Crystal Structures - SUN - Visual Numerics Java
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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