Chapter 12: Magnetic Resonance

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Welcome back to the Deep Dive.

Today we're going to try and break down one of the most powerful tools in science,

really, magnetic resonance spectroscopy.

Yeah, it pops up everywhere.

Chemistry labs, hospitals with MRI, biochemistry.

Exactly.

Yeah.

So our mission today is to sort of unpack the key ideas behind NMR and EPR.

We want to give you a solid grasp of the physics, the equations, the spectra without needing to look at any diagrams.

Right.

And it all boils down to a pretty fundamental quantum idea.

Basically, when you stick matter in a strong magnetic field, the tiny magnetic spins inside from nuclei or electrons.

They align in specific ways, creating different energy levels.

And if you hit them with just the right frequency of energy, they jump between those levels, they resonate, and we can detect that resonance, that jump, and figure out a molecule structure.

It's really quite elegant.

Okay.

Let's start with the basics then.

Spin.

What exactly is it?

Well, it's an intrinsic property like charge or mass, certain nuclei like hydrogen one, you know, a proton or carbon 13 and electrons too.

They have this property called spin angular momentum.

Which makes them behave like tiny magnets.

Precisely, little bar magnets.

And for a given nucleus, its spin is described by a quantum number, $1.

So for a proton is 12.

And that I -12 -2 means it has how many possible states in a field?

It means it can have $2 plus Stala -1 orientations.

So for I -1 -12 -2, that's two states, a lower energy one, we call it alpha aligned with the magnetic field and a higher energy one, beta aligned against the field.

Okay, so you put your sample full of these tiny magnets into a really big external magnetic field, which we call B dollars.

Yep.

And that field causes those alpha and beta states to separate in energy.

There's now an energy gap, delta Ea, between them.

And the stronger the better it'll feel, the bigger the gap.

Exactly.

The energy separation is directly proportional to the field strength.

So resonance happens when we send an electromagnetic radiation,

maybe radio waves.

For NMR, yes, typically radiofrequency.

And the frequency has to perfectly match that energy gap, delta Ea on top here, divided by Planck's constant, that's the resonance condition, delta Eheho.

And you mentioned something about precession, like a spinning top wobbling.

That magnetic moment of the nucleus doesn't just snap into alignment, it actually wobbles or precesses around the main magnetic field direction.

The frequency of that wobble is called the Larmor frequency.

Okay.

And here's the cool part.

The Larmor frequency is exactly the same as the resonance frequency needed to make the spin flip.

So you're matching the energy input to the natural wobble rate.

Right.

Now you hear about these massive multi -million dollar NMR magnets.

Why so big?

Why is field strength so critical?

Ah, sensitivity.

It comes down to population.

See, that energy gap, delta A for nuclear spins, is tiny.

So even in a huge magnetic field, the number of spins in the lower energy alpha state is only slightly more than in the beta state.

Like how slight?

We're talking maybe only a few extra alpha spins per million, literally, according to the Boltzmann distribution.

Wow.

Yeah.

And the signal intensity we detect depends directly on that tiny excess population.

Crucially, the intensity actually goes up with the square of the magnetic field strength, B022.

So double the field, quadruple the signal.

You got it.

That's why labs invest so much in higher field magnets.

You just need enough of that tiny excess population to even see a signal.

Okay.

That makes sense for NMR.

Now what about the E in EPR, electron paramagnetic resonance?

Same fundamental physics, but now we're looking at unpaired electrons.

So this is the technique for studying things like free radicals or certain transition metal ions that have unpaired electrons.

Still putting them in a magnetic field.

So looking for resonance.

Yep.

The equation for the energy difference looks similar.

Delta E equals EBB dollar B.

But the key difference is that factor mubbler, the Bohr magneton for the electron.

How different is it?

It's huge.

It's about 2000 times larger than the equivalent factor for nuclei, the nuclear magneton.

2000 times.

So the energy gap for an electron spin flip must be way bigger.

Massively bigger.

Yes.

For the same magnetic field strength.

Which means the resonance frequency needed will be much higher.

Exactly.

Instead of radio waves like an NMR, for EPR, you typically need microwaves like expand frequencies, all because the electron is a much stronger magnet than a nucleus.

That's a really clear illustration of the difference.

Okay.

Let's say we've run our NMR experiment.

We get back this spectrum, this plot with peaks.

How do we start reading it?

What's the first thing we look at?

The chemical shift.

Definitely.

This tells you about the chemical environment of the nucleus.

Right.

