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Crystal Symmetry
The concept of symmetry describes the repetition of
structural features. Crystals therefore possess symmetry,
and much of the discipline of crystallography is concerned
with describing and cataloging different types of symmetry.
Two general types of symmetry exist:
Internal or Translational symmetry : describes the periodic
repetition of a structural feature across a length or through an
area or volume.
External or Point symmetry : describes the periodic
repetition of a structural feature around a point. Reflection,
rotation, and inversion are all point symmetries.
Symmetries are most frequently used to classify the different
crystal structures.
TRANSLATION OPERATIONS
Periodic repetitions along vectors or translations of ions
can be achieved in 1-D, 2-D and 3-D producing row-lattices,
plane-lattices and space-lattices, respectively. These are
internal symmetry operations and do not occur as symmetry
element. Row-Lattices: are produced by translation along single vector. Here
ions are repeated with constant distances along a line. a
a is the repeat vector a
The origin is arbitrary
Symmetric translations involve repeat distances
Plane-Lattices: are produced by translations along two
vectors. Here ions are repeated with constant distances and
angles that produce 5 unique plane-lattices.
a
b
Lattices formed by 2-D translations
a
b
Space-Lattices: are produced by
translations along three vectors. Here
ions are repeated again with constant
distances and angles in 3-D that produce
14 unique space-lattices, which are also
known as Bravais Lattices .They are
compatible with 7 symmetry systems and
32 symmetry classes.
Crystal Systems
The space lattice points in a crystal are occupied by atoms.
The position of any unit cell in the 3D lattice can be
described by the three unit vectors, a, b, c and the angles
between them, , and
as shown by the shaded region in Fig.(a) This cell (Fig. b)
which when repeated in the three dimensions generates the
crystal structure.
(a) (b)
Crystal Systems
• Bravais Lattice
• The unit vectors a, b and c are called lattice
parameters. Based on their length equality
or inequality and their orientation (the angles
between them, , and ) a total of 7 crystal
systems can be defined. With the centering
(face, base and body centering) added to
these , 14 kinds of 3D lattices, known as
Bravais lattices, can be generated.
Crystal Systems
Cubic: a = b = c,
= = = 90o
Simple
cubic
Body-centered
cubic (BCC(
Face-centered
cubic (FCC(
Tetragonal: a = b c,
= = = 90o
Simple
Tetragona
l
Body-centered
Tetragonal (BCT(
Monoclinic: a b c , = = 90o
Orthorhombic: a b c, = = = 90o
Crystal Systems
Simple
monoclinic
Base-
centered
monoclinic
Simple Body-centered Base-centered Face-centered
Rhombohedral
a = b c
= = 90o = 120o
Hexagonal
a = b c
= = 90o = 120o
Triclinic
a b c
90o
Crystal Systems
CRYSTAL SYMMETRY Imagine a small 2 dimensional crystal composed of atoms in an ordered internal arrangement as shown below. Although all of the atoms in this lattice are the same, one of them is coloured red so we can keep track of its position.
If we rotate the simple crystals by 90o notice that the lattice and crystal look exactly the same as what we started with. Rotate it another 90o and again its the same. Another 90o rotation again results in an identical crystal, and another 90o rotation returns the crystal to its original orientation. Thus, in 1 rotation of 360o , the crystal has repeated itself, or looks identical 4 times. We thus say that this object has 4-fold rotational symmetry.
Symmetry Operations and Elements
External or Point symmetry :A Symmetry operation is an operation that can be performed either physically or imaginatively and results in no change in the appearance of an object. Note: It is emphasized that in crystals, the symmetry is internal due to an ordered geometrical arrangement of atoms and molecules in the crystal lattice. Since the internal symmetry is reflected in the external form of perfect crystals, we are going to concentrate on external symmetry, using wooden models of crystals. There are 3 types of symmetry operations:
rotation, reflection, and inversion.
