>> Last time we ended
with talking about some
of the operators that are
involved in NMR [phonetic] and,
you know, here we're
still at equilibrium,
we still have stuff
aligned along the Z axis,
but this is the starting
point for being able
to understand NMR spectroscopy
from the physical
chemistry perspective.
Okay. So I ended with this
last time, but let's go back
through it and make sure
that everything makes sense.
So, we have an operator that
corresponds to the magnetzation
on Z for a spin 1/2 nucleus
that means it's either plus
or minus 1/2 up or down.
These states are called
alpha and beta in NMR.
The eigenstates of Iz have
eigenvalues that correspond
to their spin quantum number.
So we can have plus
or minus a half here.
If we operate this operator
on its eigenstate we just get
that quantum number back
and the original state.
So, okay, not so
surprising this looks similar
to things you did last
quarter just it's a little bit
different system.
I think it would make
a lot of sense actually
to teach NMR first
when we're talking
about thermal [phonetic]
mechanics because it,
the eigenstates and
the operators
that we are using
are very simple
and it's easy to
see what they do.
So, hopefully this maybe
even clarifies some things
that were challenging
last quarter.
All right so what that
means we've written this
down in a very general way.
What that means is
that if we operate Iz
on alpha we get 1/2 alpha.
If we operated on beta,
we get minus 1/2 beta
and remember these things
make up an ortho normal set.
Alpha does not equal minus beta.
They are, in fact,
orthogonal to each other.
So if we take the integral
of alpha-alpha or beta-beta,
we're going to get 1 and if
we have a matrix [inaudible]
that looks like this or an
overlap integral that looks
like this where we have 2 of
these states that gives us 0.
So, we've seen these
things before in context
where it's really easy
to visualize what it is.
So, in this case,
what's the space
that we're integrating over?
It's spin space.
It's a two-dimensional
Hilbert space that has to do
with these two operators.
So there isn't really an
easy way to visualize it,
but fortunately mathematically
it's pretty simple.
These are the only 2
things we have going on.
We've got plus and minus
1/2 and we know that alpha
and beta are orthogonal to
each other so it's easy to set
up the overlap integrals.
Okay. So now if we want to
make matrix representations
for our spin operators,
so far the only one we really
know is Iz, let's look at how
to set up its matrix element.
Okay, so a matrix
element is, you know,
we have the two states
we're looking
at with the operator
sandwiched in between them
and this is a little
bit different way
of approaching the
problem but it's similar
to what we have already
done before
and the context is
talking about group theory.
So we made matrix
representations for operators
in a space or in a context
where we can easily visualize
what the operator does
and then we made up
the matrix that way.
Well, so now we have a situation
where we don't have an easy
spatial representation for it
and we just have to
do it mathematically
but we're doing the same thing.
So we're going to make
matrix representations
for our operators.
Okay, so if we have the
alpha-alpha matrix element,
that means that first we
operate Iz on alpha and we get,
of course, 1/2 alpha, then we
can pull the scalar out of that
and so we just get
1/2 alpha-alpha
and that gives us 1/2.
If we do the same thing for the
matrix element for alpha beta
for this Iz operator,
the first thing we have
to do is operate Iz on beta
which gives us minus 1/2 beta.
Again, you can pull the constant
out in front and you can see
that this matrix
element gives you 0.
For all our analyses
of NMR operators this is
essentially what we're going
to do .
There are cases where
we're not looking at things
that are eigenstates and we're
going to have to figure out how
to write stuff in
terms of operations
that we can trivially, you
know, find the eigenvalue for.
So, all right let's take another
look at Iz in the Hamiltonian.
So here's my matrix for Iz.
So the way I got that
is the factor of 2.
I've been sloppy about dropping
my H bars but they are in there.
The factor of 2 comes from the
fact that the eigenvalue is plus
or minus 1/2 so I just
pulled it out of the matrix
and then we said that the
alpha-alpha matrix element
of Iz is 1 and then for
alpha beta at 0 same thing
for beta alpha and then
beta-beta it's minus 1 again
with that factor of
1/2 in front of it.
