Lecture 18. Eigenstates & Eigenvalues
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Transcript: English(auto-generated)
00:07
Last time we ended with talking about some of the operators that are involved in NMR. And, you know, here we're still at equilibrium. We still have stuff aligned along the Z axis.
00:23
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
00:43
to the magnetization along Z. And for a spin 1 half nucleus that means it's either plus or minus 1 half, up or down. These states are called alpha and beta in NMR. And the eigenstates of IZ have eigenvalues that correspond
01:02
to their spin quantum number. So we can have plus or minus 1 half here. And if we operate this, if we operate this operator on its eigenstate we just get that quantum number back and the original state.
01:20
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 quantum mechanics because it, the eigenstates and the operators that we are using are very simple
01:41
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, you know, we've written this down in a very general way, what that means is that if we operate IZ on alpha we get 1 half alpha. If we operate it on beta we get minus 1 half beta.
02:05
And remember these things make up an orthonormal 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
02:26
going to get 1 and if we have a matrix element that looks like this or an overlap integral that looks like this where we have two of these states that gives us 0.
02:45
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.
03:00
So there isn't really an easy way to visualize it but fortunately mathematically it's pretty simple. These are the only two things we have going on. We've got plus and minus 1 half 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
03:20
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 we have the two states that we're looking at with the operator sandwiched in between them and this is a little bit different way
03:43
of approaching the problem but it's similar to what we've already done before in the context of 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
04:03
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
04:20
of course one half alpha and then we can pull the scalar out of that and so we just get one half alpha alpha and that gives us a half. 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 one half
04:44
beta, again you can pull the constant out in front and you can see that this matrix element gives you zero and for all our analyses of NMR operators,
05:00
this is essentially what we're going to do and 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
05:21
at IZ in the Hamiltonian. So here's my matrix for IZ and so the way I got that is the factor of two, you know, I've been sloppy about dropping my H bars but they are in there.
05:41
The factor of two comes from the fact that the eigenvalue is plus or minus one half so I just pulled it out of the matrix and then we said that the alpha alpha matrix element of IZ is one and then for alpha beta it's zero, same thing for beta alpha and then beta beta it's minus one, you know, again with that factor of one half in front of it.
06:01
Is 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 gamma omega naught IZ and so that means I'm just going
06:23
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 ket first, pulling out whatever constants I have
06:43
and then taking the overlap integral of what's left and so these are the answers that I get. And we know if we look at the matrix representation of two operators if they're diagonal
07:01
in the same basis then they commute. So this is really powerful in quantum in general because, you know, if you know that operators commute that gives you important information about the system. So if stuff commutes with the Hamiltonian then energy is
07:21
conserved and you can use this for a lot of things. But okay so so far we're just talking about, you know, when we say this is the Hamiltonian this is the Zeeman Hamiltonian, right? So we just have the spins aligned with and against the main magnetic field
07:41
and we can see the energy difference for that. You know, we know how to operate IZ on these states but we haven't learned 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 XY plane, how do we do that? So you remember from your homework assignment a long time
08:03
ago we did, you know, you proved that the angular momentum operators don't commute with each other and 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.
08:21
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 XY 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
08:40
to define some other operators to look at that. So here's what the matrix representations of IX and IY are just for reference and 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
09:03
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
09:25
and lower the states of the system but here's what they're defined as.
09:43
So we've got I plus as IX plus IIY and I minus is IX minus IIY and so we've kind of alluded to before like we're getting real and imaginary parts of our signal in the XY plane.
10:03
You can start to see how this fits together with the formalism 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
10:24
to L and M that is your eigenvalue and then the eigenstate of the raising operator is the state you have plus one so if you started with minus one-half it's going
10:42
to go to plus one-half 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 the spin system that's defined so that's going to give you zero. That's not going to be true for spins that have I greater than a half so if we had a spin one we can operate I plus
11:05
on minus one and we'll get the eigenvalue times the state with zero and then if we operate I plus on zero we can do that again and get plus one. For a spin one-half system we only have two choices, up or down, so you can either raise or lower
11:22
and if you operate the raising operator on the highest state you get zero and equivalent for the lowering operator. Okay, so let's write that out explicitly. So I plus operated on alpha gives you zero,
11:40
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
12:03
and I minus on beta is zero. 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,
12:22
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 not, they're some sort of linear combination of the Zeeman eigenstates but we don't know how to measure
12:40
that and we don't know how to operate on it. You know, 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.
