Transcript Crystal Structures - SUN - Visual Numerics Java

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:
a  2R 2
Coordination number: the number of
nearest neighbors to any atom. For FCC
systems, the coordination number is 12.
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:
4R
a
3
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 atoms
Third 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 [hkl]

Miller Indices
Crystallographic Planes
Identify the coordinate intercepts of the
plane
the 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(hkl)
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 (112)
Miller Indices