1
00:00:21,029 --> 00:00:29,050
welcome back to the lectures on chemistry
and the lectures on molecular spectroscopy
2
00:00:29,050 --> 00:00:36,239
the this the last week of this particular
course and what i propose to do is to give
3
00:00:36,239 --> 00:00:43,890
you some topics for you to think about along
with one more tutorial but one of these topics
4
00:00:43,890 --> 00:00:49,579
will definitely be also part of it be part
of the examination namely ramans spectroscopy
5
00:00:49,579 --> 00:01:01,370
ramans spectroscopy is the ah important spectroscopy
which was discovered by professor chandra
6
00:01:01,370 --> 00:01:11,590
shekara venkata raman sir c v raman he won
the noble price in nineteen thirty for his
7
00:01:11,590 --> 00:01:18,590
discovery of the effect by what is called
as the scattering effect
8
00:01:18,590 --> 00:01:27,119
it's unique in the entire molecular spectroscopy
in that it studies the scattered light whereas
9
00:01:27,119 --> 00:01:32,060
every other branch of spectroscopy that i
have taught you so far studies either the
10
00:01:32,060 --> 00:01:37,729
absorbed light or it studies the emitted light
and the scattered light is usually studied
11
00:01:37,729 --> 00:01:42,640
by passing the radiation through the sample
but looking at the lights scattered from the
12
00:01:42,640 --> 00:01:50,420
sample are ninety degrees to the ah sample
tube so we will see a little bit of that but
13
00:01:50,420 --> 00:01:56,640
in order to understand even the rudimentary
aspects of ramans spectroscopy one needs to
14
00:01:56,640 --> 00:02:06,759
understand what is called the concept of polarizability
polarizability is not easy to describe even
15
00:02:06,759 --> 00:02:13,420
in elementary terms or without null mathematical
terms therefore what i would do is that this
16
00:02:13,420 --> 00:02:18,420
segment of the lecture i will not talk about
the ramans spectroscopy but i introduce you
17
00:02:18,420 --> 00:02:24,450
to the idea what are called the tensors you
already have being using tensors in an indirect
18
00:02:24,450 --> 00:02:30,230
fashion in the sense you have being using
movement of inertia you have been using as
19
00:02:30,230 --> 00:02:36,930
three components for a symmetric top two components
of moment of inertia for symmetric top and
20
00:02:36,930 --> 00:02:43,459
one component for top but those are what are
called the principle movements if inertia
21
00:02:43,459 --> 00:02:48,840
and they are special quantities movement of
inertia as its what is called as a second
22
00:02:48,840 --> 00:02:54,870
rank tensor in mathematical sense
in the same way polarizability of a molecule
23
00:02:54,870 --> 00:03:01,409
which is the extent to which the molecular
electronic distribution gets distorted in
24
00:03:01,409 --> 00:03:08,879
the presence of an external electric field
and on top of that external electric field
25
00:03:08,879 --> 00:03:15,349
it gets further distorted when the molecule
itself is undergoing vibrational motion so
26
00:03:15,349 --> 00:03:21,450
there are two concepts which are involved
the distortion of the molecular electric field
27
00:03:21,450 --> 00:03:27,040
due to the external field and the distortion
of the molecular electric field due to the
28
00:03:27,040 --> 00:03:33,299
internal motion in a sense you have seen part
of this in a different ah concepts namely
29
00:03:33,299 --> 00:03:40,140
in the dipole movement where the charge centers
simply separate and therefore the dipole variation
30
00:03:40,140 --> 00:03:48,760
is what is studied in absorption and emission
spectroscopy on the other hand in ramans spectroscopy
31
00:03:48,760 --> 00:03:55,420
it is the polarizability which results in
what is known as an induced dipole movement
32
00:03:55,420 --> 00:04:05,310
and the extent of the induced dipole movement
is directly proportional in the linear approximation
33
00:04:05,310 --> 00:04:16,049
so what we wanted to do is basic concepts
plus ramans spectroscopy in this week's lecture
34
00:04:16,049 --> 00:04:25,300
some of these ideas are directly given as
tutorial also applied as tutorial which means
35
00:04:25,300 --> 00:04:29,710
you should study them for the exams and the
others are given to you as information
36
00:04:29,710 --> 00:04:42,750
so the first thing we will study is what is
called the polarizability and ah this leads
37
00:04:42,750 --> 00:04:53,120
to what is known as the induced dipole movement
in
38
00:04:53,120 --> 00:05:10,910
