In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912^[1] for estimating the phonon contribution to the specific heat (heat capacity) in a solid. The Debye model treats the vibrations of the atomic lattice (heat) as phonons in a box, in contrast to Einstein model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. This model correctly predicts the low temperature dependence of the heat capacity, which is proportional to T³. 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. Thermodynamics (from the Greek thermos meaning heat and dynamis meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ... Peter Joseph William Debye (March 24, 1884 - November 2, 1966) (born Petrus Josephus Wilhelmus Debije) was a Dutch physical chemist. ... In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ... The specific heat capacity (symbol c or s, also called specific heat) of a substance is defined as heat capacity per unit mass. ... In jewelry, a solid gold piece is the alternative to gold-filled or gold-plated jewelry. ... See Oscillator (disambiguation) for particular types of oscillation and oscillators. ... Rose des Sables (Sand Rose), formed of gypsum crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ... In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ... This article needs to be cleaned up to conform to a higher standard of quality. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... This article needs to be cleaned up to conform to a higher standard of quality. ... The Dulong-Petit law, found in 1819 by Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the specific heat capacity of a crystal due to its lattice vibrations. ...

Derivation

The Debye model is a solid-state equivalent of Planck's law of black body radiation, where one treats electromagnetic radiation as a gas of photons in a box. The Debye model treats atomic vibrations as phonons in a box (the box being the solid). Most of the calculation steps are identical. Black body spectrum as a function of wavelength In physics, the spectral intensity of electromagnetic radiation from a black body at temperature T is given by the Plancks law of black body radiation: where: I(ν) is the amount of energy per unit time per unit surface area per unit... Electromagnetic radiation can be conceptualized as a self propagating transverse oscillating wave of electric and magnetic fields. ... The results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number particles which do not interact with each other except for instantaneous thermalizing collisions. ... In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ...

Consider a cube of side 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 In physics the particle in a box refers to a simple mathematical exercise performed in quantum mechanics. ...

where n is an integer. The energy of a phonon is

where h is Planck's constant and ν_n is the frequency of the phonon. We make the approximation that the frequency is inversely proportional to the wavelength, giving: A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ...

in which c_s is the speed of sound inside the solid. In three dimensions we will use:

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 is one of the limitations of the Debye model. In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ...

Let's now compute the total energy in the box

where is the number of phonons in the box with energy E_n. In other words, the total energy is equal to the sum of energy multiplied by the number of phonons with that energy (in one dimension). In 3 dimensions we have:

Now, this is where Debye model and Planck's law of black body radiation differ. Unlike electromagnetic radiation in a box, there is a finite number of phonon energy states because a phonon cannot have infinite frequency. Its frequency is bound by the medium of its propagation -- the atomic lattice of the solid. Consider an illustration of a transverse phonon below. Black body spectrum as a function of wavelength In physics, the spectral intensity of electromagnetic radiation from a black body at temperature T is given by the Plancks law of black body radiation: where: I(ν) is the amount of energy per unit time per unit surface area per unit... In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ... In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ...

It is reasonable to assume that the minimum wavelength of a phonon is twice the atom separation, as shown in the lower figure. There are N atoms in a solid. Our solid is a cube, which means there are atoms per side. Atom separation is then given by , and the minimum wavelength is Download high resolution version (800x800, 60 KB)Lattice vibrations. ... In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ...

making the maximum mode number n (infinite for photons) In physics, the photon (from Greek φως, phōs, meaning light) is the quantum of the electromagnetic field; for instance, light. ...

This is the upper limit of the triple energy sum

For slowly-varying, well-behaved functions, a sum can be replaced with an integral (a.k.a Thomas-Fermi approximation)

So far, there has been no mention of , the number of phonons with energy E. Phonons obey Bose-Einstein statistics. Their distribution is given by the famous Bose-Einstein formula In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ...

Because a phonon has three possible polarization states (one longitudinal and two transverse) which do not affect its energy, the formula above must be multiplied by 3 Longitudinal waves, also referred to as compressional waves or pressure waves, are waves that have vibrations along or parallel to their direction of travel. ... A light wave is an example of a transverse wave. ...

Substituting this into the energy integral yields

The ease with which these integrals are evaluated for photons is due to the fact that light's frequency, at least semi-classically, is unbound. As the figure above illustrates, this is not true for phonons. In order to approximate this triple integral, Debye used spherical coordinates In physics, the photon (from Greek φως, phōs, meaning light) is the quantum of the electromagnetic field; for instance, light. ... In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ... Peter Joseph William Debye (March 24, 1884 - November 2, 1966) (born Petrus Josephus Wilhelmus Debije) was a Dutch physical chemist. ...

and boldly approximated the cube by an eighth of a sphere

where R is the radius of this sphere, which is found by conserving the number of particles in the cube and in the eighth of a sphere. The volume of the cube is N unit-cell volumes,

so we get:

The substitution of integration over a sphere for the correct integral introduces another source of inaccuracy into the model.

The energy integral becomes

Changing the integration variable to ,

To simplify the look of this expression, define the Debye temperature T_D -- a shorthand for some constants and material-dependent variables.

We then have the specific internal energy:

where D₃(x) is the (third) Debye function. In mathematics, the family of Debye functions is defined by Categories: Math stubs ...

