Abstract
We are now in a position to make additional important statements which contribute considerably to the understanding of the properties of crystals. For this we plot the energy versus the momentum of the electrons, or, because of (4.8), versus the wave vector, k. As before, we first discuss the one-dimensional case.
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Notes
- 1.
If two energy functions with equal symmetry cross, the quantum mechanical “noncrossing rule” requires that the eigenfunctions be split, so that they do not cross.
- 2.
A lattice is a regular periodic arrangement of points in space; it is, consequently, a mathematical abstraction. All crystal structures can be traced to one of the 14 types of Bravais lattices (see textbooks on crystallography).
- 3.
The scalar product of two vectors a and b is a · b = ab cos(ab). If i, j, and l are mutually perpendicular unit vectors, then we can write i · j = j · l = l · i = 0 and i · i = j · j = l · l = 1.
- 4.
The vector product of two vectors a and b is a vector which stands perpendicular to the plane formed by a and b. It is i × i = j × j = l × l = 0 and i × j = l and j × i = −1.
- 5.
\( {\mathbf{a}}\; \times \;{\mathbf{b}} = \left| {\left. {\begin{array}{lllllll} {\mathbf{i}} & {\mathbf{j}} & {\mathbf{l}} \\{{a_x}} & {{a_y}} & {{a_z}} \\{{b_x}} & {{b_y}} & {{b_z}} \\\end{array} } \right|} \right.. \)
- 6.
a · b = a x b x + a y b y + a z b z .
- 7.
Directions in unit cells are identified by subtracting the coordinates of the tail from the coordinates of the tip of a distance vector. The set of numbers thus gained is inserted into square brackets; see textbooks on materials science.
- 8.
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Hummel, R.E. (2011). Energy Bands in Crystals. In: Electronic Properties of Materials. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8164-6_5
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DOI: https://doi.org/10.1007/978-1-4419-8164-6_5
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