17
Lattice goes into itself through
Symmetry without translation
Operation Element
Inversion Point
Reflection Plane
Rotation Axis
Rotoinversion Axes
Rotational Symmetry The imaginary axis along which the rotation is performed is an element of symmetry referred to as a rotation axis. Rotational symmetry axes possible in crystals are 1,2,3,4,6.
1-Fold Rotation Axis - An object that requires rotation of a full 360o in order to restore it to its original appearance has no rotational symmetry. Since it repeats itself 1 time every 360o it is said to have a 1-fold axis of rotational symmetry.
2-fold Rotation Axis imagine rotation axis perpendicular to the
page If an object appears identical after a rotation of 180o, that is twice in a 360o rotation, then it is said to have a 2-fold rotation axis (360/180 = 2).
Note: A filled oval shape represents the point where the 2-fold rotation axis intersects the page.
3-Fold Rotation Axis – An object that repeats itself after a rotation of 120o is said to have a 3-fold axis of rotational symmetry, thus repeating itself 3 times in a 360o rotation (360/120 =3). Note: A filled triangle is used to symbolize the location of 3-fold rotation axis.
4-Fold Rotation Axis - If an object repeats itself after 90o of rotation, it will repeat 4 times in a 360o rotation, as illustrated. Note: A filled square is used to symbolize the location of 4-fold axis of rotational symmetry.
6-Fold Rotation Axis - If rotation of 60o about an axis causes the object to repeat itself, then it has 6-fold axis of rotational symmetry (360/60=6). Note: A filled hexagon is used as the symbol for a 6-fold rotation axis.
Is it possible to have 5, 7or 8 -fold rotation symmetry?
Objects with 5, 7and 8 or higher order symmetry do exist in
nature, e.g. star fish ( 5-fold), flowers with 5 or 8-fold
symmetry.
5 fold 8 fold
Rotation
However, these are not possible in crystallography as
they cannot fill the space completely
Mirror Symmetry
A mirror symmetry operation is an imaginary operation that can be performed to reproduce an object. The operation is done by imagining that you cut the object in half, then place a mirror next to one of the halves of the object along the cut. If the reflection in the mirror reproduces the other half of the object, then the object is said to have mirror symmetry. The plane of the mirror is an element of symmetry referred to as a mirror plane. Note: the plane is symbolized with the letter m.
inversion Another operation that can be performed through a point is inversion. In this operation lines are drawn from all points on the object through a point in the centre of the object, called a symmetry centre (Note: it is symbolized with the letter "i"). The lines each have lengths that are equidistant from the original points. When the ends of the lines are connected, the original object is reproduced inverted from its original appearance. In the diagram shown here, only a few such lines are drawn for the small triangular face. The right hand diagram shows the object without the imaginary lines that reproduced the object.
If an object has only 1 centre of symmetry, we say that it has a 1
fold rotoinversion axis. Such an axis has the symbol (i), as shown
in the right hand diagram above.
Note that crystals that have a centre of symmetry will exhibit the
property that if you place it on a table there will be a face on the
top of the crystal that will be parallel to the surface of the table
and identical to the face resting on the table.
Rotoinversion Combinations of rotation with a centre of symmetry
perform the symmetry operation of
rotoinversion. Objects that have rotoinversion
symmetry have an element of symmetry called a
rotoinversion axis. A 1-fold rotoinversion axis is the
same as a centre of symmetry, as discussed above.
Other possible rotoinversion axes are as follows:
2-fold Rotoinversion - The operation of 2-fold rotoinversion
involves first rotating the object by 180o then inverting it through an inversion centre. This operation is equivalent to having a mirror plane perpendicular to the 2-fold rotoinversion
axis. Note: A 2-fold rotoinversion axis is symbolized as a 2
with a bar over the top( ̅2), and would be pronounced as "bar 2". But, since this the equivalent of a mirror plane, m, the bar 2 is rarely used.
