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Tomorrow there will be quiz six based on homework six just on the
crystallography. There is a little bit at the end of
homework six on x-rays. And we will just start to talk
about x-rays today, so I think that we will leave that
out. It will just be based on crystallography.
Let's see. A couple of announcements.
And I have asked the recitation instructors to administer the test
at the end of the period so you can have some chance to ask questions.
Last day we talked about crystallography.
We were introduced to the seven crystal systems and the 14 Bravais
lattices which are shown up on the slide taken from the lecture notes.
And we saw that if we wanted to describe the arrangement of atoms in
a crystal, we could do so by combining the notion of the Bravais
lattice of which there are 14 distinct types with a basis.
The basis could be a single atom. Could be a group of atoms. Could
be an ion pair. Could be atom pair.
And we saw that got us through an elaborate set of crystals.
And then towards the end we focused on cubic crystals because cubic
And the reason I want to look at notation is I want to be able to
crystals are representative of a large number of metals in the
Periodic Table, and it is much more tractable
mathematically. We are going to focus on cubic
crystals, and I am going to continue along with that today.
In particular, I want to start looking at notation.
define a point, a line and a plane so that when we
start characterizing crystals, I can give you a mathematical
notation that represents the face of the cube, the body diagonal of the
cube, a line going on the body diagonal, a line going along the
edge. Instead of writing it in words, I want to be able to do so
mathematically. And so for that we are going to
turn to this handout which has the elements of crystallographic
notation. I want to quickly go through this.
I think last day we started talking about position,
so I have drawn a little sketch here. This is the unit cell.
The unit cell is the cube because we are talking about a cubic system,
That's the dimension a, the lattice constant. And all
and this is just the lattice. I have not put atoms here.
So I am just talking about lattice points in space.
And, just to remind us, a, b and c are the unit vectors,
and in a cubic system a equals b equals c. And each of them is equal
to a, the lattice constant. This is the lattice constant, a.
angles are right angles, so alpha equals beta equals gamma
equals 90 degrees. Just to practice, the origin,
What do we see? It is in the yz plane.
as I have shown here, all we do is write zero, zero,
zero, as you would in your math class. But don't put parentheses
around it because parentheses are reserved for planes.
We are going to use parentheses later on. Crystallographers have
their own notation. So here is point a.
By the way, I have chosen a set of coordinate axes that are in
conformity with the right-hand rule. I don't care which axis you choose
to be x. It doesn't matter. I generally like to make z vertical,
rule convention, you will get things identically
but I don't care, as long as you use the right-hand
backwards. The thumb is your x-axis, y-axis, z-axis,
rule. You need to observe the right-hand rule because in physics
however you want to do it. I have the x-axis coming out of the
you are going to find that there are certain vector forces that
board, and the y-axis and the z-axis are in the plane of the board.
are the cross-product. And, if you don't use the right-hand
If we come here, this point A is zero units out on the x-axis,
it is one unit out on the y-axis and one unit up on the z-axis,
so this will be zero, one, one. And b, as I've drawn it,
looks like it is one unit out on the x-axis, it is no units out on a
y-axis and a half a unit up on the z-axis. That is one,
zero, one-half. That is about the last time I am going to talk about
these things. It is more interesting to talk about lines.
For lines, I am going to say let's just follow along on the slide.
And the slide imitates what is going on here.
Let's see. I've got a couple of them I want to try.
Let's look at, for starters, OB.
I want to look at OB. I am going to start at the origin,
and I am going to come out to the point B. And I have a vector on
this. It is emanating from the origin out to B.
as written, will go from there out to one, zero, one-half.
And what does it say to do? It says define a vector from origin
But then it says no commas, enclose in brackets and clear the
to the point on a line, choose the smallest set of integers
fractions. They don't like to have fractions when you are talking about
and no commas. I put the origin,
lines, so clear the fractions and multiply through,
which conveniently here is the origin, out to the point
then that will just double it, two, zero, one. And you could say
along that direction. The vector OB,
what are they doing? If you can imagine going out one
more unit cell, this line would actually go through
the point two, zero, one. You are just looking at the
projection along here. You are just multiplying it.
