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>> Good morning everybody.
Is my mic working?
Can people hear me?
OK. Good. A couple of announcements.
First, I have unfortunately have to cancel office hours today.
I have a dentist appointment,
and I don't think I can promise to get back in time.
So sorry about that.
I just, I thought it might work, but I'm not sure.
Who knows how long it takes.
Also, I'm kind of sick, and I think nobody wants this cold.
So, you know, we'll just have office hours tomorrow.
And as always if you have more questions please post stuff
on the Facebook page.
Also I just want to comment about the PCAM seminars.
So lots of people are going.
That's really great.
People are asking questions and participating,
and that's really cool.
I just want to mention,
most people are doing what they are supposed to,
but I have noticed that, you know, it is a large group
of people in there, and there is some inappropriate side
conversations and stuff going
on during the question and answer session.
Please don't do this.
It's, you know, it's really great to have everybody there.
And it's good to ask questions and participate
in the discussion, but when everybody is talking among
themselves during the discussion or, you know,
leaving in an obtrusive manner, it's not great.
And when the speaker is making jokes
about people leaving halfway through,
like that's really not so cool.
These people are visitors to UCI,
and we want to give them a great impression.
And, again, most people are doing exactly what they're
supposed to, just, you know,
make sure that you are one of those people.
Does anybody have any questions before we get started talking
about NMR?
A couple people came and tried
to ask me stuff while I was setting up,
and I didn't have time to answer right then.
So I know there are questions.
Anybody want to ask them?
Yes?
>> When can we turn in the worksheet
for yesterday's seminar?
>> When can you turn in, you can turn them in whenever.
I mean, you can stick them under my office door.
You can give them to your TA.
Anything you want.
Ah, that's another thing that I wanted to mention.
I have most of the extra credit seminar things graded.
I know that there is a stack of the Heather Allen ones that are
in my office somewhere that I need to look for,
so if you still haven't gotten your score for that.
Sorry about that.
I will look for it.
Also, I'm a little behind on the re-grades.
I had planned to catch up on this stuff this weekend,
and I was kind of sick.
So, sorry about that.
I will get it done pretty soon.
Any other questions?
OK. Let's talk about NMR.
OK. So last time we left off talking about the Zeeman effect,
which is the condition where anything
that has a non-zero spin, so electrons
and some atomic nuclei have the degeneracy
of these states broken in magnetic field.
So if we have our little spins,
and there's no applied magnetic field.
They are just all in random orientations.
And they all have the same energy.
And if we put the sample in a magnetic field,
then now we have a quantization axis,
and the degeneracy is broken.
And I wanted to put this up here,
this is from an organic chemistry book.
And this is the explanation that you see pretty often
where you have all your little spins are in random orientation,
and then you put it in the magnetic field,
and they all magically either go into the alpha or beta state
where they're up or down.
That's not actually what happens.
They don't all have to pick one or the other of these states.
In fact, a lot of them are in different super position states.
There is still a random distribution
of the orientations of the spins.
But what this means is that's your quantization axis.
So if we measure values of the spin, we are going to be able
to measure states that are either
in the alpha or beta state.
And the rest of them are not going to be well defined.
The thing that is correct about this is that alpha
and beta have different energies as opposed to the condition
where there's no magnetic field, and they are all degenerate.
And also this change in energy
for the different spin states is really small.
And later on toward the end of the NMR discussion when we get
into talking about Boltzmann distributions and start moving
into Stat Mac, we'll see exactly how small this energy
difference is.
And it's really amazing that this works at all.
NMR depends on these very small population differences.
And when we're looking at a typical NMR sample,
most of the nuclei are not giving us any signal.
So there are almost equal numbers of spins
in the alpha and beta states.
And most of them are just cancelling each other out.
And it is amazing that it works at all.
So, OK, since the energy difference between the alpha
and beta state is really small, we want to maximize that as much
as we can in order to get more signal.
And that is one of the reasons why people like to have bigger
and bigger NMR magnets.
