and the
Spatially Periodic Structure of Solids

Solids and Crystal Structure

Solids are a state of matter that are usually highly ordered.  The chemical and physical properties of the solid depend on the detail of this ordering. Elemental carbon can have two different solid phases with differing spatial (position) ordering and vastly different solid properties. Two such allotropes of Carbon are Diamond and Graphite (sp3 and sp2)

In carbon, the bonding in the solid forms is highly directional and dictates the long ranges order. In metals, the bonding is non-directional and often the solid structure is determined by atomic 'packing'. Remember the 'electron gas' model of a metal?

The regular arrangement of atoms and molecules in matter is evident in the crystal morphology or habit of many materials from snowflakes to many common minerals.

Basic Crystallography
Close Packing of Spheres

The description of the ordering of atoms in a solid comes from simple concepts of how identical objects stack in an array. If atoms are round and they pack as close as possible, they should look like this:

The close packing of spheres in a plane leads to a repeat unit (parallelpiped) that has each edge equal in length to the diameter (twice the radius) of the spheres. The angles, edge length, and atomic positions of the repeat unit are sufficient for the visualization of the entire infinite array in the solid.

In three dimensions the repeat unit is a 3D shape called the Unit Cell. The unit cell has three uniques crystallographic axies and, in general, three edge lengths. The angles of the edges of the unit cell need not be 90 or 120 degrees. (The figure below shows a possible crystal structure and its unit cell, but it is not a closest-packed structure, like the 2D structure above)

Close packed spheres of the same size in 3D is a little complicated. This packing leads to possibility of two unique structures, depending on how planes of 2D closest packed spheres are layered. If every other layer is exactly the same then we has a so called ABABA... structure. If not, then the structure is ABCABCABC...The figures below shows the difference between these two structures:
The ABABAB structure (panel (b) in the figures above) is called the Hexagonal Closest Packed (hcp) structure. In this structure, each atom has 12 nearest neighbors and the volume of the spheres fills the maximum posssible space: 74.04%.
The ABCABC structure is called Face Centered Cubic (fcc). It also has each atom with 12 nearest neighbors and the atoms fill 74.04% of the available space. The difference in the structure is in the different long ranged order and the unit cell

Here is another way of looking at this difference:

Cubic Unit Cells

The (fcc) structure is just one of the structures that is derived from a cubic unit cell (right angles, equal length edges). (If we allow the edge lengths to be different, but keep the right angles, we create the orthorhombic cells) The Cubic cells are shown below:

The number of atoms in the unit cell is not the same as the coordination number (number of nearest neighbors). In the Body Centered Cubic (bcc) structure above the number of atoms in the unit cell is 2 but the number of nearest neightbors is 8. (The number of gray atoms in the above gives the number of atoms in the unit cell) The (bcc) structure is not as tightly packed as the (hcp) or (fcc) structures, with the atoms occupying only 68.02% of the available space.

Here is an amusing movie describing the cubic unit cells.

Binary Salts

Until now, we have 'packed' only one kind of atom, which is only relevant for the solids states of the elements. If we wish to describe more complicated solids, i.e. solids that contain more than one atom, we must 'locate' each atom in the solid. Salts are fairly easy to describe, but some molecular solids are quite complex because of all of the different kinds of unique atoms.

The NaCl crystal is face centered cubic (fcc) unit cell with the counter ion filling the octahedral holes in the structure. It does not matter which ion is taken to be at the verticies of the cell and which in the holes, the same pattern is obtained, as can be seen in the figure below:

In the face centered cubic (fcc) cell there is more than one type of 'hole'. If the octahedral holes are filled, the structure above results, with a one:one count for the two types of ions in the salt. If the terahedral holes are filled, a diffrerent structure exists, that with twice as many of one type of ion as the other. In the figure below, The left shows the structure of NaCl and the right that of CaF2.
Some salts want to use the tetrahedral holes because of the relative sizes of the positive and negative ions, but don't fill all of them to maintain stoichiometry, This is the case for ZnSe, the middle panel below. Other relative ion sizes, like CsCl, left below, are filled simple cubic cells (not fcc).

Crystal structures are determined experimentally by X-Ray Diffraction:

The position of the spots observed in X-ray diffraction are determined by Braggs Law:

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