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From my understanding, a proton's magnetic moment in an NMR experiment can undergo precession in only two discrete states: either parallel to the applied magnetic field, or antiparallel to it. Absorbing light of propper frequency allows it to go from the lower energy parallel configuration, to the higher energy antiparellel one.

This framework seems to conflict with all explanations I've found regarding pulsed (proton) NMR. Regarding 90 degree pulse experiments, I've read that they work by supplying a broadband radio frequency pulse perpendicular to the applied magnetic field, causing all protons to to absorb the light and enter an excited state. This then somehow causes them to change the precession of their magnetic moments to be along a plane perpendicular to the applied field. This precession then gradually moves back to the ground state, where it is parallel to the applied field.

I'm confused as to how the magnetic moment can ever move to be perpendicular like this, as that would mean it's in between being parallel and antiparallel to the applied field. This conflicts with my initial understanding, in which the moments can only exist in the 2 discrete states. Where am I going wrong?

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  • $\begingroup$ Well, you're right: spins don't exist only in spin-up or spin-down. This simplification is very sadly propagated in typical organic chemistry books, probably because the truth is too complicated. $\endgroup$ Jan 28, 2022 at 22:25
  • $\begingroup$ How does it really work then? $\endgroup$ Jan 28, 2022 at 22:31
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    $\begingroup$ An answer is possible, but it would be rather long, so in the meantime can I suggest reading a proper NMR book (see chemistry.stackexchange.com/questions/37303/… and Ctrl-F for Keeler or Levitt). Note, also, that if the magnetisation is parallel to the external field, it cannot precess about it. This is because precession is essentially a rotation about that axis; anything lying on the axis of rotation doesn't rotate. $\endgroup$ Jan 28, 2022 at 22:31
  • $\begingroup$ Simply speaking, the pulse brings a small fraction (equal to the difference in occupation between the two states in equillibrium) of the spins to precess in phase. That turns the static net magnetisation in $B_0$ direction into a rf ("precessing", i.e. measureable via induction) magnetisation orthogonal to $B_0$. If your book talks about "light" and "all spins", I recommend you find a nice dumpster for it. $\endgroup$
    – Karl
    Jan 28, 2022 at 23:42
  • $\begingroup$ The 90° pulse lets 50% of the spins change state. Or at least that's what the result looks like, in our simple picture. $\endgroup$
    – Karl
    Jan 28, 2022 at 23:50

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Spins don't exist only as spin-up or spin-down.

I make no apology for making this big and bold. It is so disappointing to see this misconception being propagated in so many textbooks.

Spin-up and spin-down are two of the infinite possible quantum states. In general, a spin-1/2 nucleus has a state

$$\psi = c_\alpha \psi_\alpha + c_\beta \psi_\beta$$

where $c_\alpha, c_\beta$ are complex numbers and $\psi_\alpha$ and $\psi_\beta$ represent the 'spin up' and 'spin down' states respectively. In order for the state to be 'physically sensible', we require that $|c_\alpha|^2 + |c_\beta|^2 = 1$.

A pure spin-up state would have $c_\alpha = 1$ and $c_\beta = 0$; and a pure spin-down state would have $c_\alpha = 0$ and $c_\beta = 1$.* However, there is no stipulation that these are the only possible values. It is perfectly sensible to have a linear combination, or a superposition, of spin-up and spin-down:

$$\psi = \frac{1}{\sqrt{2}} \psi_\alpha + \frac{1}{\sqrt{2}} \psi_\beta$$

It is easy to verify that $|c_\alpha|^2 + |c_\beta^2| = 1/2 + 1/2 = 1$, as required. This is a perfectly valid state for a spin-1/2 nucleus.

So, if we identify the two special states $\psi_\alpha$ and $\psi_\beta$ with "along the positive $z$-axis" and "along the negative $z$-axis" respectively, then it logically follows that all the other states can be identified with pointing in a different direction.

Formally, the way to do this is to calculate the expectation value of $x$-, $y$-, and $z$-angular momentum from each state.† I'll not go through the maths here, but essentially you will get three values out of this, often denoted $(I_x, I_y, I_z)$. Then you can plot this on what is known as the Bloch sphere. For example, if you calculate this for the spin-up state, you will get $(I_x, I_y, I_z) = (0, 0, 1)$,‡ which essentially points straight up along the $z$-axis. It follows, that different combinations of $c_\alpha$ and $c_\beta$ lead to different values of $I_x, I_y, I_z$ which in turn lead to different 'directions' on the Bloch sphere.