And then it says no commas and enclose in brackets.
This is called a bracket. These staple-like things are called
brackets. That is the direction OB. Let's see. What is the other one
we are going to do? AO. It says you have to move the
origin to the base of the vector. If I want to do AO,
I have to put the origin up here at the tail of the vector.
If this is the origin, I am moving actually what?
I am not moving anywhere in the x-axis, but I am moving minus one in
In crystallographic notation, the minus sign is represented as a
the y-axis and minus one in the z-axis in order to get this
direction. I am really going zero in the x, I am going minus one in
the y and minus one in the z. And you can see that since we don't
use commas, putting minus signs would look cumbersome.
macron. A macron is a line over. This is zero, one bar, one bar.
That is equal to minus one. Crystallographers use the term one
bar. This is 0, one bar, one bar for this line A
down to origin. That is how that is done.
Then it says we can denote an entire family of directions
by carats. Let's do that.
Suppose I wanted to say all of the body diagonals.
Well, how would I get a body diagonal? If this is the origin and
I want to go up the body diagonal, it would be out one unit, out one
unit, out one unit. So one, one, one, would be a body
diagonal. But suppose I want to say all body diagonals?
That is one, one, one this direction, one, one,
one that direction. I want to get this body diagonal.
I want to get that body diagonal. How do I do that? I can do that by
We could do a few others while we are at it. Suppose I wanted to do
putting carats. These are called carats.
all of the face diagonals like this, like this, and then over here like
Those of you who have done some editing and proofreading know these
terms. This is all body diagonals.
this, like this? Well, what is a face diagonal?
Zero, one, one. All the face diagonals. What about all cube
What does it mean? It means write this out in full.
edges? What is a cube edge? A cube edge could be this one: zero,
That is O, O, one; O, O, one bar; O, one, O. I am using O for zero.
I am cluing you into crystallographers' talk.
zero, one. This would be all cube edges.
That is how they talk. O and one bar. You are going to be really hip
if you end up in an elevator with a bunch of crystallographers.
And there are two others here. One, O, O, and one bar, O, O.
Well, how many faces are there? Six. How many face edges? Six.
And, look, just going through the permutations and combinations gives
you exactly the number that you should have. This is mathematics
imitating reality. That is good. And likewise here.
shows all of the different diagonals that are possible when you mix one
If you go through all the permutations, combinations of zero,
and one bar. That is good. So far so good. That is pretty
one and one bar, you will end up with all of the possibilities for
face diagonals, all the possibilities for body
simple. Now let's look at planes. For planes we use something called
diagonals. I think I have a cartoon here that
the Miller indices for describing planes. And they are named after
the British mineralogist William Hallowes Miller who,
back in the 1830s, gave us this means of describing a plane.
And I think, rather than writing it down, we will just follow along here
because you have the notes in front of you. You know from your math
that the equation of a plane is x over a plus y over b plus z over c
equals one, **x/a + y/b + z/c = 1** and a, b and c are the intercepts
with the x, y and z axes respectively. What Miller did is he
said let's let h be one over a, k be one over b and l equals one
over c. So this becomes the equation of the
plane, hx plus ky plus lz equals one. **hx + ky + lz = 1** And now what
Because every once in a while something like this happens.
you do is take the h, k and l and enclose those in
Here is a simple one just to get our feet wet. Here a,
parentheses. In other words, he is taking the plane index to be
b and c are the unit vectors, so this is the x-axis in the same
the reciprocals of the intercepts with the three axes.
convention I have chosen. X is coming out of the plane of the
And you will see why in a second. Why do it that way?
slide, y is moving to the right and z is moving up.
Here is a plane that is shaded in. And what do you see? It cuts the
x-axis at x equals plus one-half, it cuts the y-axis at y equals plus
one, and it never cuts the z-axis. It is parallel to the z-axis.
Mathematically, we would say the z-intercept is
infinity. And who wants to have infinity? Now you see the wisdom of
Miller. Miller said let's describe this in terms of the reciprocals.
