Debye model

In thermodynamics and solid-state physics, the Debye model is a method developed by Peter Debye in 1912 to estimate phonon contribution to the specific heat (heat capacity) in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box in contrast to the Einstein solid model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low-temperature dependence of the heat capacity of solids, which is proportional to the cube of temperature – the Debye T 3 law. Similarly to the Einstein photoelectron model, it recovers the Dulong–Petit law at high temperatures. Due to simplifying assumptions, its accuracy suffers at intermediate temperatures.

Derivation

The Debye model treats atomic vibrations as phonons confined in the solid's volume. It is analogous to Planck's law of black body radiation, which treats electromagnetic radiation as a photon gas confined in a vacuum space. Most of the calculation steps are identical, as both are examples of a massless Bose gas with a linear dispersion relation.

For a cube of side-length

L

{\displaystyle L}

, the resonating modes of the sonic disturbances (considering for now only those aligned with one axis), treated as particles in a box, have wavelengths given as

λ

n

=

2

L

n

,

{\displaystyle \lambda _{n}={2L \over n}\,,}

where

n

{\displaystyle n}

is an integer. The energy of a phonon is given as

E

n

=

h

ν

n

,

{\displaystyle E_{n}\ =h\nu _{n}\,,}

where

h

{\displaystyle h}

is the Planck constant and

ν

n

{\displaystyle \nu _{n}}

is the frequency of the phonon.