Hello everyone. Welcome to this material characterization
course. So far we have seen the scanning electron
microscopy as an instrument and its principal,
application and so on and from this class
onwards we will look at the X-ray diffraction
in much more details as we have already discussed
in the fundamentals of this, this course we
have gone through the basic aspects of electromagnetic
radiation and in that respects, we have also
looked at the indirectly the properties of
X-ray to some extent.
And this lecture we will just look at the
properties of X-ray in specific and then how
they are generated and how this X-ray diffraction
technique is exploited in understanding the
crystallography, phase identification and
quantification of phases and texture etc during
the next ten lectures.
So if you look at the, the fundamental ideas
what we have generated so far X-rays also
falls into the category of electromagnetic
radiation and then all this you know the wave
properties what we have just discussed in
terms of electrons as well as the other electromagnetic
radiation will holds good and I also assume
that before we get into the actual syllabus
content, I assume that you have enough or
a basic crystallographic knowledge to observe
this ideas whatever we are going to discuss
and I am not going to spend exclusively some
time on the crystallography.
But then I will just discuss the concepts
then and there wherever it is necessary. So
in this class I would like to just talk about
the general properties of X-rays and how is
the X-ray spectrum is going to look like and
what are the characteristic X-rays a little
more detail about that and from next lecture
onwards we will talk about production and
then how they are actually used in the practical
application and so on.
So yeah now look at this electromagnetic spectrum
which you are already familiar with. We are
now going to concentrate only on this X-rays
where you see that the range here is around
10-2 to here about 102 here and then you have
the corresponding photon, photon energy in
electron volts and you have this classification
here or the X-rays depending upon its wavelength,
whether it is hard X-rays or a soft X-rays
depending upon the, the penetrating capability
and then the wavelength.
It is all classified. We will look at the
details much more in the due course. So just
to give you an idea where these X-rays or
falling in the electromagnetic spectrum and
as I just have been mentioning all these lectures
like when you choose an electromagnetic radiation
for the material characterization you have
to be sure that you are the, the probing dimension
is equivalent to the wavelength of probing
radiation. So same thing is applicable in
X-ray of X-rays also. So in order to use X-rays
as a probe in determining the crystal structure
or any phase identification,
you have to make sure that even that material
which you are examining or the crystal system
which we are examining we have the similar
d-spacing as I mean the λ should match with
the probing dimension as well. So that is
what we are just recollecting again. For example
diffraction gratings must have the spacing
comparable to the wavelength of the diffraction
diffracted radiation, can’t resolve any,
any structure which is less than this range
of X-rays, spacing is the distance between
the parallel lines of the atoms.
So that is how the grating experiments are
I mean with these assumptions the grating
experiments are done. And we will now just
see what is the continuous spectrum there
is basic characteristic of electro I mean
X-ray spectrum is called continuous spectrum.
We will see why the word continuous. X-rays
are produced when any electrically charged
particle of sufficient kinetic energy is rapidly
decelerated. As we have already seen that
how these characteristic X-rays are produced
in, in one of the lectures in SEM where we,
we have taken up the energy dispersive spectrometry.
So the character, characteristic X-rays are
produced fundamentally in the similar manner.
The electrically charged particle of sufficient
kinetic energy how this is achieved that means
you should have an electron source and then
to accelerate its path or the moment you have
applied voltage which keeps that acceleration
and then these accelerated electrons are made
to impact on a target. These we have seen
already the same thing X-rays are generated
and their energy is rapidly decelerated.
So that is what the same thing here. We will
see in an in a systematic manner in a X-ray
diffraction meter as well. Electrons are normally
used for this purpose. X-rays are produced
at a point of impact and radiate in all direction.
You see the, the X-ray production what, what
we are going to see in a in a laboratory X-ray
diffraction, X-ray tube it is based on the
a point impact but we will also see that whether
we are only going to create X-rays by a point
impact or is there something else that is
that, that details we will see but, the characteristic
X-rays what we talk about is produced at the
point of impact and when it, I mean went it
radiates in the all direction
The kinetic energy of the electrons on impact
is given by the equation that is KE is equal
to which is in electron volts which is equal
to 1/2 m v square, e is the charge on the
electron 1.6*10-19 coulomb, m is the mass
of the electron 9.11*10-31 kilogram, V is
the voltage across the electrodes and v is
the velocity in meter per second. So this
is how the, the kinetic energy of the electron
in the X-ray tube is described and what, what
now you are seeing is a continuous spectrum.
