What is a Unit Cell

Crystals are a highly organized form of matter.  This is obvious to anyone who has looked at crystals, as they naturally tend towards particular geometric shapes.  Each crystalline compound has a characteristic form, regardless of its size.  This rigid organization continues down to the placement of individual atoms within the structure (also called a lattice).  Because the crystal lattice is such a repetitious form, it is possible to describe its structure in terms of one basic building block that is repeated throughout.  This single building block is called a unit cell.

The unit cell, unsurprisingly, has a shape similar to that of the crystal as a whole.  Crystals with perpendicular sides have unit cells with perpendicular sides, hexagonal crystals have hexagonal cells, and so forth.  There are seven different types of unit cells, each with different symmetry properties.  A quick description of each is provided at the end of this article.

For introductory purposes, it is easiest to consider a cubic unit cell.  Just like any cube, it has six perpendicular square sides (faces).  The height, width, and length of the cube are all equal.  There are many crystalline compounds which have a cubic unit cell.  Most familiar is table salt (NaCl), but the list also includes diamond, and referring to the 82nd edition of the CRC Handbook of Chemistry and Physics: ammonium chloride, barium fluoride, calcium nitrate, lithium iodide, nickel (II) oxide, silver bromide, and zinc sulfide.

Each unit cell has to preserve the ratio of elements in the compound.  For example, since NaCl has a one-to-one ratio, the unit cell must contain equal numbers of sodium and chloride ions.  Barium fluoride (BaF2) has a one-to-two ratio, so the unit cell must contain twice as many fluoride (F) ions as it does barium ions.  Notice that while the ratio has to be preserved, there is no rule on how many ions have to be present.  (The unit cell for sodium chloride requires a total of four of each ion.)

There are four types of positions an ion can occupy in a unit cell. 

1.  An ion can be somewhere entirely inside the cell.  This is called “internal”, and counts as one whole ion.  The ion might be centered in the unit cell or it might not.  All that matters is that the ion is contained within the boundaries of the unit cell.

2.  An ion may be on one face of the cell.  In this case it is half in and half out – shared with the next cell over.  “Face” ions are counted as one-half of an ion.  There will always be another ion on the opposite face, so there aren’t any unpaired halves.

3.  Ions can also be found in “edge” positions.  Sitting along one edge of the unit cell divides an ion into quarters.  (Edge ions are counted as one-fourth* of an ion.)  To make the unit cells line up properly, any time a unit cell has an edge ion, it must have matching edge ions along each of the four corresponding edges.

4.  The last place an ion may be found is at a corner of the unit cell.  “Corner” positions are counted as only one-eighth* of an ion, and that ion is shared between all eight unit cells that meet at that corner.  Again, if there is an ion at one corner position, it will have matching partners at all eight corners of the unit cell.

*These numbers are specific to cubic and other four-sided unit cells.  In a hexagonal or trigonal unit cell, the fractions are different, but they still add up to whole numbers.

The seven types of unit cells are described below.

The cubic unit cell is a cube. 

The tetragonal unit cell is like a cube that was either stretched or shortened in one direction.  It still has a pair of square faces, and the rest are matching rectangles.  A pizza box should match this geometry.

The orthorhombic unit cell has all perpendicular faces, but each side is a different length.  As a result, each pair of opposite faces is matching rectangles, and is of a different size than the other faces.  The typical shirt-box matches this geometry.

The monoclinic unit cell is just like the orthorhombic, except that it has been slanted in one direction.  One pair of faces is now matching parallelograms, while the other four faces are still rectangles.

In the triclinic unit cell, there are no perpendicular faces.  All opposing faces are matched parallelograms, and each pair is a different size from the others.

The six faces of a trigonal, or rhombohedral, unit cell are matching rhombuses.  None of the faces are perpendicular.  (Viewed from a corner, the three-fold symmetry is apparent.)

Hexagonal unit cells are regular hexagonal prisms – six equal, rectangular faces with a pair of perpendicular hexagonal faces at the top and bottom. 

An excellent depiction of the various unit cell geometries can be found at http://www.chemistrydaily.com/chemistry/Unit_cell.