Week 8: Lecture 37: Magnetic moments of particles – fundamental and composite

Uh in this lecture we will talk about magnetic moments of particles. Also one example of magnetic moment of an excited nucleus. So fundamental and composite particles. So this will be roughly the order in which I will talk about these topics. What is the magnetic moment of a particle? Why is it important? How do we measure the magnetic moment? And the thing that has excited particle physicists a lot in the last few years is the muon g-factor anomaly. So the references are the of course the particle data group references therein. And then old book by Kopfermann, nuclear moments. And Amol Dighe's NSF colloquium which is related to the muon magnetic moment measurement. Of course there's been an update now in the last year on this and I will mention that at the end. Towards the end. Okay, so what is the magnetic moment of a particle? An elementary particle with charge e and mass m has an intrinsic magnetic dipole moment which is given by mu equal to gyromagnetic ratio times the spin times eh cross by 2m. Where uh these symbols have the usual meaning. Uh so it's g times s times mu unit. Now that why have I put mu unit? Because that's of course proportional to the charge and inversely proportional to the mass. So this g is called the gyromagnetic ratio. So for the spin part of the gyromagnetic ratio this is just two. Okay, and then of course the magnetic moment because e is minus one then minus some charge number then you get the the magnetic moment is opposite to its spin. Classically, if the above charge orbits in a circle of radius R with the speed V then it's angular momentum is given just by given by MVR. This current loop produces a magnetic field corresponding to an orbital magnetic moment of L times mu unit and here the GL gyromagnetic ratio with respect to orbital angular momentum is one. Why is it important? In point particles such as the electron and muon, the deviation from the Dirac value occurs because of so-called quantum electrodynamic corrections from contributions of known particle intermediate states. So, there are these vacuum fluctuations which change the value from the Dirac value to the QED value. Uh any disagreement, of course, might indicate fear physics beyond the standard model of particle Similarly, for composite particles such as the proton, neutron, lambda, charged pion, etc. This provides a test of QCD calculation, quantum chromodynamic calculations. Uh and in the case of nuclei, of course, this tests the models for the structure of the nucleus. I'll just show one example of that. The rest will all pertain to particle magnetic moments. How to measure the magnetic moment? So, of course, you measure measure it with by making it interact with a magnetic dipole field and measure the spatial deflection in an inhomogeneous magnetic field or you measure the Larmor precession frequency. Uh There's a typo here. For example, the proton has a spin half. In the presence of a magnetic field, the two magnetic sub states m equal to plus minus half, split in energy by an amount equal to two the magnetic moment of the proton times B. Okay. So, where mu p is the magnetic moment of the proton. The Larmor precession frequency is just given by omega equal to that interaction energy divided by h cross. Now, in the early '20s, Stern and Gerlach performed an experiment to discover spin quantization. And this is from a reference of Scientific American, much later, of course, 2022, about maybe 100 years later. Uh where Stern-Gerlach measured the spin quantization of the electron. Now, Stern actually disliked quanta. And together with his friend, von Laue, he had pledged that if this nonsense of Bohr should be uh should in the end prove to be right, we will quit physics. This is a quotation in 1919. Uh this is the apparatus that he used. So, it was not actually, sorry, it was not electron, it was the odd electron in a hydrogen in a silver atom. So, he had a furnace producing these silver atoms uh and the odd the last electron actually contributed to the magnetic moment. The others were all paired off. And so, this passed through slits and through a inhomogeneous magnetic field, and then it interacted with that inhomogeneous magnetic field. And to his surprise, he actually found uh two groups of these silver atoms. So, this shows again the thing in schematic. There's a furnace which produces these silver atoms. They They interact with a inhomogeneous magnetic field. And we now, of course, know what is the reason for these. It is the 5s electron, which is unpaired, and that produces uh uh I mean, through its interaction with the inhomogeneous magnetic field, it produces these two groups. So, since it's a S, we just have uh the spin contributing, and you have two groups here. And the classical prediction, of course, was this. You should get a just a band here, whereas you actually see uh it split into two groups. So, this is a big surprise to Stern-Gerlach, and of course, they eventually got the Nobel Prize for that, but they didn't expect that to happen. Now, uh later, uh Rabi, Isidor Rabi, developed the so-called Rabi method, and then Ramsey improved on that by making a separated oscillatory field technique. So, the idea is that you have a source of whatever particles whose magnetic moment you want to measure, and then you have a magnet the two sets of magnets, and another magnet where you you can actually uh so, you you excite uh uh the