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welcome you all to the second lecture on defects
in material the first lecture we had covered
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about we had covered about one dimensional
lattice in the second ah in this present lecture
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we will talks little bit about the two dimensional
ah lattice the first thing which you have
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to look at it is how can we construct a two
dimensional lattice like we have ah like crystal
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we have considered as some motifs which is
arranged in a periodic fashion if you repeat
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this motifs in the second direction because
most of the motifs has been shown as periodicity
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in the y direction in the x direction if we
keep at some particular distance and also
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as a particular angle if we keep them then
various types of ah two dimensional lattices
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could be generated ok i had just shown some
two examples of it and in the second of some
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lattice points associated with the motifs
also what are the different types of ah if
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we consider only the lattice points ok we
generator one dimensional lattice what all
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the different types of lattices which we can
generate let us look at it ok
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here what is being done is that we have the
one dimensional lattice which is their ok
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it is ah the one dimensional lattice is kept
at regular intervals in the x direction but
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inclined at an angle with respect to the x
axis when this is being done we are able to
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generate if we look at is where type of a
parallelogram which repeats itself ok these
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parallelogram if you look at it ok ah there
are various ways in which this ah lattice
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itself could be represented one once we fix
our coordinates of the lattice ok like fixing
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the coordinate at this particular position
ok once the coordinate has been fixed at this
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particular point and at the origin of the
coordinates is fixed ok and this is the i
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x coordinate are the a coordinate and the
b a coordinate ok using the vector notation
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ok we can write it as r is equal to n one
into e a plus n two into b r as i mentioned
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if you use e v w to represent the number of
times they repeat in the x and y direction
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we can write it in the vector notation as
r is equal to u into a plus v into b ok
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and another way in which it can be represented
is that the area which is being covered by
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this unit cell if you look at it this can
be kept adjacent to each other and this way
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also it repeats itself ok and another important
aspect which you used to see that this unit
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cell how many lattice points are present per
unit cell if you look at its only one ok and
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is it the only way the unit cell can be constructed
no if you look at it here this is another
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type of a unit cell which repeats itself this
is an another way in which a unit cell is
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generated which repeats itself so the unit
cell is not unique we can have any type of
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a unit cell but what is important about all
this unit cell is that area of the unit cell
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remains the same and only one lattice point
per unit cell is being present so this type
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of lattices are called as primitive lattices
ok what we do to generate a primitive lattice
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is used the shortest translation vectors in
ah two directions and the directions which
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are not parallel to each other and similarly
we can generate non primitive lattices by
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generating a lattice of this particular type
and here the number of lattice points which
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are being present is ah essentially ah two
lattice points which are being present here
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if you wanted to represent it with this ah
vector r if you are trying to represent this
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lattice ok you find that all the lattice points
will not be represented using this ah vector
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notation this is the mathematical way of representing
the lattice
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so in the primitive lattice only using this
vector ok all the lattice points can be generated
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using this one single expression ok ah generally
as i mentioned earlier the lattices are infinite
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ok lattice points are so large than the surface
effects can be ah ignored ok ah and most of
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the time you will notice that we choose not
only a primitive lattice in many cases we
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use a non primitive lattices there is a reason
for that the reason essentially is that we
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choose the lattice which shows the full symmetry
of the crystal structure under consideration
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ok this is another type of a lattice where
if you look at it the angle between a and
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b the ah to access is ninety degree but a
and b are not equal to each other this is
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a rectangular lattice similarly we can consider
an another lattice where we find that there
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is an one more lattice point in between so
