So let's get going today. And what we were talking about
last time is the discovery that the atom was not the most basic
constituent of matter, that there was a particle that
was even less massive than an atom.
And that is the electron. But today, we're going to
discover the nucleus. And so this is 1911.
And this is Ernest Rutherford in England.
And what he was interested in doing was studying the emission
from these newly radioactive elements that were being
discovered. So the emission from radium,
for example. What he did was he got a sample
of radium bromide from his friend Marie Curie.
And what was known was that this radium bromide emitted
something called alpha particles.
And the exact nature of these alpha particles was not known.
However, what was known was that they were heavy particles,
that they were charged and that they were energetic.
And, of course, today we know what these alpha
particles are. They are helium double plus,
a helium with two electrons removed.
All right. So he got this radium bromide.
And alpha particles are being emitted from the radium bromide.
And he had some kind of detector way out here,
which detected these alpha particles.
And what he found was that at the detector there were about
132,000 alpha particles per minute.
That was the count rate that he was able to measure.
And then, because he was just trying to figure out what is the
nature of these particles, he did this experiment.
He took a gold foil, a really thin gold foil here,
really thin meaning 2^10-5 inches.
That's two orders of magnitude thinner than the diameter of
your hair. I've always wondered how he
handled that. That is pretty thin.
But he put that in there. And lo and behold,
the count rate that he measured once this gold foil was in
place, the count rate was still 132,000 alpha particles per
minute. It appeared that all of the
alpha particles that were being emitted just went through the
foil right to the detector. It didn't even seem like those
alpha particles knew the foil was there.
The same count rate. It doesn't sound like a very
interesting experiment, but at that time he was working
with a person named Geiger, a post-doc named Geiger.
The same Geiger as the Geiger counter.
And Geiger wasn't too happy about the results of these
experiments. And in addition,
Geiger had this undergraduate student hanging around the lab.
This undergraduate named Marsden.
And Marsden was really excited about doing science.
He just hung around there. He really wanted to do
something. And Geiger was,
jeez, what am I going to do with this guy?
Geiger goes to Rutherford and says I've got Marsden here,
he really wants to do something, what should I do with
him? And Rutherford said well,
okay, let's have him build a detector that can swing around
that gold sample. Let's have him build a detector
that will rotate around such that the detector can be
positioned this way, so that we can look for any
alpha particles that might be backscattered,
backscattered meaning scattered back in the direction from which
they came. And Geiger said ah-ha,
this will keep this guy busy and not in my hair.
No problem. So he goes down to Marsden,
Marsden is all excited to build this detector.
He builds the detector and gets Geiger, okay,
we're ready to go. Let's try this.
And so they put the radium bromide there in this
arrangement. They're sitting there at the
detector and they hear tick, tick, tick, etc.
Hey, particles at the detector. And they go,
oh, well, that must just be a general background.
So what they do then is they take this gold foil away,
so that presumably all the particles are going in that
direction and they listen and they hear nothing.
And they put the gold foil back.
And they listen and they hear tick, tick, tick,
etc., 20 alpha particles per minute.
And they then pick another foil, platinum foil,
basically the same result. They go up to Rutherford,
they get him down in the laboratory.
Rutherford's looking over their shoulder, tick,
tick, tick, etc. Hey, there are some alpha
particles coming off. Not many.
Look at what the probability here is of the backscatter.
Hey, that probability, we can calculate that.
The probability is simply the number of backscattered
particles, 20 (the count rate of the backscattered particles,
20), over the count rate of the incident number of particles,
132,000. That probability is 2x10^-4.
Not large, but not zero. There is something coming off.
And Rutherford was amazed. And what he wrote later on was
that this experiment was quite the most incredible event that
has ever happened to me in my life.
It was almost as incredible as if you fired a 15 inch shell at
a piece of tissue paper and it came back and bit you.
So how does he interpret that experiment?
Well, what he said is that since most of these alpha
particles went right through this foil to the detector,
that must mean that those gold atoms that make up that gold
foil must be mostly empty. So the picture is that you've
got this gold atom here. And he knew,
roughly speaking, how large that gold atom was.
He knew it to be about 10^-10 meters.