in the fixed magnetic field, you apply a RF, and so this uh spin actually precesses, and then it precesses either in the magnetic field or in the absence of a magnetic field, and then it again spins in the uh next magnet called the B magnet, and then you have a detector which detects the spin uh polarized object here. So, this is a kind of [clears throat] analyzer. This B magnet is a kind of analyzer. Okay, so uh it is akin to the uh at least the Ramsey method can be said to be akin to a two-slit experiment in which you don't know whether its spin precessed uh one way or the other in the first part of the magnetic field or in the second part, and so there is an interference between these two amplitudes, and that is what you see so there's a A region in the Ramsey method sorry the Ramsey method in the A region and the B region and C region it just drifts for a time T and so you get a the spin flip probability which you measure ultimately here at the end that keeps oscillating okay so it oscillates from basically zero to one ideally and this is for a spin half system and so of course you can measure this these oscillations especially somewhere near the place where this is changing rapidly this amplitude is changing rapidly and so this is schematically shown in this picture taken from a review article where this is the kind of plot that you get dots show the and you can see that this is this resonance is very narrow but you can actually measure the the falling part to much more accuracy than this 80 hertz width so this is what is actually done to measure the or try to measure the electric dipole moment of the neutron incidentally because of this very accurate measurement it also gives you the magnetic moment of the neutron again to several decimal places so this is of course reviewed and summarized in this annual reviews of nuclear science paper but the idea is at its minimum is just that you can flip the spin in either of those two sectors as I showed earlier and so these two amplitudes interfere and you get this resonance pattern when you actually measure the final spin of the object of the of the neutron in this case. Uh you can also measure magnetic moments of excited nuclei. If you have a nucleus existing for long, uh milliseconds or even greater than that, seconds, hours, and so on, then of course you can measure using the nuclear magnetic resonance. If it is shorter than milliseconds, uh then of course it is difficult to do that. And uh then you have to use other techniques. One of the techniques is the so-called perturbed angular correlation technique, uh PAC for short. So, there what happens is you have a Suppose you have a non-isotropic decay of an excited state. And this will happen if the angular momentum of the excited state is one or larger, uh then you can actually measure the uh perturbation of this angular distribution uh by applying a magnetic field, and then uh doing a measurement in time to see how the angular distribution changes. And so, uh if you know the B, if you know the magnetic field, the M1 moment can be extracted from the Larmor frequency. So, the first uh example of this which we discussed much earlier was uh I wouldn't say this is the first such measurement, but uh an early measurement in connection with the parity violation we have already discussed, where Garwin, Lederman, and Weinrich measured this oscillation pattern of the muon uh through its decay product, the electron. And uh because the muon comes out polarized, because when the pion, which has zero spin, decays, it decays into a mu and a nu, and the neutrino is uh polarized, so the muon is also polarized. So, if you measure the muon polar decay, uh that is asymmetric, and because it is asymmetric, you can apply a perturbing magnetic field, and then this will oscillate in time. And this was used to extract, for instance, the magnetic moment of the muon. Uh an example in nucleus is uh uh one in the case of 210 polonium, uh this is a multiplet caused by two uh protons in the H9 half orbital shell model orbital. So, you have a first gap, which is large. The 2+ state is at 1181 keV, and then this that comes down, and this can be understood within the uh nuclear structure model. Uh the point to note here is that these two states of 8+ and 4 6+ have lifetimes of the order of 99 and 43 ns. The other ones are much shorter, 1.5 ns and 6 ps before they go down to the ground state. >> [snorts] >> Now, uh of course, uh since there are two protons in the H9 half, they can at maximum produce an angular momentum of 8+, and then of course, if the angular momentum gets misaligned, if it gets antiparallel, then of course, you have the 0+ state. Uh and because of the uh nature of the uh the states involved uh and the particles involved, you can only get even uh you know, angular momentum states. Okay. So, if you looked at the uh so, this the angular distribution uh as measured from the 2+ to 0+ transition uh shows an angular anisotropy, and this changes when you apply a magnetic field. So, this was measured uh by uh in this reference, uh 1973, through the alpha 2n reaction on lead 208. And uh according to the additivity of G factors, the if you have two particles, let's say, in a H9 half orbital, and it couples to a spin, uh let's say eight or six, the G factors should be the same. This can be shown. We will probably see it in future lecture. In any case, take it for granted here that if you put two particles in a J orbital coupled to form