We are told such a number as the square root of two worried Pythagoras and his school almost to exhaustion. Being used to such queer numbers from early childhood, we must be careful not to form a low idea of the mathematical intuition of these ancient sages; their worry was highly credible.
Erwin SchrödingerIn this chapter, we introduce the metric tensor, a computational tool that simplifies calculations related to distances, directions, and angles between directions. First, we illustrate the importance of the metric tensor with a 2-D example. Then, we introduce the 3-D metric tensor and discuss how it can be used for simple lattice calculations in all crystal systems. We end this chapter with a few worked examples.
Directions in the crystal lattice
We know that a vector has two attributes: a length and a direction. By selecting a translation vector t in the space lattice, we are effectively selecting a direction in the crystal lattice, namely the direction of the line segment connecting the origin to the endpoint of the vector t. Directions in crystal lattices are used so frequently that a special symbol has been developed to describe them. The direction parallel to the vector t is described by the symbol [uvw], where (u, v,w) are the smallest integers proportional to the components of the vector t. Note the square brackets and the absence of commas between the components.
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