Rail guns!

A railgun is a fairly straightforward electromagnetic device that converts electrical potential into linear mechanical motion.

The general principle of a railgun is very simple: two rails have a voltage difference applied to them. The device being accelerated rests between the rails and has a conducting cross-link of some sort called the armature that allows a current to flow from the high rail into the low. The current through the rails produces a circulating magnetic field around the two rails, which results in a net magnetic field normal to the plane of the rails and device under acceleration. The current through the armature contains moving charges, and these are accelerated by the magnetic field according to the Lorentz force.  The result is that the armature and the affixed device experience a constant force parallel to the rails, and in the absence of other forces, the result is an acceleration in that direction. It is worth noting that no matter whether the potential difference across the rails is positive or negative, the device and armature will be accelerated away from the voltage source end of the rails.

Railguns are pretty cool devices, but they tend to be very high-current and high-power constructions, which means they have to deal with a lot of heat dissipation, not to mention finding awesome power sources that can provide enormous currents. As an example, let's look at a sample system.

But first, a bit of electromagnetism for the sake of calculations:

The magnetic field produced by a semi-infinite wire at the end of that wire is $B(x)=\frac{\mu_0 i}{2\pi r}$, where $\mu_0$ is the magnetic constant, $i$ is the current through the wire, and $r$ is the distance from the wire. Define a coordinate $x$ to start at the center of one rail and be positive in the direction of the other rail. We see that if the rails of radius $r$ are separated by a distance $d$, the magnetic field in the region between the wires is given by
\[B(x)=\frac{\mu_0 i}{2\pi}\left( \frac{1}{x}+\frac{1}{d-x} \right).\]
The force on a current-carrying wire carrying current $i$ in a perpendicular magnetic field $B$ is $F=i\ell B$, where $\ell$ is the length of the wire. We therefore see by integrating over the length of the wire that
\begin{align*}
    F &= \int_r^{d-r}i B(x)\,dx\\
    &= \frac{\mu_0 i^2 d}{2\pi}\int_r^{d-r}\frac{d}{x(d-x)}\,dx\\
    &= \frac{\mu_0 i^2}{\pi}\ln\left( \frac{d-r}{r} \right)\\
    &\approx \frac{\mu_0 i^2}{\pi}\ln\left( \frac{d}{r} \right)
\end{align*}
where we have assumed that $r\ll d$ in the last step. Based on this, we can calculate the current necessary to generate a given force in a system:
\[
    i=\sqrt\frac{\pi F}{\mu_0 \ln(d/r)}
\]
The resulting power dissipation is given by $P=i^2 R$, where $R$ is the overall resistance of the rail-armature system, which is at most $\rho(2\ell+d)$, where $\ell$ is the length of the rails and $\rho$ is the resistance per length of the wire and armature. We see
\[
    P=\frac{\pi F}{\mu_0 \ln(d/r)}\rho(2\ell+d)
\]
Let's choose some arbitrary parameters for our system:
    $\ell = 40$ cm (length of the rails)
    $d = 10$ cm (distance between the rails)
    $r = 0.5$ cm (radius of the rails and armature)
    $\rho = 1.608 \Omega/km$ (resistance per length of wire)

If we want to generate a force of $F=200$ N, then the current necessary is on the order of 13 kA (eek, that's a lot of current!), and the power generated is roughly 2 kW, which is around half the power generation of a microwave. The upside is that with this kind of wire, it takes a mere 2 V to generate these currents, so the voltage requirements are reasonable. The limiting factor in this power source will be the current draw necessary, which will be limited by any battery's internal resistance. A chief expense of this device will therefore be the use of a high-quality battery certified for absurdly high currents.

Full-wave rectifiers

I mentioned half-wave rectifiers in my last post as a pretty neat application of diodes. Unsurprisingly, it turns out we can do way better at converting alternating to direct current if we're willing to use some more diodes. Just like before, we want to input an alternating-current voltage source, which means the voltage difference between the two inputs oscillates over time. In the end, we would ideally like a completely constant voltage to be output. Well, check out this circuit:

Image adapted from electrician Joe Duncanson's highly informative blog.

(Notational note: The triangle-bar thing is the electrical symbol for a diode whose polarity will only allow current to flow in the direction the arrow points.)

As you can see, whether the top or the bottom input node is at a higher voltage, current can and will flow toward the upper output node. Then the current from the lower-potential bottom output node will flow through the diode diamond to the lower-potential of the two inputs. In this way, we ensure that the output voltage difference (voltage of the top output minus voltage of the bottom output) is always positive. More specifically, if the diodes are ideal, then the voltage we'd expect out of this circuit would be

Input and output voltages for an ideal full rectifier.