What is the reciprocal of one-half? It is two. What is the reciprocal
of one? It is one. And what is the reciprocal of
infinity? Zero. We can describe this plane as the
two, one, zero plane. Whenever you see a zero,
whatever place the zero lies that means that plane is parallel to that
axis, so right away I can say that is a plane that is parallel to the
z-axis. It does not cross the z-axis. So there it is.
See, you are excited to be back in class. There is a cool thing that
There is the formula. You take the reciprocal of the intercepts,
happens when you use Miller indices. You see what I just did? I just
put them side-by-side, no commas, and this is why we save the
drew a direction, and that direction is the two,
parentheses. That is why we don't put our position in parentheses
one, O direction. And what do you notice about the position of two,
because we are saving parentheses for the plane.
one, O versus the two, one, O plane? When you write directions,
Let's do a few more. This is a lot of fun on a Monday.
according to this formula and planes according to Miller indices,
the two, one, O direction is perpendicular to the
two, one, O plane. Two, one, O direction is
perpendicular to the two, one, O plane. Let's do another one.
This is a good one. This is a plane that cuts the y-axis at y
equals one. It is parallel to the x-axis and parallel to the z-axis.
y equals one-half, you take the reciprocal,
It never cuts the x or the z-axes, so that is an O, one, O plane. I
which is two, so O, two, O. And let's try again. Well,
mean, you can do that in your head. That is an O, one, O plane, that is
there is O, one, O direction. And what do you know? The O,
an O, one, O plane and so on and so forth. Here is an O,
one, O direction is normal to the O, one, O plane and O, two, O direction
two, O plane. Why? Well, because it cuts the y-axis at
is normal to the O, two, O plane. What do you know
about the O, one, O direction and the O,
two, O direction? They are the same direction. O anything O is going to
be along the y-axis. Now, that is an O,
two, O plane. I could shift the origin to here,
couldn't I? Then that would be an O, two, O plane and that would be an O,
two, O plane. Oh, here is some more. One, one, one.
Well, look at this plane here. It cuts the x-axis at x equals one,
the y-axis at y equals one, the z-axis at z equals one.
Shade that in. That is the one,
one, one plane, but that is also a one, one, one plane.
We can keep shifting the origin. I don't know where the origin of
the universe is, so I can move this anywhere along.
And, instead, we put the macron over the number. In that way we give the
Here is one. Here is backwards. See, this now is one,
one, one bar, because if I put the origin here I am cutting the x-axis
at plus one, I am cutting the y-axis at plus one, but I have a negative
tilt to this plane so the z-axis is being cut at minus one.
Again, we try to avoid the minus sign.
minus indication on a number, no commas in parentheses. Now,
let's do the test. Here is the one, one, one direction. And, sure
And we can do families of planes as well. Let's look at families of
enough, the one, one, one direction cuts the one,
one, one plane normally. Let's try over here. There is the one,
planes. What kind of families could we come up with if we want to do
one, one bar direction. It cuts through the one,
one, one bar plane normally. Fantastic.
families of planes with brace brackets? Families of planes use
braces. If I want to say all cubed faces that would be O, O, one.
Because O, one, O we saw cuts the y-axis parallel to
the x-z plane. That is going to tell you to roll
over zero, zero, one and one bar in this manner.
Only instead of brackets you have braces. How many faces on a cube?
There are six. How many combinations of zero,
one and one bar are there? Six. There you go. By the way,
one other thing. If you want to be hip with the
crystallographers, this is one way of writing it.
Another way of writing it is one, O, O, which is equivalent to O, O,
one. But crystallographers like to put things in ascending order.
A crystallographer would never write one, O, O.
A crystallographer would write O, O, one. That is a cool factor there.
You never know when you are going to meet a crystallographer on an
elevator. It could be the icebreaker because
they might be shy. Now, the last thing I want to do is
talk about the distance calculation, another virtue of using Miller
indices. I want to put this up here because it is on the handout,
I want to calculate the distance between adjacent planes of the same
but I want to make sure that it gets on the board. The hkl plane is
perpendicular to the hkl direction when we use such a notation.