3-fold Rotoinversion - This involves rotating the
object by 120o (360/3 = 120), and inverting
through a centre. (example, cube). A 3-fold rotoinversion axis is denoted as ( ̅3),
(pronounced "bar 3"). Note: that there are actually four axes in a cube, one running through each of the corners of the cube. If one holds one of the axes vertical, then note that there are 3 faces on top, and 3 identical faces upside down on the bottom that are offset from the top faces by 120o
4-fold Rotoinversion
This involves rotation of the object by 90o then inverting through a centre. A four fold rotoinversion axis is symbolized as ( ̅4).
Note: An object possessing a 4- fold rotoinversion axis
will have two faces on top and two identical faces upside- down on the bottom, if the ̅4 axis is held in the vertical position.
6-fold Rotoinversion A 6-fold rotoinversion axis ( ̅6 ) involves rotating the object by 60o and inverting through a centre. Note: This operation is identical to having the combination of a 3-fold rotation axis perpendicular to a mirror plane.
Combinations of Symmetry Operations
In crystals there are ONLY 32 possible combinations of symmetry elements. These 32 combinations define the 32 Crystal Classes. Every crystal must belong to one of these 32 crystal classes.
Example of how the various symmetry elements are combined in a somewhat completed crystal. If 2 kinds of symmetry elements are present in the same crystal, then they will operate on each other to produce other symmetrical symmetry elements.
The crystal shown has rectangular-shaped sides with a square- shaped top and bottom. The square-shaped top indicates that there must be a 4-fold rotation axis perpendicular to the square shaped face.
Note that the rectangular shaped face on the left side of the crystal must have a 2-fold rotation axis that intersects it. Note that the two fold axis runs through the crystal and exits on the left-hand side (not seen in the views), so that both the left- and right-hand sides of the crystal are perpendicular to a 2-fold rotation axis.
Since the top face of the crystal has a 4-fold rotation axis, operation of this 4-fold rotation must reproduce the face with the perpendicular 2-fold axis on a 90 degrees rotation. Thus, the front and back faces of the crystal will also have perpendicular 2-fold rotation axes, since these are required by the 4-fold axis.
The square-shaped top of the crystal also suggests that there must be a 2-fold axis that cuts diagonally through the crystal. This 2-fold axis is shown here in the left-hand diagram. But, again operation of the 4-fold axis requires that the other diagonals also have 2-fold axis, as shown in the right-hand diagram.
Furthermore, the square-shaped top and rectangular-shaped front of the crystal suggest that a plane of symmetry is present as shown by the left-hand diagram here. But, again, operation of the 4-fold axis requires that a mirror plane is also present that cuts through the side faces, as shown
by the diagram on the right.
The square top further suggests that there must be a mirror plane cutting the diagonal through the crystal. This mirror plane will be reflected by the other mirror planes cutting the sides of the crystal, or will be reproduced by the 4-fold rotation axis, and thus the crystal will have another mirror plane cutting through the other diagonal, as shown by the diagram on the right.
Finally, there is another mirror plane that cuts through the centre of the crystal parallel to the top and bottom faces.
Thus, this crystal has the following symmetry elements: 1 - 4-fold rotation axis (A4) 4 - 2-fold rotation axes (A2) (2 cutting the faces & 2 cutting the edges). 5 mirror planes (m) (2 cutting across the faces, 2 cutting through the edges, and one cutting horizontally through the centre). Note also that there is a centre of symmetry (i). The symmetry content of this crystal is thus: i, 1A4, 4A2, 5m
External Symmetry of Crystals, 32 Crystal Classes
Hermann- Mauguin (International) Symbols How to derive the Hermann - Mauguin symbols (also called the international symbols) used to describe the crystal classes from the symmetry content. We'll start with a simple crystal then look at some more complex examples.
The rectangular block shown here has 3 2-fold rotation axes (A2), 3 mirror planes (m), and a centre of symmetry (i) (3A2, 3m, i).
