1. Crystal Structures zTypes of crystal structures yFace centered cubic (FCC) yBody centered cubic (BCC) yHexagonal close packed (HCP) zClose Packed Structures yDifferent Packing of HCP and FCC zCrystallographic Directions and Planes ycubic systems

2. Face Centered Cubic (FCC) zAtoms are arranged at the corners and center of each cube face of the cell. yAtoms are assumed to touch along face diagonals

3. Face Centered Cubic (FCC) zThe lattice parameter, a, is related to the radius of the atom in the cell through: zCoordination number: the number of nearest neighbors to any atom. For FCC systems, the coordination number is 12.

4. Face Centered Cubic (FCC) zAtomic Packing Factor: the ratio of atomic sphere volume to unit cell volume, assuming a hard sphere model. yFCC systems have an APF of 0.74, the maximum packing for a system in which all spheres have equal diameter.

5. Body Centered Cubic zAtoms are arranged at the corners of the cube with another atom at the cube center.

6. Body Centered Cubic zSince atoms are assumed to touch along the cube diagonal in BCC, the lattice parameter is related to atomic radius through:

7. Body Centered Cubic zCoordination number for BCC is 8. Each center atom is surrounded by the eight corner atoms. zThe lower coordination number also results in a slightly lower APF for BCC structures. BCC has an APF of 0.68, rather than 0.74 in FCC

8. Hexagonal Close Packed zCell of an HCP lattice is visualized as a top and bottom plane of 7 atoms, forming a regular hexagon around a central atom. In between these planes is a half-hexagon of 3 atoms.

9. Hexagonal Close Packed zThere are two lattice parameters in HCP, a and c, representing the basal and height parameters respectively. In the ideal case, the c/a ratio is 1.633, however, deviations do occur. zCoordination number and APF for HCP are exactly the same as those for FCC: 12 and 0.74 respectively. yThis is because they are both considered close packed structures.

10. Close Packed Structures zEven though FCC and HCP are close packed structures, they are quite different in the manner of stacking their close packed planes. yClose packed stacking in HCP takes place along the c direction ( the (0001) plane). FCC close packed planes are along the (111). yFirst plane is visualized as an atom surrounded by 6 nearest neighbors in both HCP and FCC.

11. Close Packed Structures yThe second plane in both HCP and FCC is situated in the “holes” above the first plane of atoms. yTwo possible placements for the third plane of atoms xThird plane is placed directly above the first plane of atoms ABA stacking -- HCP structure xThird plane is placed above the “holes” of the first plane not covered by the second plane ABC stacking -- FCC structure

12. Close Packed Structures

13. Crystallographic Directions zCubic systems ydirections are named based upon the projection of a vector from the origin of the crystal to another point in the cell. zConventionally, a right hand Cartesian coordinate system is used. yThe chosen origin is arbitrary, but is always selected for the easiest solution to the problem.

14. Crystallographic Directions zPoints within the lattice are written in the form h,k,l, where the three indices correspond to the fraction of the lattice parameters in the x,y,z direction.

15. Miller Indices zProcedure for writing directions in Miller Indices yDetermine the coordinates of the two points in the direction. (Simplified if one of the points is the origin). ySubtract the coordinates of the second point from those of the first. yClear fractions to give lowest integer values for all coordinates

16. Miller Indices yIndices are written in square brackets without commas (ex: [hkl]) yNegative values are written with a bar over the integer. xEx: if h<0 then the direction is x

17. Miller Indices zCrystallographic Planes yIdentify the coordinate intercepts of the plane xthe coordinates at which the plane intercepts the x, y and z axes.  If a plane is parallel to an axis, its intercept is taken as . xIf a plane passes through the origin, choose an equivalent plane, or move the origin yTake the reciprocal of the intercepts

18. Miller Indices yClear fractions due to the reciprocal, but do not reduce to lowest integer values. yPlanes are written in parentheses, with bars over the negative indices. xEx: (hkl) or if h<0 then it becomes zex: plane A is parallel to x, and intercepts y and z at 1, and therefore is the (011). Plane B passes through the origin, so the origin is moved to O’, thereby making the plane the

19. Miller Indices