And then I want to go back to the previous slide.
Go back two of them because it is easier to see here.
index. We can calculate the distance between adjacent planes of
the same index. What is the distance between two
adjacent planes of some value hkl? That is given by the following
formula, which again is a testimony to Miller. The distance,
and I am going to subscript it hkl because they are all the same index,
is given by a, which is the lattice constant, divided by the square root
of h squared plus k squared plus l squared where a is the
lattice constant. Let's test it.
If hkl is O, one, O, trivially that separation should
be a, the lattice constant. And I think you can see that quite
nicely up on the screen here. It is one lattice constant between
successive O, one, O planes. Let's try it with O,
two, O. It is going to be the square root of two squared which is
a over two. And, sure enough, what is the distance
between O, two, O and the adjacent plane comparable
to it? It is only half a lattice constant.
In essence, if you look here, every O, two, O plane is only half a
lattice constant away. Every O, one, O plane is full
The lowest Miller index is O, O, one. That is a lattice constant.
lattice constant. What you are really seeing then is
that the d goes down, the distance between planes goes
down as the Miller index goes up. What is the lowest Miller index?
That is the maximum separation. And I think you can see that as you
move hkl to higher values, what you are doing is you are
cutting at a higher angular slice. And, as you cut for a higher
And why am I showing you this? Because we are going to use it in
angular slice, the distance between successive
planes is going to be reduced. Mathematically what is going on,
this is getting larger in the denominator, this is remaining the
same, so that distance becomes closer and closer together.
the very next unit. Not three years from now.
We are going to use it in the next unit when we take this information
and characterize crystals. And we are going to characterize
atomic structure by the use of something that is comparable in
dimension. If we are going to measure the dimension of the human
hair, we are not going to use a yardstick. What kind of numbers are
we looking at for these distances? We are looking at something down
around some number of angstroms. We better find some tool that gives
Today I want to talk about the
me units of some number of angstroms. And that tool will be x-rays.
Now let's look at characterization of atomic structure by x-rays.
discovery of x-rays and the underlying physics.
What does that mean? That means all of these formulas
And then next day we are going to look at application.
First of all, what are x-rays? X-rays are a form of
apply. E equals h nu equals hc over lambda equals hc nu bar.
That applies. And the special characteristic of x-rays is lambda
is very short and typically there is no hard rule on this,
I know it is not an SI unit, and you know my aversion to the
but just in round numbers we are looking at between one one-hundredth
to one hundred angstroms. It is centered at about one
angstrom where one angstrom is equal to ten to the minus ten meters.
wavelength if I use hc over lambda where lambda equals one angstrom,
nanometer, but I will put it up here because I am required by the
International Union of Pure and Applied Chemistry on the Systeme
or ten to the minus ten meters. I end up with 12,
International to put this contemptible unit here,
but I love the angstroms. This is just my personal preference.
00 electron volts per photon. And this is clearly going to be
Let's plug in a value. Suppose we have one angstrom as the
powerful radiation. Let's take a look here.
I pulled this out of the back of the text. This has ionization
energies for the first seven elements.
If you look here, in green I am showing the first
ionization energy. And this is right off your Periodic
Table. 13.6 electron volts. I have rounded it to two
significant figures. It is 13.6 for hydrogen,
it is 25.0 for helium, it is 5 for lithium and so on.
These are tiny, tiny numbers. These are electron volts. Now what
we can do is look down this row. This is the last electron to come
off. The second ionization energy of helium is for a one electron atom.
Helium has two electrons. If it loses one, it only has one
left, so this is a one electron atom, this is a one electron atom and this
is a one electron atom. And look at the energies.
This is for the inner most electron. One electron atom,
that will conform to the Bohr model. And, sure enough, look at the
ratios of these numbers. These are determined by x-ray
photoelectron spectroscopy. And roughly 14 times four is 56,
so there it is. One to four to nine to 16. And it is just a
manifestation of what you have seen as the ground state energy in the
This has the power to ionize inner shell electrons.
