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A diagram of two images is shown. In the first image, eight stacked cubes that make up one large cube are shown. Three lines that run from top to bottom, front to back and sided to side in the middle of the structure are shaded darker than the rest of the lines. The second image shows the same set of cubes, but this time spheres at the end of each line are numbered; the horizontal line that goes left to right is labeled with a “2” and a “5,” the vertical line is labeled with a “1” and a “6” and the line that goes horizontally front to back is labeled with a “3” and a “4.”
An atom in a simple cubic lattice structure contacts six other atoms, so it has a coordination number of six.

In a simple cubic lattice, the unit cell that repeats in all directions is a cube defined by the centers of eight atoms, as shown in [link] . Atoms at adjacent corners of this unit cell contact each other, so the edge length of this cell is equal to two atomic radii, or one atomic diameter. A cubic unit cell contains only the parts of these atoms that are within it. Since an atom at a corner of a simple cubic unit cell is contained by a total of eight unit cells, only one-eighth of that atom is within a specific unit cell. And since each simple cubic unit cell has one atom at each of its eight “corners,” there is 8 × 1 8 = 1 atom within one simple cubic unit cell.

A diagram of two images is shown. In the first image, eight spheres are stacked together to form a cube and dots at the center of each sphere are connected to form a cube shape. The dots are labeled “Lattice points” while a label under the image reads “Simple cubic lattice cell.” The second image shows the portion of each sphere that lie inside the cube. The corners of the cube are shown with small circles labeled “Lattice points” and the phrase “8 corners” is written below the image.
A simple cubic lattice unit cell contains one-eighth of an atom at each of its eight corners, so it contains one atom total.

Calculation of atomic radius and density for metals, part 1

The edge length of the unit cell of alpha polonium is 336 pm.

(a) Determine the radius of a polonium atom.

(b) Determine the density of alpha polonium.

Solution

Alpha polonium crystallizes in a simple cubic unit cell:

A diagram shows a cube with a one eighth portion of eight spheres inside the cube, one section in each corner. Along the bottom right side of the cube are two double ended arrows that each stretch along half of the total distance across the cube. Each arrow is labeled “r.”

(a) Two adjacent Po atoms contact each other, so the edge length of this cell is equal to two Po atomic radii: l = 2 r . Therefore, the radius of Po is r = l 2 = 336 pm 2 = 168 pm .

(b) Density is given by density = mass volume . The density of polonium can be found by determining the density of its unit cell (the mass contained within a unit cell divided by the volume of the unit cell). Since a Po unit cell contains one-eighth of a Po atom at each of its eight corners, a unit cell contains one Po atom.

The mass of a Po unit cell can be found by:

1 Po unit cell × 1 Po atom 1 Po unit cell × 1 mol Po 6.022 × 10 23 Po atoms × 208.998 g 1 mol Po = 3.47 × 10 −22 g

The volume of a Po unit cell can be found by:

V = l 3 = ( 336 × 10 −10 cm ) 3 = 3.79 × 10 −23 cm 3

(Note that the edge length was converted from pm to cm to get the usual volume units for density.)

Therefore, the density of Po = 3.471 × 10 −22 g 3.79 × 10 −23 cm 3 = 9.16 g/cm 3

Check your learning

The edge length of the unit cell for nickel is 0.3524 nm. The density of Ni is 8.90 g/cm 3 . Does nickel crystallize in a simple cubic structure? Explain.

Answer:

No. If Ni was simple cubic, its density would be given by:
1 Ni atom × 1 mol Ni 6.022 × 10 23 Ni atoms × 58.693 g 1 mol Ni = 9.746 × 10 −23 g
V = l 3 = ( 3.524 × 10 −8 cm ) 3 = 4.376 × 10 −23 cm 3
Then the density of Ni would be = 9.746 × 10 −23 g 4.376 × 10 −23 cm 3 = 2.23 g/cm 3
Since the actual density of Ni is not close to this, Ni does not form a simple cubic structure.

Got questions? Get instant answers now!

Most metal crystals are one of the four major types of unit cells. For now, we will focus on the three cubic unit cells: simple cubic (which we have already seen), body-centered cubic unit cell    , and face-centered cubic unit cell    —all of which are illustrated in [link] . (Note that there are actually seven different lattice systems, some of which have more than one type of lattice, for a total of 14 different types of unit cells. We leave the more complicated geometries for later in this module.)

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Source:  OpenStax, Chemistry. OpenStax CNX. May 20, 2015 Download for free at http://legacy.cnx.org/content/col11760/1.9
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