You said the nucleus feels the magnetic field B dollars orinus.

Yeah.

But it's not quite the full field.

Correct.

The electrons buzzing around the nucleus, they react to the external B dollar field by circulating.

And moving charges create their own magnetic field.

Okay.

Like Lentz's law, they oppose the applied field.

Exactly.

So they create a small local magnetic field that opposes B dollars right at the nucleus.

This effect is called shielding.

The nucleus feels an effective field, but a lie, which is slightly less than B dollars.

We write it as B dollars equal to one.

And sigma, sigma is the shielding constant.

Yeah.

And sigma's are the key.

It's different for nuclei in different parts of the molecule because their electron environments are different.

So different nuclei resonate at slightly different frequencies because of this shielding.

Precisely.

And that difference in resonance frequency relative to some standard reference compound like TMS is what we call the chemical shift.

And we reported on that funny delta scale in PPM, parts per million Y PPM.

Ah, that makes it independent of the spectrometer you used.

See, the actual frequency difference in Hertz does depend on beat hour, but by dividing that difference by the spectrometer's operating frequency and multiplying by a million, you get delta stat.

Which is the same value, whether you measured it on a 300 millihertz machine or a 900 millihertz machine.

Exactly.

It makes chemical shifts universally comparable.

Chemists anywhere can look at a delta value and know what kind of environment that proton is likely in.

Okay.

So what makes the shielding sigma dollar change?

You mentioned electronegativity.

Right.

Think about a proton attached to, say, an oxygen atom, like in an alcohol.

Oxygen is very electronegative.

It pulls electron density away from the proton.

Right.

So there's less electron density right around the proton to do that shielding circulation.

Less shielding means sigma is smaller.

And if sigma is smaller,

block law is closer to B dollars, meaning it resonates at a slightly higher frequency.

Correct.

Higher frequency means a larger delta value.

We call that being deshielded.

Protons near electronegative atoms are typically deshielded and appear downfield at higher delta.

And it's not just the directly attached atoms.

Neighboring groups matter, too.

Oh, absolutely.

Think about benzene, those pi electrons in the ring.

When you put benzene in the B dollar field, those electrons start circulating, creating a strong ring current.

And that current makes its own magnetic field.

A secondary field, yeah.

And depending on where proton 6 relative to that ring outside, inside, above it will feel that secondary field, which can either add to or subtract from B dollars all.

This causes significant shifts.

Protons outside the benzene ring are strongly deshielded by this effect.

So chemical shift tells us about the electronic environment.

What else does the spectrum tell us?

I see peaks that aren't just single lines.

They're split into

triplets.

Yes.

Fine structure.

Or spin -spin coupling.

This tells you about neighboring magnetic nuclei.

So the chemical shift is about the electrons, but the splitting is about nearby nuclei.

Exactly.

It's a magnetic interaction, but it's transmitted through the electrons in the chemical bonds connecting the two nuclei.

It's usually called scalar coupling or G coupling.

And unlike chemical shift, which is field dependent in Hertz, but field independent in PPM, this G coupling.

It's measured in Hertz.

And it's completely independent of the main magnetic field strength, B dollars.

The splitting pattern looks the same on any spectrometer.

Okay.

And how does the splitting work?

Is there a simple rule?

There is, for simple cases.

If a nucleus has one dollar equivalent neighboring nuclei, each with spin IO1 tend to like protons, its signal will be split into one dollar plus dollar one lines.

N plus one lines.

Okay.

Give me an example.

Classic one is chloroethane, CH3CH2Cl.

Let's look at the CH2 protons.

They have three equivalent neighbors, the CH3 protons.

So N3.

The CH2 signal should be split into three plus one equal four lines a quartet.

Exactly.

And the intensities follow Pascal's triangle.

1 .3 .3 .1.

Now look at the CH3 protons.

They have two equivalent neighbors, the CH2 protons.

N2.

So the CH3 signal is split into two plus one equals three lines.

A triplet.

Intensity is 1 .2 .1.

Perfect.

And the spacing between the lines in the quartet is the same in hertz as the spacing in the triplet.

That's the J coupling constant.

That's incredibly powerful for figuring out connectivity.

And I remember reading something about angles, the Karplus equation.

Ah, yes.

That relates the coupling constant between protons separated by three bonds, three J goller, to the dihedral angle between them.

The exact angle of twist around that central bond.

So the size of the splitting, the J value actually tells you about the 3D shape, the It does.

It was a huge breakthrough.

