Lattice Vibrational Contribution to the Heat Capacity

Lattice Vibrational Contribution to the Heat Capacity The Thermal Energy is the dominant contribution to the heat capacity in most solids. In non-magnetic insulators, it is the only contribution. Other contributions: Conduction Electrons in metals & semiconductors. Magnetic ordering in magnetic materials. The calculation of the vibrational contribution to the thermal energy & heat capacity of a solid has 2 parts: 1. Evaluation of the contribution of a single vibrational mode. 2. Summation over the frequency distribution of the modes.

Vibrational Specific Heat of Solids cp Data at T = 298 K

Historical Background In 1819, using room temperature data, Dulong and Petit empirically found that the molar heat capacity at constant volume for solids is approximately (per mole): Here, R is the gas constant. This relationship is now known as the Dulong-Petit “Law”

The Molar Heat Capacity Assume that the heat supplied to a solid is transformed into the vibrational kinetic & potential energies of the lattice. To explain the Dulong-Petit “Law” theoretically, using Classical, Maxwell- Boltzmann Statistical Mechanics, a knowledge of how the heat is divided up among the degrees of freedom of the solid is needed.

The Molar Heat Capacity The Molar Energy of a Solid The Dulong-Petit “Law” can be explained using Classical Maxwell-Boltzmann Statistical Physics. Specifically, The Equipartition Theorem can be used. This theorem states that, for a system in thermal equilibrium with a heat reservoir at temperature T, The thermal average energy per degree of freedom is (½)kT If each atom has 6 degrees of freedom: 3 translational & 3 vibrational, then R = NAk

The Molar Heat Capacity Heat Capacity at Constant Volume By definition, the heat capacity of a substance at constant volume is Classical physics therefore predicts: A value independent of temperature

The Molar Heat Capacity Experimentally, the Dulong-Petit Law, however, is found to be valid only at high temperatures.

Einstein Model of a Vibrating Solid: I In 1907, Einstein extended Planck’s ideas to matter: He proposed that the energy values of vibrating atoms are quantized & proposed the following simple model of a vibrating solid: Assumptions: Each atom is independent. Each vibrates in 3-dimensions. Each vibrational normal mode has energy:

In effect, Einstein modeled one mole of a solid as an assembly of 3NA distinguishable oscillators. He used the Canonical Ensemble to calculate the average energy of an oscillator in this model.

The Partition Function To compute the average, note that it can be written as Z has the form: Here, b  [1/(kT)] Z is called The Partition Function

With b  [1/(kT)] and En = ne, , the partition function Z for the Einstein Model is This follows from the geometric series result

Differentiating Z with respect to b gives: Multiplying by –1/Z gives: This is the Einstein Model Result for the average thermal energy of an oscillator. The total vibrational energy of the solid is just 3NA times this result.

The Heat Capacity in the Einstein Model is given by: Do the derivative & define TE  e/k. TE is called “The Einstein Temperature” Finally, in the Einstein Model, CV has the form:

The Einstein Model of a Vibrating Solid Einstein, Annalen der Physik 22 (4), 180 (1907) CV for Diamond

Thermal Energy & Heat Capacity: Einstein Model II: Another Derivation The following assumes that you know enough statistical physics to have seen the Cannonical Ensemble & the Boltzmann Distribution! The Quantized Energy of a Single Oscillator has the form: If the oscillator interacts with a heat reservoir at absolute temperature T, the probability Pn of it being in quantum level n is proportional to the Boltzmann Factor: Pn 

Quantized Energy of a Single Oscillator: In the Cannonical Ensemble, a formal expression for the average energy of the harmonic oscillator & therefore of a lattice normal mode of angular frequency ω at temperature T is given by:

QM energy of one oscillator: Thermal average energy of one oscillator: The cannonical ensemble probability Pn of the oscillator being in quantum level n has the form: Pn  [exp (-β)/Z] where the partition function Z is given by:

Some straightforward math manipulation! Thermal Average Energy: Putting in the explicit form for n gives: This sum is a geometric series which can be done exactly:

The equation for E can be rewritten: Finally, the result is:

