Linear Algebra

142 Chapter Two. Vector SpacesTopic: CrystalsEveryone has noticed that table salt comes in little cubes.Remarkably, the explanation for the cubical external shape is the simplest onepossible: the internal shape, the way the atoms lie, is also cubical. The internalstructure is pictured below. Salt is sodium cloride, and the small spheres shownare sodium while the big ones are cloride. (To simplify the view, only thesodiums and clorides on the front, top, and right are shown.)The specks of salt that we see when we spread a little out on the table consist ofmany repetitions of this fundamental unit. That is, these cubes of atoms stackup to make the larger cubical structure that we see. A solid, such as table salt,with a regular internal structure is a crystal.We can restrict our attention to the front face. There, we have this patternrepeated many times.The distance between the corners of this cell is about 3.34 Ångstroms (anÅngstrom is 10 −10 meters). Obviously that unit is unwieldly for describingpoints in the crystal lattice. Instead, the thing to do is to take as a unit thelength of each side of the square. That is, we naturally adopt this basis.( ) ( )3.34 0〈 , 〉0 3.34Then we can describe, say, the corner in the upper right of the picture above as3 ⃗ β 1 + 2 ⃗ β 2 .