Everybody okay with
how I got that?
All right good.
So now let's look
at the Hamiltonian.
So, for an NMR experiment,
the Hamiltonian is,
again, Omega not Iz.
So that means I'm just
going to do the same thing
but I have some more
constants in front of it
and we can make the matrix
elements the same way.
So here, again, I'm
operating the Hamiltonian
on the [inaudible] first pulling
out whatever constants I have
and then taking the overlap
integral of what's left.
So these are the
answers that I get.
We know if we look at the matrix
representation of 2 operators
if they're diagonal in the
same basis, then they commute.
So, this is really powerful
in quantum in general
because if you know
that operators commute,
that gives you important
information about the system.
So if stuff commutes
with the Hamiltonian,
then energy is conserved and
you can use this for a lot
of things, but okay so far we're
just talking about, you know,
when we say this is the
Hamiltonian this is the
[inaudible] on Hamiltonian,
right?
So we just have the
spins aligned with
and against the main
magnetic field
and we can see the energy
difference for that, you know,
we know how to operate
Iz on these states,
but we haven't learning anything
about the actual NMR experiment
because for that we need
to be able to apply pulses,
we need to look at stuff in the
X, Y plane, how do we do that?
So, you remember from your
homework assignment a long time
ago we did, you know, you proved
that the angular momentum
operators don't commute
with each other.
In fact, they have this cyclic
commutation relationship.
Here we're calling
them I instead of L
because we're talking
about a spin
and not an actual
angular momentum
from something rotating,
but the math is the same.
So we know that these things
don't commute with each other.
So, IX and IY, which are the
spin operators that we need
for looking at our system in the
X, Y plane, those don't commute
with Iz and that means
they don't commute
with the Hamiltonian
and it's not obvious how
to work with them.
So we need to define some other
operators to look at that.
So here's what the matrix
representations of IX
and IY are just for reference.
If you don't understand
how I got that,
that is perfectly
understandable.
We're going to go
through it in a minute,
but I want to show you what the
answer is before we get there.
Okay. So in order
to get this result,
we need to define
something called the raising
and lowering operators.
So, what the raising
and lowering operators do is
they raise and lower the states
of the system, but here's
what they're defined as.
[ Pause ]
So we've got I plus as IX plus
IIY and I minus is Iz minus IIY.
So we've kind of alluded to
before like we're getting real
and imaginary parts of our
signal in the XY plane.
You can start to see
how this fits together
with the formulism
that we're using.
Okay here's what I
plus and I minus do.
So you have this collection of
constants out in front relating
to L and M that is
your eigenvalue.
Then the eigenstate
of the raising operator is
the state you have plus 1.
So if you started
with minus 1/2,
it's going to go to plus 1/2.
So if you operate I plus
on beta, you get alpha.
If you operate I plus on alpha,
there's no state that's higher
than that in this spin
system that's defined
so that's going to give you 0.
That's not going to
be true for spins
that have I greater than 1/2.
Right? So if we had a spin 1,
we can operate 1 plus on minus 1
and we'll get the eigenvalue
times the state with 0 and then
if we operate I plus on 0, we
do that again and get plus 1.
For a spin 1/2 system, we only
have 2 choices; up or down.
So, you can either
raise or lower
and if you operate
the raise and lower,
if you operate the raising
operator on the highest state,
you get 0 and equivalent
for the lowering operator.
Okay. So let's write
that out explicitly.
So I plus operated
on alpha gives you 0.
I plus on beta gives you alpha.
Here I have dropped
the constants.
So you need an eigenvalue
in front of this.
I will try to go back and put
that in before I
post the slides.
I minus on alpha gives you,
again, your constants times beta
and I minus on beta is 0.
So, why do these
operators exist?
Why are we going to use them?
They have a bunch of uses in
quantum mechanics actually,
but in this particular context,
what we want is we don't know
how to deal with IX and IY
because if we have our
magnetization quantized along
one of those axes, you know,
now it's not, you know,
the eigenstates of IX and
IY are something else.
