NMR Theory, Relaxation and Precession

collateral

Hello everyone, I'm having a really confusing time trying to get my head around these concepts, so I will try to explain what I can...

So, in proton NMR we have nuclei that can be in either spin-up or spin-down states. Nuclei align with an external magnetic field but precess. (This precession is more towards the z axis?). Applying a second, othogonal magnetic field at the precession frequency (Larmor), will cause the precession to go in the xy plane (Im not too sure about this either - sort of like a spinning top completely on its side if i can visualise it). When this happens, the nuclei can switch between states. Because of the small difference in states (Boltzmann), this can be seen via absorption at a particular RF.

So is spin-lattice relaxation when the precessing goes from xy plane (spinning top on it side) to the z plane (spinning top is now perpendicular to the surface)?

And where does spin-spin relaxation fit into this - from what I understand, its when spins of different nuclei don't correspond to each other. How is this different to T₁ relaxation?

Thanks for any help!

marcusl

You are mixing up quantum and classical pictures of NMR. I suggest sticking with the classical picture (which has no concept of states) and stick with classical. The spins align with B0. An applied RF field B1 causes the spins to precess and increases their polar angle theta (angle away from the z axis) continuously as long as B1 is applied. The spins end up in the xy plane only if B1 is turned off when they reach theta=90°. As they continue to precess in the xy plane, a signal is induced in the pickup coil. If B1 is left on longer, the spins precess down to 180°, that is, the magnetic moment is opposite to B0.

Spin-lattice relaxation describes the transfer of energy from the spin system to the lattice. Following a 180° RF pulse, the spins are pointing along -z but they relax back to +z through this mechanism.

Spin-spin relaxation doesn't remove energy, but it changes the local magnetic field such that spins following a 90° become decoherent (neighboring spins no longer track each other). This causes the signal to disappear.

nez

Precession cone from xy plane to z axis is spin-spin relaxationT₂(exchange of energy b/w nuclie spins) after 90^o pulse applied to spin system.

This T₂relaxation is insuffecient to detect practical useful decaying signal, but when 180^o pulse is applied, in which magnetic moment will take initially -z axis direction, spin-lattice relaxationT₁(exchange of energy with suroundings"atomic vibrations or molecular tumblings") takes place.

So, this is the practical technique(Multiple-Pulse FT):
We start with T₁ process(180^o pulse) then applyT₂ process(90^o pulse) after approperiate delay time(magnetic moment in +z direction) then the entensity of signal will be proportional to magnetic moment!
repeating this several times gives us our lovely spetrum by computer

collateral

Why is there is a signal induced when it's precessing in the transverse plane? I always thought this was because the spins with B₀ all can go to spins against B₀ when they absorb photons. Also, why do we use a 90° pulse as opposed to a continuous wave (because wouldn't a continuous wave saturate the higher energy state).

Why does precession at θ=90° allow the nuclei to from spin-up to spin-down?

I understand more, but I just wish to clear some stuff up and remove confusion which is frustating. Anyway thanks guys!

DrDu

The spins cannot only bei either up or down but you can have any possible quantum mechanical superposition of these two states. The up and down states are solutions of the time independent Schroedinger equation for a spin in the magnetic field corresponding to two different energy eigenvalues. A superposition of the two will have a non-vanishing expectation value for the magentic moment having a component in the xy plane. The direction will precess with the Lamor frequency. Any rotating dipole will emit radiation which can be observed in the NMR and will drive it back to the z axis.
It is not so easy to understand this emission in the picture of photons getting absorbed or emitted. The main point is that the phase of the electromagnetic wave is due to the superposition of states with different photon number (just like the angle of the magnetic moment in the xy plane depends on the phase of the superposition of the up and down states). To get a classical picture you have to consider both superpositions of spin eigenstates and of photon number eigenstates.

collateral

OK, im a biochem, so I hope that explains my confusion. Looking at other threads, I realise that I was trying to find out why applying a B₁ field orthogonal to B₀ would cause the spins to change state. I realise that this is now in quantum territory, which I know nothing about - all I understand is that when the precession is in the xy plane, the chance that the spins can change states is increased compared to when the precession is near the z axis.

PS. what do you mean by non-vanishing expectation?

DrDu

I fear that is a term from quantum mechanics, too.
However if you are willing to forget for some time on what you know about spin up and down, NMR can be understood to a large part in purely classical terms:
http://en.wikipedia.org/wiki/Bloch_equations

DrDu

If you apply an additional field B1, the spins will rotate around the vectorial sum of B_0 and _1. How this appears in QM depends on what you measure. If you measure the projection of spin onto the z-axis, you will find that after application of B_1, a certain part of spins has flipped.

ZapperZ

There is a very old thread in which some of these issues have been discussed:

http://www.physicsforums.com/showthread.php?t=42814

Zz.

collateral

Thanks ZapperZ (and everyone else), I think I was mixing up quantum and classical too. The explanations helped as well.

I have another question now regarding T₁...When we say that the states become saturated is it that all the spin states are in the higher energy state? Because if we lowered the temp to 0K, wouldn't nearly all of the spins be at the lower state?

Secondly, I read this online here It has confused me: I thought T₁ was dependent on z magnetisation so how does fluctuations in the xy plane affect it, and how is it not spontaneous?

I was also looking at some lecture notes and another thing which caught me was linewidth. The lecture says that line width is largely independent of field strength. But on another of the notes I was reading online, it mentioned that T₂ and T₁ have an effect on line width. Is this correct and if this is, doesn't field strength have an effect on relaxation?

Again thanks a lot guys, it has cleared up a lot but replaced it with more questions!

Mike H

Saturation refers to the situation when the populations of each state have been equalized. If you apply enough strong pulses continuously, the populations will equilibrate. If you read the discussion that was linked earlier by ZapperZ, you will note that ZapperZ mentions the situation at T = 0.

It may help to think about what T₁ is - it tells you how long it takes for the system to be restored to its thermal equilibrium values. It has to interact with its surroundings to do that, as you just perturbed it with the RF pulses of an NMR experiment. The sample is in the presence of an incredibly strong static magnetic field that runs along the z-axis. Fluctuations along the z-axis are most likely going to be quite small when compared to the static magnetic field. Remember, for a 14.4 Tesla magnetic field, the proton Larmor frequency is 600 MHz - that means in one second, it precesses 600 million times about the static magnetic field. For something to be "spontaneous" - at least in a practical sense - would mean it occurs prior to a single precession.

I don't know what other notes you've been reading, but I can certainly say the following. In principle, at least for a simple enough system, one can show that the Lorentzian peak width at half-height is inversely related to T₂. Of course, in the lab, things can be more complicated - there can be effects from imperfections in the static magnetic field, magnetic susceptibility of the sample, and I believe related instrumental issues whose details I am not remembering at the moment. Clearly, in certain cases, the two times can be basically the same magnitude (I'm thinking of quadrupolar nuclei), and relaxation becomes extremely efficient. Otherwise, you're going to have to point us to these other sets of notes you've been reading.