crystal unit cell explained
A crystal is one small box copied without end in every direction of 3 dimensions. This smallest repeating unit is called the unit cell. Even one grain of salt has this box stacked regularly more times than you can count. So a crystal has smooth faces and sharp angles, and a regular pattern appears when light is sent through it. The whole solid you see is, in the end, only the repetition of one unit cell. One box is the whole solid.
The one box in the middle is the unit cell. Turn the picture and the 3 dimensional shape appears, and you can raise the tile count from 1 to 2×2×2 and to 3×3×3. The same box is copied without a gap and a regular lattice appears. The point is that fixing one box fixes the whole crystal automatically. Copying makes the crystal.
An atom on a corner or a face is shared with the neighboring boxes. A corner atom is used by eight boxes together, so one box's share is 1/8. A face atom is split by two boxes, so 1/2, and only the atom in the middle of the box is wholly 1. Turn corners, faces, and the center on and off and you can count how many actually sit in one box. Corners alone are 1, add the center and it is 2, add the faces and it is 4. The shared share makes the count.
The length of one edge of the unit cell is called the lattice constant. Change that length and the whole box grows or shrinks. Even if the number of atoms in one box and the molar mass stay the same, the volume changes as the cube of the edge, so the density = n·M / (NA·a³) is recomputed at once. Shrink the edge and the same atoms are pressed into a smaller volume, and the density growing is visible. a³ is the volume. Narrower means denser.
A crystal face carries a name tag. Read the points where the face meets the three axes, in units of the lattice constant, and take the reciprocals of those values. The (100) face cuts one axis at 1 and runs parallel to the other two, so it never meets them. The (110) face cuts two axes, and the (111) face cuts all three and becomes a triangle. Press the three faces and the cut section and the meeting points show how they change. The reciprocal is the name of the face.
The three cubic lattices differ only in where the extra atoms sit. Simple cubic, SC, has atoms only at the corners. Body-centered cubic, BCC, adds one in the middle of the box. Face-centered cubic, FCC, adds one in the middle of each of the six faces. So the atoms in one box are 1, 2, and 4. Press the three kinds and watch the atoms rearrange, and it leads naturally into the closest packing of the next lesson. The place fixes the count.