A typical X-ray spectrum what, what is plotted
here is is an X-ray intensity and in the x
axis it is the wavelength in angstroms. So
what you have to now look at is, these curves
are plotted as a function of different applied
voltage you see that 5, 10, 15, 20 and 25.
So you have so many things to observe from
this graph we will look at one by one. What
you have to see here is you see that the,
the wavelength what you are seeing in each
of this curve is not just a single wavelength
it is a range of wavelength. So what happens
is in an X-ray tube when, when they the electron
source which is accelerated and then made
in rap I mean rapidly impact on the target
the X-rays are generated.
These X-rays are not having a particular wavelength.
They will have a range of wavelengths that
is why it is coming like this. So this is
a typical X-ray signal which is coming out
of the X-ray generation tube. So you will
have a spectrum of wavelengths which is associated
with the kind of signals you get from the
target. So the first point is most of the
kinetic energy of the electrons is converted
into heat. Only less than 1% being converted
into X-rays. Please understand this that is
why the X-ray tubes are critically cooled
by the water.
It is for most important thing that this tube
is cooled continuously during its operation
because you are seeing that only 1% is being
converted into X-rays rest of them are being
converted into heat. So X-rays coming from
the target have a mixture of different wavelengths
that is what you are seeing here it is a mixture
of different wavelengths. The intensity is
zero up to certain wave length called short
wavelength limit λswl.
This is what it is. Up to certain wavelength
you do not have any intensity. Smooth curves
are called heterochromatic or continuous or
white radiation the whole spectrum this continuous
line is called white radiation as well as
heterochromatic or polychromatic radiation
which is having a mixture of wavelengths and
continuous spectrum is due to deceleration
of electrons.
You see I just said that the electrons are
accelerated and then made to impact rapidly
on the target and then it produces an X-ray.
In that process it is not going to give you
a radiation with the single wavelength or
energy it is going to give a mixture of wave
lengths that is what we have seen. Now it
is not that every impact which is being made
on the target is giving signal with one impact
there is something like you know you will
get an X-ray characteristic X-ray with the
one impact of maximum energy that may produce
a characteristic signal.
We will see what is a characteristic signal
in a new course. But you will also have an
electron which will not make one impact which
will deflect somewhere and then finally impact
the target and then produce a signal which
may have a less energy or a range of energy
like that you get signals that is why you
get a kind of a disperse wavelength signal
here and that is very important. So the one
which makes the signal with one impact which
will for example produce a maximum energy
that you make call it I mean call it as a
characteristic signal as well.
So what you are seeing here is a characteristic
radiation which is Kα and Kβ we will talk
about it little I mean in due course but before
that it is important to note that these curves
are plotted as a function of applied voltage.
What you have to appreciate here is it is
not that you know if you, you get the characteristic
signal at all the given voltage but there
is a particular threshold voltage which only
trigger the release of characteristic X-rays
that is very clear.
So as the voltage increases the intensity
of the X-ray coming out is also increases
and you can also appreciate that as the voltage
increases the wavelength of the peak intensity
also reduces. You can see that the peak is
here, peak is here, peak is here and peak
is here. As the voltage increases the peak
intensity, the wavelength correspond to the
peak intensity also moves to the left and
then at particular wave I mean applied voltage
you see that the maximum intensity with a
very narrow wavelength.
So called a characteristic wave I mean radiation
of a particular target is known. So we should
also think about what will happen if I keep
on increasing this radiation further, what
will happen that we will see. You may increase
the intensity and what will happen to this
characteristic wavelength, will it move left
or right that we will see in next few slides.
So these are all some of the important characteristic
see how do we have to observe and then and
I this schematic plot clearly shows what is
continuous spectrum.
And what is characteristic line and why do
you get a range of wavelengths all this aspects
one can understand from this X-ray spectrum
of a molybdenum here this is a molybdenum
spectrum. So continuous spectrum is due to
deceleration of electrons. Any decelerated
charge emit energy. The electrons which are
stopped in one impact will give rise to photons
of maximum energy that is X-rays of maximum
wavelength for such transition we may write
eV equal to hν max which can be written like
this λ swl equal to λ minimum which is nothing
but c by ν max which is equal to hc by eV.
