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welcome you all ah to todays class ah the
last two classes we have covered one dimensional
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lattice and then how to constrict over from
one dimensional lattice two dimensional lattice
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what are the symmetry elements associated
with it what are the types of crystal structures
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which we can have are planar ah ah groups
which we can have today we will talk about
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how to generate two dimensional lattice ah
from a two dimensional lattice how to generate
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a three dimensional lattice ok what are types
of a point groups and space groups which are
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associated with these lattices
ok ok we have to follow the same procedure
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which we have done for generating their two
dimensional lattice ok from one dimension
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that is take a two dimensional lattice keep
it on top of one another at some particular
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angle or some specific positions with respect
to the ah lattice which is kept below ok and
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try to generate a lattice and see how many
types of lattices which can be generated but
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there are many positions we can keep it one
lattice on top of each other but the distinct
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number of space lattices which will generate
are only fourteen ok
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we will take a few examples and illustrate
how this is being done one i will take with
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respect to hexagonal lattice ok and how different
types of lattices could be generated the other
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example which i will take is essentially a
ah ah a parallelogram ok that is oblique lattice
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and from that what are lattices which could
be generated what are types of ah ah how we
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can generate a ah bravais lattice from hexagonal
lattice we will consider what are we done
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here we essentially this if you look at it
it is a hexagonal lattice ok one lattice is
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ah what is where the lattice points are marked
as a ok this is the hexagonal lattice in which
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if you kept an another lattice on top of it
ok the lattice points which it can come is
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that this lattice can be called as the b lattice
which is kept at this particular position
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that is on a hexagonal lattice if i consider
it here this is one particular position where
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the next layer can come or this is an another
position where the next layer can come ok
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if you keep at this particular position b
layer position and the next layer on c layer
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position and the third layer if you keep it
it will be kept on this one a b layer c layer
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then the a layer will come
so this is how the layering sequences that
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is first layer second layer third layer fourth
layer so if we consider it this way we can
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generate a space ah lattice can be generated
ok while depending upon the distance which
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we choose it from here if we consider it this
is ah this distance it is called as ah s here
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ok and then at what height the next layer
is being kept ok depending upon that various
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types of lattices can be generated ok if we
keep the b layer at a distance which is h
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is equal to two s by root six ok then this
stacking sequence will generate an f c c lattice
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ok if we keep the b layer and all the successive
layers at a distance which is h is equal to
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ah like s by root six ok then we can generate
a simple cubic
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if we keep the distance between the layers
ok such that h the height is equal to s into
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s by two into root six then we can generate
your body centered lattice if we choose a
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value of h which is different from any of
this value then we can we generate a trigonal
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lattice now you can understand that using
the same hexagonal lattice at what positions
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that is the stacking sequence is a b c a b
c type of a stacking sequence but the height
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at which these lattices are this each of planar
lattice is being kept with respect to one
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another we can generate either trigonal simple
cubic or body centered or face centered bravais
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analysis can be generated if we keep on top
of an a layer and another a layer ok we generate
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h hexagonal lattice so all the all these lattices
could be generated from just a two dimensional
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hexagonal itself ok
suppose we take a ah lattice which is an oblique
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lattice ok its nothing but a unit cell is
a parallelogram if we keep this parallelogram
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one just on top of the another then what we
generate and keep the distance such that it
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is not equal to neither the translation vector
a nor the translation vector b then what we
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generate essentially is nothing but a monoclinic
lattice if we keep the next layer so that
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this ah ah lattice point is at some position
ok the projection of it at some position which
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corresponds to a random point in the lattice
and the lattice parameter in that direction
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ok is not equal to neither a and b we generate
a triclinic lattice ok and if we keep these
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lattices in such a way that we kept at some
particular one the next lattice but at a position
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which is halfway on the x axis ok just above
it then this is you can see that this ah the
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next layer which is being kept ok and then
the third layer which is being kept is right
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on top of a ah the first layer if you repeat
it like this now what we have generated is
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a face centered monoclinic lattice
so by keeping either at symmetry points or
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at random point we can generate different
type of lattices but finally if you try to
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look at how many lattices which we can generate
there are going to be only fourteen ok these
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fourteen lattices are represented here ok
so generally the way these lattices are represented
