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welcome back to the lectures in ah chemistry
and the introduction to molecular spectroscopy
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i just finished in the last lecture with a
very simple and a quick definition of what
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is known as a tensor and i think it is important
for you to understand even in a rudimentary
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form the idea of a tensor because the molecular
spectroscopy a spectroscopy quantities is
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that you deal with some of them have components
from the ah higher rank tensors and some of
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them are vectors the dipole moment is a vectors
for example the polarizability is a second
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rank tensor as i told you earlier the high
per polarizability which is often measure
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for non linear spectroscopic systems is a
third rank tensor and then there are higher
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rank tensodial properties which are often
study
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in the introductory course we don't know to
go as far as the but in even the study of
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an elementary account of raman spectroscopy
it is important for us to have an appreciation
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of the idea of tensor now if you remember
we will start with the polarzibaility tensor
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first ah in this lecture and if there is enough
time i will also introduce you to the idea
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of moment of inertia being the second rank
tensor and what the components are ok so first
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let us write the relation that we left in
between namely the induced dipole moment mu
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which is a vector in terms of a second rank
tensor alpha
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which changes the electric field to give you
the induced dipole movement what exactly this
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picture suppose you have electric fields in
this direction and you have a molecule for
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example say n o lets consider n o to be in
this direction suppose the field is in this
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direction is moving in this right field is
pointing in this direction and n o molecule
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is sitting in a perpendicular direction you
know that the bonding electron of the n o
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is between the two nitrogen and oxygen the
electron density is concentrated there and
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the electron density will try to adjust itself
with the positive side of the electric field
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and the negative side of the electric field
in such a way that the molecule will have
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a slight orientation and there may be a slight
deformation of the electron density if the
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fields are ah very large
to what extent the molecule is able to polarize
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itself to adjust itself to the external field
and undergo simple changes in its geometry
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that number is given by the ah the number
ah is by this quantity in a certain form if
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the molecule is like say h c m it has ah the
bending vibrational modes it has stretching
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vibrational modes if has a what is also known
as the ah i mean a symmetrical in the sense
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both the hydrogen and nitrogen atom going
one way and then the hydrogen and nitrogen
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atom going in the opposite direction and so
on that's a normal motion now if you put such
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a molecule which is undergoing vibration in
an electric field the electric field may induce
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an additional dipole movement and that induce
to dipole movement is directly a function
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of the property of the electric field and
remember i put a double arrow on top of it
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instead of writing it simply as alpha e because
its possible that induced dipole movement
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of say n o in the field in the presence of
the field see n o itself is undergoing some
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vibrations so if i have to write n double
bond o there is a mu and now if this molecule
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is in the electric field e there may be additional
induced dipole movement and that may not be
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in the direction of the electric field and
therefore you have this quantity call that
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the second rank tensor
exactly how do we write this is a vector therefore
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you have three components mu induced x then
mu induced y and mu induced z in the axis
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system which defines also the electric field
if the electric field is in the z direction
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then you know in a plane perpendicular to
the z direction is where the x and y axis
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are and with respect to that axis system the
mu induced is a vector therefore it has three
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coordinates and if you put this together and
the electric field itself has three components
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e x e y and e z in an arbitrary direction
and suppose we will put this as three components
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then alpha is given by a three by three matrix
and the components of alpha the sub the indexes
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the indices that you see here is not one one
one two one three but they are related to
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the direction on this side as well as the
direction of the component on this side which
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is the effect this is the cause this is the
effect and the alpha connects the cause to
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the effect and therefore the directions of
the cause and the effect are written as x
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x ok you write all of these as capitals x
y z
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then alpha x x times e x meaning that this
is the component which tells you that the
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x component of the induced dipole movement
with respect to the x component of the electric
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field and likewise alpha x y alpha x z and
alpha y x alpha y y alpha y z alpha z x alpha
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z y alpha z z what is this mean this essentially
means that the induced x component dipole
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movement mu induced x is given by the effect
of the e x alpha x x the e y alpha y y and
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the e z alpha sorry alpha ah x y and alpha
x z ok this relation that the induced dipole
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movement is actually a function of all the
three components of the applied electric field
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and the respective co efficient connect them
to give you the total value this is the tonsorial
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relation what about mu induce to y this is
capital x this is capital y its all for y
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x e x plus alpha y y e y plus alpha y z e
z and likewise mu induced z is alpha z x e
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x plus alpha z y e y plus alpha z z e z therefore