In general, you can convert freely between these two representations of the spin states: you can either use the $(c_\alpha, c_\beta)$ representation, or you can use the $(I_x, I_y, I_z)$ representation.§


NMR on one spin

If we start with a single spin pointing along the $+z$-axis, i.e. spin-up, then the effect of a 90° pulse would be to tilt it to somewhere in the $xy$-plane. It could be along the $x$-axis (in which case we'd have $I_x = 1$ and $I_y = I_z = 0$), or it could be along the $y$-axis, etc. etc. In general, such states are superpositions of $\psi_\alpha$ and $\psi_\beta$. The exact state you get depend on the phase of the pulse, which isn't super important for now.

Let's say, for argument's sake, that you get a state along the $x$-axis. It turns out that this state is not actually 'stable'; if you leave it to be, it will evolve in time. How does it evolve? It turns out that if you work through the mathematics of time-dependent quantum mechanics, this state rotates about the $z$-axis with a characteristic frequency $\omega$. We can express this in terms of the coefficients:

$$\begin{pmatrix}I_x \\ I_y \\ I_z \end{pmatrix} = \begin{pmatrix}\cos(\omega t) \\ \sin(\omega t) \\ 0 \end{pmatrix}$$

You can verify that if we plug in $t = 0$, we get back $(I_x, I_y, I_z) = (1, 0, 0)$, which corresponds to "immediately after the pulse".

Fundamentally, NMR is based on detecting these oscillations. The electronics are set up to detect terms which behave like $\cos(\omega t)$ as a function of time; Fourier transformation of these terms allows us to extract the frequencies $\omega$ and hence plot an NMR spectrum.

Now here comes the issue when discussing NMR on single spins: we can't actually continually measure $I_x$ or $I_y$ on a single spin. This is not only technologically impossible, but also highly problematic considering that the act of quantum measurement leads to 'wavefunction collapse'.

This is the point at which we need to consider many spins, or macroscopic properties.


NMR on many spins

We now consider an ensemble of many (identical) spins. For example, we could have a sample of water, in which all of the $\ce{^1H}$ spins are identical.

Are all of the spins going to point up? Well, if the energy difference between spin-up and spin-down was large enough, then maybe they would. However, NMR energy differences are extremely tiny! As a result, there isn't enough bias towards the lower-energy states. What you get is that all the spins point in essentially random directions, with a slight overall bias towards the $+z$-axis. We call this a polarisation, in that the spins generally lean towards the $+z$-axis.

The 90° pulse acts on each of these spins differently. If one of those spins happens to start off along the $+z$-axis, it will be tilted into the $xy$-plane correctly. If one of the spins happens to already be in the $xy$-plane, then it may end up somewhere else entirely.

What can be shown (using a formalism called density matrices) is that the overall polarisation, which starts off as $+z$, mimics the single-spins case. That is to say, the 90° pulse brings the net polarisation into the $xy$-plane, after which it starts evolving (or precessing) at its characteristic frequency $\omega$. At this point we tend to call these "coherences" instead of "polarisations", which is a slightly complicated term to define properly, but essentially it revolves around the idea that: even though each individual spin is doing its own thing, collectively, they act in tandem to provide a net effect that can be detected.

If you have different types of spins, like in a more complicated organic molecule, then you would get have multiple types of polarisation which precess at different rates. Again, this can be Fourier transformed to extract the individual frequencies of each component.

Now, the fact that we are dealing with many spins helps us to avoid the traps of measurement previously mentioned. The first thing to note is that we have many spins, so the signal that we detect is likely to be rather bigger than just that from a single spin.# The second thing to note is that we no longer care about each individual spin's state: we're not measuring each spin's wavefunction individually, but rather we are measuring how the entire ensemble is biased or polarised. The net bias of the ensemble can be constructed in many ways; by measuring it, we don't actually enforce any requirement on the individual spin's wavefunctions. This means that we don't run into awkward issues with wavefunction collapse.


Further reading

The NMR books by Keeler (Understanding NMR Spectroscopy, 2nd ed., Wiley) and Levitt (Spin Dynamics, 2nd ed., Wiley) explain this more thoroughly than I can here.


Footnotes

* More precisely, one is equal to $0$ and the other is equal to some arbitrary complex number with magnitude $1$, which can be represented as $\mathrm{e}^{\mathrm{i}\theta}$ for any angle $\theta$.

† Note that the Heisenberg uncertainty principle (HUP) doesn't stop you from doing this. The HUP doesn't tell you anything about the actual expectation value itself, it only tells you about the uncertainties in them. For example, consider position and momentum: the HUP says that you cannot measure both simultaneously and with perfect accuracy. It doesn't say that you cannot measure these simultaneously, nor does it say that you cannot measure them at all: it is only the combination of simultaneously and perfect accuracy that is forbidden.