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is with respect to a b and c the lattice parameter
and with respect to a angle between them ok
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actually that is essentially a consequence
of the type of symmetry which is associated
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with these lattices strict classification
is on the basis of symmetry that is if we
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take triclinic it has no symmetry associated
with it if we like monoclinic it has one twofold
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axis rotation axis if you take at orthorhombic
perpendicular to each of the axis there is
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a twofold axis ok if you take triclinic there
is only one fourfold axis which is there if
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we look at a cubic systems there are ah four
threefold axis which are present ok if you
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take trigonal there is only one threefold
axis hexagonal is one which has got a six
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ah section hexagonally has got one sixfold
axis
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so essentially what we have now is the seven
crystal systems which we call it is based
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on these types of symmetry these are all the
minimum symmetries which will be associated
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with these crystal systems ok they can have
a maximum symmetry that will come to later
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what is the maximum symmetry which we have
ok and we should always remember that hexagonal
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close pack system is not a bravais lattice
bravais lattice is only hexagonal ok its a
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one which contains two atoms ah two lattice
points per unit cell hexagonal close pack
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lattice ok and the symbols which we use here
are primitive lattice is represented by p
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i for body centered f for face centered r
rhombohedral a b c for the different face
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centering ok but all are capital letters which
are used ah unlike in the case of ah two dimensional
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lattice ok
in the next slide in this particular one all
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the fourteenth ah ah bravais lattices are
being shown what x i have try to show here
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is that ah um in addition to the this one
bodies the space group symmetry which is associated
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with this i will come back to it later what
these space group symmetries are but essentially
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what the p represents is what is the type
of a lattice which we have ok and what are
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the symmetry elements which are associated
with it that is what it represents generally
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the lattices as i mentioned earlier exhibits
the full symmetry of the lattice ok the maximum
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symmetry will be exhibited by all the lattices
ok that way here all the maximum symmetry
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which is associated is exhibited one thing
which one should always remember is that earlier
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case when we considered how if you put a motive
around different ah ah planar lattices ok
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we are not able to we are able to join only
uniquely some of them that is if we put your
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motive asymmetric motive around whether it
is a square lattice or whether it is a ah
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rectangular lattice we generate only an onefold
symmetry ok that gives an indication ok that
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irrespective of the value of a and b they
the symmetry is the one which besides the
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what type of a specific space group which
it has ok
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on that basis that when we write a not equal
to b not equal to c it does not mean that
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it is not equal to a necessarily not equal
to b not necessarily not equal to c that is
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what in crystallography term it means not
equal to is doesnt mean not equal to necessarily
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not equal to it can be also ok
so now let us look at what are the types of
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symmetry elements which ah can be that is
ah before going into the details we will just
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list these are all the symmetry element and
then we will try to see how ah go into a details
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about the symmetry elements they are one translational
symmetry element will always be there with
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any lattice then we have a rotational symmetry
element ok which we have already seen what
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are rotational symmetry elements reflection
symmetry element which we had seen it in addition
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to it here we will have an inversion symmetry
element also will come ok then in addition
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to this this is all with respect to consistent
with translation but we are considering it
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around a point in addition combination of
rotation and translation combination of rotation
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and reflection combination of rotation and
inversion all the three are possible they
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will also we can consider cases so these are
all the cases which we look at it
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in fact the combination of rotation plus translation
gives raise to screw combination of rotation
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and translation gives raise to screw axis
reflection and translation gives raise to
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glide ok reflection and translation will give
rise to ah one is a glide which you get it
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ok ah then the other is a rotation and translation
gives a screw axis ok then ah ah rotation
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and a reflection perpendicular to the rotation
axis if we consider that will give rise to
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an inversion we will complete later ok ok
in this particular slide we have just shown
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the ah three dimensional unit cell which is
essentially a ah like a triclinic structure
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and any vector in this ah can be represented
by a vector ok r is equal to u into a plus
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v into b plus w into c where u v w are the
ah ah integers and a b c represents the translation
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symmetry this way in the vector notation all
the lattice points can be generated by taking
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various combinations of u v and w ok and what
i have shown here is that various types of
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this one because different types of notations
are used different ways it is represented
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in the books ok about the rotational symmetry
i had already explained ok how this ah rotational
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ah rotation consistent with translation put
