By Blumler P.

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5] (see also eq. 45], factor 3) To test the results of Einstein’s theory, we want to calculate the limits at T = 0K and T → ∞. ~ ~ We expect for T → 0, that CVE → 0 and for T → ∞, that CVE → 3R (Dulong-Petit’s rule). Furthermore, experimental data (see later this section) reveal a T 3 dependence when approaching 0K. a) Einstein’s theory in the low limit (T → 0) 6→ 7∞8 64→ 7∞4 8 2 Θ Θ E exp E we can argument that the denominator approaches infinity T ~ T faster than the nominator, hence the limit is 0 (as lim CVE = lim 3R required).

It’s neighbours are fixed (no vibrational coupling) 3) The vibrational displacements are very small (Hooke’s law is valid) Hence, each atom in the lattice has 3 degrees of vibrational freedom. A crystal of N atoms can therefore be described as 3N harmonic oscillators, which should have a single characteristic (Einstein-) frequency ν E. 8: From eq. 42]: and from eq. 44] © Dr. 2] University of Canterbury page:46 PH 605: Thermal & Statistical Physics 4. Quantum Statistics When we also define a characteristic temperature, Θ E , the Einstein-temperature, via (cf.

053 He (liquid) λ *) conductive electrons in Cu We clearly see that for light materials or low temperatures (like electrons or liquid helium) the classical condition 3 VN λ−1 >> 1 (right column in table) is no longer fulfilled. 1, the de Broglie wavelength, λ, becomes about the size of the inter-atomic distance, d = 3 VN . Hence, we must include interference effects of the particle wavefunctions. As a result of this, the occupation numbers of different energy states can no longer be chosen arbitrarily.