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.

 

[charng]

As a scientist in the field of solid state physics, I can provide some insight into the concept of harmonicity in phonon transport.

1. Phonons can be considered as harmonic vibrations when the interatomic forces between atoms in a crystal lattice are linear and the potential energy is quadratic. This means that the force between atoms is directly proportional to the displacement, and the potential energy of the atoms is directly proportional to the square of their displacement. In this case, the lattice vibrations can be described by simple harmonic motion, similar to an elastic spring.

2. The assumption that phonons are harmonic vibrations is based on the general behavior of materials at low temperatures. At low temperatures, the thermal energy is not sufficient to cause significant anharmonic effects, and the interatomic forces can be approximated as linear. This allows for the simplification of the lattice dynamics to a harmonic system. However, at higher temperatures or for certain materials with strong interatomic interactions, anharmonicity can play a significant role in phonon transport.

In summary, the assumption of harmonic phonons is a simplification that is valid under certain conditions, but it is important to recognize that anharmonicity can also play a significant role in phonon transport in some cases. Further research and understanding of the interatomic forces and potential energy in different materials can help us better understand the behavior of phonons and their role in thermal transport.