2D Packing Examples
Answer the following questions for both
packing diagrams.
hole
1. Find the smallest unit that, if repeated,
would give you the entire structure. This
is the unit cell. Draw a box on the
diagram representing the unit cell.
2. How many total atoms are inside the unit
cell? Remember that only fractions of
some atoms lie inside the box you drew.
hole
3. Accurately draw boxes exactly 3 atoms
wide. How many total atoms are in each?
Which displays the greater packing
density?
SIMPLE CUBIC PACKING
A
TOP VIEW
AA
SCP 1 | Flash Anim | Jmol 1
A
unit cell: simple cubic
BODY CENTERED CUBIC PACKING - ABAB
A
B
Unit cell: body centered cubic
Do not touch
BCP 1 | Flash Anim | Jmol 1
HEXAGONAL CLOSEST PACKING - ABAB
B
A
1200
Unit cell: hexagonal
CUBIC CLOSEST PACKING
B
A
C
A
7
6
A
8
4
3
5
C
B
1
2
B
A
C
Unit cell: face centered cubic
6
7
8
A
4
3
5
CUBIC CLOSEST PACKING
FCC = CCP: Jmol
CCP (FCC) 1 | Flash Anim | Jmol 1
2
1
Counting atoms in a unit cell
1. What fraction of the white colored sphere is part of a unit cell in each of the
lattices shown below.
A face atom? ________
An edge atom? ________
A corner atom? ________
2. From your answers above, how many spheres below to the unit cell shown.
Simple Cubic ________
Face centered cubic ________
Body centered cubic ________
Now to some real structures. How do we know the arrangement of ions and
atoms?
X-Ray Crystallography:
• Na – bcc – what’s up with it’s 3s1 electron
• NaCl
• Bragg’s Law and diffraction
• Xray diffraction (Laue transmission) from simple cubic, bcc and fcc lattices
How do you describe salts with more than one atom type? By unit cells and holes
NaCl
Space fill
Types of Holes
109.50
3 atoms
e clouds
in plane
Tetrahedron – 4 identical
equilateral triangles
5 atoms/
e clouds
in plane
•What's a tetrahedron
•What's an octahedron
•Holes: Cubic, tetrahedral, and Octahedral
•CCP and holes
SF6
Octahedron – 8 identical
triangles
Determining the empirical formula from unit cell data:
A salt must be electrically neutral. If you know the charge on the cations and
anions, you can determine MxNy.
Alternatively, you can use crystal data to determine the structure of the unit cell
and hence the empirical formula. Determine the number of each different ion in
the following unit cells. Does it correspond to the formula from charge balance?
1. Copper (I) chloride
Cl
Cu
4. Based on Clalone, what type of
unit cell is this?
What type of “hole”
is occupied by Cu+?
2. Barium Chloride
5. Based on just Clalone, what type of
unit cell is NaCl?
What type of “hole”
is occupied by Na+?
3. Rhenium oxide
Calculation of the atomic radius of a polonium atom, which forms a simple cubic
cell, from macroscopic parameters. How many P atoms are in the unit cell?
Given Polonium Data
• density 9.23 g/cm3 macroscopic meas.
• molar mass: 208.98 g
• Avogadro’s #: 6.022 x 1023 atoms/mol
r
Top
Side
views
2r
Use 2 simple equations
Solve for the following:
Vatom/Veff =
2r
Veff (from density data) and dimensional analysis
Veff =
Vatom =
rPo =
Body-Centered Cubic Unit Cell
unit
cell
a
4r
a
√2 a
a
x2 = a2 + a2
x = √2 a
2
( 2 a)  a
a 
2
 ( 4r )
4r
3
Vunitc ell  (
4r
3
)
3
2
Edge sphere
Do how many unit
cells does the
yellow sphere
belong?
Corner sphere
Do how many unit
cells does the
yellow sphere
belong?
Face sphere
Do how many unit
cells does the
yellow sphere
belong?
CUBIC CLOSEST PACKING
ABC ABC
B
A
C
A
C
B
A