12 Lattice Vibrations and Thermal Properties 3

Mahavir Singh

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The objective of the module is to understand the following

  • Reviewing the concept of specific heat
  • Concept of lattice specific heat in preview of lattice dynamics
  • Reviewing the explanation of specific heat on the basis of classical model given by Dulong and Petit’ law.
  • Concept of quantization of elastic waves and phonons
  • Understanding of the shortfall of classical theory and the need of the other.

   1.Lattice Specific Heat : An Overview

 

Since heat is not a thermodynamic quantity, however it becomes so under the constraints of constant volume and constant pressure i.e,

 

dQ|p = dH

 

and

 

dQ|v = dE

 

where H and E are enthalpy and internal energy respectively.

 

The specific heat is defined as the amount of heat energy required to raise the temperature of a unit mass of solid by one degree. Also it is the heat capacity per unit mass. Consequently

 

(∂H/∂T)p = Cp and (∂E/∂T)v = Cv are then the expressions for heat capacity at constant pressure and heat capacity at constant volume respectively.

 

For solids and liquids, Cp ~ Cv, especially at low temperatures (~0-20K), but even at higher temperatures (~300K), the difference is not more than 2%.

 

With the supply of heat energy to a solid, there is an increase in its internal energy. The increase in the internal energy is manifested as an increase in the vibrations of the atoms about their mean positions and also as an increase in the kinetic energies of the free electrons. The specific heat corresponding to lattice energy is called lattice specific heat. Table below shows specific heats of a few materials.

 

There are various theories of lattice specific heat. In all the theories the vibrational energy of a crystal containing N number of atoms is considered equivalent to the energy of a 3N harmonic oscillator. The distinction in various theories is the difference in proposed frequency spectrum of the oscillators and the problem regarding calculations of wavelengths and frequencies of the possible modes of vibration in the crystal remains the centric concern.

 

In order to understand the lattice dynamics and the phenomenon that occur when heat or thermal energy is supplied to a solid and is consequently raised to a temperature (it occurs even at absolute zero), it will be easier to begin with the first law of thermodynamics which states that whenever some amount of heat dQ is given to a system, it results in the increase in energy dE of the system plus the amount of work done, i.e

 

dQ = dE+ pdV —————(1)

 

Considering that the work done by the system is of a mechanical nature only.

Now, E is determined uniquely by the temperature and volume of the system. Hence

 

 

 

 

 

 

 

 

 

dQ/dT is the general expression for specific heat. Our interest remains in Specific heat at constant volume Cv and specific heat at constant pressure Cp.

 

From equation (2)

 

 

 

The second law of thermodynamics is stated as relation between Cp and Cv in the form

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

As depicted from fig (1) at low temperatures their difference becomes very small and both Cp & Cv go to zero at T = 0.

 

In general, the variation of Cv with time is studied. We assume that a change in volume is not much with a small increase in temperature.

 

As the temperature increases from absolute zero, there is a rapid increase in specific heat and it finally levels off at a nearly constant value (6 cal/mole) at high temperature as shown in figure(2) below. This is the classical approach put forward as in Dulong Petit’s Law.

 

2. Dulong and Petit Law (Classical overview)

 

The physical properties of the solids is roughly assumed to be

  • due to the contributions resulting from the atomic vibrations (as in crystals)
  • due to additional contribution to specific heat from electronic system (as in metals and semiconductors). Although this contribution is relatively small to that of lattice vibrations.

    Taking these assumptions into consideration and assuming the behaviour of the atoms of the crystal as to be independent classical harmonic oscillator, the heat capacity of the solid crystal can be calculated simply by finding average thermal energy of a single oscillator and then multiplying it by total number of oscillators, N.

 

In comparison to the translational motion of the ideal gas molecules, the constituent atoms in a crystal , have almost fixed positions which vibrate about their mean positions executing simple harmonic motion. The total energy of the oscillator at an instant is a composition of its instantaneous potential energy and kinetic energy. Classically a harmonic oscillator vibrating with its natural frequency ω has energy expression as

 

E = P2/2m + ½ mω2(x2+ y2 + z2)

E = P2/2m +mω2q2/2 ———————–(6)

 

where p is the momentum in kinetic energy term and q is the displacement from equilibrium position in potential energy term .

 

The average energy of each harmonic oscillator as given by Plank’s distribution law is

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This gives the vibrational energy of one harmonic oscillator. To obtain total vibrational energy of the crystal whish has N independent harmonic oscillators in three dimensions, we have

 

 

 

 

 

 

This is called Dulong Petit’s classical law of specific heat.

