I hope you had a good break.

Can you remember back to when we, before break, what we were

talking about?

The last thing we did was this gyroscope bicycle-wheel

precession to show what would happen to a nucleus that was

spinning in a magnetic field, or an

electron for that matter.

Just to rehearse it a little bit, remember the idea of a

pulse, a 90 degree pulse, that if you have a big magnetic

field-- the blue one there, really enormous--

and then the little magnet of the nucleus precesses at 100

MHz, for example, in a certain field.

And that gives rise to a constant vertical field, but a

rotating horizontal field from that one.

So the question is whether that rotating horizontal

field, which as you see it will be going back and forth

and back and forth, will act as an antenna

and give you a signal.

You should be able to pick up a radio

signal at 100 MHz.

Indeed, you should be able to, except that there's not just

one proton.

There are lots of protons, and they're in different phases of

precession.

So although they all add vertically, and you have a

substantial vertical magnetism from those, their horizontal

components cancel, so you don't see anything.

In fact, the energy is so small, of the interaction of

each of these magnets with the field, that there are ones

pointing the opposite direction, with higher energy,

almost exactly the same population.

Just a tiny, tiny difference.

But we'll look just at the excess.

Obviously, ones down will cancel ones up.

But if we look just at the net ones up, even they cancel,

because they're at different phases of rotation or

precession about that axis.

But you can do a trick.

We could consider that we're rotating with them, so that they

look like they're standing still.

And then in our frame we'll put on a little bit of a

magnetic field that's horizontal, a very weak field.

And now, as far as we can see, we don't care about the big

field anymore, because we've compensated for it by orbiting

around this thing as we're looking at it.

But what we see is that these will begin to process in our

frame around that horizontal field, so they'll do a slow

precession, all of them, in that direction.

So they'll be going like this.

They'll start here, they'll go there, then they'll go down,

then they'll go back, and then back up again.

But we're going to put on a pulse only long enough, of

this field, so that they go down to here.

That's called a 90 degree pulse, and they'll

rotate down like that.

And now forget the rotating frame.

They'll look like they're just pointing out toward us in the

rotating frame.

But if we go back to the real frame, the laboratory frame,

what we see is this whole bunch

precessing around the field.

And now as they precess around the field in the laboratory

frame, there will be a net horizontal field that they're

generating, a magnetic field.

And that will be the antenna that broadcasts a signal that

we can hear.

And what will determine the frequency of that signal

that's going to be coming out from going back and forth, and

back and forth or around?

What will its frequency be?

How rapidly are they precessing?

It says 100 MHz.

And what determined the 100 MHz?

The strength of this big magnetic field.

Remember, the more you twist on something,

the faster it precesses.

So we could make it 100 MHz.

If we had half as big a big field, it'd be 50 MHz.

Or twice the field, it would be 200 MHz, and so on.

So we get a signal that's 100 MHz radio frequency in

the laboratory frame, that we could detect with an antenna.

But in time it will relax.

This is a non-equilibrium situation when we put the

energy in to make it go down.

And in time it'll come back to equilibrium.

And that process is called relaxation.

And there are various things that control

how fast that happens.

But it will reestablish equilibrium, and it will be

very important later in the lecture about this relaxation,

and you'll see why.

So a 90 degree pulse makes the spinning nuclei, protons, or

C-13s broadcast a frequency that tells what their local

magnetic field is.

The higher the field, the faster they precess, the

higher the frequency.

Now let's first look at this as it arises in magnetic

resonance imaging, where the purpose is to locate protons

within the body using a non-uniform magnetic field.

Now, the idea of tomography is important here.

I've taken this from a Colorado Physics 2000-page.

So this is a slice through somebody's body, show their

rib cage, spine, and so on, and they're wearing some sort

of jacket that's opaque to X-rays.

And we're interested in finding what it looks like

inside, where these things are.

So what we do is we take X-rays, and send an X-ray beam

straight through and see how much gets through.

And we scan the X-ray from top to bottom, and see how much

gets through at every different x-coordinate.

So we do a scan, and it starts at the top, but oops, there's

something there.

And there's even more there.

And lots, then a little bit more.

And for when we hit the spine there's going to be quite a

bit, and so on.

But that's just a one-dimensional

picture of the density.

Now what we're going to do is take that same picture and

just smear it out to the right.

