Crystal symmetry is defined by repeated patterns of atoms in a crystal structure. There are different types of symmetry operations including planes of symmetry, axes of symmetry, and centers of symmetry. Planes of symmetry divide the crystal into mirror images, axes of symmetry involve rotational symmetry, and centers of symmetry involve equidistance from a point. The six main crystal systems are defined by their unique combinations of symmetry elements and are classified from highest to lowest symmetry as isometric, tetragonal, orthorhombic, hexagonal, monoclinic, and triclinic. Miller indices and Weiss parameters are used to describe the orientation of crystal planes.

1. Crystal symmetry

2. Introduction
Crystallography is a natural science with the
scope of investigating matter in the crystalline
state. In modern era, this mainly implies
determining the arrangement of atoms in
crystals. The word "crystallography" derives
from the Greek words crystallon “cold drop,
frozen drop", with its meaning extending to all
solids with some degree of transparency,
and graphic "to write".

3. Crystal Symmetry It explains how similar atoms or group of
atoms (motif) repeated symmetrically in
space to produce ordered structure. In
many crystals, regularity of arrangement
of these plane faces occurs, and a careful
study of such a crystal will reveal
elements of symmetry.

4. Symmetry Operation
Symmetry operation: Is an operation that
can be performed either physically or
imaginatively on crystal with reference to
Plane, Axis, and Point within its mass

5. Symmetry Operation
Achieved By • Select the plane which shows
mirror image
• Rotating the crystal in particular
axis
• Select the point which shows
equidistance

6. Plane of symmetry
It is an imaginary plane, which passes
through the centre of a crystal can,
divides it into two equal portions, which
are exactly the mirror images of each
other

7. Axis of Symmetry
An axis of symmetry or axis of rotation is
an imaginary line, passing through the
crystal such that when the crystal is rotated
about this line, it presents the same
appearance more than once in one complete
revolution i.e., in a rotation through 360̊.

8. Four Axis of Symmetry
• On the rotation about the axis, if the
same faces or same view occurs 2
times, the axis termed as Diad axis
i.e. 3600/1800= 2 rotations
• On the rotation about the axis, , if the
same faces or same view occurs 3
times, the axis termed as Triad axis
i.e. 3600/1200= 3 rotations

9. Four Axis of Symmetry
• On the rotation about the axis, if the same faces
or same view occurs 4 times, the axis termed as
Tetrad axis or four fold axis. i.e. 3600/900= 4
rotations
• On the rotation about the axis, , if the same
faces or same view occurs 6 times, the axis
termed as Hexad axis or six fold axis. i.e.
3600/600= 6 rotations.

10. Centre of Symmetry It is an imaginary point in the crystal
that any line drawn through it
intersects the surface of the crystal at
equal distance on either side

11. Isometric system
Isometric system has highest degree of
symmetry and having total 23crystal
symmetry among which 9 plane of
symmetry and 13 axis of symmetry
present i.e. this having 6-II fold axis,4-
IIIfold axis, 3-IV fold axis no VI fold axis
and also centre of symmetry is present.
Symmetrical characteristics of six crystal system

12. Tetragonal system • Tetragonal system having total 11
symmetry among which, 5 plane of
symmetry and axis of symmetry
present i.e. this having 04-II fold
axis, no III-fold axis, 01-IV fold
axis no VI fold axis and also centre
of symmetry is present

13. Orthorhombic system
Orthorhombic system having total 7
symmetry among which , 3 plane of
symmetry and axis of symmetry
present i.e. this having 3-II fold axis,
no III-fold axis, no IV fold axis no VI
fold axis and also centre of symmetry
is present.

14. Hexagonal system Hexagonal system having total 15
symmetry among which, seven plane
of symmetry and axis of symmetry
present i.e. his having 6-II fold axis, no
III-fold axis, and no IV fold axis 01-VI
fold axis and centre of symmetry is
present

15. Monoclinic system Monoclinic system having total 3
symmetry among which, 1 plane of
symmetry and a axis of symmetry
present i.e. this system having 01-II
fold axis, no III-fold axis no IV fold
axis no VI fold axis and also centre of
symmetry is present.

16. Triclinic System
This system has lest degree of
symmetry and having only centre of
symmetry. Plane of symmetry and
Axis of symmetry is absent

17. Crystallographic Axes
A set of reference axes in a crystal that
are used to describe the crystal systems.
These are distinct and different from the
classic Cartesian axis x, y, z are used the
angles are denoted as α, β, ϒ and the
exception of the hexagonal system the
axes are designated a, b and c

18. Crystallographic Planes
Crystal plane may be defined by
considering how the plane intersects
the main crystallographic axes of the
crystal.

19. Crystallographic notation
Crystallographic notation is the symbolic
representation of relationship of any
crystal face to crystallographic axes. This
crystallographic notation system can be
explained by two methods.

20. WEISS Parameter
Weiss parameter is the relative
numbers of at which given crystal face
cuts the crystallographic axesThe most
general expression for Weiss
parameter is
na:mb:pc
Where n, m, p are the length cut off by
the face on the a, b, & c axes
respectively

21. Miller Indices
Miller Indices are a symbolic
representation for the orientation of
plane in a crystal lattice & are defined
as the reciprocals of the fractional
intercepts, which the plane makes with
the crystallographic axes
Most common Millerian symbol is
h,k,l

22. Miller indices
• To find the Miller indices of a plane,
take the following steps:
• Determine the intercepts of the plane
along each of the three crystallographic
directions.
• Take the reciprocals of the intercepts.
• If fractions result, multiply each by the
denominator of the smallest fraction

23. Conclusion
Crystals have an ordered internal
arrangement of atoms. This ordered
arrangement shows
symmetry. These crystal faces
reflect the ordered internal
arrangement of atoms and thus
reflect the symmetry of the crystal
lattice.

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