this is called as a centered lattice here
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again the a and b access the lengths are not
equal but that angle between them equals ninety
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degree then we have a ah another type of a
periodic cell where a and b the length of
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a and b are equal and the angle between them
is also ninety degree this is then thats a
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square lattice and then angle between a and
b when it is ah equal to one twenty degree
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and a equals b then be generate essentially
like hexagonal lattice ok
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so essentially if you look at it what are
the types of lattices which we have for can
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be generated by keeping ah one dimensional
lattice ok adjacent to each other maintaining
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periodicity in two directions ok maintaining
a periodicity along the x as well as the ah
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y axis we have only five types of ah lattice
planar lattices are possible these five planar
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lattices of this nothing but in geometry we
have studied essentially what we can have
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is ah a parallelogram a rectangle a rhombus
a square and the hexagon ok these are all
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the basic five planar lattices which possible
in these planar lattices ok if you try to
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put an atom are the motifs around each of
the lattice points then we can generate two
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dimensional crystal structures ok you are
seeing it here assume that one atom has been
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put around and put on top of each of the lattice
points now we are able to generate a crystal
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with one atom per unit cell ok
you look ah here what we have done it is that
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at the center of this each of this unit cell
and another type of atom has been placed here
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ok so if you look and this one we have two
types of atoms which are being present ok
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but what is the motifs which is getting generated
if you look at it these two together can be
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considered as a motifs ok and a lattice point
can be generated anywhere that mine gets repeated
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itself so the unit cell dimensions if you
look at it they remain that same so in this
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particular unit cell we have two different
types of atoms per unit cell is there but
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still if you look at the lattice this is again
your primitive lattice this is an another
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example where you can see that the same type
of atom is being present but in this particular
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case we have ah essentially two atoms per
lattice point but this again is a primitive
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lattice ok area under the unit cell can be
calculated by finding out the magnitude of
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the axis times the sin of angle between the
axis
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before we go further into this one as i had
mentioned and given some idea of what the
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symmetry is let me talk about what is the
type of symmetry which ah how symmetry is
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defined mathematically symmetry is defined
as a type of an invariance that is it is invariant
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the property that something does not change
under a set of transformations the transformation
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could be either rotation translation our reflection
whatever be the operation it brings you to
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a position which appears as if it is the same
as the original position ok what are the types
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of symmetry elements which we can have in
two dimensional lattices ok ah one like what
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we mentioned in the case of ah one dimensional
lattice in two dimensional lattice we can
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have a translational symmetry rotational symmetry
reflection symmetry and inversion which is
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nothing but a rotation and a glide we will
go into a detail ah shortly but what we should
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ah remember is that in ah one dimensional
lattice we have only one type of a rotational
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symmetry only a twofold rotation
let us look at what are the different types
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of other symmetries which are possible in
ah two dimensional lattice suppose we assumed
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that ah we wanted to find the symmetry around
any point which is not consistently like this
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is a ah lattice which i am showing it here
ok around this lattice ok if i take any ah
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ah point here and this point if i rotate it
ok by some arbitrary angle this the entire
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lattice will move and come to an another position
which is distinct from the earlier one so
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that means that it is not consistent with
translation ok that type of a symmetry if
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you look at it infinite symmetry which is
possible around a point but if we look for
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a symmetry which is consistent with translation
there are some certain restrictions are there
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these restrictions we can easily find out
how exactly its been this year looking at
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this ah figure in this figure if you look
at it that is ah the lattice vector here the
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lattice point here and the lattice point is
here is rotated by an angle t where the translation
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periodicity is t ok
