Crystal Structures

The regular geometric shapes of crystals reflect the orderly arrangement of the atoms, ions, or molecules that make up the crystal lattice. Many different types of compounds will form crystals when solid; metals and ionic compounds are almost always in solid crystal form at room temperature. Only a few covalently bonded compounds are solids at room temperature. Of those covalent compounds that are solid, not all form crystals. The ones that do not are referred to as amorphous. Many covalently bonded compounds which are liquids or gases at room temperature will crystallize at lower temperatures (i.e. water forming ice).

In this experiment, you will be using styrofoam spheres as models of atoms or ions to gain insight into the ways in which metallic or ionic crystals are formed. You will investigate three basic crystal structures which can form if all of the particles are the same size, as would happen in a pure metal: simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC). structures. Using the styrofoam models, you will determine the number of nearest neighbors (the coordination number) of the particles in each of these structures. In a simple crystal structure, all atoms or ions of the same type should have the same coordination number; in other words, any atom or ion of the same element is identical with respect to size and position relative to its neighbors. When ions form crystals, the cations and anions are normally different sizes. Often, the resulting lattice (a term that refers to the repeating layout of particles in the crystal) resembles one of the basic structures, however the difference in size will cause slight alterations in the packing arrangement. The coordination number in an ionic compound refers to the number of nearest neighbors of opposite charge. Another important concept in the study of crystals is the “unit cell”. The unit cell is the smallest unit that contains all of the elements of the crystal. When repeated (or “translated”) in all directions, the repeating crystal lattice is obtained. The study of the entire crystal can be performed by an analysis of the unit cell.

Objectives:
- To gain familiarity with the geometry of crystal structures.
- To determine coordination numbers and calculate packing densities of crystal structures.

CAUTIONS: - Toothpicks can be hazardous to your eyes and to balloons. Be extra careful when trying to spear grape tomatoes with a toothpick... wear gloves and aprons and be sure to immobilize the tomato so it does not roll off your plate. - Old styrofoam spheres may look like matzo balls, but they are easily lodged in the throat and cause death by asphyxiation. Do NOT ingest them even if you are extremely hungry.

Pre-Lab:
1. Looking at the layers used to build each crystal structure below, which structure seems to have the most closely-packed atoms? The least closely-packed?
2. How does the definition of coordination number differ in simple crystals (made of one type of atom) vs. ionic crystals?
3. Add several more atoms to each layer in FCC and compare. What do you see? Does this also hold true for SC and BCC? Explain.

Directions:
• Do not fool around or “play” with the spheres. Make only the structures you are directed to in the lab. Do not toss spheres. DO NOT MAKE SNOWMEN. Violation of any of these rules will result in dismemberment and/or correctional treatment with the fraction hammer!
• Each crystal structure will be formed in several layers. Unless told to, it is better to not connect the three different layers in any model together.
• Remember, crystals are repeating structures: in a real crystal each layer you build would contain many, many more atoms. Also, many more layers like those you build would be found on top of and below those of your model.
• Use toothpicks to connect the spheres (pipe cleaners or coffee stirrers may serve as alternates)
• Disassemble all structures and remove all toothpicks when you are done.

Basic lattices (pure metals): Look at the three basic lattices shown in your text on page 245-246 and at the end of the lab. The unit cells is the cube in each picture. Build a model of each lattice for your group according to the instructions on the next page. Answer the questions that follow (the first seven by completing the table):

Very Important Hints: (all unit cells are cubic for this lab) 1a) the nuclei of the atoms or ions can be found in one of four spots in a cubic unit cell: they may be exactly on a corner, in the center of the body (meaning inside of the cube, called the “body center” position), the center of a face of the cube (“face center” position), or the middle of an edge of the cube. 1b) there is an atom or ion centered at all corners of all of our cubic cells 2) remember that atoms or ions on the outside of the unit cell are shared equally by two or more unit cells... only the fraction inside the cell counts as part of that cell. For example, if an atom is shared by 2 cells, then 1/2 of the atom is in each cell. 3) unit cells stack and repeat in all directions to create the entire crystal 4) In styrofoam or “space-filling” models, most “nearest” neighbors are actually touching.

