topic- Crystallography M.Sc. (chemistry),3rd semester. Op Jindal University, Raigarh, chhattisgarh.

1. A seminar presentation
on
“Study of Symmetric Properties of Crystal System”
Presented by-
Srishti Gupta
M.Sc.(chemistry)3rd semester

2. Content
• Introduction
• Symmetry Elements
• Symmetry operations
• Constructing Point Groups
• Symmetry Notation
• Seven Crystal System

3.  Introduction -
• Symmetry, in crystallography, fundamental property of orderly arrangement of atoms found in crystalline solid. Each
arrangement of atoms has a certain number of element of symmetry; i.e., changes in the orientation of the
arrangement of atoms seem to leave the atoms unmoved.
• The term crystallography, crystallography is the experimental science of determining the arrangement of atoms in
crystalline solids. In crystallography terms, lattice system and crystal, the system are associated with each other with
a slight difference. Based on their point groups crystals and space groups are divided into seven crystal system.
Symmetry Elements –
• A Symmetry element is a geometrical entity about which a symmetry operation is performed.
• A symmetry element can be point, line, or plane.
Symmetry elements :- n-fold, mirror plane, point.

4.  Symmetry Operation –
A symmetry operation is an action that leaves an object looking the same after it has been carried out. A symmetry operation
is a moment of a molecule such that after moment the molecule appears the same as before.
1. Rotation – The presence of rotation is established by rotating a motif about an imaginary axis that intersects the crystal
center.
Axis of Rotation – It is an imaginary line through the center of the crystal about which a specific crystal face repeats
itself in appearance. Rotation is denoted as ‘A’.
Eg. - A4 , here A is rotation and 4 is one-four fold rotation axis.

5. 2. Reflection or Mirror Plane -
In perfectly developed crystal a mirror plane, m is an imaginary plane that divides a crystal into two halves, each of
which, is the mirror image of each other.
A single mirror in crystal is called as Symmetry Plane.
Eg.- Mirror plane in cubic crystal;

6. 3. Center of symmetry (i) -
It is present in crystal if an imaginary line can be passed fro any point on its surface through its center and the same point
is found on the line at an equal distance beyond the center on the opposite side.
• Also known as an inversion, produces an inverted object through an inversion center.
• It is denoted by i or i bar.

7. 4. Translational -
Translational symmetry in a crystal refers to the repeating pattern of the crystal lattice, where each unit cell is ident.

8. 5. Rotoinversion – To generate rotoinversion, rotate motif through angle, then invert the motif through the center. An axis of
rotoinversion within a crystal is an imaginary line that relate rotation about an axis with inversion.( bar A2, bar A3, bar A4, bar A6)
 Constructing Point Group -
• Point group show all symmetry relationships in a set of points that don’t move. It turns out that in 3 dimensions, there
are only 32-point groups in 3 dimensions.
Eg :- methane (4C3, 3C2, 6σ) – set of points.

9. • First, we can fit each point group to a crystal system; triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.
• When naming the point groups, we typically write the highest order symmetry on each perpendicular axis. The “/”
sign indicates that 2 operations are present in the same axis.
Eg :- 6/m,
• Obviously, a six fold axis of rotation also includes 3-fold, 2-fold, and 1-fold axes of rotation, so we don’t also need to
include those if we already know there’s a 6-fold axis of rotation.
• If we follow these rules, we can create the 32 point groups with “Hermann-Mauguin” Notation.

10.  Symmetry Notation -
The notation of symmetry is simplified by the use of symbols.
3A4 – three four fold rotation axis,
6A2 - six two fold rotation axis.
Overall symmetry content of a crystal are referred as the Hermann-Mauguin Notation.
Symmetry Elements Symmetry Operation Symmetry Symbol HM Notation
1. Rotation axis Rotation, A1, A2, A3, A4, A6 1, 2, 3, 4, 6
2. Mirror plane Reflection m m
3. Center of symmetry Inversion i i bar
4. Rotoinversion R+ I A + i A bar

11.  Seven Crystal System -
Cubic crystal Orthorhombic Tetragonal Rhombohedral Hexagonal Triclinic
a = b = c a ≠ b ≠ c a = b ≠ c a = b = c a = b ≠ c a ≠ b ≠ c
α = β = γ α = β = γ = 90º α = β = γ α = β = γ ≠ 90º α = β = 90º α ≠ β ≠ γ ≠ 90º
γ = 120º
Monoclinic
a ≠ b ≠ c
α = γ = 90º
β ≠ 90º

12. 1. Cubic Crystal System -
There are 5 point groups in cubic crystal
In a cubic crystal 2-fold axis of rotation, 3-fold axis of rotation, 4-fold axis of rotation, and 3 rectangular
plane and 6 diagonal plane of symmetry and one inversion center.

