1. Chapter 2. Reciprocal Lattice Issues that are addressed in this chapter include: - Reciprocal Lattice - Packing Factor - Bragg law - Scattered wave amplitude - Brillouin Zones - Fourier analysis of the basis

2. zThe set of all waves vectors k that yield plane wave with the periodicity of a given Bravais lattice. Reciprocal lattice zA diffraction pattern is not a direct representation of the crystal lattice zThe diffraction pattern is a representation of the reciprocal lattice

3. Reciprocal Lattice/Unit Cells We define a plane and consider some lattice planes (001) (100) (002) (101) (102)

4. 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

5. ATOMIC PACKING FACTOR zFill a box with hard spheres yPacking factor = total volume of spheres in box / volume of box yQuestion: what is the maximum packing factor you can expect? zIn crystalline materials: yAtomic packing factor = total volume of atoms in unit cell / volume of unit cell y(as unit cell repeats in space)

6. 1 atom/unit cell (8 x 1/8 = 1) 2 atoms/unit cell (8 x 1/8 + 1 = 2) 4 atoms/unit cell (8 x 1/8 + 6 x 1/2 = 4) 1 atom/unit cell (8 x 1/8 = 1) coordination number 12coordination number 8coordination number 6

8. APF for a simple cubic structure = 0.52 ATOMIC PACKING FACTOR contains 8 x 1/8 = 1atom/unit cell Adapted from Fig. 3.19, Callister 6e. Lattice constant close-packed directions a R=0.5a

9. 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

10. APF for a Face-centered cubic structure =  /(3  2) = 0.74 (best possible packing of identical spheres) Adapted from Fig. 3.1(a), Callister 6e. ATOMIC PACKING FACTOR: FCC

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

12. APF for a body-centered cubic structure =  3/8 = 0.68 Adapted from Fig. 3.2, Callister 6e. ATOMIC PACKING FACTOR: BCC

13. 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