so this generates a new point one here and
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another new point here if it is consistent
with the translational symmetry ok the distance
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between these two points should be again you
have multiple of the translation vectors that
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is what is it turn us empty so this empty
is nothing but t plus two times t cos theta
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and if you do all the mathematical operations
then we can find out that the number of rotations
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which are possible which are consistent with
translational symmetry is one fold two fold
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three fold four fold and six fold these are
all the ah only type of translation symmetry
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ah rotational symmetry which is possible which
is consistent with translational symmetry
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ok what is the other type of symmetry which
we talked about which of you mentioned earlier
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also is a reflection symmetry
so in a reflection symmetry if you put a mirror
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ok in front of an object the mirror image
will be generated on the other side of it
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so where can we place that mirror if its a
like here it is a one dimensional periodic
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lattice which is being shown the mirror can
be kept either halfway between the lattice
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points or it can be kept on the lattice point
itself these are all the two options which
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we have for keeping the mirror symmetry ok
how can we represent this mirror this is essentially
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basically ah the trying to understand qualitatively
how a mirror symmetry is generated but mathematically
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when we look at it essentially by this operation
of the symmetry there point which has got
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some coordinates x y z it gets repeated and
generates new point ok what should be the
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coordinates of those points that is given
for a mirror reflection if we consider around
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the x ah where the mirror plane is lying on
the x axis then it will be x y zero if the
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coordinate of this motifs then it gets repeated
to x minus y zero y had chosen zero because
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that ah in a two dimensional lattice its only
the x y plane so the value of e z becomes
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zero otherwise if you consider three dimension
this will become x y e z turning to x minus
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y e z this can itself can be also represented
in the form of a matrix form where x is a
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column vector x y is a and x dash is the ah
new vector column vector no x ok and a is
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the transformation matrix which transforms
from one coordinate system to the another
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coordinate system
this way mathematically we can find out by
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an our symmetry operation using the specific
transformation breaking ah matrix which is
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associated with it that all the the coordinates
of all other positions could be generated
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ok similarly for if it is an inversion symmetry
which is equal to a twofold rotation if he
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considered here its a x y turns to minus x
minus y ok and for a three dimensional one
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its x y e z will turn to minus x minus y minus
z like i mentioned for a reflection symmetry
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we can write a similar expression ah transformation
ah expression for transforming from one coordinate
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that is how the coordinates of that other
lattice points could be generated using the
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ah transformation matrix associated with the
inversion operation ah in addition to these
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symmetry elements another symmetry element
which comes like which mentioned in the case
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of one dimensional crystal is a glide symmetry
ok
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the glide symmetry essentially as i mentioned
earlier is that ah like for this the motifs
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are is shifted by a distance some particular
distance are t by two and the then it is reflected
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and again you shifted by another t by two
again you reflect it then one can generate
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your particular type of a ah pattern ok this
is called as a glide ok this is something
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like when we walk ok on this ah beach sand
our ah body itself is across the body thats
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a mirror symmetry associated with it so when
we walk that our right leg and the left leg
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as we move the periodicity is from right leg
how much the first position to the next position
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which it takes ok but half the distance is
the one where that left leg comes so this
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is a perfect example of a glide symmetry so
as i mentioned ah the glide symmetry this
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can be represented by this sort of a transformation
if it is a coordinates are x y z these changes
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to minus x ah y plus half to z and the this
ah coordinate transformation takes place only
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for glide when it is taking plus on ah why
there glide is present in the y direction
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ok
what are the types of symmetry elements for
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two dimensional lattice which we are seeing
one rotation consistent with translation reflection
