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M1L1g.txt
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M1L1g.txt
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#
# File: content-mit-8-421-1x-subtitles/M1L1g.txt
#
# Captions for 8.421x module
#
# This file has 107 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
OK.
With that motivation, we're now talking
about magnetic resonance.
And we will later do a fully quantum-mechanical description,
but to get the concepts and also understand the analogies
to a classical system, we want to understand
what happens when we have a classical magnetic moment
in magnetic fields.
And that includes static fields, but then we
want to excite the system.
We want to drive the system.
And this is time varying fields.
And we realize soon that the fields are spatially uniform.
So let me just remind you of the obvious equations of motion
that also allows me to introduce the nomenclature.
The interaction energy between a classical magnetic moment
mu and the magnetic field is mu dot B.
The force is the gradient of the interaction energy.
But it is 0 for uniform fields.
So therefore, we don't need to look at the force,
but the next thing which we then have to consider is the torque.
And when we think about the classical magnetic moment,
you can think about a compass needle,
but magnetic materials are complicated.
If I think about the simplest magnetic moment,
I think about a loop of current I and area A.
And that's sort of the classical model for magnetic moment.
So we have a magnetic moment mu.
And if we now add a magnetic field, which is at an angle,
we have a torque.
But just to be sure, to make sure,
the torque is something which is nothing else than the Lorentz
force on the electrons.
But since the electron is forced to go in a circle,
we don't have to look at the Lorentz force microscopically,
we just immediately jump to the torque,
and the torque is what describes the dynamics of the system.
OK.
So we have torque.
When we have torque, we want to formulate the problem
in terms of angular momentum.
And our equation of motion is the classical equation
of motion that the derivative of angular momentum
is given by that.
Now, what makes those equations immediately solvable and find
a very easy limit is that the magnetic moment of this system,
we assume, is proportional to its angular momentum.
Well, if you have a mechanical object which
goes in a-- If you have a charged object which circles
around a central potential, then you, of course,
find immediately that if it moves faster,
it has more angular momentum.
It has a larger magnetic moment.
So we use that as the defining equation for what
is called the gyromagnetic ratio, which, of course, is
very closely related to g-factors, which
we define later on for atoms.
The gyromagnetic ratio is the ratio between magnetic moment
and angular momentum.
And then, we find that the derivative of angular momentum
is given by the situation.
And this is now an equation which
you've seen in classical mechanic
and in many situations the solution of that
is a pure precession.
The motion is pure precession of the angular momentum
around the axis of the magnetic field.
So in other words, we have the axis of the magnetic field.
We have the angular momentum, and at a constant tipping
angle, we have the tip of the angular momentum precesses
around the magnetic field.
And the precession happens with an angular frequency, which
is called the Larmor frequency.
The frequency, the Larmor frequency,
the frequency of precession is proportional
to the magnetic field and the gyromagnetic ratio.
So let me give you an example for an electron.
The gyromagnetic ratio is 2 pi times 2.8 megahertz per Gauss.
And we've discussed last class what
it means when I take out 2 pi, because the Larmor frequency is
an angular frequency, and angular frequency is not
measured in hertz, because there's the 2 pi factor,
and I just make it obvious where the 2 pi factor is hidden.
Now this is for the electron.
But if you have an ensemble of classical charges,
an arbitrary distribution of classical charges,
where with the same charge-to-mass ratio,
you find that the gyromagnetic ratio is 1/2 of that.
And this here is the Bohr magneton,
which we will use quite often in this course.
The third example is the proton.
The proton is heavier.
It has a heavier mass, about 1,000 times heavier
than the electron, and therefore the Larmor frequency is not
megahertz per Gauss, it is kilohertz per Gauss.