They're some sort of
linear combination
of the Zeeman eigenstates, but
we don't know how to measure
that and we don't know
how to operate on it.
They're not, if you
measure, you know,
your normal spin
states alpha and beta
when they're quantized along IY,
you'll just get alpha and beta
with 50% probability and
that experiment doesn't tell
you anything.
We need a way to deal with it.
So these operators are defined
in terms of IX and IY in a way
that they give us something
that we know how to take,
that we know how to find the
eigenvalue of that's going
to give us a while to find
answer and we're going to go
through it right now so
you'll see what I mean.
Okay so let's find the
matrix elements of I plus
and I minus in this basis.
So, again, we're still
in an examine basis,
we have the spin 1/2.
So if we operate I plus
on alpha, we get 0.
Now if we operate
I plus on beta,
that gives us our
constant times alpha
and then we take the
integral of alpha-alpha,
which gives us 1 and, again,
that should be times
the eigenvalue.
Again, operating I plus
and alpha gives us
0 so that one is 0.
If we operate I plus on beta,
that gives us a constant
times alpha
but then the integral
of beta alpha is 0.
So that's our matrix for I plus.
We can do the same
thing for I minus.
We operate I minus on alpha.
That gives us beta but the
integral of beta alpha is 0.
I minus operated on beta is 0.
I minus operated on alpha
gives us a constant times beta
and then beta-beta gives us 1.
So we get the matrix
for I minus.
So, again, these are
just convenient operators
that we can work with
in the Zeeman basis
and they give us matrix
representation that makes sense.
Now we're going to be
able to use the definition
of these things in terms of
IX and IY to enable us to work
with the eigenstates
of those operators,
which again why do
we want to do that?
Because that's the signal
that we can actually
measure in the experiment.
Okay. So, here's how
these work and I'm going
to let you use this to verify
the matrix representations
for IX and IY.
It's tedious but
it's good practice.
So, you know, you can go through
and operate these things and,
you know, like I showed you in
the previous couple of slides
and once you work it through
once then I think you'll be
pretty comfortable dealing
with these kinds of operators
and you know what
the answers are
because I showed you
earlier in the lecture.
Again, it's tedious so
if you do one of them
and you think you totally
get it, that's good enough,
but if you need extra practice,
work through them both.
So, again here are the
answers that you get.
So now we're really taking
our knowledge that we learned
from looking at group
theory and being able
to make matrix representations
of operators and work with them
and now we can apply that to
a quantum mechanical system
where the transformations that
we're doing are not obvious,
you can't really visualize
it in Cartesian space
because it doesn't live
in Cartesian space;
it lives in spin space, but
because we have these skills
of being able to put
operators in terms of matrices,
we can use all of that
same formulism to do stuff
where it's not so
easy to visualize.
So now hopefully the
point of being able
to do that becomes clear.
So we practiced on systems where
it's easy to verify the answer
because you can visualize it.
Now we can do these things where
it's a little bit more abstract.
Okay. As I have sort of
hinted at along the way,
we can have spins in a super
position of alpha and beta.
We don't have to
have everything just
in one eigenstate or the other.
So again this is where
the basic textbook picture
of NMR goes wrong.
You get this idea that
everything is either
in the alpha state
or the beta state.
Well, it's not.
You can have these
super positions.
Okay, so you can have a
way function for your spin,
and again, it's a
funny wave function.
It's not, you know, it's
not a function in the sense
that we're used to looking at.
It's a probability mass
on either alpha or beta
or some combination of the two.
So, our spin state can
have a super position
where we have some
amounts of alpha and beta.
How much is described by these
constants and we can write
that down as a vector.
So in that notation, here's
alpha and here's beta.
Why is that useful
to be able to do?
Because we have all of
our operators written
out in terms of matrices.
These things are normalized as
I said when we talked about how
to do the matrix elements.
Again, it's kind of hard
to picture these functions
and how they're orthogonal
to each other.
They're in spin space.
It is pretty abstract,
but they are.
So, they are orthogonal
and they're normalized.
All right.