some restrictions on the type of a rotations
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which are possible so i will not go into the
detail
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but in this slide what i had shown is there
how they are represented on the ah stereography
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projection i had just given a brief idea without
going to a detail anything about the stereography
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projection but that i will do it now what
a stereography projection is how important
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it is ok because when we have to represent
in three dimensions what we would like to
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do is that when we talk of symmetry angular
relationships have to be ah specified or the
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angular relations that be shown on two dimensions
because three dimensions we can view it but
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when we have to ah put a projection ok it
is always in two dimensions we deal with ok
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how do we go about and do it ok there are
various types of projections are there the
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simplest is a perspective projection in a
perspective projection we can see it ok but
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the distance unless we give more information
we do not know how far an object is far away
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ok in a stereography projection is a projection
where it is from the angular relationship
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in three dimensions could be represented in
two dimensions ok here i had just given what
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are the applications of stereography projection
one in the ah ah crystallography ok for may
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may showing angular relationship between different
planes and directions we can use stereography
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protection in x ray diffraction ok when we
have to represent texture we require it in
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electron microscopy different orientation
of crystals their planes and directions ok
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between different interfaces when we wanted
to find out orientation relationship in all
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these cases stereography ah projection is
important then when we reforming a single
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crystal how the different ah slip planes are
rotating all these information we can get
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it using a stereography projection because
there its at a three dimension what is happening
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but we wanted the same angular relationship
to be projected in two dimensions so that
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it is easy to do the calculation ah what is
a stereography projection the way in which
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we can understand it is is that ah you take
a sphere ok in a sphere as we know for a globe
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when we consider there are latitudes longitudes
are there or we can have it similar to that
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we can have it in a globe we assume that there
is nothing else is there ok fix the coordinate
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system at the center ok x y and e z coordinates
are fixed and then what we do is that ah take
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a point which is on the surface of the sphere
with respect to the e z coordinate it makes
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an angle theta are the e z coordinate we considers
the pole ok and if you look at the if you
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view this from the other end of the pole from
the south pole ok
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the ray which connects our eye to this pole
cuts the equatorial plane at some particular
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position so this distance o p if you look
at it on the equatorial plane ok it is given
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by the formula r is the radius of the sphere
into tan theta by two what the angle which
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this point p makes with respect to the s axis
ok so essentially using this relationship
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all the points which are there on the sphere
can be represented on the equatorial plane
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ok if this point p rotates around this pole
making an angle theta it will be a circle
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and the projection of it will be nothing but
this o p will take a rotation you take it
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a circle so it will generate a circle ok
similarly you take ah this particular ah ah
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this one longitude ok this longitude all the
points on this longitude the final projection
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will be nothing but it will be a line passing
through the center this will be clear from
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this particular slide this slide here if you
look at it these are called as the great circles
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are nothing but in the ah ah geography if
we look at it we call it as the longitude
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ok this ah longitude essentially you can see
that all these projections pass through the
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center and this is a line which passes through
ok this is nothing but a diameter ok and these
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circles which are the latitudes their project
it has a concentric circles ok if you assume
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that on the surface of the sphere we have
marked at every ah one degree two degree or
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five degrees or so some specific ah angular
separation latitudes and longitudes ok and
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then view it from the south pole then how
will it look like we will be generating projection
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this projection is called as the polar projection
in this projection if i consider this particular
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one it makes an angle thirty degree these
are latitude which it represents so if i take
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any particular position if i fix the coordinates
x y and this will be the normal to the screen
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is there e z coordinate now we know that if
i mark a particular position i know what is
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the position of this on the surface of the
sphere ok so essentially all the angular relationship
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in three dimensions are projected in this
two dimensional we essentially for stereography
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projection this is the polar projection which
is used ok this is but only thing which is
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important which we should ah and which we
should note is that if we consider ok in this
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particular ah projection if we consider any
point on the sphere if it is on an equatorial
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equator where will it get projector it will
get projector onto the circumference of the
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equator correct right if it is going to be
anywhere on the top hemisphere the projection
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will be inside the ah circle equatorial plane