But what he was proposing is that these helium ions going
through, these alpha particles are just going right through
that volume because most of the atom, --
-- and he knew that there were electrons in this atom because
he knew about the electrons already, most of that atom is
empty. However, occasionally,
once in a while, not very often but occasionally
these helium ions, these alpha particles,
what happened? Well, occasionally they hit
something very massive in that atom.
And when it hit that massive part of the atom,
it reflected, it scattered back into the
direction from which it came. And from knowing what these
scattering probabilities are and from knowing roughly the size of
this atom and the thickness of the gold foil,
he could calculate what the diameter of this nucleus was.
And that number came out to be about 10^-14 meters,
the diameter of this heavy massive part of the atom.
Now, he called that massive part, he called it the nucleus.
He called it a nucleus in analogy to the massive center of
a living cell. That's where that name comes
from. And this number of 10^-14
diameter, that's a good number to know, the relative size of
the nucleus. All MIT students should know
that. He also knew that that nucleus
really had to be positively charged.
Why did it have to be positively charged?
Well, because he knew that these atoms were neutral,
he knew already that an electron was part of an atom.
And so this nucleus had to be positively charged.
And, in addition, he made some very sophisticated
measurements of the angular distribution of the
backscattered alpha particles. And from those measurements of
the angular distribution, he was able to back out the
fact that this nucleus had a charge of plus Z,
where Z is the atomic number, times unit charge e.
So his model is the Z electrons filled this volume,
this sphere effectively of 10^-10 meters in diameter.
The electrons filled that up, but somewhere in the center
there is something that is very small and very massive that has
a diameter of 10^-14 meters. Because occasionally that alpha
particle hit that massive part so most of the mass of the atom
is in this nucleus. Now, isn't this a great UROP
project for Mr. Marsden?
What did he do? He discovered the nucleus.
Fantastic. I also want to tell you that
this kind of Rutherford backscattering experiment here
is also the basis of the experiment that was done to
discover the quark. The quark which is the
elementary particles that make up the proton,
the neutron. Same idea.
There is a high energy particle that comes into the proton and
backscatters from the quark essentially.
That is an experiment carried out by one of my colleagues in
the Physics Department, Jerry Friedman and Henry
Kendall. Henry Kendall has since passed
away. But the same idea.
It's really the Rutherford backscattering experiment that
was done. All right.
Well, now it is time for us to do our own Rutherford
backscattering experiment. And the way we are going to do
that is we're going to be scattering off of this lattice
here, this lattice of these Styrofoam balls.
These Styrofoam balls are gold atoms, gold nuclei,
and then the space around them is where the electrons are.
So this is one monolayer of gold atoms.
Now, this scale is a little wrong.
The real scale would be if we had a pinhead in the center of
this room and then the rest of the room was the atom and then
there would be another room 10-251 with a pinhead in the
center of that. That's the real scale,
but the fact that the scale isn't right, that's okay.
We are going to be able to understand the principle here.
And what we want to do is we want to be able to measure the
diameter of these Styrofoam balls or the diameter of those
gold nuclei. And we're going to measure it
by measuring the probability of backscattering.
And the way we're going to measure the probability of
backscattering, the way we are going to do
that, is we're going to take these ping-pong balls,
and they are going to be our alpha particles.
And we are going to aim them at the lattice.
And we are going to have a total of 287 ping-pong balls,
or 287 alpha particles. And so the probability of
backscattering will be simply the number backscattered over
the total number incident, which will be 287.
So this is what we are going to measure.
Now, the next thing we have to do is that we've got to relate
this probability of backscattering.
We are going to have to relate it to the diameter of those
nuclei. How do we do that?
Well, the probability of a particle backscattering is
simply going to be the total area of this crystal,
there's our crystal, the total area of this.
Now, you've got to ignore these aluminum blocks here because
this piece of equipment comes from my laboratory.
And I pressed it into service for this demonstration.
This has nothing to do with the demo that we're going to put
here. It's just structural because
this is my manipulator puller-outer from our machine.
So you've got to ignore those, but that whole area there is
equal to 2,148 square inches. And now the probability of that
backscattering is this total area of the nuclei to the total
area of the crystal. And the total area of the
nuclei then is just the sum over all nuclei, and there are 119 of
them here. So i=1 to 119 of the area of
each one of the nuclei. And so that is just 119 times
the cross-sectional area of these nuclei actually,
which is pi d squared over 4, and this is all over 2,148.
