You will learn about crystal lattices and unit cells of some common solids. Upon completion of this section you should also be able to do some calculations involving these unit cells (review your geometry and get out your marbles and building blocks).
Synopsis
The solid state consists of a much more fixed arrangement of atoms and molecules than the gas or liquid state. The atoms will arrange themselves in a manner to maximize the volume occupancy (called close packing) subject to any restrictions such as charge and bonding requirements. The resulting arrangement is a three-dimensional lattice with atoms at the lattice sites. There is no unique way of categorizing crystal structures. If we categorize the crystals by bonding types we will have four categorizes: metals, ionic, van der Waals, and covalent. The last category is different in that the bonds are directional, which puts additional restrictions on the crystal structure. In this section we will examine the metallic and ionic crystals.
A unit cell is the smallest repetitive pattern that has all of the symmetry properties of the macro arrangement. Be sure to go to the CD for some rather nice rotating drawings that help to visualize the geometries of these crystals. It also helps to have a bag of marbles! If you have some marbles (all of the same size) or a lot of pennies, you can do an experiment to visualize these structures yourself. Take your marbles or pennies and put them into a close packed arrangement as indicated below. You can see that each marble is in contact with six nearest neighbors.
Now let's put the second layer on by putting each marble in the space formed between the marbles on the first layer.
So far no choices are possible when we are achieving close packing. Now let's put on the third row and you now see that a choice is possible. We can repeat the first layer so that we form the arrangement xyxyxyxyxyxy… or we can put the third row into the spaces formed by the second layer which are not over the first layer and we have an arrangement xyzxyzxyzxyz…
The first type is hexagonal close-packed (hcp) and the second is cubic close packed (ccp) which is also called the face-centered cubic (fcc) structure. In each structure each marble is in contact with 12 nearest neighbors. These are both a cubic structure, which is one of the seven crystal classes shown in Figure 13.27 in your text. Another important cubic structure is the body centered cubic (bcc) structure. The unit cells for the simple cubic, the bcc, and the fcc structures are seen in Figure 13.28 from page 613.
Now let's put the second layer on by putting each marble in the space formed between the marbles on the first layer.
Now we need to discuss how atoms are shared between neighbor unit cells. Obviously the center atom in a bcc crystal is not shared with any other unit cells. However, the corner atoms are shared with eight total unit cells and the face-centered atoms are shared with two other unit cells (see Figure 13.29 on page 613).
We can now calculate the total number of atoms in a particular type of unit cell. A simple cubic cell has 8 atoms (one on each corner of the cell) but each is shared with 8 total unit cells, so there is a net of only one atom per unit cell in the simple cubic structure. The bcc structure also has one atom on the each of the 8 corners also, but it also has an unshared atom in the center so the bcc structure has 2 atoms per unit cell. The fcc structure has one atom on each of the 8 corners and one atom on each of the 6 faces, each of which is shared with two other unit cells. So the fcc structure has 4 atoms per unit cell. Experimentally we use x-ray crystallography to determine the precise structures of the various solids.
Ionic solids have the additional requirement of satisfying the zero-charge requirement for the solid. The structures are often a bit more complicated and it would help to have some building blocks to visualize the structures. For example, consider the NaCl structure. There is an octahedral hole in a fcc cubic structure as seen in Figure 13.31c on page 619.
The Na+ ions are inserted in the octahedral holes with a resulting interlocking fcc structure for the NaCl structure. There are four Na+ ions and four Cl- ions per unit cell.
You now have the background so that you can work some interesting problems relating to these solid structures. Remembering that density is mass/volume and some basic geometry will allow you to work these problems. Remember to not try to memorize an example from the book, but rather just write down what you are given and intuit the solution. You should work through all example problems (remember that following an example is not the same thing as understanding). You will know if you really understand when you sit down with a problem, no notes, no books, and no help from people and see if you can work the problem. You may be given the density and some geometry data and asked to determine the molecular weight of the species. You may be given the species and some geometry data and asked to calculate the density. There are many different permutations of questions for this section! In all questions you will need to know how to calculate the mass of the unit cell (number of atoms per unit cell * gram molecular weight divided by No) and the volume of the unit cell (for a cubic cell this is just the cube of the edge length, but sometimes you have to use some geometry to determine that edge length).
Review Question
1.) Cesium (atomic radius 266 pm) crystallizes in the bcc structure. What is the volume of the cube in cubic centimeters? In a bcc structure the atoms touch along the body diagonal. Sketch the cube with the atoms touching. How many Cs radii are included in the diagonal? You will need to use the Phythagorean theorem (a2 + b2 = c2).
2.) What is the mass of the unit cell for Cs (a bcc structure)?
3.) What is the density of Cs? Compare with an experimental value.
Web Author: Dr. Leon L. Combs
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