Bohr model goes as the square of the atomic number.
Everything is making sense, but what I draw your attention to is,
look, this is the innermost held electron of nitrogen.
It is 668 electron volts and I have 12,400. This has enormous power.
This is like a bowling alley. All the pins are going flying here.
It was a dark and stormy night, really. It was a dark and stormy
This is incredible energy. What are the underlying principles here?
Well, I think we have to start with a history lesson.
We will go back to November 8, 1895.
night on November 8, 1895 in WŸrzburg, Bavaria,
which is now part of Germany. There was a physics professor there
by the name of Wilhelm Roentgen. And he was doing research,
as many people were at that time, on gas discharge tubes. We have
seen gas discharge tubes a number of times in 3.091,
and Roentgen was part of that army of researchers.
That is what his take was. He wanted to look at high voltage
And what Roentgen's specialty was, he was pursuing gas discharge
spectral analysis under reduced pressure. He was pumping down with
a vacuum pump to get the gas pressure very,
very low, and very, very high voltage.
and very, very low pressure. And let's take a look at what his
And down in the basement he had a bunch of batteries.
apparatus looked like. Well, we will start with the tube.
Chromic acid in open beakers down in the basement.
As you have seen many times, we will put the cathode to the left and
Let's indicate that with the classical symbol.
we will put the anode to the right. And he had the laboratory on the
And this altogether put out about 20 volts. And he was a professor of
second floor. His apartment was on the first floor.
physics who understood some electrical engineering. He
had a choke coil here. The choke coil would take the
current, store up, and eight times a second it would
discharge. And it would discharge at 35,000 volts.
He could take 20 volts and up it to 35,000 by the use of this choke coil.
Eight times a second he had 35, 00 volts coming off the cathode here.
You have bam, bam, bam. But speed that up about eight times.
That is what is going on. And so with 35,000 volts he launches
electrons off the cathode, and they go zooming across what is
And so he had a detector. His detector was over here off to
very, very low pressure gas and crash into the anode.
In the meantime, to keep this thing dynamically at low pressure,
this goes to a vacuum pump to keep this thing pumped down as low as he
can. And he was measuring what was going on inside this gas tube.
the side. The detector consisted of either a piece of cloth or a paper
towel which had been painted with a chemical that would glow when it was
struck by something. What would the something be?
The something would be electrons or photons that have an energy high
enough to cause excitation of something in the paint and then
reemission in the visible. It doesn't do any good to excite
electrons in the detector to have them reemit outside the visible.
And this is called a scintillation screen. It comes from the Latin
That is what is going on. This was paper or cloth,
and it was painted with an aqueous solution of barium plantinocyanide
which glows green when it is excited.
word for spark. Scintilla means spark in Latin.
And a streetlight was coming in. Oh, by the way, Roentgen suffered
When the screen is struck by some form of energy it will glow.
And so he had dinner. He was up on the second floor cranking away after
dinner. It was dark and raining. He had the windows open.
from Daltonism. He was red-green colored blind.
And this was a green screen. And with 35,000 volts,
he decides to darken the room so he puts a cardboard box over the tube.
even at reduced pressure, given the capacity of those days for
Now the tube is enclosed in a cardboard box so it is not glowing
vacuum pumps, he still had a lot of glow from the tube.
visibly to him. It is glowing inside the box.
The tube is glowing, he has a streetlight in the
And he continues to see scintillation,
background, so he decides to close the drapes and take out
continues to see things glowing. He says that is something
the streetlight. And then, furthermore,
interesting. The other thing that happened which was kind of
interesting, his graduate student who had prepared some
of the apparatus -- I guess he was practicing painting
the scintillation screen. What the student had done was to
paint the letter A on a paper towel, and he left the paper towel on the
lab bench. Now, Röntgen has the tube enclosed in
cardboard, this thing is glowing and he looks down and sees the letter A
and the letter A is glowing. He is going, wait a minute, this is
crazy. He takes a piece of black paper and
puts it here in between the tube and the screen, and the screen continues
to glow. He had a deck of playing cards in his lab.