13.  m𝟑 point group has a mirror plane and 3-fold roto-inversion.
 432: point group has a four-fold axis of rotation,
3 fold axis of rotation, and 2-fold axis of rotation.
 𝟒3m : point group 43m has a 4-fold roto inversion,
3-fold axis of rotation and a mirror plane.
 m𝟑m : point group has a mirror plane, 3-fold roto inversion and another mirror plane.
2. Hexagonal Crystal System –
There are 7 hexagonal point groups: 6 : point group has a 6-fold axis of rotation.
 𝟔 : point group has a 6-fold roto inversion.
 6/m : point group has a 6-fold axis of rotation perpendicular to a mirror plane.
 622 : point group has 6-fold axis of rotation perpendicular to a mirror plane.
 6mm: point group has a 6-fold axis of rotation perpendicular to
the normal of two mirror plane.
 𝟔2m : point group has a 6-fold roto inversion,
a 2-fold axis of rotation and a mirror plane.
 6/mmm : point group has a 6-fold axis of rotation perpendicular to a mirror plane,
and 2 other perpendicular mirror planes.

14. Here, below illustrate some symmetry operations in hexagonal crystal:

15. 3. Tetragonal Crystal System -
Tetragonal Crystal systems have one 4-fold axis of rotation.
There are 7 tetragonal point groups:
 4: point group has a single 4-fold axis of rotation.
 𝟒 : point group has 4-fold roto inversion.
 4/m: point group has 4-fold rotation perpendicular to a mirror plane.
 422: point group has 4-fold rotation and two 2-fold rotations,
all perpendicular to each other.
 4mm: point group has 4-fold rotation and two mirror planes.
 𝟒2 : point group has 4-fold roto inversion and perpendicular to 2-fold rotation.
 4/mmm : point group has 4-fold rotation perpendicular to a mirror plane,
and two more mirror planes all perpendicular to each other.

16. 4. Orthorhombic Crystal System -
Orthorhombic crystal systems can have three 2-fold axes of rotation or one 2-fold axis of rotation with two mirror
planes.
There are 3 orthorhombic point groups:
 222: point group has three perpendicular 2-fold axes of rotation.
 mmm: point group has three mirror plane, perpendicular to each other.
 mm2: point group has two perpendicular mirror plane and one 2-fold axis of rotation,
all three operations are perpendicular to each other.

17. 5. Monoclinic Crystal System -
The monoclinic crystal system can have at most one 2-fold axis of rotation or one mirror plane.
There are 3 monoclinic point groups:
 2: point group has a 2-fold axis of rotation.
 m : point group has one mirror plane, m stands for mirror plane.
 2/m : point group has a 2-fold axis of rotation perpendicular to a mirror plane.
 “/” sign indicates parallel elements.

18. 6. Triclinic Crystal System -
The triclinic crystal system has the lowest symmetry.
There are 2 triclinic point groups:
 1: point group has symmetry operations, 1 stands for 1-fold rotation(360º).
 𝟏 : point group has a center of inversion, but still no mirror planes and only the 1-fold axis of rotation,

19. 7. Rhombohedral or Trigonal Crystal System -
There are 5 trigonal point groups:
 3: point group has 3-fold axis of rotation.
 𝟑 : point group has a 3-fold roto inversion.
 32: point group has a 3-fold axis of rotation and 2-fold axis of rotation in different axis.
note- it may sometimes be written as point group “321” to completely the third axis,
although of course “1” doesn’t convey any extra information.
 3m : point group has a 3-fold axis of rotation, and a mirror plane.
 𝟑m : point group has a 3-fold roto inversion, and a mirror plane.

20. Thank
You.