Chapter 35: Paramagnetism & Magnetic Resonance

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Welcome back to the Deep Dive.

Today we're really getting into the quantum weeds looking at Chapter 35 of the Feynman Lectures, Volume 2.

It's all about paramagnetism and this fascinating magnetic resonance.

Our plan is to sort of unpack how these tiny little magnetic moments at the atomic level actually behave, how they react to fields.

And we want to connect that fundamental quantum stuff, well, connect it to real world things like getting things incredibly cold or even medical tech.

Yeah, and this chapter, it's a big one conceptually, because Feynman really forces you to ditch the classical way of thinking.

How so?

Well, the idea that atomic magnets can just point any which way, that's out.

Quantum mechanics steps in with this really weird rule.

Ah, yes, quantization.

Exactly, angular momentum quantization.

It's the key reason why atoms don't just act like tiny bar magnets you can spin around freely.

Okay, let's start there then.

This quantization rule, it says angular momentum dollars isn't continuous.

Right.

If you measure its component along some axis, say the z -axis, you don't get just any value.

You only get specific discrete steps,

multiples of shed, Planck's constant over 2 pi, either integers or half integers.

Precisely, no smooth gradients allowed, and that has direct consequences for energy.

Okay.

If you put an atom that has a magnetic moment, let's call it the LeBouce into a magnetic field, bit of dollars, its energy changes.

Right, the potential energy shifts.

And that energy shift, delta u -bells, it's proportional to the moment's component along the field.

Specifically, its moves be dull.

So if meanwhile's itself is quantized.

Then the energy shift is quantized too.

Right.

For the simplest case, like an electron spin, which is spin 12, well, moves can only be up or down relative to the field.

Two options.

Just two.

Which means only two possible energy states in the field.

And the sort of standard unit for measuring these electron moments is the Bohr magnet, moves dual.

That's the fundamental chunk of magnetic moment for an electron.

Two states.

That sounds almost too simple.

How on earth did they prove something that's strange?

It feels very counterintuitive.

It is counterintuitive.

The proof was a landmark experiment back in 1922.

Stern -Gerlach.

Ah, the Stern -Gerlach experiment.

I remember reading about the setup.

A beam of silver atoms, right?

Neutral silver atoms, yes.

Yeah.

And they shot this beam through a magnetic field that wasn't uniform.

It was inhomogeneous.

Meaning the field strength changed rapidly across the beam path.

Exactly.

They used specially shaped magnet poles, maybe one with a sharp edge, to create a strong gradient.

And the fact they used neutral atoms was crucial, wasn't it?

Absolutely critical.

If they were charged ions,

the main force deflecting them would be the Scandered Lorentz force, totally obscuring the tiny magnetic effect.

Okay.

By using neutral atoms, any deflection had to come from the interaction of the atom's internal magnetic moment with that changing field.

So classically, if those moments could point in any random direction,

what should have happened?

You've sure seen a smear.

A continuous band on the detector screen.

Atoms pulled up or down by varying amounts depending on their random orientation.

A smooth distribution.

But that's not what they saw.

Not even close.

This was the shocker.

The beam split cleanly into two distinct spots.

Two sharp lines.

Wow.

Just two.

Just two.

No smear, no in between.

It completely destroyed the classical picture.

And that proved that the angular momentum component along the field axis was quantized.

It had to be, what, plus bar two or the bar two?

That's it.

And since the magnetic moment comes from that angular momentum, it meant the magnetic moment component, moose, could only be plus or minus the Bohr magneton.

Up or down.

Binary.

Incredible experimental proof.

Irrefutable.

Okay.

So Stern -Gerlach showed quantization is real.

But just seeing two spots doesn't tell you the exact value of the Bohr magneton or measure moments precisely.

No.

For precision, you need something more sophisticated.

And that brings us to Is it a Robbie and his molecular beam method?

Right.

Robbie's technique.

How does that let you measure the moment so accurately?

It involves flipping the spins, doesn't it?

Exactly.

It's based on resonance.

You start, like before, with atoms in a strong, constant magnetic field.

Dollar dollars.

This sets up those two distinct energy levels.

The up and down states.

Yeah.

Uh -huh.

Then you add a second magnetic field.

But this one is weak and it's oscillating, usually at radio frequencies, and it's applied perpendicular to the main field.

Okay.

An oscillating field.

Like sending in radio waves.

Pretty much.

Yeah.

Now think about tuning an old radio.

You turn the dial, changing the frequency.

If the frequency of that oscillating field, let's call it a magi, hits exactly the right value.

