1. PH0101 UNIT 4 LECTURE 3 CRYSTAL SYMMETRY CENTRE OF SYMMETRY
PLANE OF SYMMETRY
AXES OF SYMMETRY
ABSENCE OF 5 FOLD SYMMETRY
ROTOINVERSION AXES
SCREW AXES
GLIDE PLANE
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2. CRYSTAL SYMMETRY Crystals have inherent symmetry.
The definite ordered arrangement of the faces
and edges of a crystal is known as `crystal
symmetry’.
It is a powerful tool for the study of the internal
structure of crystals.
Crystals possess different symmetries or
symmetry elements.
PH UNIT 4 LECTURE 3

3. CRYSTAL SYMMETRY What is a symmetry operation ?
A `symmetry operation’ is one, that leaves
the crystal and its environment invariant.
It is an operation performed on an object or pattern
which brings it to a position which is absolutely
indistinguishable from the old position.
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4. CRYSTAL SYMMETRY The seven crystal systems are characterised by
three symmetry elements. They are
Centre of symmetry
Planes of symmetry
Axes of symmetry.
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5. CENTRE OF SYMMETRY
It is a point such that any line drawn through it will
meet the surface of the crystal at equal distances
on either side.
Since centre lies at equal distances from various
symmetrical positions it is also known as `centre
of inversions’.
It is equivalent to reflection through a point.
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6. CENTRE OF SYMMETRY A Crystal may possess a number of planes or
axes of symmetry but it can have only one centre
of symmetry.
For an unit cell of cubic lattice, the point at the
body centre represents’ the `centre of
symmetry’ and it is shown in the figure.
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7. CENTRE OF SYMMETRY
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8. PLANE OF SYMMETRY A crystal is said to have a plane of symmetry, when
it is divided by an imaginary plane into two halves,
such that one is the mirror image of the other.
In the case of a cube, there are three planes of
symmetry parallel to the faces of the cube and six
diagonal planes of symmetry
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9. PLANE OF SYMMETRY
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10. AXIS OF SYMMETRY
This is an axis passing through the crystal such that if
the crystal is rotated around it through some angle,
the crystal remains invariant.
The axis is called `n-fold, axis’ if the angle of rotation is
If equivalent configuration occurs after rotation of
180º, 120º and 90º, the axes of rotation are known as
two-fold, three-fold and four-fold axes of symmetry
respectively.
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11. AXIS OF SYMMETRY
If equivalent configuration occurs after rotation of 180º,
120º and 90º, the axes of rotation are known as two-
fold, three-fold and four-fold axes of symmetry .
If a cube is rotated through 90º, about an axis normal to
one of its faces at its mid point, it brings the cube into
self coincident position.
Hence during one complete rotation about this axis, i.e.,
through 360º, at four positions the cube is coincident
with its original position.Such an axis is called four-fold
axes of symmetry or tetrad axis.
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12. AXIS OF SYMMETRY
If n=1, the crystal has to be rotated through an angle =
360º, about an axis to achieve self coincidence. Such an
axis is called an `identity axis’. Each crystal possesses an
infinite number of such axes.
If n=2, the crystal has to be rotated through an angle =
180º about an axis to achieve self coincidence. Such an
axis is called a `diad axis’.Since there are 12 such edges in
a cube, the number of diad axes is six.
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13. AXIS OF SYMMETRY If n=3, the crystal has to be rotated through an
angle = 120º about an axis to achieve self
coincidence. Such an axis is called is `triad
axis’. In a cube, the axis passing through a
solid diagonal acts as a triad axis. Since there
are 4 solid diagonals in a cube, the number of
triad axis is four.
If n=4, for every 90º rotation, coincidence is
achieved and the axis is termed `tetrad axis’.
It is discussed already that a cube has `three’
tetrad axes.
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14. AXIS OF SYMMETRY If n=6, the corresponding angle of rotation is
60º and the axis of rotation is called a hexad
axis. A cubic crystal does not possess any
hexad axis.
Crystalline solids do not show 5-fold axis of
symmetry or any other symmetry axis higher
than `six’, Identical repetition of an unit can take
place only when we consider 1,2,3,4 and 6 fold
axes.
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15. SYMMETRICAL AXES OF CUBE
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16. SYMMETRICAL ELEMENTS OF CUBE
(a) Centre of symmetry
(b) Planes of symmetry
(Straight planes -3,Diagonal planes -6)
(c) Diad axes
(d) Triad axes
(e) Tetrad axes
----
Total number of symmetry elements = 23
Thus the total number of symmetry elements of a cubic structure is 23.
PH UNIT 4 LECTURE 3

17. ABSENCE OF 5 FOLD SYMMETRY
We have seen earlier that the crystalline solids show only
1,2,3,4 and 6-fold axes of symmetry and not 5-fold axis of
symmetry or symmetry axis higher than 6.