so that is essentially and when we consider
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in a space lattice ok if we consider any planar
directions ok its going to be in three dimensions
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so anything that is in a two dimensional lattice
when we considers only the equatorial projection
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which you have to consider that is why the
motifs are always placed on the circle here
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since it is a three dimensional case the motifs
are all kept ah right inside the circle ok
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so if you take a motif like this here if a
mirror operation is there we put a mirror
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ah along this x axis then itll be getting
reflected and this is how the mirror will
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be reflected in a three dimensional projection
in a two dimensional projection this point
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would have been put here and this point would
have come here thats all the difference is
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ok mirror perpendicular to it if it is there
now we can see that the circumference of the
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circle is made thick to show that this is
a mirror ok if it is an inversion operation
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ok thats the point on the upper hemisphere
invert it through the center it will come
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into the lower hemisphere
so anything which is on the lower hemisphere
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when we show in the projection the closed
circle represents always motifs on the upper
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hemisphere and the open circle shows motifs
on the lower hemisphere ok so in an inversion
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operation this is how it will take place and
if it is a fourfold rotation now we can see
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00:22:49,230 --> 00:22:55,560
that a motif which is being kept here a fourfold
rotation this is how in a stereogram their
198
00:22:55,560 --> 00:23:02,400
various types of rotation and reflection and
inversion symmetry operation could be are
199
00:23:02,400 --> 00:23:09,150
ah rotation and a mirror which is perpendicular
to it all these can be represented all the
200
00:23:09,150 --> 00:23:16,930
examples are present here in this slide
ok so this is about a symmetric representation
201
00:23:16,930 --> 00:23:22,650
but when we have to find out the coordinates
associated with any one of them ok all the
202
00:23:22,650 --> 00:23:27,980
motifs which are generated if we are given
a coordinate x y e z we should know what is
203
00:23:27,980 --> 00:23:33,410
the transformation matrix which is associated
with it each of this ah symmetry operation
204
00:23:33,410 --> 00:23:40,410
and then we can find out because of using
the formula which i had given earlier in the
205
00:23:40,410 --> 00:23:49,660
last class one can generate various types
of a a points ok all the corresponding symmetry
206
00:23:49,660 --> 00:23:55,780
related points could be generated ok whether
it is for a rotation or whether it is for
207
00:23:55,780 --> 00:24:02,810
a reflection or a whether its for inversion
ok here for different operation this is for
208
00:24:02,810 --> 00:24:08,720
a one fold rotation the matrix this is the
matrix which will be there if it is ah one
209
00:24:08,720 --> 00:24:14,910
which is a twofold rotation ok then this is
the type of a transformation matrix which
210
00:24:14,910 --> 00:24:20,260
we will use it if it is a fourfold rotation
that this is the sort of a transformation
211
00:24:20,260 --> 00:24:25,560
matrix which you have to but on also we have
to define that whether its a clockwise rotation
212
00:24:25,560 --> 00:24:33,000
and anti crosswise rotation thats also very
important with these what we have ah ah looked
213
00:24:33,000 --> 00:24:39,130
at it is what are the various types of rotations
which are possible and how they are represented
214
00:24:39,130 --> 00:24:47,010
now can we have a combination of this just
the rotation itself axis suppose i take a
215
00:24:47,010 --> 00:24:52,800
two dimensional three dimensional lattice
like for example i have taken a ah a orthorhombic
216
00:24:52,800 --> 00:25:01,410
structure ok in this along each of this axis
along this axis or along this axis or along
217
00:25:01,410 --> 00:25:10,230
this axis all these axis we can have a twofold
rotation ok so what is being done is that
218
00:25:10,230 --> 00:25:16,850
if i represent a twofold axis on the surface
of a sphere because this is the origin o ok
219
00:25:16,850 --> 00:25:25,160
so around this direction or around this direction
around this direction along two directions
220
00:25:25,160 --> 00:25:32,270
i find out some sort of a rotational symmetry
what is the third direction in identify an
221
00:25:32,270 --> 00:25:37,330
another third direction where what is the
type of symmetry which exists ok these are
222
00:25:37,330 --> 00:25:44,050
all the combination of symmetry elements which
are possible so if i take ah that is two and
223
00:25:44,050 --> 00:25:52,530
two ok and then you find the some other direction
i find the twofold rotation ok then what is
224
00:25:52,530 --> 00:25:59,050
going to happen is that between the two axis
which have two fold rotation all the axis
225
00:25:59,050 --> 00:26:03,990
each of them taken separately two at a time
we find that angle between them is ninety
226
00:26:03,990 --> 00:26:16,820
degree and then that structure is orthorhombic
suppose the two operations ok it is there
227
00:26:16,820 --> 00:26:24,120
in another direction you find that a threefold
then the crystal system becomes their trigonal
228
00:26:24,120 --> 00:26:29,880
one then the angle between the two axis twofold
will be ninety degree and the other one is
229
00:26:29,880 --> 00:26:36,300
turning out to be sixty degree this way various
combinations are possible on that basis we
230
00:26:36,300 --> 00:26:42,000
can have different types of crystal structures
also ok this is one way in which various types
231
00:26:42,000 --> 00:26:47,200
of crystal systems could be classified ok
so essentially what means that these are all
232
00:26:47,200 --> 00:26:59,380
the unique combinations which are possible
by combination of just rotation alone right
233
00:26:59,380 --> 00:27:07,750
ok now in this particular slide what i had
shown here is essentially that the different
234
00:27:07,750 --> 00:27:14,850
crystal ah systems ok based on how they are
defined based on symmetry and what is the
235
00:27:14,850 --> 00:27:24,690
minimum symmetry which is necessary for this
bravais lattices ok and also the for a conventional
236
00:27:24,690 --> 00:27:31,760
cell if we consider what are the relationship
between the ah angle between the axis and
237
00:27:31,760 --> 00:27:38,600
the translational vectors ok this all of you
have studied also earlier itself you are very
238
00:27:38,600 --> 00:27:44,559
well aware of it ok
now about reflection i had already explained
239
00:27:44,559 --> 00:27:49,480
so i will not go into the details now let
us look at the inversion symmetry ok so far
240
00:27:49,480 --> 00:27:54,651
i am not talked about the inversion symmetry
there is if we take with respect to the unit
241
00:27:54,651 --> 00:28:02,430
cell where or the crystal lattice if you take
any point r ok by an inversion the r will
242
00:28:02,430 --> 00:28:08,300
become minus r so the all the coordinates
will become the x y z will become minus x
243
00:28:08,300 --> 00:28:17,080
minus y minus z so for this the ah operation
the transformation matrix which will be is
244
00:28:17,080 --> 00:28:24,180
this particular type of a transformation matrix
which has to be used ok
245
00:28:24,180 --> 00:28:30,160
so essentially if you now look at it what
are the types of point group symmetry operations
246
00:28:30,160 --> 00:28:38,600
which are there in three d lattices ok rotation
which you have considered reflection which
247
00:28:38,600 --> 00:28:47,780
we have considered inversion is one ok then
we can have a combination of rotation its
248
00:28:47,780 --> 00:28:56,170
possible then rotation and an inversion we