So now we get to what are
the eigenstates of IX and IY?
And I'm not going to
prove this, you know,
as to how those are
the, how we get to those
as the eigenstates just because,
you know, there's a limit
to how many NMR core
dumps we can do,
but this is what they are.
So, if we have the
eigenstate for plus X, so,
that's your spin quantized
along the positive X direction,
it has this particular
eigenstate of alpha and beta.
So, our constants for
each of them is one
over square root of 2.
So now if we're in this state,
the X component is sharp
and Y and Z are not.
So we can measure along X,
we're in this plus X eigenstate.
Every time we measure along X
we're going to get that value.
If we measure along Y and Z,
it's going to be ill defined.
It's not far up in that case.
[ Pause ]
Okay, so again, we're
going to apply our matrices
that we have been writing down.
To operate your operator
on the ket [phonetic] write your
spin state in vector notation
and then multiply the
appropriate matrix by it.
So here's what we get for IX
operated on the plus X state.
That gives us if we
simplify the constants
that gives us a half X,
which makes sense, right?
That's what we expect.
We get the original
state back as the,
we said it's an eigenstate
and then we get its value
of its spin quantum
number which is plus 1/2.
All right so that's
one of these things.
Let's look at the
value for minus Y.
So similarly we can write it
out and, again, I'm not going
to show you how we get
this as we eigenstate.
We're just going to
look at the result.
Here it is in vector notation.
We can also operate
IY on it and we see
that we get minus 1/2 minus IY.
Sorry, minus Y. This
is what's detected
in a typical pulse
NMR experiment.
So, if we do even some
complicated pulse sequences
where we flip the spins
through all kinds of gymnastics
and make them do different
things, at the end we have
to end up with minus
Y as an eigenstate
because this is what
we can detect.
Okay, so I'm going to show
you how to operate some
of these things on our
spin states and look
at what a realistic
NMR experiment does.
So here are our rotation
operators for RX, RY and RZ.
This is rotating about the X,
Y or Z axes by some angle beta
and when we get into this
it should look familiar
because they're the same
as the rotation matrices
that we've been making
for rotating some physical
object about an axis.
All right.
So if we have, so
beta is the angle.
So, if we have an
operator RX pi over 2,
that rotates the magnetization
90 degrees about the X axis.
A rotation operator commutes
with the angular momentum
operator about the same axis.
So RX commutes with IX but it
doesn't commute with IY and Iz.
So for a different angular
momentum operator we have this
kind of a relationship.
[ Pause ]
So there's a lot of math and
it's a little abstract but stick
with me because we're
going to get back
to how this actually works in
the real pulse NMR experiment.
Okay so we want to apply
a pulse with phase X. So,
we have our spins, they're
aligned along the Z axis,
they're quantized along
Z, they're in the alpha
and beta states, and
we want to put them
into a state that
we can measure.
So, we apply an X plus.
That's going to take us
from whatever our starting
state was to a final state.
So we're going to do that
by operating our operator
on the initial spin state
and so that means, you know,
we're going to take whatever
spin state we started in,
write it in vector notation
and then multiply the rotation
matrix for the pulse by it and,
again, this is called
the pulse propagator.
This beta P is the flip
angle of the pulse.
So, pi over 2 in the example
that we're talking about.
All right.
So, let's back up and talk
about that for a minute.
This is something that is, it's
treated in your book in kind
of a hand wavy way and I want
to really show you how it works.
It's important to
understanding this.
All right.
So, we talk about how the pulse
NMR experiment works and we say
that we have our
spins along the Z axis
and then we apply a
pulse that's on resonance
so we have the right
amount of energy
and it flips the
magnetization into the XY plane.
Well, how do I get it
to actually be exactly
perpendicular
with the main magnetic
field, right?
So you can imagine if say if
it's off resonance a little bit
or if the pulse just isn't
strong enough we can tip it part
way down and we won't
see very strong signal
because we can only measure the
projection along the XY plane.
If it's too strong, say, and
it rotates it farther down,
then we're going to see a
weaker signal there, too.
