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A mapping from the Brillouin zone to a sphere cannot be continuous and invertible (i.e. a homeomorphism) because continuous, invertible maps preserve topological properties, and the torus and the sphere are topologically distinct. More specifically, the 2-sphere is simply connected, which means that any curve drawn on the surface can be shrunk to a point ...

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The chemical potentials used in the equation for the defect formation energy are the energies at which atoms are taken from / given to reservoirs to form the defect in the solid. The electronic chemical potential similarly is the energy at which electrons are exchanged to form charged defects (i.e. the Fermi level, which is mid-band gap for insulators and ...

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It is easier to think of the shortest wavelength for phonons (the largest wavevector). In the Debye model, it is assumed that there is no dispersion up to that wavevector, that the proportionality is given by the velocity of sound in the crystal. This is not true, the Debye frequency overestimates the highest frequency. Often the Debye frequency is ...

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A lot of work has been done since the 1984 review by Moriya and Takahashi, both theory and experiments. On the theoretical side, calculations have been refined a lot. The effect of the on-site $dd$ correlation energy $U$ was taken into account first by LSDA+U, then by LSD+DMFT (a dynamic mean field theory), for example this paper from 2001 about high ...

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The purpose of drawing the dispersion relation like this is to facilitate the visualization of a 3-dimensional function in a 2-dimensional plot by traveling in lines along high symmetry points. Traveling along 2 points, say from $\Gamma$ to $X$ means to travel from k value [0,0,0] to k value [1,0,0] with points of the form [x,0,0] in between.

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What is the difference between Permanent set and Elastic limit? Permanent set is the deformation that stays in the material after the applied stress is removed. The elastic limit is the maximum stress that is still within the materials elastic range. If you apply a greater stress then the elastic limit, you get permanent set in the object - it gets ...

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