Heat capacity of an Einstein solid as a function of temperature. Experimental value of 3Nk is recovered at high temperatures.
Reference: Disturbance Theory
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Einstein Solid – Wikipedia
The Einstein solid is a model of a solid based on two assumptions:
- Each atom in the lattice is an independent 3D quantum harmonic oscillator
- All atoms oscillate with the same frequency (contrast with the Debye model)
While the assumption that a solid has independent oscillations is very accurate, these oscillations are sound waves or phonons, collective modes involving many atoms. In the Einstein model, however, each atom oscillates independently. Einstein was aware that getting the frequency of the actual oscillations would be difficult, but he nevertheless proposed this theory because it was a particularly clear demonstration that quantum mechanics could solve the specific heat problem in classical mechanics.
A 3D quantum harmonic oscillator would be made up of high frequency, compacted cycles of electromagnetic field that have slowed down considerably due to their high inertia. Their motion is no longer linear, but a combination of circular, rotational and linear, which shows up as oscillatory.
Einstein treated each atom as an independent 3D harmonic oscillator, whose energy could only increase in quantum intervals of ‘hω’. Einstein assumed the same frequency for all atoms for the sake of simplicity.
The original theory proposed by Einstein in 1907 has great historical relevance. The heat capacity of solids as predicted by the empirical Dulong-Petit law was required by classical mechanics, the specific heat of solids should be independent of temperature. But experiments at low temperatures showed that the heat capacity changes, going to zero at absolute zero. As the temperature goes up, the specific heat goes up until it approaches the Dulong and Petit prediction at high temperature.
The classical mechanics predicts the heat capacity of solids to be independent of temperature. It did not explain the observed dependence at lower temperatures. Einstein could show this dependence with his quantum model even if not very accurately.
By employing Planck’s quantization assumption, Einstein’s theory accounted for the observed experimental trend for the first time. Together with the photoelectric effect, this became one of the most important pieces of evidence for the need of quantization. Einstein used the levels of the quantum mechanical oscillator many years before the advent of modern quantum mechanics.
The continuous change in properties, which is a feature of classical mechanics, is seen in context of normal dimensions. When we view properties at atomic dimensions, as in the case of black body radiation, the quantum effects of frequency cycles become prominent.
In Einstein’s model, the specific heat approaches zero exponentially fast at low temperatures. This is because all the oscillations have one common frequency. The correct behavior is found by quantizing the normal modes of the solid in the same way that Einstein suggested. Then the frequencies of the waves are not all the same, and the specific heat goes to zero as a T3 power law, which matches experiment. This modification is called the Debye Model, which appeared in 1912.
Einstein demonstrated the quantum effects on specific heats of solids at low temperatures which classical mechanics could not explain. Einstein did make simplified assumptions as regard the frequency of atoms, which were modified later in Debye model.
When Walther Nernst learned of Einstein’s 1906 paper on specific heat, he was so excited that he traveled all the way from Berlin to Zürich to meet with him.
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