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Crystallography
A crystal
is a solid object with a geometric shape that reflects a “longrange” regular
internal structure (commonly with some element of symmetry).
I Space lattices
•
Definition
: The regular internal structure of a mineral is manifested by the existence
of
a space lattice
, which is "
an array of points in space that can be repeated
indefinitely"
. All "points" in a lattice have identical environments (Fig. 1). In the
case of a mineral or crystal, these "points" (also known as
motifs
) may be considered
atoms, ions, or groups of atoms/ ions. Note that a lattice has no origin.
•
A unit cell
(Fig. 1) is the smallest number of "points" which completely define the
space lattice. The repetition of those points or unit cells in a space lattice is
performed by certain
operations
which build the space lattice.
Criteria used for the selection of unit cells:
1 The smallest sized unit that retains the characteristics of the space lattice (Fig.
1b).
2 Edges of the cell should coincide with symmetry axes (see below).
3 Edges of the cell related to each other by the symmetry of the lattice.
•
Building a space lattice: from motifs to lattices:
Motif
→
Line lattice
→
Plane lattice
→
Space Lattice
•
Operations
: (a) elements of symmetry; (b) translations; (c) glide planes; (d) screw
axes
A Elements of symmetry
:
Types:
i Axes of rotation (
1, 2, 3, 4 or 6
)
: If during the rotation of a crystal around an
axis one of the faces repeats itself two or more times, the crystal is said to have an axis
of symmetry. Symmetry axes may be two fold (digonal) if a face is repeated twice every
360°, three fold (trigonal) if it is repeated three times, four fold (tetragonal) if it is
repeated four times, or six fold (hexagonal) if that face is repeated 6 times. Figure 2a
shows these relations.
ii
Center (
n
or
i
)
:
If two similar faces lie at equal distances from a central
point, the crystal is said to have a centre of symmetry (Fig. 2e).
iii
Planes (
m
)
:
When one or more faces are the mirror images of each other, the
crystal is said to have a plane of symmetry (Fig. 2f). Motifs related to each other by
mirror planes are known as “enantiomorphs” (Fig. 4).
iv Axes of rotary inversion (
1
,
2
,
3
,
4
or
6
)
:
When two similar faces are
repeated 2, 3, 4 or 6 times when the crystal is rotated 360° around an axis, but in such a
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way that these faces appear inverted. Therefore, if the face is repeated 2 times during a
full rotation, the axis is known as a 2fold rotary inversion axis, 3 times
→
3fold rotary
inversion, .
.... etc. Figure 3 shows the types of rotary inversion axes. Note that axes of
rotary inversion can also produce “enantiomorphs” (Fig. 4).
Equivalence of some symmetry elements:
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 Spring '09
 Staff

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