Hello friends, in the previous lecture we
discussed how the atomic and molecular spectrum
is generated. Now, based on the theory of
quantum mechanics that we discussed in the
previous lectures, the transitions either
electronic transitions, vibrational transitions
or rotational transitions, they give rise
to emission or absorption of photon at a single
wavelength. That means, the energy absorbed
should be monochromatic, that means at a single
wavelength. However, in practice, it has been
observed that radiation cannot be emitted
or absorbed at a single wavelength. So, let
us understand this concept which is called
line radiation.
So, we basically studied that we have a energy
level E1 and we have a energy level E2. Now,
this energy level E1 and E2 may be electronic
levels as in the case of atoms or it may be
vibrational levels or rotational level in
the case of molecules. When an energy photon
is emitted, then the level drops down E1 to
E2. So, this gives through photon emission.
When a photon is absorbed, the level changes
from lower level to higher level and a photon
is basically absorbed by the gas.
This is what we have learned in the previous
lecture. And what it basically gives you is,
if you look at the spectrum and light absorption
coefficient or emission coefficient, it should
be a single line. That means, a monochromatic
line. So, monochromatic means single wavelength.
But what is observed in actual practice is
that this is a not a single line, but rather
it is a spread of line like this. That means,
the wavelength is not sharp.
And what basically decides this sharpness
is called broadening. So, broadening is a
concept related to both atomic and molecular
lines. And let us first discuss the nomenclature
of this. So, we have let us say a any line
we are considering, it may be a rotational
line, electronic line or vibrational transition.
This line will be not sharp. The intensity,
let us say this is s, the wavelength is lambda,
it is called s lambda.
So, s lambda is called line strength or the
magnitude of the intensity. It is called line
strength. Let us call this line strength.
Okay. Now, what we see is that this cannot
be sharp. Okay. We will see what are the reasons.
It cannot be monochromatic. Although, from
this we see that E 1 is a fixed value, E 2
is fixed value and lambda is given by E1 – E2
upon hc. So, as per this relation, lambda
should have a unique value.
Because E1 and E2 are unique value. But we
also understand that E1 an E2 is subjected
to uncertainty as per Heisenberg’s uncertainty
principle. E1 and E2 are not sharp. So, there
may be some variation in E1 and E2 as well.
So, if E1 and E2 are subjected to change,
then definitely lambda is subjected to change.
But that is just 1 cause of uncertainty. There
are other causes of line broadening also.
That we will discuss.
So, finally the line looks like this. So,
this is the sharp line that is theoretically
predicted. But actual line looks like this.
So, the peak value will be less and the line
will be broadened like a typical Gaussian
profile. It is Gaussian actually, when we
talk about Doppler broadening that we will
discuss. And it is Lorentzian in shape, when
we talk about collision broadening. So, the
line will look like this.
So, a single transition from 1 electronic
level to another electronic level will actually
lead to wavelengths. Not a single value of
wavelength, but over a range of wavelengths.
And we represent this just like we represent
Gaussian distribution by mean and standard
deviation. We represent this line by what
we called full width at half maximum. That
means, at the maximum, half of that value
and the width of this line. So, we represent
the line with what we call full width at half
maximum also called broadening width. Okay.
So, we will discuss how this is formed.
So, there are many types of broadening mechanisms
that leads to spread of lines. The first one
is the uncertainty principle of Heisenberg.
It is called natural broadening. So, what
uncertainty principle says that, you cannot
determine the energy of a state electronic
or otherwise sharply. So, there will be always
some uncertainty in its energy value. And
this energy uncertainty leads to uncertainty
in wavelength.
So, this is called natural broadening. That
means, a single monochromatic photon will
have spread out and line will be broadened.
The second one is collision between molecules.
It is called collision broadening. When we
have molecules, let us say this is the molecule,
maybe oxygen or some other molecule. And it
is emitting some photon let us say. And in
the mechanism, when it is emitting photon,
it is collided by another molecule.
Then the collision between these two molecules
basically leads to change in the wavelength
of photon. This is called collision broadening.