So that's the profile, top to bottom, or stomach to back, of

this particular slice through the body of bones

or whatever it is.

Now the neat trick is that you rotate that,

rotate it by 15 degrees.

And now do the same thing again, and superimpose the new

one on the old one.

And now rotate another 15 degrees and do the same trick,

scan top to bottom and add it up.

And do it again, and again, and again, and again, and

again, again, again, and again.

And see what you got now?

When you superimpose all those, you get what it looked

like, the two-dimensional slice through the thing.

So that's called tomography.

If you can get a one-dimensional projection,

and do it in lots of different directions and add them

together, you can get the two-dimensional, or in fact, a

three-dimensional picture of what's going on inside.

So that's the trick that's used, except you want to do it

for protons, not for bone.

So we want to find protons in the body.

For example, let's find where there's fluid

water in the body.

So there's a body, and we put it inside this cylinder and

wrap the cylinder with special wire, that if we cool it to

liquid helium temperature is superconducting.

So essentially, we've made a big solenoid magnet that goes

along the body's axis.

Now, what will happen?

Well, suppose that field is 1.5 Tesla, which

means 15,000 Gauss.

A Gauss, you remember, is about the size of the earth's

magnetic field more or less, so 15,000 times as strong as

the earth's magnetic field.

So what will happen to the protons in there?

Well, they're going to precess.

And in that field, at 15,000 Gauss or 1.5 Tesla, they'll

precess at 63 MHz.

So if we put an antenna in there and give a 90 degree

pulse, we're going to hear a signal at 63 MHz.

Our radio will pick that up.

So we know there are protons in the body.

Surprised?

No.

The question is, where are the protons in the body?

Now here's an analogy to figure this out.

Suppose we had a cricket in this room, and

wondered where it was.

But I'm blind.

I can hear, but with only one ear, so I can't hear it.

I don't have spatial resolution with my ear.

How can I find out where the cricket is in this room?

Anybody got an idea?

Well, I have one bit of control over the room.

I can make a temperature gradient in the room, make it

cold in front and warm behind.

Suppose I can do that.

Now how can I find the cricket?

What?

Derek?

STUDENT: Crickets chirp at different speeds at different

temperatures. and different intervals.

PROFESSOR: You hear it chirping.

You count how many there are in 13 seconds and add 40.

You establish a temperature gradient and you

listen with a stopwatch.

And if you go to Snopes, you can see that this is not an

urban legend or a rural legend.

It's true that you can do that.

And they actually show this picture of Doctor LeMone from

Boulder, Colorado, who actually did this with

crickets, and showed that it gives a very good measure of

the temperature.

So if I could establish a temperature gradient from

front to back, and count for 13 seconds and add 40, and

knew what the temperature was, I'd know how far

from front to back.

What would I do next if I want to find the cricket?

I'd fiddle with the air conditioning controls and make

a temperature gradient from left to right.

I could even make one from top to bottom, and then I'd

know what its x-, and its y-, and its z-coordinates are.

So I could find the cricket that way.

So we're going to do the same thing with the protons in the

water in the body.

So back to the body here.

What I need is not a uniform field where all the protons

are going at 63 MHz.

I want to make them faster in some regions than others.

So what I do is I put two coils around this solenoid.

And in the one near the head, I make the current go that

way, which reinforces the field.

And in the one near the feet I make it go that way, which

subtracts from the field.

So now I've generated a gradient along the body.

And it turns out to be that what's actually used is about

40 microtesla per millimeter.

So if I went, like, 25 millimeters, about an inch,

that would be 1000 microtesla, that is a millitesla.

So it'd be about one part in 15,000.

Or, pardon me, one part--

it's a millitesla out of a 1.5 tesla, so about a part per

1000 difference.

So that means if I slice the body there or there, on the

first slice--

remember, there's is a gradient from foot to head--

so there we have the average.

63 MHz is going to be a signal coming out from protons

that are in that first slice.

But in the second slice they're going to be at 63.05,

about a part per 1000, because there's a higher

field near the head.

And if I did another slice another inch or so along, it'd

be 63.1 MHz.

So if I had an antenna and could hear all these different

frequencies and how strong the signal was, I'd get a profile

of the proton distribution from foot to head.

So just like we did with the X-rays scanning down.

I now know how much water there is at different places

along the height of the body, or the length of the body.