the molecule mu induced this is due to the
presence of the external electric field
39
00:05:10,910 --> 00:05:17,430
so what is the relation the simple relation
is that the induced dipole movement which
40
00:05:17,430 --> 00:05:29,060
is a vector is proportional to the applied
electric field which is also a vector ok but
41
00:05:29,060 --> 00:05:37,190
the important thing to note is that the two
vectors are proportional that means the proportionality
42
00:05:37,190 --> 00:05:45,770
constant can be a constant lets assuming that
the induced dipole movement is in the same
43
00:05:45,770 --> 00:05:51,940
directions as the electric field therefore
the two vectors are sort of collinear so they
44
00:05:51,940 --> 00:05:57,759
are in the same directions if it is a constant
the proportionality concepts unfortunately
45
00:05:57,759 --> 00:06:03,910
this a proportionality concepts but in the
next step i will right this as alpha e that
46
00:06:03,910 --> 00:06:16,389
alpha called the polarizability of the molecule
ok
47
00:06:16,389 --> 00:06:23,150
now if the alpha is a scalar quite obviously
the induced dipole movement and the electric
48
00:06:23,150 --> 00:06:27,949
field are in the same direction but it doesn't
have to be the induced dipole movement need
49
00:06:27,949 --> 00:06:34,569
not be in the direction of the applied field
then what happens then how can two vectors
50
00:06:34,569 --> 00:06:41,070
be related to each other that's when you need
idea what is known as a second rank tensor
51
00:06:41,070 --> 00:06:56,570
alpha is actually a second rank tensor and
the symbol for that is to write two arrows
52
00:06:56,570 --> 00:07:01,789
and in order to make you feel comfortable
with that terminology i must tell you that
53
00:07:01,789 --> 00:07:27,800
vectors that we have here or first rank tensors
and scalars or zeroth rank tensors
54
00:07:27,800 --> 00:07:34,610
so in this mathematical representations of
the polarizability as well as the movement
55
00:07:34,610 --> 00:07:40,800
of inertia you will realize when you studying
more and more that the polarizability and
56
00:07:40,800 --> 00:07:46,860
movement of inertia and later when we study
the chemical shift in the nuclear magnetic
57
00:07:46,860 --> 00:07:52,690
resonance when a related time that i will
give you a course on the chemical shift is
58
00:07:52,690 --> 00:07:58,830
also known as is also related to what is called
the chemical shielding tensor which is also
59
00:07:58,830 --> 00:08:03,639
your second rank tensor ok
therefore things gets a little more complicated
60
00:08:03,639 --> 00:08:09,349
as we go higher up in the understanding of
this concept and we start exploring them but
61
00:08:09,349 --> 00:08:16,250
you don't need to get very ah um worried about
what is a tensor how do i understand it ok
62
00:08:16,250 --> 00:08:21,580
so the rest of this five are ten minutes that
i have i will explain to you what is meant
63
00:08:21,580 --> 00:08:28,129
by the tensor not what how do we get it get
to a tensor and so on just to give you an
64
00:08:28,129 --> 00:08:41,180
introductory ah concept ok all of you know
that in two dimensions you have two mutually
65
00:08:41,180 --> 00:08:51,140
perpendicular axis x and y and ah any point
in the co ordinate plain that is in the two
66
00:08:51,140 --> 00:08:58,270
dimensional plain represents a vector which
is pointing in this direction and if you write
67
00:08:58,270 --> 00:09:08,890
to the coefficients as or x and or y ok the
x coordinate is the projection of the vector
68
00:09:08,890 --> 00:09:19,459
or which is from zero this the zero zero the
origin the projection of r on to the x axis
69
00:09:19,459 --> 00:09:30,569
gives you the length or x and the projection
of or on the y axis gives you r y those are
70
00:09:30,569 --> 00:09:34,530
the coordinate points that you have being
clotting into dimensions
71
00:09:34,530 --> 00:09:44,600
now if instead of this x y axis suppose we
use a slightly rotated axis by an angle t
72
00:09:44,600 --> 00:09:53,230
top that is x y axis the the axis system is
rotated by an angle t term which means this
73
00:09:53,230 --> 00:10:09,090
new y prime and the x prime ok or about or
an angle theta from the old x and y now what
74
00:10:09,090 --> 00:10:20,049
is the relation between x prime x y prime
x and y and y prime x and y what is the relation
75
00:10:20,049 --> 00:10:30,150
between x prime and and x and y these are