Suddenly NMR could give detailed conformational information about molecules in solution, which was incredibly difficult before.

But there's a catch, right?

What about coupling between nuclei that are identical, like the two protons in CH2Cl2?

Right.

Chemically equivalent nuclei.

Even though they are coupled to each other, that coupling doesn't cause any observable splitting in the spectrum.

Their transitions happen at the same energy.

So you just see one signal, a singlet.

That's a key rule.

What about when things are moving?

Molecules aren't static.

They twist, bonds rotate, maybe protons jump between molecules.

Good point.

NMR is fantastic for studying dynamics.

Let's say a proton can exist in two different environments, maybe due to rotation or chemical exchange, like an alcohol proton swapping with water.

So it should have two different chemical shifts.

If the exchange is slow, yes, you'll see two separate peaks.

But if the exchange happens really fast, faster than the frequency difference between those two peaks in hertz.

What happens then?

The spectrum can't resolve the two separate environments anymore.

The two peaks broaden, move towards each other, and eventually merge or coalesce into a single sharp peak at the average chemical shift.

So by looking at the shape of the peak, or the temperature where it coalesces, you can actually measure the rate of the exchange process.

Precisely.

You can get kinetic information directly from the NMR spectrum.

What about solids?

I've heard solid -state NMR is much harder.

The peaks are often really broad.

They are, naturally.

In solution, molecules are tumbling rapidly, and that averages out a lot of interactions that depend on orientation relative to the magnetic field, like direct magnetic dipole interactions between nuclei.

And chemical shift anisotropy.

That's the shielding being orientation dependent.

Exactly.

In a solid, molecules are locked in place, so you see the full effect of all these

interactions, and they all contribute to broadening the lines massively.

Plus, nuclei with spin I12R22 have quadrupole moments which interact strongly with electric field gradients in solids.

So how do you get sharp spectra from solids?

The trick is called magic angle spinning, or MAS.

You pack the solid sample into a rotor, tilted at a very specific angle relative to the main magnetic field beer of dollars.

The magic angle?

Yes.

$54 .74.

And then you spin the thousands of times per second.

Why that specific angle?

Because at that angle, the mathematical term describing those orientation -dependent interactions, which often involves $1 .3 data, averages to zero.

Spinning effectively mimics the tumbling motion in solution.

Wow.

So MAS basically averages out the broadening effects and gives you sharp solution -like spectra from solid samples.

That's the goal, yes.

It revolutionized solid -state NMR.

Okay, let's shift gears slightly.

We've talked about the spectrum, but how is it actually acquired these days?

It's not usually by slowly sweeping the frequency anymore, right?

No, not for many years.

The standard now is pulsed Fourier Transform NMR, or FTNMR.

Right.

Instead of a gentle sweep, you hit it hard and fast.

Kind of, yeah.

Think of it like hitting a bell with a hammer versus carefully finding its resonant frequency.

You give the sample a short, intense pulse of radio frequency energy, covering a wide range of frequencies all at once.

Typically a 90 -degree pulse, what does that mean?

It means the pulse is strong enough and lasts just long enough to tip the net magnetization vector, a nuller which normally points along the main field axis, the z -axis, by 90 degrees, so it ends up lying entirely in the horizontal plane, the xi plane.

So you knock the overall magnetization sideways?

Then what?

Then the pulse turns off, and that magnetization vector, now in the xi plane, starts precessing around the z -axis at its characteristic Larmor frequency.

Or rather, all the different nuclei with different Larmor frequencies start precessing.

And that precessing magnetization is what we detect.

Exactly.

A precessing magnet induces an oscillating current in the detector coil surrounding the sample.

That oscillating signal decays over time as the spins relax.

We call this decaying signal the free induction decay, or FID.

But the FID is messy, right?

It contains all the frequencies mixed together.

It's in the time domain.

Correct.

It's a complex interference pattern over time.

To get our familiar spectrum intensity versus frequency, we need to mathematically disentangle those frequencies.

And that's where the Fourier transform comes in.

Precisely.

It's a mathematical algorithm that converts the time domain signal, the FID, into the frequency domain signal, the spectrum.

It's incredibly efficient and allows us to get spectra much, much faster than the old scanning methods.

Okay, you mentioned relaxation.

The FID decays because the system returns to equilibrium.

There are two main relaxation times.

T $1 and $2, $2, $2.

Why two?

They describe different aspects of the return to equilibrium.

T $1 is the longitudinal, or spin lattice, relaxation time.

