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Crystals are solid material whose constituents, such as atoms, molecules or ions, are *arranged in a highly ordered microscopic pattern*, a crystal lattice that extends with regularity in all directions. Use for all crystallography and ordered structure topics.

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Ashcroft Mermin Solid State Eq. 22.15

I recently read a paper on Specific heat of a Classical Crystal: Dulong-Petit law. In Eq. 22.15, I don't understand why the ionic displacement $\mathbf{u}$ and momentum $\mathbf{P}$ become independen …
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