Don't laugh. That is how Mendeleev discovered periodicity.
He takes a playing card and puts it in front and the tube continues to
glow. And then he takes a book and puts it in between and the tube
continues to glow. He is going wait a minute.
Then he grabs a piece of lead foil. He puts the lead foil in here.
And what happens is that where the lead foil is the screen is not
glowing, but then he sees the outline of the bones of his hand on
the lead holding the lead foil. All of this stuff is going on, and
he just goes to town. He says, you know what,
this cannot be electrons because electrons cannot live in air.
They will be stopped by air. And he knew about radio waves from
Hertz and said could this be some form of mysterious radiation that is
capable, and he was really afraid to say this, of penetrating matter?
It is 1895. Are you going to come out publicly and say I have a form
of radiation that can penetrate matter? He gets a magnet out.
He moves a magnet and it doesn't change, so this stuff is not
sensitive to magnetic field. But, if he puts a magnet here,
of course he gets changes. He cannot see this stuff.
He can only see evidence of it. It is mysterious. He calls it x,
the unknown quantity. It is x radiation. This is November 8,
And then around Christmas time of 1895 he sends off a manuscript.
1895. And, to show you what a great scientist he was,
rather than rushing to publication, he spends all of November and
December repeating the experiment and trying it in different ways
until he is convinced that the effect is real.
And it is published in the first week of January of 1896 and takes
the world by storm. He is announcing that there is a
form of radiation that can penetrate matter. And, furthermore,
he has an x-ray of a human hand. This is the 1890s and people are
very prude, very private, and he has a form of radiation that
can look inside the human body. This is really shocking stuff.
Already on January 16, 1896 there is an article in The New York Times
He had his photographic film stored in a cabinet on the other side of
about this stuff, a mysterious form of radiation.
By the way, when he was doing these experiments how was he recording his
results? He was recording his results on photographic film,
and his photographic film was fogging.
the lab, and half of the time he would take the photographic film out
and it was fogged. He said whatever this stuff is,
In London there was a manufacturer who announced,
it is penetrating everything. It is penetrating the doors of his file
cabinet. They weren't steel cased. They were probably wooden file
cabinets. So, widespread adoption,
some of it silly. What are some of the silly ones?
remember this is prudish Victorian England, that he could sell you
x-ray proof ladies underwear. In France, there was no such thing.
The French don't care about such matters. Remember,
the French Daguerre had invented halide photography.
So someone in France offered to x-ray the human soul.
This is very good. I am ashamed to say what happened in
students, thereby creating, and I quote, an enduring impression.
the United States. The New York City College of
Physicians and Surgeons announced that they would use x-rays to
project anatomical diagrams from textbooks onto the brains of medical
In Iowa, somebody offered to x-ray copper pennies and thereby turn them
into gold. That is what happened. But, seriously,
what really did happen very quickly was the use of x-rays as a
diagnostic tool. Already in February of 1896 there
was a Scottish physician who used x-rays. It is simple.
You could do this in a high school. I mean what is it? It is just a
gas discharge tube, a pair of electrodes with a feed
through, and he has 35, 00 volts, I mean that is the kind of
stuff that comes off the chassis of your television set.
This is trivial stuff. People were putting these things
everywhere. They were sitting in offices in open
view. Nobody knew. This physician had a seamstress
come to him. She was involved in an industrial accident in a mill.
A needle broke off in her hand. Now, how would you find the needle
in a hand? You cannot. X-ray, they identified the location
of the needle, performed the surgery and boom.
For dental applications it was immediately adopted.
They were already using it in 1899 as a form of treatment for cancer.
Very interesting. It took over. What is the relative
physics? What is going on here? Where do we have to go? I say we
have to look at the anode because that is where the electrons are
Let's just see what might be going on. I want to show you the power of
crashing into. That is the site of the collision.
Something is happening at that anode. Let's look inside the anode.
