# Nuclear Magnetic Resonance (NMR) Explanation

1. Oct 10, 2009

### mrjeffy321

I am trying to understand the theory behind nuclear magnetic resonance. I have been reading various explanations (some more detailed than others) but I still have several holes in my understanding and I hope that some of you can help clear them up.

I realize there are quite a lot of questions here, and this is a rather long post, but even if you could address only a few of these that would be helpful.

First, let me explain what I do know in order to give you my background before I ask my questions.
-------------------------------------------------------
In an NMR apparatus one applies a strong, uniform, static magnetic field to a sample along what we will define to be the z-direction. This magnetic field causes the nuclear magnetic dipoles / spins to align either parallel, or anti-parallel, to the field (with a slight preference in the parallel direction) which leads to a net magnetic dipole moment in the z-direction. The individual nuclear magnetic dipoles are not, however, completely aligned with the static magnetic field (which has to due with the uncertainty principle), but rather they precess around the field at a particular frequency.

While applying the static magnetic field in the z-direction, causing the nuclear spins to align in the z-direction, one then applies a radio-frequency (RF) pulse to the sample in a direction perpendicular to the static magnetic field. The RF pulse’s magnetic field causes the nuclear dipoles to begin to tilt down away from the z-axis since they are now also precessing around the magnetic field perpendicular to the first. The amount which the nuclear dipole rotates is proportional to the length of time the second magnetic field is applied to the sample, allowing one to rotate it by a certain angle (such as a “90° pulse”). Once the RF pulse is discontinued the nuclear dipoles return to their original orientation, in the process emitting an RF signal of their own which can be detected by a wire coil (oriented perpendicularly to the static field) and this is the data one records during the measurement.

Alternatively, one could explain the second magnetic field / RF pulse quantum mechanically in the form of photons which are absorbed by the nucleus and which possess just the right energy to cause the “spin-up” dipoles to flip to “spin-down”, and then the signal one collects is the emission of photons as the spin-down dipoles return to their lower energy spin-up orientations.
-------------------------------------------------------

Now onto my questions….

The quantum mechanical description of the process make perfect sense to me in terms of how one ‘excites’ a sample into its spin-down state, against the magnetic field (higher energy) by the absorption of a photon is appropriate energy. Afterward, one watches as the nuclei return to their preferred spin-up orientation by re-emitting a photon. One may easily calculate the frequency of light needed to induce a spin-flip. This turns out to be the Larmor frequency. I do not understand why. Surely this is not just some happy coincidence that the photon has the same frequency as the precessing dipole moment, but I do not understand the connection between the two.

As the net dipole rotates around, is there still some “wiggle” in it even when it is at 90° from the z-axis? For example (if one is sitting in a coordinate system rotating about the z-axis such that the x, y directions are fixed) as the net dipole tilts over to the XY plane, does it still wiggle up and down in the z direction just as it wiggled side-to-side when the dipole was oriented along the z-axis? In other words, is the dipole “cone” tilting over or are is it just expanding its radius and shrinking in height? (Am I describing this clearly enough for people to know what I mean?)

The emitted signal varies from sample to sample, even when measuring the same nuclei due to the chemical / electronic environment, I am not talking about lattice interactions, etc. between two molecules, but rather the nature of the chemical bonds within a molecule. For example, a hydrogen in a C-H bond will emit differently than one in a O-H bond. Could someone point me toward an source which will explain this in more detail?

The secondary magnetic field from the RF pulse should vary in time (and space), which means that the torque experienced by the dipoles should also vary in time and lead to a non-constant precession frequency about the perpendicular axis. And yet in the explanations I have read it seems to always be assumed that the pulsed magnetic field is constant in time. How is this assumption justified? Is it frequency of the field so much slower than the precession time it is virtually constant? Are we assuming some root-mean-square value? What about what happens when the field reverses directions during the second half of the period?