In fact, we can rotate
180 degrees
and just invert the
magnetization relative
to how it was at the beginning
and then we won't see anything
because it'll just be
along the negative Z axis.
So how do we know that our pulse
is actually 90 degree pulse?
The answer is we typically
measure this experimentally.
I mean you can calculate
it and get close,
but we optimize this
experimentally.
The flip angle depends on the
mutation frequency of the RF
so that is, you know,
you can imagine the RF
as so it's an oscillating
electric field in, you know,
in a direction that's orthogonal
to the main magnetic field,
but we can also imagine it
as inducing oscillations
in the magnetization.
So we start along
Z if we apply a pi
over 2 pulse we tip it this way.
If we go too far and give it 180
degree pulse, it goes like this,
and you can imagine if we're
looking at the signal for that,
we get oscillatory behavior and
so we can express that frequency
in frequency units and so
we're measuring the strength
of the magnetic field in a
way but in frequency units
and that tells us about
how much power we have
to flip these pulses.
So this flip angle is
that field strength
in kilohertz times the time,
the length of the pulse.
So, you know, we have an angular
frequency times the time and so
that gives us an angle.
Let's look at what
that looks like.
So here's a rotation matrix for
a pulse of flip angle beta and,
again, notice how it looks
just like the rotation matrix
that we used for looking at
physical rotations of molecules
in a particular coordinate
system.
So, these things that we
have learned are definitely
applicable to this system
that's a bit more abstract.
So, again, let's see
what this looks like.
So, we have our magnetization
vector initially at 1Z.
We turn on the pulse and we
have calculated the flip angle
and the mutation frequency
to be exactly right
so that it's a 90-dgree pulse
and that's going to give us.
Our magnetization in the minus
Y eigenstate it also picks
up a phase factor, which is
this extra little E to the I pi
over 4, which we shouldn't
worry about right now,
but this tells us
about, you know,
how we can actually
experimentally make these
spins flip.
Could you guys please shut
down the side conversations?
It's very distracting.
It's distracting to me
and I think it's distracting
to other people too.
Okay so let's look at
experimentally how you do this.
So this is something
that my lab does.
We build NMR probe so
we build the RF circuits
that produce these
radio frequency pulses
that flip the spins and it
turns out that there's lots
of experiments that we
can do that you have
to develop special hardware
to do and grad students
and actually a few undergrads
in my lab have worked on this.
So, you know, that means that
we work in the machine shop,
we build electronics, it's
pretty interesting stuff.
So here's what the
probe looks like.
So it's really long because
it's inside the magnet
so when you see pictures
of NMR magnets or you go
to the NMR Lab, you know,
to run your experiment.
If you're just a casual user
and you don't build the stuff,
you don't see what's
actually doing most
of the interesting stuff.
So inside the magnet there's a
little coil that is the thing
that delivers the pulses to
the sample and it also listens
to the signal that comes
back and, you know,
that's just represented as the
inductor here, but you know,
the devices is long because
that coil has to be located
in the very center of
the magnetic field.
So it's inside the magnet.
The actual business end of
it is relatively simple.
So we have this parallel
resonance circuit
so that's the inductor
in parallel
with the tune capacitor
that's called CT.
By adjusting that variable
capacitor we can change the
resonant frequency
of the circuit
and then you notice there's this
other little capacitor in series
with it, that's the match
capacitor, we can adjust
that to zero out the imaginary
part of the incoming RF.
So that matches the
signal to 50 ohms.
We have to have the
pulse that's coming
in impedance match to the load.
So, this is how we
experimentally deliver the
RF pulses.
So what I wanted to show
you is we've been talking
about mutation frequencies
and how we can measure
that here are some that
are experimentally measured
for some real probes.
So one thing to notice is
that we have three of these
so we're looking at
protons, carbon and nitrogen.
So when we're doing a
multidimensional NMR experiment,
you know, we talk about, okay,
we can look at proton carbon,
nitrogen, phosphorous for every
nucleus that we're looking
at we have to have a
separate channel of the probe.