But as usual, the real world isn't quite so friendly. In reality, there's a bit of internal resistance in the diodes, so they don't start letting current through until there's enough voltage across them. As a result, the output voltage looks more like this:

Input and output voltages for a (slightly) more realistic full rectifier.

We're still not quite at the constant output voltage we set out to find, but we're a good deal closer than when we stuck with just one diode. Hopefully I can write a bit about capacitive filtering, which goes a long way towards smoothing this curve out, sometime in the near future.

Diodes

I got the chance to play with some diodes recently, and I came away with the fact that diodes are really cool. Abstractly, a diode is simply an electric device (not electronic, I believe, because it doesn't involve transistors or logic gates or any fun stuff like that) which prefers for electric current to pass through it in just one direction. In theory, if you apply a forward bias to an ideal diode, that is, you apply a voltage to it such that electric current "wants" to flow in the preferred direction, then the diode should behave just like a wire. But if you apply a reverse bias, trying to force electric current through the other way, then the diode opposes you with (theoretically) infinite resistance. Naturally, nothing behaves quite that nicely, and in real life, diodes can be broken down if you apply too strong a reverse bias. (The exception is Zener diodes, which are designed to be reverse biased, and they have the cool property that when you run current through them in reverse bias, the voltage across the diode itself is constant - weird, right?) The idea remains, though, that you can use a diode as a sort of one-way gate in electric circuits, and this turns out to be a very handy property.

For example, one application of diodes is in rectifier circuits, which convert alternating current (AC) into direct current (DC). In other words, they can be used to allow current to flow in only one direction in a circuit. The simplest type is a half-wave rectifier, which consists of one boring diode connecting your input voltage (AC) to your output voltage (DC). In the ideal case, this results in an output voltage identical to the input, except never negative, like this:

Input and output voltages for an ideal half-wave rectifier.

But, as usual, I don't have access to the magical ideal physics stockroom, so I have to make do with real-life, physical diodes. And those come with a little bit of internal resistance. That means that it takes a non-zero positive voltage across them to actually get current to flow, and when current does flow, some of the voltage is dissipated into heat inside the diode, leaving only a fraction of it to contribute to the voltage across the diode. So in reality, the output voltage from a half-wave rectifier looks more like this:

Input and output voltages for a real, physical half-wave rectifier.

There's a lot more you can do with diodes, but even this simple example is pretty neat. Someday, I might write up full-wave rectifiers and capacitive filter circuits. Those are pretty cool, too. Conclusion: physics and electronics are awesome.

Galaxy spirals

Galaxies fall into several categories, which I think I've discussed before. The simplest two are elliptical and spiral. Those classifications are fairly self-explanatory; an elliptical galaxy is a fuzzy blob of stars, typically in the form of a three-dimensional ellipsoid, while a spiral galaxy has those distinctive spiral arms we all know and love from Hubble pictures. If you want to get even more specific, there are two forms of spiral galaxy: the simple spiral galaxy and the barred spiral. The barred spiral galaxies have a distinctive-looking bar in their centers, and while they look pretty strange, we're actually living in one right now! Yep, the Milky Way is a barred spiral galaxy.

In both kinds of spirals, stars orbit roughly circularly around the center, just like our planets orbit the Sun. But because the galaxy is roughly a disk with mass, the galaxy's stars also exhibit up-down oscillation around the center of the disk.

In an elliptical galaxy, on the other hand, there isn't a nice, uniform central gravitational potential, so stars, especially those far from the center, trace out bizarre three-dimensional shapes. It's a really fascinating problem to analyze the orbits of both individual stars and general classes of stellar orbits in elliptical galaxies.

Cepheid variables!

It's been a while since I wrote anything about astrophysics, so it's high time I came back to a really cool topic: Cepheid variable stars. These provide a so-called standard candle that allows astronomers to determine extraordinary long distances to far-away galaxies.

At its root, a Cepheid variable star is one that breathes in and out either in its first mode of oscillation (just expanding and contracting spherically) or its second (becoming slightly tubular, then bouncing back to a pancake shape, and so on). As it contracts, its surface temperature increases, so its luminosity increases. In most cases, these sorts of oscillations would damp out pretty quickly, but in Cepheid variables, there's a shell of partially ionized helium inside the star. As the star compacts, the internal temperature increases, and this results in extra ionization in the helium layer. As a result, the opacity of that thin shell increases, so the radial pressure from inside increases, pushing the helium layer (and thus the rest of the star) back outwards. At this point the helium ionization decreases thanks to a decrease in temperature, and the whole cycle begins again. To make a long story short, the partially ionized helium layer allows the star to expand and contract with very little damping pretty much indefinitely. Most importantly, there is a set relationship between the period of oscillation and the star's luminosity, which means that by observing how long a brightening and dimming period takes, we know exactly how bright the star really is. Based on this and how bright we see it, we can determine its distance from us!