unique vector you can write x is a unique
76
00:10:30,150 --> 00:10:36,750
vector and r x is the number that multiplies
the unit vector to give the component in the
77
00:10:36,750 --> 00:10:42,320
x corrections and then to that you add the
vector that is along the y axis by multiplying
78
00:10:42,320 --> 00:10:47,850
the unit vector along the y axis which the
magnitude r y and you add that the parallel
79
00:10:47,850 --> 00:10:53,089
the triangle the parallelogram what tells
you or the triangle whatever it is all these
80
00:10:53,089 --> 00:10:56,790
things they tell you essentially what is the
vector or this ok
81
00:10:56,790 --> 00:11:05,170
so if you write the vector or in this way
notation it is r x times x unit vector plus
82
00:11:05,170 --> 00:11:15,800
r y y unit vector ok now x prime is now the
unit vector in this direction and obviously
83
00:11:15,800 --> 00:11:22,270
it has components in both the directions of
x and y therefore x prime has components in
84
00:11:22,270 --> 00:11:28,500
both this directions therefore the best way
to write x prime is to write the scalar product
85
00:11:28,500 --> 00:11:40,030
of x prime on to the x axis times the unit
vector x plus the scalar product of x prime
86
00:11:40,030 --> 00:11:50,170
on to the y axis times the unit vector y exactly
the same way you would write to the or quantity
87
00:11:50,170 --> 00:11:59,500
this way so in a co ordinate rotation this
is nothing but cost theta times x plus sin
88
00:11:59,500 --> 00:12:07,299
theta times y the scalar product between these
two vectors x and x prime unit vectors is
89
00:12:07,299 --> 00:12:11,919
of course a magnitude of times the magnitude
of x prime times are cost theta the angle
90
00:12:11,919 --> 00:12:17,030
between the two were given that since the
magnitudes are all one you have cost theta
91
00:12:17,030 --> 00:12:24,510
times x and sin theta times y and likewise
you write y prime as minus sin theta times
92
00:12:24,510 --> 00:12:34,630
x plus cost theta times y
now this is the property of the rotated coordinate
93
00:12:34,630 --> 00:12:43,560
system and its relation to the un rotated
coordinate system ok that is under coordinate
94
00:12:43,560 --> 00:12:49,880
rotations the quantity x prime and y prime
the two unit vectors are connected to the
95
00:12:49,880 --> 00:12:59,310
two un rotated coordinate system by this matrix
relation minus sin theta cost theta times
96
00:12:59,310 --> 00:13:11,250
x y all are unit vectors so yeah you have
only ok this is called the rotation matrix
97
00:13:11,250 --> 00:13:16,540
actually about the z axis which doesn't exist
in this plain but you know it is an access
98
00:13:16,540 --> 00:13:21,230
about a which is perpendicular to that because
the rotation is happening in plain and the
99
00:13:21,230 --> 00:13:27,669
rotation basically tells you that the rotation
access is perpendicular to the plain of rotation
100
00:13:27,669 --> 00:13:33,929
therefore normally we write this as or z theta
but the z coordinate does not exist in two
101
00:13:33,929 --> 00:13:38,550
dimensional plain its basically perpendicular
to that plain
102
00:13:38,550 --> 00:13:48,290
therefore what is a vector any pair of quantity
is x and y now will have a new component or
103
00:13:48,290 --> 00:14:01,360
x prime and or y prime because the axis is
now the x prime and y prime the relation between
104
00:14:01,360 --> 00:14:16,250
r x prime r y prim and r x and r y if this
relation is the same relation that you have
105
00:14:16,250 --> 00:14:26,480
there cost theta sin theta minus sin theta
cost theta then the two quantities r x and
106
00:14:26,480 --> 00:14:32,670
r y are said to be components of the vector
this is the definition of vector not something
107
00:14:32,670 --> 00:14:37,560
that you call as a quantity with the direction
and the quantity with the magnitude that's
108
00:14:37,560 --> 00:14:41,819
how you will get introduced to the elementary
idea but a proper mathematical definition
109
00:14:41,819 --> 00:14:49,549
of a vector is vectors are the number of components
of the given quantity whose transformation
110
00:14:49,549 --> 00:14:59,189
property from one coordinate system to another
coordinate system is identical this may exists
111
00:14:59,189 --> 00:15:06,860
the transformation matrix that takes r x r
y to r x prime r y prime this transformation
112
00:15:06,860 --> 00:15:14,419
property is identical to the transformation
property of the coordinate system themselves
