Chapter 34: Magnetism of Matter – Diamagnetism & Paramagnetism

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Welcome back to the Deep Dive.

Today we're digging into a really fundamental piece of physics.

We're looking at the atomic origins of magnetic behavior, specifically through the lens of Feynman's lectures, chapter 34, the magnetism of matter.

Our mission really is to figure out why materials react to magnets the way they do.

We want to go beyond just saying it attracts or it repels.

Yeah, get down to the nitty gritty, the atomic scale where electricity and magnetism, you know, really make things happen.

Exactly.

This chapter is a key transition point.

We move from looking at magnetic fields in a sort of macroscopic way to understanding that magnetism itself is born from the behavior of electrons inside atoms, their motion, their intrinsic properties.

And maybe right off the bat, we should define the two main weak magnetic types we see everywhere.

Good idea, because most things aren't like iron, right?

They don't stick strongly to a fridge magnet.

Not at all.

These effects are tiny, maybe a million times weaker than that strong ferromagnetism you mentioned.

So most materials fall into two camps, diamagnetic, meaning they're slightly repelled by a magnetic field.

Or paramagnetic, meaning they're slightly attracted.

That's the core distinction.

Paramagnetism happens when atoms in the material have their own permanent magnetic moments.

Think of them as tiny, built -in compass needles.

These moments often come from, say, electron spin or orbital motion that hasn't perfectly canceled out within the atom.

And when you apply an external field.

Well, these little atomic magnets tend to weakly align with the field.

Just a slight preference, giving you that small net attraction.

Aluminum is a good example.

Okay, so that's attraction from permanent moments, repulsion, diamagnetism.

Diamagnetism is different.

It's an induced effect.

It happens in all substances, even paramagnetic ones, though it might be overshadowed.

When you put any material in a field, the field itself causes the atoms to generate their own new magnetic moment.

And crucially, this induced moment always points opposite to the direction of the applied field.

So it pushes back.

It pushes back, exactly.

Leading to that weak repulsion, Vismuth is a classic example where diamagnetism is quite noticeable.

And you can actually see this, right?

If you hang a little piece of something near a strong magnet pole where the field lines are concentrated,

a paramagnetic material gets pulled towards the pole, but a diamagnetic one gets pushed away towards where the field is weaker.

Precisely.

It's a subtle but real force.

You mentioned temperature earlier, too.

How does that play in?

Right.

Temperature is important for paramagnetism.

Heat means more random motion, more thermal jostling of the atoms.

That random energy fights against the weak alignment caused by the external field.

So generally, paramagnetism gets weaker as things heat up.

Makes sense.

Little compass needles get shaken around more.

Yeah.

But diamagnetism, because it's an induced reaction to the field itself, it's much less sensitive to temperature changes.

It's always there, opposing the field.

Okay.

So we have these two effects.

Now let's get into the origin.

How does an atom even get a magnetic moment?

We start with the classical picture, right?

An electron orbiting the nucleus.

Yes.

The classical model is our starting point, even though we'll see its limitations.

Imagine an electron, a little charge, $2, zipping around the nucleus.

That moving charge is essentially a tiny current loop.

And any current loop generates a magnetic field, or a magnetic moment, next spirit.

Okay.

At the same time, that orbiting electron has mass and velocity, so it also has angular momentum, math BFJU.

Like a spinning planet.

Sort of, yeah.

And there's a direct mathematical link between the magnetic moment and the angular momentum, math BFJ.

The ratio, math BFJ, turns out to be simply two -meter dollars, where Tuller's is the charge, and Ahler's is the mass of the electron.

Just depends on the particle's basic properties.

Exactly.

And since the electron's charge, Tuller's, is negative, this is important, the magnetic moment vector, Laguler's, actually points in the opposite direction to the angular momentum vector, math BFJ.

Ah, okay.

So if the angular momentum points up, the magnetic moment points down.

You got it.

That negative sign is key.

But just modeling orbits isn't the whole story, is it?

There's this other weird property,

electron spin.

Right.

This was a huge discovery.

Electrons have an intrinsic angular momentum, called spin, which isn't really about them physically spinning like a top, but it behaves mathematically like angular momentum.