either J or J prime, then these G factors G factors are equal. And so this experiment actually aimed to look for small deviations. Indeed, they found such a deviation. You can see these are the angular distributions at plus and minus 45° and by taking the difference and dividing by the sum, you can get this beautiful oscillatory pattern. And so from the initial phase you can actually in you know, infer the difference in the G factors divided normalized to G one of the G factors and get the G factor separately as well, G8 and G6. And you can see that these G factors are very close to each other within a couple of percent. So this additivity actually holds, but the small difference actually cannot be understood within the nuclear models at that time. Perhaps by now it is understood. I don't know. I don't have the update on this, but in any case at that time, this was a very beautiful measurement of difference in a multiplet in the nucleus which is close to the shell model doubly closed nucleus of 208 lead. Okay, so I should make probably some remarks on this that the magnetic moment of fundamental particles is reasonably well understood and this allows for searching for beyond standard model effects with more precise experiments. Composite particle magnetic moments of course have a long way to go. So although we know some of these magnetic moments is that of the proton and neutron to great precision, we do not really understand it at better than a few percent level in terms of theory. So, in this case at least the experiment is far far ahead of theory. And that is mainly has to do with the fact that these are strongly interacting systems of quarks and we don't understand the same way or with the same precision as quantum electrodynamics. Then of course the magnetic moments of atoms as nuclei, they probe the internal structure and they enable refinements in the relevant models. We will now come to the gyromagnetic ratio of the electron and then later the muon. So, the gyromagnetic ratio of the electron has been measured very precisely uh uh to you know, one part in a trillion, one part in 10 to the 12 uh by capturing single electrons in a Penning trap and looking at them oscillating in the magnetic field uh doing a procession uh up to you know, months or even year uh these electrons can be trapped and they survive that. And so, you can make this measurement to great precision, which has already been done. Uh again, here the uh precision with respect to experiment is much more than the precision with the uh in the theoretical calculations. However, this might change with the advent of better and better computers uh uh and so on. Anyway, the point is that these agree uh within error bars of theory and experiment to you know, to a very good accuracy. Uh as I said, this accuracy is limited by the theoretical accuracy. Okay. So, now what about uh in the case of the muon, the gyromagnetic ratio of the muon? Now, that is a little harder problem because the electron is stable and therefore you can look at it, as I said, for months or years. Uh muon only lives for 2.2 microseconds, and it'll decay before you measure its magnetic moment, so that becomes difficult. So, of course, you can use relativity, and that comes to your rescue. So, if you have a fast-moving muon, then the lifetime in the lab uh in the in the frame of reference where the muon is stationary, of course, it still has a lifetime of 2.2 microseconds, but in the lab, there is a time dilation factor, and so this tau lab is increased by gamma factor, which is just 1 by 1 - v squared by c squared. And depending on how much this is, if this is a factor of 100, then or 1,000, let's say, then this lives for uh milliseconds before it decays. So, you have to uh accelerate the muons, uh and that is done in, for instance, uh if you can uh produce it uh by a proton hitting a target, then it produces pi pluses, and then this decays into muons, so you can actually uh cool these muons in a uh ring, and then store them in a ring uh for as long as the uh they uh they can live, and uh so, this is a uh cartoon of a ring uh where you uh cool and store these muons, and here you can see that the spin can be aligned uh with respect to the momentum, and uh because, of course, the spin is precessing in the magnetic field. So, if uh uh you have to have a magnetic field, of course, to uh store these ion these muons, but this magnetic field also serves to precess the spin about that. Now, if you choose the energy uh of the muon uh carefully and the magnetic field, then you can arrange such that to first order the muon spin is aligned with respect to the momentum all the time. Uh and its small deviation would then tell you about deviations of that from the uh expected gyromagnetic ratio. So, you have to have a uniform magnetic field, which is perpendicular in this case to the figure. The muon spin precesses about the magnetic field, and the frequency of precession gives you the magnetic moment of the uh muon. Then the muon spin precession frequency omega A is just Q by uh mu into A mu times B uh minus A mu times a gamma by gamma plus one times a V.B factor times V squared by C squared V by C squared uh times uh and uh of course there's another term here A mu minus one by gamma squared minus one times a V cross E, where this A mu is just G minus two. Uh I hope I have not forgotten a factor of two. If so, I'll correct it. In any case, this measures the deviation of the gyromagnetic ratio from