So, as a result of collision, the photon wavelengths
shifts little bit. The third mechanism of
broadening is Doppler Shift. You all have
observed that the frequency of sound changes
when the source, let us say a electric, a
railway engine is moving towards or away you
from you; that the frequency of sound changes.
Same thing happens in the case of stars. A
star that is moving away from you, you will
see flicker in its color. That means, the
wavelength appears to be changing for that
star. So, here, the molecules which are emitting
or absorbing radiation, they are also moving.
So, they are moving source. So, we expect
that there will be a Doppler Shift applied
to them. And the amount of radiation emitted
from this moving atoms and molecules will
be shifted by Doppler broadening.
The third effect is Stark effect. In high
temperature plasmas, where we have significant
amount of electrons, there is the formation
of electric field. And this electric field
basically perturbs the internal structure
of the atom or molecule and leads to what
we call Stark broadening. Now, this is only
relevant for high temperature applications
like plasma. It is not important for the typical
combustion applications, while these two are
important for combustion applications.
Out of this, Doppler broadening is particularly
important in low pressure combustion. So,
if we have low pressure combustion, this will
be particularly important. And natural broadening
is mostly insignificant. So, we do not have
much importance to natural bonding. So, we
will consider only collision broadening and
Doppler broadening in this lecture. There
are different mathematical functions used
to represent it. So, natural broadening, as
I explained is based on the principle of Heisenberg
uncertainty. It is generally not important.
So, we will not discuss this in detail.
So, the first thing we will discuss is collision
broadening. As I explained, if the molecule
which is part of a gas mixture is collided
with another molecule, the wavelength of transition,
the emission or absorption is shifted or spread
out. So, we represent absorption coefficient
by a function. It is called Lorentz function.
We will derive this function in an example.
And this is an example of collision broadening.
So, what basically it says is, that if you
have a line of line strength S, lambda nought
is its center wavelength or and eta nought
is wavenumber at the center where lambda nought
is predicted as per the theory we discussed.
It is sharp. So, we know that this will not
be sharp, it will be spread out. The in; the
strength of single line without perturbation,
without dispersion is S. And due to broadening,
this will become like this.
This will be broadened out. Okay. So, this
is the coordinate eta. And on this, we have
what we called absorption coefficient, kappa
eta or kappa lambda. Okay. The expression
that I have given is in term of wavenumber
eta. And we also understand that the total
strength of the lines remain the same. That
means, when we integrate, that means the area
under this curve should be = S.
So, S is basically nothing but integrated
absorption coefficient over the entire width
of this line. So, we call this line strength.
And the expression bc is basically half-width,
line half-width. Okay. So, we define sometimes
the line as FWHM. That means, full width at
half maximum. So, that means at the half peak
we define the full width. And sometimes we
define as bc or half width. So, sometimes
we define full-width, sometimes we define
half-width, just to represents the spread
of this line.
Now, this is how this Lorentz shape looks
like this. It is basically a function which
has a form 1 over 1 + x square. So, this function
when plotted looks like this. And the broadened
line based on collision broadening exactly
has the same shape, where the half-width bc
is calculated. So, it depends on temperature
and pressure. So, the collision width bc;
that means, how much wide this line will be.
So, this is full-width and this is half-width.
This is the collision width, half-width. So,
this collision width, how much the line has
broadened depends on number of parameters.
It depends on diameter D of the molecule.
Depends on the mass of the molecule, pressure
and temperature. So, these parameters, this
width depend on. We will take 1 example and
see how and where this broadening may be important.
And the absorption coefficient as we have
already discussed is given by this expression.
kappa eta is = S upon pi bc eta – eta nought
square + bc square, where eta nought is the
center line wavenumber and bc is the collision
half-width.
So, let us do 1 problem. What we are basically
interested in is understanding how this Lorentz
profile is fitted for this type of problem.
Now, this is more of a physics kind of thing.
I am just doing this example so that you understand
why this mechanism of broadening is coming
into picture. So, what we will do is, we will
take two atoms or a molecule. Let us say we
have a molecule. And this molecule is vibrating.
