>> 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.
[ Background Discussion ]
[ Silence ]
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