- What is a crystal and what is the meaning of crystal structure?
- What are X-rays and how they help us investigating crystals?
- What are closest packed structures?

As the temperature increases, a solid usually melts at some temperature.
During the melting process, the temperature does not change as long as
there are liquid and solid present. The energy absorbed by a solid during
the melting process is called **enthalpy of fusion**. When all solid
is melt, the liquid temperature will increase as it absorbs heat.
Of course, vapour pressure of the liquid increases with temperature.
When the vapour pressure is the same as the atmosphere, the liquid
boils. The temperature at which the liquid and gas phases are in
equilibrium is called the **boiling point**. A solid also sublimates
into a vapour. The changes among vapour, liquid, and solid are called
**phase transitions**

A **glass state** is also seen as a solid, but from structural point of
view, a glass is really a frozen liquid, because the molecules do not
have long range regular and periodic arrangement. On this web page,
we concentrate on crystalline solids.

We encounter solids all the time, and solids are used both in everyday life as well as technology. Metals, oxides, cement (concrete), diamond, minerals, and silicon devices are some of the important materials, and we must equip ourselves to deal with and understand them.

Representation of 3-dimensional structures should usually be made using models, and their representation on a flat surface is difficult. For simplicity, we use a 2-dimensional pattern (plane) to illustrate a 2-dimensional (planar) crystal structure.

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Thus, if we know the arrangement of a unit cell, we can use our imagination
to build a crystal sturcture, or use symbols or models to represent a crystal
structure. For a more formal representation of a crystal structure, consult
A HREF=http://www.physics.purdue.edu/~vogelges/diamond.html>
Introduction to Solid State Physics by Charles Kittel. This
visualization interface gives many animated diagrams of crystal patterns.
Since each pattern has features shared by many structures, often such a
pattern is called a **lattice**. For example, the diamond, zinc blende,
wurtzite, and NaCl structures has been called lattices, however, the word
lattice has a more formal definition by crystal physics and chemists.
The above site gives a gallery of lattices.

Inorganic Chemistry by Swaddle defined crystals as packed regular array
of atoms, ions, or molecules in a pattern repeated periodically *ad
infinitum*.

However, it is important for you to understand what X-rays are by working
on a problem related to X-rays. You are asked to calculate the frequencies,
and photon energies of X-rays of certain wavelengths. These are compared to
photons of ordinary light. The equation required for this comes from
Max Planck's postulates that lights are emitted one bundle energy at a
time. The energy *E* of the photon is proportional to the frequency
*v* of the photon, and the proportional constant is *h*, now known
as the Planck constant.

In a web page, Dr. S.J. Heyes gives some nice diagrams for
Structures of Simple Inorganic Solids. This compliments the description
of close packing concept of *Inorganic Chemistry* by Swaddle
very well.

When you put spheres on a flat surface, you have got a layer as shown on the right here. Note that an atom is surrounded by six atoms on the same layer.

On top of this layer, you can pack a second layer. As you do so, please note
that you have created two typical sites or holes called **octahedral**
and **tetrahedral** sites. These sites accommodates spheres up to
0.414 and 0.225 times the radii of the sphere or atom, r_{atom}.

A side view showing the spheres (P), tetrahedral sites (T) and octahedral (O) sites are also shown here.

When you put the third layers on, there are two possible options.
The 3rd layer can go directly on top of the first layer. In such an
arrangement, you produce an ABAB... type of sequence. Such a sequence
is called a **hexagonal closest packing (hcp)**.
A second option is to place a layer that were not directly on top of
A nor on top of B. We call such a location C. Thus, we produced an
ABCABC... sequence. Such a sequence belongs to the cubic type, and
is usually called **face-centered closest packing (fcc)**.

We demonstrated these type of packing in lectures, and asked you to
evaluate the radii of the small balls that fits tightly in the octahedral
and tetrahedral sites respectively in terms of r_{atom}.

If you do not have a set of models, take a good look at a pile of oranges in a super market on your next shoping trip. Dr. Heyes has an illustration shown here. This pile is the cubic arrangement.

The Crystal Lattice and Structures gives a dynamic view on these structures.

**
Example 1
**

*Solution*

If we assume the 6 balls forming the octahedral site to be located on
the two directions of a set of Cartesian coordinates in such a manner that
each ball touches four others, the coordinates will be
(*r+R*, 0, 0), (*-r-R*, 0, 0), (0, *r+R*, 0),
(0, -*r-R*, 0), (0, 0, *r+R*), and (0, 0, -*r-R*).

The ball touches along the edges. Therefore, we have

*Discussion*

Show that *r* = 0.414*R*

**
Example 2
**

*Solution*

A tetrahedral site can be modeled by having the four balls of radius *R*
located at alternate corners of a cube. The balls touch each other along
the faces of the cube, whose edges has a length of *a*.
Thus,

and therefore

*Discussion*

Show that *r* = 0.215*R*

**
Example 3
**

*Solution*

Let the edge of a unit cell of the fcc closest packing be *a*, and the
radius of the spheres be *R*. Then *a* = 8^{1/2}*R*.
There are four spheres per unit cell, and the volume occupied by each sphere
is V_{sphere}. The volume of the unit cell is V_{cell}.
We have these relationships

V

V

Fraction of volume occupied by spheres = 4*V

= p/(3*2

= 0.74 or 74%.

*Discussion*

Show that the Fraction of volume occupied by spheres in an hcp arrangement
is 0.74.

**
Example 4
**

*Solution*

The hcp unit cell is outline by the two top diagrams of the figure below.

A unit cell can be constructed in such a way that it has two base vectors of
*a* = 2*R*,
extended by 120 degrees (2p)/3), *R* being the
radii of the spheres. The third *c* axis is perpendicular to both
*a*s. It can be shown that for ideal *hcp* structures,
*c* = Ö(8/3) *a*.
The volume of the unit cell

= 2

= 8Ö(2)*

V

Fraction of volume occupied by spheres = 2*V

= p/(3*2

= 0.7405 or 74%.

© cchieh@uwaterloo.ca