Now, we discussed that molecules should have
dipole moment. For example, carbon and oxygen
atom tied in this carbon monoxide molecule,
they have a dipole moment. Because, number
of electron in oxygen is more than the number
of electrons in carbon. So, there will be
a permanent dipole. While oxygen and oxygen
atom will not have a dipole, because same
number of electrons in both the atoms, it
will not have dipole.
So, this will; no permanent dipole. While
carbon monoxide will have permanent dipole,
nitric oxide will have permanent dipole and
so on. So, only a dipole will basically be
able to emit and absorb radiation. Based on
this analysis, we put some charge q and negative
charge on this – q. The mass is m, let us
say, in this case we have assumed the mass
to be same for both the atoms. It can be different
also.
As in the case of carbon monoxide, the mass
will be different. In the case of nitric oxide,
there will be, mass will be different. But
we have simplified the analysis and I have
assumed that the masses are same. Now, we
write down the harmonic oscillator equation.
So, harmonic oscillator equation for this
can be written as; now, this coordinate is
x. So x, or we can write down this as d square
x by dt square.
That means acceleration + gamma times dx by
dt. Okay. Now, what basically, I will explain
why this gamma is coming in, + omega nought
square + x is = 0. Okay. So, this is a modified
harmonic oscillator equation. Normally, you
will not have this term. The harmonic oscillator
equation will be d square x by dt square is
= – omega nought square x. This is the harmonic
oscillator equation, right.
But, we have added this term. This is basically
for damping, when we have damping or capacitance
coming into play. So, what happens is, when
the molecules are subjected to collision,
a damping term basically appears. So, let
us put a damper here, between the molecules.
So, we have a spring, as we have in simple
harmonic motion, we have a spring, mass spring
system. But there is a damper here.
So, we define gamma as c by m, where c is
the damping coefficient and m is the mass
of the molecule. And we define the natural
frequency omega nought square as spring constant
k by mass m. So, this is something I think
you might have studied how to define spring
mass system with capacitance or damper and
write down the harmonic oscillator equation.
Now, we can solve this using any method on
differential equation.
This is an ordinary differential equation.
We can solve it and the solution can be written
as x is = x nought e power – gamma by 2t
e power – i omega nought t. This is a solution
of this equation. Okay. Now, we find acceleration
x double dot. That is, d square x by dt square
is = x nought gamma by 2 + i omega nought
square e power – gamma by 2t e power – omega
nought t. So, this is the acceleration of
the system of the mass that we have calculated
using this solution.
Now, we will do a force balance. So, force
balance. So, we can write force as = q times
the electric field E. So, we have a dipole.
So, it will exert a force which is = q times
e, q is the charge and E is the electric field.
And we have, force can also be written as
mass time acceleration. So, we can also write
down force as m x dot. So, this spring system,
mass spring system, what is the driving force
for this mass spring system?
The dipole is the driving force. So, as we
see that it can only vibrate when we have
a dipole, permanent dipole. Without permanent
dipole, it cannot emit or absorb radiation.
So, the driving force for emission and absorption
in this is basically the dipole which has
electric field E. The charge is q and electric
field E. And this electric field E is related
to emission and absorption of photon. And
we are interested in finding E because we
have already solved for force, which is = mass
times acceleration. We can find electric field
E as;
So, E is = m by q x dot is = m by q. We substitute
for acceleration expression x nought gamma
by 2 + i omega nought square e power – gamma
by 2t e power – i omega nought t. So, this
is the expression for the electric field that
we have obtained. Now, what we do is, we can
just write down this E as E nought. So, we
just put it like E nought. This is the magnitude.
So, E nought e power – gamma by 2t e power
– i omega nought t.
So, this is the expression for the electric
field. Now, we have to find out, what we do
is, we apply the Fourier transform. So, what
does Fourier transform does? It basically
takes you from time to frequency domain. So,
we take to frequency domain. We write E omega.
Now, E omega is the frequency. So, E omega
becomes 1 upon 2 pi. So, some of the steps
I am just skipping. So, we apply the Fourier
transform.