That's the probability.
This is the diameter of the nuclei.
And now I am going to rearrange that and solve for the diameter.
And when I do that I get 4.79 times P to the one half.
If I measure the probability of
backscattering by this, measuring the number of
backscattered balls over the total number we throw at it,
then I can calculate the diameter of the nuclei,
the diameter of those Styrofoam balls.
That is the idea. Now, you're going to do this
experiment. And so now the TAs are going to
give you all a ping-pong ball or two.
Come on. We need to distribute the
ping-pong balls. Here's the deal.
What you have to do is you have to aim to hit this lattice.
That's all you've got to do. Then you have to watch your
ball, and you have to see if it scatters back to you.
Now, what doesn't count is if you hit these steel bars and it
scatters back. That doesn't count,
okay? It also doesn't count if the
ball hits the Styrofoam but it continues going in that
direction. Your ping-pong ball has to
actually scatter back at you to count.
And then, when we get all done, I am going to ask whose ball
backscattered. That's what we've got to do.
Now, some of you aren't in the optimal position to hit the
lattice. I invite you to come down and
get a little closer. That is okay.
Now, are you roughly ready? Yeah.
All right. But there is one other piece of
information that is very important that I have got to
tell you. And that is that only fools aim
for their chemistry professor. [LAUGHTER] Okay,
go to it. Watch your ball.
Okay. Have all the alpha particles
been launched? Yes.
Fantastic. All right.
Now comes the moment of truth. How many of you had an alpha
particle that backscattered? Okay, so now I've got to count.
Keep your hands up because there are some lights that make
it a little hard for me to see. Anybody in this section here?
Yes. One, two, three,
four, five. Six.
Anymore? Did I get everybody?
That's it? Six alpha particles that
backscattered. Okay, so now we have to
calculate some probabilities here.
The probability of backscattering is the number
that backscattered. 6 divided by 287 and that gives
me 0.021, carrying one extra significant
figure. And if I now take that
probability and I plug this into here, I've already done a
calculation. I got a table here.
I find that that diameter to the correct number of
significant figures is 0.7. And the actual diameter of
these balls is 0.8. So you guys did a very great
job. [APPLAUSE] All right.
This works. That's how they calculated the
diameter of the nucleus. That's it.
Nothing else involved in this. That is the principle behind
how it was done. Very simple.
That is the principle behind how the quark was discovered and
the size of the quark was revealed.
That is terrific. Now, we've got the problem that
the scientific community had in 1911, 1912, right after the
discovery of the nucleus. And the problem is,
now we know the atom has got a nucleus, we know it has got an
electron, what is the structure of the atom?
How does the electron and the nucleus hang together?
In particular, what we have to do is we have
to ask what is the force of attraction that keeps the
nucleus and the electron together?
We are going to talk, here, about this classical
description of the atom. And we've got to talk about the
force of the interaction first. Well, there are four known
fundamental forces. What is the force that is the
weakest force? Gravity.
Absolutely. What is the force that is the
next stronger force? Electromagnetic.
Absolutely. What is the next stronger
force? Weak force.
And the next force? Strong force.
All right. The weak and the strong forces
here, these are intranuclear forces within the nucleus.
That's what keeps the protons and neutrons and the quarks
together. For the most part,
the weak and the strong force don't have an effect in
chemistry, with the exception for beta decay,
for some radioactive elements. That's where you need to think
about, in particular, the weak force.
Gravity also -- Well, gravity actually has no
known consequence for chemistry. It doesn't mean that there
isn't one, but it has no known consequence for chemistry.
All of chemistry is tied up in this force, the electromagnetic
force. Now, what I am going to do is I
am going to simplify it just a little bit and just call this
the Coulomb force. And why I am going to do that
is going to be a little more obvious when you get to 8.02.
But I am going to call it the Coulomb interaction.
And we know what the Coulomb force law looks like.
The Coulomb force law is the following.
Say I have a positive charge, which is my nucleus.
I am going to call that force plus e.
