Question

determine the crystallographic directions of vectors b, c, and d shown in the following cubic unit cell: example: direction a = -4 3 0 (use this notation: square brackets, spaces between indices, and a minus sign for a negative index) direction b: direction c: direction d:

Explanation:

Step1: Find vector - end points for B

Assume origin \((0,0,0)\) and end - point of B is \((\frac{2}{3},0,0)\). Subtract origin coordinates from end - point coordinates: \(\frac{2}{3}-0=\frac{2}{3}\), \(0 - 0=0\), \(0 - 0=0\). Multiply by 3 to get integer indices \([2\ 0\ 0]\).

Step2: Find vector - end points for C

Assume origin \((0,0,0)\) and end - point of C is \((0,1,\frac{1}{3})\). Subtract origin coordinates: \(0 - 0=0\), \(1 - 0 = 1\), \(\frac{1}{3}-0=\frac{1}{3}\). Multiply by 3 to get integer indices \([0\ 3\ 1]\).

Step3: Find vector - end points for D

Let start - point be \((\frac{1}{2},\frac{1}{2},0)\) and end - point be \((\frac{2}{3},\frac{1}{3},\frac{2}{3})\). Subtract start - point from end - point: \(\frac{2}{3}-\frac{1}{2}=\frac{4 - 3}{6}=\frac{1}{6}\), \(\frac{1}{3}-\frac{1}{2}=\frac{2 - 3}{6}=-\frac{1}{6}\), \(\frac{2}{3}-0=\frac{2}{3}\). Multiply by 6 to get integer indices \([1\ - 1\ 4]\).

Answer:

Direction B: [2 0 0]
Direction C: [0 3 1]
Direction D: [1 - 1 4]