Einstein solid

The Einstein solid is a model of a solid based on two assumptions: The Einstein solid is a model of a solid based on two assumptions: 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. 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. 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. 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 T 3 {displaystyle T^{3}} power law, which matches experiment. This modification is called the Debye model, which appeared in 1912. 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. The heat capacity of an object at constant volume V is defined through the internal energy U as T {displaystyle T} , the temperature of the system, can be found from the entropy To find the entropy consider a solid made of N {displaystyle N} atoms, each of which has 3 degrees of freedom. So there are 3 N {displaystyle 3N} quantum harmonic oscillators (hereafter SHOs for 'Simple Harmonic Oscillators').