And it also creates a magnetic moment.

It absolutely does.

And here's the kicker.

If you calculate that same ratio, math BFJ, for the spin part, it's almost exactly twice the value you get for the orbital motion.

Twice why?

Well, why is a deep quantum mechanical question involving relativistic effects?

But the fact is, it's double.

We capture this with something called the g -factor.

For orbital motion, the g -factor is basically one.

For electron spin, it's very close to two.

So spin contributes proportionally more magnetism for its amount of angular momentum.

Precisely.

The overall magnetic character of an atom then depends on how all the orbitable and spin angular momenta add up, considering their different g -factors.

It gets complicated quickly for multi -electron atoms.

Okay, so atoms can have these moments from orbit and spin.

That helps explain paramagnetism, the attraction.

But how does the repulsion dimagnetism actually work mechanically?

You put an atom with a moment, boot in a field, math BFBD, what does the field do to it?

You'd think the field just grabs the moment and tries to yank it into alignment, right?

Like a compass needle snapping to north.

Yeah.

But it's not that simple because the magnetic moment is tied to angular momentum, math BFJ.

Think about trying to tip over a spinning gyroscope.

It doesn't just fall over, it sort of swings around sideways.

Exactly.

It precesses.

The torque exerted by the magnetic field on the magnetic moment doesn't cause alignment.

It causes the angular momentum vector, math BFJ, and therefore the magnetic moment to rotate or precess around the direction of the magnetic field, math BFBD.

Ah, the wobble.

That wobble, yes.

And this is formalized by Larmor's theorem.

It tells us that the magnetic field effectively adds an extra bit of rotational speed to the electron's motion.

This extra angular velocity is the Larmor frequency, omega, and its value is QB,

$200.

It's directly proportional to the field strength dollars.

So the stronger the field, the faster the wobble.

Right.

Now think about what that extra rotation means.

It changes the electron's motion slightly.

Which changes its angular momentum.

Precisely.

It induces a change in angular momentum, delta, math BFJ.

And because magnetic moment is tied to angular momentum,

this induced change in angular momentum creates an induced magnetic moment, delta mu.

And this is the diamagnetic moment.

This is the diamagnetic moment.

And because of the way the physics works out with the negative charge of the electron and the direction of the torque, this induced moment, delta mu, always points opposite to the applied field, math BFB.

Always opposing.

That's the repulsion.

That's the universal repulsion of diamagnetism, explained classically by Larmor precession.

Okay, that classical picture seems quite neat.

Orbiting electrons, spin, precession.

It paints a nice picture.

It does.

And it's incredibly useful for intuition.

Feynman uses it beautifully.

But, and this is a huge but, the classical picture ultimately fails.

It breaks down.

Fails how?

We just used it to explain diamagnetism.

Well, there are a couple of critical failure points.

The first one is a bit subtle.

When you the classical calculation really carefully, including energy considerations,

it turns out that the Larmor precession is more like just watching the whole atom rotate slightly differently.

It doesn't robustly predict the persistent change in magnetic properties or energy needed to explain the observed diamagnetism under all conditions.

It shows the motion, but the magnetic consequence isn't fully guaranteed classically.

Okay, a bit shaky then.

But you said a couple of failures.

What's the big one?

The big one is catastrophic for classical physics.

It involves thermal equilibrium.

Think about any normal material at room temperature.

Its atoms are constantly moving, vibrating, colliding.

They're bathed in thermal energy.

Or at the heat.

If you apply classical statistical mechanics, the rules for how large collections of classical particles behave with temperature to these atomic magnets, you reach a stunning conclusion.

In thermal equilibrium, the average magnetic moment should be zero.

Completely zero.

Meaning no net magnetism at all.

None.

Classically, the randomizing effect of heat should totally overwhelm any tendency for moments to align—paramagnetism—or for that induced Larmor effect to create a stable opposing

moment—diamagnetism.

So classical physics predicts that the magnetism we routinely observe in materials shouldn't exist at room temperature.

Pretty much, yes.

The very fact that you can pick up a paperclip with a magnet or see Bismuth repelled fundamentally contradicts classical physics when thermal energy is properly accounted for.

Wow.

Okay.

