Bragg's Law for X-Ray Diffraction Video Lecture - Civil Engineering (CE)

okay so we're going to look at x-ray
diffraction specifically we want to
derive Bragg's law okay that's what
we're going to look at so let's um let's
start here with a little sketch so we
know that there's going to be an x-ray
source often copper source there's a
electron transitions and energy and when
there's a going to be some radiation
released from that and it's a sample and
so I was going to draw over your sample
and it's this sample and that reflects
off the sample or interact with the
sample a little bit and goes to an x-ray
detector detect or there we go okay so
it comes in instant there and there's
actually an angle and it hits the sample
at does something with the sample so
we're going to want to understand what
exactly is going on there and then the
detector picks up the radiation and
tells us the intensity of the radiation
and it gives us this plot of intensity
that is how bright is the radiation
versus the angle and the angle actually
pay for historical reasons is plotted as
to theta so what happens is there's some
radiation that's quite low intense not
very much as getting to the detector and
then at certain values of 2 theta
certain angles we get these peaks and it
might look something like this so
there's a certain these particular
angles that are giving us these peaks
and so what we'd like to do is we like
to understand what it is that's causing
these peaks why what's what's going on
what is it it's happening something has
to be happening here in the sample to
cause that
so what we should do is look at more
detail here this little region of the
sample to try to understand that okay so
I'm doing a little call-out here as best
I can
my limited artistic abilities and here's
a couple of planes on the surface of the
sample and what I'll do is I'll just
drawing cartoon depiction of some some
atoms so what we can see now is at the
surface of the sample
there's atomic planes that are revealed
to the surfaces certain planes that are
parallel to the surface okay so these
are and like I said these are atomic
planes and what you might also know from
from crystallography and our
crystallographic planes and directions
is we could we could name these sort of
a mathematical notation using Miller
indices H K and L there's a certain
specific plane that we're looking at
HTML and then what I mean here by this
next one down is that that the next
plane is the same plane it's it's
parallel to the first but just
translated in space so for for example
example only we could be looking at say
if this is a cube cubic system well we
could be looking at the top plane of a
cube and the next plane down just if a
simple simple cubic situation here this
plane alright those could be the planes
that we're looking at and so of course
you'd realize that this is the 0 0 1
that is HK and L are 0 0 and 1 and this
is again 0 0 1 as well this is an
example so don't really get too hung up
over where these atoms are in space I'm
just drawing a cartoon depiction of the
atoms and then some radiation comes in
and hits the sample and some of the
radiation though bypasses that first
layer
and it's the second plane down and what
we what we know is that the radiation
coming coming out of the source here as
a wavelength and these two beams that
I've depicted beam one and we'll call it
beam two are in phase with with one
another so here let me just define the
wavelength for you okay
wavelength they have a defined
wavelength lambda and the two of them
are in phase with one that means the
peak of one corresponds to the peak of
another and the trough of one
corresponds to the trough another to
worthwhile just to spend a moment more
time describing that so here we go if we
have beam one and beam 2 and clearly
they're starting off at the same point
like this well if we sum those two up
the result will be a waveform with
should be the same wavelength so I tried
as best I could to draw the same
wavelength lambda there that's a
bad-looking lambda let's let's stay
erased man try to do better lambda there
we go
and the amplitude though amplitude a is
now doubled so now the amplitude is 2a
and this is what we call um constructive
interference and this occurs when the
two waves are in phase with one another
Peaks correspond to the peaks and the
troughs to the troughs in fact this
scanning is going to give us a high
amplitude this gives us a high intensity
so really what we're saying is our peak
are occurring when we have constructive
interference so what we'd really like to
do is we'd like to understand when these
two waves here are going to be in phase
with one another and give constructive
interference and we can figure that out
with a little bit of simple geometry if
I draw a perpendicular line there to the
beam number two when I draw identify a
point here call it point a well by the
time being two is travel to point a it's
traveled the same distance as being
number one they're parallel to each
other at the same wavelength so they
will be in phase with one another but
then beam two you'll see has to travel
this additional distance to point B and
then again additional distance to point
C and of course geometry will tell us
that line segment a B is equal to BC so
wave number two or rate beam number two
is that to travel the distance ay B
twice longer than being number one and
we want to know when that extra distance
is going to leave beam number one and be
number two in phase with one another so
let's take a look at another situation
actually let me just draw this over here
for you so say this is beam number one
now and I'm going to draw a bit number
two so be number one again there it is
come along and what about though if I
hold back be number two so I get it to
start back here
well what if if it has been held back by
this certain value specifically the
wavelength well then it starts its next
cycle exactly the same place is the
first and they're back in phase with one
another so when beam number two is held
back or is out of phase by a multiple of
the wavelength whether it's one
wavelength or you could even have two
wavelengths if you continued it out to
the left all right you continued you
said well instead we started it back
here and then or we could start it out
here three times lambda
by the time it's this point they will be
in-phase so what we need to do is we
need to say okay when this extra
distance ay B plus BC is equal to a
multiple of the wavelength they will be
in phase with one another so let's look
at this little triangle here
let's get that for you right here and we
can just about wrap up our discussion so
here's point a point a here's point B
and we said BC is the same as a B so
what we need to do is we need to
discover how long a B is well if we look
at our sketch here we see that this
angle is Theta
well if that's this is going to be 90
minus theta so therefore this has to be
theta again so I can add that to this
angle this is Theta and then of course
you can see well if we only knew this
distance here the hypotenuse we would
know that the hypotenuse times sine
theta would be equal to the line segment
a B but we do know that distance or we
can describe it because we know it's the
distance between planes with Phil Miller
indices H K and L so it's a distance
we're going to call it D it's the HKL
right that plate spacing between planes
but the Miller indices H K and L so then
like I said this is going to be sine
theta times I'll write it up this way
actually D H K L times sine theta that's
what this extra disk thats what that
distance ay B is equal to so see if we
can wrap this up here beam two travels
two times the line segment a B further
then beam one well if that extra
distance that beam two travels right
here is equal to a multiple of the
wavelength if two and a B equals an
integer of the wavelength
then it will be constructive and if it's
constructive like your set up here it
will give us a peek so what we're doing
is we're saying well the condition for
that peak is the special geometry or
special angle giving the extra distance
an equal to a multiple of the wavelength
and that is essentially Bragg's law so
let's just summarize it that's an
integer
okay so Bragg's law then just equates
those it says an integer multiple of the
wavelength let's try that again have a
lot of troubles lambda today is going to
be equal to two times a B which we said
was D H K now sine theta fantastic
so that is in fact Bragg's law and just
based on the condition for constructive
interference