You need a separate RF strike to
be able to interact with that.
Particularly when we get
into talking about, you know,
okay we're going to look
at proton and detect,
we're going to look at
proton and decouple carbon,
you have to have 2 channels to
be able to interact with that
and that means that you need,
2 of these are RF circuits
and they're all coupled.
So, why am I showing you this?
Just to give you a feel for we
can talk about all the stuff
in theory and, you know, it's
neat, it works out really nicely
but it just doesn't give you a
feel for what you actually do.
So, you're getting that flavor
for it in the case of NMR
because that's what
my lab does, you know,
if I did something
else, you know,
then you might be hearing
more about IR spectroscopy
or something like that,
but again so this is how you
experimentally measure the flip
angle that a pulse is going
to have on your signal.
So you can see like we
have the RF field strength
at some constant value.
So here for, you know,
this is what B1 refers to.
So, for proton we had
132 kilohertz carbon.
It's 71.4 nitrogen and
86.2, that frequency refers
to the frequency at which the
magnetization is going around
and around in these sign waves
and in that case it's a measure
of the amplitude of the
field that's being applied.
It's one of these things where
the units are very weird.
It's strange to think of
a magnetic field in units
of frequency but we do
this in NMR all the time.
So, we're talking about
the main magnetic field
and frequency units.
Like we usually say we have a
500 megahertz magnet, you know,
rather than an 11.7 Tesla magnet
which would be the appropriate
SI unit for magnetic field.
The reason we do
that is we're saying
that the procession
frequency for protons
in that magnet is 500 megahertz
and that's something
that's convenient to talk
about in terms of NMR.
Same thing here.
We're talking about the
amplitude of the RF field
that we're applying not in units
of Tesla or something else,
but in terms of how much can we
actually influence the spins.
So, again, that mutation
frequency times the time
that the pulse is applied
gives you the flip angle.
So, if you look at these plots
in the case of the proton,
each one of these steps
is .5 microseconds.
So if we apply the pulse
for .5 microseconds,
you see the first point doesn't
tip the magnetization very much
at all and we get a weak signal.
The second one after, you know,
1 microsecond tips it a
little bit more and then we go
up to 90 degrees and, you know,
so on as the magnetization
goes around and around.
I want to point out
that if experimentally
if stuff were perfect,
this should look
like a perfect sign wave and the
magnetization should go around
and around forever and
there should be no limit.
That's not how things
actually work.
So, it turns out that your coil
is not perfect that you're using
to apply the field, and
if you look at, you know,
especially the proton channel
here you can see if you look
at the third or the fourth
maximum the overall amplitude is
a little bit lower.
This thing is starting to decay.
That's because stuff
starts to lose coherence.
As you apply the
field for longer
and longer, it's not perfect.
One reason for that is that your
coil is not an infinitely long
solid wave where the
magnetic field is the same
in all parts of it.
It's higher in the middle and
it falls off toward the end.
We actually can do things
to try to make it better
when we're engineering
these things
so for a solenoid
[phonetic], for example,
if we just have a coil
that's literally wound
on a cylindrical form,
which sometimes we do use,
you can make it stretched in the
center and squished on the edges
to try to even out
the magnetic field.
That's something that we do.
So, here's a kind of a funky
looking coil that was built
in my lab and you can see
that one of the things,
one of the properties it has is
that it has a really
nice magnetic field right
in the center.
This plot is the magnetic
field as a function of distance
from the inside of the coil.
So, right in the center it has
a really nice magnetic field
and it falls off very
quickly at the edges
and that gives us these very
nice looking mutation curves.
Now, they look so
nice because in
that experiment the sample
is restricted to only be
in the region where
the coil looks perfect.
Okay. So, we talked about how
to deal with our spin operators.
You got some homework as
far as applying the raising
and lowering operators,
you know, which is just
to give you the experience
of working with, you know,
how do you apply these
operators to stuff
and be able to make this work.
We related that to pulse NMR and
how we actually see a signal.
Now I want to talk a little
bit about relaxation and relate
that to actual experimental
factors.