So what you are trying to see here is the
electrons which are stopped in one impact
how that is being visualized in terms of energy
and then indirectly the λ. What is that short
wavelength limit and how to find out that,
this is the simple expression. And if you
put all this units in it I mean and its values
constants in place and then you get a final
expressions like this as a function of applied
voltage λ short wave length is equal to 12.40
into 103 divided by V. This equation gives
the short wave length limit in angstroms as
a function of applied voltage.
So this is about continuous spectrum. Few
more marks, the total X-ray energy emitted
per second which is proportional to the area
under one of the curves also depends on the
atomic number Z of the target and on the tube
current I, the later being the measure of
number of electrons per second striking the
target. So we are now talking about the energy
of the characteristic X-rays or the X-rays
which is coming out of their target in X-ray
tube and what is the kind of energy we are
interested in.
So the total X-ray intensity is given by I
continuum spectrum which is equal to A multiplied
by i multiplied by Z times V to the power
m. A is proportionality constant and m is
a constant with the value of about 2. So you
get a kind of value for a continuous spectrum
in terms of intensity using this expression
which mainly depends upon the atomic number
and the voltage.
So another important aspect of this continuous
spectrum is characteristic spectrum as I said.
You can look at the schematic which is again
plotted versus intensity of the X-rays versus
wavelength in angstrom. When the voltage is
raised above a certain critical value characteristic
of the target metal, sharp intensity maxima
appear at certain wavelengths superimposed
on the continuous spectrum. So this is a continuous
spectrum and it is being superimposed on it
and it has got a very sharp intensity signal
with a narrow wave length.
These lines are narrow and since their wavelengths
are characteristic of a target metal used
they are called characteristic lines. So now
you have a basic explanation for what is characteristic
lines and you have to appreciate one more
thing here. If you for example this spectrum
of molybdenum is obtained at 35 KV and with
that we have the normally the Kα is resolved,
if it is not that it may appear as a single
line.
The another important thing is the as the
voltage is increased then the you may get
the, the intensity of the continuous spectrum
also will go up and also you will have the
higher intensity of your characteristic peak
however the wavelength will not change. You
have to understand that. If the voltage is
increased you may get higher intensity in
the continuous spectrum as well as the characteristic
lines but the wavelength is always a constant
very narrow range. So that is the characteristic
spectrum we talk about. Only K lines are useful
in X-ray diffraction as the longer wavelength
line being easily absorbed.
You see you have a range of signals and out
of this range of signals only K lines are
useful and you may get other signals like
you know L M and N shell so on but they will
have a very high wavelength and since they
are getting easily absorbed they typically
they are not being used in X-ray diffraction.
So only the typically only K lines are being
used we will see how this K lines are defined
and produced and typical K lines are given
here only three strongest are observed in
normal diffraction work. So you see that Kα1,
Kα2 and Kα3 these are the typical signals
you get from the K shell which is being used
for the X-ray diffraction in a normal diffraction
work.
The some more marks on the characteristic
spectrum, the intensity of the continuous
spectrum depends both on the tube current
and the applied voltage. So we can write I
K line this is equal to B times i into V minus
Vk to the power n where B is the proportionality
constant, Vk the K excitation voltage and
n a constant with a value of about 1.5. And
you have another relation called Mosley relation
where the square root of ν is equal to C
into Z minus σ. The wavelength of any particular
line decreased as the atomic number of the
emitted increased.
Where C and σ are the constants. And if you
want to just appreciate this the continuous
spectrum and its origin of this continuous
spectrum you can look at the basic all the
possible electronic transitions and this is
just brought back to you again we have already
gone through this. Just for your reference
you see the all K shell, L shell, M shell,
N shell and so on with different different
possibilities and various possibilities of
this electronic transitions and which forms
the basis for this continuous spectrum.
The difference in two shell energies equals
the energy of the characteristic X-ray. This
point we have already seen we all know if
we fill a K shell hole from an L shall we
get Kα X-ray but if we fill it from the M
shell we get Kβ X-ray. The α1 X-ray is from
the outermost shell that is LIII or MV and
the α2 is from the next innermost shall LII
or MIV and so on so that is how the, the energy
is being defined based upon which kind of
shell and which level it is coming from and
they the difference of the two shell energy
is the energy of the characteristic X-ray.
K excitation voltage is necessary to excite
K characteristic radiation. You see from the
very beginning we have seen we are seeing
that in the continuous spectrum only at particular
value of the voltage the characteristic signals
are appearing otherwise you get only a continuous
spectrum or white radiation only we are seeing.
