Nuclear Magnetic Resonance

Having brought in a huge new audience at the end of last week– partly through the “framing”/”screechy monkeys” things, but mostly because my What Everyone Should Know About Science post hit the front page on Reddit– I figured I should take this opportunity to… Well, drive them all right the hell away again with a peer-reviewed physics post.

Unfortunately, I seem to have misplaced the papers I was going to write about, on experiments with qubits in diamond. They’re probably on my desk at work, doing me no good at all. That’s OK, though, because it would probably benefit from a little bit of background. So let me first write about Nuclear Magnetic Resonance (NMR), which is the “M” and the “R” in “Magnetic Resonance Imaging”– they ditched the “N” because “nuclear” is scaaaary, and doctors are wusses.

NMR is somewhat unusual, in that the basic physics can be explained almost entirely in classical terms, which is how it’s introduced to most physicists. You can even buy a demonstration apparatus that will let you simulate NMR using a billiard ball with a magnet in it. It’s a great explanation, and provides a lot of the language we use to talk about resonant interactions more generally.

Unfortunately, while it provides a clear picture for physicists and physics majors, it requires you to be familiar with angular momentum and torque and some properties of magnetic fields, and thus, is not terribly illuminating to the general public. Ironically, it’s actually easier to understand NMR if you just start with quantum ideas.

The whole foundation of NMR is that the nuclei of atoms are made up of protons and neutrons, which have a property called “spin.” The name is a little unfortunate, in that they’re not really spinning balls of charge (you can easily calculate how fast they would need to be spinning for that to be the case, and it’s not remotely realistic), but in many respects they do behave as if they were spinning balls of charge. They have angular momentum, and they have a “magnetic moment,” which means they interact with magnetic fields as if they were little loops of current.

When you put a proton or neutron in a magnetic field, it will act like any other sort of magnet, and want to align with the field. Because these are quantum particles, though, they have to have discrete states, so a particle can either have its magnetic moment aligned with the magnetic field, or opposite the magnetic field. If you put a bunch of protons and neutrons together to make a nucleus, things get a little more complicated, as they tend to pair up in oppositely orientations, cancelling the field out, but in very rough sense, if you have an odd number of protons and neutrons, the nucleus as a whole will have a magnetic moment that behaves in the same way that a single particle does– it can be aligned with the field, or opposite to the field.

This means that such a nucleus has two distinct possible internal states. The energies for these states differ by a tiny amount, and the energy difference depends on the strength of the magnetic field that the nucleus is in– the higher the field, the larger the difference. And any time you have distinct states separated by some energy, you can use light to drive transitions between those states. In the case of NMR, the “light” you use is in the radio frequency range– a few megahertz to hundreds of megahertz.

So, the essence of NMR is that you take a bunch of atoms, put them in a big magnetic field, and apply a RF field to them and see if you can make them flip their nuclei between the two possible energy states. You can detect when they do by using an antenna to monitor the strength of the RF field– when you hit the right frequency, the nuclei will absorb energy, and change states, and you can pick that up.

This may not seem like a terribly interesting thing to do, but it’s an extremely useful tool because the energy difference between the states depends very sensitively on the local conditions. A hydrogen atom all by itself will flip its spin at one frequency, but a hydrogen atoms bound to a carbon atom will flip at a different frequency, and one bound to an oxygen atom at yet another frequency. Given a sample of organic molecules, you can use NMR spectroscopy to determine the arrangement of atoms by looking at the different frequencies at which the hydrogen atoms flip their nuclear spins back and forth. Any synthetic chemistry talk will almost certainly include a couple of NMR spectra of whatever it is they’re supposed to be making– these are readily identifiable by the fact that they work very hard to make them look like chart recorder traces.

This works for lots of different atoms. Hydrogen is the easiest, but any atoms whose nucleus has a magnetic moment can, in principle, be used for NMR spectroscopy. Of course, most of these aren’t really much use, but there are a few other atoms that are important. Carbon-13 is used quite a bit for NMR studies– it’s not all that common, but there’s enough of it around that a macroscopic quantitity of some compound will happen to contain large numbers of molecules with carbon-13 in place of the usual carbon-12 in each of the places where you would normally find a carbon atom. Chemists can use carbon-13 NMR to deduce the arrangement of carbon atoms, which provides another important tool for figuring out molecular structure.

