1. DESCRIBING CRYSTALS MATHEMATICALLY Three directional vectors Three distances: a, b, c Three angles:   ll points can be described using coordinate system in terms of the three vectors qa, rb, sc In cubic systems, symmetry makes a = b = c and  = 90 o Describe points using (q, r, s)

2. CRYSTAL DIRECTIONS Directions are represented by arrows Labeled as [hkl] where h,k,l represent the coordinates of the point they pass through, when they originate from the origin Directions in crystals are like compass points; they don’t depend on where they start from All directions parallel to [111] are all [111], similar to north being north regardless of where you are standing.

3. CRYSTAL PLANES Planes are described using Miller Indices They are the reciprocals of the intercepts on the major axes, reduced to the lowest possible integers Represented by (hkl) In (001) the intercepts are ∞ for the x and y axes as they are parallel; and 1 for the z axis All parallel planes have the same Miller Indices

4. CRYSTAL PLANES In (110) There are intercepts on the x and y axes The plane is parallel to the z axis Intercepts are for h for k for l

5. CRYSTAL PLANES For (111) There are three intercepts All at one unit vector FOR FCC THIS IS THE HEXAGONAL PLANE IT IS THE CLOSEST PACKED PLANE

6. SUMMARY OF MEANINGS OF PARENTHESES (q,r,s) represents a point – note the exclusive use of commas [hkl] represents a direction represents a family of directions (hkl) represents a plane {hkl} represents a family of planes