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 Phonon s 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

:

\lambda_n = {2L\over n}

where

n

is an integer. The energy of a phonon is

:

E_n\ =h
u_n

where

h

is Planck's Constant and

u_n

is the frequency of the phonon. We make the approximation that the frequency is inversely proportional to the wavelength, giving:

:

E_n=h
u_n={hc_s\over\lambda_n}={hc_sn\over 2L}

in which

c_s

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

:

E_n^2=E_{nx}^2+E_{ny}^2+E_{nz}^2=\left({hc_s\over2L}
ight)^2\left(n_x^2+n_y^2+n_z^2
ight)

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 Phonon s.) This is one of the limitations of the Debye model.

Let's now compute the total energy in the box

:

U = \sum_n E_n\,\bar{N}(E_n)

where

\bar{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:

:

U = \sum_{n_x}\sum_{n_y}\sum_{n_z}E_n\,\bar{N}(E_n)

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

\sqrt atoms per side.  Atom separation is then given by L/\sqrt[3 {N}

, and the minimum wavelength is

:

\lambda_{
m min} = {2L \over \sqrt

{Link without Title} {N}}

making the maximum mode number

n

(infinite for Photon s)

:

n_{
m max} = \sqrt

{Link without Title} {N}

This is the upper limit of the triple energy sum

:

U = \sum_{n_x}^{\sqrt

{Link without Title} {N}}\sum_{n_y}^{\sqrt {Link without Title} {N}}\sum_{n_z}^{\sqrt {Link without Title} {N}}E_n\,\bar{N}(E_n)

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

:

U \approx\int_0^{\sqrt

{Link without Title} {N}}\int_0^{\sqrt {Link without Title} {N}}\int_0^{\sqrt {Link without Title} {N}} E(n)\,\bar{N}\left(E(n) ight)\,dn_x\, dn_y\, dn_z

So far, there has been no mention of

\bar{N}(E)

, the number of phonons with energy

E

. Phonons obey Bose-Einstein Statistics . Their distribution is given by the famous Bose-Einstein formula

:

\langle N
angle_{BE} = {1\over e^{E/kT}-1}

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

:

\bar{N}(E) = {3\over e^{E/kT}-1}

Substituting this into the energy integral yields

:

U  = \int_0^{\sqrt

{Link without Title} {N}}\int_0^{\sqrt {Link without Title} {N}}\int_0^{\sqrt {Link without Title} {N}} E(n)\,{3\over e^{E(n)/kT}-1}\,dn_x\, dn_y\, dn_z

The ease with which these integrals are evaluated for Photon s 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 Phonon s. In order to approximate this triple integral, Debye used spherical coordinates
:

\ (n_x,n_y,n_z)=(n\cos 	heta \cos \phi,n\cos 	heta \sin \phi,n\sin 	heta )

and boldly approximated the cube by an eighth of a sphere
:

U \approx\int_0^{\pi/2}\int_0^{\pi/2}\int_0^R E(n)\,{3\over e^{E(n)/kT}-1}n^2 \sin	heta\, dn\, d	heta\, d\phi

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,

:

N = {1\over8}{4\over3}\pi R^3

so we get:

:

R = \sqrt

{Link without Title} {6N\over\pi}

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

The energy integral becomes

:

U = {3\pi\over2}\int_0^R \,{hc_sn\over 2L}{n^2\over e^{hc_sn/2LkT}-1} \,dn

Changing the integration variable to

x = {hc_sn\over 2LkT}

,

:

U = {3\pi\over2} kT \left({2LkT\over hc_s}
ight)^3\int_0^{hc_sR/2LkT} {x^3\over e^x-1}\, dx

To simplify the look of this expression, define the Debye temperature

T_D

-- a shorthand for some constants and material-dependent variables.

: $T_D\equiv {hc_sR\over2Lk} = {hc_s\over2Lk}\sqrt = {hc_s\over2k}\sqrt[3$

We then have the specific internal energy:

: $rac{U}{Nk} = 9T \left({T\over T_D}
ight)^3\int_0^{T_D/T} {x^3\over e^x-1}\, dx = 3T D_3 \left({T_D\over T}
ight)$

where

D_3(x)

is the (third) Debye Function .

Differentiating with respect to

T

we get the dimensionless heat capacity:

: $rac{C_V}{Nk} = 9 \left({T\over T_D}
ight)^3\int_0^{T_D/T} {x^4 e^x\over\left(e^x-1
ight)^2}\, dx$

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

:

n \sim {1 \over 3} 
u^3 V F

in which

V

is the volume and

F

is a factor which he calculated from Elasticity Coefficient s 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

:

U = \int_0^\infty \,{h
u^3 V F\over e^{h
u/kT}-1}\, d
u

if the vibrational frequencies continued to infinity. This form gives the

T^4

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

u_m

chosen so that the total number of states is

3N

:

:

3N = {1 \over 3} 
u_m^3 V F

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:

:

U = \int_0^{
u_m} \,{h
u^3 V F\over e^{h
u/kT}-1}\, d
u

= V F kT (kT/h)^3 \int_0^{T_D/T} \,{x^3 \over e^x-1}\, dx

::where

T_D

h
u_m/k

.

::

= 9 N k T (T/T_D)^3 \int_0^{T_D/T} \,{x^3 \over e^x-1}\, dx

= 3 N k T D_3(T_D/T)

where

D_3

is the function later given the name of third-order Debye Function .

LOW TEMPERATURE LIMIT

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

T \ll T_D

, leading to

:

rac{C_V}{Nk} \sim 9 \left({T\over T_D}
ight)^3\int_0^{\infty} {x^4 e^x\over \left(e^x-1
ight)^2}\, dx

This definite integral can be evaluated exactly:

: $rac{C_V}{Nk} \sim {12\pi^4\over5} \left({T\over T_D}
ight)^3$

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

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Debye Model