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
• Linear Density of Atoms  LD =
Number of atoms
Unit length of direction vector
[110]
ex: linear density of Al in [110]
direction
a = 0.405 nm
# atoms
a
Adapted from
Fig. 3.1(a),
Callister &
Rethwisch 8e.
LD =
length
2
= 3.5 nm-1
2a
Chapter 3 - 2
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]
Chapter 3 -
Planar Density of (100) Iron
Solution: At T < 912ºC iron has the BCC structure.
2D repeat unit
(100)
Planar Density =
area
2D repeat unit
1
a2
=
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=
1
4 3
R
3
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
atoms above plane
atoms below plane
h=
3
a
2
2
atoms
2D repeat unit
 4 3  16 3 2
2
area = 2 ah = 3 a = 3 
R  =
R
3
 3

1
atoms =
= 7.0
2
Planar Density =
area
2D repeat unit
16 3
3
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