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consistent with translation and reflection
plus translation glide ok these are all the
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type of point group symmetries which are present
for a two dimensional lattice ok essentially
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the same thing is being ah ah defined in this
way that if we forget the translational symmetry
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then the remaining symmetry elements rotation
reflection and the center of inversion consistent
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with translational symmetry can be arranged
in to distinct groups and each group is called
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as a point group ok what are the rotational
symmetry elements which are ah possible one
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two three four and six fold rotation and in
addition to it reflection is the one ah which
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is possible
so we have essentially ah five plus one six
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types of ah ah point groups are six types
of symmetry operations are present we can
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choose various combinations of these ah symmetry
operations ok we can choose and ah you can
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calculate how many combinations which are
possible out of which we will find that only
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ten point groups which are present which are
consistent with the translation ok why this
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study of point group is very important because
whenever we look at a property of a material
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any sample it is the property which can vary
for example thermal expansion electrical resistivity
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elastic constants optical property ok if the
crystal structure external a symmetry of the
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crystals when we look at it all these properties
what essentially is that we can visualize
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it by macroscopic experiments and look at
it and decide how the properties are changing
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in various directions ok like electrical conductivity
we can measure it at different directions
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and find out how ah in which all directions
they are identical which are directions are
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different on that basis we can find out what
all the types of symmetry elements which are
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associated with this these symmetry elements
when we look at all of them will pass through
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a point in the case of a ah intersection of
different types of symmetry element ok other
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ways if it is only as ah ah rotation which
is being present only a twofold rotation if
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we consider or fourfold rotation which is
being considered then around an axis ok it
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is invariant ok
so because of this this is called as a point
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group symmetry ok the rotation reflection
and inversion these operations are called
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macroscopic symmetry because by simple experiments
macroscopic experiments like measuring properties
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ok we can identify these symmetry elements
ok
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now coming back to what all the types of ten
i mentioned that there can be ah a point group
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symmetries associated in two dimensional lattice
what all the point group symmetry which can
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have which is consistent with the translation
periodicity in two dimensions when so here
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what i have done it is i have shown taken
an asymmetric motifs ok then in an art form
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there is different types of ah motifs which
have for asymmetric sanctity associated with
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it and then chemistry a type of molecules
which exhibit the ah corresponding symmetry
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like if you take a own fold symmetry this
is bromochlorofluoroethine ok this if you
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look at it it has got only own fold symmetry
otherwise way whatever operation which we
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do other operations twofold rotation it will
not be brought back to its identical version
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the same is true for this particular motifs
also for r ok
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if a mirror is being ah present then what
we do is that the the how its getting repeated
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is that r across the mirror there will be
a mirror reflection we can generated this
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is the symbol with which it is ah expressed
and amidst the symbol which is used to represent
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that mirror and if you look at this picture
ok this picture has got around this ah axis
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ok vertical axis we have a mirror symmetry
which is being present so like difluroethene
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if you look at it this as a twofold ah mirror
symmetry which is associated within and if
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you look at this structure ok its not this
is a twofold symmetry which is associated
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with it and trans difluroethene is one which
if you look at it it has a by twofold rotation
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the hydrogen it will come here so we will
not be able to make out whether it is the
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00:21:48,780 --> 00:21:55,010
position which we generate new position is
identical with the ah earlier one know whether
201
00:21:55,010 --> 00:22:01,020
it is ah undergone a transformation or not
will not be able to make out hence we say
202
00:22:01,020 --> 00:22:06,070
that this has got a twofold symmetry associated
with it ok
203
00:22:06,070 --> 00:22:11,419
similarly we have examples for threefold symmetry
how exactly and if you look at the center
204
00:22:11,419 --> 00:22:16,370
we are showing what all the different symbols
