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Close Packing

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay

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Packing and Interstices

As shown here,

identical atoms (or

any round object, like

coins or pool balls)

can be arranged in a

hexagonal pattern

where each is

surrounded by sixneighbors. This is the

tightest possible

packing.

The left diagram shows what happens when we place a second layer (light gray) on top of a lower layer

(dark gray). Each upper atom nestles into a pocket formed by three other atoms, enclosing a small space

called an interstice. In this case, the four atoms occupy the corners of a tetrahedron, so the openings are

called tetrahedral interstices. There are two sets. One is enclosed by an upper atom resting on three below

(shown in purple). Another consists of a cluster of three upper atoms resting on one below (shown in blue)In both cases the centers of the four atoms are the vertices of a tetrahedron (shown in yellow) but in one

case the tetrahedron points up, in the other it points down.

It turns out there are much larger openings between the sheets. A cluster of three upper atoms can also

rest on a cluster of three lower atoms (shown in red). These six atoms lie at the corners of an octahedron,

so these openings are called octahedral interstices. In real minerals the close-packed sheets are often

anions (usually oxygen) with cations in the interstices. Some of the tetrahedral and octahedral interstices

are shown between the atoms (blue and yellow, respectively)

On the right is a representation of a pair of close-packed sheets just showing the shapes of the interstices

Each vertex is the site of an atom. We have a sheet of octahedra (blue top faces) separated by tetrahedra.The top faces of the tetrahedra are omitted and only the downward-pointing tetrahedra are shown. The

upward-pointing tetrahedra are hidden; their top vertices are located where three octahedra join.

We can see each octahedron has six atoms, but each atom is shared with three octahedra, so for each

octahedron there are 6(1/3)=2 atoms. Each upward-pointing tetrahedron can be pictured as sharing the

three base atoms with three neighbors plus having the top atom to itself, or 3(1/3)+1=2 atoms. The same i

true for downward-pointing tetrahedra. Thus there are equal numbers of octahedra, upward-pointing

tetrahedra and downward-pointing tetrahedra, and two atoms for each polyhedron. There are two

tetrahedra for each octahedron, one upward and one downward.

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Hexagonal Close Packing

Top: Identical atoms can be

packed into a sheet with a

hexagonal pattern. Anotherlayer can be placed on top in

one of two ways: over the

upward-pointing gaps (blue) or

the downward-pointing gaps

(yellow). Layers B and C show

layers in these positions.

Bottom: If layers alternate

(left) or are randomly stacked

(right) the overall structure has

only the hexagonal symmetry

of the individual sheets. The

alternating pattern is called

Hexagonal close packing. The

oxygen atoms in corundum and

hematite have this packing.

In real crystals the anions often have close packing with the interstices between layers occupied by cation

Since the cations attract the anion sheets and repel nearby cations, electrical forces tend to produce

symmetrical arrangements. Thus the regular alternating stacking is favored.

Cubic Close Packing

Top: The three possible

arrangements of close-packed

atomic layers are shown. Note

how the three layers also form

a square lattice of atoms. The

lattice plane is tilted in this

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view.

Bottom: If the layers repeat

cyclically, cubic close packing

results. The two right diagrams

show how a cube can be

formed by this packing. Oxyge

atoms in spinel and magnetite

have this arrangement, as dothe chlorine atoms in halite,

the sulfur atoms in sphalerite

and the calcium atoms in

fluorite.

The face-centered cubic unit

cell (F cell) has cubic close

packing.

Return to Crustal Materials Index

Return to Crystal Structures Index

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Created 18 September 1998, Last Update 22 September 1999

Not an official UW Green Bay site

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