Suppose it is copper, what could we say about it? This is where the
electron collision is. That is the maximum impact.
estimation. I have shown you that with nitrogen it is 668 electron
volts to get to the inner shell electrons. What about in the case
of the anode? What is going on there? Well, I can say what is the
energy of the 1s electron in copper 28 plus? That is a one
electron system. And you know that in real copper
with all of its electrons this value is going to be an over-estimate
If I take 13.6 electron volts times 29 squared, I get 11.
because the presence of all the other electrons is going to sap some
thousand electron volts, which is still less than the 35
of the protonic Coulombic force. This will be an upper bound of what
thousand electron volts that Roentgen had. Roentgen is running a
that energy could be. And we know that is simply equal to
huge bowling alley here. He is able to knock out 1s
the energy of the 1s in hydrogen times z squared.
electrons from copper. And what happens when you knock out
inner shell electrons? What is the next thing that happens?
Let's look at an energy level diagram.
Let's suppose that this is the anode. This is the anode in the gas tube.
This is an energy level diagram. This is going to be n equals
infinity. And we will just do a few of them. This is n equals four,
n equals three, n equals two and n equals one. Not to scale.
I am trying to show that they are a little bit farther apart as you go
lower and lower. And so this is ground state energy
E1, this is n equals two shell, n equals three shell, n equals four
shell. And out here it is zero.
And so what do I have? I have some incident electron that comes zooming
in here. This is incident. This is the one that has been
And we have just shown that it has enough energy that it can even knock
accelerated. The energy of this electron is all kinetic.
And that is equal to product of the charge on the electron which is
elemental e times a plate voltage which is 35 thousand volts.
That is how I get 35 thousand electron volts as the energy.
out an inner shell electron. It can even dislodge an inner shell
electron. Let's see what would happen if that were the case.
It dislodges an inner shell electron. What is the next thing?
Well, if it falls from n equals two to n equals one,
I have a vacancy down here. What will happen? One electron will fall
from n equals two to n equals one. And when it does so it gives off
there is a slim chance it might even fall from n equals three to n equals
You have expulsion of k shell all of the electrons,
one or maybe even n equals four to n equals one. Well,
heck, if it can go from two to one, there is a vacancy in two. I might
get three to two, I might get four to two,
and you get the picture. You get a whole cascade here.
Each one of these gives off photons, photon emission.
but you can go all the way down to the k shell electrons and then
cascade and accompanied by photoemission.
different photons. And, first of all,
The difference is that instead of being in the visible or the infrared,
the spectroscopists don't like the numbers. This is the chemist
these energy levels are so, so tightly bound that this is in the
notation, n equals one, n equals two, n equals three,
x region of the spectrum. And we have labels for these
n equals four. The spectroscopists call the ground state k shell,
this is m shell, this is n shell. I left out a letter in the alphabet.
K, l shell, m shell, n shell. We can label this as a photon that came
And that is the subscript alpha. This means the cascade down to k
from a cascade down to the k shell. And to distinguish this k shell
electron from the k shell electron involving a transition from n equals
three down to n equals one, we have a second indicator.
shell away. That means from n equals two to n equals one,
shell and the alpha represents how far the electron traveled from one
This is like a Lyman series. This is like a Balmer series
this is a k beta photon, this is a k gamma photon.
Let's take a look at what that spectrum might look like.
because it always ends at n equals two. N equals three to n equals two
gives us a photon, and that photon is going to be
called L alpha. N equals four to n equals two will
give us a photon, and that is going to be called L
beta. I am going to get a whole spectrum.
I could calculate any one of these. I could say lambda of copper k
We know h and c. We can calculate what those energy
alpha would equal what? It is the same as you have seen
before, hc over what? The energy of n equals two shell to
energy of n equals one shell. **lambda Cu K alpha = hc/(E2-E1)
What can we do? We can measure this.
levels are. Let's plot this out. I could make a plot of wavelength
intensity and I will get an entire spectrum of lines.
This is high wavelength so that means energy is increasing from
right to left. The highest energy is over here.
What's the highest energy up there? The highest energy looks like n
equals four down to n equals one. And that is going to be over here.
