Crystal field theory wave numbers

The interpretation of Eq. (4-3) is direct and the formula could almost be written down immediately in terms of Fermi s familiar Golden Rule of quantum theory (see, for example, Schiff, 1968, p. 314). The absorption of light arises from the coupling, caused by the electric field, of occupied states with unoccupied states, However the absorption can only occur if the two states differ in energy by hco, so that energy can be conserved if the transition is made and one photon is absorbed. It should also be noted that in a perfect crystal the matrix elements will vanish unless the wave numbers of the coupled states in the Brillouin Zone are the same. That is, the transitions must be between states that are on the same vertical line in a diagram such as Fig. 4-7. We shall return to a more complete description of absorption later. 

Th.e refinements of the theory, which have been worked out in particular by Houston, Bloch, Peierls, Nordheim, Fowler and Brillouin, have two main objects. In the first place, the picture of perfectly free electrons at a constant potential is certainly far too rough. There will be binding forces between the residual ions and the conduction electrons we must elaborate the theory sufficiently to make it possible to deduce the number of electrons taking part in the process of conduction, and the change in this number with temperature, from the properties of the atoms of the substance. In principle this involves a very complicated problem in quantum mechanics, since an electron is not in this case bound to a definite atom, but to the totality of the atomic residues, which form a regular crystal lattice. The potential of these residues is a space-periodic function (fig. 10), and the problem comes to this— to solve Schrodinger s wave equation for a periodic poten-tial field of this kind. That can be done by various approximate methods. One thing is clear if an electron... 

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