There's another reason also that has to do
with chemical shift dispersion and being able to separate
out nuclei that are
in chemically different environments.
So having a higher field magnet gives you both greater
sensitivity and greater resolution.
And here's a plot of what that looks like.
So the energy differences between the spin states
for a particular nucleus or for an electron are determined
by the strength of B-not, the main magnetic field.
[ Silence ]
And so here are just some pictures of instruments
that we have at UCI that we have 300-megahertz instruments.
We also have a 600.
And there is an 800-megahertz magnetic,
which is the large one here.
OK. So just to show you some
of the high-end instruments that people use.
This thing that looks like it lands
on Mars is the Oxford 900 megahertz magnet.
It just has a bunch of fancy packaging
that you have the platform around the top
and everything is just for show.
But, you know, it is important to make really big magnets
to get higher resolution of the chemical shift.
And then the lower picture is a high field MRI scanner
for medical diagnostics.
And the same thing applies there.
So in imagine, instead of looking at local differences
in the magnetic field from the local chemical environment
of the nuclei, what we're looking
at is essentially all water.
And magnetic field gradients are applied in order
to make apparent chemical shift differences
that are spatially encoded.
And it's desirable to have bigger and bigger magnets
for that too because the larger the magnetic field,
the higher your signal is.
And if we apply larger gradients we can get finer
and finer resolution.
But the problem with that is that the magnetic fields
and in particular the RF that we have to use start
to actually interact with your brain at these levels.
So we have to be careful about applying to much power
and heating tissue up, and also
if you apply very strong magnetic field gradients,
it can actually induce electrical signals
in your brain, and you see flashes of light.
And it's kind of interesting, but not what most people want
to experience when they go in for an MRI.
So this picture of a brain is actually my brain.
I had it scanned at UC Berkeley while I as a post-doc
because one of my friends does this kind of research.
And so, you know, I got to experience fun things
like turning the gradients up really high and seeing,
you know, flashes of light in there.
So, you know, it's neat.
And in these research instruments people use really
high fields.
But for the ones that are actually in the clinic,
you have to be a little bit careful because, you know,
random sick people are not to, are not interested
in experiencing these things.
OK. So back to talking about the Zeeman effect, let's put this
in terms of quantum mechanical things that we're seen before.
OK. So we mentioned that spin up is called alpha,
and spin down is called beta.
Alpha does not equal minus beta.
We have gotten into this when we're talking about the,
you know, doing term symbols and looking at electronic states,
the individual electrons are interchangeable.
And that goes for nuclei as well.
But, you know, if you have an alpha
and a beta they don't cancel each other out except
in the sense that if you have equal numbers
of them you are not going to see an NMR signal.
All right.
So this energy difference, the difference between beta
and alpha, again, is directly proportionally to B-not.
So this gamma here is the gyromagnetic ratio, which is,
you know, we can, it has to do
with the structure of the nucleus.
We can take it as pretty much a fundamental physical constant
for a particular kind of nucleus.
And that is something that we look up.
So for a particular type of nucleus whether it's a proton
or a C13 or whatever, we have this gyromagnetic ratio.
We have a factor of H-bar and B-not.
So if we want to increase our signal at this point,
really all we can do is increase the strength
of the magnetic field.
It turns out there are other things that we can do
to increase the polarization difference.
We could use what's called hyperpolarization.
And if we have time maybe I'll talk a little bit more
about that later.
But in terms of traditional NMR and EPR techniques
for increasing the sensitivity,
all that you have is increasing the number of spins
or making the magnetic field bigger.
And, again, it's nuclear magnetic resonance.
So the resonance condition is that you're,
the energy of the RF that you put in has to be equal
to the energy difference between these two states,
or you're not going to see a signal.
OK. So here's our nuclear spin Hamiltonian.
And just like we talked about in electronic spectroscopy,
we are going to treat the nuclei and the electrons separately.
And so here we are worried about the nuclear spin Hamiltonian.
And so we're going to ignore the electrons except
as a time averaged local magnetic field
that the nuclei see.