(1) This is the Mean Phonon Energy. The first term in (1) is called the Zero-Point Energy. As mentioned before, even at 0 K the atoms vibrate in the crystal & have a Zero Point Energy. This is the minimum energy of the system. The thermal average number of phonons n(ω) at temperature T is given by The Bose-Einstein (or Planck) Distribution, & the denominator of the second term in (1) is often written:

<E> = ћω[n() + ½] The number of phonons at temperature (1) (2) By using (2) in (1), (1) can be rewritten: <E> = ћω[n() + ½] In this form, the mean energy <E> looks analogous to a quantum mechanical energy level for a simple harmonic oscillator. That is, it looks similar to: So the second term in the mean energy (1) is interpreted as The number of phonons at temperature T & frequency ω.

ħω << kBT High Temperature Limit: <E> Temperature dependence of the mean energy <> of a quantum harmonic oscillator. Taylor’s series expansion of ex  for x << 1 <E> High Temperature Limit: ħω << kBT <E> At high T, <E> is independent of ω. This high T limit is equivalent to the classical limit, (the energy steps are small compared to the total energy). So, in this case, <E> is the thermal energy of the classical 1D harmonic oscillator (given by the equipartition theorem). <E>

ħω > > kBT “Zero Point Energy” Temperature dependence of the mean energy <> of a quantum harmonic oscillator. Low Temperature Limit: ħω > > kBT “Zero Point Energy” At low T, the exponential in the denominator of the 2nd term gets larger as T gets smaller. At small enough T, neglect 1 in the denominator. Then, the 2nd term is e-x, x = (ħω/(kBT). At very small T, e-x  0. So, in this case, <E> is independent of T: <E>  (½)ħω <E>

Einstein Heat Capacity CV The heat capacity CV is found by differentiating the average phonon energy: Let

Einstein Heat Capacity CV The specific heat CV in this approximation vanishes exponentially at low T & tends to the classical value at high T. These features are common to all quantum systems; the energy tends to the zero-point-energy at low T & to the classical value at high T. where Area =

Dulong-Petit law. The specific heat at constant volume Cv depends qualitatively on temperature T as shown in the figure below. For high temperatures, Cv (per mole) is close to 3R (R = universal gas constant. R  2 cal/K mole). So, at high temperatures Cv  6 cal/K-mole The figure shows that Cv = 3R at high temperatures for all substances. This is the classical Dulong-Petit law. This states that specific heat of a given number of atoms of any solid is independent of temperature & is the same for all materials!

Einstein Model for Lattice Vibrations in a Solid Cv vs T for Diamond Einstein, Annalen der Physik 22 (4), 180 (1907) Points = Experiment. Curve = Einstein Model Prediction

Einstein Model of Heat Capacity of Solids The Einstein Model was the first quantum theory of lattice vibrations in solids. He made the assumption that all 3N vibrational modes of a 3D solid of N atoms had the same frequency, so that the whole solid had a heat capacity 3N times In this model, the atoms are treated as independent oscillators, but the energies of the oscillators are the quantum mechanical energies. This assumes that the atoms is an isolated oscillator, which is not at all realistic. In reality, there are a huge number of coupled oscillators. Even this crude model gives the correct limit at high temperatures, where it reproduces the Dulong-Petit law of 3R per mole.

At high temperatures, all crystalline solids have vibrational specific heat Cv = 3R = 6 cal/K per mole; they require 6 calories per mole to raise the temperature 1 K. This agreement between observation & classical theory breaks down if the temperature is not high. Observations show that at room temperatures and below the specific heat of crystalline solids is not a universal constant. In each of these materials (Pb,Al, Si,and Diamond) the specific heat approaches a constant value asymptotically at high T. But at low T, the specific heat decreases towards zero which is in a complete contradiction with the above classical result.

The Einstein model also correctly gives a specific heat tending to zero at absolute zero, but the temperature dependence near T= 0 does not agree with experiment. Taking into account the actual distribution of vibration frequencies in a solid this discrepancy can be approximately accounted for using a model first proposed by Debye.