can take that combination rotation and reflection
249
00:28:56,170 --> 00:29:01,780
also we have considered right that is either
that rotation axis is there the mirror is
250
00:29:01,780 --> 00:29:09,870
a ah reflection is parallel to it or perpendicular
to it that combination can be chosen ok then
251
00:29:09,870 --> 00:29:16,890
we can have a rotation and a reflection ok
that is rotation and a mirror perpendicular
252
00:29:16,890 --> 00:29:22,010
to rotation axis that gives rise to inversion
that is why sometimes inversion is called
253
00:29:22,010 --> 00:29:30,230
as is it a new operation or is it a combination
of two operations of a rotation and reflection
254
00:29:30,230 --> 00:29:38,040
ok then in these things what we have to look
at it is that if we take only a rotation ok
255
00:29:38,040 --> 00:29:41,940
whats going to happen is that
suppose i take this object ok by a rotation
256
00:29:41,940 --> 00:29:46,270
it will come like this it will go like this
always it creates one particular type of an
257
00:29:46,270 --> 00:29:50,680
object that is if i take a right hand i rotates
it like this all the directions its only the
258
00:29:50,680 --> 00:29:56,640
right hand gets rotated but during this rotation
operation right hand motif can never become
259
00:29:56,640 --> 00:30:02,180
like a left hand reflection and inversion
of the operations where that enantiomorphic
260
00:30:02,180 --> 00:30:09,950
structures where the by this operation the
right hand is generated into a left hand operation
261
00:30:09,950 --> 00:30:14,890
ok this has some significance in the behavior
of the material which i will not go into the
262
00:30:14,890 --> 00:30:19,930
detail because that is not particularly for
the defection material one doesnt have to
263
00:30:19,930 --> 00:30:25,060
bother about it ok
so as i have mentioned using stereography
264
00:30:25,060 --> 00:30:31,080
projection we can represent it here all the
basic symmetry elements especially the point
265
00:30:31,080 --> 00:30:37,540
group symmetry elements or so the combination
of a rotation reflection and inversion and
266
00:30:37,540 --> 00:30:46,140
roto inversion these together constitute the
a point group symmetry ok so this can be represented
267
00:30:46,140 --> 00:30:55,140
in a stereography projection ok in this its
for one fold ok and then twofold ok that symbol
268
00:30:55,140 --> 00:31:00,260
which is used to indicate that it is a twofold
rotation ok and the three fold rotation we
269
00:31:00,260 --> 00:31:07,930
can see that a motive which is being placed
here ok that a threefold rotation it generates
270
00:31:07,930 --> 00:31:13,740
like that various symbols are being used ah
i will not go into details of any of these
271
00:31:13,740 --> 00:31:18,440
symbols because a internet in our crystallography
table if you look at it all the symbols are
272
00:31:18,440 --> 00:31:25,060
explained very nicely ok and in many of the
books which i had mentioned earlier also in
273
00:31:25,060 --> 00:31:30,940
those books also all these symbols are explained
essentially what we can make out is that these
274
00:31:30,940 --> 00:31:41,070
are all the ah just pure rotation and this
corresponds to roto inversion the this layer
275
00:31:41,070 --> 00:31:46,710
and this layer if you look at it it is mirror
which is parallel and the another is perpendicular
276
00:31:46,710 --> 00:31:53,550
ok then this is just a simple inversion
so these are all the basic operations and
277
00:31:53,550 --> 00:31:57,980
various combinations which we can choose of
all of them then you can imagine how many
278
00:31:57,980 --> 00:32:04,040
combinations which we can have ok generate
many but how many distinct ones finally we
279
00:32:04,040 --> 00:32:09,600
find it like in the case of ah two dimensional
lattice we did the exercise then we find that
280
00:32:09,600 --> 00:32:15,690
only ten are going to be there point groups
correct plane or point group similarly here
281
00:32:15,690 --> 00:32:21,400
it is going to be only thirty two distinct
point groups are possible so essentially what
282
00:32:21,400 --> 00:32:27,950
is ah point group is essentially the various
a symmetry operations like rotation reflection
283
00:32:27,950 --> 00:32:35,450
and center of inversion consistent with translational
symmetry ok and roto in the and their combination
284
00:32:35,450 --> 00:32:40,630
roto in the combination is called as a point
group symmetry ok many as i mentioned many
285
00:32:40,630 --> 00:32:49,140
ah possibilities exist but distinct ones are
only thirty two ok why do we require a study
286
00:32:49,140 --> 00:32:58,140
of this point group symmetry or why it is
very important this is because when we look
287
00:32:58,140 --> 00:33:06,200
at the properties of the material ok change
in different directions are on different surfaces
288
00:33:06,200 --> 00:33:11,740
if you look at it if a crystal grows you find
that it grows with some particular morphology
289
00:33:11,740 --> 00:33:15,490
earlier all the point group symmetries where
found out by looking at the morphology of
290
00:33:15,490 --> 00:33:23,580
the crystals similarly electrical conductivity
ok thermal expansion all these are and different
291
00:33:23,580 --> 00:33:27,860
directions can change depending upon the crystal
structure they are related to a point group
292
00:33:27,860 --> 00:33:34,010
symmetry how experimentally we can find out
is by measuring these properties in different
293
00:33:34,010 --> 00:33:40,090
directions we can determine what are symmetry
elements are associated with it ok are essentially
294
00:33:40,090 --> 00:33:44,340
the point group symmetries can be done but
the space group symmetry if we look at it
295
00:33:44,340 --> 00:33:50,860
which involves others like a screw axis which
i will come shortly its going to be externally
296
00:33:50,860 --> 00:33:55,190
it is very difficult to see because it is
only associated with that only a translation
297
00:33:55,190 --> 00:34:00,570
which is very so the translation it will not
shows a change in property in a particular
298
00:34:00,570 --> 00:34:05,030
direction ok
so what is important is that study of point
299
00:34:05,030 --> 00:34:10,869
group symmetry is important because that gives
information about the properties in various
300
00:34:10,869 --> 00:34:23,779
directions in crystal structures ok in short
what we can have is that ah thirty two point
301
00:34:23,779 --> 00:34:29,629
groups symmetries ok just the combination
of all the symmetry elements which consider
302
00:34:29,629 --> 00:34:37,399
they are eleven are there roto inversion gives
another five ok then combination of proper
303
00:34:37,399 --> 00:34:45,279
and improper ah rotation axis if you consider
all together another sixteen so in short thirty
304
00:34:45,279 --> 00:34:52,730
two our generator ok in this ah slide i am
just showing how they are in a stereography
305
00:34:52,730 --> 00:35:00,059
projection they are represented ok i will
give some assignments one can work it out
306
00:35:00,059 --> 00:35:06,500
how to generate given the point group symmetry
which is given how these a stereography projection
307
00:35:06,500 --> 00:35:14,529
ah they are generated ok the how they look
like ok then one will understand how this
308
00:35:14,529 --> 00:35:19,670
has been generated actually and ah presented
in various books
309
00:35:19,670 --> 00:35:25,890
now we will ah going to the international
in our crystallography table if you look at
310
00:35:25,890 --> 00:35:30,579
it one a stereography projection which is
being given this is the a case which he is
311
00:35:30,579 --> 00:35:40,609
considered is four m m point group ok in this
particular one in addition to giving this
312
00:35:40,609 --> 00:35:50,309
ah ah a ah the position of the motif ok the
general motif what are other positions if
313
00:35:50,309 --> 00:35:57,390
the motif we keep it also the same symmetry
element will be a can be represented like
314
00:35:57,390 --> 00:36:04,440
for example if i put one at that center that
has got a fourfold symmetry so i dont require
315
00:36:04,440 --> 00:36:11,740
four motif to present them only one at the
center is good enough ok that is how what
316
00:36:11,740 --> 00:36:17,170
is being essentially explained here like the
way i explained earlier for two dimensional
317
00:36:17,170 --> 00:36:24,670
lattice ok here also what is the side symmetry