And this is why NMR is useful to chemists.
We have these local magnetic fields that are,
that depend on the distribution of electrons around the nucleus,
which of course are primarily due to electrons
in the chemical bonds.
And that's what enables us to find
out things about structures.
So if you go back in the early, early literature, 50 years ago,
physicists, you know, discovered NMR, and, you know,
they discovered the effect.
And they were really excited about it.
In the original paper where this is described, they are kind
of speculating about what it's useful for.
And they said, "Well, maybe it would be useful
as a really accurate means of measuring the strength
of magnetic fields except
that there's this crappy thing called the chemical shift
where a proton doesn't just behave like a proton.
It's different depending
on the chemical environment that it's in.
So that makes it less useful.
And, of course, that's the whole reason that this is useful
as an analytical technique because we do have differences
in the local chemical environment that have to do
with the molecular structure.
So the lesson there is, you know, the application
that you think might be most useful
for something isn't necessarily what it will end
up being used for, you know,
if you're lucky you publish something,
and people in different fields pick it up
and find other stuff to do with it.
And, you know, also it's good to do basic research.
You never know what applications things will have.
OK. So when we're talking
about our spin Hamiltonian there are all kinds
of terms that go on in here.
And here's a graphical representation
of what the different interactions are in NMR.
OK. So and notice we have different plots
for solids and liquids.
So in organic chemistry,
and I am pretty sure you've mostly just seen solution-state
NMR, that's most of what we are going to talk about in PCAM too,
but we'll talk about solids a little bit
because they have a lot interesting effects
that are not present in solution.
And also that's what I do, so you get to hear
about solid state NMR.
All right.
So in this Hamiltonian for your nuclear spins,
we have all these different terms.
And here the size of the circles is proportional
to the relative sizes of the interaction.
So it's just to give you an idea.
So the first term is the Zeeman interaction.
So that has to do with what kind of nucleus is it?
And how big is the magnetic field?
And that is, under normal experimental conditions,
that is almost always going to dominate.
So then the next term here is the RF.
That's the radiofrequency pulse.
So, again, remember we put our spins
in the big magnetic field, and they line up.
But that's boring.
That doesn't give us a signal.
We have to change their quantization access and get them
to release some energy that we can measure.
And that's done with the radiofrequency field.
And I am going to tell you some details about how we do that.
And, you know, equally for solids and liquids,
this is the next most important term in the Hamiltonian.
Did you talk about perturbation theory last quarter?
So who knows what I'm talking
about when I say perturbation theory?
Sort of?
[ Inaudible ]
OK. So you can think about he NMR Hamiltonian here
as your unperturbed term is the Zeeman interaction.
And then the first order of perturbation to that is the RF.
And then we have all this other stuff going on.
OK. So the next thing involved is the dipolar interaction.
And so this is a special interaction
between the nuclear spins.
So we can treat them like little magnetics.
And these little dipoles interact
with each other through space.
And that interaction goes as one over R cubed,
and it also has an orientation dependence.
And you can imagine that this is really useful
in solving molecular structures.
You know, we have an orientation dependence,
and we have a distance dependence
for these little dipoles.
And in solid state NMR, this is, in fact, where we get a lot
of our structural information,
but it also makes the spectra more complicated.
Notice that it's not therein liquids.
That's because in solution,
the molecules are tumbling isotropically.
They are moving around really fast
on the time scale of the experiment.
And so anything that has an orientation dependence is going
to get averaged out.
OK. So the next thing down here is the chemical shift.
So for solids this is quite a bit smaller
than the dipolar interaction.
But for liquids, this is the next largest term
in the Hamiltonian.
A chemical shift is, again, this interaction
between the nuclear spin and the local magnetic field that's
there as a result of interactions with the electrons.
And we're treating the electrons as just this smeared
out time averaged magnetic field that the nuclei see.
Notice that the chemical shift is larger for solids
than it is for liquids.
That's because there is an isotropic part
and an anisotropic part.
And, again, in liquids, everything is moving
around really fast, and it gets averaged out.
