The Einstein approximation

As shown above, the heat capacity Cy (or at any rate that part of it which is due to the vibrations) may be expected to have a value of 3Jt whenever hvjkT 1. This would be so, even at the lowest temperatures, if Planck s constant h were zero, and this is the case in the classical or pre-quantum mechanics. In fact, classical theory leads to the expectation that, for any crystalline substance, Cy has the constant value of 3R per mole. This is contrary to experiment, and it is known that Cy usually diminishes below 3A, with fall of temperature, and seems to approach zero at the absolute zero. One of the early successes of the quantum theory consisted in finding the reason for this decrease in Cy which is quite inexplicable in classical theory. The explanation is implicit in the previous equations and is due to the fact that the oscillators can only take up finite increments of energy. When a system of oscillators is held at low temperature, most of them are in their lowest energy level, and a small rise oftemperature is insufficient to excite them to the next higher level. Therefore Cy, which measures the intake of energy per unit increase of temperature, is smaller than at higher temperature. 

In order to make use of equations such as (13 37) and (13 38) some assumption must be made with regard to the frequencies. As early as 1907 Einstein used the approximation of assuming that all of the 3 frequencies are equal. This is equivalent to supposing that each of the N atoms in the lattice makes quantized harmonic oscillations in three dimensions, and these oscillations are quite imaffected by the motion of the neighbouring atoms. This supposition of atomic independence cannot be correct, but nevertheless it leads to an 

If a suitable value of is used, (13 60) fits the experimental data quite well at temperatures at which Cy is more than half of its maximum value, 3R. But at low temperatures there is a serious discrepancy, and the Einstein equation predicts a value of Cy which approaches zero too rapidly. This is due to the error in treating all the normal modes as if they have the same frequency Vg. In the actual crystal there are no doubt a large number of modes which have quite a small frequency and therefore a small separation of their energy levels. At low temperatures these modes will have a much greater probability of becoming excited, with absorption of heat, than would be expected on the Einstein model. 

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Comment on the choice of representative values of Vj for the 12 vibrational modes of the crystal. How much would reasonable changes (say, 10 to 20 percent) in these values affect the results of the calculations If possible, conunent on the effect of using the Debye approximation for the acoustic lattice modes instead of the Einstein approximation. 

Distribution of vibralioiial frequencies in GaA.s. The frequencies of the two main peaks arc identified with characteristic frequencies of the spectrum. The figure suggests the Einstein approximation of replacing the distribution by two sharp peaks. [After Dolling and Waugh, 1965, p. 19.]... 

The g((o) can be used to provide a simple approximation to the lattice vibrations, the Einstein approximation. Let us begin by agreeing that single characteristic frequencies, could be chosen to individually represent each of the six types (three translational and three rotational) of external mode. The ( 6)y value is the density of states weighted mean value of all of the frequeneies over which that external mode, j, was dispersed. Normalising as discussed above ... 

We invoke the Einstein approximation, see Eq. (2.57), and assume that (a/Oext is the displacement that stems from Thus there is only a single external vibrational mode and so is isotropic. In a manner analogous to Eq. (2.44) we write ... 

The discussion in this section has only been concerned with the enthalpy term. In order to determine the free energy, which is necessary for a calculation of the equilibrium defect concentration, the standard entropy change for the formation of a mole of defects may be estimated as follows. In the simplest case of the Einstein approximation for the limiting case of Dulong-Petit behaviour, the crystal with Nq lattice atoms is considered to be a system of... 

Having adopted this convention we may ask. Does there exist a physical state of a substance for which the conventional entropy is actually zero Now perfect crystals are known to have a very orderly structure, and at very low temperatures the lattice vibrations will all be in their lowest states which correspond to the zero-point energy. Therefore it may be expected that a crystal will have a very low entropy at temperatures approaching the absolute zero, and in one of the original forms (Planck s version) of the third law it was asserted that the entropy of a pure substance is actually zero under such conditions. On the other hand, from (13 51), based on the Einstein approximation, it is seen that... 

At high temperatures (T 0 ) we obtain the classical value (3.64a), namely, C (T) = 3nNkg, while at low temperatures (T 0 ) the Einstein approximation gives... 

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