So that clearly tells that there is a excitation
critical voltage which only can excite the
for a given shell in this case since we are
using only K shell electrons or K shell lines
we talk about K excitation voltage.
So your critical voltage is necessary to excite
K characteristic radiation and increasing
the voltage above the critical voltage increases
the intensities of the characteristic lines
relative to the continuous spectrum but does
not change their wavelengths. This also we
have just seen. You have the characteristic
lines at a critical or I would say the excitation
voltage of K shell.
As the voltage increases the intensity will
increase but not the wavelength. And these
are all the, some of the application of this
mostly relation between the frequency and
the atomic number of two characteristic lines
where K α1 and L α1 are shown, how this
frequency and atomic numbers are related with
respect to these two levels and also the wavelength.
These curve shows that L lines are not always
of longer wave lengths. The L α1 line of
a heavy metal like tungsten they have the
same wavelength like K α1.
Critical excitation voltage is required for
a characteristic radiation that we have seen.
For example K radiation cannot be excited
unless the tube voltage is such that the bombarding
electrons have enough energy to knock an electron
out of the K shell of the atom. So we will
not talk about the work. The work required
to remove a K electron then the necessary
kinetic energy of the electrons is given by
WK equals 1by 2 mv square. So WK will determine
the energy required to knock out an electron
from the K shell of the atom.
So similarly you will have WL, WM and so on
depending upon the amount of energy required.
So you can guess that since K is K shell is
very close to the nucleus which will require
highest energy to remove the electron as compared
to M, N, L and so on because they are further
away from the nucleus so you will me you may
require less work as compared to, to remove
an electron from K shell as compared I mean
compared to other M, N and L shell and so
on.
So that is the idea one should get from this.
So when X-rays encounter any form of matter
they are partly transmitted or partly absorbed.
See now we talk about the properties of X-rays
and its interaction with matter. So in order
to appreciate this characteristic spectrum
it is not only important to understand the
interaction of electrons and, and matter and
you, you have to understand the interaction
of X-rays with the matter has well. So in
that context we, we talk about little bit
about this absorption of the X-rays.
And the first point is this. So when the X-rays
are in when the X-rays encounter any form
of matter they are partly transmitted or partly
absorbed. The fractional decrease in the intensity
I of an X-ray beam as it passes through any
homogeneous substance is proportional to the
distance traversed, that is minus dI by I
equal to μ dx, where the proportionality
constant μ is called linear absorption coefficient
and is dependent on the substance, its density
and the wavelength of the X-rays, where if
you can integrate this equation you can write
Ix equal to Io times e to the power minus
μx, where Io is the intensity of the incident
X-ray beam and Ix is the intensity of transmitted
beam after passing through a thickness x of
the material. The linear absorption coefficient
μ is proportional to the density ρ which
means that μ by ρ is a constant of a material
and it is independent of physical state whether
it is a liquid solid or a gas. μ by ρ is
called mass absorption coefficient.
So if you consider this into account the above
equation and can be rewritten like this Ix
equal to Io into e to the power minus μ by
ρ into ρx. Whether the substance is a mechanical
mixture a solution or a chemical compound
and whether it is a solid liquid or gaseous
state it's the absorption coefficient is simply
the weighted average of the mass absorption
coefficients of its constituent elements.
Suppose w1, w2 etc are the weight fractions
of the elements 1, 2 etc in the substance,
And (μ by ρ)1, (μ by ρ)1 and (μ by ρ)2
is the most absorbent coefficients then the
mass absorption coefficient of the substance
is given by μ by ρ = w1 into (μ by ρ)1
+ w2 into (μ by ρ)2 + and whatever the number
of constituents there in the substance depending
upon that this entity also will continue like
this. So this particular slide shows the,
the way in which the absorption coefficient
varies with the wavelength gives a clue to
the interaction of X-rays and the atoms. You
see this schematic plot where you have the
energy per quantum versus wavelength, as well
as μ by ρ, this mass absorption coefficient
versus λ. So you have this two similar branches
separated by a sharp
discontinuity called absorption edge. This
is one branch and this is another branch which
is being separated by a sharp discontinuity
called absorption edge. Here it is belong
to K shell so it is called K absorption edge
and you can see the corresponding critical
energy to eject the electron from the K shell
of the nickel here which is clearly shown
here. So along each branch the absorption
coefficient varies with the wave length approximately
according to the form μ by ρ equal to k
λ to the power 3 Z to the power 3, where
k is a constant with a different value for
each branch of the curve and Z equal to atomic
number of the absorber.
