u4lecture 3(Crystal Physics)

8/17/2019 u4lecture 3(Crystal Physics)

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PH 0101 UNIT 4 LECTURE 3

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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


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CRYSTAL SYMMETRY Crystals have inherent symmetry.

The definite ordered arrangement of the faces

and edges of a crystal is known as `cry!"#

y$$%!ry’.

It is a powerful tool for the study of the internal

structure of crystals.

Crystals possess different symmetries or

symmetry elements.


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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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CRYSTAL SYMMETRY

The seven crystal systems are characterised by

three symmetry elements. They are

C%&!r% '( y$$%!ry

P#"&% '( y$$%!ry

A)% '( y$$%!ry*


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CE!TRE "# 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.

ince centre lies at equal distances from various

symmetrical positions it is also known as `c%&!r%'( +&,%r+'&’.

It is equivalent to reflection through a point.


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CE!TRE "# SYMMETRY

! Crystal may possess a number of planes or

a"es of symmetry but it can have only one centre

of symmetry.

#or an unit cell of cubic lattice$ the point at the

body centre represents’ the `c%&!r% '(

y$$%!ry’ and it is shown in the figure.


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CE!TRE "# SYMMETRY


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$LA!E "# SYMMETRY

! 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 si"

diagonal planes of symmetry


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$LA!E "# SYMMETRY


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This is an a"is passing through the crystal such that if

the crystal is rotated around it through some angle$

the crystal remains invariant.

The a"is is called `&-('#./ ")+’ if the angle of rotation

is .

If equivalent configuration occurs after rotation of

%&'($ %)'( and *'($ the a"es of rotation are known as

two+fold$ three+fold and four+fold a"es of symmetry

respectively.

A%&S "# SYMMETRY

n

,-'


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A%&S "# SYMMETRY

If equivalent configuration occurs after rotation of %&'($

%)'( and *'($ the a"es of rotation are known as two+

fold$ three+fold and four+fold a"es of symmetry . If a cube is rotated through *'($ about an a"is normal to

one of its faces at its mid point$ it brings the cube into

self coincident position.

ence during one complete rotation about this a"is$ i.e.$through ,-'($ at four positions the cube is coincident

with its original position.uch an a"is is called four+fold

a"es of symmetry or !%!r". ")+.


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A%&S "# SYMMETRY

If n/%$ the crystal has to be rotated through an angle /

,-'($ about an a"is to achieve self coincidence. uch an

a"is is called an `+.%&!+!y ")+’. 0ach crystal possesses an

infinite number of such a"es.

If n/)$ the crystal has to be rotated through an angle /

%&'( about an a"is to achieve self coincidence. uch an

a"is is called a `diad a"is’.ince there are %) such edges ina cube$ the number of diad a"es is si".


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A%&S "# SYMMETRY

If n/,$ the crystal has to be rotated through an

angle / %)'( about an a"is to achieve self

coincidence. uch an a"is is called is `!r+".")+’. In a cube$ the a"is passing through a

solid diagonal acts as a triad a"is. ince there

are 1 solid diagonals in a cube$ the number of

triad a"is is four.

If n/1$ for every *'( rotation$ coincidence is

achieved and the a"is is termed `!%!r". ")+’.

It is discussed already that a cube has `three’

tetrad a"es.


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A%&S "# SYMMETRY

If n/-$ the corresponding angle of rotation is

-'( and the a"is of rotation is called a he"ad

a"is. ! cubic crystal does not possess anyhe"ad a"is.

Crystalline solids do not show 2+fold a"is of

symmetry or any other symmetry a"is higherthan `si"’$ Identical repetition of an unit can take

place only when we consider %$)$,$1 and - fold

a"es.


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SYMMETR&CAL A%ES "# C'(E


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SYMMETR&CAL ELEME!TS "# C'(E

3a4 Centre of symmetry %

3b4 5lanes of symmetry *

3traight planes +,$6iagonal planes +-4

3c4 6iad a"es -3d4 Triad a"es 1

3e4 Tetrad a"es ,

++++

Total number of symmetry elements / ), ++++

Thus the total number of symmetry elements of a cubic structure is

),.