‡ There is a constant factor of $\hbar/2$ which I've neglected. It's not hugely important. The important part is that $I_x = I_y = 0$ and $I_z \neq 0$.

§ Going deeper, this is a result of a homomorphism between the SU(2) and SO(3) groups, which has quite a bit of significance in spin physics.

Yes, technically, there's also going to be $\ce{H3O+}$ and $\ce{OH-}$.

# Because the energy gap is small, though, the net polarisation is still generally pretty small: NMR is considered an insensitive technique.

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I will "cheat" by circumventing a detailed attempt to describe how a macroscopic magnetization arises from an ensemble of spins in the presence of a static external magnetic field, or how such an ensemble responds to perturbations in the form of applied RF fields.

Consider instead that when we probe a spin and say that it points up or down relative to a magnetic field, what we are saying is that one specific component of the spin angular momentum - that along the field - points up or down, and that the magnitude of that component (the projection of the spin angular momentum onto the coordinate axis defined by the field) is fixed (it is a quantum property). There will be an additional component in the plane perpendicular to the field, and which according to the Uncertainty Principle may point anywhere on that plane, the direction being indeterminate, provided the total angular momentum vector retains the right magnitude.

Next, it is worth distinguishing between quantum and macroscopic properties. Spin angular momentum is a quantum property and when attempting to describe it the Uncertainty Principle must be kept in mind at all times. Precession on the other hand is a macroscopic (classical E&M) concept that can be used to describe the motion of a very large collection of spins about an external magnetic field. This ensemble property can be dealt with classically without giving the Uncertainty Principle much thought.

One way to bridge the two perspectives is to consider that an applied RF field perturbs the quantum state of a spin and can result in a mixed state in which it is no longer up or down, it is in a superposition of those. An ensemble of spins in such a state is referred to in NMR as a "quantum coherence", to imply that all of the spins are behaving the same way (they are coherent), but so as not to forget the QM nature of the individual spins.

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From my understanding, a proton's magnetic moment in an NMR experiment can undergo precession in only two discrete states: either parallel to the applied magnetic field, or antiparallel to it.

You have to distinguish bulk magnetization, which you can treat classically, with the magnetic moment of a single proton, which is governed by quantum mechanics. The precession picture is for the classic view.

Absorbing light of propper frequency allows it to go from the lower energy parallel configuration, to the higher energy antiparellel one.

When you look at the classic NMR spectrometer (scanning), switching spin states is a good picture. If you scan too slowly, you reach saturation (same number of spin up and spin down) and lose the signal. You would have to wait (with kinetics determined by the so-called T1 parameter, several seconds for a 600 Mhz instrument) until you are back at the equilibrium situation, where a small excess of spins is in the lower energy state.

This framework seems to conflict with all explanations I've found regarding pulsed (proton) NMR.

In pulsed NMR, you start things off with a 90 degree pulse, resulting in half the spins up and half down. The reason you still get a signal is that there is a coherence (or spin polarization). This coherence goes away after a while (the protons from different molecules experience a slighlty different field because of tumbling and conformational differences), with kinetics determined by the so-called T2 parameter. T2 is typically much shorter than T1, on the order of tens of milliseconds. So once you lose coherence, you have to wait out T1 before you can run the next scan.

Regarding 90 degree pulse experiments, I've read that they work by supplying a broadband radio frequency pulse perpendicular to the applied magnetic field, causing all protons to to absorb the light and enter an excited state.

No, the pulse results in a coherent state for a 90 degree pulse. For a 180 degree pulse applied to the equilibrium state, you would get what you describe. However, most pulse sequences start with a 90 degree pulse (because that is how you get a signal).

This then somehow causes them to change the precession of their magnetic moments to be along a plane perpendicular to the applied field. This precession then gradually moves back to the ground state, where it is parallel to the applied field.

For this picture to be almost correct, you need two things: First, you have to look at bulk magnetization rather than single protons. Second, you have to be in a rotating frame. In the laboratory frame, the magnetization always precesses around the B0 field (the constant, much stronger field) and it would be difficult to describe what the pulses do.

I'm confused as to how the magnetic moment can ever move to be perpendicular like this, as that would mean it's in between being parallel and antiparallel to the applied field. This conflicts with my initial understanding, in which the moments can only exist in the 2 discrete states. Where am I going wrong?

As orthocresol states in the comments, you can have a superposition of states (up and down) if you want to explain the magnetization perpendicular to the B0 field at the level of single protons. This is explained handwaivingly here (with a pinch of quantum mechanics if you click on the advanced explanation).

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