113
00:15:14,419 --> 00:15:20,270
the un rotated coordinate system to the rotated
coordinate system whatever matrix that transforms
114
00:15:20,270 --> 00:15:27,870
the coordinate system if it also gives you
the new vector quantities from the old vector
115
00:15:27,870 --> 00:15:35,209
quantities of any pair of numbers then those
pair of numbers form the vector this is the
116
00:15:35,209 --> 00:15:38,620
definition
in three dimension you have to add the z axis
117
00:15:38,620 --> 00:15:44,059
to it and then what is meant by a rotation
in three dimension is slightly more complicated
118
00:15:44,059 --> 00:15:48,919
than your rotation into dimension because
your rotation in two dimension involves only
119
00:15:48,919 --> 00:15:54,470
one angle your rotation in three demission
involves three angles and these are called
120
00:15:54,470 --> 00:16:06,520
oilers angles
therefore let's get to the point even with
121
00:16:06,520 --> 00:16:14,299
this two dimensional representation what is
meant by a second rank tensor in a two dimensional
122
00:16:14,299 --> 00:16:25,370
system
in a two coordinate system x and y the second
123
00:16:25,370 --> 00:16:34,919
rank tensor is essentially four components
and in the case of coordinates the components
124
00:16:34,919 --> 00:16:45,610
are written as xx xy yx yy they are the products
the regular multiples and these four components
125
00:16:45,610 --> 00:16:51,420
under coordinate rotation by the same value
of theta these four components become the
126
00:16:51,420 --> 00:17:04,750
four coordinates becomes x prime x prime x
prime y prime y prime x prime y prime y prime
127
00:17:04,750 --> 00:17:16,990
and that is a rotation matrix
and these are called the unit tensors in the
128
00:17:16,990 --> 00:17:23,230
ah second rank ah in in the unit tensors of
the second rank in the two dimensional system
129
00:17:23,230 --> 00:17:30,520
and therefore any quantity which has four
components whose properties under coordinate
130
00:17:30,520 --> 00:17:39,220
rotation follow from the four components in
the un rotated directions to the four components
131
00:17:39,220 --> 00:17:47,110
in the new rotated direction and connected
by the same rotation matrix as the rotation
132
00:17:47,110 --> 00:17:52,520
matrix that connects the coordinate systems
those four quantities are called components
133
00:17:52,520 --> 00:18:12,030
of a second rank tensor these four the same
rotation matrix ditto and they are connected
134
00:18:12,030 --> 00:18:21,220
to the un rotated components x y a x and y
y ok these are called the second rank tensor
135
00:18:21,220 --> 00:18:26,760
and what is this rotation matrix and this
rotation matrix i mean what what is it compared
136
00:18:26,760 --> 00:18:31,010
to the rotation matrix two by two that we
talked about there is something called the
137
00:18:31,010 --> 00:18:38,830
direct product of let me go to the side something
called the direct product of these rotation
138
00:18:38,830 --> 00:18:47,750
matrixes i will be done in another three minutes
so that you should get a picture that if you
139
00:18:47,750 --> 00:18:59,280
take a matrix a one one a one two a two one
a two two the direct product of the matrix
140
00:18:59,280 --> 00:19:08,890
with for example itself let's write another
one b one one b one two b two one b two two
141
00:19:08,890 --> 00:19:18,140
is equal to a one one multiplying all the
four as four elements b one one a one one
142
00:19:18,140 --> 00:19:26,820
b one two a one one b two one a one one b
two two let us put a patrician so that we
143
00:19:26,820 --> 00:19:35,550
know what we are writing and this set of the
four elements are obtained by multiplying
144
00:19:35,550 --> 00:19:42,640
a one two with all the four elements here
therefore you get a one two b one one a one
145
00:19:42,640 --> 00:19:54,290
two b one two a one two b two one and a one
two b two two and the lower two elements are
146
00:19:54,290 --> 00:20:01,760
b a two one
the second row multiplying all the four elements
147
00:20:01,760 --> 00:20:11,190
a two one b one one a two one b one two a
two one b two one a two one b two two and
148
00:20:11,190 --> 00:20:20,110
the last set is the second row second column
a two two b one one a two two b one two a
149
00:20:20,110 --> 00:20:29,770
two two b two one and a two two b two two
this is called the direct product this symbol
150
00:20:29,770 --> 00:20:42,340
represents direct product of two matrixes
