Debye frequency

In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the 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 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, which is proportional to T 3 {displaystyle T^{3}} – the Debye T3 law. Just like the Einstein model, it also recovers the Dulong–Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures. l x 2 + l y 2 + l z 2 = 1. {displaystyle l_{x}^{2}+l_{y}^{2}+l_{z}^{2}=1.}     (1) 2 l x L x λ = n x ; 2 l y L y λ = n y ; 2 l z L z λ = n z {displaystyle {frac {2l_{x}L_{x}}{lambda }}=n_{x};{frac {2l_{y}L_{y}}{lambda }}=n_{y};{frac {2l_{z}L_{z}}{lambda }}=n_{z}}     (2) N ( ν ) = 1 8 4 π 3 ( 2 ν c s ) 3 L x L y L z = 4 π ν 3 V 3 c s 3 , {displaystyle N( u )={frac {1}{8}}{frac {4pi }{3}}left({frac {2 u }{c_{mathrm {s} }}} ight)^{3}L_{x}L_{y}L_{z}={frac {4pi u ^{3}V}{3c_{mathrm {s} }^{3}}},}     (3) 3 N = N ( ν D ) = 4 π ν D 3 V 3 c s 3 {displaystyle 3N=N( u _{ m {D}})={frac {4pi u _{ m {D}}^{3}V}{3c_{ m {s}}^{3}}}} .    (4) N ( ν ) = 3 N h 3 ν 3 k 3 T D 3 , {displaystyle N( u )={frac {3Nh^{3} u ^{3}}{k^{3}T_{ m {D}}^{3}}},}     (5) d U ( ν ) = ∑ i = 0 ∞ E i 1 A e − E i / ( k T ) {displaystyle dU( u )=sum _{i=0}^{infty }E_{i}{frac {1}{A}}e^{-E_{i}/(kT)}} .    (6) θ D = ℏ k B ω D , {displaystyle heta _{ m {D}}={frac {hbar }{k_{ m {B}}}}omega _{ m {D}},} In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912 for estimating the 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 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, which is proportional to T 3 {displaystyle T^{3}} – the Debye T3 law. Just like the Einstein model, it also recovers the Dulong–Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures. The Debye model is a solid-state equivalent of Planck's law of black body radiation, where one treats electromagnetic radiation as a photon gas. The Debye model treats atomic vibrations as phonons in a box (the box being the solid). Most of the calculation steps are identical as both are examples of a massless Bose gas with linear dispersion relation. Consider a cube of side L {displaystyle L} . From the particle in a box article, the resonating modes of the sonic disturbances inside the box (considering for now only those aligned with one axis) have wavelengths given by where n {displaystyle n} is an integer. The energy of a phonon is where h {displaystyle h} is Planck's constant and ν n {displaystyle u _{n}} is the frequency of the phonon. Making the approximation that the frequency is inversely proportional to the wavelength, we have: in which c s {displaystyle c_{s}} is the speed of sound inside the solid.In three dimensions we will use: in which p n {displaystyle p_{n}} is the magnitude of the three-dimensional momentum of the phonon. The approximation that the frequency is inversely proportional to the wavelength (giving a constant speed of sound) is good for low-energy phonons but not for high-energy phonons (see the article on phonons.) This disagreement is one of the limitations of the Debye model, and corresponds to incorrectness of the results at intermediate temperatures, whereas both at low temperatures and also at high temperatures they are exact.