mirror is essentially shown with a line thickline
205
00:22:16,370 --> 00:22:22,680
and ah ellipses which is being used to show
a twofold ah equilateral triangle is used
206
00:22:22,680 --> 00:22:29,760
to represent the threefold symmetry ok a square
is used for a fourfold symmetry to represent
207
00:22:29,760 --> 00:22:37,390
it and then hexagon is used for representing
a sixfold symmetry you can go through this
208
00:22:37,390 --> 00:22:45,840
slide and you will be able to make out that
around your point ok these are all the types
209
00:22:45,840 --> 00:22:54,360
of ah symmetry elements which we can have
which is consistent with a lattice translation
210
00:22:54,360 --> 00:23:03,630
ok because of that ok this puts a restriction
and hence we have only ten plane groups are
211
00:23:03,630 --> 00:23:08,540
there i will give you some ah assignments
later you can go through the assignments and
212
00:23:08,540 --> 00:23:13,680
then you will notice it that whatever be the
combination you choose finally turn out to
213
00:23:13,680 --> 00:23:19,250
be only ten distinct ones are possible there
is an another way in which this is represented
214
00:23:19,250 --> 00:23:23,809
this is represented under stereogram ok
now about the how that stereogram is being
215
00:23:23,809 --> 00:23:30,410
constructed ah it will come into the ah next
lecture but i just wanted to mention here
216
00:23:30,410 --> 00:23:35,470
that this is something like an stereographic
sub projection ok of what is happening in
217
00:23:35,470 --> 00:23:45,330
is ah three dimension ok are on the on a sphere
onto a two dimension and the advantage of
218
00:23:45,330 --> 00:23:50,950
a stereographic projection is that all the
three dimensional angular relationships are
219
00:23:50,950 --> 00:24:02,990
being maintained in two dimension in the case
of ah ah in the case of a ah two d lattice
220
00:24:02,990 --> 00:24:09,400
essentially it is essentially a ah projection
two dimensional projection of a two dimensional
221
00:24:09,400 --> 00:24:15,430
ah two dimensional lattice because of it is
much simpler so its essentially a circle which
222
00:24:15,430 --> 00:24:22,010
represents ah the stereographic projection
and the central point all the angular relationship
223
00:24:22,010 --> 00:24:27,929
which are ah present between the different
lattice points can be represented ok all the
224
00:24:27,929 --> 00:24:33,900
symmetry points by points on the circumference
of the ah circle ok
225
00:24:33,900 --> 00:24:42,440
the two ways in which a stereographic projection
is represented in international union of crystallography
226
00:24:42,440 --> 00:24:50,480
one is a general point how by a symmetry operation
its getting represented on the stereogram
227
00:24:50,480 --> 00:24:57,760
and another is what are the ah symmetry elements
which are associated with the point group
228
00:24:57,760 --> 00:25:02,890
representation these are all the two ways
in which is being represented in addition
229
00:25:02,890 --> 00:25:09,870
to it corresponding to the general point what
all the positions which are possible ok the
230
00:25:09,870 --> 00:25:15,429
coordinates of them that is also being represented
that i are not given here but you can go through
231
00:25:15,429 --> 00:25:21,980
the book on international of crystallography
for ah planar groups where you will get all
232
00:25:21,980 --> 00:25:27,080
the information ok for example here if you
look at it the mirror which is to be there
233
00:25:27,080 --> 00:25:33,290
around this plane that is under y axis your
motifs which is being present generally it
234
00:25:33,290 --> 00:25:40,340
is represented by a dot on the circumstances
this is getting reflected and it comes here
235
00:25:40,340 --> 00:25:45,880
and if he what is the mi mirror ah symmetry
element if you want to represent take a circle
236
00:25:45,880 --> 00:25:51,549
and on this ah same axis as the thick line
which is being drawn that of the that mirror
237
00:25:51,549 --> 00:25:57,720
ok if you look at a two m m symmetry here
three is motifs which is there by a twofold
238
00:25:57,720 --> 00:26:05,679
rotation this gets ah ah reflected here ok
we can see the position of the motifs and
239
00:26:05,679 --> 00:26:13,800
now if i put a mirror in this direction ok
this axis this will get reflected new positions
240
00:26:13,800 --> 00:26:19,669
are generated but when this has been generated
we can see that it is equivalent to putting
241
00:26:19,669 --> 00:26:24,060
an another mirror here if you put it then
also these positions will be generated which
242
00:26:24,060 --> 00:26:28,900
are identical to each other so here now we
show the twofold rotation which is there at
243
00:26:28,900 --> 00:26:40,210
the center and that the mirrors are also being
represented similarly um ah for four m m we
244
00:26:40,210 --> 00:26:50,710
can represent it that is what is given here
ok if you look into that the type of point
245
00:26:50,710 --> 00:26:55,830
groups which are present two d point groups
we have pure rotation which are five numbers
246
00:26:55,830 --> 00:27:04,090
are there one two three four and six fold
and then reflection and in mirror is one ok
247
00:27:04,090 --> 00:27:11,820
then the combination of rotation and reflection
if you take it there is two m m three m are
248
00:27:11,820 --> 00:27:20,440
three m m four m m and six m m these are all
the ah so essentially ten distinct our point
249
00:27:20,440 --> 00:27:28,679
groups are possible ok ah
so far we have talked about only point groups
250
00:27:28,679 --> 00:27:34,871
ok these point groups are generated around
a lattice point ok if you put this point group
251
00:27:34,871 --> 00:27:40,980
around each lattice point the five lattices
which we we said that we can have in ah two
252
00:27:40,980 --> 00:27:49,490
dimensional lattice then what are the type
of ah ah ah planar groups which we can generate
253
00:27:49,490 --> 00:27:57,210
that is what we will look at it ok here it
is an ah oblique that is only one unit cell
254
00:27:57,210 --> 00:28:04,830
which is being shown ok here since it has
got only a twofold rotation ok that is around
255
00:28:04,830 --> 00:28:11,340
this point which is ah the origin of the coordinate
system if you take it by a twofold rotation
256
00:28:11,340 --> 00:28:18,340
this point itself will rotate and come somewhere
here ok then this point will rotate it and
257
00:28:18,340 --> 00:28:23,539
come here so that way this lattice will be
generated so essentially around each lattice
258
00:28:23,539 --> 00:28:30,990
point we have a twofold axis which is being
present but once this ah twofold axis if we
259
00:28:30,990 --> 00:28:36,789
look at it so now we can notice that around
this point there is another twofold axis at
260
00:28:36,789 --> 00:28:45,250
the center also another twofold axis is generated
and at the center of the x axis also did another
261
00:28:45,250 --> 00:28:54,070
twofold axis is generator
so by having a ah planar lattice ok where
262
00:28:54,070 --> 00:29:01,500
we have a ah twofold rotation around the lattice
point we notice that there are other positions
