Reciprocal Space and the k Quantum Number

Looking at the exponential arguments in Equations (2.23) or (2.26), it is clear that the quantum number k must have the dimension of an inverse length it is a resident of reciprocal space. Reciprocal space is an ingenious invention in order to simplify life with infinite objects such as crystals, so let us introduce it now, rather belatedly [53]. 

Any position in real space is given by the real-space vector R which is a linear combination of the three basic vectors i, 2/ arid 3, namely 

Likewise, any given reciprocal lattice vector K is constructed from the reciprocal basic vectors g, g, and 3 according to 

The latter definition (which would include an additional In in the exponential argument if written down by a crystallographer) is equivalent to the orthogonality relations of the basis vectors which are 

The important term electronic band structure is identical with the course of the energy of an extended wave function as a function of k, and we seek for E ip k,r)), the crystal s equivalent to a molecular orbital diagram. As stated before, the fc-dependent wave function ip k,r) is called a crystal orbital, and there may be many one-electron wave functions per k, just as there may be several molecular orbitals per molecule. Due to the existence of these periodic wave functions, there results stationary states in which the electrons are travelling from atom to atom the Bloch theorem thereby explains why the periodic potential is compatible with the fact that the conduction electrons do not bounce against the ionic cores. 

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