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 mode).

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

(500) |

(501) |

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
would imply
, 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

(502) |

giving

(504) |

Here,

(506) |

(507) |

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
approximation
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).
The molar
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
particularly
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:

(509) |

(510) | |||

(511) | |||

(512) |

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 spaced. For given values of and , the number of allowed values of which lie in the range to is given by

(513) |

where 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

(515) |

Consider an isotropic continuous medium of volume . According to the above
relation, the
number of normal modes whose frequencies lie between and
(which is equivalent to the number of modes whose values lie in the range
to
) is

The Debye approach consists in approximating the actual density of normal modes
by the density in a continuous medium
, not
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

(518) |

Substituting Eq. (516) into the previous formula yields

(519) |

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 is , 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.

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

(521) |

(522) |

(523) |

(524) |

(525) |

(526) |

We have also defined the

(528) |

Consider the asymptotic limit in which . For small , we can approximate
as in the integrand of Eq. (527), so that

(529) |

(530) |

(531) |

in the limit :

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 to obtain this asymptotic behaviour. Theoretically, should be calculable from Eq. (520) 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.

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).