associated with it ok then the wyckoff position
318
00:36:24,670 --> 00:36:30,380
and the multiplicity corresponding to the
particular side symmetry and here it is not
319
00:36:30,380 --> 00:36:36,710
represented in x y z it is given in terms
of a planes because most of the symmetry elements
320
00:36:36,710 --> 00:36:42,799
ah earlier pin when the plane groups ah point
group symmetry if try to look at it ok some
321
00:36:42,799 --> 00:36:47,160
directions we are looking at it the planes
which are perpendicular to them that is what
322
00:36:47,160 --> 00:36:52,999
its being represented ok
so the symbol which is used to represent is
323
00:36:52,999 --> 00:37:01,270
planes ok then in this side what i had shown
it is that for the general one ok position
324
00:37:01,270 --> 00:37:07,779
exactly if we put one motif around it how
the other motifs will be generated for this
325
00:37:07,779 --> 00:37:14,859
particular point group symmetry one fourfold
rotation which is taken here ok that is this
326
00:37:14,859 --> 00:37:23,359
point is rotated from here it comes here then
from here now from here to here to here to
327
00:37:23,359 --> 00:37:29,680
here it comes and then what we do it is that
put the symmetry elements mirror symmetry
328
00:37:29,680 --> 00:37:36,980
you consider it then you find that that gets
reflected ok suppose on the symmetry axis
329
00:37:36,980 --> 00:37:45,279
ok on this if i place it only four points
have to be placed here ok still that fourfold
330
00:37:45,279 --> 00:37:53,609
symmetry and mirror all the symmetry elements
if i place it around this axis ok the symmetry
331
00:37:53,609 --> 00:37:58,799
element mirror symmetry elements which are
there then i have to place at only four positions
332
00:37:58,799 --> 00:38:04,569
ok if i place it at the center only at one
position so this is essentially what is being
333
00:38:04,569 --> 00:38:10,799
given one ok this is corresponding to general
this is corresponding to one mirror this is
334
00:38:10,799 --> 00:38:15,539
corresponding to another mirror how the various
planes will come this is corresponding to
335
00:38:15,539 --> 00:38:24,700
the third one fourth one ok
ok these various ah thirty two crystallography
336
00:38:24,700 --> 00:38:33,309
point groups ok they correspond to seven crystal
ah systems which are there we can route them
337
00:38:33,309 --> 00:38:41,940
ok so triclinic has got one and one bar ok
monoclinic if you look at it it is two m and
338
00:38:41,940 --> 00:38:47,190
two by m these are all the point groups which
are associated with it in this if you look
339
00:38:47,190 --> 00:38:54,890
at here what is essentially is being given
here is the minimum symmetry elements in this
340
00:38:54,890 --> 00:39:00,839
side what is marked with the pink color shows
the lattice which will have the maximum symmetry
341
00:39:00,839 --> 00:39:08,140
elements ok thats how i have just identify
marked it ok
342
00:39:08,140 --> 00:39:13,720
so trigonal will have three ok but if you
take us the trigonal as a lattice it can have
343
00:39:13,720 --> 00:39:21,829
three bar m that is if we have a crystal at
least you should have a one threefold axis
344
00:39:21,829 --> 00:39:26,670
is it clear ok and then another is that here
some symbols are being when we use m three
345
00:39:26,670 --> 00:39:33,960
bar m this is essentially is a short form
it actually corresponds to four by m three
346
00:39:33,960 --> 00:39:39,619
bar and two by m ok then the next question
comes is that what is the convention which
347
00:39:39,619 --> 00:39:43,920
is being used to represent these symbols is
their any convention which is being followed
348
00:39:43,920 --> 00:39:50,540
otherwise we dont in one case we triclinic
use only one monoclinic we use two ok here
349
00:39:50,540 --> 00:39:57,109
three symbols are being used in which directions
are where which the direction along which
350
00:39:57,109 --> 00:40:02,609
the symmetry operations are performed and
that is given in this slide triclinic if you
351
00:40:02,609 --> 00:40:08,180
look at it we dont have any symmetry operation
ok there is nothing like a primary or a secondary
352
00:40:08,180 --> 00:40:14,859
if you take monoclinic ok always that zero
one zero the b axis is chosen further show
353
00:40:14,859 --> 00:40:23,680
the twofold symmetry which is present there
ok orthorhombic if is see a b c all have got
354
00:40:23,680 --> 00:40:27,839
a twofold ok
so essentially what it represents is these
355
00:40:27,839 --> 00:40:36,460
are all the symmetry along various axis when
we say two m m that means that twofold along
356
00:40:36,460 --> 00:40:44,390
the a axis that is primary is a then secondary
is mirror along this direction then another
357
00:40:44,390 --> 00:40:50,369
is along this direction mirror so that will
be so two two two and here if we look at it
358
00:40:50,369 --> 00:40:58,049
tetragonal ok zero zero one is shown along
the ah primary one so the first letter represents
359
00:40:58,049 --> 00:41:06,099
a fourth ok then the next one represents a
secondary that is what is the symmetry along
360
00:41:06,099 --> 00:41:11,880
one zero zero or zero one zero and the third
one the third letter represents symmetry along
361
00:41:11,880 --> 00:41:20,309
one one zero direction like cubic if you see
it again that ah on the primary one represents
362
00:41:20,309 --> 00:41:29,230
the symmetry which is along the ah ah x y
are z axis and the second letter represents
363
00:41:29,230 --> 00:41:35,810
the symmetry along one one one direction and
the third letter represent a represents the
364
00:41:35,810 --> 00:41:41,249
symmetry which is along the one one zero direction
ok if you take four three two that means that
365
00:41:41,249 --> 00:41:50,089
fourfold along any one of the a b or c axis
and three is along the one one one direction
366
00:41:50,089 --> 00:41:54,900
and the twofold along one one zero direction
right
367
00:41:54,900 --> 00:42:00,380
so from this one ah very clearly understand
how what is the sort of coordinate system
368
00:42:00,380 --> 00:42:07,529
which is being used to represent the various
symbols which are given here for the thirty
369
00:42:07,529 --> 00:42:19,940
two point groups ok having looked at this
ah thirty two point groups and their representation
370
00:42:19,940 --> 00:42:26,390
how it is done in stereogram ok if you try
to generate a space group what we have to
371
00:42:26,390 --> 00:42:31,160
do it is that point group is around a point
which we are considering it space group with
372
00:42:31,160 --> 00:42:36,010
that around the lattice if you are trying
to put motifs having this sort of a point
373
00:42:36,010 --> 00:42:42,079
groups around each of them what are the distinct
types of a crystals which could be generated
374
00:42:42,079 --> 00:42:49,410
with having specific symmetries associated
with them ok is there here again the combinations
375
00:42:49,410 --> 00:42:56,599
if we try many are possible distinctly finally
we find that only thirty two space groups
376
00:42:56,599 --> 00:43:01,099
are possible ok
we will not be going into all of them but
377
00:43:01,099 --> 00:43:07,410
before going into them so far we consider
only point group if we consider as a lattice
378
00:43:07,410 --> 00:43:16,229
ok or what we have not done is that we have
considered between rotation and reflection
379
00:43:16,229 --> 00:43:21,849
which is consistent with translation but we
are not combined in translation we can combine
380
00:43:21,849 --> 00:43:32,849
rotation and translation ok and we can combine
mirror and translation if you do it in the
381
00:43:32,849 --> 00:43:39,039
lattice they generate some special symmetry
elements ok for example if we combined rotation
382
00:43:39,039 --> 00:43:45,410
and translation it generates a lattice which
we call it the symmetry which it generates
383
00:43:45,410 --> 00:43:57,170
is called as the screw axis correct for example
here if we look at it ok example if you take
384
00:43:57,170 --> 00:44:04,210
this is a six fold ok you know that sixfold
means that sixty degree rotation has to be
385
00:44:04,210 --> 00:44:15,079
given ok and then a translation by some vector
or some pitch you that is you rotate it and
386
00:44:15,079 --> 00:44:25,220
then ah translate it along the screw axis
by ah ah some vector of magnitude you take
387
00:44:25,220 --> 00:44:35,869
tau ok or a pitch which is tau then after
n rotations ok we should be able to come back
388
