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Consider a simple solid containing atoms. Now, atoms in solids cannot
translate (unlike those in gases), but
are free to vibrate about their equilibrium positions.
Such vibrations are called lattice vibrations, and can be thought of
as sound waves propagating
through the crystal lattice. Each atom is specified by three independent position
coordinates, and three conjugate momentum coordinates. Let us
only consider small amplitude vibrations.
In this case, we can expand the potential energy of interaction between the atoms
to give an expression which is quadratic in the atomic displacements
from their equilibrium positions. It is always possible to perform a
normal mode analysis
of the oscillations. In effect, we can find independent modes of oscillation of the solid.
Each mode has its own particular oscillation frequency, and its own particular pattern
of atomic displacements.
Any general oscillation can be written as a linear combination of these
Let be the (appropriately normalized) amplitude of the th normal mode,
and the momentum conjugate to this
coordinate. In normal mode coordinates, the total energy of the lattice vibrations takes the
particularly simple form
Specific heats of solids
where is the (angular) oscillation frequency of the th normal mode. It is
clear that in normal mode coordinates, the linearized lattice vibrations are equivalent to
independent harmonic oscillators (of course, each oscillator corresponds to a different normal
The typical value of is the (angular) frequency of a sound wave
propagating through the lattice. Sound wave frequencies are far lower than the
typical vibration frequencies of gaseous molecules. In the latter case, the mass involved in the
vibration is simply that of the molecule, whereas in the former case the mass involved is that
of very many atoms (since lattice vibrations are non-localized). The strength of
interatomic bonds in gaseous molecules is similar to those in solids, so we can use the estimate
( is the force constant which measures the strength of
interatomic bonds, and is the mass involved in the oscillation) as proof that the typical
frequencies of lattice vibrations are very
much less than the vibration frequencies of simple molecules.
It follows from
that the quantum energy levels of lattice vibrations are
far more closely spaced than the vibrational energy levels of gaseous molecules. Thus, it is
likely (and is, indeed, the case) that lattice vibrations are not frozen out at room temperature,
but, instead, make their full classical contribution to the molar specific heat of the solid.
If the lattice vibrations behave classically then, according to the equipartition theorem,
each normal mode of oscillation has an associated mean energy in equilibrium at
temperature [ resides in the kinetic energy of the oscillation,
and resides in the potential energy].
Thus, the mean internal energy per mole of the solid is
It follows that the molar heat capacity at constant volume is
for solids. This gives a value of joules/mole/degree. In fact, at room temperature most
solids (in particular, metals)
have heat capacities which lie remarkably close to this value. This
fact was discovered
experimentally by Dulong and Petite at the beginning of the nineteenth century, and was used to
make some of the first
crude estimates of the molecular weights of solids (if we know the molar heat capacity
of a substance
then we can easily work out how much of it corresponds to one mole, and
by weighing this amount, and then dividing the result by Avogadro's number,
we can obtain an estimate of the molecular weight). Table 4 lists the experimental
capacities at constant pressure for various solids. The heat capacity at constant
volume is somewhat less than the constant pressure value, but not by much,
are fairly incompressible.
It can be seen that Dulong and Petite's law (i.e., that all solids have a molar heat capacities
close to joules/mole/degree) holds pretty well for metals.
However, the law fails badly for
diamond. This is not surprising. As is well-known,
diamond is an extremely hard substance, so its intermolecular bonds must be very strong, suggesting
that the force constant is large.
Diamond is also a fairly low density substance, so the mass involved in
lattice vibrations is comparatively small. Both these facts suggest that the typical lattice vibration
frequency of diamond (
) is high. In fact, the spacing between
the different vibration energy
levels (which scales like
) is sufficiently large in diamond for the vibrational
degrees of freedom
to be largely frozen out at room temperature. This accounts for the anomalously low
heat capacity of diamond in Tab. 4.
Values of (joules/mole/degree) for some solids at K. From Reif.