One drawback of these candles is that they tend to be large (apparently 4-20 solar masses), and therefore short-lived, stars. That means it's hard to find Cepheids in particularly elderly galaxies. But hey, they're one of the coolest (hottest?) standard candles in the astrophysics arsenal, so I'm not complaining!

The foundations of statistical mechanics

Many problems that students solve in introductory physics classes have to do with one or two interacting bodies. The two-body problem, in which two objects interact with a force dependent on the distance between them, is often exactly solvable. Take the Earth and the Moon, or the Earth and the Sun, and you can exactly solve the simple equations of motion by simplifying the two-body system into a one-body system. The three-body problem, on the other hand, is intractable. Add Jupiter or another planet into the bargain and suddenly all the math in the world won't help you except to numerically approximate the system's behavior. There are all sorts of approximations that can help you out, but the exact solution for arbitrary conditions is elusive. This, of course, makes analysis of complicated systems...well, complicated. Luckily for physicists, while the 3-body, 4-body, and 5-body systems are difficult or impossible mathematically, the $10^{23}$-body system is actually solvable in a statistical sense. The examination of systems with many many constituent parts (atoms, for instance) forms the basis of statistical mechanics, an incredibly powerful framework that describes large systems and explains tons of other physics into the bargain.

Statistical mechanics is based on just two simple hypotheses. As far as I know, these two hypotheses aren't provable from more basic principles, but taking them as axioms gives all sorts of useful and accurate results. In order to discuss these hypotheses, I first need to define a couple of terms. Any large system can be described in terms of a (very large) collection of microparameters. These do things like saying exactly where each particle is and what its momentum is, or the energy level of each particle, and so on. The exact microparameters will depend on the system we're examining. In general, it's very difficult (not to mention time-consuming) to measure microparameters. Instead, we define macroparameters to describe the system as a whole. Things like temperature, pressure, or the total number of spin-up atoms are all macroparameters, and again, we tend to define different macroparameters for different systems.

With those definitions out of the way, let's take a look at the first hypothesis of statistical mechanics: the ergodicity hypothesis. It states that for any system, there is an equilibrium state, in which the macroparameters describing the system (no matter which ones we choose) are constant, and that furthermore, the values of these macroparameters are given by the unweighted statistical average of all the microstates of the system.

This is a very profound statement, on a variety of levels. The existence of an equilibrium state for any old system is moderately intuitive, but it seems a little strange to have such an overarching statement of its existence. More than that, though, it gives an extraordinarily helpful way to find the equilibrium values of all the macroparameters: just take the average of all the microstates! Naturally, the math behind such averages is a little messy at times, but it's of critical importance that we can so easily find a description of equilibrium.

Alright, time for one more definition: let's call any system in which the microstates change faster than macroscopic time scales an ergodic system. In order for a system to experience this flavor of fast evolution, it is necessary to have interactions between particles in the system - imagine taking a collection of noninteracting gas particles and squishing them into a small ball at the center of a room; in the absence of interactions, nothing will ever happen and the system won't evolve at all.

The second critical hypothesis of statistical mechanics is called the relaxation hypothesis, and it states that any ergodic system always evolves towards equilibrium. This translates roughly into the third law of thermodynamics, which states that entropy always increases.

Based on these, we can derive (with sufficient approximations) things like the ideal gas law, rigorous definitions of temperature and pressure, and in general the behavior of very large systems. 

Cloud chambers!

One of the original detectors in physics was the cloud chamber. The basic idea is that you have a very pure gas with high humidity and cool it off or reduce its pressure until it's on the verge of condensing. Since you have essentially a pure gas without any impurities or dust particles, there's nothing for the water to condense on. As a result, the air in the chamber is supersaturated, and it will condense at the introduction of any small impurity. Luckily for physicists, the chamber can be tuned so that a charged particle passing through the detector is just enough to cause condensation. So a charged particle leaves a track of condensation as it passes through the gas. If you're lucky enough to have a particle decay or otherwise interact inside the detector, you can see all the (charged) tracks involved in the interaction.

Once you have that set up, you can produce a nice, uniform magnetic field throughout the detector, and then a low-momentum particle will spiral in a characteristic way, while a higher-momentum particle will curve only slightly. At this point, the early physicists took pictures of the chamber for later analysis (often by human "computers"). The resulting images are entrancing and emblematic of the early days of particle physics.

