Label each unit cell as simple cubic, body-centered cubic, or face-centered cubic.

In this activitiy, we will be using the crystal visualization tool from Cal Poly located here (Links to an external site.)

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.

How to turn in the lab

Submit your answers to the questions below to canvas as a .pdf

Be sure to answer the questions in order and to number each question. If the question requires you to do a calculation, show all of your work

You can type your lab or hand write it. If you do the lab by hand, I recommend using the adobe scan app (Links to an external site.) to scan your assignment and convert it to a .pdf

Lattice Structures of Atomic Solids

Use the crystal visualization tool (Links to an external site.) to answer the questions below

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.” If the first, second, and third layers are all staggered relative to eachother (none are stacked directly above the other), this stacking pattern is described as "ABC". You can explore this layering effect by selecting Layering on the left of the visualization tool and using the Expansion slider.

lab1_fig1.pngFigure 1. Square and rhombic unit cells in 2D layers.

For each of the lattices (simple cubic, body-centered cubic, face-centered cubic, and HCP), answer the following questions. Use the visualization tool to help.

1. What type of layer, square or rhombic, exists in each type of unit cell? (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 (simple cubic, body-centered cubic, face-centered cubic), answer the following questions.

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

for example, the simple cubic cell has 8 atoms and each atom is located at the corners

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

for example, for simple cubic - each one of the corner cells is shared between 8 neighboring unit cells. Therefore, each corner atom contributes 1/8 of an atom to the unit cell

you can see this more clearly in the program if you use the expansion slider in the unit cell view

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

We can use the answers from 3 and 4 to answer this question. Each simple cubic unit cell has 8 atoms touching it, but only 1/8 of each of those atoms belongs to that one unit cell.

So the total number of atoms per unit cell = 8 * 1/8 = 1 atom

For the HCP unit cell, answer the following questions.

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

use similar logic as above to answer this question and consider how atoms are shared between neighboring unit cells

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.

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

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

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

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

11. 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.

LaTeX: Packing\:Efficiency\:=\:\frac{V_{occupied}}{V_{total}}\times100P a c k i n g E f f i c i e n c y = V o c c u p i e d V t o t a l × 100

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

lab1_fig2.pngFigure 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.

Using figure 2, we can calculate the length of each unit cell edge in terms of the atomic radius, r.

For example, for FCC the length of the face diagonal is 4r. Using trig, we can solve for the length, l

LaTeX: l\:=\:4r\cos(45^\circ ) = 4r(\frac{\sqrt{2} }{2} )=2\sqrt{2}rl = 4 r cos ⁡ ( 45 ∘ ) = 4 r ( 2 2 ) = 2 2 r

The edge length of each unit cell is shown in the table below

Unit CellEdge lengths in terms of radius Simple cubicLaTeX: l\:=\:2rl = 2 rBody-centered cubicLaTeX: l\:=\:\frac{4r}{\sqrt{3}}l = 4 r 3Face-centered cubicLaTeX: l\:=\:2\sqrt{2}rl = 2 2 r

The volume occupied by atoms is calculated as the number of atoms times the volume of a sphere

LaTeX: V_{occupied}=(\# atoms)\times \frac{4}{3} \pi r^3V o c c u p i e d = ( # a t o m s ) × 4 3 π r 3

The total volume of the cube is calculated as

LaTeX: V_{total}=l^3V t o t a l = l 3

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

12. Assume an atom is a perfect sphere. In terms of r, what volume of the simple cubic unit cell is occupied by atoms?

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

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

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

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

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

Summary

18. Fill out this summary table with your answers from above

2D layer pattern (square vs. rhombic)

Stacking pattern (e.g. abab)

Number of atoms per unit cellCoordination NumberPacking Efficiency Simple CubicBody-Centered CubicFace-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

19. Determine the number of sodium ions per unit cell.

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

21. What is the empirical formula of sodium chloride based on the relative number of each ion in the unit cells?

22. Is the empirical formula determined from the lattice structure in agreement with the one predicted by the typical ion charges?

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

Calcium Fluoride

24. Determine the number of calcium ions per unit cell.

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

26. What is the empirical formula of calcium fluoride based on the relative number of each ion in the unit cells?

27. Is the empirical formula determined from the lattice structure in agreement with the one predicted by the typical ion charges?

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

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