263
00:29:01,500 --> 00:29:08,679
where either the same symmetry element or
some other symmetry element could be generated
264
00:29:08,679 --> 00:29:14,770
like in ah this particular case of a rectangular
lattice if we consider here if we notice it
265
00:29:14,770 --> 00:29:25,200
we can see that only twofold ah ah rotation
has been ah twofold axis is present on all
266
00:29:25,200 --> 00:29:31,850
the lattice points now if we look at it ah
the presence of them has generated another
267
00:29:31,850 --> 00:29:42,220
three more ah positions where ah twofold rotation
axis is present in addition to it along that
268
00:29:42,220 --> 00:29:47,539
ah lying on the x axis as well as the y axis
we have mirror planes which are being present
269
00:29:47,539 --> 00:29:53,970
similarly at halfway between gender is in
this direction as well as in this direction
270
00:29:53,970 --> 00:30:03,179
we have again mirrors are being present
if you look very carefully ok by ah putting
271
00:30:03,179 --> 00:30:10,640
ah motifs are looking at the symmetry around
each of the lattice points we can see that
272
00:30:10,640 --> 00:30:16,909
at other points also different types of symmetries
are generated here we have a ah rotation and
273
00:30:16,909 --> 00:30:26,300
a mirror and if you look at the square lattice
ok here we can see that these are all the
274
00:30:26,300 --> 00:30:30,299
atom points around each of this point we will
have a fourfold rotation which is being present
275
00:30:30,299 --> 00:30:35,370
so when we put a fourfold rotation around
this point then we can see that at the center
276
00:30:35,370 --> 00:30:42,240
also there is a fourfold rotation is present
and if you look at the ah midway bit in the
277
00:30:42,240 --> 00:30:48,440
x axis between the lattice translation vector
we have a twofold rotation which is present
278
00:30:48,440 --> 00:30:55,110
similarly here if we look at it another twofold
rotation ok and by lattice translation vectors
279
00:30:55,110 --> 00:31:01,510
we can generate the other two fold rotations
so essentially for this this the symmetry
280
00:31:01,510 --> 00:31:11,210
which it has got is a fourfold rotation then
in addition to it along the x and y axis mirror
281
00:31:11,210 --> 00:31:20,799
symmetry is there then at passing through
this ah center point we have again mirror
282
00:31:20,799 --> 00:31:27,980
symmetry is there then along the body diagonals
we have mirror symmetry is present ok so essentially
283
00:31:27,980 --> 00:31:34,730
what we have is that the ah the actual symmetry
representation which we call it as four m
284
00:31:34,730 --> 00:31:44,970
m but if you look at it there are the fourfold
rotation twofold rotation then symmetry along
285
00:31:44,970 --> 00:31:52,850
x and y axis ok symmetry shifted from that
by some distance ok and then symmetry passing
286
00:31:52,850 --> 00:31:57,299
through the body diagonal not symmetry the
mirror passing through the so three types
287
00:31:57,299 --> 00:32:12,490
of mirror symmetries two types of a ah fourfold
rotation and ah one type of a twofold rotation
288
00:32:12,490 --> 00:32:25,230
is possible ok
how are these represented in ah international
289
00:32:25,230 --> 00:32:32,190
inverse cryptographic table thats what is
shown in this slide if you look ah when we
290
00:32:32,190 --> 00:32:40,929
try to represent a planar lattice ok the lattice
can be represented by a unit cell ok these
291
00:32:40,929 --> 00:32:46,270
what essentially is being shown ok is unit
cell which is being shown the angle between
292
00:32:46,270 --> 00:32:57,290
the unit cell is not equal to ninety degree
ok these two sides do not have the ah ah they
293
00:32:57,290 --> 00:33:05,429
are not equal ok if i put take a motifs generally
the motifs is represented by an open circle
294
00:33:05,429 --> 00:33:11,500
if i keep this motifs at a point which does
not have any symmetry element associated within
295
00:33:11,500 --> 00:33:17,429
ok its an one fold symmetry then what is going
to happen is that only by own fold rotation
296
00:33:17,429 --> 00:33:22,630
this will repeat itself so essentially we
wont fold what are the coordinates of these
297
00:33:22,630 --> 00:33:32,600
points these coordinates of the points could
be x y ok and in the ah international union
298
00:33:32,600 --> 00:33:37,960
of crystallography table there are two things
which are important as far as when we wanted
299
00:33:37,960 --> 00:33:47,409
to generate ah the crystal structures ok two
dimensional structures essentially what we
300
00:33:47,409 --> 00:33:53,290
should know is that what all the special and
general positions for the motifs which have
301
00:33:53,290 --> 00:34:00,710
to be placed ok how are they graphically represented
ok these are all the two important information
302
00:34:00,710 --> 00:34:09,049
which we require
let us look at a a case with a the plane group
303
00:34:09,049 --> 00:34:14,679
which has a ah fourfold symmetry associated
with it only a fourfold symmetry is present
304
00:34:14,679 --> 00:34:24,520
ok then this is how the graphical representation
of the symmetry is given all the lattice points
305
00:34:24,520 --> 00:34:29,450
we have symmetry element are the fourfold
symmetry is there at the center of the lattice
306
00:34:29,450 --> 00:34:36,310
again another fourfold symmetry then at the
edges at the midway there is a twofold symmetry
307
00:34:36,310 --> 00:34:45,139
is percent if we take a motifs and put it
around any point in this random point ok at
308
00:34:45,139 --> 00:34:50,190
this point if we try to look at it what is
the type of symmetry which it will have if
309
00:34:50,190 --> 00:34:58,440
i do around this point you rotate this lattice
this lattice will rotate and come back to
310
00:34:58,440 --> 00:35:06,620
its original position only for one fold rotation
for all other one they will generate a position
311
00:35:06,620 --> 00:35:13,510
which is not identical with the original position
so this is called as a general point or the
312
00:35:13,510 --> 00:35:19,070
symmetry associated with this one is only
one fold rotation we have put a motifs around
313
00:35:19,070 --> 00:35:26,000
that position and when we try do the symmetry
operation ok this has to be repeated itself
314
00:35:26,000 --> 00:35:30,920
by a ninety degree we rotate it we generate
an another position then by another ninety
315
00:35:30,920 --> 00:35:36,970
degree then by another ninety degree so the
full symmetry operation is completed now we
316
00:35:36,970 --> 00:35:49,550
can see that around each that is point ok
if you put our motifs at a position ok which
317
00:35:49,550 --> 00:35:58,460
is not a symmetry ah any symmetry or special
position then four points have to be generated
318
00:35:58,460 --> 00:36:04,839
so which show ah only then the crystal will
exhibit the fourfold symmetry
319
00:36:04,839 --> 00:36:09,060
now we can see that around each lattice point
we have a fourfold symmetry if you look with
320
00:36:09,060 --> 00:36:14,700
respect to this center again we will see that