So, we get 0 to infinity e power i omega t.
Okay. So, this is the expression for applying
the Fourier transform. And what we get here
is, E omega is = E nought upon 2 pi by just
substituting the value of E here and integrating.
We get E nought upon 2 pi 1 upon 2 omega – omega
nought. Sorry, this is i, where i is the root
of – 1. And this will be = – gamma by
2. So, this is the expression for electric
field in terms of the frequency domain.
Now, line strength S. The line strength is
the line strength of emission or absorption
is proportional to the electric field square.
The magnitude of electric field square. So,
we get line strength as proportional to 1
upon omega – omega nought square + gamma
by 2. So, this is basically what we call Lorentz
profile, that we have discussed earlier also.
Omega you can write down in terms of frequency
or wavenumber. So, omega is 2 pi the frequency
nu.
And we can convert it into wavenumber or wavelength,
whatever units we want. So, what we observe
here is that the collision broadening, the
reason of collision broadening is basically
the damping effect that is coming into the
harmonic oscillator model. And also, the collision
width bc is basically proportional to gamma.
Gamma, we have already defined as damping
coefficient c by m. So, this damping coefficient
is basically leading to what we called broadening
due to collision.
So, the, so this was the collision broadening.
The other mechanism of broadening is the Doppler
broadening. We discussed that when a sound
producing source is coming towards you or
going away from you, you observe a change
in frequency. So, same thing goes with the
light when we have a light source or a source
that is emitting or absorbing photon is moving,
then we observe a different wavelength or
different frequency.
And this change or observed frequency or wavelength
can be written. It depends on the velocity
of the source. That is, the molecule that
is basically emitting, so velocity of the
source. And s is basically the direction in
which the photon is emitted. We have already
discussed this many times that emission is
a directional property. So, s is the direction
of emission. So, with this, we can find out
observed frequency, wavenumber or wavelength
of the photon. And this is leads to what we
call broadening.
The Doppler broadening depends on temperature.
And the profile is Gaussian in nature. This
is Gaussian in nature. And we can plot the
profile, the broadened profile of the line
which depends on line strength, Doppler width
and Gaussian profile. So, this is Gaussian
profile of the Doppler broadening.
Now, this picture basically explains everything.
If we have a what we called Lorentz profile,
this is the Lorentz profile. When we add both
Lorentz and Doppler, what we get Doppler void
profile. So, pure Doppler, this is pure Doppler.
This is pure Lorentz. And if we have both
Doppler broadening and collision broadening
taking place in our example or in our problem,
we get a combination of Doppler broadening
and collision broadening.
That is a combination of Doppler which is
Gaussian. And combination of collision broadening
which is Lorentzian. The combined profile
can be represented with a mathematical function
given by void. This is called void profile.
And the void profile is given by this integral
relation which is very difficult to evaluate.
But it is available in books. So, one can
use this to find out the profile.
So, the this profile is void profile and this
profile is Doppler. And the combination between
the two is Lorentzian, sorry void profile,
which is a sum of Gaussian function and Lorentzian
function. A combined function is called void
function.
Now, what is the effect of this broadening
on the spectrum? We discussed this. So, if
we have no broadening, lines will be discreet
at a single wavelength. But due to broadening,
lines tends to merge, as we have seen here
in this picture. This is a spectrum of atomic
nitrogen that I showed you earlier also. I
have taken a small part of that spectrum,
only 10 to 12 angstrom wide.
And you see that lines, lot of lines have
merged together due to broadening. So, they
have merged into each other due to the effect
of this Doppler and collision broadening.
And the spectrum becomes relatively smooth.
So, this is a smooth part of the spectrum
between the lines. Otherwise the lines are
very erratic. And you see that, the calculation
of absorption coefficient is very difficult
for this type of lines.
Finally, I will take 1 more problem where
we will compare the broadening for different
type of applications. So, discuss the importance
of various broadening mechanism, that is collision
broadening and Doppler broadening that we
are mainly interested in, in various type
of applications. So, the first example is
a temperature of 300 kelvin. And we have pressure
which is = 1 atmosphere.