Differentiating with respect to T we get the dimensionless heat capacity:

These formulae give the Debye model at all temperatures. The more elementary formulae given further down give the asymptotic behavior in the limit of low and high temperatures.

Debye's derivation

Actually, Debye derived his equation somewhat differently and more simply. Using the solid mechanics of a continuous medium, he found that the number of vibrational states with a frequency less than a particular value was asymptotic to Solid mechanics is the branch of physics and mathematics that concern the behavior of solid matter under external actions (e. ...

in which V is the volume and F is a factor which he calculated from elasticity coefficients and density. Combining this with the expected energy of a harmonic oscillator at temperature T (already used by Einstein in his model) would give an energy of Elasticity has meanings in two different fields: In physics and mechanical engineering, the theory of elasticity describes how a solid object moves and deforms in response to external stress. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

if the vibrational frequencies continued to infinity. This form gives the T⁴ behavior which is correct at low temperatures. But Debye realized that there could not be more than 3N vibrational states for N atoms. He made the assumption that in an atomic solid, the spectrum of frequencies of the vibrational states would continue to follow the above rule, up to a maximum frequency ν_m chosen so that the total number of states is 3N:

Debye knew that this assumption was not really correct (the higher frequencies are more closely spaced than assumed), but it guarantees the proper behavior at high temperature (the Dulong-Petit law). The energy is then given by: The Dulong-Petit law, found in 1819 by Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the specific heat capacity of a crystal due to its lattice vibrations. ...

where T_D is hν_m / k.

= 3NkTD₃(T_D / T)

where D₃ is the function later given the name of third-order Debye function. In mathematics, the family of Debye functions is defined by Categories: Math stubs ...

Low temperature limit

The temperature of a Debye solid is said to be low if , leading to

This definite integral can be evaluated exactly:

In the low temperature limit, the limitations of the Debye model mentioned above do not apply, and it gives a correct relationship between (phononic) heat capacity, temperature, the elastic coefficients, and the volume per atom (the latter quantities being contained in the Debye temperature).

High temperature limit

The temperature of a Debye solid is said to be high if T > > T_D. if | x | < < 1, leads to

This is the Dulong-Petit law, and is fairly accurate although it does not take into account anharmonicity, which causes the heat capacity to rise further. The total heat capacity of the solid, if it is a conductor or semiconductor, may also contain a non-negligible contribution from the electrons. The Dulong-Petit law, found in 1819 by Pierre Louis Dulong and Alexis Thérèse Petit, states the classical expression for the specific heat capacity of a crystal due to its lattice vibrations. ... Electrical conduction is the current (movement of charged particles) through a material in response to an electric field. ... A semiconductor is a material with an electrical conductivity that is intermediate between that of an insulator and a conductor. ...

Debye versus Einstein

Debye vs. Einstein. Predicted heat capacity as a function of temperature.

So how closely do the Debye and Einstein models correspond to experiment? -- Surprisingly close, but Debye is correct at low temperatures whereas Einstein is not. Debye vs. ... Debye vs. ...

How different are the models? To answer that question one would naturally plot the two on the same set of axes... except one can't. Both the Einstein model and the Debye model provide a functional form for the heat capacity. They are models, and no model is without a scale. A scale relates the model to its real-world counterpart. One can see that the scale of the Einstein model, which is given by

is ε / k. And the scale of the Debye model is T_D, the Debye temperature. Both are usually found by fitting the models to the experimental data. (The Debye temperature can theoretically be calculated from the speed of sound and crystal dimensions.) Because the two methods approach the problem from different directions and different geometries, Einstein and Debye scales are not the same, that is to say

which means that plotting them on the same set of axes makes no sense. They are two models of the same thing, but of different scales. If one defines Einstein temperature as

then one can say

and, to relate the two, we must seek the ratio

The Einstein solid is composed of single-frequency quantum harmonic oscillators, . That frequency, if it indeed existed, would be related to the speed of sound in the solid... even though there is no sound in Einstein solid. If one imagines the propagation of sound as a sequence of atoms hitting one another, then it becomes obvious that the frequency of oscillation must correspond to the minimum wavelength sustainable by the atomic lattice, λ_min. This article needs to be cleaned up to conform to a higher standard of quality. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...

which makes the Einstein temperature

and the sought ratio is therefore

Now both models can be plotted on the same graph. Note that this ratio is the cube root of the ratio of the volume of one octant of a 3-dimensional sphere to the volume of the cube that contains it, which is just the correction factor used by Debye when approximating the energy integral above.

Debye temperature table

Even though the Debye model is not completely correct, it gives a good approximation for the low temperature heat capacity of insulating, crystalline solids where other contributions (such as highly mobile conduction electrons) are negligible. For metals, the electron contribution to the heat capacity goes as T, which at low temperatures, dominates the debye T^3 result for lattice vibrations. In this case, the debye model can only be said to approximate the lattice contribution to the specific heat. The following table lists Debye temperatures for several substances:

See also

• Bose gas
• gas in a box

An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. ... The results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number particles which do not interact with each other except for instantaneous thermalizing collisions. ...

References

• ↑ 'Zur Theorie der spezifischen Warmen', Annalen der Physik 39(4), p. 789 (1912)
• CRC Handbook of Chemistry and Physics, 56th Edition (1975-1976)

External links

• Experimental determination of specific heat, thermal and heat conductivity of quartz using a cryostat.