A somewhat stylized cloud chamber image, courtesy of the CERN Courier.

The trouble with gluons...

If you've ever heard of Feynman diagrams, you've probably seen the simplest sort - the ones with the fewest possible interactions. What you may not have seen much of are loop diagrams - Feynman diagrams with the same starting and ending states but with extra, seemingly-extraneous interactions in the middle. The funky thing about the Feynman path integral formulation of particle physics is that in order to calculate the amplitude (related to the probability) for an interaction, you need to account for all possible diagrams with start and end states you care about. That means, yes, all of those loops on loops on loops, which we call higher-order diagrams. Luckily, for forces like the weak force and electromagnetism, the amount that higher-order diagrams contribute to the amplitude decreases with complexity. As a result, it's often a reasonable approximation to use only the simplest diagram to describe an interaction, and even professionals often content themselves with fourth- or fifth-order diagram contributions.

Less fortunately, the strong nuclear force is less obliging. Rather than higher-order diagrams contributing less and less to the total amplitude, they contribute increasingly large amounts. This trouble with gluons (those are the force carriers for the strong force) is one of a couple of traits that makes the strong nuclear force such a pain to work with for theorists.

An awesome tidbit about particles

A fun fact recently came up in a discussion about particle physics: the fact that elementary particles can have mass but take up no space. This seems a little weird at first. Based on our everyday experience, volume and mass seem to come hand in hand. But like so many other classical ideas, this preconception breaks down in the subatomic world.

How does that work? Well, volume and mass are actually very different concepts. Mass determines how a particle reacts to an external force, according to $F=ma$ (it also has to do with gravity based on its gravitational mass, but that's a whole separate story - one of these days I'll figure out what's up with general relativity, and in any case, subatomic particles don't interact with gravity enough to worry about), and in the world of elementary particles, a particle's mass is determined by its interaction with the Higgs field.

Volume, on the other hand, gives some measure of how much physical space an object occupies. In a particle like a proton, which has three quarks bound together in a complicated way by the strong nuclear force, that volume is determined by the quarks' interactions. It isn't the quarks that take up space, as much as the fact that they can't all occupy the same space, by the laws of quantum mechanics and inter-particle interactions. Similarly, the "space" that an atom takes up is dictated not by the space taken by protons and electrons, but by their electromagnetic attraction and the necessary space for electron orbitals and clouds. In a sense, we define the volume of an object in terms of its internal interactions. But if you take a look at a single particle like an electron, there aren't (as far as we know) any constituent parts. So what would it mean for there to be internal interactions to give it volume? It's practically nonsensical.

This is a really strange concept in the already-slightly-weird world of particle physics, but like so many such issues, it's a lot of fun to wrestle with.

Mushroom clouds

I suspect that if you're reading this blog right now, you've seen the fear- and awe-inspiring pictures of mushroom clouds over nuclear tests. What on earth causes those? Part of it is a result of all the dust and debris that was surely churned up by such an explosion. The ground underneath the explosion is partially vaporized, but the rising plume of hot air pulls in nearby air (and dust and debris) as it heads upwards. (As an aside, some fireballs can rise as fast as 300 miles per hour!) At a certain height, the air stops uniformly rising and starts falling, creating convection plumes that form the donut-shaped top of the mushroom. Note that all this happens a while after the initial detonation, after the fireball has done some expanding and shock-wave generation of its own.

Another interesting issue is the nice circular rings that sometimes form around the stem of the mushroom cloud. These are actually partial condensation clouds, which are formed in the negative phase of the shock wave, behind the obvious high-pressure shock front. Previously, I've discussed complete condensation clouds, which appear as continuous surfaces behind, for instance, an airplane in supersonic flight. The basic idea here is the same: as the front of lower-pressure air expands outward, the air inside it expands and therefore cools, so if there's enough humidity, the water in the air condenses into a cloud. The difference is that in supersonic flight, the shock wave is localized enough that the whole lower-pressure region condenses. In atmospheric nuclear weapon detonations, though, the shock wave is large enough that the structure of the atmosphere comes into play. The rings you see, then, indicate high-humidity layers in the atmosphere, which condense in rings as the negative phase of the shock wave intersects them.

Thank you to Jonathon Vigh's thesis for much of the information contained here.