Rule for the derivation of the Hermann-Mauguin symbol are as follows: Write a number representing each of the unique rotation axes present (A unique rotation axis is one that exists by itself and is not produced by another symmetry operation). Eg. 1- In this case, all three 2-fold axes are unique, because each is perpendicular to a different shaped face, so we write a 2 (for 2-fold) for each axis 2 2 2
Next we write an "m" for each unique mirror plane. (A unique mirror plane is one that is not produced by any other symmetry operation). In this example, we can tell that each mirror is unique because each one cuts a different face. So, we write: 2 m 2 m 2 m If any of the axes are perpendicular to a mirror plane we put a slash (/) between the symbol for the axis and the symbol for the mirror plane. In this case, each of the 2-fold axes are perpendicular to mirror planes, so our symbol becomes: 2/m2/m2/m
Contains 1 4-fold axis, 4 2-fold axes, 5 mirror planes, and a centre of symmetry (1A4, 4A2, 5m, i). Note that the 4-fold axis is unique. There are 2 2-fold axes that are perpendicular to identical faces, and 2 2-fold axes that run through the vertical edges of the crystal. Thus there are only 2 unique 2 fold axes, because the others are required by the 4-fold axis perpendicular to the top face. So, we write: 4 2 2 Although there are 5 mirror planes in the model, only 3 of them are unique. Two mirror planes cut the front and side faces of the crystal, and are perpendicular to the 2-fold axes that are perpendicular to these faces. Only one of these is unique, because the other is required by the 4-fold rotation axis.
Eg. 2.
Another set of 2 mirror planes cuts diagonally across the top and down the edges of the model. Only one of these is unique, because the other is generated by the 4-fold rotation axis and the previously discussed mirror planes. The mirror plane that cuts horizontally through the crystal and is perpendicular to the 4-fold axis is unique. Since all unique mirror planes are perpendicular to rotation axes, our final symbol becomes: 4/m2/m2/m
If the total symmetry of a crystal is ascertained, the table below can be used to determine to which crystal class and system the crystal belongs
Crystal System Crystal Class Name of Class Symmetry Content
Triclinic
1
Pedial none
Triclinic
1
Pinacoidal
i
Triclinic System
Crystal System Crystal Class Name of Class Symmetry Content
Monoclinic 2 Sphenoidal 1A2
Monoclinic m Domatic 1m
Monoclinic 2/m Prismatic i, 1A2, 1m
Crystal System Crystal Class Name of Class Symmetry Content
Orthorhombic 222
Rhombic-disphenoidal 3A2
Orthorhombic mm2 Rhombic-pyramidal 1A2, 2m
Orthorhombic 2/m2/m2/m Rhombic-dipyramidal i, 3A2, 3m
Monoclinic System
Orthorhombic System
Crystal System Crystal Class Name of Class Symmetry Content
Tetragonal 4 Tetragonal-pyramidal 1A4
Tetragonal 4 Tetragonal-
disphenoidal
1A2
Tetragonal 4/m Tetragonal-
dipyramidal
i, 1A4, 1m
Tetragonal 422 Teragonal-
trapezohedral
1A4, 4A2
Tetragonal 4mm Ditetragonal-
pyramidal
1A4, 4m
Tetragonal 42m Tetragonal-
scalenohedral
3A2, 2m
Tetragonal 4/m2/m2/m Ditetragonal-
dipyramidal
i, 1A4, 4A2, 5m
Tetragonal System