Lattice meaning the molecular surroundings.

Yeah, the rest of the molecule and the solvent.

T $1 measures how quickly the magnetization along the z -axis recovers back to its equilibrium value after we've disturbed it, like with that 90 -degree pulse or even a 180 -degree pulse, which inverts it.

It's about energy exchange between the spins and their surroundings.

And it tells you something about how fast the molecule is tumbling.

It does.

Faster tumbling in small molecules leads to efficient T $1 relaxation.

Very slow tumbling in large molecules can make T $1 long.

Okay, so that's T $1, recovery along z.

What's T $2?

T $2 is the transverse, or spin -spin relaxation time.

This describes the decay of the magnetization in the xi plane, the plane where it's precessing and creating the FID signal.

Why does that decay?

Because the individual nuclear spins, which were initially precessing together in phase right after the pulse, start to lose that phase coherence.

They fan out in the xi plane.

This loss of phase causes the net xi magnetization to drop to zero.

So T $2 is about loss of phase, while T $1 is about returning to equilibrium energy population.

You got it.

And T $2 is directly related to the line width of the NMR peak.

A short T $2 means fast decay of the signal, which corresponds to a broad line in the frequency spectrum.

Specifically, line width is proportional to one T $2.

How do you measure T $2 so cleanly?

Doesn't the magnet not being perfectly uniform also cause spins to get out of phase?

Uh, good point.

Yes, magnetic field in homogeneity contributes to the observed decay, T $2.

To measure the true molecular T $2, we use a clever trick called the spin echo sequence.

What does that do?

It involves a 90 -degree pulse, then a short delay, tau, then a 180 -degree pulse, then another delay, tau, before detection.

The 180 -degree pulse acts like a mirror, refocusing the dephasing caused by static field in homogeneities.

So it flips the fan out, spins around, and they naturally come back together.

Exactly.

Any decay you still observe at the time of the echo, two tau, is due only to the irreversible random T $2 processes within the molecule itself, not imperfections in the magnet.

Clever.

Okay.

Pulsed NMR also enables some really powerful manipulation techniques.

Let's talk about spin decoupling.

You use this a lot in carbon -13 NMR.

Oh yes, routinely.

Carbon -13 nuclei are not very abundant, and they have a small magnetic moment, so signals are weak.

Plus, they're coupled to all the attached protons, which splits the signals into complex multiplets.

Making it hard to see and interpret.

Right.

So while we're observing the carbon -13 signals, we simultaneously irradiate the sample with a broad band of radio frequencies covering all the proton resonances.

What does that do to the proton?

It causes the proton spins to flip back and forth so rapidly that, from the carbon's perspective, the magnetic effect of the proton spin averages out to zero over the time scale of the measurement.

So the J -coupling effectively disappears.

Correct.

The splitting collapses.

Each unique carbon atom now appears as a single sharp line, a singlet.

It massively simplifies the spectrum and also enhances the signal intensity through mechanisms like the NOE we'll discuss next.

Okay.

The nuclear overhouser effect.

NOE.

This sounds important for structure.

It's absolutely crucial, especially for determining the 3D structures of large molecules, like proteins or nucleic acids, and even smaller organic molecules.

How does it work?

What are you actually measuring?

The NOE is observed when you saturate or strongly irradiate the resonance of one specific nucleus, say proton A, and you see an intensity change in the resonance of another nucleus, say proton X.

Why would irradiating A affect X?

It only happens if A and X are relaxing each other through the dipole mechanism.

This mechanism depends very strongly on the distance between the two nuclei.

It falls off incredibly fast, proportional to 100R66 numbers, where R is the distance between them.

Wow.

One over R to the sixth.

So it's a really short range effect.

Extremely short range.

If you see an NOE enhancement between proton A and proton X, it's definitive proof that those two protons are very close to each other in space, typically less than about 0 .5 nanometers or five angstroms.

Even if they're far apart in the chemical structure, connected by many bonds.

Doesn't matter how many bonds separate them, the NOE goes through space.

It tells you about spatial proximity, which is why it's indispensable for figuring out folding patterns and 3D structures.

Amazing.

Okay, let's wrap up by revisiting EPR quickly.

What are the key features in an EPR spectrum that give us chemical information?

Two main things, analogous to NMR, the G value and the hyperfine structure.

The G value is like the chemical shift for electrons.

Sort of, yeah.

For a free electron, the G value is about 2 .00023.

But inside a molecule, the electron's magnetic moment is affected by its orbital motion and the local electronic environment.

