Waves of Electrons in Three-Dimensional Space

FIGURE 3.21 A real, three-dimensional monoclinic unit cell (heavy lines) and its corresponding reciprocal unit call (thin lines). The unique b and b axes (unique because of the single twofold axis along b) are parallel for the monoclinic system, but the relationship of the a and a to the c and c axes depends on the angle between a and c. Note the inverse relationship between the real and reciprocal edge lengths. 

CRYSTALS AS WAVES OF ELECTRONS IN THREE-DIMENSIONAL SPACE 

The important thing is that what is true about one concept must be true, and have some analogue, for the other. 

The converse of what has just been described must also be true. If a complex wave can be synthesized from its harmonic components, then one must also be able to break down, or analyze, a complex wave into spectral components. That is, given the electron density within a unit cell, one should be able to derive in some fashion its component waves. What, in the real crystal, corresponds to the harmonic components we have been discussing The families of planes, each characterized by a set of Miller indexes, are the spectral components of the unit cell and its contents, the electron density. 

We might look at this in yet another way. If you systematically cut, or sectioned a unit cell in a manner corresponding to a particular family of planes, say the h = 3, k = 5, l = 7 family, then you could examine what lay on each of the planes in the family and then have 

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