2D Packing Examples

hole

hole

Answer the following questions for both packing diagrams.

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.

3. Accurately draw boxes exactly 3 atoms wide. How many total atoms are in each? Which displays the greater packing density?

A

TOP VIEWAA

SIMPLE CUBIC PACKING

A

SCP 1 | Flash Anim | Jmol 1

unit cell: simple cubic

BODY CENTERED CUBIC PACKING - ABAB

BCP 1 | Flash Anim | Jmol 1

Unit cell: body centered cubic

A

B

Do not touch

A

B

HEXAGONAL CLOSEST PACKING - ABAB

1200

Unit cell: hexagonal

A

B C

CUBIC CLOSEST PACKING

A

B

C

A

6

A

A

B

C

CUBIC CLOSEST PACKING

CCP (FCC) 1 | Flash Anim | Jmol 1

Unit cell: face centered cubic

3

6

4

1

7

2

5

8

12

3 4

5

7

8

FCC = CCP: Jmol

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 corner atom? ________

A face atom? ________ An edge atom? ________

2. From your answers above, how many spheres below to the unit cell shown.

Simple Cubic ________

Body centered cubic ________

Face 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

NaCl

Space fill

C

H H

H H3 atomse clouds in plane

109.50

Tetrahedron – 4 identicalequilateral triangles

SF6

5 atoms/e clouds in plane

Octahedron – 8 identical triangles

How do you describe salts with more than one atom type? By unit cells and holes

Types of Holes

• What's a tetrahedron• What's an octahedron• Holes: Cubic, tetrahedral, and Octahedral• CCP and holes

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

2. Barium Chloride

3. Rhenium oxide

4. Based on Cl- alone, what type of unit cell is this? What type of “hole” is occupied by Cu+?

Cu

Cl

5. Based on just Cl- alone, what type of unit cell is NaCl? What type of “hole” is occupied by Na+?

TopSideviews 2r

r

Given Polonium Data• density 9.23 g/cm3 macroscopic meas.• molar mass: 208.98 g • Avogadro’s #: 6.022 x 1023 atoms/mol

Use 2 simple equations

Solve for the following:Vatom/Veff =

Veff (from density data) and dimensional analysisVeff =

Vatom =

rPo =

Calculation of the atomic radius of a polonium atom, which forms a simple cubiccell, from macroscopic parameters. How many P atoms are in the unit cell?

2r

x2 = a2 + a2

x = √2 a

a4r

√2 a

Body-Centered Cubic Unit Cell

unitcell

a

a

3unitcell

222

)3

r4(V

3

r4a

)r4(a)a2(

Corner sphereDo how many unit cells does the yellow sphere belong?

Face sphereDo how many unit cells does the yellow sphere belong?

Edge sphereDo how many unit cells does the yellow sphere belong?

A

B C

CUBIC CLOSEST PACKINGABC ABC

A

B

C

A