Now, at 300 kelvin and pressure is 1 atmosphere,
we expect that the pressure is reasonably
high. We expect collision broadening will
be important. Now, temperature is low. We
know Doppler broadening basically takes place
mainly it, mainly at high temperature. So,
in this type of problem, we expect that the
collision broadening will be important. So,
the width, the collision width for this can
be calculated as the collision width.
And these are rough calculations only. The
collision width broadening is calculated as
0.1 centimeter inverse. This is in wavenumber.
So, the collision width is in 0.1 centimeter
inverse and the Doppler width for this problem
is 0.04 centimeter inverse. Okay. So, we see
that the Doppler width is less than the collision
width. So, the collision broadening is important
for this application. Although, the Doppler
width, Doppler broadening is not negligible.
So, if you have an application where temperature
is of the order of 300 kelvin and pressure
is 1 atmosphere; so, this example is from
atmospheric radiation. So, atmospheric solar
radiation, both Doppler broadening and collision
broadening may be important. Albeit, collision
broadening will be more important than Doppler
broadening. The second example is, so this
was the application where we have; now, we
will take another example, temperature is
2700 kelvin and the pressure is 1 atmospheric
pressure. Okay.
So, I will take a now two cases. Case 1 at
wavelength of around 600 nanometer. So, this
will be roughly 60 angstrom or 6000 angstrom.
Now, this is possible with electronic transition.
Now, for this particular case, the Doppler
broadening b d comes out to be 0.1 centimeter
inverse. Roughly, Doppler broadening comes
out to be 0.1 centimeter inverse. And the
collision broadening comes out to be 0.03
centimeter inverse.
So, what we see here is that, for this type
of application, when we are talking about
electronic transitions and the wavelength
range is in visible or vacuum ultraviolet,
the temperature is high. Both Doppler and
collision broadening are important, but the
Doppler broadening is more important than
collision broadening. The b part is basically
vibrational rotational spectrum. That means,
in the wavelength region of 6 micron, which
is 60,000 angstrom.
That is far infrared. In far infrared, we
have Doppler broadening which is = 0.01 centimeter
inverse, same temperature and pressure. Temperature
is 2700 kelvin, pressure is 1 atmosphere,
so Doppler broadening is 0.01 centimeter inverse
and collision broadening for this case comes
out to be 0.03 centimeter inverse. So, that
means collision broadening does not change
much or in this case, has not changed with
the wavelength.
But, Doppler broadening has reduced. So, where
we have now almost similar magnitude of the
two broadening. Although, now in this case,
collision broadening has a slightly less value
than the Doppler broadening. And the final
example that I will take is vibrational rotational
transition at lambda is again 60,000 angstrom.
That means, far infrared. Now, the temperature
is 300 kelvin and pressure is 1 atmosphere.
So, that means atmospheric radiation but far
infrared we are considering. So, Doppler broadening
here is 0.004 centimeter inverse. And collision
broadening is 0.1 centimeter inverse. So,
here Doppler broadening not important. So,
Doppler broadening is not important. So, what
are the gases here? If you have this first
application, the first application where again
this is 6,000 angstrom.
The first application atmospheric radiation
6,000 angstrom is the application where we
have basically ozone and some amount of absorption
by H2O. We may have to consider both Doppler
broadening and collision broadening. But in
this case, we have far infrared. So, some
CO2 is absorbing. So, we have only collision
broadening. So, this is CO2, which is mainly
collision broadening.
So, we may neglect Doppler broadening in this
particular example. So, this is basically
the end of topic on this line radiation. So,
we have found how to construct the spectrum
of various atomic and molecular gases based
on the theory of quantum mechanics of rotational
and vibrational transition using Boltzmann
distribution of molecular species or population
and applying the broadening mechanism such
as Doppler and void profile for collision
broadening.
The lines becomes broad, the spectrum becomes
complicated and the lines are spread out over
a range of wavelength. Next, we will consider
approximate methods to deal with this complicated
spectrum.
