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Phonon II Thermal Properties

Physics, UCF

OUTLINES

Phonon heat capacityPlanck distributionNormal mode enumerationDensity of states in one dimensionDensity of states in three dimensionDebye Model for density of statesDebye T3 lawEinstein model of the density of statesGeneral result for D(ω)

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Phonon heat capacity

The heat capacity is defined as the energy needed to increase a material (solid) by one degree centigrade.

We are dealing with heat capacity at constant volume, which is more fundamental than the heat capacity at constant pressure. We are interested in the contribution to the heat capacity from the lattice.

The form of is given by the Planck distribution.

Eq. (1)

Eq. (2)

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Experimental observations

1. At room temperature, almost all monatomic solid have

2. At low temperature,

2a. Insulators

2b. Metals

2c. Glasses

~

~ ~or

~

3. Magnetic solid at low temperature, there are large contributions to from the ordering of spin.

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To find heat capacity, we need to find the energy of the system, for example, for an Einstein solid:

where is the occupation number. In Einstein model, . In general is a function of k.

There are three different kind of distribution function (occupation number).

1. Boltzmann Distribution

∝ ∝ Eq. (3)

This is the probability of a system will be in certain state as a function of that state’s energy and the temperature of the system.

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Maxwell-Boltzmann distribution at different temperatures

∝

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Bose-Einstein distribution

B-E distribution is also called Planck distribution. It was proposed initially by Max Planck to explain the deficiency in the Rayleigh-Jeans law (ultraviolet catastrophe). Planck found that if the energy is a discrete quantity, he will be able to explain the experimental data. This was the starting of quantum physics.

At low temperature, B-E distribution approaches Boltzmann distribution.

∝

∝ ∝ Eq. (4)

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The expression in Eq. (2) in the previous page can be derived as follow: Start with Boltzmann factor and a system of SHO and assume the energy is quantized.

# of particles in nth energy state

/

/

/

The ratio of number of oscillators at different energy (with ≡ )

The average occupation number at temperature T is

∑ · /

∑ /Eq. (5)

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Eq. (5) can be solved this way. Since for small x,

·

and

/

/ /

This is eq. (4) on page 7. This is the probability distribution of bosons, such as phonons or photons. Next we will see how to utilize this distribution to calculate certain physical quantities.

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To find the average energy of a collection of oscillators with frequencies , we can use the following expression

It is common to replace the above summation equation with an integration over the angular frequency

Here is called density of modes or density of states. Density of states is an important concept in Solid State Physics.

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Density of States

Density of states defined as the number of states per unit energy (angular frequency)

So when we calculate the total energy of a system, we can either (a) find the energy per state and then sum over all states; (b) or we can calculate the number of states per unit energy and then integrate over all allowed energy.

Method (a)

Method (b) · · ,

≡ Eq. (6)

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Density of States in 1D

Consider the boundary value problem for vibrations of a 1D line of length L with N+1 particles at a separation a. The boundary condition is that at both ends, the particles are held fixed. The displacement of the particle is given by

The wavevector K is restricted by the boundary conditions

and can be see above that , , , ⋯ ·

· · Eq. (7)

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From the figure in k-space, we can see that there are one point (mode) for every ∆ (this is volume in 1D),

therefore the density of mode is

· ·

··

Where is the group velocity, defined as .

For example, if we use eq. (9) (Chap. 4) as an example:

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The dispersion relation is given by

· · ·/

· Eq. (8)

.

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For small

~/

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Density of State in 3D

First, we will use the periodic boundary such that the last point (N) is wrapped around just like in the 1D case, so

≡

This leads to the following

, , ; ; ; ;

To count the number of states is the same as to count the number of points in the k-space

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So there is one state per volume

or density of points in k-space is

Total number of states (points) in a sphere with radius k

·

And the density of states per polarization is

· Eq. (11)

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Debye model for density of states

In Debye model, the velocity of sound is taken as a constant for each polarization.

Debye model

and Eq. (11) becomes

In the Debye model, the density of states depends on the energy (frequency) quadratically.

· Eq. (12)

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To find the cut-off frequency,

Similarly the cutoff wave-vector, can be expressed this way

/

The sphere in the reciprocal space is called the Debye

sphere.

Eq. (13)

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To find the energy in the system, we used the method (b) discussed in page 11.

/ (Let

The specific heat is given by

Debye integral

/

(14)

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In the previous slide, we let and at cutoff frequency

, we define

≡ ≡ (15)

This defines the Debye temperature in terms of , using eq. (13) we obtain

·

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Re-arrange the previous equation, we end up with

or

At high T, ≫

lim→

lim→

/(15)

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At low T, ≪ ( ≫

.

~ As T →

This is lattice contribution to the specific heat at low T.

Almost all insulators at low temperature obey law.

Exceptions

1. Metals ~2. Amorphous materials ~

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In Debye model, the density of states depends on the dimensionality of the space.

3-D ·

2-D ·

1-D ·

For phonons, sound velocity is a constant, therefore

∝ in 3D

∝ in 2D

∝ in 1D

· (16)

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Now if we want to know the density of state of electrons in solid, then we can see that the dispersion relation is different.

Substitute into eq. (16), we obtain

· ∝ ∝3D

· .2D

· ∝ ∝1D

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Einstein model of the density of state

In Einstein model, it is assumed that all oscillators have the same frequency, so the total energy is

/(17)

The heat capacity is given by

/

/ (18)

At high T, ≫

≅

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At low T, ≪

(19)

However this does not agree with experimental data of ~ at low temperature. This is due to the simplicity of

the model. However, this model can be applied to the optical branch of the phonon, since optical mode does not depend on k that much.

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Specific heat of free electron gas

In unit#1, we mentioned that the Drude model could not explain why the classical prediction of the specific heat was never realized.

Drude Model

Experimentally, specific heat of most metals are

~ . ·

This is due to the fact that electrons are fermions and follow Fermi-Dirac statistics

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Namely the electrons fill up to the Fermi level, . At temperature T most of the states are filled and only the electrons within of will be excited by the thermal energy.

· ·

For most metals, , so at room Temp, 1% of the electrons will be excited.

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General result for the density of state

Eq. (12) on page 18 gives us the expression for the density of states in Debye approximation. Here we want to derive a general expression for the density of state without any approximation.

First ·

Volume in the reciprocal space

Density of states

(20)

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·

Where is the constant energy surface in the reciprocal space and is the distance between surfaces and

The gradient of ω is also normal to the constant ω surface,

(21)

Substitute back into (21) and (20)

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