Cl
A body-centered cubic unit cell has atoms on each of the eight corners and one atom in the center.
A face-centered cubic unit cell has atoms on each of the eight corners and an atom in the middle of each of the six faces of the cube.
There are 8 corner atoms in this unit cell.
There are 2 face atoms in this unit cell (yellow).
There are 4 edge atoms in this unit cell (yellow).
There are 1 body atom in this unit cell (yellow).
To determine the number of atoms in the unit cell, sum the contributions from the 4 different lattice positions:
The bromide ions are in the face-centered cubic lattice positions, one in each of the 8 corners and 6 faces.
The lithium ions occupy the holes in between the bromide ions.(depicted as the yellow atoms in the diagram below)
Almost all ionic and metallic compounds can form crystals when in the solid state. The characterization of the crystal involves the identification of its unit cell.
Under certain conditions, elemental gold, silver and copper form into beautiful crystals which are highly prized by mineral collectors around the world. Shown here is crystalline silver on copper.
Photograph by John A. Jaszczak. A.E. Seaman Mineral Museum collection
A crystalline solid is composed of repeating patterns of atoms or ions. This repeating pattern is called a unit cell. A two-dimensional example will help us begin to identify a unit cell.
One possible selection for the unit cell is shown. Using the unit cell, we can re-create the pattern.
A slightly different unit cell can also be chosen. We can also re-create the pattern using this unit cell.
Let's look at another example.
The items at the corners of the unit cell are shared by four unit cells.
Each unit cell is said to contain one-fourth of the item. Since there are four corners and each corner has one fourth of the item, the unit cell contains 1 entire item. The other item in the center of the unit cell is not shared. The ratio of item A to item B in each unit cell is 1:1.
A crystal is a three-dimensional arrangement of ions or atoms called a lattice.
The positions occupied by atoms or ions are called lattice points.
The unit cell for this crystal is called simple cubic.
The three common cubic systems are simple cubic, body-centered cubic, and face-centered cubic.
Simple cubic
Body-centered cubic
Face-centered cubic
Atoms or ions are shared between adjacent unit cells. The lattice position of the atom or ion determines the number of unit cells involved in the share. There are four different lattice positions an atom or ion can occupy.
A compound that exhibits a simple cubic crystal structure is cesium chloride, CsCl. The chloride ions occupy the corner positions of the unit cell and the cesium ion occupies the hole in the middle.
Why are there 8 chloride ions and only 1 cesium ion?
Each corner position in a unit cell is shared equally by eight unit cells.
So, for one unit cell, each corner position counts for 1/8 of the ion. There are 8 chloride ions, each counting 1/8 for a total of 1 chloride ion per unit cell.
CsCl is simple cubic, not body-centered cubic. To determine the type of unit cell, only consider one type of ion (Cl in this example). The other ion fills the hole(s) in the unit cell.
Sodium chloride (NaCl) exhibits a face-centered cubic arrangment. Using the chloride ions in yellow, the face-centered cubic unit cell is constructed as shown here.
The smaller sodium ions fit into the holes between the chloride ions.
What is the ratio of sodium ions to chloride ions in the unit cell?
Looking at the chloride ions, there are 8 at the corner positions and 6 at face positions.
Looking at the sodium ions, there are 12 at edge positions and 1 in the body position.
So there is a 1:1 ratio of sodium to chloride, as the formula NaCl indicates.
We can use X-ray diffraction data coupled with the composition of the unit cell to predict the density of the compound. X-ray diffraction data provides the edge length of the unit cell, from which we can calculate its volume.
Using the atomic masses of the elements in the compounds we can determine the mass of the unit cell.
Previously, we determined there were four chloride ions in a unit cell of NaCl.
Looking at the chloride ions
Previously, we determined there were four sodium ions in a unit cell of NaCl.
Looking at the sodium ions
The measured density of NaCl at 25 °C is 2.18 g/cm³
In the absence of X-ray diffraction data, we can approximate the length of the unit cell using the ionic radii of the ions. Looking at the space-filling model of the NaCl unit cell, we see that each edge is composed of one sodium ion sandwiched between two chloride ions. The length of the edge is therefore the radius of the two chloride ions plus the diameter of the sodium ion.
The ionic radius for Na = 98 pm
The ionic radius for Cl = 181pm
= 181 pm + 2 (98 pm) + 181pm
= 558 pm
The length of the unit cell =
Instructions: Place the atoms or the ions into the unit cell by clicking on the position in the unit cell. Click Check Answer when you are finished placing atoms.
Question 1:
Crystalline iron is body-centered cubic. Place the iron atoms into the unit cell.
Instructions: Place the atoms or the ions into the unit cell by clicking on the position in the unit cell. Click Check Answer when you are finished placing atoms.
Question 2:
Crystalline copper is face-centered cubic. Place the copper atoms into the unit cell.
Instructions: Place the atoms or the ions into the unit cell by clicking on the position in the unit cell. Click Check Answer when you are finished placing atoms.
Question 3:
LiBr crystallizes into a face-centered cubic arrangement, with the lithium ions in the holes.
a. Place the bromide ions in to the unit cell.
Question 4:
a. Click on the atom(s) in the corner position(s).
Question 5:
From X-ray diffraction data, it is determined that iron crystals have a unit cell length of 286.8 pm.
What is the density (in g/cm3) of iron if it is body-centered cubic?
g/cm³
To determine the density we need the mass of the unit cell and the volume of the unit cell.
Volume = (edge length)3
= (286.8 pm)3
= (268.8 × 10−10 cm)3
= 2.359 × 10−23 cm3
There are 2 iron atoms per unit cell in a body-centered cubic structure.
2 Fe atoms × (
1 mol
1 mol
) = 1.855 × 10−22 g
6.022 × 1023 atoms
) × (
55.85 g
Determining a crystal's unit cell structure is a key step in characterizing the crystal. Using the edge length of the unit cell from X-ray diffraction experiments and the structure of the unit cell, you can calculate the density of the compound.
Congratulations, you have reached the end of the unit cell exercise.
Using the ionic or atomic radii to calculate unit cell volume produces an approximate value. An ion or atom is not a hard sphere and its size can therefore vary according to the space available.
LiCl, NaCl, KCl, and RbCl are all face-centered cubic. Why is CsCl simple cubic?