Is the classical description of what happens only valid with large numbers of nuclei? Or does the same explanation extend down to few (or one) nuclear magnetic moment? I would think not since the spin is quantized and can only be spin-up or spin-down, not spin-sideways for some brief time as it rotates around. So is the seemingly continuous rotation of the net magnetic dipole from the (large) sample really quantized into discrete changes?

The description of how the magnetization returns to normal by spiraling back up to the z-axis explains the decaying oscillatory signal one sees in an NMR experiment. How is this explained in terms of photons emitted by the nucleus?

2. Oct 14, 2009

### mrjeffy321

Can anyone shed some more light on this topic for me and address a couple of my questions?

3. Oct 18, 2009

### conway

I'm going to give this a try. First of all, do you know how to calculate the resultant dipole moment of a superposition of the up and down states? You should understand that a combination such as

A|up> + B|down>

where |A| = |B| = sqrt(1/2)

gives you a magnetic moment pointing somewhere in the xy plane, depending on the relative phase of A and B.

4. Oct 18, 2009

### Naty1

Have you read wikipedia on NMR: (a pretty thorough discussion)

http://en.wikipedia.org/wiki/NMR

Note further references at the bottom including Larmor equation.....Larmor discovered the relationship between dipole precessions and applied external magnetic fields....so his discovery describes the physical phenomena....

5. Oct 18, 2009

### Ygggdrasil

Well, in the classical picture of NMR, the Larmor frequency depends on an empirically measured fudge factor called the gyromagnetic ratio. It's this fudge factor that captures the quantum mecahnical behavior of the system. Otherwise, the equation for the Larmor frequency just states that the frequency of light emitted is proportional to the applied magnetic field which is also true in the quantum mechanical picture of NMR.

It would make sense that this cone also rotates into the x,y-plane, but I'm not sure if this is the case.

Basically, electrons nearby a nucleus will shield the nucleus from the applied magnetic field. This shielding will reduce the magnetic field felt by the nucleus, lowering the energy difference between the two spin states, and reducing the energy of the photons emitted from the spin flips. For example, in an OH bond, oxygen is much more electronegative than the carbon in a CH bond, meaning that the oxygen pulls the electrons in the OH bond toward itself, lowering the electron density around the hydrogen atom. In the case of the OH bond, the proton will be much more de-sheilded and emit light at a higher frequency of the proton in the CH bond.

These types of effects are extremely important to chemists as they allow us to deduce the structure of organic molecules from NMR spectra. Any introductory organic chemistry book should have a good discussion of how to interpret NMR signals from different types of chemical groups. When NMR was first developed, physicists hated NMR because no two protons in a molecule resonated at the same frequency, so the technique produced such messy spectra. On the other hand, chemists loved it for exactly this reason. Just another example of how in science, one person's noise may be another person's data.

You have to remember that the spins are precessing around the z-axis during the excitation with the secondary RF pulse. Now, what's the frequency of the secondary RF pulse? It corresponds to the Larmor frequency, meaning that the oscillations between the precession and the magnetic field should cancel. In doing classical NMR calculations, sometimes we play a trick and move into a "rotating frame" (as opposed to the "lab frame") where the x,y-axes spin at the Larmor frequency. In this frame of reference, the magnetic field from the RF pulse is constant.

In the classical picture, you visualize the nuclear magnetic moment as a freely rotating dipole that can occupy any arbitrary angle. This is obviously incorrect once we know about quantum mechanics, but the classical picture of NMR does have a strinkingly similar quantum analog: the idea of the Bloch sphere, which I briefly describe in this other thread on NMR (https://www.physicsforums.com/showpost.php?p=2321099&postcount=11). So, even considering quantum mechanics, this picture of NMR is still valid for a single spin.

T1 relaxation is due to the stimulated emission of photons (because for emission in the radio frequency, spontaneous emission is negligible). Emission is stimulated by interactions with the environment surrounding the spin (hence the alternative name of spin-lattice relaxation).