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How is a Fermi surface (a surface in reciprocal space separating the occupied electron states from unoccupied states at $T=0$) different from a Fermi sphere? Is Fermi sphere a special case of Fermi surface?

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Yes, a Fermi sphere is just a special case (and almost always an idealization). The shape of a ``real'' Fermi surface is discussed in Mermin and Ashcroft, but a quick search on google image gives examples of what they really look like in solids.

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When the system is perfectly periodic, the Fermi surface is the level set associated to the Fermi energy $E_{\mathrm{F}}$, i. e. consisting of those Bloch momenta $k$ for which $E_n(k) = E_{\mathrm{F}}$ holds for some band index $n$. Note that this definition also works for insulators where the Fermi energy lies in a spectral gap. But then the Fermi energy is the empty set.

For homogeneous systems where the Hamiltonian is just the free Hamiltonian $H = \frac{1}{2m} \bigl (- \mathrm{i} \hbar \nabla \bigr )^2$, then the only energy band is $E(k) = \frac{\hbar^2}{2m} k^2$ so that the Fermi surface is a perfect sphere.

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