13:00
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 is going to give us a well-defined 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
13:23
and I minus in this basis. So again, we're still in the Zeeman basis. We have the spin 1 half. 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
13:44
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 on alpha gives us 0 so that one's 0. If we operate I plus on beta,
14:01
that gives us a constant times alpha but then the integral of beta alpha is 0 so that's our matrix for I plus and we can do the same thing for I minus. So we operate I minus on alpha. That gives us beta but the integral of beta alpha is 0.
14:26
I minus operated on beta is 0. I minus operated on alpha gives us a constant times beta but and then beta beta gives us 1
14:41
and 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 a matrix representation that makes sense. And 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
15:06
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
15:24
this to verify the matrix representations for Ix and Iy. It's tedious but it's good practice so, you know, you can just go through and operate these things and you know, like I showed you in the previous couple of slides
15:44
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. And, you know, again, it's tedious so if you do one of them and you think you totally get it,
16:01
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
16:22
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
16:40
of matrices we can use all of that same formalism to do stuff where it's not so easy to visualize. So now, you know, 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.
17:01
Okay, as I have sort of hinted at along the way, we can have spins in a superposition of alpha and beta. We don't have to have everything just in one eigenstate or the other. And 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.
17:22
Well, it's not. You can have these superpositions. Okay, so you can have a wave 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.
17:44
So our spin state can have a superposition 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.
18:06
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.
18:25
And these things are normalized, as I said when we talked about how to do the matrix elements. And again, it's kind of hard to picture these functions and how they're orthogonal to each other.
18:41
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
19:02
of Ix and Iy? And I'm not going to prove this 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.
19:21
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 1 over square root of 2.
19:47
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. You know, we're in this plus x eigenstate. Every time we measure along x, we're going to get that value.
20:01
If we measure along y and z, it's going to be ill-defined. It's not sharp in that case. Okay, so again, we're going to apply our matrices that we have, that we've been writing down.
20:23
To operate your operator on the ket, 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.
20:45
And 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
21:03
of its spin quantum number, which is plus 1 half. All right, so that's one of these things. Let's look at the value for minus y. So similarly,
21:24
we can write it out, and again, I'm not going to show you how we get this as the eigenstate. We're just going to look at the result. And here it is in vector notation. We can also operate iy on it, and we see that we get minus 1 half minus iy, or sorry, minus y.
21:45
And this is what's detected in a typical pulsed 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,
22:04
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.
22:22
So here are our rotation operators for rx, ry, and rz. And 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
22:41
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.
23:03
And 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. And so for a different angular momentum operator,
23:23
we have this kind of a relationship. 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
23:41
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
24:04
that we can measure. So we apply an x pulse, and that's going to take us from whatever our starting state was to a final state. And so we're going to do that by operating our operator
24:24
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.
24:47
And this beta p is the flip angle of the pulse, so pi over 2 in the example that we were talking about. All right, so let's back up and talk
25:01
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,
25:21
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,
25:43
we can tip it partway down, and we won't see a very strong signal then, 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. And, in fact, we can rotate it 180 degrees and just invert the magnetization relative
26:02
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 a 90-degree pulse? The answer is we typically measure this experimentally. I mean, you can calculate it and get close,
26:21
but we optimize this experimentally. The flip angle depends on the nutation frequency of the RF, so that is, you know, you can imagine the RF as, you know, so it's an oscillating electric field in, you know,
26:41
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 a 180-degree pulse, it goes like this. And you can imagine if we're looking at the signal for that,
27:01
we get this oscillatory behavior. And so we can express that frequency, you know, 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, you know, how much power we have to flip these pulses.
27:21
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 a 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.
27:48
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.
28:01
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 along Z. We turn on the pulse.
28:21
We have calculated the flip angle and the mutation frequency to be exactly right so that it's a 90-degree pulse. And that's going to give us our magnetization in the minus Y eigenstate. It also picks up a phase factor,
28:41
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?
29:00
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 probes. So we build the RF circuits that produce these radio frequency pulses that flip the spins. And it turns out that there are lots of experiments
29:21
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.
29:40
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 this 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
30:03
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 device is long because that coil has to be located in the very center of the magnetic field.
30:21
So it's inside the magnet. The actual business end of it is relatively simple. So we have this parallel resonant circuit. So that's the inductor in parallel with the tune capacitor that's called CT. By adjusting that variable capacitor,
30:42
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
31:03
in impedance match to the load. So this is how we experimentally deliver the RF pulses. And 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
31:21
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, you know, okay, we can look at proton, carbon, nitrogen, phosphorus. For every nucleus that we're looking at,
31:40
we have to have a separate channel of the probe. You need a separate RF circuit to be able to interact with that. And 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 two channels to be able to interact with that.