The resonance frequency.

The resonance frequency.

The magi dog equals the magi guy.

The atoms in the lower energy state can absorb energy from the oscillating field and flip.

They transition to the higher energy state.

And that right frequency corresponds exactly to the energy difference between the states.

Precisely.

The energy of the absorbed photon from the RF field has to perfectly match the energy gap, delta u on, between the spin up and spin down states, delta u.

So if you can measure that frequency and make it really accurately.

You can calculate delta u really accurately and from that determine the magnetic moment, Rillemer energy and related factors like the g factor with incredible precision.

That's clever.

Turning a frequency measurement, which can be done very precisely, into a measure of a fundamental particle property.

How does the machine actually detect the flip?

The apparatus is ingenious.

It uses magnets kind of like filters.

First, a set of magnets selects atoms in only one state, say spin up and guides them towards the detector.

Okay.

So only spin up atoms are heading for the detector.

Right.

Then they pass through the region with the oscillating RF field.

You slowly sweep the frequency of that field.

When you hit the resonance frequency mogapark, bam.

The spin up atoms flip to spin down.

They flip.

Now they're spin down.

And the final set of magnets is designed to deflect these newly flipped spin down atoms away from the detector path.

Ah, so when you hit resonance, the atoms that were going to hit the detector suddenly get kicked out of the beam.

Exactly.

You see a sharp dip, a big drop in the detector current right at that specific resonance frequency.

And the center of that dip tells you Brilliant.

Okay.

We've got the quantum rules for single atoms nailed down.

Now scale up.

What happens with, you know, billions and billions of these things in a chunk of material?

That brings us to paramagnetism.

Right.

Now we need to think statistically.

We define magnetization, capital Nullar, as the net magnetic moment per unit volume of the material.

And we have to consider temperature now, don't we?

Thermal energy catered dollars.

Crucially,

because there's this constant battle going on.

The external magnetic field wants to align all the little atomic moments, forcing them into the lower energy state.

That's order.

But heat means jiggling randomness.

Exactly.

The thermal energy catered dollar constantly tries to knock the spins randomly into the higher energy state.

It promotes disorder.

Chaos versus order.

So the final magnetization depends on who's winning that fight.

Pretty much.

The probability of finding an atom in the lower state versus the higher state depends on the Boltzmann factor, delta U T.

It's all about the ratio, the magnetic energy splitting, delta U tier to the thermal energy.

Take your dollars.

And the classical calculation, just assuming random angles.

Completely failed.

It just didn't match experiments for how magnetization changed with field and temperature.

But the quantum approach, using just those two allowed states.

It works beautifully.

For spin 12 systems, the calculation gives the magnetization dollars as being proportional to something called the the tan function of the ratio, the mub, BKT.

Okay.

The tan function sounds a bit mathematical, but what's the physical meaning?

What does it tell us?

It basically describes a saturation effect.

If you make the field bubbly dollars really strong or the temperature dollar really low, that ratio bunny BKPT gets large.

The tan function then approaches one, meaning the magnetization flattens out.

Right.

It saturates pretty much all the spins are aligned with the field.

You can't get any more aligned than that.

But under normal conditions, room temperature, reasonable fields.

Usually the magnetic energy Moby D is much, much smaller than the thermal energy K dollars.

The ratio is small.

And what does the tan function do then?

For small values of X, passes are approximately just X.

So the magnetization dollar simplifies.

It becomes directly proportional to the magnetic field dollars and inversely proportional to the absolute temperature dollars.

Ah, that's Curie's law, famous result for paramagnetism.

It comes directly out of the quantum calculation in this common limit.

It shows how the bulk material responds, how alignment fights thermal randomness.

That's a fantastic connection from the weird two state quantum rule all the way up to a measurable macroscopic law.

Okay.

Let's talk applications.

This stuff gets used in some pretty extreme ways.

First, cooling things down, adiabatic demagnetization.

Yeah, this is seriously cool physics, literally.

It's a method to reach temperatures incredibly close to absolute zero, like thousands of a Kelvin.

How does removing a magnet make something colder?

It's a two -step process all about controlling entropy or disorder.

Step one, isothermal magnetization.

Okay.

You take a special paramagnetic salt crystal, cool it down as much as you can, maybe with liquid helium.

Then you put it in a really strong magnetic field.

Forcing the spins to align.

Forcing them into that low energy, highly ordered, low entropy state.

Now forcing water like that actually releases a bit of heat.