ok matrixes now what you have to do in order
151
00:20:42,340 --> 00:20:47,760
get this rotation matrix is to do the direct
product of the two rotation matrixes that
152
00:20:47,760 --> 00:20:53,040
you have mainly the one rotation matrix cost
theta sin theta that you have so you would
153
00:20:53,040 --> 00:21:05,100
write this as cost theta sin theta minus sin
theta cost theta direct product of cost theta
154
00:21:05,100 --> 00:21:14,200
sin theta minus sin theta cost theta ok
therefore if you took if you took the direct
155
00:21:14,200 --> 00:21:21,410
product of this you will get a four by four
matrix of which the first two the this square
156
00:21:21,410 --> 00:21:26,490
is cost theta multiplying all these four elements
and next is the sin theta applying all the
157
00:21:26,490 --> 00:21:33,250
four minus sin theta multiplying so that is
known as the direct product of rotation matrixes
158
00:21:33,250 --> 00:21:51,130
and this direct product defines the components
of a second rank tensor
159
00:21:51,130 --> 00:21:58,600
under coordinate rotation the unit tensors
the coordinate depend the coordinator system
160
00:21:58,600 --> 00:22:03,970
defining these unit tensors they will undergo
transformation to give you this and therefore
161
00:22:03,970 --> 00:22:09,120
any pair of four quantities that you have
vectorial in one dimension the two quantities
162
00:22:09,120 --> 00:22:15,580
can be vector therefore any pair of four quantities
which follow the same multiplication property
163
00:22:15,580 --> 00:22:19,670
for coordinate transfer is known as a second
rank tensor
164
00:22:19,670 --> 00:22:25,320
the same idea when it is extended to three
dimension with the help of a three by three
165
00:22:25,320 --> 00:22:33,640
rotation matrix because you have x y z it
is known as a second rank tensor in three
166
00:22:33,640 --> 00:22:42,470
dimensions so what i would suggest you is
to ah try out as the as an example what is
167
00:22:42,470 --> 00:22:53,950
a direct product of
one this one what is the result do this multiplication
168
00:22:53,950 --> 00:23:02,750
and i will give you one more which is cos
theta sin theta zero minus sin theta cos theta
169
00:23:02,750 --> 00:23:16,310
zero zero zero one with itself the same matrix
put put the same matrix here so with a simple
170
00:23:16,310 --> 00:23:24,030
practical exercise of this kind will give
you some confidence in the elementary mathematics
171
00:23:24,030 --> 00:23:28,640
associated with what are known as tensors
and therefore now you see is in three dimensions
172
00:23:28,640 --> 00:23:34,620
the instead of four quantities that we talked
about in two dimensions for the second rank
173
00:23:34,620 --> 00:23:42,260
it will be three by three multiply multiplying
three by three and therefore you will have
174
00:23:42,260 --> 00:24:03,910
nine quantities namely xx xy xz yx yy yz zx
zy zz how do they transfer form during coordinator
175
00:24:03,910 --> 00:24:12,090
rotation to the nine quantities x prime x
prime x prime y prime x prime z prime y prime
176
00:24:12,090 --> 00:24:24,010
x prime y prime y prime y prime z prime and
z prime x prime z prime y prime z prime z
177
00:24:24,010 --> 00:24:28,059
prime
and correspondingly like movement of inertia
178
00:24:28,059 --> 00:24:35,900
the polarizability mechanical the shielding
tensor all have nine components but they have
179
00:24:35,900 --> 00:24:40,060
additional special properties called they
are all symmetric and therefore what happens
180
00:24:40,060 --> 00:24:46,770
is certain components are equal and finally
you realize that the concept of tensor is
181
00:24:46,770 --> 00:24:52,020
ultimately to find the correct direction in
which we have the minimum set of measurable
182
00:24:52,020 --> 00:24:58,960
experimentally measurable quantities and those
are called principle movements of inertia
183
00:24:58,960 --> 00:25:06,870
principle values for polarizability and principle
value for chemical ah shielding tensor ok
184
00:25:06,870 --> 00:25:12,670
we will use some of these ideas in the next
lecture in determining what is known as the
185
00:25:12,670 --> 00:25:19,720
ah raman effect therefore this is a preliminary
lecture this is not part of the examination
186
00:25:19,720 --> 00:25:24,700
for this course but this is something important
for you to understand if you want to know
187
00:25:24,700 --> 00:25:31,990
more about spectroscopy in the future
thank you very much