So where you have the shortwave length X-rays
they are characterized as a hard X-rays where
the long-wavelength X-rays they are called
soft X-rays. So these two classifications
in fact even in the very first slide which
was marked on the electromagnetic spectrum
how these X-rays are classified as a hard
as well as a soft X-rays. So you see that
the mass absorption coefficient clearly shows
to characterize this I mean the μ/ρ - λ
plot clearly characterizes the, the edge absorption
edge of a given element. So this is the experimental
arrangement for measuring absorption where
you have the source and you have the detector
and you have slits and this is a absorber
and then you see that intensity before reaching
the sample and Ix is the after the transmission.
So the scattered radiation that the dashed
line does not represent energy absorbed in
the specimen but it constitutes energy removed
from the beam and accordingly forms part of
the total absorption represented by the absorption
coefficient μ/ρ. So we will now just rewrite
this the work that is energy required to remove
an electron from the K shell WK in terms of
photon hνK which can be written as hc by
λK.
So you now arrive at you can get the characteristic
wavelength which correspond to a K shell can
be obtained from this relation. So you see
that another typical example for absorption
coefficient of lead where you have the K edge
and L edge, LI edge LII edge and LIII edge
they are all shown having a sufficiently higher
wavelength. You can see that the kind of range
of wavelengths has as compared to the K absorption
edge in this sample.
So in order to again appreciate this characteristic
spectrum, we will just talk about a little
bit of the, the emission process that is the
energy of the atom vs the transition between
different shells here you can talk we can
just look at it to appreciate the, the characteristic
wavelength which is arises because of the
electron being removed from the particular
shell. So you see that you have this energy
of the atom where you have different shell
WK, WL, WM, WN and you have the K excitation,
L excitation and you have Kα emission K β
emission and then Lα emission and so on.
And then you see that the, the valence electron
removed from the neutral atom then you, you
also see the corresponding the other states
of energy like M and N and so on. So for all
each of these transitions all possible transition
we can write the corresponding the W that
is work required to knock out that particular
electron from the same given shell where the
subscripts K and LIII refers to the absorption
edges and the subscript Kα1 to the emission
line. So this is all summarized in one graph
and this is how the, the energy being worked
out and then the λ is being calculated using
this simple relation.
So this is a schematic plot where one can
just characterizes the the, the wavelength
corresponding to shortwave length limit, you
can find out for a given voltage, you can
characterize this plot where you have this
for example if you take electron volts for
a K shell which is WK h νK is equal to hc
by λK & VK is equal to hc by e λK and this
also we have already seen where voltage required
to remove the electron from the K shell is
equal to12.40 into 10 to power 3 by λK. So
for a given voltage you will find the λ which
is the absorption edge wavelength or I would
say the short wavelength limit or absorption
edge wavelength. In this particular case VK
is the K excitation voltage λK is the K absorption
edge wavelength in angstrom. So similarly
you can find the, the absorption edge for
the given shell from this a plot.
So we will see a few more remarks. An atom
with the K shell vacancy is an ionized state
where high energy state. It can lose this
excess energy and return to its normal state
in two ways. By emitting K radiation or by
emitting an electron which is called Auger
effect. In the Auger process a K shell vacancy
is filled from say the LII level and the resulting
K radiation does not escape from the atom
but ejects an electron from L level, the ejected
electron called Auger electron has kinetic
energy related to the energy difference between
K and LII states.
The likelihood of the Auger process can be
found from the fluorescence yield which is
defined by omega K is equal to number of atoms
that emit k radiation / number of atoms with
a K shell vacancy. See you have to remember
the Auger electron again every surface phenomenon
this is also a part of a characteristic emission
that is why it has come under this category
of you know whether when the electron is removed
from the K shell it can either it can remove
a characteristic X-rays or it can further
remove an electron from the outermost shell
like an LIII or LII, then the electron which
is coming out of that process is called if
Auger electron and this is again a characteristic
I mean characteristic signal which is also
being used to characterize the material that
we will see it in a separate lecture. But
you should know this kind of signals also
associated with when we talk about characteristic
radiation that is all I want to mention here
and then that Auger process can be found from
this fluorescence shield which is given by
this relation. So next we will talk about
the production of X-rays and its equipment
details and then how exactly the instrument
are operated those details we will start our
lectures in the next class. Thank you for
listening.
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