Crystal Structure Activity Learning Objectives After this activity, students should be able to

1. Identify different layering patterns that lead to the cubic unit cells, determine coordination

numbers, and compute packing efficiencies for atomic solids.

2. Determine the empirical formula of an ionic compound from its crystal structure.

Overview Use the visualization tool found at https://atom.calpoly.edu/crystal/ and answer the following

questions. Many of the functions in the simulation are bound to keys; look at Key Controls for the list.

The simulation starts by default with the Simple cubic lattice screen. The drop-down menu allows you

to view other lattice structures. You can rotate the structure and view it from different sides by holding

the mouse and dragging the structure. You can also zoom in and out with the mouse wheel. There are

two important modes that are controlled with the Expansion slider at the bottom of the screen. In

Layering mode, you can see how the 3D crystal lattice can be made by stacking layers of atoms. In Unit

Cell mode, you can see how the 3D lattice is composed of repeating unit cells with fractional atoms.

Lattice Structures of Atomic Solids

Layering We will begin this activity by looking at the layering pattern of particles that gives rise to each of the

cubic unit cells. A unit cell is the smallest unit in a repetitive pattern that makes the 3-dimensional lattice

structure. As shown in Figure 1, there are two basic 2D patterns for layers of atoms. The atoms in each

layer can be packed in a square array, or “close-packed” with a rhombus representing the simplest

repeating pattern. When multiple layers of a particular 2D pattern are stacked together, they can

generate a variety of 3D patterns, depending on how the layers are shifted relative to each other. If the

layers repeat identically as they stack, this can be described as “aa” stacking. If the second layer is

staggered relative to the first layer, but the third layer is stacked directly above the first layer, this

stacking pattern is described as “aba.” You can explore this layering effect by selecting Layering on the

left of the visualization tool and using the Expansion slider.

Figure 1. Square and rhombic unit cells in 2D layers.

For each of the cubic lattices (simple cubic, body-centered cubic, and face-centered cubic), answer the

following questions. Use the visualization tool to help.

1. What type of 2D unit cell exists in each layer, square or rhombic? (See Figure 1).

2. What is the stacking pattern in the corresponding lattice structure? (use letters a, b, c, etc. to label

different layers).

Unit Cells Once atoms are stacked into a 3D crystal lattice, the simplest repeating geometric pattern—the unit

cell—will usually contain fractions of atoms. While only whole atoms exist in the crystal, the geometric

representation of the unit cell will have atoms split between multiple neighboring unit cells. To find a

unit cell, we take the smallest repeating pattern and “slice” the shared parts off, to make it look like a

cube (here we are exploring cubic unit cells, but there are shapes for unit cells as well). With Unit Cell

selected on the left, use the Expansion slider to see how multiple unit cells together makes up an entire

lattice. To highlight a single unit cell within the crystal lattice, press “t” on the keyboard to toggle the

translucency.

For each of the cubic lattices, answer the following questions.

1. Which part(s) of a 3D unit cell do the atoms occupy (corner, edge, center, face)?

2. What fraction of an atom does each contribute to the unit cell?

3. What is the total number of atoms per unit cell?

Coordination Number The coordination number is the number of closest neighbors an atom has in the lattice, including atoms

in the adjacent unit cells. For the following questions, you can use the Coordination mode in the

visualization tool to verify your answer.

1. Determine the coordination number for the simple cubic lattice.

2. Determine the coordination number for the grey atoms in a body-centered cubic (bcc) lattice.

3. Determine the coordination number for the red atoms in a bcc lattice.

4. Explain why the coordination number for all the atoms in the bcc lattice is the same.

5. Determine the coordination number for the face-centered cubic lattice.

Packing Efficiency Since the layering pattern in all of the lattices leaves empty space between the particles, the unit cell is

not completely occupied by atoms (here we are treating atoms like hard spheres). The packing

efficiency, which is the percentage of occupied space in the cube, is not 100%. The packing efficiency is

not the same for all 3 cubic lattices. A more densely packed unit cell will have a higher packing efficiency

than a less densely packed one. The packing efficiency of a lattice structure measures how well the

space inside of a unit cell is utilized. It is the percent ratio of volume occupied by the particles in a unit

cell to its total volume.

=

× 100

The occupied volume is related to the number of particles occupying the cell and their location within

the cell. The edge length of each unit cell is derived using the trigonometric relationships shown in

Figure 2.

Figure 2. Geometric relationships showing how the edge length is related to the atomic radius for simple cubic, body-centered cubic, and face-centered cubic unit cells.

Unit Cell Edge length in terms of radius

Simple cubic = 2

Body-centered cubic = 4

√3

Face-centered cubic = 2√2

= (# ) × 4

3 3

= 3

Answer the following questions. Assume that the lattice consists of only one type of atom, and the

radius of this atom is r.

1. Assume an atom is a perfect sphere. In terms of r, what volume of the simple cubic unit cell is

occupied by atoms?

2. What is the total volume of the simple cubic unit cell?

3. Determine the packing efficiency of a simple cubic unit cell. Use your answers from the previous

two questions.

4. Determine the packing efficiency for a body-centered cubic unit cell.

5. Determine the packing efficiency for a face-centered cubic unit cell.

6. Observe the difference in stacking patterns of the unit cells and note how they are related to the

packing efficiency.

Summary

2D layer pattern

(square vs rhombic)

Stacking Pattern (e.g.

aba)

Number of Atoms per Unit Cell

Coordination Number

Packing Efficiency

Simple Cubic

Body-Centered Cubic

Face-Centered Cubic

Lattice Structures of Ionic Compounds Now we will look at a few examples of ionic solids. The Legend button will show the ion coloring

scheme. The ions are roughly scaled to their relative ionic radii within each of the lattices.

Sodium Chloride 1. Determine the number of sodium ions per unit cell.

2. Determine the number of chloride ions per unit cell.

3. What is the empirical formula of sodium chloride based on the relative number of each ion in

the unit cells?

4. Is the empirical formula determined from the lattice structure in agreement with the one

predicted by the typical ion charges?

5. Are either of the ions arranged in one of the basic cubic unit cells (simple, body-centered, face-

centered)?

Calcium Fluoride 1. Determine the number of calcium ions per unit cell.

2. Determine the number of fluoride ions per unit cell.

3. What is the empirical formula of calcium fluoride based on the relative number of each ion in

the unit cells?

4. Is the empirical formula determined from the lattice structure in agreement with the one

predicted by the typical ion charges?

5. Are either of the ions arranged in one of the basic cubic unit cells (simple, body-centered, face-

centered)?