And at some distance R, R is the distance now,
between that nucleus and the negatively charged particle,
which is my electron. I can describe the force of
attraction between this electron in the nucleus by the following
expression. That force, which is a function
of the distance between the two charges, is just the magnitude
of the charge on the negatively charged particle times the
magnitude on the positively charged particle over 4 pi
epsilon nought r squared.
Now, I am going to treat this
force, just for simplicity purposes right now,
as a scalar. I am not going to talk about
the direction, but because this is a force
between a positive and negative charge, this is an attractive
force. This epsilon down here is the
permittivity of the vacuum. It is in there for doing our
unit conversion correctly. That is all I will say about
it. Maybe you will talk a little
bit more about it in 8.02. And then, finally,
here is the r squared. That is the important part.
That is the distance between the electron and the nucleus.
And what I want you to notice here-- I don't need the screen
down yet. Hello.
She just switched it? Oh, all right.
Sorry. It looks like I'm not going to
get to it, actually. Thank you.
What I want you to notice here is that when r goes to infinity,
what is the force going to do? Zero.
When those two particles are infinitely far apart,
you've got an infinity in the denominator, this force is zero,
there is no attraction between them.
But when r goes to zero, what does that force do?
Infinity. When r is zero,
there is an infinity large force.
And in between, what is happening is that the
closer the particles get to each other, the larger the force.
The more attraction there is. The more they want to be even
closer to each other. All right.
Now, what this equation is telling me is simply,
what is the force of attraction between the nucleus in an
electron for a fixed distance r? In other words,
if I were holding the electron and the nucleus in my hands at
some distance r, that equation would tell me
what the force was. And you can bet that in order
to keep them apart, I would really have to be
pulling like this. You'd feel it.
But you also know that if you let them go, what would happen
is that they would move toward each other.
What this equation does not tell us is how they move toward
each other. It does not tell us how r
changes with time. It doesn't tell us anything
about r as a function of time. Although we know the force law,
we don't know anything from that force law about how those
two particles move under influence of that force.
And so what we need is we need an equation of motion.
And the equation of motion that was known at that time is,
of course, Newton's equations of motion, Newtonian mechanics,
classical mechanics, the mechanics that governs the
motion of all bodies around us that we see easily and
astronomical bodies. In particular,
Newton's second law, F equals ma,
this force equals the mass of the particles times the
acceleration. And, of course,
in this acceleration term, well, I think you know what's
in there. In that acceleration term,
let me just write it out, the acceleration then is the
change in velocity with respect to time and the velocity is the
change in the position r with respect to time.
So we have a second derivative here.
What is in this force law is a way to get r as a function of
time under influence of this force.
So we take this force and we substitute it in here and we
would be able to solve a differential equation for how
those two bodies moved under influence of that force.
And when we solved this equation, we would know exactly
where the two bodies would be at all times.
As long as we know where the bodies were when they started
out, using that force law and this equation of motion,
we could tell you what r would be for all future times.
That's called deterministic. That's classical mechanics.
Well, we're not going to do it, but we could actually set up a
differential equation and solve for what r(t) is using that
force law. We could do that.
And if I did that and if I started out the electron and the
nucleus at one angstrom from each other, like they are,
roughly speaking, in an atom, what would happen
is that that electron would plummet into the nucleus in
10^-10 seconds. If I solved these equations,
I would find that r would be equal to zero and t equal to
10^-10 seconds. We've got a problem here.
The problem is that this classical physics,
Newton's second law and the force law, it is predicting that
after 10^-10 seconds the electron and the nucleus are on
top of each other, they've neutralized each other.
We no longer have an atom that has a diameter of 10^-10 meters.
That's contrary to our observations.
This is saying the atom collapses in 10^-10 seconds.
But what we see is that the atom seemingly lives forever.
So we've got a big problem here, as they did in 1911,
1912. And the problem is that this
kind of classical way of thinking does not predict our
observations. It does not predict the fact
that the atom lives together, it lives forever.
It says the atom falls apart. So it could be that the Coulomb
force law is wrong, or it could be that this
equation of motion is incorrect. And of course what's going to
turn out is that this equation of motion does not work for
electrons in atoms. You cannot use classical
mechanics to describe the electron within the atom.
We need a new kind of mechanics, and that new kind of
mechanics is going to be quantum mechanics.
That is where we will pick up on Monday.
Have a nice weekend.