Lecture8.pdf0

Crystal Symmetry
Crystal Symmetry
The external shape of a crystal reflects the
The external shape of a crystal reflects the
presence or absence of translation-free
symmetry elements in its unit cell.
y y
While not always immediately obvious, in
While not always immediately obvious, in
most well formed crystal shapes, axis of
rotation, axis of rotoinversion, center of
, ,
symmetry, and mirror planes can be
spotted.

All di d i b bi d b h
All discussed operations may be combined, but the
number of (i.e. unique) combinations is limited,
to 32 Each of these is known as a point group
to 32. Each of these is known as a point group,
or crystal class.
The crystal classes may be sub-divided into one of
6 crystal systems
6 crystal systems.
Space groups are a combination of the 3D lattice
Space groups are a combination of the 3D lattice
types and the point groups (total of 65).

Each of the 32 crystal classes is unique to one
of the 6 crystal systems:
Triclinic, monoclinic, orthorhombic, tetragonal,
hexagonal and isometric (cubic)
hexagonal and isometric (cubic)
Interestingly, while all mirror planes and poles
Interestingly, while all mirror planes and poles
of rotation must intersect at one point, this
point may not be a center of symmetry (i).
point may not be a center of symmetry (i).

Crystallographic Axes
y g p
The identification of specific symmetry operations
enables one to orientate a crystal according to
an imaginary set of reference lines known as the
t ll hi
crystallographic axes.
ff f
These are distinct and different from the classic
Cartesian Axes, x, y and z, used in other
common day usage such as plotting graphs
common day usage, such as plotting graphs.

With the exception of the hexagonal system, the
axes are designated a b and c
axes are designated a, b, and c.
The ends of each axes are designated + or - This
The ends of each axes are designated + or -. This
is important for the derivation of Miller Indices.
The angles between the positive ends of the axes
are designated α, β, and γ.
α lies between b and c.
β lies between a and c.
γ lies between a and b.

Quantities can also be applied to further describe
vectors and planes relative to a b and c
vectors and planes relative to a, b, and c
These are u, v, w:
u: projection along a
v: projection along b
p j g
w: projection along c

Quantities can also be applied to further describe
vectors and planes relative to a b and c
vectors and planes relative to a, b, and c
These are h, k, l:
h: information relative to a axis
v: information relative to b axis
w: information relative to c axis
[ ] ith (hkl)
[uvw] with (hkl)
(hkl) f b
(hkl) faces on a cube

Axial Ratios
With the exception of the cubic (isometric) system,
h ll h d ff l h
there are crystallographic axes differing in length.
I i i l it ll d i th
Imagine one single unit cell and measuring the
lengths of the a, b, and c axes.
To obtain the axial ratios we normalise to the b axis.
These ratios are relative.

Unique crystallographic axes of the 6
crystal systems
crystal systems
Triclinic: Three unequal axes with oblique angles.
Monoclinic: Three unequal axes, two are inclined to
one another, the third is perpendicular.
Orthorhombic: Three mutually perpendicular axes of
different lengths.
Tetragonal: Three mutually perpendicular axes, two
are equal, the third (vertical) is shorter.
Hexagonal: Three equal horizontal axes (a1, a2, a3)
and a 4th perpendicular (vertical) of different length.
Cubic: Three perpendicular axes of equal length.

Triclinic: Three unequal axes with oblique
angles.
angles.
• To orientate a triclinic crystal
the most p ono nced one
the most pronounced zone
should be vertical.
c
• a and b are determined by
the intersections of (010) and
(100) ith (001)

 b
(100) with (001).
• The b axis should be longer


a
• The b axis should be longer
than the a axis.

The unique symmetry operation in a triclinic
The unique symmetry operation in a triclinic
system is a 1-fold axis of rotoinversion(equivalent
to a center of symmetry or inversion, i).
to a center of symmetry or inversion, i).
All forms are pinacoids – therefore must consist of
p
two identical and parallel faces.
Common triclinic rock-forming minerals include
microcline, some plagioclases, and wollastonite.

Monoclinic: Three unequal axes, two are
inclined with oblique angles, the third is
perpendicular.
O i t ti f t l h
• Orientation of a crystal has
few constraints – b is the
only axis fixed by
t
c
symmetry.
• c is typically chosen on the
basis of habit and b

cleavage.
• α and γ = 90 °.
• There are some very rare


• There are some very rare
cases where b equals 90°
giving a pseudo-
orthorhombic form
a
orthorhombic form.

