1. Einstein Model for the Vibrational Heat Capacity of Solids

2. Vibrational Heat Capacity of Solids R is the Ideal Gas Constant!
Einstein Model:
Vibrational Heat Capacity of Solids
Starting as early as the late 1600’s, people started
measuring the heat capacities Cp (or Cv) of various
solids, liquids & gases at various temperatures.
1819: Dulong & Petit observed that the molar
heat capacity for many solids at temperatures
near room temperature (300 K) is approximately
independent of the material & of the temperature
& is approximately
R is the Ideal Gas Constant!
This result is called the “Dulong-Petit Law”

3. Cv = AT3 (A is a constant) Early 1900’s: A theory explanation for this
Late 1800’s & early 1900’s: experimental capabilities
became more sophisticated, & it became possible to
achieve temperatures well below room temperature.
Then, measurements clearly showed that the heat
capacities of solids depend on what the material is &
are also strongly dependent on the temperature. In fact,
at low enough temperatures, the heat capacities of many
solids were observed to depend on temperature as
Cv = AT3
(A is a constant)
Early 1900’s: A theory explanation for this
temperature dependence of Cv was one of the
major unsolved problems in thermodynamics.

4. Molar Heat Capacity of a Solid
Assume that the heat supplied to a solid is
transformed into kinetic & potential
energies of each vibrating atom.
To explain the classical Dulong
Petit Law, use a simple classical
model of the vibrating solid as in
the figure. Assume that each
vibrating atom is an independent
simple harmonic oscillator & use the
Equipartition Theorem to calculate the Thermal
Energy E & thus the Heat Capacity Cv

5. The Thermal Average Energy for each degree of freedom is (½) kT
Molar Thermal Energy of a Solid
Dulong-Petit Law: Can be explained using the
Equipartition Theorem of Classical Stat Mech:
The Thermal Average Energy for each degree of freedom is (½) kT
Assume that each atom has 6 degrees of
freedom: 3 translational & 3 vibrational, the
Thermal Average Energy (per mole) of the
vibrating solid is:
R  NA k

6. The Molar Vibrational Heat Capacity In agreement with Dulong & Petit!
Using this approximation, the Thermal Average
Energy (per mole) of the vibrating solid is:
(1)
By definition, the Heat Capacity of a substance
at Constant Volume is
(2)
Using (1) in (2) gives:
In agreement with Dulong & Petit!

7. Measured Molar Heat Capacities
Measurements on many solids have shown that their
Heat Capacities are strongly temperature dependent
& that the Dulong-Petit result Cv = 3R is only valid
at high temperatures.
Cv = 3R

8. Einstein Model of a Vibrating Solid
1907: Einstein extended Planck’s quantum ideas to solids:
He proposed that energies of lattice vibrations in a solid
are quantized simple harmonic oscillators. He proposed the following model for the lattice vibrations in a solid:
Each vibrational mode is an independent oscillator
Each mode vibrates in 3-dimensions
Each vibrational mode is a quantized oscillator with energy: or
En = (n + ½ )ħ

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

10. Einstein Model of a Vibrating Solid
To compute the average energy
for one oscillator, note that
it can be written as:
where
and b = 1/kT
Z is the partition
function

11. Einstein Model For b = 1/kT & En = ne, the partition function
function for one oscillator in the Einstein model is:
which, follows from the result

12. Einstein Model
Differentiating with respect to b:

13. Einstein Model …and multiplying by –1/Z: This is the Einstein result
for the average energy
of one oscillator of
energy .

14. Einstein Model Einstein’s result for the average energy
of one oscillator of
energy .
Einstein then made the simple (&
absurd & unphysical!) assumption
that each of the 3NA oscillators
vibrates at the same frequency so
each has the same energy !
With this assumption, the mean
vibrational energy of the solid
is just 3NA times the above result!

15. “The Einstein Frequency”
So, what is the frequency at which each of
the 3NA oscillators is vibrating?
In the Einstein Model this is an adjustable
parameter which is fit to data & which
depends on what solid is being considered.
It is usually denoted as E & is called
“The Einstein Frequency”

16. the “Einstein temperature”.
So, the molar heat capacity in Einstein’s model is:
Here
TE  (ħE)/kB is called
the “Einstein temperature”.
Its not a true temperature, of course, but its an
energy in temperature units.

17. Einstein Model: CV for Diamond
Einstein, Annalen der Physik 22 (4), 180 (1907)

18. The Debye Model Vibrating Solids

19. Summary
The Dulong-Petit law is expected from the equipartition theorem of classical physics. However, it fails at temperatures low compared with the Einstein temperature
If the energy in solids is assumed to be quantized, however, models (such as Debye’s) can be developed that agree with the observed behavior of the heat capacity with temperature. Einstein was the first to suggest such a model.