Atomic Arrangement

Objectives:

Assignments: 3.3, 5, 9, 11, 15, 17, 19, 21, 23 (a, c, g, k), 35, 39, 41(a, c, e ), 45, 51

Reference: §3.1 - 3.9 of The Science and Engineering of Materials, 3rd Edition, Askeland, PWS Publishing Co.

Because of the length of this document, you may want to access each topic using the links within the table below:
 

Atomic Order Unit Cells Points, Directions and Planes
Interstitial Sites Crystal Structures X-Ray Diffraction

Long and Short Range Order:

The behavior of materials depends greatly on how the individual atoms are arranged. Atoms arranged complete randomly are considered to have no order. If the atoms form organized molecules but the molecules are arranged randomly this is called short range order or amorphous. In a crystalline or semi-crystalline material the atoms form a repetitive structure over a relatively large portion of a material. This is referred to as long range order.

The discovery of quasicrystals in 1994 opened the way to working with a new and puzzling class of materials. These materials have neither regular order nor randomness. Mathematicians describe the patterns as quasiperiodic. See Science News, vol 155 Jan 23, 1999 (page 60). Typical materials exhibiting this structure are aluminum mixed with cobalt, manganese, and nickel.

Here are some web sites that you may want to investigate:

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Unit Cells:

The unit cell is a geometric shape that, when repeated in all directions, represents the crystal structure, or lattice of the material. The points where atoms, or groups of atoms exist are calledlattice points. Some atoms exist entirely inside a unit cell but others are located at lattice points that are shared by more than one unit cell.

Example: In a simple cubic crystal structure each corner lattice point is occupied by an atom but this corner is shared by seven other unit cells. Therefore only one eighth of the atom is inside the cell. Since the cell has eight corners there is a total of one atom inside.

By identifying the direction that atoms touch inside the unit cell the atomic radius (Radii) can be mathematically related to the lattice parameter.

Using this understanding of the unit cell, the following quantities can be calculated:
 

· volume density (details) · packing factor (details)
· linear density (details) · linear packing fraction (details)
· planar density (details) · planar packing fraction (details)
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Points, Directions and Planes (Miller indices):

Miller indices are used to describe the geometric aspects of a unit cell.

Points in a lattice structure are defined by their location on in reference to a 3-dimensional right-hand coordinate system places any at any lattice point.
 

Directions can be determined and are written in square brackets - [x y z]
Families of directions (Example) are written using angle brackets - <x y z>
Here is a simple list of steps you can use to draw a direction arrows on a unit cell:

Drawing Direction arrows:

1. Start at the bottom back left corner of the unit cell. This is the best starting point if all three indices of the direction are positive. Example : [101]  But if an index is negative, a better starting point can be chosen (see Example 3-9, page 55).
2. Move the distances of the three indices in the x, y and z directions to locate the finishing point of your direction line. If you want you can multiply the x,y and z values by any number to change the distance between the points. Example: [½ 1 2] is equivalent to [1 2 4]
3. Draw an arrow from the starting point to the finish.
 

Planes can be determined and are written in rounded brackets - (x y z)
Families of planes (Example) are written using these brackets - {x y z}


Here is a simple list of steps you can use to draw a planes on a unit cell:
Drawing Planes:

1. Start at the bottom back left corner of the unit cell. This is the best starting point if all three indices of the direction are positive, but see Example 3-9 on page 55 for the method to use for negative indices.  In general, take reciprocals of the indices to obtain the intercepts.
2. If any of the intercepts are negative, move one unit in the positive direction in each negative intercept direction. This now will be the starting point of the direction arrow.
3. Move the distance of the x coordinate and place a point.
4. Move the distance of the y coordinate and place a second point.  If it is zero, the plane is parallel to the y-axis; do not mark a point.
5. Move the distance of the z coordinate and place a third point.
6. Connect the three points and shade the plane that they form.

Here's an example that was prompted by a question one of the students asked.

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Interstitial Sites:

The atoms in a crystalline structure do not fuill the entire volume.  That's the meaning of the Packing Factor (PF) - a PF of 0.725 (see example 3-13 on page 63) means that the atoms fill only 72.5 % of the volume of the unit cell.  If another atom is small enough, it can fit in between the atoms of the cell.  Usually, this causes stress since the remaining volume is not exactly "empty"; the space is filled with electromagnetic fields.  The addition of another atom perturbs those fields and therefore the atomic arrangements.  For example, the addition of less than 0.5 % carbon to iron results in steel, a structure which is often much harder than pure iron!

Interstitial sites may have various coordination numbers, as shown in Figure 3-21 and Table 3-6 on pages 60 and 61.

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Crystal Structures:

There are fourteen types of unit cells (Figure 3-4, page 42).  These are also called Bravais Lattices.  While most of the discussions will focus on simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC), it is occasionally necessary to consider others such as hexagonal (magnesium) and body-centered tetragonal (martensitic phase for steel).  The cesium chloride structure (see page 63) is simple cubic, whereas sodium chloride (see page 64) is FCC.  Metals are mostly BCC and FCC, with some hexagonal (or hexagonal close-packed; HCP); no metals are simple cubic.  Ionic structures are easier to visualize than covalent (see the diamond cubic structure on page 66).  Crystalline polymers (Figure 3-30, page 67) are messy!

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X-Ray Diffraction:

Figure 3-31, page 68, shows how the X-Rays interfere constructively (b) and destructively (a).  You may recall from Physics that when visible light passes through a diffraction grating the same interference happens and the familiar pattern of lines is produced.  The usual line separation for a diffraction grating is 12500 lines per inch, or about 2 micrometers separation between the lines.  Since the wavelength of visible light varies from 400 to 700 nm (0.4 to 0.7 micrometers), the line separation is on the order of the wavelength being used.  It's the same for X-Rays; since it's impossible to make a grating with line separations on the order of X-Ray wavelengths, to "see" (it's "seen" using XRay film) diffraction patterns, we use crystals.  The separation between crystalline planes is about 4 Angstroms (0.4 nm) for aluminum (see your text's front cover).  That's a thousand times smaller than the separation for diffractrion gratings!

At one time (and perhaps still), single crystals were grown to be used for X-Ray diffraction.  But it's much easier to obtain the material in powdered form, and powder X-Ray diffraction is used for almost all analyses of crystal structure.  You should study Example 3-17 (page 70) carefully.

Diffraction pattern for Penrose tiling

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