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Linear and Planar Atomic Densities

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.

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Linear and Planar Atomic Densities

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  1. 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.

  2. [110] # atoms 2 a = = 1 - LD 3.5 nm 2 a length FCC: Linear Density Number of atoms Unit length of direction vector • Linear Density of Atoms  LD =  ex: linear density of Al in [110] direction a = 0.405 nm Adapted from Fig. 3.1(a), Callister & Rethwisch 8e.

  3. P 3.53 (a): Linear Density for BCC • Calculate the linear density for the following directions in terms of R: • [100] • [110] • [111]

  4. 4 3 = a R 3 atoms 2 2D repeat unit 1 a 1 atoms atoms 12.1 = 1.2 x 1019 = = Planar Density = 2 nm2 m2 4 3 R 3 area 2D repeat unit Planar Density of (100) Iron 2D repeat unit Solution:  At T < 912ºC iron has the BCC structure. (100) Radius of iron R = 0.1241 nm Adapted from Fig. 3.2(c), Callister & Rethwisch 8e.

  5. 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.

  6. atoms in plane atoms above plane atoms below plane 2 æ ö 4 16 ç  = = = = 2 2 area 2 ah 3 3 3 3 a R R ç  3 3 atoms è ø 2D repeat unit 1 atoms Planar Density = atoms = = 7.0 0.70 x 1019 m2 nm2 16 3 2 R area 3 2D repeat unit Planar Density of BCC (111) Iron 1 atom in plane/ unit surface cell a 2 Solution (cont):  (111) plane 2D repeat unit 3 = h a 2

  7. P 3.54 (a): FCC Derive planar density expressions for FCC (100), (110), and (111) planes.

  8. 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)

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