So the classical models are helpful analogies, but they fundamentally can't be the true story.

They can't.

The existence of stable magnetism in matter is, at its core, a purely quantum mechanical phenomenon.

We have no choice but to bring in quantum rules.

And the key quantum rule here is about angular momentum, right?

It's not just any old vector anymore.

Exactly.

In quantum mechanics, angular momentum is quantized.

It can't point in any arbitrary direction, and its magnitude isn't continuous.

Specifically, the component of angular momentum along a chosen axis, like the direction of our magnetic field—let's call it zeobar.

That component Gajd can only take on specific, discrete values.

These values are given by zeo, where greater r is the reduced Planck constant.

The fundamental unit of action in quantum mechanics.

Right.

An mjjwar is a quantum number that can be an integer or a half -integer, depending on the total angular momentum of the system, like an electron or an atom.

Its values range from day or to plus jj in integer steps.

So for a single electron, where we know its intrinsic spin angular momentum is $12, too.

For spin jwars, knee's lacta is a $12, and the only possible values for knee jday are plus $12 and one total one.

Just two possibilities.

Just two.

So the spin component along the field, jjwar, can only be Dylos -Frolidobar or Frelendor.

That's it.

No values in between.

This is the origin of spin -up and spin -down.

And since magnetic moment is tied to angular momentum,

does this quantization lead to a fundamental unit of magnetic moment, too?

It absolutely does.

We define the Bohr magneton, moob dollar, as the natural unit for magnetic moments arising from electrons.

Its value is moob or bar, two -may, where a megatude of the electron charge and moob dollars is its mass.

So the electron's magnetic moment components are basically plus or minus some factor times this Bohr magneton.

Essentially, yes.

For spin, it's roughly PM, remembering the g factor is close to two.

For orbital contributions, it relates directly to moobium moode.

It provides that fundamental non -zero quantized magnetic building block that classical physics lacked.

And what does this quantization mean for the energy of an atom in a magnetic field?

This is the crucial payoff.

The energy of a magnetic moment moob priors in field math BFB day depends on its orientation relative to the field, specifically on the component moves.

Since moos is proportional to J's values, and J's can only take discrete values, the energy of the atom in the field can also only take on discrete values.

The energy levels split.

They split.

For that simple electron spin case, tree 12 -gear doll, what was one energy level without a field becomes two distinct energy levels in the field, corresponding to never dollars plus 12 to bad of spin up, say, and never dollars equals 12 to pole two.

And these states are stable.

They don't get wiped out by heat like the classical prediction.

They are quantized stable states.

Thermal energy can cause transitions between these levels, but it can't destroy the levels themselves.

This inherent stability arising from quantization is why magnetism persists even in thermal equilibrium.

Quantum mechanics saves the day.

Okay, let's try and wrap this up for everyone listening.

We started by seeing that most matter shows weak magnetism.

Either diamagnetism, this induced repulsion.

Present in everything.

Or paramagnetism, a weak attraction due to permanent atomic magnetic moments, usually from electron orbits or spin.

Right, and we saw the classical picture tries to explain these using orbiting electrons and Larmor precession that wobble in a magnetic field.

That classical picture gives great intuition, especially for diamagnetism's repulsion.

But it fundamentally fails when you consider thermal energy.

Classical physics predicts magnetism should vanish at room temperature, which it clearly doesn't.

The solution lies in quantum mechanics.

Angular momentum isn't continuous, it's quantized.

Its component along a field takes discrete values.

Which leads directly to the quantization of magnetic moments, with the Bohr magneton as the fundamental unit.

And crucially, it causes atomic energy levels to split into discrete stable states within a magnetic field.

It's this quantization that provides the stability against thermal randomness that classical physics just couldn't explain.

The very existence of everyday magnetism is proof of quantum mechanics at work.

So here's a final thought to leave you with.

We just saw how a seemingly simple phenomena, weak attraction or repulsion by a magnet, required a complete shift from classical intuition to the weird rules of quantum mechanics for a real explanation.

It makes you wonder, what other everyday things, things we think we understand perfectly well are actually governed by deep, non -intuitive quantum principles hiding just beneath the surface.

Something to ponder.

Thank you for joining us on this deep dive.