00:44:35,869 --> 00:44:41,190
to original position what is the position
which well be coming it could be either equal
389
00:44:41,190 --> 00:44:47,950
to the lattice translation vector if it is
t or it could be some multiple of t that is
390
00:44:47,950 --> 00:44:57,359
what essentially return n rho equals p into
t ok this is equivalent to like a example
391
00:44:57,359 --> 00:45:02,950
which we can think of in real life is spiral
staircase when we go on a spiral staircase
392
00:45:02,950 --> 00:45:09,089
that is a pitch with which its being the staircase
it rotates and finally afterwards it will
393
00:45:09,089 --> 00:45:14,640
be coming back to original position again
it rotates correct
394
00:45:14,640 --> 00:45:20,009
so depending upon the how many times it gives
we have different types of combinations which
395
00:45:20,009 --> 00:45:28,190
are possible one here the pitch which is taken
is if we take one sixth of the lattice translation
396
00:45:28,190 --> 00:45:38,890
vector for a sixfold but after every rotation
by sixty degree we move it by ah one by six
397
00:45:38,890 --> 00:45:46,640
of t then we find that after six rotations
and a in combination with that translation
398
00:45:46,640 --> 00:45:50,930
we will be coming we will be able to come
back to identical position original lattice
399
00:45:50,930 --> 00:45:58,319
point we are able to reach it but it has been
shifted by a lattice translation vector ok
400
00:45:58,319 --> 00:46:03,239
that is how it can be done
so this also if you take a twofold rotation
401
00:46:03,239 --> 00:46:11,309
its possible in a lattice because this is
not possible in a ah point group but its in
402
00:46:11,309 --> 00:46:16,489
a space group when we consider positions of
atom these sort of translations this sort
403
00:46:16,489 --> 00:46:22,400
of symmetry is also possible ok if it consider
for twofold there is only one is possible
404
00:46:22,400 --> 00:46:28,999
two one that means that one eighty degree
rotation and translation by t by two threefold
405
00:46:28,999 --> 00:46:35,800
if we consider it ok there is one twenty degree
rotation and a translation then another one
406
00:46:35,800 --> 00:46:41,359
twenty degree rotation so three so it can
be three one or three two there are many combinations
407
00:46:41,359 --> 00:46:45,819
which are possible because i dont want to
go into a detail of this one because if it
408
00:46:45,819 --> 00:46:49,329
has to be done it should be done in a separate
crystallography class where all these things
409
00:46:49,329 --> 00:46:55,150
could be explained at length ok but essentially
these are all the symbols which are being
410
00:46:55,150 --> 00:47:02,079
used
so in this slide we can see that if it is
411
00:47:02,079 --> 00:47:07,539
just a mirror which is their twofold we can
see that its a seven which is getting just
412
00:47:07,539 --> 00:47:14,900
reflected this is how at different lattice
points the motifs will be kept now if we look
413
00:47:14,900 --> 00:47:23,339
at a ah two one rotation ok this one is rotated
by one eighty degree it comes here another
414
00:47:23,339 --> 00:47:32,440
rotation and a translation it is brought to
that point ok similarly for threefold three
415
00:47:32,440 --> 00:47:39,539
one as well as three two how a motif will
be rotated and translated around the screw
416
00:47:39,539 --> 00:47:49,160
axis it is depicted in this figure ok from
this we can i understand that the three one
417
00:47:49,160 --> 00:47:56,869
and three two one it will be a ah clockwise
rotation and a translation another is an anticlockwise
418
00:47:56,869 --> 00:48:04,720
rotation and a translation both of them ok
this is the difference between these two though
419
00:48:04,720 --> 00:48:09,250
the pitch remains that same the angle of rotation
is also the same but the sense in which it
420
00:48:09,250 --> 00:48:18,140
is being rotated is different ok
glide if we consider it ok this has mirror
421
00:48:18,140 --> 00:48:24,790
plus translation which is their glide is nothing
but a mirror plus translation i had already
422
00:48:24,790 --> 00:48:29,750
explained glide ok but what is essentially
important is that if the glide is along the
423
00:48:29,750 --> 00:48:38,749
a axis the symbol which is being used to represent
in the crystallography table is a ok and the
424
00:48:38,749 --> 00:48:45,920
translation which is associated with the glide
is a by two b itll be b by two similarly we
425
00:48:45,920 --> 00:48:51,579
can have a glide along the face diagonal are
on the body diagonal which is called as a
426
00:48:51,579 --> 00:48:57,589
diamond glide ok if there are no translation
which is involved then it becomes a mirror
427
00:48:57,589 --> 00:49:03,740
operation ok all these ah various types of
operations and symbols what is the translation
428
00:49:03,740 --> 00:49:13,280
vector associated with different types of
glides is given in this transparency ok
429
00:49:13,280 --> 00:49:18,130
so so far what we have considered is different
types of symmetry operations which are that
430
00:49:18,130 --> 00:49:27,979
is ah glide and screw which are symmetry operations
which involve either rotation and translation
431
00:49:27,979 --> 00:49:36,140
or reflection and translation ok now like
we have represented in planar lattice how
432
00:49:36,140 --> 00:49:42,960
do we represent all these symmetry elements
ok ah so first what we have to do it is that
433
00:49:42,960 --> 00:49:48,790
some projection will be required ok suppose
we take the example of an orthogonal the example
434
00:49:48,790 --> 00:49:54,760
of a orthorhombic lattice how are we going
to present ah orthorhombic lattice if you
435
00:49:54,760 --> 00:50:01,490
look at projection of one particular plane
ok that completely doesnt represent the ah
436
00:50:01,490 --> 00:50:09,250
orthorhombic lattice right at least two projections
are minimum required to complete it if it
437
00:50:09,250 --> 00:50:15,869
is a cube one projection is good enough ok
so mono depending upon the type of crystal
438
00:50:15,869 --> 00:50:24,130
structure different projections are required
ok graphical projections are required to show
439
00:50:24,130 --> 00:50:31,420
that this is what essentially is the ah planar
lattice corresponding to that in a particular
440
00:50:31,420 --> 00:50:37,430
direction correct are the units corresponding
planar unit cell that is essentially what
441
00:50:37,430 --> 00:50:44,579
is being when we represent it how are we going
to show at what position they are going to
442
00:50:44,579 --> 00:50:50,499
be there and ah the so that is what essentially
is being shown here like if we take in this
443
00:50:50,499 --> 00:51:00,940
particular one where it is ah nothing but
a tetragonal lattice in this particular tetragonal
444
00:51:00,940 --> 00:51:08,050
lattice the projection if we see this is the
in this plane which we show the atom at the
445
00:51:08,050 --> 00:51:14,630
next plane will be projected to the middle
and the next plane is essentially identical
446
00:51:14,630 --> 00:51:21,690
to this one
so this can be represented as having ah coordinates
447
00:51:21,690 --> 00:51:27,980
half half half are in this projection it can
be just shown at all these x y positions are
448
00:51:27,980 --> 00:51:32,710
there only that e z position is half this
way also it can be shown this is what is being
449
00:51:32,710 --> 00:51:40,119
generally used these sort of projection in
the case of ah space group symmetry when we
450
00:51:40,119 --> 00:51:47,790
represent it ok now just let us have a look
at the different type of ah ah how it is a
451
00:51:47,790 --> 00:51:54,180
space group is presented in the international
union of crystallography table ok one if you
452
00:51:54,180 --> 00:52:00,519
look at the table at the right hand side they
will show what is the crystal system then
453
00:52:00,519 --> 00:52:06,110
what is the point group symmetry which is
associated with it ok then the show symbol
454
00:52:06,110 --> 00:52:10,989
also will be shown which i have not just shown
here because that is the symbol which we see
455
00:52:10,989 --> 00:52:17,369
the this particular type of a space group
symbol is essentially ah is the one which
456
00:52:17,369 --> 00:52:22,779
is now conventionally adapted in crystallography
but still people who use crystal chemistry
457
00:52:22,779 --> 00:52:27,440
they use the symbol because that's one easier
to work with when we consider it as different