How does this lead to big, noisy, claustrophobia-inducing machines that make pretty pictures of your insides? Well, an MRI machine is basically just an apparatus for doing NMR spectroscopy of hydrogen, which it basically uses to detect water. The difference between the research NMR machines that chemists use to determine molecular structure and the MRI machines that doctors use to diagnose knee injuries is that the imaging machines deliberately change the resonance frequency at different positions, rather than relying on natural processes to shift the frequency.

They do this by applying a magnetic field gradient. Rather than applying a uniform magnetic field across the whole sample (say, the inured knee of an athlete), they put on a field that’s larger on one side of the sample than the other. That means that the hydrogen atoms in a water molecule on one side of the kneecap will flip its spin at a different frequency than a hydrogen atom in a water molecule on the other side of the kneecap. If you know the field very well, you know exactly what frequency will resonate at each position, so you can get a cross-section of how many water molecules are at different positions just by varying the RF frequency across the whole range of possible frequencies, and looking at how much the power changes.

You can turn this into a three-dimensional picture of the distribution of water molecules in your sample by rotating the field gradient and repeating the scan. For example, first you set the field so it’s higher on the left than on the right, which gives you the distribution of water from left to right. Then you shift the field so that it’s greater on the top than on the bottom, and do the scan again, which gives you the distribution from top to bottom. Putting those two scans together gives you a crude two-dimensional picture– if there’s a big spike in the signal at the frequency corresponding to 2cm from the left edge in the first scan, and a big spike at 3cm down from the top in the second, you can be pretty sure that there’s a high concentration of water at a point 2cm from the left and 3cm from the top.

You can repeat this many times for many different gradients, and build up a picture of the distribution of water in three dimensions, with very good resolution. Because different types of tissues in the body contain different amounts of water, this allows you to make pictures showing the internal structure of the body in detail.

This has a number of advantages over other imaging techniques, chief among them being that it doesn’t require blasting a patient with X-rays that can do damage to tissues. It also has a number of disadvantages, chief among them being that it requires a whopping huge magnetic field, which is why NMR machines are big and bulky and confining.

The high field requirement is not actually intinsic to the NMR process, but is a problem resulting from statistical mechanics. In a very rough sense, the problem is that in order to measure the flipping of nuclear spins, you need to know where you start– if the nuclei start in the low-energy state, and move to the high-energy state, that gives a different signal than nuclei that start in a high-energy state and move to the low-energy state. In a naturally occurring sample, you tend to get nearly equal numbers of high- and low-energy nuclei, and those signals tend to cancel each other out.

The numbers aren’t exactly equal, though– the fact that the higher energy state is, well, higher in energy means that there will be slightly fewer nuclei in that state, and slightly more in the low-energy state. That “polarization” gives you just enough of an excess signal that you can do the detection you need to do. The signal you get out is greatly reduced, though– it’s as if the only atoms you had were the tiny excess fraction in the low-energy state, so rather than dealing with 10²³ water molecules, you’re dealing with 10¹⁸ of them (or whatever).

The population difference between high- and low-energy states depends on the energy difference, so you maximize the excess population by using a really huge field. Which means big magnets, which means cramped and noisy NMR machines.

There is another solution to this, which is to polarize the sample ahead of time. This is a little tricky to do with the water in your body, but there are a number of research groups who have demonstrated the use of polarized gases (xenon or helium, usually) to do MRI with low fields– see, for example, the Walsworth group at Harvard. The idea there is that you use some other process to make a sample of gas with a very large fraction of the nuclei in one of the two possible states. Then you put this gas into your system, say by having a patient breathe it, and do your MRI looking for that gas. Because you start with a huge difference in populations between the two states, you don’t need a big magnetic field to get a good signal– the Walsworth group page I linked shows good images using a field of just 21 gauss, instead of the 50,000 of a standard system for looking at water.

This technique has been used to produce high-resolution MRI images of lungs, and has some promise as a research tool. It’s not going to see commercial application any time soon, for annoying reasons having to do with patents and profits, but it’s an extremely cool trick.

This problem of signal size is one that applies to NMR experiments outside the medical imaging field, as well. It is possible to do single-molecule NMR (just ask Google), but it’s very hard to get good signals from samples involving small numbers of spins. Which is the problem that leads to the research paper I’m going to write about, as soon as I find where I left it…