Okay, so where we're going with
this is we've talked about,
you know, when you, you know,
you have your perfect 90-degree
pulse, you plus magnetization
in the XY plane and it's going
to relax back to equilibrium
and end up back along Z. So, so
far all I've told you about how
that process works is that it's
not emission of an RF photon.
Our system does not
spit out a packet of RF
and come back to equilibrium.
It does something else.
So, what? Let's talk about that.
Okay so if I put the
sample in the magnet.
So, you know, I go to
the liquids machine
and put my little NMR tube
in the top of the magnet
and let it sit there,
how long does it take
for my spins to align along Z?
What do you think?
Anybody want to take a guess?
Twenty minutes?
If you're looking at like
Silicon 29 or, you know,
maybe a carbon that's not close
to anything at all like a carbon
in a perfect diamond, that
might be a good guess.
If it's a liquid, it's
a couple of seconds.
But here's what it's not.
It's not nanoseconds.
So when you put your
sample in, you know,
one guess that people
often make is, well,
I have a 500 megahertz magnet
so take 1 over 500 megahertz
and that's how long it
takes the spins to align.
It's not. It's independent
of the mutation frequency;
it's a different effect.
So what's happening is you have
all your little spins in there
and they get bumped into
by other molecules and so
when the molecules move there
are other little oscillating
fields and they get bumped and
they eventually end up aligning
with the field and that
process takes a little while,
but it depends on the spin,
it depends on the local
chemical environment, you know,
what's actually causing the
relaxation and, you know,
it's on the order of half
a second or a few seconds
for some typical samples.
So, you hear about this when
you're talking about pulse NMR
in kind of a practical
context because you know
that if you give a pulse and
you wait for the magnetization
to relax back if you
don't wait long enough,
the second scan you're not
going to get very much signals.
So if I only wait for
it to come halfway back
and then pulse again, I'm going
to get a smaller
signal the second time.
Of course, that's not
what we want to do.
We want to signal average over a
long time and add up many scans.
So we have to wait long enough
for that magnetization
to come back.
So this relaxation
that we're talking
about is called longitudinal
relaxation
and that's along
the Z direction.
So, we're decoupling the
interaction in the XY plane
from this relaxation at
this point in the discussion
and that's fair to do.
So this is what relates to, you
know, the spins losing energy
to the surroundings and
coming back to equilibrium.
So, here's what that looks like.
Here's the functional
form of that.
So, we have our MZ as a
function of time minus M not
so M not is the initial value.
That depends on minus
T over the constant T1.
So T1 is the longitudinal
relaxation constant;
this relaxation along Z. Notice
that this is just an
exponential decay.
There's no oscillatory
component here.
We're just talking about the
relaxation in the Z direction.
Okay so let's talk
about what causes it.
So in organic molecules or
proteins or things like that,
a lot of what causes
it is methyl rotation.
So, in the context of other
kinds of molecular motions,
we've said a bunch of
times that, you know,
methyl groups are freebies
spinning all the time.
Methyl groups have a
carbon which could be C13,
usually it's not but it has
3 protons that have a nice,
strong local magnetic field
and they're spinning around.
This is something that
can cause relaxation.
Another thing that can cause
it is segmental motions.
So, if we have a
chain, you know,
again stuff rotates
freely about strong bonds.
So in this particular liquid
crystal this is a spectrum
that I took.
These chains rotate around
and that causes relaxation.
Another thing that can cause
relaxation is chemical shift
and anisotropy.
So, if we have an
anisotropic chemical shift,
we have an electron distribution
that's shaped, you know,
like say it's shaped like
a football, as that moves
around there's a
locally changing
and little magnetic
field that's going
to induce relaxation
in nearby things.
There's also dipole-dipole
relaxation.
So that's like, you know,
again we're talking about,
we've talked about dipolar
coupling as the little spins act
like bar magnets and they
interact with each other
with a 1 over R cubed dependence
they can also relax each other.
So that's why I said that if
we're talking about something
like if we want to come up
with an example of something
that has a long relaxation
time, you know, on the order
of 20 minutes or something.
