Crystals
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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