must be taken into account for some phenomena. For instance, you can not explain the thermal expansion of solids in the harmonic approximation (you get the wrong answer that the expansion coefficient = zero).

But for many cases where the amplitude of oscillations of each lattice point is small compared to the lattice spacing, a harmonic approximation works pretty well. Incidentally, in an ideal elastic spring, the force varies linearly with distance, the force constant does not (it is a constant). And in the spring the potential energy goes like x^2. This x^2 dependence is a pretty good approximation to the interatomic potential in the vicinity of the equilibrium spacing.

See here: http://www.doitpoms.ac.uk/tlplib/stiffness-of-rubber/images/image01.gif

The bottom portion of the potential is pretty close to parabolic. Hence the justification for the approximation.

Alternatively, you can Taylor expand the potential about the equilibrium spacing, using the condition that the force (-dU/dx) is zero at the equilibrium position. This throws away the linear term and leaves you with quadratic and higher terms. For sufficiently small displacements, you can throw away terms of third order or higher and you are left with a quadratic (or harmonic) potential in the vicinity of the equilibrium spacing.