That would be a nucleus
that's very isolated.
It doesn't have any of these
mechanisms for relaxation.
So something that would
take a really long time
to relax would be
like a C13 carbon
at natural abundance
in a diamond.
So C13 is normally
1% natural abundance.
So that one little C13
in a sea of C12 is going
to take a really,
really long time to relax
because it has none
of these mechanisms;
it doesn't have any magnetically
active nuclei near it
to interact; it's going
to have to give energy
to its environment through lotus
vibrations and things like that
and it'll take a lot longer.
This can be a huge
pain for some samples
that people are interested
in looking at.
So, for instance, if you
have an organic molecule
that has a bunch of protons
and carbons, if it has a lot
of quaternary carbons that
aren't attached to any protons,
they can take a really
long time to relax
and you waste all your
time experimentally
because remember you have to
wait for the magnetization
to relax all the way back along
Z and you'll have a few nuclei
in your sample that are
really stubborn about this
and it takes a really long time.
Okay, so here's a pulse
sequence for measuring that.
So, let's talk about
pulse sequences.
Have you seen any of this
in organic chemistry?
So raise your hand if
this looks familiar.
Okay. How many musicians
do we have?
People play music?
Quite a few.
Okay, so NMR pulse sequences
are like musical scores.
So this particular one
is only 1 dimensional.
We are talking about
protons or C13 or N15,
one kind of nucleus at a time.
When we look at pulse sequences
that are more realistic,
we're going to see a
whole bunch of lines
where the protons
are doing one thing
and the carbons are
doing something else
and the N15s are
doing something else
and they're all synchronized
and it's a lot
like the musical score.
So, like a musical score
there's specialized notation.
We're not going to
get into it too much.
I mean it's fun but
basically what we need to know
about this is that if we have
a pi pulse that's 180 degree
pulse, that's written as
a pulse of longer duration
and a lot times it's
the square is open.
A pi over 2 pulse is written
as shorter duration and a lot
of times the square is black
and then the free
induction decay is clear.
This single headed arrow
tau means that we're going
to do this experiment
but we're not just going
to do it once we're going to
repeat it over and over again
and we're going to make
tau longer each time
and that's called arrayed
[phonetic] experiment.
Here's what the results
of that experiment look
like for an organic molecule.
So, this is called the
inversion recovery experiment
and I'll just show
you in the sort
of finger pointing explanation
why it's called that.
So, the pi over 2 pulse at
the beginning inverts it.
Now I wait some time tau
and it starts to relax back
to the equilibrium position,
but then before it gets there
I pulse it again and detect it.
So, you know, at the
very beginning it's going
to be almost all the way
along the minus Z axis
so I'll get a strong signal
when I pulse it back
along the X direction
but then the next time I
make tau a little bit longer
so it has a little bit more time
to recover before I measure it.
Then as we get to the point
where it crosses through 0,
then I'm not going to see
any signal when I pulse it
and it's going to
start coming back.
So we will see this logarithmic
dependence where we start
with a negative signal and
then it slowly comes back
and then levels off because
it's never going to get higher
than the original
equilibrium value, right?
So, here's what that looks
like for this organic molecule.
So, we have for these
different carbon atoms
that are labeled here, we see
in spectrum number 1 here that's
our time that we're waiting,
everything is down
along the minus Z axis
because everything is inverted.
Then as we wait longer
and longer times some
of these things start
to recover and we see
that as we would expect from
what I just said the CH3
and the CH2 recover first.
These are things that have a lot
of motion, they're protonated,
they've got dipolar
interactions with the protons
and then the things that
are in the phenyl ring
that have less mechanisms
for relaxation take a
little bit longer to relax.
Let's quite there for now.
What I want you to get
about, to understand
about this is what causes
the longitudinal relaxation?
Conceptually what are the
molecular factors causing it
and also I would like you
to understand conceptually
the experiment that we do
to measure this, the
inversion recovery experiment,
and you should practice
operating your matrix operators
on the spin states.
That is it for today.
Have a good weekend.
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