Muon g-2: The inflector

As I mentioned previously, muons that are injected into our favorite ring need to enter it in a region with no magnetic field. In particular, every magnetic field has some sort of fringe region, and especially in this high-precision experiment, passing through this region (not to mention the possibility of entering the storage field too early) has the potential to deflect the mouns, which is no good for the ring's acceptance (how many of the 'injected' muons get stored). In a run-of-the-mill accelerator experiment, this issue is avoided by simply injecting particles in gaps between the magnets. In Muon g-2 (remember, that's pronounced gee-minus-two), we need an incredibly precise, fantastically uniform magnetic field in the muon storage region, which means we need a single continuous magnet all the way around the ring. That's the thing that just got transported, and the result is that the typical injection design just doesn't cut it.

The solution, at least for us, is what's called an inflector, which serves the dual purpose of injection and deflection (thus the name). It's responsible for allowing the muons to enter the ring tangent to the eventual orbit, and it does so by producing a region with essentially no magnetic field. As a result, the muons can simply drift into the ring without having to deal with the deflection from the magnetic field. But like so many things in the experiment, the inflector design had some serious constraints. They boil down to the critical issue of the constant magnetic field around the muon orbit - the inflector can't contain any ferromagnetic material or time-varying fields, as those would affect the magnetic field in the storage region. For the same reason, it cannot leak the magnetic field that it contains into the area the muons inhabit. Apart from that, the hope is to accept as many muons as possible, which means having a large opening for them to pass through.

That's a tall order, but all the requirements are satisfied by a superconducting inflector. It's roughly a cylinder that needs to cancel the ring's storage field, and it does so by generating a field of its own in the opposite direction. To get the muons to circulate, the storage field is vertical, so to cancel that, the (superconducting) coils in the inflector have to circulate length-wise, rather than just spiraling around the central axis of the cylinder (think right-hand rule). Much geometry and engineering later, we end up with a funky thing called a "truncated double cosine theta magnet," and it produces a highly uniform, roughly 1.5 T field in the region the muons occupy, along with a roughly 1.5 T field next door as return flux. When this is put into the magnetic field of the ring, the result is a net zero magnetic field in the muon's region, and an almost 3.5 T field for the return flux.

Even with a magnet design chosen to reduce the magnetic field flux that leaked into the storage region of the ring, this inflector still produced a high-gradient fringe field (meaning that it changes very quickly - so quickly, in fact, that the probes designed to map the field wouldn't be able to detect its change accurately). The solution the previous experiment arrived at was the use of a superconducting shield. Superconductors have this really cool property called the Meissner effect in which they expel external magnetic fields. As a result, getting the magnets cooled and running in the right order (first the storage field, then the superconducting shield, then the inflector) allows the shield to trap the main storage field while preventing any change in the flux from the inflector's field. This very effectively prevents any flux leakage, which allows the storage field to be very nearly uniform, and my understanding is that we'll be using the same sort of design in the new g-2 experiment at Fermilab.

There were a few issues with the previous experiment's inflector, though. The most crucial one was that in order to better trap its own magnetic field, the inflector had closed ends. That is, the superconducting wires actually cover both the entrance and exit of the inflector. This wreaked havoc with the muons being injected - it unsurprisingly turns out that running them into the wires decreases the number that make it into the ring. Luckily, the new experiment has invested a good deal of engineering in the design of an improved inflector, and I believe we're planning to open the ends of the inflector to allow muons to pass through unimpeded.

If you're interested in more information about the Muon g-2 experiment, you should check out its website. More g-2-related posts can be found here.

Muon g-2: The anatomy of the ring

While the large ring that was transported contained mostly the superconducting coils and their cryostat, there's a lot more to the final ring than that. We have to add quite a bit of material: vacuum chambers, yokes and poles, and so on. And in the end, of course, we also need to put our detectors into place. To complicate matters, the electrostatic quadrupoles take up their own fair share of space, and the inflector and kicker are necessary to allow muons into the ring in the first place. All this gets rather complicated fairly quickly, so before I get any further into the details of the experiment, I wanted to present a simplified picture of what's involved in the ring.

 

This is a somewhat simplified and very much not-to-scale sketch of where various components go in the ring. It's a view from above, and from here, we'll see the muons whiz around clockwise. Once any muon decays, some of its momentum is carried off by neutrinos, and the resulting positron ends up with lower total momentum. Since it's in the same magnetic field as before, this causes it to curl inwards, where the experiment detects it using calorimeters (energy-measurers) and trackers (which, as their name suggests, help to track particles and figure out their trajectories). In the case of the detectors (possibly including the not-yet-shown fiber harp) and inflector, I will hopefully write a bit about each component at another point in time, but here's a brief overview:

• Inflector: The issue with having an enormous ring with a strong magnetic field is that it's hard to get muons (or any particle, for that matter) into it in the first place. They are affected by the field too early, and end up on all sorts of funky trajectories. That's where the inflector comes in. It's another magnet, and it generates a magnetic field opposite to that in the ring, so that the muons can really just drift inside, blissfully unaware of the extreme bending forces within.
• Kicker: Well, now we have another issue: the inflector injects particles at a bit of an angle compared to the ideal trajectory. This can be viewed as the particles moving along a circular trajectory (since they're in a uniform magnetic field) with its center offset somewhat from the actual center of the ring. What the kicker does is produce a time-dependent burst of magnetic field that jolts the muons into the orbit we want them in. Of course, time dependent fields bring in their own sorts of problems, so the kick is an interesting area of study.
• Quads: I've previously mentioned the importance of quadrupole magnets (also known as strong-focusing magnets) for vertical focusing of a beam. Since a magnetic quadrupole would disrupt our beautiful, uniform-to-within-a-few-parts-per-million magnetic field, Muon g-2 actually uses electrostatic quadrupoles, which operate on the same principle but do so using electric fields. These extend for long sections of the ring, and (fun fact alert!) one of the quad plates on the first quadrupole (the one right after the inflector) is actually in the way of muons being injected! 
• Calorimeters: In any particle physics experiment, calorimeters, which measure the energy of incident particles, play a central role in identifying and describing particles. This is especially true in Muon g-2, since the critical measurement is the energy of decay positrons over time, which gives us a clue to the wiggle (precession) of the muons that produce them. We have 24 calorimeter stations, positioned in order to catch as many of the decay positrons as possible. They're outside the vacuum chamber (shown in thin black lines above), but only barely.
• Trackers! New in this iteration of the measurement, the Muon g-2 experiment has two tracker stations in front of calorimeters. These are useful in identifying pileup events, when two low-energy positrons hit the calorimeter at nearly the same time, and thus masquerade as a single higher-energy one. They also allow us to trace the trajectory back to the decayed muon, and that gives us a pretty good look at how the beam is behaving, a valuable asset in an experiment in which beam dynamics crucially affect the measurement.

Well, that's the whirlwind summary of the structures in the ring. I haven't mentioned things like the fiber harps, which monitor the beam, and collimators, which filter out muons that get too far away from the ideal parameters, and all of the magnetic field measurements. They're essential to the experiment, but I'm not even nearly an expert on them, so I'll leave their explanation to people more experienced than I.

Muon g-2: The muon

I've previously discussed quite a few topics in particle physics, including a brief introduction to leptons and a discussion of parity-violating weak decays. Now I'd like to go into a little more depth regarding my current favorite elementary particle: the muon.

As mentioned previously, the muon is a sort of strange second cousin of the electron. The two interact with other particles and forces very similarly, so the two major differences are the muon's mass (207 times that of the electron) and its lifetime (a mere 2.2 microseconds - yes, that's millionths of a second). Like the electron and quarks, the muon is a 'spin-one-half' particle, which for our purposes means it has a spin of either 1/2 (unitless) or $\hbar/2$ (unitful) either along or against any axis you choose to measure. But of course, the muon doesn't actually take up any space (it's a point particle), so it's not really spinning...that's an interesting quantum phenomenon that will have to await another post for a deeper explanation.

One interesting property of many subatomic particles is called the magnetic moment, generally denoted as $\mu$. Macroscopically, a loop of current has a magnetic moment, which determines the amount of torque it will feel from a magnetic field. Similarly, a muon's magnetic moment determines its behavior in a magnetic field. Just as a top spinning on a table spins a bit (precesses) as it starts to slow down, a muon in a magnetic field experiences a precession of its spin direction. The magnetic moment of a particle is determined by a particle-specific constant called $g$, which is short for the gyromagnetic ratio (thus the 'g-2' in the experiment's name), and this determines how fast the particle's spin precesses in a magnetic field. We can talk specifically about the muon's gyromagnetic ratio as $g_\mu$. Another consequence of throwing muons at a magnetic field is that they end up on a curved path, as charged particles in magnetic fields are wont to do. In the Muon g-2 experiment, we inject muons into a very uniform magnetic field. It's tuned just right so that the muons go around in a circle inside our storage ring, and we can (indirectly) measure the spin angle of the muon when it decays. By studying a lot of muons, we can see how the muon's spin compares to its momentum over time, and after 20-30 turns, it's overtaken the momentum by a full turn. Well, that's pretty neat, but I can very clearly hear you asking "so what?"

