How do particles—whether atoms, ions, or molecules—pack together in crystals?

Let’s look at metals, which are the simplest examples of crystal packing because the individual atoms can be treated as spheres. Not surprisingly, metal atoms (and other kinds of particles as well) generally pack together in crystals so that they can be as close as possible and maximize intermolecular attractions.

If you were to take a large number of uniformly sized marbles and arrange them in a box in some orderly way, there are four possibilities you might come up with.

◦ One way to arrange the marbles is in orderly rows and stacks, with

the spheres in one layer sitting directly on top of those in the previous layer so that all layers are identical (Figure 10.20a). Called simple cubic packing, each sphere is touched by six neighbors—four in its own layer, one above, and one below—and is thus said to have a coordination number of 6. Only 52% of the available volume is occupied by the spheres in simple cubic packing, making inefficient use of space and minimizing attractive forces. Of all the metals in the periodic table, only polonium crystallizes in this way.

10.8 Unit Cells and the Packing of Spheres in Crystalline Solids

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◦ Alternatively, space could be used more efficiently if, instead of stacking the spheres directly on top of one another, you slightly separate the spheres in a given layer and offset alternating layers in an a-b-a-b arrangement so that the spheres in the b layers fit into the depressions between spheres in the a layers, and vice versa(Figure 10.20b). Called body-centered cubic packing, each sphere has a coordination number of 8—four neighbors above and four below—and space is used quite efficiently: 68% of the available volume is occupied.

◦ Iron, sodium, and 14 other metals crystallize in this way. The remaining two packing arrangements of spheres are both said to be closest packed. The hexagonal closest-packed arrangement (Figure 10.21a) has two alternating layers, a-b-a-b. Each layer has a hexagonal arrangement of touching spheres, which are offset so that spheres in a b layer fit into the small triangular depressions between spheres in an a layer. Zinc, magnesium, and 19 other metals crystallize in this way.

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The cubic closest-packed arrangement (Figure 10.21b) has three alternating layers, a-b-c-a-b-c. The a-b layers are identical to those in the hexagonal closest packed arrangement, but the third layer is offset from both a and b layers. Silver, copper, and 16 other metals crystallize with this arrangement.

In both kinds of closest-packed arrangements, each sphere has a coordination number of 12—six neighbors in the same layer, three above, and three below—and 74% of the available volume is filled. The next time you’re in a grocery store, look to see how the oranges are stacked in their display box. Chances are good they’ll have a closest-packed arrangement.

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Having now taken a bulk view of how spheres can pack in a crystal, let’s also take a close-up view. Just as a large wall might be made up of many identical bricks stacked together in a repeating pattern, a crystal is made up of many small repeat units called unit cells stacked together in three dimensions.

Fourteen different unit-cell geometries occur in crystalline solids. All are parallelepipeds—six-sided geometric solids whose faces are parallelograms. We’ll be concerned here only with those unit cells that have cubic symmetry; that is, cells whose edges are equal in length and whose angles are 90°.

Unit Cells

There are three kinds of cubic unit cells: primitive-cubic, body-centered cubic, and face-centered cubic. As shown in Figure 10.22a, a primitive-cubic unit cell for a metal has an atom at each of its eight corners, where it is shared with seven neighboring cubes that come together at the same point. As a result, only 1/8 of each corner atom “belongs to” a given cubic unit. This primitive-cubic unit cell, with all atoms arranged in orderly rows and stacks, is the repeat unit found in simple cubic packing.

Unit Cells

Unit Cells

A body-centered cubic unit cell has eight corner atoms plus an additional atom in the center of the cube (Figure 10.22b). This body-centered cubic unit cell, with two repeating offset layers and with the spheres in a given layer slightly separated, is the repeat unit found in body-centered cubic packing.

A face-centered cubic unit cell has eight corner atoms plus an additional atom on each of its six faces, where it is shared with one other neighboring cube (Figure 10.23). Thus, 1/2 of each face atom belongs to a given unit cell. This face-centered cubic unit cell is the repeat unit found in cubic closest-packing, as can be seen by looking down the body diagonal of a unit cell (Figure 10.23b). The faces of the unit-cell cube are at 54.7° angles to the layers of the atoms.

Unit Cells

A summary of stacking patterns, coordination numbers, amount of space used, and unit cells for the four kinds of packing of spheres is given in Table 10.10. Hexagonal closest-packing is the only one of the four that has a non-cubic unit cell.

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Unit Cells

TABLE 10.10 Summary of the Four Kinds of Packing for Spheres

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WORKED EXAMPLE 10.7Q. How many atoms are in one primitive-cubic unit cell of a metal?STRATEGY AND SOLUTION

As shown in Figure 10.22a, there is an atom at each of the eight corners of the primitive-cubic unit cell. When unit cells are stacked together, each corner atom is shared by eight cubes, so that only 1/8 of each atom “belongs” to a given unit cell. Thus there is1 X 1/8 = 1atom per unit cell.

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WORKED EXAMPLE 10.8Q. Silver metal crystallizes in a cubic closest-packed

arrangement with the edge of the unit cell having a length d = 407 pm What is the radius (in picometers) of a silver atom?

STRATEGY AND SOLUTIONCubic closest-packing uses a face-centered cubic

unit cell. Looking at any one face of the cube head-on shows that the face atoms touch the corner atoms along the diagonal of the face but that corner atoms do not touch one another along the edges. Each diagonal is therefore equal to four atomic radii, 4r:

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Because the diagonal and two edges of the cube form a right triangle, we can use the Pythagorean theorem to set the sum of the squares of the two edges equal to the square of the diagonal , and then solve for r, the radius of one atom:

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WORKED EXAMPLE 10.9Q. Nickel has a face-centered cubic unit cell with a

length of 352.4 pm along an edge. What is the density of nickel in :

STRATEGYDensity is mass divided by volume. The mass of a

single unit cell can be calculated by counting the number of atoms in the cell and multiplying by the mass of a single atom. The volume of a single cubic unit cell with edge d is

=

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SOLUTIONEach of the eight corner atoms in a face-centered cubic unit cell

is shared by eight unit cells, so that only atom belongs to a single cell. In addition, each of the six face

atoms is shared by two unit cells, so that atoms belong to a single cell. Thus, a single cell has 1 corner

atom and 3 face atoms, for a total of 4, and each atom has a mass equal to the molar mass of nickel (58.69 g/mol) divided by Avogadro’s number We can now calculate the density:

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