PH 0101 UNIT 4 LECTURE 31 CRYSTAL SYMMETRY CENTRE OF SYMMETRY PLANE OF SYMMETRY AXES OF SYMMETRY...

PH 0101 UNIT 4 LECTURE 3 1

PH0101 UNIT 4 LECTURE 3CRYSTAL SYMMETRY

CENTRE OF SYMMETRY

PLANE OF SYMMETRY

AXES OF SYMMETRY

ABSENCE OF 5 FOLD SYMMETRY

ROTOINVERSION AXES

SCREW AXES

GLIDE PLANE


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.


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.


CRYSTAL SYMMETRY

The seven crystal systems are characterised by three symmetry elements. They are

Centre of symmetry

Planes of symmetry

Axes of symmetry.


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.


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.


CENTRE OF SYMMETRY


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


PLANE 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.

AXIS OF SYMMETRY

n

360


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.


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.


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.


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.


SYMMETRICAL AXES OF CUBE


SYMMETRICAL ELEMENTS OF CUBE

(a) Centre of symmetry 1(b) Planes of symmetry 9 (Straight planes -3,Diagonal planes -6)(c) Diad axes 6(d) Triad axes 4(e) Tetrad axes 3

----Total number of symmetry elements = 23

----Thus the total number of symmetry elements of a cubic structure is 23.


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.


MATHEMATICAL VERIFICATION

Let us consider a lattice P Q R S as shown in figure

P Q R S

a a a



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.

n

o360



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



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.

22

1 Nm



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.


N N/2 cos (degrees)

-2 -1 -1 180 2

-1 -1/2 -1/2 120 3

0 0 0 90 4

+1 +1/2 +1/2 60 6

+2 +1 +1 360 (or) 0 1

0360n=

ROTATION AXES ALLOWED IN A LATTICE


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.


ROTO INVERSION AXES

X1

X

1

2

3

4


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.


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.


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.



SCREW AXES

AB

C

T

θ



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.



GLIDE PLANE

m m1

a

a / 2


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’.


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’.

PH 0101 UNIT 4 LECTURE 31 CRYSTAL SYMMETRY CENTRE OF SYMMETRY PLANE OF SYMMETRY AXES OF SYMMETRY...

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