It's Magnetic Moment Season: Measuring Various g-Factors

Among the articles highlighted in this week's Physics is one about a new test of QED through a measurement of the g-factor of the electron in silicon ions. This comes on the heels of a measurement of proton spin flips (this includes a free PDF) a couple of weeks ago, and those, in turn, build on measurements of electrons from a few years back, which Jerry Gabrielse talked about at DAMOP. Evidently, it's magnetic moment season in the world of physics.

The media reports on the proton experiment tend to be a little garbled in a way that reveals the writers don't quite understand what's going on. So let's take a crack at explaining this in ResearchBlogging form, in the usual Q&A format.

So, what's the deal with all this stuff? These look like very different experiments, and I don't see how they're related. Well, they're all using different systems, but in the end, they're after the same thing: measurements of what's known as the "g-factor" for some particle or another. This is a dimensionless number that relates the spin of a particle to its magnetic moment.

OK, what? Well, fundamental particles like electrons have a property known as "spin," which is an intrinsic angular momentum, as if the particles were spinning. They're not literally spinning balls of charge, and the behavior is somewhat strange compared to ordinary spinning objects (as explained with SteelyKid's help last summer), but in many ways they behave as if they were little spinning charges.

One of the things that spinning charges do is they produce a small magnetic field. For a literal spinning ball of charge, you can understand this by thinking of the charge on the surface as a little loop of current, which produces a magnetic field characterized by the "magnetic moment." If electrons are going to have the properties of a spinning thing, then, they need to produce a small magnetic field of their own.

If the electron was really a spinning charged ball, then the conversion from the rate of spin to the size of its magnetic moment would be really simple. Since the electron isn't really a spinning ball, though, there's a numerical factor that enters into the calculation. This is generally given the symbol "g," and thus is known as the "g-factor" for the electron.

And this thing has been measured? And calculated theoretically. The best current measurement of the electron g-factor was made by the Gabrielse group at Harvard, and gives a value of:

g/2 = 1.001 159 652 180 73 (28)

This agrees perfectly with a theoretical calculation of the expected value using quantum electro-dynamics (QED), which is why I tend to say that QED is the most precisely tested theory in the history of science.

How do they measure all those digits? It's a really impressive trick, involving holding a single electron in a "Penning trap," which confines it with the help of a strong magnetic field.

One of the fundamental properties of a charged particle in a magnetic field is that it experiences a force perpendicular to both its velocity and its magnetic field. This tends to push the particle into a circular orbit with a frequency that depends on the strength of the magnetic field. As this property was the basis for an early type of particle accelerator called a cyclotron, the frequency at which a charged particle goes around its little loop in the magnetic field is called the "cyclotron frequency."

If you stick an electron in a Penning trap, then, and cool it down to its lowest trap state, it will be making a little loop inside the trap. If you use a radio-frequency light field to add a little energy to the system, it will jump up to a higher-energy state making a slightly bigger loop at a higher average speed. The energy separation between these states is equal to Planck's constant times the cyclotron frequency, so measuring the radio frequency you need to drive one of these transitions tells you about the motion of the electron. And we can measure radio frequencies with exceptional precision.

Yeah, but what does this have to do with the magnetic thingy? I'm getting to that.

Because the electron has its own magnetic moment, its total energy depends on the orientation of that magnetic moment with respect to the trap field, giving it two possible energy states, traditionally called "spin-up" and "spin-down." That means there are two sets of cyclotron states, and those with the spin up have a higher energy than those with the spin down.

If the g-factor were exactly 2, the energy increase from flipping the electron spin would be exactly equal to the difference between the cyclotron levels. Since g is a little bit bigger than 2, though, there is a small difference between the energy of the spin-down state kicked up by one cyclotron level and the spin-up state in the lowest cyclotron level. That frequency difference is the thing that they measure to determine the electron g-factor: they put the electron in a known spin state, and then apply fields at different frequencies to try to cause it to flip its spin, and change its cyclotron level. since these are just RF frequency measurements, they can be done with exceptional precision, giving you all those digits in g.

