# How is the electron $g$-factor determined?

I found a description of the experiments at CERN where a Penning trap is used to to determine the g-factor but I could not find the value of the radius of the circle of precession, nor could I (surely my bad!) find a clue in the wikipedia article, they say:

The orbital motion of ions in the radial plane is composed of two modes at frequencies which are called the magnetron and the modified cyclotron frequencies. These motions are similar to the deferent and epicycle, respectively, of the Ptolemaic model of the solar system.The sum of these two frequencies is the cyclotron frequency, which depends only on the ratio of electric charge to mass and on the strength of the magnetic field. This frequency can be measured very accurately and can be used to measure the masses of charged particles.

We also know from another article that if B is one Tesla, the frequency of precession (gyromagnetic ratio) is 1.76*10 ^11 radians per second, but we do not know how long is one radian. In the same article the orbit is depicted as circular, whereas in tne Penning trap article the path is somewhat different.

Or, can you please explain if the value of $r$ (radius and radian) is irrelevant and, if it is not, what is the radius of the precession circle when B= 1 tesla? Can you explain what the value of 1.76 = 1.75882*1.00115965 tells us?

• radian is a unit for measuring angles, used instead of degrees. The angle made by taking the radius and wrapping it round the circle. 1 radian is about 57.296°. mathsisfun.com/definitions/radian.html – anna v Feb 26 '18 at 7:59
• here is a talk that includes the electron in atoms cds.cern.ch/record/1712509?ln=en and here are a series of measurements gabrielse.physics.harvard.edu/gabrielse/overviews/… – anna v Feb 26 '18 at 8:13
• @annav, sure , I was asking how many cm is the radius (radian) long, isn't radius a main factor that determines the precession rate? – user157860 Feb 26 '18 at 9:29
• in my "introduction to high energy physics" by perkins, second edition, the basics are described simply in 8.3 experimental determination of the g-factor of electron and muon. The radius is estimated from the momentum, p=Ber. – anna v Feb 26 '18 at 10:03

As it happens, I used to work on such electron-$$g$$-factor experiments in Penning-traps, so let me try to shed some light on the issue.

Typially, the symbol $$\omega$$ in a formula tells you that the corresponding quantity is given in "radians-per-second". This is simply how fast something rotates. You could also use Hertz for such a quantity (typical symbol: $$f$$), but theoreticians like radians-per-second because otherwise they have to put $$\cdot 2 \pi$$ everywhere. There is a simple conversion between radians-per-second and Hz:

$$\omega = 2 \pi f$$ or, for a concrete example, $$2 \pi \, \frac{\text{rad}}{s} = 1 \, \text{Hz}$$

The symbol $$\text{rad}$$ in this formula stands for "angle measured in radian", not for radius.

If you want to be super picky, you should describe $$\omega$$ as an "angular velocity" (how much angle per second), and only $$f$$ as a "frequency", but most people are lazy and write "the frequency $$\omega$$", which is technically wrong but understood by all.

The precesion of the electron's magnetic moment in a Penning trap happens at an angular velocity of

$$\omega_\text{L} = \frac{g}{2}\frac{q}{m}B$$

or, in other words, at a frequency of

$$f = \frac{g}{4\pi}\frac{q}{m}B \quad .$$

If the electron were a tiny ball (which it is not, but let's imagine it anyways), this would be how fast the ball is spinning around its own axis.

Unfortunately, there are technical (and physical) limits on how "still" or "motionless" we can keep an electron in a trap. The electron will always move a bit. It turns out, it's best to live with this motion and construct traps in a way that makes the motion as predictable as possible. In the electron-as-a-ball picture, this would be the ball moving in some predictable fashion through your trap (all the while it is of course spinning around its own axis as much as it wants to).

A Penning trap is a special trap where the motion of the electron is a superposition of three simultaneous oscillations: The reduced cyclotron oscillation, the magnetron oscillation, and the axial oscillation. Typically, you try to keep the amplitude of these oscillations small. This helps with measurement accuracy. I don't have values for typical trapped electrons handy right now, but for trapped ions, each oscillation has an amplitude of a few 10 µm.

There is an upside to this electron motion: You can excite the motion to detect the electron's presence and to measure the strength of the magnetic field that the elctron experiences.

This does not yet fully answer your question, and I apologize for that. The full measurement scheme on how to determine the $$g$$-factor can get arbitrarily complex, depending on how much detail you are interested in. But I hope this answer helps with some of the underlying concepts.

• thankyou Martin, if you find any data please update (btw, could you upvote the question if you think it is useful?) – user157860 Aug 27 '19 at 10:43