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I'm trying to wrap my head around Larmor precession and NMR pulses, but there's something that seems totally confusing and contradictory to me, and I haven't been able to find a definitive answer yet.

So, when I read about Larmor precession, everywhere it is stated that magnetic moments 'tend to' align with a (static) magnetic field, but they won't actually align, and just precess around an axis perpendicular to the magnetic field. I think I get this (based on this handout: https://www.physics.rutgers.edu/grad/506/Pulsed_NMR.pdf).

But when I read about NMR pulses, it is also stated that after an applied pulse of a time varied magnetic field, which de-aligns the magnetic moments, they will go through an exponentially decreasing relaxation, and finally be aligned with the static magnetic field. And somehow when the explanations come to this point, there isn't a word about precession anymore, and the whole problem is treated like the magnetic moments do actually get fully aligned and perpendicular to the static magnetic field. But I would think this should not be the case, I would expect them to return to their precessing state, not a fully aligned state.

Am I interpreting this the wrong way?

Benjamin Márkus
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    NMR measures the classical expectation value of a very large number of spins (certainly something like $10^{10}$ or more in the most sensitive experiments, I would say, and typically more on the order of $10^{16}-10^{22}$) that are almost in thermodynamic equilibrium with their environment. In a typical experiment the number of microscopic spins that are in the |up> vs. |down> state only differs by something like one part in a million. So while you are correct about the behavior of the single spin, the measured ensemble average behaves differently. – CuriousOne Jun 09 '16 at 20:07
  • So if I'd be able to observe just a single spin, I would see it precess, but the average of the ensemble will be aligned? Is this the case because the individual spins are precessing with different phases, and the whole thing just averages out? – Benjamin Márkus Jun 09 '16 at 20:10
  • If you were to observe single spins, you would see them flip at time dependent average rates and the averages would resemble the ensemble average that we observe in an NMR experiment. Instead of doing one at a time, we are doing a very large number at once. If you want to see single flips, you can do an ODMR experiment where a spin flip, which has a tiny energy difference, is "amplified" by an atomic photon emission process. I find that technique to be one of the most amazing experiments I have ever done. – CuriousOne Jun 09 '16 at 20:19
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    Ok, so there's still one thing I do not get: if we don't see precession when only the static field is applied, because the collection of a lot of out-of-phase spins produce a z axis ensemble average, then why do we see a precession signal after applying a 90° pulse? Does this bring the spins in-phase? If not, why is there a net precession? If yes, how does it bring them in-phase? This is still unclear to me. – Benjamin Márkus Jun 09 '16 at 20:53
  • Just curious, and pedantic as well I know, but in theory, because of the uncertainty principle, I don't know if QM allows "complete" alignment of the spins of the electron and external field. I appreciate that's not what you are saying here, just wondering. –  Jun 09 '16 at 20:55
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    UPDATE: This seems helpful, hopefully it's correct as well. – Benjamin Márkus Jun 09 '16 at 21:01
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    We are seeing a signal because the distribution of up and down spins is not the same after we turn the magnetic field on. Because there is an energy gap, the thermal equilibrium leaves slightly more spins in the lower energy state than the upper one. While $kT>>\hbar \gamma B_0$, there is still a tiny imbalance. That imbalance remains even after the 90 degree pulse and we can measure an effective magnetization. The signal is very small. In a typical MRI experiment we have to apply hundreds of Volts to the excitation coils for the 90 degree pulse, but we only get $nV-\mu V$ in signal back. – CuriousOne Jun 09 '16 at 21:28

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In thermal equilibrium, we can work out the bulk magnetisation (i.e. how many particles are likely to be in the aligned vs antialigned eigenstates) based on the energy difference between those states and the temperature. In other words, this gives us the net magnetic moment along the axis of the applied field, and we presume (from random phase incoherence of a thermal state) there will be no net magnetic moment in any perpendicular direction.

If you somehow knew the exact quantum state of an individual proton spin, then you would find that the expectation value for the component along any axis perpendicular to the applied field is oscillating at the Larmor frequency. But there is no precession per se of the bulk magnetisation, because it is already aligned exactly with the external field.

If you perturb this system, with a weak perpendicular field oscillating at a resonant frequency, then the individual protons will Rabi cycle between their two energy eigenstates, or all their spins will nutate in synchrony, or the net magnetic moment will tilt and spiral (whichever picture you prefer). If you turn off this perturbation at the right time, you can leave the net magnetic moment pointing along one of the axes that is perpendicular to the static external field. Now the net magnetic moment will be precessing (because it is no longer parallel with the field).

Note that the perturbation, and likewise the subsequent precession, has not increased the magnetisation of the bulk, just rotated it. We presume the perturbation phase was adiabatic; there is still the same amount of disorder among the individual spins, but the excess net magnetic moment is now stranded pointing perpendicular to the applied field (and precessing equatorially at the Larmor frequency) rather than parallel to it.

However, the system is no longer in thermal equilibrium. (If you measured the individual spins you would at this moment find equal numbers in both energy eigenstates, as there is no longer any net magnetic moment along the axis of the external field.) This means that random thermal fluctuations will now, statistically, tend to push the system back towards its thermal equilibrium. One process is that the precessing spins can randomly lose phase-coherence with each other (in the equatorial plane), broadening their distribution and diluting (toward zero) the net magnetisation (in that plane). Another process is that the axial magnetisation component can be re-established (dissipating energy into the thermal bath as some of the spins realign with the applied field). In NMR, there are techniques to separately measure the rates of either of these two relaxation processes.

benjimin
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