But you do this isothermally, meaning you let that heat flow out into the surrounding helium bath, keeping the salt's temperature constant.

So end of step one, spins are ordered, low entropy, but still at the starting temperature of say one Kelvin.

What next?

Step two, adiabatic demagnetization.

First, you thermally isolate the salt sample completely, no heat allowed in or out.

Adiabatic means quarter dollars dollar.

Got it.

Insulated.

Then you slowly, very slowly, turn the magnetic field off.

Okay.

The aligning force is gone.

Right.

Now the spins want to go back to their natural, random, disordered, high entropy state.

But to increase their disorder, to randomize themselves, they need energy.

And since no heat can come from outside.

You have to steal that energy from the only place available, the internal thermal energy, the vibrations of the crystal lattice itself.

So the randomization of the spins sucks energy out of the crystal's heat.

Exactly.

This internal energy drain causes the temperature of the salt itself to plummet dramatically, down to maybe fractions of a degree above absolute zero.

That is genuinely clever.

Using magnetic order disorder to pump heat out.

Amazing.

Okay.

Final topic and maybe the most widespread application,

NMR nuclear magnetic resonance.

Yes, NMR.

And MRI in medicine is basically NMR in action.

The core principle is identical to Robbie's method.

Using an oscillating field to flip spins between energy levels split by a constant field.

Exactly the same physics, but here we're targeting the magnetic moments of the atomic nuclei, not the electrons.

Things like the proton, which is just a hydrogen nucleus.

And nuclear moments are different from electron moments.

Hugely different.

They are much smaller, typically thousands of times weaker than the

So the energy level splitting delta UE in the same magnetic field will be much smaller.

Much, much smaller.

Which means the resonance frequency omega plus needed to cause the flip is also much lower, often down in the radio wave part of the spectrum, even for strong magnets.

Okay.

Weaker signals, lower frequencies.

But what makes NMR so incredibly powerful, especially in chemistry and medicine?

It's something called the chemical shift.

It turns out the exact resonance frequency of a specific nucleus, like a proton, isn't just determined by the external magnetic field by dollars.

Oh.

What else affects that?

It's local electronic environment.

The electrons buzzing around that nucleus and its chemical bonds slightly shield it from the external field.

So the actual field at the nucleus is a tiny bit different depending on what kind of chemical bond it's in.

Precisely.

A proton in water feels a slightly different field than a proton in fat.

Or a proton attached to an oxygen versus one attached to a carbon.

And that tiny difference in the local field causes a tiny difference in the resonance frequency.

Yes.

A tiny but measurable shift in the mega ballers.

So different chemical environments give rise to slightly different NMR frequencies.

Which means you can use these frequency shifts to figure out the structure of molecules.

Exactly.

You get a whole spectrum of frequencies and each peak tells you about a specific type of nucleus in a specific chemical environment.

It's an incredibly powerful tool for figuring out molecular structure, identifying compounds.

And in MRI, mapping the locations of, say, water molecules in the body?

Precisely.

MRI maps the density and environment of hydrogen nuclei, protons in the water, and fats within tissues, giving doctors detailed images.

What a journey.

Okay, let's recap.

We started with that foundational, kind of weird quantum fact.

Angular momentum is quantized.

Which Stern and Gerlach spectacularly proved with their two spots.

Then Rabi showed us how to measure the resulting magnetic moments with amazing precision using resonance.

Flipping the spins.

We saw how that same quantum rule, scaled up with statistics,

explains the paramagnetism of bulk materials through Curie's law, that Mind -Dollar -Propto -BT relationship.

The battle between alignment and thermal chaos.

And finally, we saw how these principles lead to amazing technologies.

Adiabatic demagnetization to reach ultra -low temperatures.

An NMR, an MRI, using nuclear spin flips to analyze everything from molecules to human bodies.

The really striking thing for me is how this non -intuitive quantum behavior, that a moment must be up or down, no other choice, is not just some abstract idea.

It's the very mechanism that underlies both creating the coldest temperatures we can reach and performing incredibly detailed chemical analysis.

It really is fundamental.

And here's something to think about.

That energy needed to flip a nuclear spin in NMR, it's tiny.

A single radio frequency photon carries almost no energy compared to, say, visible light.

Vanishingly small.

So think about the enormous amounts of energy your brain and body use every single day, just living and thinking.

And compare that to these minuscule discrete quantum jumps in energy.

The flip of a single proton spin that formed the basis for so much of modern chemistry and medical imaging.

It really highlights the incredible information density and power packed into those tiny quantized energy steps.