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. FCC: Linear Density 3.5 nm a 2 LD = Linear Density of Atoms  LD =
Number of atoms
Unit length of direction vector a
[110]
ex: linear density of Al in [110] direction  a = nm
# atoms
length 1
3.5 nm a 2 LD - =
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. Planar Density of (100) Iron
Solution:  At T < 912ºC iron has the BCC structure.
2D repeat unit R 3 4 a =
(100)
Radius of iron R = nm
Adapted from Fig. 3.2(c), Callister & Rethwisch 8e. =
Planar Density = a 2 1
atoms
2D repeat unit
nm2
12.1 m2
= 1.2 x 1019 R 3 4
area

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

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 nm)