This causes the measured G value to shift slightly.

So the G value tells you something about the radical's environment.

It does.

For most organic radicals, the electron is in an orbital with not much angular momentum, so the G value stays very close to 2 .00023.

But for transition metal complexes, where D orbitals are involved, the coupling between spin and orbital angular momentum can be significant, leading to G values that can vary widely, maybe from 0 up to 6 or even more.

So a G value far from 2 is a clue you might have a metal involved.

Often, yes.

Or significant spin orbit coupling.

The second key feature is hyperfine structure, HFS.

Which is the EPR equivalent of J -coupling and NMR splitting patterns.

Exactly.

It's the splitting of the EPR signal caused by the magnetic interaction between the paired electron and any nearby magnetic nuclei, like protons, nitrogen -14, etc.

Is there a simple splitting rule, like N plus 1?

There is.

A single nucleus with spin a dollar will split the EPR signal into $2 plus i -tolerant lines.

Importantly, these lines usually have equal intensity.

So a proton splits the signal into?

212 plus 1 equals 22 lines.

A doublet of equal intensity, a nitrogen -14 nucleus, splits it into 2 honors plus 1 equals 33 lines.

A triplet of equal intensity.

What if you have multiple equivalent nuclei coupling to the electron?

Then you use the same N plus 1 rule as in NMR for i -off -or -i -1, 1202 nuclei.

So overall equivalent protons will split the signal into $1 plus 1 lines, and the intensities will follow Pascal's triangle ratio.

Like the benzene radical anion we mentioned earlier.

6 equivalent protons.

Yep.

And 40 similar others.

So you expect $6 plus 1 up to 7 lines, with intensities 1 .6, but from 49 to 20 knot,

and that's exactly what you see.

It's quite beautiful.

And can you get quantitative information from this splitting?

Like the J -coupling giving angles?

Yes, absolutely.

The magnitude of the hyperfine splitting, called the hyperfine coupling constant, is directly proportional to the amount of time the unpaired electron spends near that nucleus.

There's a famous relation called the McConnell equation, rho -arrow.

Rho is the spin density.

Like the probability of finding the unpaired electron at that nucleus.

Essentially, yes.

Particularly the density in an s -orbital at the nucleus, which gives rise to the isotropic Fermi contact interaction seen in solution.

Q is a proportionality constant.

So by measuring the splitting A, we can map out the distribution of the unpaired electron across the molecule.

You can literally see where the radical character is located.

Exactly.

It gives you a picture of the molecule's electronic structure.

Okay, this has been a fantastic tour.

Let's try to summarize the big picture for everyone listening.

Sounds good.

I think there are maybe three core ideas we've hit on.

First, the whole basis is resonance.

Spins have different energies in a magnetic field, and we measure the exact frequency needed to flip them.

Right.

NMR for nuclei, EPR for electrons, different energy scales, different frequencies.

Second, the details of the spectrum chemical shifts and coupling constants, or G -values, and hyperfine structure in EPR.

Tell us about the local environment and connectivity, basically the molecular structure.

Yeah.

Shielding reveals the electronic neighborhood.

Coupling reveals the nuclear neighbors through bonds.

And third, modern pulse techniques, like FTNMR, let us not only get spectra quickly, but also study dynamics through relaxation, a T -Doron TW2, and determine 3D structure through space using things like the NOE.

Couldn't have said it better.

Those relaxation times and the NOE are really key for understanding molecular motion and shape, especially for bigger systems.

Yeah.

And you mentioned earlier that for really big molecules, like proteins, their slow tumbling makes T -Doron TW2 very short.

That's right.

Short T -Doron TW2 means broad lines, which is a major challenge for getting high resolution NMR data on large biomolecules.

So that leads to a final thought for you to ponder.

We know slow tumbling shortens T -Doron TW2 and broadens lines.

We use high fields and magic angle spinning.

But what else?

What other properties of the molecule or its environment could we perhaps manipulate or exploit to try and overcome this fundamental limitation?

How might we trick physics into giving us sharper lines, better resolution for these massive, complex, but incredibly important biological machines?

That's a great question to leave people with.

It's where a lot of cutting edge research is focused.

New pulse sequences, sample preparation methods, ways to control molecular motion.

Food for thought indeed.

Well, thank you so much for guiding us through the world of magnetic spin today.

My pleasure.

It's a fascinating topic.

Thank you all for tuning in to the Deep Dive.

We'll catch you next time.