This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.
 

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In crystals, a periodic potential appears because the ions are arranged with a periodicity of their Bravais lattice. The periodic potential can be given by,

          U(r + R) = U(r) ----------------------------------- [2204a]

where,
          R -- The lattice vector.

This potential can be applied into the Schrödinger equation,

           ------------------- [2204b]

The eigenstates ψ of the Hamitonian Ĥ in Equation 2204b can be chosen to have the form of a plane wave times a function with the periodicity of the Bravais lattice so that each eigenstate ψ satisfies,

           ψ_nk(r) = e^ik•ru_nk(r) ------------------- [2204c]

where,
          ħk is the crystal momentum (but not true momentum);
          n -- The quantum number (also called the band index, are equal to 1, 2, 3, . . .);
          u_nk(r + R) = u_nk(r) -------------------- [2204d]

The quantum number (n) corresponds to the appearance of independent eigenstates of different energies but with the same k. Equation 2204d means that u_nk(r) has the same periodicity as the Hamitonian Ĥ for any lattice vector R of the Hamitonian.

An alternative formulation of Bloch’s theorem is that the eigenstates of a periodic Hamitonian Ĥ can be chosen so that for any lattice vector R of the Hamitonian, associated with each ψ  is a wave vector k and thus each eigenstate satisfies,

           ψ_nk(r+R) = e^ik•Rψ_nk(r) ------------------- [2204e]