you have got these nine components x x x y
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all the way down to z z these nine components
in the definition of the tenser that i gave
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you in the last lecture
these nine components behave under coordinate
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rotation the same way the unique diverts the
x x x y x z there are nine unit these are
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these nine how do the transform under the
coordinate rotation in the same way these
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quantities transform in the four way become
they the quantities of a second rank the component
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of a second rank tensor ok
now one interesting thing is the fundamental
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property of the microscopy system appears
to tell us now the x y and y x see the correction/
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connection connection between them the they
are equal first of all that's a result that's
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a result that comes from the microscope consideration
first of all you see the x y is a component
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due to the y component of the electric field
inducing a change in the x there is a mu induce
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to dipole moment but in x component that connects
x y connects e y to mu induce to x y x connects
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the x component of the electric field lead
in to your y component of the mu induced they
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are switched around there are beautiful principals
known as there is a positive relation in ah
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basic physics i don't want to say much about
them
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but in this case the property is that such
quantities which are connected with the reversal
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of directions of the electric field and the
outcome they are both equal by geometrical
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as well as the physical consideration in the
same way alpha x z and let me write that alpha
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x z and alpha d x are identical alpha y x
and alpha z y are identical therefore what
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we have here is not nine independent components
but only six independent components because
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all this of diagonal components this is the
diagonal these are the three of diagonal components
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this is the same as alpha x y this is the
same of alpha x z diagonally opposite diagonally
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opposite it is alpha y z
so what you have is essentially x x x y y
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z z x y x z y z so six independent components
for the polarizability tensor that's one more
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let me start this is with respect to the coordinate
system x y z ok but all the matrix that we
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have here ah the alpha x x alpha x y alpha
x z alpha x y alpha y y alpha y z alpha x
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z alpha y z alpha z z this matrix is called
the second rank tensor matrix of alpha and
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it is a symmetric matrix symmetric matrices
have the property that they can be diagonalized
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to give only what are known as diagonal elements
that means this can be diagonalized
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to give the final form of sum alpha x x prime
alpha prime alpha y y prime this this is not
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the same as but it comes after that diagonalizing
this matrix we get alpha z z prime with zeros
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every were else ok this property of transformation
defines three angles or three real coordinates
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and therefore of this six quantity that you
have six independent quantities that you have
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you reduce in to only three independent quantities
by removing the three quantities constrains
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as coordinates as rotation axis or rotation
angles therefore what is left over is three
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but there is one more property such a diagonalization
also ensures that the trace of the matrices
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alpha x x plus alpha y y plus alpha z z is
always equal to alpha prime x x plus alpha
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prime y y plus alpha prime z z that case of
the matrices the trace is some of the diagonal
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elements of the matrices the some of the diagonal
elements of the matrices the trace of the
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matrices is invariant to the process of diagonalization
in this case this is a similarity transformation
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and this being a real matrices and symmetric
matrices this similarity transformation is
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also known as an orthogonal transformation
therefore what happen is that not only do
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we have six quantities combined to what is
called three quantities with respective some
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new axis system ok so better would be to write
this as x prime x prime hm y prime y prime
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z prime z prime to indicate that we have a
new axis system this is some fixed axis system
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and in the new axis system we have only three
quantities and therefore accordingly this
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will also be x prime x prime y prime y prime
and z prime z prime this axis system is called
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the principal axis system therefore our idea
in doing experiment is to find that particular
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axis system known as the principal axis system
in which we have only three miserable quantities
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x x y y z z but they are connected by the
fact that this is a constant because if you
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don't choose the principal axis system but
some other axis system you still have the
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relation that the sum of the three is the
same as the sum of the three in that axis
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system
so there is a constrain that there is a constant
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the constant is the sum of the three therefore
we have three independent quantities but with
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one constrain three variable three quantities
with one constrain therefore finally we have
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only two independent quantities these are
called the polarizability parallel and polarizability
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perpendicular that we will talk about in raman
spectroscopy in the next lecture therefore
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what you see here is the property of what
is known as a polarizability as here tensor
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and later the tensor being transformed to
something called a principal system and then
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removing the constancy of the sum of the three
components we see that there were two independent
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polarizability components are we can measure
and raman spectroscopy gives beautiful results
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about the experimental from the experiment
it gives you beautiful information about the
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molecular polarizability will continue now
with the raman spectroscopy in the next lecture
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thank you