That's pretty clever. So, these new measurements improve that old measurement? No, the new measurements do the same sort of thing, just in different systems. In the proton case, a German group has just demonstrated that they can detect spin flips of a single proton in a Penning trap, which is an essential preliminary step toward measuring the proton g-factor. In the silicon measurement, a different German group has made a measurement of the electron g in a single silicon ion held in a Penning trap.

What's the point of that, though? If they're measuring the same thing less well, why do it at all? Well, because it's not the same thing. The value they measure in silicon is:

g = 1:995 348 958 7(5)(3)(8)

(The numbers in parentheses are the uncertainty in the last digit from various sources.) This agrees very well with the theoretical value of 1.995â348â958â0(17).

Wait, that can't be right. You said the g-factor was slightly bigger than 2. This is smaller. Right, because it's a different sort of system. In QED, the g-factor is a result of interactions involving "virtual particles" popping into existence from the vacuum. For a free electron, the total effect of all those interactions increases g slightly, but an electron inside an atom is also interacting with the protons in the nucleus, which changes things.

Doesn't that make the calculation harder? Very much so. Which is why there aren't as many digits in those g values as those for the single electron. The result still agree beautifully, though, reinforcing QED's status as--

"The Most Precisely Tested Theory in the History of Science." Yes, yes, we get it. What about the proton thing, though? Why don't they have a value of g to report? In the case of the proton, the measurement is much harder to do, because the magnetic moment decreases as the mass of the particle increases, and the mass of a proton is around 1836 times the mass of the electron. Which means that the energy difference between the states of a spin-up proton and a spin-down proton is ridiculously tiny-- something like 0.05Hz out of 674,000 Hz. Sorting that out from all the other noise sources that cause tiny shifts in the trap properties is a major challenge. They see a clear shift, though, which is the first step toward being able to measure the proton g directly (the current value of 5.585 694 706 (56) is laboriously extracted from measurements with a hydrogen maser).

Why is the proton g value so different? Well, because it's a completely different type of particle. Electrons are leptons, and interact with the world via the electromagnetic and weak forces. Protons are made up of three quarks each, and quarks also interact via the strong force. This changes a bunch of things in the calculation, giving a very different result. That's also why the g-factor for the proton is an interesting thing to look at-- being heavier, a composite particle, and subject to the strong force allows you to test a wider range of physics.

So, the news stories about this all talk about antimatter, which you haven't mentioned at all. What's the deal with that? Well, if you can make these sorts of measurements for electrons and protons, you should also be able to make similar measurements for positrons and antiprotons. Now, in theory, there shouldn't be any difference between the g-factors for a particle and its antimatter equivalent, but then, in theory, there should've been nearly equal amounts of matter and antimatter created in the Big Bang, but everything we see, no matter where we look, is made of amtter, not antimatter.

So, a measurement of the proton g-factor compared to a measurement of the antiproton g-factor might turn up a tiny difference. And if it did, that could be an important clue as to why there's so much matter and so little antimatter in the visible universe.

Isn't that sort of comparison a really long way off, though? Absolutely. But that doesn't stop PR offices from mentioning it in press releases, or news writers from picking up that aspect of the experiment and running with it.

Sturm, S., Wagner, A., Schabinger, B., Zatorski, J., Harman, Z., Quint, W., Werth, G., Keitel, C., & Blaum, K. (2011). g Factor of Hydrogenlike ^{28}Si^{13+} Physical Review Letters, 107 (2) DOI: 10.1103/PhysRevLett.107.023002

Ulmer, S., Rodegheri, C., Blaum, K., Kracke, H., Mooser, A., Quint, W., & Walz, J. (2011). Observation of Spin Flips with a Single Trapped Proton Physical Review Letters, 106 (25) DOI: 10.1103/PhysRevLett.106.253001

Hanneke, D., Fogwell, S., & Gabrielse, G. (2008). New Measurement of the Electron Magnetic Moment and the Fine Structure Constant Physical Review Letters, 100 (12) DOI: 10.1103/PhysRevLett.100.120801

Odom, B., Hanneke, D., D'Urso, B., & Gabrielse, G. (2006). New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron Physical Review Letters, 97 (3) DOI: 10.1103/PhysRevLett.97.030801