Let me take a brief step back in time. Once upon a time, many years ago, a brilliant physicist named Paul Dirac came up with a very elegant equation describing the behavior of a free charged particle, which we now (very creatively) call the Dirac Equation. In this model, the value of the gyromagnetic ratio was exactly two. Well, it turns out that nothing in particle physics is so simple, and so the value of $g$ is actually a tiny bit higher. This is a result of these crazy things called virtual particles. They're sort of like nature's own particle accelerator, and basically pairs of particles are constantly popping into and out of existence - quarks, bosons, muons, electrons - all the time, everywhere. It turns out that while these little fellows may be short-lived, they have an effect on the value of the gyromagnetic ratio $g$. As a result, the value that we're measuring isn't $g$ so much as $g-2$, which is important enough that it gets its own name: the anomalous magnetic moment of the muon ($a_\mu$).

The interesting thing about muons is that because of their higher mass, they're more sensitive to these virtual particles than the less massive electron, and they live long enough to be a pretty good candidate for study. Furthermore, the last time an experiment (E821 at Brookhaven National Lab) measured their precession rate and gyromagnetic ratio, what they found disagreed pretty substantially from what theory predicted at the time. In mathematical parlance, there was a $3\sigma$ (so three standard deviations) disagreement, which isn't enough to claim a discovery (in particle physics, we require $5\sigma$), but it's certainly enough to label the issue as interesting. If we make the same measurement again (but better, of course, thanks to improved detector technology and vastly improved beam quality at Fermilab) and that difference persists, it will be a good indication that something that we don't understand is going on at the most fundamental level. That could mean the existence of new particles, and the really exciting thing about the measurement is that it could find traces of new particles that even the Large Hadron Collider in Switzerland couldn't produce! And all from a very precise measurement of a single property of a single particle. Wow - physics is awesome.

If you want to learn more about the Muon g-2 experiment, some of my other blog posts can be found here, or you can check out the experiment's webpage here.

Muon g-2: What's that thing for, anyway?

A natural question to ask in light of my recent post is what exactly the enormous Muon g-2 (remember, it's pronounced gee minus two) electromagnet is good for. But first, a point of clarification: the electromagnet wasn't on while it was being transported; as an electromagnet, it has to be plugged in before it becomes magnetic. And before we can really plug it in, we need to cool it down. A lot. Along with the superconducting coils, the cryostat for the experiment also couldn't be safely disassembled, so the two were transported together. With the aid of the cryostat and liquid helium, the temperature of the coils will be reduced to just 5 Kelvin (around -450 degrees Fahrenheit), a hair above absolute zero, which chills them enough to be able to superconduct; that is, conduct electricity with exactly zero resistance. It's a fascinating physical phenomenon that falls squarely in the realm of "post some other day," but suffice it to say that once we cool this thing down and plug it in (slightly more complicated than your standard wall socket, but the same general idea), it generates a pretty uniform magnetic field of around 1.5 T (tesla) inside the storage area for the muons. As far as magnetic fields go, 1.5 T is pretty strong - the magnetic field of the Earth is on the order of a few dozen millionths of a tesla at the surface, and a standard refrigerator magnet has a field strength of a few thousandths of a tesla. That said, it is fairly similar to the magnet in an MRI, which tends to generate a magnetic field of one to three tesla. This is moderately unsurprising, as both MRIs and the g-2 ring make use of superconducting coils to produce their magnetic fields. (This, by the way, is why it is more expensive to turn an MRI off overnight than to leave it running - the coils just keep conducting in any case, and the costs and danger of releasing all that liquid helium in gas form are pretty high.)

Okay, we now have a magnetic field...so what? We rely on the fact that a magnetic field bends the path of a charged particle. In this experiment, we inject a very pure muon beam into this ring, and if we've calibrated the magnetic field just right, it bends these muons into a circular path so that they whiz around the ring in the region we want them to. Once they're injected, we essentially leave the muons alone - we're not pushing them to higher and higher energies like they do in the enormous ring at the Large Hadron Collider at CERN. Instead, we just let the muons circle around a few thousand times in just a few millionths of a second and wait for it to decay. Thus this particular ring is classified as a storage ring rather than a cyclotron, synchrotron, or synchrocyclotron. That's the general story behind what our giant electromagnet will be used for - I'll post more about the science behind it and some of the fascinating techniques used in the next few days.

If you want to check out other posts on Muon g-2, they can be found here.

Muon g-2: The big move

If you live in or around Chicago, chances are good that you've heard about the big move of a giant electromagnet to its new home at Fermilab. The 50-foot ring was designed and constructed at Brookhaven National Lab, on Long Island, in the early 1990s for a Muon g-2 (pronounced gee minus two) experiment going on there. It took data through about 2001, after which the experiment published a bunch of results and shut down. Luckily for the new experiment at Fermilab, it was cheaper to just let it sit around than to disassemble it, so it sat in a very large garage for a little over a decade until it began its journey to the Midwest.

