##### Exercise8

Crystallographers find it convenient to use coordinate systems that are adapted to the specific geometry of a crystal. As a two-dimensional example, consider a layer of graphite in which carbon atoms are arranged in regular hexagons to form the crystalline structure shown in Figure 12.

The origin of the coordinate system is at the carbon atom labeled by "0". It is convenient to choose the basis $$\bcal$$ defined by the vectors $$\vvec_1$$ and $$\vvec_2$$ and the coordinate system it defines.

1. Locate the points $$\xvec$$ for which

1. $$\coords{\xvec}{\bcal} = \twovec{1}{0}\text{,}$$

2. $$\coords{\xvec}{\bcal} = \twovec{0}{1}\text{,}$$

3. $$\coords{\xvec}{\bcal} = \twovec{2}{1}\text{.}$$

2. Find the coordinates $$\coords{\xvec}{\bcal}$$ for all the carbon atoms in the hexagon whose lower left vertex is labeled "0".

3. What are the coordinates $$\coords{\xvec}{\bcal}$$ of the center of that hexagon, which is labeled "C"?

4. How do the coordinates of the atoms in the hexagon whose lower left corner is labeled "1" compare to the coordinates in the hexagon whose lower left corner is labeled "0"?

5. Does the point $$\xvec$$ whose coordinates are $$\coords{\xvec}{\bcal} = \twovec{16}{4}$$ correspond to a carbon atom or the center of a hexagon?

in-context