Simple Cubic (SC)

Body-Centered
Cubic (BCC)

(layers 1 and 3 do not touch)

Face-Centered
Cubic (FCC)

Put answers to the following questions in the table below:

1. How many total atoms are inside the boundaries of the unit cell (count parts of atoms as fractions and only add up the parts inside- some parts of many atoms will be outside and not count!)
2. For each cube, let the length of one side = s. What is the volume of the cell in terms of s? NOTE: the value of s varies with every crystal, but knowing the value is not needed here.
3. Along what part of the unit cell are atoms continuously in contact (edge, diagonal of a face, diagonal of the cell)
4. Express the radius of an atom (r) in terms of the length of a side of the cell (s). (see p. 246… but WATCH OUT!... the order of the structures is switched!)
5. In terms of s, what is the volume of one atom? (Vsphere = 4πr3/3) Write as a decimal.
6. What percent of the volume of the unit cell is filled by the atoms (often called the “packing density”). Remember to consider only parts of atoms inside the cell. Base your answer on your work in questions 1-5 and show the calculation for BCC only below your table in the write-up.
7. What is the coordination number of an atom in this structure?
| |SC |BCC |FCC |
|1. atoms inside cell (total of parts) | | | |
|2. cell volume | | | |
|3. continuous contact | | | |
|4. radius, r= | | | |
|5. volume of one atom | | | |
|6. packing density | | | |
|7. coordination number | | | |

8. Comment on the relationship between coordination number and density of crystal structures based on your answers from #6 and #7.

9. The FCC structure is made of close packed planes (see diagram below). Sketch an FCC unit cell and shade in a close packed plane (hint the plane should contain 6 atoms). Use your model to help. If you do not get this one immediately, finish the lab then go back to this one.

Close-packed plane: (extends for many atoms in x and y directions)

Ionic lattices: Look at the three ionic lattices shown in your text on page 248. The unit cell for each of these structures would be defined by the centers of the eight green atoms found in the corners of each structure. Answer the following questions (the first seven by completing the table below). If you need to, you may build a model to visualize the structure.

10. In what general pattern are the cations arranged? (SC, BCC, FCC, or other?) Hints: • In the case of ions, the cations or anions may be moved apart so they are not touching, but still considered to have the same general structure. • Don’t copy answers out of the book for these questions; the book makes different interpretations in many cases.
11. In what general pattern are the anions arranged? (SC, BCC, FCC, or other?)
12. How many total cations are found inside each unit cell? (calculate as in question 1)
13. How many total anions are found inside each unit cell? (calculate as in question 1)
14. What is the coordination number for a cation in this structure (# closest anions)?
15. What is the coordination number for an anion in this structure (# closest cations)?
16. Along what part of the unit cell are ions (regardless of type) continuously in contact (edge, diagonal of a face, diagonal of the cell)
| |LiCl |NaCl |CsCl |
|10. cation pattern | | | |
|11. anion pattern | | | |
|12. cations inside cell (total of parts) | | | |
|13. anions inside cell (total of parts) | | | |
|14. cation coordination number | | | |
|15. anion coordination number | | | |
|16. continuous contact | | | |

17. Comment on how your answers to 12 to 15 relate to the ideas of a unit cell and the formula unit for these compounds.

18. The coordination number of calcium in calcium fluoride is 8. What is the coordination number of fluoride? Explain.
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Extra credit: The unit cell for the Hexagonal Closest Packed (HCP) structure can made from these three layers (the hole in layer 2 goes directly over the center atom of layer 1 and under the center atom of layer 3). One version of the unit cell is defined by the centers of the outer atoms in layers 1 & 3.

Answer the following questions (you may build a model if you have extra time in class, after school, or any time with your own materials):
- What is the coordination number of an atom in HCP? Explain.
- What do you think is the packing density of this structure? Explain.
- Draw a unit cell and show a calculation of the density.
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[pic] [pic] [pic] Simple Cubic Body-Centered Cubic Face-Centered Cubic
[pic] [pic] [pic]

Top pictures show exactly one unit cell, cutting off parts of atoms in adjacent cells. Bottom pictures show positions of atoms even though parts of some atoms are in adjacent unit cells.
Lab group members (all must sign):

| |SC |BCC |FCC |
|total atom parts inside the cell | | | |
|cell volume | | | |
|continuous contact | | | |
|radius, r= | | | |
|volume of one atom | | | |
|packing density (show 1 ex. below) | | | |
|coordination number | | | |

| |LiCl |NaCl |CsCl |
|cation pattern | | | |
|anion pattern | | | |
|cations / cell | | | |
|anions / cell | | | |
|cation coordination number | | | |
|anion coordination number | | | |
|continuous contact | | | |

----------------------- layer 2

layer 1

layer 1

layer 3

layer 2

layer 3

layer 2

layer 1

layer 1

layer 3

layer 2