The seven crystal systems
Chemistry 4000/5000/6000
Chemical Crystallography
The unit cells of the primitive lattices
Trigonal vs. Hexagonal
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This figure highlights the
relationship between the
rhombohedral primitive and the
conventional hexagonal triple
unitit cells
ll for
f the
th trigonal
ti
l crystal
t l
system.
It also illustrates the special
character of “hexagonal” where
the primitive unit cell is also one
third of the full hexagon that
describes the 6-fold rotation axis
of hexagonal crystals.
Why does the hexagonal triple of
the trigonal system not have
hexagonal, but instead trigonal
(i.e. 3-fold rotation) symmetry?
The hexagonal lattice
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A hexagonal lattice can most
easily be visualized in the ab
plane
This reduces the problem to 2
dimensions which is easier to
visualize
It is called hexagonal because
of the six-fold axis described by
the faint black lines
y
z
x
1
The hexagonal lattice
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A hexagonal lattice can most
easily be visualized in the ab
plane
This reduces the problem to 2
dimensions which is easier to
visualize
It is called hexagonal because
of the six-fold axis described by
the faint black lines
BUT the hexagon is NOT the
unit cell, which is given in red
Th unitit cells
The
ll mustt stack
t k
together to give the entire
lattice
Hence the green lozenges
represent the multiple unit cells
The hexagonal lattice
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y
z
x
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Trigonal vs. hexagonal lattices
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The difference between the
hexagonal and the trigonal (in
hexagonal setting) is subtle
It is hexagonal if the full
contents of the unit cell retain
the 6 or -6 symmetry shown by
the hexagon
This is shown by the purple
lines in the diagram here
But if the contents of the cell
reduce the symmetry to a 3-fold
axis then it is by definition a
axis,
trigonal lattice (in the hexagonal
setting)
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z
x
y
z
x
An example of a Trigonal lattice
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y
A hexagonal lattice can most
easily be visualized in the ab
plane
This reduces the problem to 2
dimensions which is easier to
visualize
It is called hexagonal because
of the six-fold axis described by
the faint black lines
BUT the hexagon is NOT the
unit cell, which is given in red
Th unitit cells
The
ll mustt stack
t k
together to give the entire
lattice
Hence the green lozenges
represent the multiple unit cells
The dashed line is NOT the
next unit cell!
In this cell, there is “something”
at 1/3,2/3,0
This has the effect of lowering
the symmetry around the corner
lattice point to a 3-fold axis
Hence this lattice which
otherwise looks “hexagonal” is
actually classified as trigonal
y
z
x
2
Lattices and lattice types
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It is useful to consider a crystal structure in terms of a lattice of points
An entirely abstract construction; no need for atoms to be located at the lattice points
Only certain types of lattices are able to describe real crystals
These are described by the different types of unit cells:
The smallest repeating unit of a lattice which reflects the full symmetry of the whole lattice
Pictures of the basic lattice types
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Different Lattice Types
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- primitive
P
a cell which has lattice points only at the vertices
(and therefore "contains" 8 ×1/8 = 1 lattice point only)
- face-centred
F
a cell which has points in the middle of each face
- side-centred
A (or B or C) a cell which has points centred on two opposite faces only
- body-centred
I
a cell which has one additional point in the very centre
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The 14 Bravais lattices
Three of these were encountered in
Chemistry 2810 to describe the metallic
l tti
lattices
Elemental polonium is thought to be
primitive cubic
Elemental sodium has a body-centred
cubic structure, with a Na atom at each
lattice point
Elemental magnesium has a face-centred
cubic structure with an atom at each lattice
point
But the lattice types are not just of cubic
symmetry (despite the diagrams!)
We apply these types to the 7 crystal
systems
Do not get 7 × 4 = 28 lattices; only 14!
Side-centred A (or B or C)
Alternate figures of the Bravais lattices
This is Fig. 2.8 from
“the Massa”!
I will try to use these so
called “Pearson” labels for
the Bravais lattices.
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