there are four points which are associated
321
00:36:14,700 --> 00:36:23,230
with it ok so from this what we can make out
is that if the point which is occupied by
322
00:36:23,230 --> 00:36:31,140
the motifs has a own fold symmetry then its
multiplicity is four four more points have
323
00:36:31,140 --> 00:36:41,150
to be generated their coordinates are given
in this fashion ok suppose i assume that this
324
00:36:41,150 --> 00:36:48,349
motifs i am trying to put it around this particular
point ok then by lattice translation i can
325
00:36:48,349 --> 00:36:56,560
generate ah the unit cell ok then the motifs
will come at this point this point and this
326
00:36:56,560 --> 00:37:06,450
point and this is being shown here ok so now
we can see that this is the coordinate of
327
00:37:06,450 --> 00:37:14,410
the origin so by putting a motifs around these
points now we are able to generate ok so what
328
00:37:14,410 --> 00:37:24,380
is the ah symmetry of this point it has a
fourfold symmetry so since it has got ah you
329
00:37:24,380 --> 00:37:31,119
know fourfold symmetry ok only one lattice
point ah motifs has to be placed around each
330
00:37:31,119 --> 00:37:36,500
lattice point ok and if you look at the number
of lattice points per unit cell it is only
331
00:37:36,500 --> 00:37:41,270
one which is required and that it satisfies
that condition ok
332
00:37:41,270 --> 00:37:48,070
so in the wyckoff table if you look at it
this is how it is being represented the symmetry
333
00:37:48,070 --> 00:37:53,270
around the lattices fine this called as a
special position where it has got a fourfold
334
00:37:53,270 --> 00:37:59,160
symmetry there the multiplicity one means
that we have to keep only one motifs at that
335
00:37:59,160 --> 00:38:08,200
point ok if you look at this position this
position essentially yes got a half of zero
336
00:38:08,200 --> 00:38:17,839
is there ah the coordinates are in two dimensional
lattice it will only half half half here again
337
00:38:17,839 --> 00:38:23,990
ok since it as a fourfold symmetry associated
with it we have to put only one motifs that
338
00:38:23,990 --> 00:38:31,290
is what essentially is represented as the
multiplicity here ok if you look at this point
339
00:38:31,290 --> 00:38:38,490
this particular point has got a twofold symmetry
associated with it ok if you put a motifs
340
00:38:38,490 --> 00:38:45,890
around this point the coordinate of this will
ah motifs is going to be zero half ok that
341
00:38:45,890 --> 00:38:50,770
what essentially is given here and by a lattice
translation we can generate the another which
342
00:38:50,770 --> 00:38:58,710
is going to be present on the unit cell and
then along this axis is another there is at
343
00:38:58,710 --> 00:39:06,339
opposition half zero your motifs has to be
kept so by lattice translation vector we can
344
00:39:06,339 --> 00:39:15,050
generate ah the other position of this position
of this ah motifs in the unit cell
345
00:39:15,050 --> 00:39:27,270
now if you look at it that number of motifs
ok which are a number of motifs which correspond
346
00:39:27,270 --> 00:39:35,280
only to the unit cell is two ok and the way
in which these motifs are being placed that
347
00:39:35,280 --> 00:39:42,339
satisfies the fourfold symmetry ok that is
essentially what is being given here is the
348
00:39:42,339 --> 00:39:51,740
symmetry around this point is a twofold symmetry
ok and what are the coordinates of those points
349
00:39:51,740 --> 00:39:58,089
that is being given ok and then these are
all that this is the total two points are
350
00:39:58,089 --> 00:40:03,250
supposed to be there thats what is being shown
here and general point as i had mentioned
351
00:40:03,250 --> 00:40:11,740
earlier so around each ah lattice point if
you are placing a motifs at a point which
352
00:40:11,740 --> 00:40:15,900
doesnt have any symmetry associated with it
so that is called as a general point then
353
00:40:15,900 --> 00:40:22,690
we have we should have four points which be
generated for each of the motifs ok this way
354
00:40:22,690 --> 00:40:31,060
if you look at it essentially in the cryptographic
table they had given the on the unit cell
355
00:40:31,060 --> 00:40:40,800
what are the types of a ah symmetry elements
which are present ok if you are trying to
356
00:40:40,800 --> 00:40:48,950
place a motifs around ah motifs at different
points ok how many motifs have to be placed
357
00:40:48,950 --> 00:40:56,800
ok so that the symmetry the full symmetry
of the lattice is ah the crystal is satisfied
358
00:40:56,800 --> 00:41:05,220
ok that information is given so if we have
this information this can be used to in fact
359
00:41:05,220 --> 00:41:15,900
ah generate ah two dimensional ah crystals
this is an another example which is being
360
00:41:15,900 --> 00:41:26,920
taken where it is ah we have fourfold ah symmetry
along with it ah mirror symmetry is also there
361
00:41:26,920 --> 00:41:33,820
that is essentially p four m m ok here if
we look at it we have ah one general point
362
00:41:33,820 --> 00:41:40,650
and then other symmetry positions are one
two three four five six special positions
363
00:41:40,650 --> 00:41:50,780
which are possible ok so totally essentially
if you look at it ah the
364
00:41:50,780 --> 00:41:56,940
motifs could be placed at any of these particular
positions and then we can generate the various
365
00:41:56,940 --> 00:42:05,461
ah ah types of ah the ah we could generate
atoms occupying different positions in the
366
00:42:05,461 --> 00:42:13,950
unit cell but having that same type of a ah
symmetry element associated with it ok and
367
00:42:13,950 --> 00:42:17,500
only thing which you should notice is that
here when there is a mirror reflection is
368
00:42:17,500 --> 00:42:23,609
being present the symbol which is being used
is within a circle like comma is being inserted
369
00:42:23,609 --> 00:42:28,829
ok what are the symbols which are used to
represent symmetry elements in two d lattices
370
00:42:28,829 --> 00:42:36,320
one twofold rotation and threefold fourfold
mirror with a line glide with a dashed line
371
00:42:36,320 --> 00:42:41,800
and a symmetric motifs is represent with an
open circle and mirror image of a symmetric
372
00:42:41,800 --> 00:42:45,619
motifs is represented with a circle with a
comma in say
373
00:42:45,619 --> 00:42:53,730
now what i have shown is all the seventeen
ah plane ah groups in two dimension ok how
374
00:42:53,730 --> 00:43:02,050
are they represented ok we show one of the
the that we show the unit cell and then how
375
00:43:02,050 --> 00:43:10,120
if we take a motif and put it in a general
point ok how the motifs are generated ok that
376
00:43:10,120 --> 00:43:15,420
is the only structure which is being shown
so this is how ah ah oblique motifs looks
377
00:43:15,420 --> 00:43:22,070
like we put one of the symmetric motif here
so this is how it will appear when it has
378
00:43:22,070 --> 00:43:27,849
got only one fold symmetry when the same structure
there is a twofold symmetry is associated
379
00:43:27,849 --> 00:43:31,730
with it corresponding to this there will be