This could be k gamma, then k beta, k alpha. And then the L lines are
three to two, four to two, so they are over here. This is L
alpha, L beta and so on. And these different wavelengths are
associated with different energies. And these different energies are
associated with what? They are associated with the binding
inside the atom, inside the target.
Every target has its own number of protons, its own number of electrons,
And I will know on my deathbed that the wavelength of copper k alpha is
so this set of lines is characteristic of the target.
And I had to do a whole year of x-ray crystallography as a junior
back at the University of Toronto. We used copper as the target.
1.5418 angstroms. If I walk into the room and see 1.
418 angstroms, that is copper. It comes from copper.
And any other element is going to give me a different set of lines.
What I am learning here is I have a way of identifying elements.
This is now turning into something that could be a technique
of chemical analysis. And we will return next day to see
what else we can do with it. For the last five minutes, I am
going to show you a little bit more about Roentgen and his discovery.
But I wanted to just draw attention to our winner from Friday who was
unable to attend the lecture. And so I want to ask Nivair Gabriel
Now, here is the dark and stormy
to stand and accept her acknowledgment for the fine job she
did on the Actinides thing. And she is wearing her scarf.
Why don't you come down and people can see you wearing the really super
This is Roentgen's laboratory on the second floor of his flat in WŸrzburg.
hot American Chemical Society scarf. [APPLAUSE]
night the following morning. It is sunny.
It is a museum today. And you can see the tubes and the
lead wires going down to the basement and so on.
Hold it. I don't want a lot of noise. This is the first radiograph
ever. This is Roentgen's wife Bertha. He noticed that there was a
shadow with the lead. Roentgen was smart.
He had to share the glory so he said, honey, would you mind coming
upstairs, I would like to take your picture. And so Bertha put her hand
This gave birth to medical radiography. And I would,
in the path of the x-ray, and for 15 full minutes waited while
those x-rays went through. You can see the ring on her finger
and so on. She endured in that 15 minutes greater than a lifetime dose
of what we would consider safe today. People just did not know. Anyway.
just out of curiosity, like to know if there is anybody in this room who
has never had a medical or dental x-ray. I see very few.
I see a few hands. To you I say congratulations and I hope you keep
up the good work. Here is the second thing.
We have radiography that came out of Roentgen's discovery.
Roentgen had a torsion balance. Like the Justice,
she has the two scales. On one scale you put the unknown
and the other scale you put the calibrated mass.
Whereas, the brass is less so. You have a differentiation. The
He had a set of brass weights here in a wooden box,
and he closed the wooden box and irradiated the box with the x-rays
and took this photograph. And what do you see? You see that
different materials have different electron densities and,
therefore, the wood is transparent, more or less, to x-rays.
way we see things is by differentiation.
Here you have a differentiation. He could identify what was in the
box when the box was closed. That gives rise to security
technology. Every time you go to the airport you are subjected to
this, and it started with Roentgen. Roentgen also had a gun.
It was a musket. He wasn't holding up convenience stores.
What are you laughing at? He had a gun. He went hunting.
He took an x-radiograph of his gun. And this is used today. It gives
birth to the technology of failure analysis. In fact,
there are certain types of weld, typically involving very dangerous
circumstances such as pressure vessels that are going to contain
electron density. And so you will see contrast in the
steam at very, very high pressures. The code is that every inch of the
welded seam must be inspected by x-radiation in order to find what?
Internal cracks. Because, if there is a crack, there is a difference in
x-ray photograph. From that experiment come three
technologies. And for his work,
Roentgen was the recipient of the very first Nobel prize in physics.
It was awarded in 1901. He was the unanimous choice of the world
He donated them to the University of WŸrzburg to be made an endowment for
physics community for what he had done. And you are going to see next
student scholarships. And, to this day, physics students
day more that the discovery of x-rays enabled.
at the University of WŸrzburg are recipients of scholarships that were
But one last thing, what a fine human being he was.
endowed by Roentgen with his Nobel prize winnings in 1901.
You know what he did with his winnings?
I will see you on Wednesday.