In solids that isn't true.
OK. So the next item down is the quadrupolar interaction,
which is, it can be quite large in solids.
And what that is is the interaction that's due
to nuclei, it only exists for nuclei
that have spin greater than a half.
And in liquids this is also averaged out.
So nuclei would spin greater than a half include deuterium,
nitrogen-14, lots of metals, lots of things like sodium.
We'll see some examples of that later on, but, again,
we don't have to worry about it in liquids.
And then the last small interaction here is
the J-coupling.
That's the scalar coupling.
It's this interaction between the nuclei
that is transmitted through the bonds.
And as the name implies, it's a scalar,
so it stays unchanged regardless of the motions of the molecule.
And so it is there in both solids and liquids.
And it's something that we can use to tell us something
about the structures of the molecules
as you've most likely seen in organic chemistry.
OK. So that's kind of an overview of what the terms
in the Hamiltonian look like.
And we'll see this picture again as we go
through the different interactions.
Let's go through and talk about how this experiment works.
OK. So if we have our pulsed NMR experiment,
this is a little bit different
from other types of spectroscopy.
So, again, if you open up your organic chemistry book,
depending on which one it is, it might have an explanation
of NMR that's not quite right.
So a lot of them, I was horrified to discover recently,
have this picture where you put in the radiofrequency pulse,
and your spin state goes from alpha to beta,
and then a photon gets emitted, and you detect it.
That's not actually how it works.
I mean, that's analogous to other types of spectroscopy,
but that is not really what is going on in NMR.
So remember we talked about what happens
when you have some excitation.
You put energy into a system,
and there are all these different mechanisms
by which it can relax back, some of which we can measure
and some of which we can't.
In NMR, the relevant relaxation mechanisms are all kinds
of other things other than your system spitting
out an RF photon.
That is not really, a stimulated emission is not really an
important effect here.
So instead what we see is we deliver a 90-degree pulse
and put our quantization axis into the XY plane,
and then we see this free induction decay.
Remember we have the magnetization relaxing back
to the equilibrium position after we release the plus.
And it has this dependence
because we're detecting any XY plane.
So we get a decaying exponential convoluted
with a cosign function.
And, you know, again, remember our Fourier transforms.
So the FID has this kind of a functional form.
And then the Fourier transform of that is a Lorentzian,
which we approximate with this first term.
And there is an inverse relationship between the length
of the FID and the time domain and the width
of the Lorentzian in frequency domain.
So if we have a signal that takes a long time to die away
that is going to give us nice narrow lines.
If it dies away quickly, then we have broad peaks,
and we're going to talk about the things that might dictate
that a little bit later on.
And as a result of this, we get a spectrum.
So, you know, again, it's a completely different mechanism
from the CW case where we sweep the frequency
and see how the sample responds at different energy levels.
We're putting in a pulse exciting the whole thing
at the same time and then taking the Fourier transform.
OK. So the information that you get is
on the basic level largely independent
of whether you're doing CW or pulsed center marks up the,
in the pulse center mark case it works a lot better,
but the information that we're getting
in the chemical sense is essentially the same.
So here we're just looking at protons, but this holds true
for any kind of nucleus that has a non-zero spin,
and we can see an NMR signal.
So protons in a particular kind
of chemical environment are going
to have a characteristic chemical shift.
And so this tells us a lot about what kinds
of functional groups are present in the molecule
and what kinds of structure we have.
And so this table is something that I'm sure you've seen before
in organic chemistry books.
And these are useful things to know.
It's good to know where different types of protons show
up roughly in terms of chemical shift.
[ Silence ]
When I say it's good to know that means there are likely
to be exam questions where you have to sketch the spectrum
of some molecule, and I will give you some kind
of basic rudimentary chemical shift table, but it's good
to have a general idea about how this stuff works.
So, you know, in organic chemistry
to get really complicated spectra, and you have to figure
out the structure of molecules.
For PCAM I'm likely to have you do it the other way.
I'll give you a molecule, and you'll have
to predict was the NMR spectrum looks like because that's,
you know, that's really what it's about.
