13 Lattice Vibrations and Thermal Energy 4

Mahavir Singh

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   The objective of the module is to
  • Acquire understanding of the basic Einstein’s quantum model.
  • Observe the acceptance of Einstein’s theory over Classical theory.
  • Understand the need of further modifications in the basic quantum theory.
  • Get an understanding of concept of Density of States and modes of vibration.
  • Appreciate the success of Debye’s theory over the shortfall of Einstein’s theory Mathematically arrive at Debye’s T3 law at low temperatures.

    1.Introduction

 

At low temperatures, the specific heat Cv falls below the limiting value of 3NR, where R is universal gas constant. It was assumed that some of the oscillators stop vibrating as the temperature is lowered, and Einstein supported the logic on the basis of Plank’s hypothesis on quantization of energy. As a consequence, the there is a decrease in the number of oscillators oscillating at lower temperatures (below room temperatures).

 

An oscillator must possess at least one quantum of energy (phonon) to oscillate at all. But the oscillators which do not have even single quantum will not oscillate and have zero energy according to Plank’s hypothesis.

 

But Einstein visualized some ingenuity in this and rather assumed a rigid lattice made of identical oscillators and all of them to be oscillating with same frequency.

 

2.  Einstein’s Theory

 

To understand the dip in specific heat curve at low temperatures Einstein employed a physical model (although oversimplified to arrive at desired results), but it indicated that the problem had its solution in quantum approach.

 

The suggested model assumes the following

  • lattice containing N atoms is equivalent to 3N harmonic oscillators
  • atoms vibrate independently of each other
  • all atoms vibrate with same frequency ( as all have same environment)
  • harmonic oscillators have discrete quantized energy levels represented as

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Now for T=0,  hν /2 and not zero, so the first term in equation (8) is referred to as ‘zero point energy’ and according to quantum mechanics the atoms possess vibrational energy even at absolute zero.

 

Recalling the question that why did the Dulong and Petit’s law fail at lower temperatures?

 

The energy hν of the oscillator indicates the difference between the allowed energy states of the given oscillator. In case the difference is small, ie hν /kT is negligible in comparison with unity then Cv will be equal to 3R at all finite temperatures.

 

However at low temperatures, unity is negligible as compared to hν /kT and Cv becomes proportional to.

e-hv/KT

Heat capacity decreases exponentially at low temperatures.

 

This is in fact a indirect evidence of quantization of energy  as proposed by Plank’s hypothesis. At kT>>hν, equation (8) reduces to classical result value, however at lower temperatures specific heat decreases.

 

It is better understood by rewriting equation (6) in terms of θE, (Einstein temperature) as

 

 

 

Where FE  is Einstein’s function determining the ratio of specific heat at temperature T and classical value 3R.

 

For T >> θE,

 

Cv =3R,

 

ie Dulong Petit law is obeyed.

 

For T << θE,

 

,  Cv = 3R(θE/)2

 

the specific heat is proportional to θE/T .

 

This is in good agreement with experimental results at higher temperatures but fails to give exact values at very low temperatures. The reason for this discrepancy must be sought in the oversimplification of the model assumed by Einstein considering the vibrations to be independent of each other.

 

A modification suggesting a solution is made by Debye known as Debye’s approximation theory.

 

3.  Debye’s Theory

 

Although Einstein’s assumption that the atoms vibrate independently of each other in the lattice with a constant frequency had been very successful in explaining the exponential decrease in heat capacity at low temperatures but the predicted decrease was much sharper than the experimentally observed values. Further Einstein’s model failed to explain and fit T3 dependence

 

To explain the variation of specific heat of solids with temperature, Debye relied on the following assumptions

  • the vibrations of all atoms are coupled together
  • solid acts as an isotropic elastic continuum
  • atomic vibrations produce a continuous spectrum of frequencies
  • the frequency is same for longitudinal and transverse vibrations produced in solids
  • beyond a certain frequency VD (Debye frequency), no vibrations are produced

    Consider an elastic wave u(x, y, z, t) propagating through a medium. Writing its wave equation as

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Eqn (12) shows that the different values of frequencies are quantized.

Different no. of modes of vibration can be determined using this equation.

 

Let Z(ν) is the number of modes of vibration in frequency interval ν and ν+ dν

Therefore, the number of modes of vibration is

 

 

 

 

 

 

Equation (13) gives the possible modes of vibration also known as Density of states of elastic continuous medium.

In deriving equation (12) it has been assumed that the velocity v is irrespective of longitudinal or transverse nature of the wave propagating in the medium. But actually the frequency is associated with one longitudinal mode and two transverse modes; hence equation (13) gets modified as

 

 

 

Equation (14) gives the total number of modes of vibration in elastic continuum medium.

 

Debye proposed that a solid is a continuously vibrating medium giving rise to a spectrum of frequencies (wavelengths comparable to inter-atomic separations) called Debye’s continuum. Then the total energy expression becomes

 

 

 

As there are N atoms therefore the frequency spectrum has to comply with 3N modes of vibration i.e there must be a maximum frequency VD beyond which no frequencies are possible such that the internal energy expression becomes,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The following conclusions are drawn

 

1)  for high temperature range T>>θD, equation (16) reduces to classical limit i.e,

 

 

 

 

 

 

 

 

 

 

 

This is Debye’s T3 law which holds for sufficiently low temperatures for long wavelength excitations, i.e the modes that may be treated in elastic continuum with macroscopic elastic constants.

 

Except for the extreme situation of temperatures, Debye’s approximation falls back to classical model.

 

you can view video on Lattice Vibrations and Thermal Energy 4

    4. Summary:

   After the completion of this module we are able to

  • Understand the lapse in explaining dip in specific heats at low temperatures which is not explained as in classical theory.
  • Study a basic quantum model designed by Einstein in accordance with Plank’s hypothesis.
  • Relate lattice behavior to independent harmonic oscillators vibrating with same frequency.
  • How Einstein’s model successfully arrived at experimental results at higher temperatures but at the same time failed to explain the same at low temperatures. We have come across the limitations of the most accepted theories.
  • Understand the need of modifications in the basic quantum model to explain dip in specific heats at low temperatures.
  • How Debye’s modified model with assumption of the coupled vibrations of  lattice over a Continuous spectrum of frequencies removed this discrepancy.

    Value Addition:

 

Do You Know?

 

Debye’s approximations leading to accurate measurements in the low temperature region still showed deviations from theoretical predictions. As we observed that Debye’s theory suggests thst T3 law should hold in low temperature regions i.e, T≤0.1 θD .

 

But it deviated from actual results from the data produced by Blachman Paper.

 

It seems that these deviations still point out doubts at continuum approximations. The deficiencies in the results were further taken up by Blackman and Kellermann that expected T3 law to hold for temperature region T≤ θD/50, i.e, at considerably lower temperature than predicted by Debye in his approximation.

 

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

 

Adrianus J Dekker,Solid State Physics

Charles Kittel, Introduction to Solid State Physics.

James D Patterson Bernard C Bailey, Solid State Physics, Introduction To Theory

    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.

 

Density of modes:

 

It’s the number of modes of vibration per unit interval. The density of modes per unit volume is a constant independent of the magnitude or the shape of periodicity of the volume.

 

Debye frequency:

 

It is a critical frequency chosen so that the density of normal modes is 3N. Debye frequency is dependent only on the velocity of sound in solid.

 

Debye temperature:

 

When the temperature greatly exceeds θD =hν/k ,(the debye temperature), Dulong petit law is recovered.