Dulong and Petite's law is essentially a high temperature limit. The molar heat capacity cannot
remain a constant as the temperature approaches absolute zero, since, by
Eq. (488), this
, which violates the third law of thermodynamics. We can make
a crude model of the behaviour of at low temperatures by assuming that all the normal
modes oscillate at the same frequency, , say. This approximation was first employed by
Einstein in a paper published in 1907. According to Eq. (499),
the solid acts like a set
of independent oscillators which, making use of
Einstein's approximation, all vibrate at the same frequency.
We can use the quantum mechanical result (479) for a single
oscillator to write the mean energy
of the solid in the form
The molar heat capacity is defined
which reduces to
is called the Einstein temperature. If the temperature is sufficiently
, and the above expression reduces to
, after expansion of the exponential functions. Thus, the law of Dulong and
Petite is recovered for temperatures significantly in excess of the Einstein temperature.
On the other hand, if the temperature is sufficiently
low that then the
exponential factors in Eq. (505) become very much larger than unity, giving
So, in this simple model the specific heat approaches zero exponentially as
In reality, the specific heats of solids do not approach zero quite as quickly as
suggested by Einstein's model when
. The experimentally observed low temperature
behaviour is more like
(see Fig. 6). The reason for this discrepancy is the crude
that all normal modes have the same frequency. In fact, long wavelength modes have lower frequencies
than short wavelength modes, so the former are much harder to freeze out than the latter
(because the spacing between quantum energy levels, , is smaller in the former case).
heat capacity does not decrease with temperature as rapidly as suggested by Einstein's model
because these long wavelength modes are able to make a significant contribution
to the heat capacity even at very low
temperatures. A more realistic model of lattice vibrations was developed by the Dutch physicist
Peter Debye in 1912.
In the Debye model, the frequencies of the normal modes of vibration are estimated by treating
the solid as an isotropic continuous medium. This approach is reasonable because the only modes
which really matter at low temperatures are the long wavelength modes: i.e., those whose
wavelengths greatly exceed the interatomic spacing. It is plausible that these modes are not
sensitive to the discrete nature of the solid: i.e., the fact that it is made up of atoms
rather than being continuous.
Consider a sound wave propagating through an isotropic continuous medium.
The disturbance varies with position vector and time like
, where the wave-vector and
the frequency of oscillation satisfy the dispersion relation for sound waves in
an isotropic medium:
Here, is the speed of sound in the medium. Suppose, for the sake
of argument, that the medium is periodic in
the -, -, and -directions with periodicity lengths , , and , respectively.
In order to maintain periodicity we need
where is an integer. There are analogous constraints on and . It follows that
in a periodic medium the components of the wave-vector are quantized, and can only take the
where , , and are all integers. It is assumed that , , and are
macroscopic lengths, so the allowed values of the components of the wave-vector are very closely
For given values of and , the number of
allowed values of which lie in the range to is given by
It follows that the number of allowed values of (i.e., the number of allowed
modes) when lies in the range to ,
lies in the range to , and
lies in the range to , is
is the periodicity volume, and
. The quantity is called the density of modes.
Note that this density is independent of , and proportional to the periodicity
volume. Thus, the density of modes per unit volume is a constant independent of the magnitude
or shape of the periodicity
volume. The density of modes per unit volume when the magnitude of lies in
the range to is given by multiplying the density of modes per unit volume by
the ``volume'' in -space of the spherical shell lying between radii and .
Consider an isotropic continuous medium of volume . According to the above
number of normal modes whose frequencies lie between and
(which is equivalent to the number of modes whose values lie in the range
The factor of comes from the three possible polarizations of sound waves in solids.
For every allowed wavenumber (or frequency) there are two independent torsional modes,
where the displacement is perpendicular to the direction of propagation, and one
longitudinal mode, where the displacement is parallel to the direction of propagation.
Torsion waves are vaguely analogous to electromagnetic waves (these also have two
independent polarizations). The longitudinal mode is very similar to the compressional
sound wave in gases. Of course, torsion waves can not propagate in gases because gases
have no resistance to deformation without change of volume.