We want to understand how the spectroscopy works.
So here's a spectrum of a molecule,
and you can see the methyl groups show up between one
and two PPM as we expect.
And then the methyl group that's attached to the oxygen is,
has an increased chemical shift.
So does everybody remember what the chemical shift is
from organic chemistry?
I'm not, I realize I'm not going over this,
but I think it's review for everyone.
Is that true?
Yeah? OK. So, I will just say it has
to be defined relative to some reference.
That is usually a TMS, tetramethylsilane,
so it's just a silicon atom with atom groups all around it.
That is defined as being zero PPM.
So, you know, if you go measure an NMR spectrum
without having it referenced, if you have an old instrument
like the one in my lab, you will get this axis
in kilohertz basically.
So you just have a frequency scale,
and the PPM scale is parts per million.
So it's kind of like a percent but it's out of a million,
and that is relative to the main magnetic field and relative
to what do the protons in TMS generally.
There are other references that you can use
for different things, but that is standard
for a lot of organic molecules.
OK. So that's the chemical shift from kind
of a practical perspective, you know, how do we want to use this
to see what structure molecules have.
Let's look at it a little bit more as far
as where it comes from.
So what we're looking at here is the electron cloud
around a particular spin.
And the electrons are making a local magnetic field depending
on their distribution.
And that causes the nuclei to see this local effect
that either adds to or subtracts from the main magnetic field.
So here's a molecular model
of glycine just so you can see this.
And I'm showing you a C13 spectrum just
to remind everybody that we don't have to look
at protons all the time.
There are lots of other nuclei
that give interesting NMR spectra.
And if we look at the molecular model and, you know,
picture the electron clouds, it's really clear
that the carbon that's attached to,
that's the carbonyl carbons attached to two oxygens,
is going to have a very different distribution
of electrons than the methylene.
And so, you know, here I've labeled these at the two carbons
in red and blue schematically just to indicate
that this has the same general trend as protons.
So methyl carbons are going to be,
methyl or aliphatic carbons are going to be,
you'll have lower values of chemical shift and, you know,
things that are attached to something
like a carbonyl are going to be
at higher chemical shift values just as in the proton spectrum.
OK. So typically what people do with this
in a synthetic context is get more
or less a fingerprint of a molecule.
So you have one-dimensional proton spectra, and, you know,
they get to more and more messy.
And organic chemists are really good at looking at these things
and pulling out structures.
So now I know Professor Nowak [phonetic] teaches a graduate
NMR class that's all about this kind of stuff.
So it's all about, you know,
how to interpret really complex spectra
and get structures of organic molecules.
I also teach a graduate NMR class that is all
about Hamiltonians and, you know,
how do write your own pulse sequences
and really developing the spin physics of NMR.
They are very different skills.
You know, and we have joked
that we couldn't pass each other's final, which, you know,
may or may not be true.
But there really are very different ways to approach it.
And what I am going to try to give you
in this class is a little bit
of the physical chemist perspective on NMR.
So, you know, don't lose sight of the fact
that you can use this to solve the structures of molecules,
and it's fantastically useful in the synthetic context,
but there's a whole field of NMR research
where we do something else.
OK. So back to talking about chemical shift.
Let's look at this, what this looks like in the solid state.
So so far we've talked about chemical shift
as though it's just a number.
So we have a different distribution
of electrons around the nuclei.
And as a result of that they experience a magnetic field
that is adding to or subtracting from the main magnetic field.
And they show up in a different place on this spectrum.
Well, that's only true if your molecules are moving
around really quickly in the timescale of the experiment
and averaging out orientation effects.
If we have something that's in a solid, so say we have a protein
in a crystal, and let's say it's a single crystal
so that it has a really well defined orientation,
if we look at a carbonyl carbon in the protein backbone,
if we look at that double bond between the carbon
and the oxygen and think about the local field
that the carbon is experiencing as a result of those electrons,
if it is staying still we can easily imagine
that this is not isotropic.
