# Chapter 3: The Structure of Crystalline Solids Linear and Planar Atomic Densities Linear Density: Directional equivalency is related to the atomic linear density in the sense that equivalent directions have identical linear densities. The direction vector is positioned so as to pass through atom centers. The fraction of line length intersected by these atoms is equal to the linear density. Planar Density: Crystallographic planes that are equivalent have the same atomic planar density. The plane of interest is positioned so as to pass through atom centers. Planar density is the fraction of total crystallographic plane area that is occupied by atoms. Linear and planar densities are one- and two-dimensional analogs of the atomic packing factor. Chapter 3 - FCC: Linear Density Number of atoms Linear Density of Atoms LD = Unit length of direction vector 

a Adapted from Fig. 3.1(a), Callister & Rethwisch 8e. ex: linear density of Al in  direction a = 0.405 nm # atoms LD length 2 2a 3.5 nm 1 Chapter 3 - 2 P 3.53 (a): Linear Density for BCC

Calculate the linear density for the following directions in terms of R: a.  b.  c.  Chapter 3 - Planar Density of (100) Iron Solution: At T < 912C iron has the BCC structure. 2D repeat unit (100) Planar Density = area 2D repeat unit 1 a2 = 1

4 3 3 4 3 R 3 Radius of iron R = 0.1241 nm Adapted from Fig. 3.2(c), Callister & Rethwisch 8e. atoms 2D repeat unit a R atoms atoms 19 = 1.2 x 10 2 = 12.1 2

nm m2 Chapter 3 - 4 P 3.55 (a): Planar Density for BCC Derive the planar density expressions for BCC (100) and (110) planes in terms of the atomic radius R. Chapter 3 - Planar Density of BCC (111) Iron Solution (cont): (111) plane 1 atom in plane/ unit surface cell 2a atoms in plane nit atoms above plane

r ep ea tu atoms below plane 2D h 3 a 2 2 4 3 16 3 2 area 2 ah 3 a 3 R R 3

3 2 atoms 2D repeat unit Planar Density = area 2D repeat unit 1 16 3 3 atoms = = 7.0 2 R 2 nm

0.70 x 1019 atoms m2 Chapter 3 - 6 P 3.54 (a): FCC Derive planar density expressions for FCC (100), (110), and (111) planes. Chapter 3 - P 3.56 3.56 (a) Derive the planar density expression for the HCP (0001) plane in terms of the atomic radius R. (b) Compute the planar density value for this same plane for magnesium. (atomic radius for magnesium is 0.160 nm) Chapter 3 - 8