two. Uh So, of course first of all you have to put make make sure that V is exactly perpendicular to B, so that V.B is zero, then this term vanishes. Uh If you choose a energy such that A mu is equal to one minus gamma squared minus one, then of course this term also vanishes, and this happens at this magic energy where gamma is 29.3 or the energy is about 3 GeV. Okay, so of course this is known, and so uh in that case these terms vanish, and then omega A is just dependent on the A mu and the magnetic field. And if the B is about 1.45 Tesla, then the radius is manageable is about 7 m. So, you have a muon which decays by an electron so and uh uh of course it emits a electron neutrino and a muon antineutrino uh but you actually measure the charged particle here. You can't measure the neutrinos event by event. So anyway, you can do this and so uh you basically measure the uh anisotropy of the decay electron with respect to muon spin and uh uh as is stated here the positron is emitted preferentially in the direction of the mu plus spin and uh this is what you measure. So you have to detect electrons for measuring the muon spin and so uh this is a cartoon shows that the muons are injected here with a spin which is parallel to its momentum. It goes around and then it comes back and so uh at some stage you measure the direction of the electron with respect to the muon momentum. So you need a very precisely known magnetic field. You need a magnetic field which is mapped in terms of XYZ using NMR probes. So you move this probe around uh all around and make sure that you understand the magnetic field very precisely. So using this technique the uh in the past uh these uh g mu minus two measurements have been made and uh this g mu is basically two times a factor here and this uh the error in this delta uh a mu which is was the difference uh this is just about 0.004. Uh Uh the next round uh came about with using storage rings. So the first storage ring experiment was done at CERN where this uh uh delta a mu the error was of the order of, uh, uh you know, 27 parts in about 10,000. Uh, this was reduced further, eight parts in, uh, about, uh, a million. And then the Brookhaven one, uh, improved this by another factor of 20. This The Brookhaven experiment was done between 1990 and 2001. Now, the Fermilab measurement, which is, uh, which builds on the Brookhaven measurement, in fact, used the same, uh, magnet that was used in Brookhaven, made some changes so that the stability and the uniformity improves. And, uh, this is the published result. So, in time, uh, time after injection, you see so many of these oscillations in the magnetic field. And, uh, these are the number of electrons that you see. So, you can observe a decay, for instance, uh, over several orders of magnitude. And, uh, this is the, uh, analysis of, uh, Fourier analysis of this, uh, this data in terms of, uh, so, this is a Fourier fast Fourier transform as a function of frequency. So, you measure the main frequency and then there are some side bands. And using that, you can actually infer about the, uh, the Larmor frequency and hence the gyromagnetic ratio of the, uh, muon using this parametrization, uh, a decaying term and then a oscillating frequency uh, with some corrections to it. So, the first published result from them in 2021 actually agreed with the Brookhaven result. That was done to about 46 parts per million accuracy. A subsequent measurement published in 2025 improves that, uh, so, this is like 460, uh, parts per billion. And so this improved that by a factor of another three and a half or so in 2025. So, uh, since I have borrowed this slide from Amol Dighe, uh, this is the earlier 2021 measurement. Uh, so it says that the, uh, the experiment gives a certain number of for the G factor. The theory gives a certain number for the G factor. They agree to almost the, you know, eighth or ninth decimal place. Uh, and this is depicted here in terms of a graph, a mu, which is g mu minus two. Okay, so I was wrong in that slide. I'll correct it. Uh, it was g mu minus two by two. I had forgotten the two there. So, in any case, this, uh, deviation from the, of the gyromagnetic ratio from two, uh, normalized to two, uh, this is plotted here, uh, a mu into 10 to the nine minus some constant so as to be able to see it clearly. So, because this is a very high precision measurement. So, the standard model prediction then was here and the measurement, uh, the Fermilab measurement agreed with the Brookhaven, uh, measurement within errors and the, uh, the, you know, the deviation from theory was about 4.2 sigma. However, subsequently, uh, the theory was, uh, improved upon because mainly it comes about from QCD corrections and these were improved upon and, uh, the claim now is that this agrees with the experiment. In fact, uh, the 2025 result, uh, gives this for a mu. Uh, so it is about an error of, uh, 124 parts per billion. Very precise measurement. Okay. So, in summary, we have seen some of the measurements of magnetic moments of fundamental particles and nuclei. And the precision muon M1 moment shows a discrepancy with theory. However, at the present time, there seems to be an agreement between theory and experiment. I have not given the reference for that, but this does appear to be so. So, anyway, right now there seems to be an agreement. So, I'll stop here. I will, of course, correct the expression the expression for AMU a little later. Thank you. >> [music] [music]