So that carbon sees a different local magnetic field
in the X, Y, and Z directions.
And you'll see a signal for each of those,
and it gives this funny line shape.
And that's called chemical shift and isotropy.
Again, it's averaged out in liquids.
We only see the isotropic value,
which is essentially the average value.
But in solids this is really important.
And as with many of these things it's a double-edged sword.
It contains a lot of information.
So we can fit this line shape
and get very detailed information about exactly how
that carbonyl is oriented relative to the rest
of the protein, certainly relative
to the main magnetic field.
This is really useful in context like looking at a peptide
and a membrane protein where you want
to get the relative orientation of that carbonyl
with respect to the membrane.
However, if you have a whole protein worth of line shapes
that look like this and they are all overlapping,
that's a little bit hard to deal with because it's difficult
to separate out the signals because they're all overlapping.
And a lot of solid state NMR methods development is
about how we deal with this.
You know, putting in these interactions selectively during
the times that we want to see them,
and that can be done either with selective labeling, you know,
involving putting C13 in specific places in the sample,
or it can be don spectroscopically.
OK. So chemical shift, you know, as I alluded
to on the previous slide, is not in solids.
It's not a number.
It's a tensor.
And so we can show, you know, we can make matrix representations
of the Zeeman effect, which here I have omitted the gamma
and H-bar.
And then our chemical shift is a tensor in three dimensions.
And you don't really have to worry about this except
on the conceptual level.
I am not going to ask you to do anything with it.
But I do want you to know that it exists and that there is more
to the picture than just the solution state idea
where we have just the isotropic value.
All right.
So here are some pictures
of actual chemical shift tensors depending on the shape
of the electron, the electron density around the nucleus.
And you can see they look really different depending
on whether you have a prolate or oblate ellipsoid
or if you have something that is centrosymmetric versus something
that is something that is completely asymmetric.
And so there is this orientation dependence
that can be fantastically useful or it can be a nuisance
if you have a bunch of these things on top of each other.
OK. So that's sort of the run down of the chemical shift
and everything that's associated with that.
We will come back to it and talk about it some more.
Let's talk about, let's go back to our organic chemistry picture
of structure elucidation with NMR.
So if we're talking about protons or carbon or N-15
or anything like this, there are some features
that tell us something about the structure.
So the number of signals is the first thing that gives us a clue
about what's going on.
That tells us about the number
of chemically inequivalent nuclei.
The position of the signals, the chemical shift,
tells us exactly what functional groups are present.
The intensity of the signals if we integrate the area
under all the peaks tells us
about the relative number of protons.
We have to be very careful about using that for heteronuclei,
things that aren't protons.
And the reason is because magnetization gets transferred
from proton to C13 in the course of a lot of the experiments
that people typically use.
And so you can't just take a C13 spectrum
under typical experimental conditions and assume
that it's quantitative because you are also going
to be seeing information about which carbons are closer
to the protons than others.
But for protons that is a good assumption.
You can integrate things
and find the relative numbers of them.
The last thing that's important is the spin spin splitting.
So this, in the solution context, this is mostly going
to be due to J-coupling.
And this can be, again, between,
it can be homonuclear or heteronuclear.
So it can be between protons or if you,
depending on how you do the experiment it can be
between protons and C13, protons and N15, and there's,
that gives you information about coactivity
of chemical environments.
And I will also add, if we're talking about solids,
dipolar couplings are very important
in learning about the structure.
These give us long-range distances.
OK. So let's look at some practical examples.
And the goal here is to tie together what you already know
from organic chemistry.
You know, how to look at these spectra in a practical way
with the underlying physical chemistry concepts
of what's going on.
And, you know, if that's not happening please feel free
to ask questions.
All right.
So let's look at some typical examples.
So just a reminder, in order for protons
to give different NMR signals they have
to be chemically inequivalent.
So protons that are occupying sites that are the same
in molecular that look the same
when things are motionally averaged will show
up at the same place.
So for this particular molecule, the methyl protons,
they're labeled in blue.