The unique symmetry operation in a monoclinic
The unique symmetry operation in a monoclinic
system is 2/m – a twofold axis of rotation with a
mirror plane.
b is the rotation, while a and c lie in the mirror
l
plane.
Monoclinic crystals have two forms: pinacoids and
Monoclinic crystals have two forms: pinacoids and
prisms.
Common monoclinic rock-forming minerals include
clinopyroxene, mica, orthoclase and titanite.

Orthorhombic: Three mutually
perpendicular axes of different
perpendicular axes of different
lengths.
• Convention has it that a crystal is
oriented such that c > b > a.
c
• Crystals are oriented so that c is 
parallel to crystal elongation.
• In this case the length of the b axis
a


• In this case the length of the b axis
is taken as unity and ratios are
calculated thereafter.
b

The unique symmetry operation in an orthorhombic system is
The unique symmetry operation in an orthorhombic system is
2/m 2/m 2/m – Three twofold axis of rotation coinciding with
the three crystallographic axes.
Perpendicular to each of the axes is a mirror plane.
The general class for the orthorhombic system are rhombic
dipyramid {hkl}.
There are three types of form in the class: pinacoids, prisms,
and dipyramids.
Common orthorhombic rock-forming minerals include andalusite
and sillimanite, orthopyroxene, olivine and topaz.

Tetragonal: Three mutually perpendicular
axes, two are equal, the third (vertical) is
shorter.
• The two horizontal axis in a
tetragronal mineral are oriented in
the plane of the horizontal
c
the plane of the horizontal.
Therefore, if a = b, c must be in
the vertical. a2
• There is no rule as to whether c is
greater or less than a
greater or less than a.
a1
 
   

The unique symmetry operation in a tetragonal system is 4/m 2/m
2/m – The vertical axis (c) is always a fourfold axis of rotation
2/m – The vertical axis (c) is always a fourfold axis of rotation.
There are 4 two-fold axis of rotation: 2 parallel to the
crystallographic axes a and b the others at 45°
crystallographic axes a and b, the others at 45 .
There are 5 mirror planes.
The general class for the orthorhombic system is known as the
ditetragonal-dipyramidal class.
There are four types of form in the class: basal pinacoids,
tetragonal prisms, tetragonal dipyramids, and ditetragonal
prisms.
Common tetragonal rock-forming minerals include zircon, rutile and
anatase, and apophyllite.

Hexagonal: Three equal horizontal axes (a1, a2, a3)
and a 4th perpendicular vertical axis of different
length.
• The three horizontal axis of a
he agonal mine al a e o iented in
c
hexagonal mineral are oriented in
the plane of the horizontal, with c
in the vertical.
• Unlike the other systems the
B i Mill l t f
a3
Bravais-Miller nomenclature for
crystal faces is given by 4 numbers
(i.e. {0001})
a2
• The first three numbers are listed
in order of a1, a2, a3. a1
 90°
 =  = 90°
 = 120°

The unique symmetry operation in the hexagonal system is a six-
fold axis of rotation, and the most common space group is 6/m
, p g p
2/m 2/m.
There vertical axis is the six-fold rotational operation, while there
f th 6 t f ld i f t ti i th h i t l l (3
are a further 6 two-fold axis of rotation in the horizontal plane (3
coincide with the an axes).
There are 7 mirror planes
There are 7 mirror planes.
The general class for the orthorhombic system is known as the
dihexagonal-dipyramidal class
dihexagonal-dipyramidal class.
There are five types of form in the class: pinacoids, hexagonal
prisms hexagonal dipyramids dihexagonal prisms and
prisms, hexagonal dipyramids, dihexagonal prisms, and
dihexagonal dipyramids.
Common hexagonal minerals include beryl and apatite.
Common hexagonal minerals include beryl and apatite.

Isometric (cubic): Three equal length axes that
intersecting at right-angles to one another
intersecting at right angles to one another.
• The axes are indistinguishable, as
a e the inte secting angles As
are the intersecting angles. As
such all are interchangable.
a3
• There are 15 isometric forms, but
the most common are: a2
3
– Cube
– Octahedron
– Dodecahedron
a1
– Tetrahexahedron
– Trapezohedron
– Trisoctahedron
Trisoctahedron
– Hexoctahedron
 =  =  = 90°

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