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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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MAT+EMAT&CAL ER&CAT&"!
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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MAT+EMAT&CAL ER&CAT&"!
#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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MAT+EMAT&CAL ER&CAT&"!
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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MAT+EMAT&CAL ER&CAT&"!
#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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TRA!SLAT&"!AL SYMMETRY
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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TRA!SLAT&"!AL SYMMETRY
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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