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Chapter 3 - 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.

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Chapter 3 -2 ex: linear density of Al in [110] direction a = 0.405 nm FCC: Linear Density Linear Density of Atoms LD = a [110] Unit length of direction vector Number of atoms # atoms length 1 3.5 nm a2 2 LD Adapted from Fig. 3.1(a), Callister & Rethwisch 8e.

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

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Chapter 3 -4 Planar Density of (100) Iron Solution: At T < 912ºC iron has the BCC structure. (100) Radius of iron R = 0.1241 nm R 3 34 a Adapted from Fig. 3.2(c), Callister & Rethwisch 8e. 2D repeat unit = Planar Density = a 2 1 atoms 2D repeat unit = nm 2 atoms 12.1 m2m2 atoms = 1.2 x 10 19 1 2 R 3 34 area 2D repeat unit

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Chapter 3 - 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.

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Chapter 3 -6 Planar Density of BCC (111) Iron Solution (cont): (111) plane 1 atom in plane/ unit surface cell 33 3 2 2 R 3 16 R 3 4 2 a3ah2area atoms in plane atoms above plane atoms below plane ah 2 3 a 2 2D repeat unit 1 = = nm 2 atoms 7.0 m2m2 atoms 0.70 x 10 19 3 2 R 3 16 Planar Density = atoms 2D repeat unit area 2D repeat unit

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Chapter 3 - P 3.54 (a): FCC Derive planar density expressions for FCC (100), (110), and (111) planes.

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Chapter 3 - P 3.56 8 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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