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A(SE!CE "# ) #"L* SYMMETRY

7e have seen earlier that the crystalline solids show only

%$)$,$1 and -+fold a"es of symmetry and not 2+fold a"is of

symmetry or symmetry a"is higher than -. 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 and -+fold a"es.


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MAT+EMAT&CAL ER&CAT&"!

8et us consider a lattice 5 9 : as shown in figure

θ θ

P Q R

S

a a a


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8et this lattice has n+fold a"is of symmetry and the

lattice parameter be equal to ;a’.

8et us rotate the vectors 9 5 and : through an

angle θ / $ in the clockwise and anti clockwise

directions respectively.

!fter rotation the ends of the vectors be at " and y.

ince the lattice 59: has n+fold a"is of symmetry$

the points " and y should be the lattice points.

n

o

,-'


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#urther the line "y should be parallel to the line 59:.

Therefore the distance "y must equal to some integral

multiple of the lattice parameter ;a’ say$ m a.

i.e.$ "y / a < )a cos θ / ma 3%4

ere$ m / '$ ±%$ ±)$ ±,$ ..................

#rom equation 3%4$

)a cos θ / m a = a


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i.e., )a cos θ / a 3m + %4

3or4 cos θ / 3)4

ere$

> / '$ ±%$ ±)$ ±,$ .....

since 3m+%4 is also an integer$ say >.7e can determine the values of θ which are allowedin a lattice by solving the equation 3)4 for all values

of >.

))

% N m

=

−


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#or e"ample$ if > / '$ cos θ / ' i.e.$ θ / *'o ∴ n / 1.In a similar way$ we can get four more rotation a"es

in a lattice$ i.e.$ n / %$ n / )$ n / ,$ and n / -.

ince the allowed values of cos θ have the limits =%to ? ).

Therefore only %$ )$ ,$ 1 and - fold symmetry a"es

can e"ist in a lattice.


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! !-.cos

/de0rees1

- -1 -1 10

-1 -12 -12 10 3

0 0 0 0 4

1 12 12 0

1 1 30 6'r7 0 1

θ

',-'n=

R"TAT&"! A%ES ALL"WE* &! A

LATT&CE


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R"T" &!ERS&"! A%ES

:otation inversion a"is is a symmetry element which

has a compound operation of a proper rotation and

an inversion.

! crystal structure is said to possess a rotation =

inversion a"is if it is brought into self coincidence by

rotation followed by an inversion about a lattice point

through which the rotation a"is passes.


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R"T" &!ERS&"! A%ES

%2

%

1

2

3

4


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R"T" &!ERS&"! A%ES

8et us consider an a"is ""′$ normal to the circle passing

through the centre.

8et it operates on a point 3%4 to rotate it through *'o to the

position 314 followed by inversion to the position 3)4$ this

compound operation is then repeated until the original position is again reached.


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R"T" &!ERS&"! A%ES

Thus$ from position 3)4$ the point is rotated a further *'o

and inverted to the position 3,4@ from position 3,4$ the point

is rotated a further *'o

and inverted to a position 314@ from position 314$ the point is rotated a further *'o and inverted

to resume position 3%4.

Thus if we do this compound operation about a point four

times$ it will get the original position. This is an e"ample

for 1+fold roto inversion a"is. Crystals possess %$)$,$1

and -+fold rotation inversion a"es.


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TRA!SLAT&"!AL SYMMETRY

SCREW A%ES

A

C

T

B


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3L&*E $LA!E

This symmetry element also has a compound

operation of a reflection with a translation parallel

to the reflection plane.

#igure shows the operation of a glide plane

If the upper layer of atoms is moved through a

distance of a)$ and then reflected in the plane

mm%$ the lower plane of atoms is generated.


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3L&*E $LA!E

! !1

a

a " 2


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C"M(&!AT&"! "# SYMMETRY

ELEME!TS

!part 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 `8'+&! 9r':8’.


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C"M(&!AT&"! "# SYMMETRY

ELEME!TS

In crystals$ ,) point groups are possible.

The combination of ,) point groups with %1

Aravais lattices lead to ),' unique

arrangements of points in space.

They are called as `8"c% 9r':8’.


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PH 0101 UNIT 4 LEC 34

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u4lecture 3(Crystal Physics)