We'll think of it as a plain wave.
Mathematically, we could keep track of it as an array of numbers.
And each pixel within this plane wave,
each pixel would have an amplitude and it will have a phase.
Now in a plane wave, all the amplitudes across the array are equal.
So let's just say that all the amplitudes are one.
And let's say, at a particular position in space in this plane wave,
all of the pixels have the same phase and let's say that phase is zero degrees.
Then as the wave moves downwards through the electron microscope,
we might draw it later.
Okay.So this is the same plane wave.
But after it's moved downward in the electron microscope some time later.
Now, all of the pixels still have the same amplitude, but
now their phases have advanced maybe we'll say to 90 degrees.
So just as a wave passes through a position with time,
the phase would change.
So to this wave as it passes down the microscope,
the phase of each pixel moves around and around a phase circle.
So for instance,
then the time later as this plane wave passes lower into the microscope.
[SOUND] Now again, all the amplitudes of all the pixels might be one.
But now, the phase might be a 180 degrees.
And sometime later, the phases would advance to 270 degrees.
Sometimes later, they would come back to 360 degrees and
the cycle would start again.
And so as this plain wave moves down through space, propagates through space,
the amplitudes of all the pixels stay the same.
But their phases move forward around and around,
the face circle from 0 to 360 degrees and back.
Okay.
Now let's step back and think about instead of the plain wave traveling
through a vacuum, what if the plane wave hit a sample and interacted with it?
So I'm going to draw a little sample here and let's suppose that there was
an object here in the sample that started to scatter the electron wave.
Now, in this way, scattering can be thought of an analogy to this
scattering would be like having a flat pond and taking a pebble and
throwing it into the middle of the pond and as it hit the pond, you would
see a little circular ripple expand away from the place that the pebble had hit.
In somewhat similar manner, when a plane electron wave
interacts with a scattering center like an atom.
The atom causes the phase of that part of the wave to be slowed down.
And as a result, there's a little phase ripple that emal,
that emanates from that scattering center, so that later on in the plane wave,
let's say, here, you would see a ripple in the phases.
If you were to plot a surface of constant phase,
it would have little ripples in it and also perturbs the amplitudes.
At, and as this wave propagated further down the microscope,
these ripples would expand and start to perturb the amplitude and
phase throughout of greater portion of that plane length.
Now, if there was another scattering center in that object say,
like another atom, it would also induce phase ripples.
And these ripples here in that plane wave right there, might be expanding from
that position and they would start to interfere with the ripples from the first.
Just like if you threw multiple pebbles in the lake at the same time,
each would have these circular ripples that eventually interfere with each other.
So later on down the column,
you would have these phase ripples interfering with each other and
so on if you had yet another scattering center here for
instance, it would have it's own phase ripples and
all of these would interfere together later on down the column.
Now, one of the important principles that this
analogy conveys is that just like a single flat lake,
each electron feels each atom in the sample simultaneously.
And the scattering that each produces interferes with the scattering
from all the other perturbations of that wave.
And so, each electron is by itself in the microscope.
It's not that you have one electron in the microscope that's scattered and another
scattered from a different part of the sample and these two electrons interfere.
No, if you look at how many electrons are used to form a typical image and
how fast their moving down the column, and how long the column is.
It turns out, that on average,
each electron is produced by the gun passes through the column,
scatters from the sample and contributes to the image all by itself and
then there's a long period of nothing before the next electron comes.
And individually, fields the entire object is scattered by many scattering centers.
That scattering is collected by the lenses and
contributes to the image at the bottom.
In fact, this calls to mind the famous double slit experiment
that first proved that electrons have a wavelike nature.
In the experiment,.
Electrons were fired at a small slit and
produce a particular pattern behind that slit.
And then when two slits were introduced, the pattern was different.
And the pattern evidenced that each electron as it came thru had to be
feeling both slits simultaneously.
It had to go thru both slits simultaneously and
interfere with itself to produce the pattern that it did.
Like manner an electron microscope is like a billion slit experiment.
Each electron individually fills all the atoms with the sample and
collects all of their scattering to produce an image.
It's an, it's an amazing thing.
Now, I told you that what the scattering centers did is produce a phase ripple.
There's one context in which this is easier to understand,
which is the scattering of an X-ray by the electrons in an atom.
