Debye model

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.

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

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. Making the approximation that the frequency is inversely proportional to the wavelength, we have:

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

in which 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 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.

Let's now compute the total energy in the box,

where N ¯ ( E n ) 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.

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 N 3 atoms per edge. Atom separation is then given by L / N 3 , and the minimum wavelength is

making the maximum mode number n (infinite for photons)

This is the upper limit of the triple energy sum

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

So far, there has been no mention of N ¯ ( E ) , the number of phonons with energy E . Phonons obey Bose–Einstein statistics. Their distribution is given by the famous Bose–Einstein formula

Because a phonon has three possible polarization states (one longitudinal, and two transverse which approximately do not affect its energy) the formula above must be multiplied by 3,

(Actually one uses an effective sonic velocity c s := c e f f , i.e. the Debye temperature T D (see below) is proportional to c e f f , more precisely T D − 3 ∝ c e f f − 3 := ( 1 / 3 ) c l o n g − 3 + ( 2 / 3 ) c t r a n s − 3 , where one distinguishes longitudinal and transversal sound-wave velocities (contributions 1/3 and 2/3, respectively). The Debye temperature or the effective sonic velocity is a measure of the hardness of the crystal.)

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

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 x = h c s n 2 L k T ,

To simplify the appearance of this expression, define the Debye temperature T D

Many references describe the Debye temperature as merely shorthand for some constants and material-dependent variables. However, as shown below, k T D is roughly equal to the phonon energy of the minimum wavelength mode, and so we can interpret the Debye temperature as the temperature at which the highest-frequency mode (and hence every mode) is excited.

Continuing, we then have the specific internal energy:

where D 3 ( x ) is the (third) Debye function.

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

These formulae treat 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. As already mentioned, this behaviour is exact, in contrast to the intermediate behaviour. The essential reason for the exactness at low and high energies, respectively, is that the Debye model gives (i) the exact dispersion relation E ( ν ) at low frequencies, and (ii) corresponds to the exact density of states ( ∫ g ( ν ) d ν ≡ 3 N ) , concerning the number of vibrations per frequency interval.

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

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

if the vibrational frequencies continued to infinity. This form gives the T 3 behavior which is correct at low temperatures. But Debye realized that there could not be more than 3 N 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 3 N :

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:

where D 3 is the function later given the name of third-order Debye function.

Another derivation

First we derive the vibrational frequency distribution; the following derivation is based on Appendix VI from. Consider a three-dimensional isotropic elastic solid with N atoms in the shape of a rectangular parallelepiped with side-lengths L x , L y , L z . The elastic wave will obey the wave equation and will be plane waves; consider the wave vector k = ( k x , k y , k z ) and define l x = k x | k | , l y = k y | k | , l z = k z | k | . Note that we have

Solutions to the wave equation are

and with the boundary conditions u = 0 at x , y , z = 0 , x = L x , y = L y , z = L z , we have

where n x , n y , n z are positive integers. Substituting (2) into (1) and also using the dispersion relation c s = λ ν , we have

The above equation, for fixed frequency ν , describes an eighth of an ellipse in "mode space" (an eighth because n x , n y , n z are positive). The number of modes with frequency less than ν is thus the number of integral points inside the ellipse, which, in the limit of L x , L y , L z → ∞ (i.e. for a very large parallelepiped) can be approximated to the volume of the ellipse. Hence, the number of modes N ( ν ) with frequency in the range [ 0 , ν ] is

where V = L x L y L z is the volume of the parallelepiped. Note that the wave speed in the longitudinal direction is different from the transverse direction and that the waves can be polarised one way in the longitudinal direction and two ways in the transverse direction; thus we define 3 c s 3 = 1 c long 3 + 2 c trans 3 .

Following the derivation from, we define an upper limit to the frequency of vibration ν D ; since there are N atoms in the solid, there are 3N quantum harmonic oscillators (3 for each x-, y-, z- direction) oscillating over the range of frequencies [ 0 , ν D ] . Hence we can determine ν D like so:

.

By defining ν D = k T D h , where k is Boltzmann's constant and h is Planck's constant, and substituting (4) into (3), we get

;

this definition is more standard. We can find the energy contribution for all oscillators oscillating at frequency ν . Quantum harmonic oscillators can have energies E i = ( i + 1 / 2 ) h ν where i = 0 , 1 , 2 , … and using Maxwell-Boltzmann statistics, the number of particles with energy E i is

The energy contribution for oscillators with frequency ν is then

.

By noting that ∑ i = 0 ∞ n i = d N ( ν ) (because there are d N ( ν ) modes oscillating with frequency ν ), we have

From above, we can get an expression for 1/A; substituting this into (6), we have

= d N ( ν ) ( 1 − e − h ν / ( k T ) ) ∑ i = 0 ∞ h ν ( i + 1 / 2 ) e − h ν i / ( k T ) = d N ( ν ) h ν ( 1 2 + ( 1 − e − h ν / ( k T ) ) ∑ i = 0 ∞ i e − h ν i / ( k T ) ) =

Integrating with respect to ν yields

Low temperature limit

The temperature of a Debye solid is said to be low if T ≪ T D , 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 . Using e x − 1 ≈ x 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.

Debye versus Einstein

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.

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, ϵ = ℏ ω = h ν . That frequency, if it indeed existed, would be related to the speed of sound in the 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 m i n .

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.

Alternately the ratio of the 2 temperatures can be seen to be the ratio of Einstein's single frequency at which all oscillators oscillate and Debye's maximum frequency. Einstein's single frequency can then be seen to be a mean of the frequencies available to the Debye model.

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 is proportional to 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 for the lattice contribution to the specific heat. The following table lists Debye temperatures for several pure elements:

The Debye model's fit to experimental data is often phenomenologically improved by allowing the Debye temperature to become temperature dependent; for example, the value for water ice increases from about 222 K to 300 K as the temperature goes from Absolute zero to about 100 K.

Extension to other quasi-particles

For other bosonic quasi-particles, e.g., for magnons (quantized spin waves) in ferromagnets instead of the phonons (quantized sound waves) one easily derives analogous results. In this case at low frequencies one has different dispersion relations, e.g., E ( ν ) ∝ k 2 in the case of magnons, instead of E ( ν ) ∝ k for phonons (with k = 2 π / λ ). One also has different density of states (e.g., ∫ g ( ν ) d ν ≡ N ). As a consequence, in ferromagnets one gets a magnon contribution to the heat capacity, Δ C V | m a g n o n ∝ T 3 / 2 , which dominates at sufficiently low temperatures the phonon contribution, Δ C V | p h o n o n ∝ T 3 . In metals, in contrast, the main low-temperature contribution to the heat capacity, ∝ T , comes from the electrons. It is fermionic, and is calculated by different methods going back to Arnold Sommerfeld.

Extension to liquids

It was long thought that phonon theory is not able to explain the heat capacity of liquids, since liquids only sustain longitudinal, but not transverse phonons, which in solids are responsible for 2/3 of the heat capacity. However, Brillouin scattering experiments with neutrons and with X-rays, confirming an intuition of Yakov Frenkel, have shown that transverse phonons do exist in liquids, albeit restricted to frequencies above a threshold called the Frenkel frequency. Since most energy is contained in these high-frequency modes, a simple modification of the Debye model is sufficient to yield a good approximation to experimental heat capacities of simple liquids.