We have free rotation around single bonds in solution.
You know, everything is isotropically averaged.
And all those methyl groups show up in the same place.
The same thing for the two methylene protons here.
Now, again, this is something
that wouldn't necessarily be true in a solid.
If we had this molecule crystalized
and things were really rigid, it's possible that the way
that the particular crystal structure worked out that some
of these protons could be closer to other things than others,
and we would see splittings.
In solution that's definitely not going to happen.
You have to assume that everything is moving freely.
OK. So the number of NMR signals is going to be equal
to the number of chemically inequivalent types
of protons in your compound.
So here are just some examples where you have different numbers
of different kinds of protons.
[ Silence ]
And, you know, again, here are some examples that are going
to give slightly more complicated spectra.
And we'll revisit some of these molecules as we talk
about drawing these kinds of spectra yourself.
And, you know, again, you have to take
into account the rigidity of the molecule.
So in this cyclopropane with a chlorine in one site, you know,
this thing can't flex very much.
So the ones on the bottom are not equivalent to the ones
on the top even if they otherwise look symmetric.
OK. So the intensity
of the signals also tells you something assuming
that we're talking about protons.
And you can't just measure the height.
You have to integrate it because peaks might have different
widths even in the same spectra.
You know, you can have, again, the peak width depends
on the relaxation time.
And that could be different for different types
of protons even in the same sample.
And we will see how that works.
So, again, this gives you a ratio, not an absolute number
of protons that we have.
But it does give us a good relative idea of how many
of each type there are in the sample.
OK. So getting back to the quantum mechanical underpinnings
of this stuff, we've mostly been talking about spin one half.
And I'm sure that's pretty much what you've seen
in your previous work on these things.
There are also nuclei that spin greater than have.
And I alluded to this a little bit talking
about the quadrupolar interaction.
And these things are important, and we are going
to do some problems pertaining to them later on in the class.
So, for example, in organic chemistry you assume
that if protons on a molecule are deuterated
that you're not going to see any signal from the deuterium.
And that's true if you're looking
at the proton resonance frequency.
So one thing that's nice about NMR is
that it's incredibly specific in terms
of the resonant frequency of the nuclei.
If you are looking at protons you are not going
to see interference from other kinds
of nuclei except indirectly through the J-couplings
if the coupling is strong enough.
And it turns out that the J-coupling between deuterium
and anything else that you're going
to see is sufficiently weak
that you often don't have to worry about it.
But deuterium is a perfectly fine NMR nucleus with spin one.
And in my lab, for instance, we look at it all the time.
And lots of NMR labs do that.
So just to give this in a more general way.
Here's the spin quantum number for a nucleus.
So it's the same as other type
of angular momentum that we've looked at.
There is an overall angular momentum,
and there's also a Z-component of the angular momentum.
So, again, the math works out just
like orbital angular momentum and other things
that you have seen in a physics context,
but here we're talking about nuclear spin.
So what is nuclear spin or electron spin?
Nobody really knows.
It's an intrinsic property of these objects that happens
to obey the same mathematical formalism as spinning charges.
But, you know, it's really convenient to understand how
to do the math, but that doesn't necessarily mean we
understand it.
OK. So, again, you know, if we go back to the spectra,
if we have spins that are greater than a half,
we need to worry about the quadrupolar interaction.
We can look at our nuclear angular momentum in the same way
as some of these other things we've seen
in the electron angular momentum.
You have seen this before, the cyclic commutation relationship
between angular momentum operators.
That was, that came up in a homework assignment.
And previously we didn't really use it for anything.
It was just an example of finding commutators and things
like that that we needed to do to look
at matrix representations of operators.
Well, now we're going to use it for something.
So these values of the spin angular momentum,
as you can image, are pretty useful in NMR because we have,
you know, IZ is the eigenvalue of the Zeeman interaction.
So the eigenvalue is plus and minus one-half.
That corresponds
to the eigenstates being alpha and beta.
And IX and IY are what we can measure in the XY plane.
So it's good to review your angular momentum operators
because we're about to use them.
