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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
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Face Centered Cubic (FCC)
Atoms are arranged at the corners andcenter of each cube face of the cell.
Atoms are assumed to touch along facediagonals
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Face Centered Cubic (FCC)
The lattice parameter, a, is related to theradius of the atom in the cell through:
Coordination number: the number ofnearest neighbors to any atom. For FCCsystems, the coordination number is 12.
22Ra =
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Face Centered Cubic (FCC)
Atomic Packing Factor: the ratio of atomicsphere 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 allspheres have equal diameter.
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Body Centered Cubic
Atoms are arranged at the corners of thecube with another atom at the cube
center.
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Body Centered Cubic
Since atoms are assumed to touch alongthe cube diagonal in BCC, the lattice
parameter is related to atomic radiusthrough:
3
4R
a =
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Body Centered Cubic
Coordination number for BCC is 8. Eachcenter 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
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Hexagonal Close Packed
Cell of an HCP lattice is visualized as a topand bottom plane of 7 atoms, forming a
regular hexagon around a central atom.In between these planes is a half-hexagonof 3 atoms.
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Hexagonal Close Packed
There are two lattice parameters in HCP,aand c, representing the basal and height
parameters respectively. In the idealcase, the c/aratio is 1.633, however,deviations do occur.
Coordination number and APF for HCP areexactly the same as those for FCC: 12 and0.74 respectively.
This is because they are both consideredclose acked structures.
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Close Packed Structures
Even though FCC and HCP are closepacked structures, they are quite different
in the manner of stacking their closepacked planes.
Close packed stacking in HCP takes placealong the cdirection ( the (0001) plane). FCCclose packed planes are along the (111).
First plane is visualized as an atomsurrounded by 6 nearest neighbors in both
HCP and FCC.
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Close Packed Structures
The second plane in both HCP and FCC issituated in the holes above the first plane ofatoms.
Two possible placements for the third planeof 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 firstplane not covered by the second plane
ABC stacking -- FCC structure
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Close Packed Structures
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Crystallographic Directions
Cubic systems
directions are named based upon the
projection of a vector from the origin of thecrystal 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 theproblem.
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Crystallographic Directions
Points within the lattice are written in theform h,k,l, where the three indices
correspond to the fraction of the latticeparameters in the x,y,z direction.
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Miller Indices
Procedure for writing directions in MillerIndices
Determine the coordinates of the two pointsin the direction. (Simplified if one of thepoints is the origin).
Subtract the coordinates of the second pointfrom those of the first.
Clear fractions to give lowest integer valuesfor all coordinates
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Miller Indices
Indices are written in square bracketswithout commas (ex: [hkl])
Negative values are written with a bar overthe integer.
Ex: if h
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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
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Miller Indices
Clear fractions due to the reciprocal, but donot reduce to lowest integer values.
Planes are written in parentheses, with barsover the negative indices.
Ex: (hkl) or if h
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Miller Indices