Crystal Symmetry

Crystal Symmetry 

The external shape of a crystal reflects the 

presence or absence of translation-free 

symmetry y elements in its unit cell. 

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 discussed d 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, 

or crystal class. 

The crystal classes may be sub-divided into one of 

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) 

Interestingly, while all mirror planes and poles 

of rotation must intersect at one point, this 

point may not be a center of symmetry (i).

Crystallographic Axes 

The identification of specific symmetry operations 

enables one to orientate a crystal according to 

an imaginary set of reference lines known as the 

crystallographic axes. 

These are distinct and different from the classic 

Cartesian Axes, x, y and z, used in other 

common day usage, such as plotting graphs.

With the exception of the hexagonal system, the 

axes are designated a, a b, b and c. c 

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 

These are u, v, w: 

u: projection along a 

v: projection along b 

w: projection along c

Quantities can also be applied to further describe 

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 

[uvw] with (hkl) 

(hkl) faces on a cube

Axial Ratios 

With the exception of the cubic (isometric) system, 

there are crystallographic axes differing dff in length. 

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 

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 (a 1 , a 2 , a 3 ) 

and a 4 th perpendicular (vertical) of different length. 

Cubic: Three perpendicular axes of equal length.

Triclinic: Three unequal axes with oblique 

angles. 

• To orientate a triclinic crystal 

the most pronounced nced zone 

should be vertical. 

c 

• a and b are determined by 

the intersections of (010) and 

(100) with (001). 

 

 

 

b 

• The b axis should be longer 

than the a axis. 

a

The unique symmetry operation in a triclinic 

system is a 1-fold axis of rotoinversion(equivalent 

to a center of symmetry or inversion, i). 

All forms are pinacoids – therefore must consist of 

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. 

• Orientation ti of a crystal has 

few constraints – b is the 

only axis fixed by 

symmetry. 

• c is typically chosen on the 

basis of habit and 

cleavage. 

• α and γ = 90 °. 

• There are some very rare 

cases where b equals 90° 

giving a pseudo- 

orthorhombic form. 

a 

 

c 

 

 

b

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 

plane. 

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 

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 

is taken as unity and ratios are 

calculated thereafter. 

b 

 

 

 

a

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. 

Therefore, if a = b, c must be in 

the vertical. a 2 

c 

• There is no rule as to whether c is 

greater or less than a. a 

a 1

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. 

There are 4 two-fold axis of rotation: 2 parallel to the 

crystallographic axes a and b, 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 (a 1 , a 2 , a 3 ) 

and a 4 th perpendicular vertical axis of different 

length. 

• The three horizontal axis of a 

hexagonal mineral are oriented in 

the plane of the horizontal, with c 

in the vertical. 

c 

• Unlike the other systems the 

Bravais-Miller nomenclature for 

crystal faces is given by 4 numbers 

(i.e. {0001}) 

• The first three numbers are listed 

in order of a 1 , a 2 , a 3 . 

a 3 

a 2 

a 1 

90° = = 90° 

= 120°

The unique symmetry operation in the hexagonal system is a sixfold 

axis of rotation, and the most common space group is 6/m 

2/m 2/m. 

There vertical axis is the six-fold rotational operation, while there 

are a further 6 two-fold axis of rotation ti in the horizontal plane (3 

coincide with the a n axes). 

There are 7 mirror planes. 

The general class for the orthorhombic system is known as the 

dihexagonal-dipyramidal class. 

There are five types of form in the class: pinacoids, hexagonal 

prisms, hexagonal dipyramids, dihexagonal prisms, and 

dihexagonal dipyramids. 

Common hexagonal minerals include beryl and apatite.

Isometric (cubic): Three equal length axes that 

intersecting at right-angles to one another. 

• The axes are indistinguishable, as 

are the intersecting angles. As 

such all are interchangable. 

• There are 15 isometric forms, but 

a 3 

the most common are: 

a 2 

– Cube 

– Octahedron 

– Dodecahedron 

– Tetrahexahedron 

– Trapezohedron 

– Trisoctahedron 

– Hexoctahedron 

= = = 90° 

a 1

Crystal Symmetry

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