Crystal System Crystal Class Name of Class Symmetry Content
Trigonal or
Rhombohedral
3
Trigonal-pyramidal 1A3
Trigonal or
Rhombohedral
3 Rhombohedral
i, 1A3
Trigonal or
Rhombohedral
32 Trigonal-trapezohedral
1A3, 3A2
Trigonal or
Rhombohedral
3m
Ditrigonal-pyramidal 1A3, 3m
Trigonal or
Rhombohedral
32/m Trigonal scalenohedral i, 1A3, 3A2, 3m
Trigonal System
Crystal System Crystal Class Name of Class Symmetry Content
Hexagonal system 6 Hexagonal-pyramidal 1A6
Hexagonal system
6 Trigonal-dipyramidal 1A3, 1m
Hexagonal system 6/m Hexagonal-
dipyramidal
i, 1A6, 1m
Hexagonal system 622 Hexagonal-
trapezohedral
1A6, 6A2
Hexagonal system 6mm Dihexagonal-
pyramidal
1A6, 6m
Hexagonal system 62m Ditrigonal-
dipyramidal
1A3, 3A2, 4m
Hexagonal system 6/m2/m2/m Dihexagonal-
dipyramidal
i, 1A6, 6A2, 7m
Hexagonal System
Crystal System Crystal Class Name of Class Symmetry Content
Isometric 23 Tetartoidal 3A2, 4A3
Isometric 2/m3 Diploidal li, 3A2, 3m, 4A3
Isometric 432 Gyroidal 3A4, 4A3, 6A2
Isometric 43m Hextetrahedral 4A3, 3A2, 6m4/m
Isometric 32/m Hextetrahedral i, 3A4, 4A3, 6A2, 9m
You should note from the table there are certain associated symmetry functions as:
1. a crystal which has an A6 cannot have an A4 and vice versa
2. a crystal with only 1A4 is tetragonal, and if more present, has to have 3A4 and
must belong to a specific isometric crystal class
3. a crystal with an A6 cannot have an A3 and must be in the hexagonal system
4. a crystal with only 1A3 must be in the trigonal system, and if more present there
has to be exactly 4A3 and belong to the isometric system
Isometric System
Crystal System Crystal Class Symmetry Name of Class
Triclinic
1 none Pedial
i Pinacoidal
Monoclinic
2 1A2 Sphenoidal
m 1m Domatic
2/m i, 1A2, 1m Prismatic
Orthorhombic
222 3A2 Rhombic-
disphenoidal
mm2 (2mm) 1A2, 2m Rhombic-
pyramidal
2/m2/m2/m i, 3A2, 3m Rhombic-
dipyramidal
Tetragonal
4 1A4 Tetragonal-
Pyramidal
4 Tetragonal-
disphenoidal
4/m i, 1A4, 1m Tetragonal-
dipyramidal
422 1A4, 4A2 Tetragonal-
trapezohedral
4mm 1A4, 4m Ditetragonal-
pyramidal
2m 1 4, 2A2, 2m Tetragonal-
scalenohedral
4/m2/m2/m i, 1A4, 4A2, 5m Ditetragonal-
dipyramidal
3 1A3 Trigonal-pyramidal
1 3 Rhombohedral
32 1A3, 3A2 Trigonal-
trapezohedral
3m 1A3, 3m Ditrigonal-
pyramidal
Hexagonal
2/m 1 3, 3A2, 3m Hexagonal-
scalenohedral
6 1A6 Hexagonal-pyramidal
1 6 Trigonal-dipyramidal
6/m i, 1A6, 1m Hexagonal-
dipyramidal
622 1A6, 6A2 Hexagonal-
trapezohedral
6mm 1A6, 6m Dihexagonal-
pyramidal
m2 1 6, 3A2, 3m Ditrigonal-
dipyramidal
6/m2/m2/m i, 1A6, 6A2, 7m Dihexagonal-
dipyramidal
Isometric 23 3A2, 4A3 Tetaroidal
Isometric
2/m 3A2, 3m, 4 3 Diploidal
432 3A4, 4A3, 6A2 Gyroidal
3m 3 4, 4A3, 6m Hextetrahedr
al
4/m bar3
2/m
3A4, 4 3, 6A2,
9m
Hexoctahedra
l
32 crystal classes are divided into 7 crystal systems
DIAGNOSTIC SYMMETRY System
1-fold or 1-fold rotoinversion axes/ centre Triclinic
mirror plane or a single 2-fold axis Monoclinic
three 2-fold axes /or 3 mirror planes Orthorhombic
single 4-fold axis Tetragonal
3-fold axis Trigonal
one 6-fold Hexagonal
four 3-fold axes Isometric
Next lecture will begin with the description of all the crystal
classes