 

This turns out to be that the heat capacity is independent of temperature and has a constant value at room temparature.

 

 

3. Shortfall of Classical Theory:

 

The classical theory gives a good idea of energy contributions to the total energy of the system as long as the volume remains constant, which is given in equation (9).

 

The specific heat at constant volume as obtained for a solid containing N number of atoms is given as

 

 

 

This is the level off value of the experimental results at high temperature.

 

Equation (8) is a frequency independent expression and also it depends only on temperature. The results obtained here are in quantitative agreement with those obtained from experiments but at higher temperatures only.

 

This does not explain the decrease of specific heat at low temperatures as observed for all solids.

 

The discrepancy to a certain extent is removed by using quantum theory but before that it is important to understand the concept of quantization of elastic waves and the concept of phonons.

 

4. Quantization of Elastic Waves:

 

As is a well understood concept that the atoms in solids vibrate about their equilibrium positions and these lattice vibrations can be expressed in the form of waves, like repetitive and systematic sequence of atomic displacements that can be longitudinal or transverse or even a combination of both.

 

The energy of a lattice vibration is quantized and the quantum of energy is called ‘Phonon’, analogous to quanta of electromagnetic energy – photon.

 

We have assumed vibrations of a linear lattice connected through springs and their particle motion can be quantized in a similar way as that for a harmonic oscillator or a combination of coupled harmonic oscillators. The Hamiltonian for a harmonic oscillator is

 

H = (1/2M) p2 + ½ Cx2

 

With energy eigen values for n = 1, 2 , 3 ,….

 

Hence average thermal energy of the oscillator is

 

Єn= (n + ½)ħω

 

Where ω is the angular frequency.

 

The term ½ ħω is called zero point energy of the mode

 

 

Thus thermal lattice vibrations are thermally excited phonons. Thermal conduction in non metallic crystals is a consequence of annihilation or creation of a phonon. The energy of phonons is ~0.1 ev.

 

We know that if the particles have zero spin then these follow Bose – Einstein statistics and the probability of such particles having energy E is given as

 

 

 

 

 

 

 

 

 

The following observations confirm the experimental evidence for the quantization of lattice vibrational energy:

  • at absolute zero the lattice contribution of heat energy always approaches zero
  • the neutrons and X rays are scattered inelastically by crystals and the changes in energy and momentum correspond to creation or absorption of phonons.

    (Which is discussed in detail in the later section of the modules.)

 

5. Summary:

 

After the completion of this module we are able to understand and correlate the following

  • Heat capacity, a thermodynamic concept as a function of temperature.
  • Classical theory of specific heat capacity and discrepancy with the experimental results obtained at lower temperatures.
  • Shortfall of the Dulong Petit’s classical theory and need of new (quantum) approach.
  • Quantization of elastic waves, its physical significance and mathematical treatment.
  • Introduction of concept of phonons and its relevance in lattice dynamics.
you can view video on Lattice Vibrations and Thermal Properties 3

    Value Addition:

 

Do You Know?

 

The relation between specific heat at constant pressure and at constant volume in terms of coefficient of volume expansion αv and the compressibility K is given as

 

Cp – Cv = αv2TV/K,

 

Cp – Cv ≥ 0                                             αv and K being positive quantities,

 

And in general the relationship employed is

 

Cp = Cv (1+ γαvT)

 

γ   = αv V / KCv, is Grüneisen constant and is independent of temperature.

 

By calculation of γ at any arbitrary temperature, an approximation for Cv is obtained.

 

As already discussed that a part of energy given to a system results in the increase in its internal energy which is associated with more vigorous notion of the atoms. In metals and semiconductors the remaining contribution in the specific heat comes from the electronic system, although this contribution is very small as compared to that of the lattice.

 

Csolid =Clattice + Celectronic

 

Anomalies have been observed in specific heat curves for ferromagnetic metals and e.g, nickel, iron and cobalt. A prominent peak appears near Curie temperature. This is correlated with transition from ordered state to disordered state. The alloys show more peaks in specific heat curves that exhibit similar transitions.

 

For More Details ( on this topic and other related topics ) See

 

Adrianus J Dekker, Solid State Physics

 

Glossary:

 

Elastic vibrations:

 

Waves when require a material medium to travel (unlike e.m waves), are called elastic vibrations.

 

These can have both longitudinal and transverse modes of vibration.

 

Crystal momentum:

 

Phonons do not carry any momentum on the lattice but a phonon wave vector k interacts with the crystal constituents and other particles as if its momentum is ħk, more precisely called Crystal momentum