The reason is that, a crystal is a one in which the atoms or
molecules are internally arranged in a very regular and
periodic fashion in a three dimensional pattern, and
identical repetition of an unit cell can take place only
when we consider 1,2,3,4 and 6-fold axes.
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18. MATHEMATICAL VERIFICATION
Let us consider a lattice P Q R S as shown in figure  
P Q R S
a a a
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19. MATHEMATICAL VERIFICATION
Let this lattice has n-fold axis of symmetry and the
lattice parameter be equal to ‘a’.
Let us rotate the vectors Q P and R S through an
angle  = , in the clockwise and anti clockwise
directions respectively.
After rotation the ends of the vectors be at x and y.
Since the lattice PQRS has n-fold axis of symmetry,
the points x and y should be the lattice points.
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20. MATHEMATICAL VERIFICATION
Further the line xy should be parallel to the line PQRS.
Therefore the distance xy must equal to some integral
multiple of the lattice parameter ‘a’ say, m a.
i.e., xy = a + 2a cos  = ma (1)
Here, m = 0, 1, 2, 3,
From equation (1),
2a cos  = m a – a
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21. MATHEMATICAL VERIFICATION
i.e., 2a cos  = a (m - 1)
(or) cos  = (2)
Here,
N = 0, 1, 2, 3, .....
since (m-1) is also an integer, say N.
We can determine the values of  which are allowed in a lattice by solving the equation (2) for all values of N.
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22. MATHEMATICAL VERIFICATION
For example, if N = 0, cos  = 0 i.e.,  = 90o
 n = 4.
In a similar way, we can get four more rotation axes
in a lattice, i.e., n = 1, n = 2, n = 3, and n = 6.
Since the allowed values of cos  have the limits –1
to +1, the solutions of the equation (2) are not
possible for N > 2.
Therefore only 1, 2, 3, 4 and 6 fold symmetry axes
can exist in a lattice.
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23. ROTATION AXES ALLOWED IN A LATTICE
cos 
(degrees) -2 -1
180 2
-1/2
120 3 90 4 +1
+1/2 60 6 +2
360 (or) 0 1 n=
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24. ROTO INVERSION AXES
Rotation inversion axis is a symmetry element which
has a compound operation of a proper rotation and
an inversion.
A crystal structure is said to possess a rotation –
inversion axis if it is brought into self coincidence by
rotation followed by an inversion about a lattice point
through which the rotation axis passes.
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25. ROTO INVERSION AXES X1 X 1 2 3 4
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26. ROTO INVERSION AXES
Let us consider an axis xx, normal to the circle passing
through the centre.
Let it operates on a point (1) to rotate it through 90o to the
position (4) followed by inversion to the position (2), this
compound operation is then repeated until the original
position is again reached.
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27. ROTO INVERSION AXES
Thus, from position (2), the point is rotated a further 90o
and inverted to the position (3); from position (3), the point
is rotated a further 90o and inverted to a position (4); from
position (4), the point is rotated a further 90o and inverted
to resume position (1).
Thus if we do this compound operation about a point four
times, it will get the original position. This is an example
for 4-fold roto inversion axis. Crystals possess 1,2,3,4
and 6-fold rotation inversion axes.
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28. TRANSLATIONAL SYMMETRY
SCREW AXES
This symmetry element has a compound operation of
a proper rotation with a translation parallel to the
rotation axis
This is shown in the figure.In this operation, a rotation
takes place from A to B by an amount of  and it
combines with a translation from B to C by an amount
of T, which is equivalent to a screw motion from A to C.
The symmetry element that corresponds to such a
motion is called a screw axis.
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29. TRANSLATIONAL SYMMETRY
SCREW AXES A B C T θ
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30. TRANSLATIONAL SYMMETRY
GLIDE PLANE
This symmetry element also has a compound
operation of a reflection with a translation parallel
to the reflection plane.
Figure shows the operation of a glide plane
If the upper layer of atoms is moved through a
distance of a/2, and then reflected in the plane
mm1, the lower plane of atoms is generated.
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31. TRANSLATIONAL SYMMETRY
GLIDE PLANE m m1 a
a / 2
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32. COMBINATION OF SYMMETRY ELEMENTS
Apart from the different symmetry elements different
combinations of the basic symmetry elements are also
possible.
They give rise to different symmetry points in the
crystal.
The combination of symmetry elements at a point is
called a `point group’.
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33. COMBINATION OF SYMMETRY ELEMENTS
In crystals, 32 point groups are possible.
The combination of 32 point groups with 14
Bravais lattices lead to 230 unique
arrangements of points in space.
They are called as `space groups’.
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34. Physics is hopefully simple but Physicists are not
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