So, I'm going to draw an atom, it has a nucleus and it has an electron
cloud surrounding it and will have a single electron in the cloud.
And now, I'll draw an incoming X-ray.
Now the X-ray can be understood as an oscillating electric field.
So this is an electric field that is pointing in one direction and
then the other direction and the other direction.
And that electric field is oscillating strong to weak, to strong to weak and
changing direction.
And as it passes by one of the electrons in a sample,
because electrons experience a force when they're in an electric field.
One might imagine that this electron starts balancing up and down and
up and down.
A moving electron as it experiences this,
this oscillating electric field going by it.
As the photon fly's by it,
you might imagine the electron is oscillating up and down.
Now we know from physics that any charge that's oscillating,
moving up and down and up and down is going to create
an oscillating electric field around it that then propagates outwards.
So you have an oscillating electric field in response to its own movement.
And this is a way to think about scattering.
The energy coming in as the alos, oscillating electric field
of the photon is converted by the movement of the electron up and
down and spread into all these directions, scattered.
It's scattered and
spread into these directions by the movement of the electron.
Now this is an analogy, what happens when the incoming particle
is actually an electron like in an electron microscope?
We don't actually know what it is about the electron that is oscillating,
if anything is oscillating.
And so we don't know exactly what it means when that electron is
scattered by some other electron in an atom of the sample.
But we believe that it's in some similar fashion that
the energy of this incident electron is scattered into
all directions as it interacts with an electron of the sample.
Now this idea leads us to the second way to understand how the scattering
from all the different scattering centers interferes.
Imagine that the incoming electron wave,
now I'm going to draw it as a series of straight lines, moving in this direction.
You can think of these lines ass wavefronts where the wave all has
a uniform phase like where the phase is all zero, say.
And as it moves forward again, here again the phases are all zero and
here again the phases are all zero, phases are all zero.
So, as this wave propagates forward,
the phases are advancing around the phase circle.
And then if it impinges upon the sample,
it hits a large number of gathering centers.
So these dots are supposed to represent individual scattering centers within
the sample.
And because the wavefront hits them all at the same time, they're all,
they're scattering is induced in phase.
And so then, they start producing phase ripples away from their centers.
That one does and this one does.
This one does as well, I was just skipping it to, for speed.
Each induces phase ripples emanating from its center.
Now, if you follow these ripples as they expand outwards and
the ripples from each of these scattering centers.
What you find is that there's some directions where they all
constructively interfere.
So for instance all of these phase ripples you'll
see start to be in phase in a forward direction.
And the further out you go, the flatter that line
of ripples that are all in phase will appear.
And so you can see that the incoming wave is essentially transmitted
through it into another wave that's headed straight on forward.
But in addition to that, there's other directions where the scattering occurs,
you can see that this, these phase ripples are in this direction.
And the phase ripple from this scattering center has a front moving
in that direction as does this one moving in this direction.
And so you can see that there's going to be wavefronts moving in this direction
as well, where there's significant constructive interference.
So we can think of this as a scattered ray moving at a scattering angle and
we can call that theta.
Some scattering angle and it moves forward and
it turns out the angle there depends critically on the distance between these
individual scattering centers as we'll see later.
But of, of course, there's other directions as well.
In this case, you know, the ripples from this scattering center and
this ripple from that scattering center and this one from that scattering center.
These all build up to create another set of
another series of wavefronts that's moving in this direction.
So this is yet, another scattered wave.
So, in this way, we can think of scattering from a large
number of centers as resulting in separate scattered
beams each going in a particular direction.
Now letâs take it one step more complicated and imagine that instead
of our sample being a series of equidistant scattering centers,
instead letâs imagine that our sample is a set of proteins.
A very small crystal of proteins.
Letâs imagine that these proteins have some alpha helices.
Okay.
And then they're connected by a loop and then another alpha helix and, and
that's it.
And in our sample, there's one of the proteins is here.
Another of the proteins is there.
Another of the proteins are there.
And they're spaced at a particular distance.
Now, as the incoming electron wave, impinges on this sample.
There are a number of scattering centers around here in this protein and
a number of scattering centers here.
And a number of scattering centers here.
And each one of these will cause phase ripples to emerge away from itself.
And as these fav, phase ripples move forward, it turns out that there is
a particular direction where the scattering will be constructive.