OK. So as I said, the eigenstates of IZ are specified
by these quantum numbers, and we can write them
as a ket [phonetic] like this.
And that's useful when we're talking about nuclei
that have spins greater than one-half.
So for a spin one-half there's only two states,
and you can call them up and down or alpha and beta.
The ones for a nuclei with larger values
of I don't have nicknames.
So you have to represent them using this kind
of a ket [phonetic].
So if we operate IZ on the state with the values of L and M,
we get M back as the eigenvalue.
And the eigenstate is this original state.
And so in the Zeeman basis, you know, again,
our sample is aligned along Z, and we have well defined values
of IZ that we can measure.
Here our eigenstates are alpha and beta.
And here's what those look like if we write them
out as these kets [phonetic] with values of L and M. OK.
So all that means is that if we measure,
if we have our spins aligned in the magnetic field along IZ,
and we measure the values of IZ, we're going to get alpha
and beta in some well-defined ratio that depends
on the relative populations.
If we measure IX or IY, it's,
we're not quantized along that axis.
So we'll get random proportions of these states.
OK. I also want to point out that the Hamiltonian
and IZ are both diagonal in the Zeeman basis.
And that means they commute.
So I have a little bit of animation fail in here.
So I am just going to put everything up and talk about it.
All right.
So here's our matrix representation for IZ.
So our spins were in the Zeeman basis.
And so what that means is we have this H-bar over two
out in front which has to do
with the particular energy values,
but more important is looking
at what the eigenstates are at this point.
So everything is in the, either the alpha or beta spin state.
And, you know, again the one-half has been pulled
out in front that since the values
of M are plus and minus one-half.
And so we have values on the diagonal
and nothing off the diagonal, which tells us
that everything is either in alpha or beta.
And we know what our Hamiltonian is.
This is gamma, omega-not times IZ.
And so that means if we measure the energy of alpha,
we get back one-half gamma omega-not alpha.
And similarly for beta we get minus one-half gamma
omega-not beta.
This is the same thing that we've already seen.
And so we can use that to construct the matrix
representation of the Hamiltonian.
So we're just applying these operators the same way
that we have before with things that were more concrete.
You know, now we are applying this to the spin states.
And so we can make these matrix representations
of both IZ and the Hamiltonian.
And since they are both diagonal in this basis,
they commute with each other.
And if that's not 100% clear, that's fine.
We're going to spend more time talking about it next time.
I just wanted to introduce it so that people have something
to think about for the next class.
I'm going to post this lecture plus some practice problems
for the NMR part later today.
Does anybody have any questions before we quit?
Yes?
>> So I just want to make sure I understand the NMR [inaudible],
you put in a pulse, is that a pulse magnetic,
are you [inaudible] magnetic field direction?
What is the pulse [inaudible]?
>> That's a really good question.
OK. So the pulse is a radiofrequency field
that is essentially producing a magnetic field that's orthogonal
to the main magnetic field.
And that should be really weak, right?
Because like my big magnetic field is, you know,
if we talk about it in frequency units,
in my lab it's 500 megahertz.
The RF field, again to give a typical value,
is maybe 140 kilohertz.
So it should be way weaker than the main magnetic field,
so it's amazing that it does anything.
The only reason that it does anything is
because it's on resonance.
So our nuclei are processing
about the main magnetic field really fast,
and that applied field that you're adding is following it
around for thousands of revolutions [multiple speakers].
Exactly. And so you tip only the ones that are on resonance.
And so that's why we don't randomly see carbon signals
when we're looking at a proton spectrum because the carbons are
at a way different frequency,
and they're not interacting with that RF.
>> And so your readout thing is whether it's resonating
with that frequency or not?
>> Right [multiple speakers].
And you control the frequency that you apply.
That's an experimental parameter that you control.
And one of the main things that we do
in my lab is build probe circuits to apply RF pulses
in different ways and change the experimental conditions.
I'll show you guys a little bit of that probably later on.
All right.
We are done for today.
See you next time.
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