Crystal Structures

Types of crystal structures Face centered cubic (FCC) Body centered cubic (BCC) Hexagonal close packed (HCP)

Close Packed Structures Different Packing of HCP and FCC

Crystallographic Directions and Planes cubic systems

Face Centered Cubic (FCC)

Atoms are arranged at the corners and center of each cube face of the cell. Atoms are assumed to touch along face

diagonals

Face Centered Cubic (FCC)

The lattice parameter, a, is related to the radius of the atom in the cell through:

Coordination number: the number of nearest neighbors to any atom. For FCC systems, the coordination number is 12.

22Ra

Face Centered Cubic (FCC)

Atomic Packing Factor: the ratio of atomic sphere volume to unit cell volume, assuming a hard sphere model. FCC systems have an APF of 0.74, the

maximum packing for a system in which all spheres have equal diameter.

Body Centered Cubic

Atoms are arranged at the corners of the cube with another atom at the cube center.

Body Centered Cubic

Since atoms are assumed to touch along the cube diagonal in BCC, the lattice parameter is related to atomic radius through:

3

4Ra

Body Centered Cubic

Coordination number for BCC is 8. Each center atom is surrounded by the eight corner atoms.

The lower coordination number also results in a slightly lower APF for BCC structures. BCC has an APF of 0.68, rather than 0.74 in FCC

Hexagonal Close Packed

Cell of an HCP lattice is visualized as a top and bottom plane of 7 atoms, forming a regular hexagon around a central atom. In between these planes is a half-hexagon of 3 atoms.

Hexagonal Close Packed

There are two lattice parameters in HCP, a and c, representing the basal and height parameters respectively. In the ideal case, the c/a ratio is 1.633, however, deviations do occur.

Coordination number and APF for HCP are exactly the same as those for FCC: 12 and 0.74 respectively. This is because they are both considered close

packed structures.

Close Packed Structures

Even though FCC and HCP are close packed structures, they are quite different in the manner of stacking their close packed planes. Close packed stacking in HCP takes place

along the c direction ( the (0001) plane). FCC close packed planes are along the (111).

First plane is visualized as an atom surrounded by 6 nearest neighbors in both HCP and FCC.

Close Packed Structures

The second plane in both HCP and FCC is situated in the “holes” above the first plane of atoms.

Two possible placements for the third plane of atomsThird plane is placed directly above the first

plane of atoms • ABA stacking -- HCP structure

Third plane is placed above the “holes” of the first plane not covered by the second plane

• ABC stacking -- FCC structure

Close Packed Structures

Crystallographic Directions

Cubic systems directions are named based upon the

projection of a vector from the origin of the crystal to another point in the cell.

Conventionally, a right hand Cartesian coordinate system is used. The chosen origin is arbitrary, but is

always selected for the easiest solution to the problem.

Crystallographic Directions

Points within the lattice are written in the form h,k,l, where the three indices correspond to the fraction of the lattice parameters in the x,y,z direction.

Miller Indices

Procedure for writing directions in Miller Indices Determine the coordinates of the two

points in the direction. (Simplified if one of the points is the origin).

Subtract the coordinates of the second point from those of the first.

Clear fractions to give lowest integer values for all coordinates

Miller Indices

Indices are written in square brackets without commas (ex: [hkl])

Negative values are written with a bar over the integer.Ex: if h<0 then the direction is

][ klh

Miller Indices

Crystallographic Planes Identify the coordinate intercepts of the

planethe coordinates at which the plane intercepts

the x, y and z axes.If a plane is parallel to an axis, its intercept is

taken as .If a plane passes through the origin, choose

an equivalent plane, or move the origin

Take the reciprocal of the intercepts

Miller Indices

Clear fractions due to the reciprocal, but do not reduce to lowest integer values.

Planes are written in parentheses, with bars over the negative indices.Ex: (hkl) or if h<0 then it becomes

ex: plane A is parallel to x, and intercepts y and z at 1, and therefore is the (011). Plane B passes through the origin, so the origin is moved to O’, thereby making the plane the

)( klh

)121(

Miller Indices