458
00:52:27,440 --> 00:52:32,910
group symmetry operations ok
then that one number is given this gives the
459
00:52:32,910 --> 00:52:39,799
what is the space group number ok and then
what is called as a patterson symmetry which
460
00:52:39,799 --> 00:52:46,950
thought something about that ah ah diffraction
it is something related to a diffraction symmetry
461
00:52:46,950 --> 00:52:51,249
ok these are all the information which is
being given then next what is the information
462
00:52:51,249 --> 00:53:01,900
which is being given is essentially the ah
unit cell ok and this is the unit cell one
463
00:53:01,900 --> 00:53:08,420
projection ok and in this projection the what
are the symmetry elements this is for a p
464
00:53:08,420 --> 00:53:13,049
two means that only one twofold rotation is
there and the detail symbol if you look at
465
00:53:13,049 --> 00:53:21,349
it p two is one two one that means that along
x axis one fold rotation two fold is along
466
00:53:21,349 --> 00:53:28,960
the y axis and onefold along the ah ah z axis
ok
467
00:53:28,960 --> 00:53:37,519
now this is the and then it is being marked
zero to a this is that the ah x direction
468
00:53:37,519 --> 00:53:43,269
and this zero to c ok b direction is perpendicular
to it in this specific case and then not only
469
00:53:43,269 --> 00:53:49,450
this is being shown in the other two directions
also how the units cell looks like and the
470
00:53:49,450 --> 00:53:55,229
symbols which are associated within because
this symbol which is being used is represents
471
00:53:55,229 --> 00:54:08,720
that there is a screw axis ok and generally
afterwards you show a unit cell ok how the
472
00:54:08,720 --> 00:54:15,630
atoms are place ok in this particular case
ok since it is only one fold axis is there
473
00:54:15,630 --> 00:54:24,349
the origin could be chosen anywhere ok if
different symmetry elements intersect origin
474
00:54:24,349 --> 00:54:29,219
gets automatically fixed like if only twofold
rotation is there where do you fix the origin
475
00:54:29,219 --> 00:54:36,519
arbitrarily we have to fix it so essentially
the origin is fixed ok and done that the projection
476
00:54:36,519 --> 00:54:46,489
of that the ah ah a unit cell is being shown
now when you place a motif the motif is placed
477
00:54:46,489 --> 00:54:55,569
where is it ah being placed at some position
ok above it that is why it is being shown
478
00:54:55,569 --> 00:55:04,130
o plus plus indicates that its at some particular
value its above this ah plane of the unit
479
00:55:04,130 --> 00:55:14,430
cell and from here by around this axis is
a twofold rotation ok so whats going to happen
480
00:55:14,430 --> 00:55:19,069
is that it gets twofold rotation and these
said it comes again o plus this is how the
481
00:55:19,069 --> 00:55:22,959
general position is being shown this is how
the graphical representation which is given
482
00:55:22,959 --> 00:55:33,349
for a monoclinic lattice ok then as we can
see here there are many symmetry elements
483
00:55:33,349 --> 00:55:37,630
which are associated with it what are the
symmetry elements which are this point has
484
00:55:37,630 --> 00:55:43,709
a one unique symmetry this is another one
this is another one this is another one because
485
00:55:43,709 --> 00:55:48,519
from this point to this point it is a lattice
point so thats the identical but these are
486
00:55:48,519 --> 00:55:54,599
all the new symmetry elements which are generated
in the unit cell so we can put atom or the
487
00:55:54,599 --> 00:55:59,630
motive is at this position and this position
or at this position so essentially thats what
488
00:55:59,630 --> 00:56:06,729
is being shown one this corresponds to zero
y zero is its being placed on this axis and
489
00:56:06,729 --> 00:56:18,549
the another is zero y half ok half position
this is corresponding to ah here and this
490
00:56:18,549 --> 00:56:21,339
one corresponds to no this is
491
00:56:21,339 --> 00:56:32,289
ya ah these position and this corresponds
to one at the and this corresponds to coordinates
492
00:56:32,289 --> 00:56:37,859
where the motifs are being placed with respect
to a general point ok so this is the table
493
00:56:37,859 --> 00:56:44,589
which is very important for constructing the
crystal structures ok along with it we should
494
00:56:44,589 --> 00:56:50,519
know what the lattice parameters are if this
information is available we can construct
495
00:56:50,519 --> 00:56:59,329
the complete crystal structure ok
if we consider in this one thats a screw axis
496
00:56:59,329 --> 00:57:06,239
also is possible that two instead of a mirror
rotation associated with there can be a translation
497
00:57:06,239 --> 00:57:11,479
so it could be a screw axis then the symbol
which is being used is this particular symbol
498
00:57:11,479 --> 00:57:15,819
again if you look at the symmetry elements
which are associated with it its an identical
499
00:57:15,819 --> 00:57:23,269
type of a symmetry element which you see it
ok and in this particular case ok the symbol
500
00:57:23,269 --> 00:57:28,259
which is been general point that is at where
there is no symmetry is associated with it
501
00:57:28,259 --> 00:57:38,910
if you put a motive at a height some height
y ok then then after one eighty degree rotation
502
00:57:38,910 --> 00:57:45,130
it will come at a position that but it is
shifted up by plus half that is exactly what
503
00:57:45,130 --> 00:57:49,640
is being shown and then the coordinates if
you see it here we can generate represent
504
00:57:49,640 --> 00:57:56,579
the coordinates this correct these coordinates
also there is another representation in which
505
00:57:56,579 --> 00:58:03,450
it is being done in crystallography table
ok i will not go into the detail of it
506
00:58:03,450 --> 00:58:10,390
now let us look at a p four m m ok here it
is a fourfold symmetry two mirrors are also
507
00:58:10,390 --> 00:58:17,339
associated with it ok the way in which it
is represented is that these are all the positions
508
00:58:17,339 --> 00:58:26,220
of a fourfold symmetry lattice points then
here we have twofold then mirrors which are
509
00:58:26,220 --> 00:58:33,440
there the motifs these are all points if we
keep your motive at this particular point
510
00:58:33,440 --> 00:58:38,119
it is lying on a mirror so it is called a
special position if you put it here it has
511
00:58:38,119 --> 00:58:44,339
a symmetry which is two m m so this is a special
point and if i put a motif here it is lying
512
00:58:44,339 --> 00:58:49,769
at a general point so each of the position
what will be the coordinates that is the information
513
00:58:49,769 --> 00:59:01,339
which is given in this particular table ok
using this table ok and essentially here again
514
00:59:01,339 --> 00:59:06,230
like as i mentioned for two dimensional lattice
what are the various positions at which the
515
00:59:06,230 --> 00:59:11,170
motifs will be kept for different associated
with different symmetry elements ok
516
00:59:11,170 --> 00:59:19,900
so in short if you look at it ok the general
position where we put it has got onefold symmetry
517
00:59:19,900 --> 00:59:28,859
the special positions in the unit cell ok
there we have many symmetries which are associated
518
00:59:28,859 --> 00:59:35,019
with it depending upon that the multiplicity
will change ok so the crystallography table
519
00:59:35,019 --> 00:59:44,029
if you look at it the graphical representation
shows the unit cell ok associated with all
520
00:59:44,029 --> 00:59:51,229
the symmetries associated with it and then
a general position which is represented then
521
00:59:51,229 --> 00:59:58,779
in addition to it in an another table all
the positions of the special positions and
522
00:59:58,779 --> 01:00:06,589
the general positions are also given ok so
the later part of that information is what
523
01:00:06,589 --> 01:00:13,910
is necessary for constructing a three dimensional
crystal ok this i have explained it with a
524
01:00:13,910 --> 01:00:22,920
few examples but ah when we have to look for
in the actual crystal structure all these
525
01:00:22,920 --> 01:00:29,999
things have to be specific positions have
to be considered in the next class we will
526
01:00:29,999 --> 01:00:37,859
take some examples and explain ok how different
types of structures can be constructed using
527
01:00:37,859 --> 01:00:40,829
the information which is given in the space
group table