an another point which has to be done which
380
00:43:31,730 --> 00:43:37,180
is ah one eighty degree rotation around each
of the lattice point this is how it is generated
381
00:43:37,180 --> 00:43:43,609
if we look at it it becomes very clear that
is now if we look at it though the motifs
382
00:43:43,609 --> 00:43:49,920
are placed around each of the lattice point
now symmetry elements are generated here here
383
00:43:49,920 --> 00:43:58,010
as well as at this point correct and similarly
in this particular case where we have considered
384
00:43:58,010 --> 00:44:02,040
a mirror which is associated with it here
the symbol which is being used that the this
385
00:44:02,040 --> 00:44:10,660
is the motif ok a symmetric motif and that
symbol essentially is that ah with the open
386
00:44:10,660 --> 00:44:16,680
circle in which we put a comma that shows
that it is a reflected image this way all
387
00:44:16,680 --> 00:44:22,900
i will not go into a detail and explain all
of them but you can go through it and find
388
00:44:22,900 --> 00:44:29,450
out the all the different types of planar
lattices how they are represented here its
389
00:44:29,450 --> 00:44:33,210
only a graphical representation which has
been shown ok
390
00:44:33,210 --> 00:44:40,060
in addition to this graphical representation
ok there is something else also is there the
391
00:44:40,060 --> 00:44:45,130
coordinates of the various positions are also
shown how these coordinates of the various
392
00:44:45,130 --> 00:44:52,800
positions are shown that i will tell you if
we look at the square lattice p four ok we
393
00:44:52,800 --> 00:44:58,890
can put a motif here at this particular general
point which has been placed which is what
394
00:44:58,890 --> 00:45:11,329
is shown in that we can put a motif at the
lattice point where fourfold symmetry is there
395
00:45:11,329 --> 00:45:18,940
we can put a motifs at the center where fourfold
symmetry is there we can put a motifs at these
396
00:45:18,940 --> 00:45:23,970
points also where twofold symmetry is there
then also we can generate a lattice so there
397
00:45:23,970 --> 00:45:28,550
are many possibilities are there all of them
are finally going to generate a lattices though
398
00:45:28,550 --> 00:45:36,180
these positions are distinct were motifs are
being placed but all these structures will
399
00:45:36,180 --> 00:45:44,130
show the same p four symmetry correct this
is what is being given by the wyckoff position
400
00:45:44,130 --> 00:45:50,410
what we call it the positions where they can
be present coordinates of the wyckoff position
401
00:45:50,410 --> 00:45:55,990
the site symmetry which are associated with
it that table gives all this information ok
402
00:45:55,990 --> 00:46:04,490
like ah twelve planar groups can be generated
by keeping motifs representing different two
403
00:46:04,490 --> 00:46:10,710
d group symmetry around each lattice point
that is one and then the glide symmetry is
404
00:46:10,710 --> 00:46:16,690
generated when motifs are placed in a special
way ok in the planar lattice there are four
405
00:46:16,690 --> 00:46:22,060
additional planar groups are generated then
the three m group symmetry there are two orientations
406
00:46:22,060 --> 00:46:26,160
are possible
so they give rise to one additional planar
407
00:46:26,160 --> 00:46:40,220
group so totally we have seventeen these seventeen
are represented ah so here what i have shown
408
00:46:40,220 --> 00:46:49,079
is just only the graphical representation
of ah both the planar lattice and the general
409
00:46:49,079 --> 00:46:57,650
point ok the other information which we require
are essentially the ah site symmetry ok wyckoff
410
00:46:57,650 --> 00:47:06,390
position and the multiplicity associated with
in the coordinates of the motifs that i think
411
00:47:06,390 --> 00:47:12,830
one can look into the international of crystallography
and then that all this ah information and
412
00:47:12,830 --> 00:47:17,570
in this particular view graphs what i have
done it is ah whatever what are the types
413
00:47:17,570 --> 00:47:26,500
of symmetry elements that is if you mentioned
as p one ok this is how ah the different ah
414
00:47:26,500 --> 00:47:31,050
that is ah this one space group are the planar
space group
415
00:47:31,050 --> 00:47:35,270
so that means that there is only one fold
symmetry is associated with it what all what
416
00:47:35,270 --> 00:47:40,640
all the symmetry elements which are associated
with this p one is given here p one one one
417
00:47:40,640 --> 00:47:49,359
and p two is a twofold rotation along z axis
ok and and one fold on the other two directions
418
00:47:49,359 --> 00:47:56,170
similarly for various like for p two m m there
is a twofold rotation along the z axis and
419
00:47:56,170 --> 00:48:01,710
the mirror plane perpendicular to y and x
axis so like this i had just given explanation
420
00:48:01,710 --> 00:48:07,010
for what all the various types of symmetry
elements present which are present the same
421
00:48:07,010 --> 00:48:14,510
symmetry elements if you try to put a motifs
around each of their points the seventeen
422
00:48:14,510 --> 00:48:21,140
plane groups which are generated that is being
shown here ok especially here one should look
423
00:48:21,140 --> 00:48:27,440
at it that with respect to that coordinate
axis which is being chosen x and y in this
424
00:48:27,440 --> 00:48:35,819
particular case three m symmetry ok
the threefold axis is always ah ah normal
425
00:48:35,819 --> 00:48:46,790
to the plane of the ah page ok that is along
the z axis and the x axis is in this direction
426
00:48:46,790 --> 00:48:54,040
so the mirror is along the x axis ah this
particular axis ok and then if you look perpendicular
427
00:48:54,040 --> 00:48:59,500
to each we have only a one fold rotation which
is being present in this particular structure
428
00:48:59,500 --> 00:49:06,490
if we look at it in this particular that is
threefold axis remains that same along the
429
00:49:06,490 --> 00:49:15,250
z axis but along the x axis we have only a
one fold rotation and perpendicular to that
430
00:49:15,250 --> 00:49:20,620
the mirror is there so these two if we look
with respect to a coordinate system which
431
00:49:20,620 --> 00:49:25,530
has been chosen they are two distinct one
so they are given as two different plane groups
432
00:49:25,530 --> 00:49:34,130
ok this is a tabular form in which the various
types of ah two dimensional lattices point
433
00:49:34,130 --> 00:49:41,020
group space group ah ah there is the full
as well as the short symbols for the point
434
00:49:41,020 --> 00:49:45,450
group underplaying groups and what is the
space group symbol which is being given for
435
00:49:45,450 --> 00:49:50,470
them and how this is being represented in
that international of crystallography table
436
00:49:50,470 --> 00:49:57,030
is given in this with this i had given some
brief idea about the different types of two
437
00:49:57,030 --> 00:50:01,710
dimensional lattices now we will go into a
three dimensional lattices in the next class