Two-Dimensional Periodicity and Vectors in Reciprocal Space

In multidimensional systems we must keep track of k values for waves oriented along different directions, and we need to calculate each CO energy at a point corresponding to k values for each of these directions. This makes treatment of such problems considerably more complicated. It is convenient and conventional to handle all this in terms of vectors. We will outline the vector treatment here, using the two-dimensional rectangular lattice shown in Fig. 15-27a as an example. 

For a three-dimensional crystal, we label the translation vectors for a unit cell in real space ai, a2, and as. Any point r in the unit cell can then be expressed in terms of vectors a. For the vectors in reciprocal space, we use the symbols bi, b2, and bs. The position k in reciprocal space can then be expressed in terms of vectors b. Vectors a have dimensions of length, and b of reciprocal length. 

The vectors b in reciprocal space are defined according to the formula 

Chapter 15 Molecular Orbital Theory of Periodic Systems 

We have seen that one-dimensional systems have degenerate COs for equal values of k and so, if we wish to portray only the unique energies of the system, we need consider only the range from 0 to n/a. The analogous situation in two or three dimensions is that symmetrically equivalent positions k give degenerate COs. Hence we need consider only a symmetrically unique portion of the FBZ—the reduced first Brillouin zone (RFBZ). Since the FBZ is symmetric for the reflections mentioned above, the RFBZ is the quadrant shown in Fig. 15-27c. 

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Section 15-15 Two-Dimensional Periodicity and Vectors in Reciprocal Space... 

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