The Debye approach consists in approximating the actual density of normal modes
by the density in a continuous medium
only at low frequencies (long wavelengths) where these should be nearly the same, but
also at higher frequencies where they may differ substantially. Suppose that we are
dealing with a solid consisting of atoms. We know that there are
only independent normal modes. It follows that we must cut off the
density of states above some critical frequency, say, otherwise we
will have too many modes. Thus, in the Debye approximation the density
of normal modes takes the form
Here, is the Debye frequency. This critical frequency is chosen such that
the total number of normal modes is , so
Substituting Eq. (516) into the previous formula yields
This implies that
Thus, the Debye frequency depends only on the sound velocity in the solid and the number
of atoms per unit volume. The wavelength corresponding to the Debye frequency
, which is clearly on the order of the interatomic spacing
It follows that the cut-off of normal modes whose frequencies exceed the Debye frequency
is equivalent to a cut-off of normal modes whose wavelengths are less than the interatomic
spacing. Of course, it makes physical sense that such modes should be absent.
The true density of normal modes in diamond compared with
the density of normal modes predicted by Debye theory. From C.B. Walker, Phys. Rev. 103, 547 (1956).
Figure 5 compares the actual density of normal modes in diamond with
the density predicted by Debye theory.
Not surprisingly, there is not a particularly strong resemblance
between these two curves, since Debye theory is highly idealized.
Nevertheless, both curves exhibit sharp cut-offs at high frequencies, and
coincide at low frequencies. Furthermore, the areas under both curves are
the same. As we shall see,
this is sufficient to allow Debye theory to correctly account for
the temperature variation of the specific heat of solids at low temperatures.
We can use the quantum mechanical expression for the
mean energy of a single oscillator, Eq. (479), to calculate the mean
energy of lattice vibrations in the Debye approximation. We obtain
According to Eq. (503), the molar heat capacity takes the form
Substituting in Eq. (517), we find that
in terms of the dimensionless variable
According to Eq. (520), the volume can be written
so the heat capacity reduces to
where the Debye function is defined
We have also defined the Debye temperature as
Consider the asymptotic limit in which . For small , we can approximate
as in the integrand of Eq. (527), so that
Thus, if the temperature greatly exceeds the Debye temperature we recover the law of
Dulong and Petite that . Consider, now, the
asymptotic limit in which . For large ,
The latter integration is standard (if rather obscure), and can be looked up in any
(large) reference book
on integration. Thus, in the low temperature limit
in the limit : i.e., varies with temperature like .
Comparison of Debye temperatures (in degrees kelvin) obtained from the
low temperature behaviour of the heat capacity with those calculated from the
sound speed. From C. Kittel, Introduction to solid-state physics, 2nd Ed. (John Wiley & Sons, New York NY, 1956).
|| from low temp.
|| from sound speed
The fact that goes like at low temperatures is quite well verified experimentally,
although it is sometimes necessary to go to temperatures as low as
this asymptotic behaviour. Theoretically, should be calculable from
in terms of the sound speed in the solid and the molar volume. Table 5 shows a
comparison of Debye temperatures evaluated by this means with temperatures obtained
empirically by fitting the law (532) to the low temperature variation of the
heat capacity. It can be seen that there is fairly good agreement between the theoretical and
empirical Debye temperatures. This suggests that the Debye theory affords a good, thought not
perfect, representation of the behaviour of in solids over the entire temperature range.
The molar heat capacity of various solids.
Finally, Fig. 6 shows the actual temperature variation of the molar heat capacities
of various solids as well as that predicted by Debye's theory. The prediction of Einstein's theory
is also show for the sake of comparison. Note that 24.9 joules/mole/degree is about
6 calories/gram-atom/degree (the latter are chemist's units).
Next: The Maxwell distribution
Up: Applications of statistical thermodynamics
Previous: Specific heats of gases