32:00
And that means that you need two of these 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 this 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. And so, you know, you get, you're getting that flavor
32:23
for it, you know, with, 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, this is, so this is how you experimentally measure the flip
32:43
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, for, and this is what B1 refers to. So for proton we had 132 kilohertz carbon.
33:01
It's 71.4 nitrogen 86.2. That frequency refers to the frequency at which the magnetization is going around and around in these sine waves. And in that case it's a measure of the amplitude of the field that's being applied. And it's one of these things where the units are very weird.
33:21
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 in 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.
33:41
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
34:00
or something else, but in terms of how much can we actually influence the spins. And 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.
34:23
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, one microsecond tips it a little bit more, and then we go up to 90 degrees and, you know, and so on as the magnetization goes around and around.
34:43
I want to point out that if experimentally if stuff were perfect, this should look like a perfect sine 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
35:03
that you're using to apply the field. And if you look at, you know, especially the proton channel here, you can see, you know, 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.
35:22
As you apply the field for longer and longer, it's not perfect. One reason for that is that your coil is not, you know, an infinitely long solenoid 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. And we actually can do things to try to make it better
35:41
when we're engineering these things. So for a solenoid, 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. And that's something that we do. So here's a kind of a funky looking coil that was built
36:04
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.
36:20
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.
36:44
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
37:02
and be able to make this work. We related that to pulsed 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
37:21
about, you know, when you, you know, you have your perfect 90 degree pulse, you pulse the magnetization into 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
37:44
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.
38:01
How long does it take for my spins to align along Z? What do you think? Anybody want to take a guess? 20 minutes? If you're looking at like silicon 29 or, you know, maybe a carbon that's not close to anything at all,
38:22
like if it's like a carbon in a perfect diamond, that might be a good guess. If it's, 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,
38:41
I have a 500 megahertz magnet, so take one 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
39:00
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, it depends on the spin, it depends on the local chemical environment, you know, what's actually causing the relaxation.
39:21
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 pulsed NMR in kind of a practical context because you know that, you know, if you give a pulse and you wait for the magnetization to relax back, if you don't wait long enough, the second scan,
39:42
you're not going to get very much signal. 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
40:02
about is called longitudinal relaxation and that's along the Z direction. And so, we're decoupling the interaction in the XY plane from this relaxation at this point in the discussion.
40:20
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 naught.
40:45
So, M naught is the initial value. That depends on minus T over this constant T1. So, T1 is the longitudinal relaxation constant. It's relaxation along Z. And notice that this is just an exponential decay.
41:04
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
41:22
of molecular motions we've said a bunch of times that methyl groups are freely spinning all the time. Methyl groups have a carbon which could be C13. Usually it's not. But it has three protons that have a nice strong local magnetic field and they're spinning around. This is something that can cause relaxation.
41:42
Another thing that can cause it is segmental motion. So, if we have a chain, you know, again, stuff rotates freely about single bonds. So, in this particular liquid crystal, this is a spectrum that I took, these chains rotate around and that causes relaxation.
42:05
Another thing that can cause relaxation is chemical shift 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.
42:20
As that moves around, there's a locally changing 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
42:40
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 wanted to come up with an example of something that has a long relaxation time, you know, maybe 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.
43:03
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 percent natural abundance. So, that one little C13 in a sea of C12 is going to take
43:22
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 lattice vibrations and things like that and it will take a lot longer. This can be a huge pain for some samples
43:41
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
44:01
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.
44:23
Have you seen any of this in organic chemistry? So, raise your hand if this looks familiar. Okay, how many musicians do we have? Like, do people play music? Quite a few, yeah, okay. So, NMR pulse sequences are like musical scores.
44:42
So, this particular one is only one dimensional. We're talking about protons or C13 or N15, you know, 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 for, you know, the protons are doing one thing and the carbons are doing something else and the N15s are doing something else
45:02
and they're all synchronized and it's a lot like a musical score. And so, you know, 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
45:25
of longer duration and a lot of 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.
45:40
And 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 an arrayed experiment.
46:01
And 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
46:22
and 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
46:42
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. And then as we get to the point where it crosses through zero, then I'm not going to see any signal when I pulse it. And then it's going to start coming back.
47:00
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, you know, for these different carbon atoms
47:20
that are labeled here, we see in spectrum number one here, that's our time that we're waiting. Everything is down along the minus Z axis. So everything is inverted. And then as we wait longer and longer times, some of these things start to recover and we see
47:41
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.
48:04
Let's quit there for now. What I want you to understand about this is, you know, okay, what causes the longitudinal relaxation? You know, conceptually, what are the molecular factors causing it? And also, I would like you to understand conceptually the experiment that we do
48:23
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.