The vast majority of its trip was by barge - around a month spent traveling south along the Atlantic seaboard and all the way around Florida, then up a series of rivers, from the Tombigbee River and the Tennessee to the Ohio River and eventually the Mississippi. It ended its barge trip on the Illinois and eventually the Des Plaines Rivers, and it was unloaded just last weekend (July 21-22) in Lemont. From there, it went on a three-night journey to Fermilab, reaching the site successfully on Friday, July 26th. 

While it is a pretty large piece of equipment, this move would have been fairly straightforward if it hadn't been for the incredible sensitivity of the ring. It's a superconducting coil that is all one piece, so it couldn't be disassembled at all. As if that weren't enough, flexing just 3 millimeters (around a tenth of an inch) out of the plane of the ring could be enough to damage some of the superconducting coils within, which would make the whole magnet essentially an oversized doorstop. This added complication made the ring's "Big Move" an engineering marvel, and Emmert International really rose to the challenge, custom designing a fantastic (bright red) frame to support the ring and transporting it on a hydraulically balanced barge and truck.

Some of the transportation details are pretty neat, too. For instance, the truck trailer that supported the ring had a whopping 64 wheels on 16 independent axles, and those axles could be remotely controlled to help with maneuvering around trees and backing up the truck. That turned out to be very useful, as the ring's route required that the truck back up several highway ramps in order to avoid tollbooths. Oh, and one other fun fact: when it went through an open-road tolling arch, it had just six inches of clearance on either side!

If you're interested in more information about the ring's move, check out its website, complete with twitter feed, pictures, and a map showing its route.

I'm hoping to write some more posts about Muon g-2 in the near future, in particular about what that ring is useful for and some of the cool things that could come out of its use.

If you want to check out other posts on Muon g-2, they can be found here.

Condensation clouds

Condensation clouds seems like a strange name for an interesting phenomenon - after all, aren't all clouds formed by condensation? Well, it turns out that condensation clouds, also known as Wilson clouds, are actually a somewhat different phenomenon than your run-of-the-mill precipitation source.

They appear in shock wave situations, like super-sonic aircraft and large explosions, but only in particularly humid air (or in some underwater nuclear tests - vaporize enough water and you end up with really humid underwater bubbles). The obvious part of any shock wave is the high pressure at the 'positive phase' of the shock wave - this is what causes buildings to implode near a large explosion (the remnants are then blasted with winds of several hundred miles per hour in the shock's wake). But there's more to a shock wave than that. There's also a negative phase, in which the air is much lower-pressured. As the shock front expands, the positive and negative phases experience very little air flow with the rest of the world. As a result, the positive phase's pressure decreases as it expands, so its destructive power diminishes with distance. The negative phase of the shock wave similarly experiences a decrease in pressure as the shock radiates outward, and because air and heat are not being readily exchanged between the negative phase and the rest of the world, this manifests as a decrease in temperature. In sufficiently humid air, that decrease in temperature is enough to cause water vapor in the negative phase of the shock wave to condense. When large explosions are involved, the result is a roughly spherical-looking cloud around the explosion that has little or nothing to do with the debris and fireball inside.

Similar effects can be observed when airplanes fly really fast in humid conditions or do high-acceleration maneuvers - the curves of the wings and the body of the plane cause local areas of low and high pressure, and the low-pressure areas often experience condensation. That is why you often see a conical cloud behind planes in pictures of supersonic flight; the plane is moving pretty darn fast, so there's a lot of opportunity for low-pressure areas.

As for the eventual fate of the condensation cloud, when the extenuating circumstances that created it wear off, the water re-evaporates and the cloud seems to disappear. In some cases, parts of the condensation cloud can stabilize, which is apparently the source of those rings you sometimes see around the stems of mushroom clouds.

Ionization-based detectors

Ionization detectors were some of the earliest detectors developed in the study of high energy physics. While they have fallen out of widespread use in favor of more recent technology such as silicon detectors, they remain a simple, easy to maintain, and inexpensive option for tracking charged particles. Furthermore, descendents of the original ionization detectors, like the multi-wire proportional chamber and the time projection chamber, continue to push the boundaries of particle detection and are now being used in dark matter searches and collider detectors. 

The simplest ionization detector looks a bit like a straw. The straw has a wire fed through its center, and a high voltage is applied. That central wire is the anode, and conducting material along the inside of the straw itself is connected to ground and serves as the cathode. The straw is filled with gas, which generally cycles through the straw. The basic principle is that when a charged particle passes through, it ionized some of the gas molecules inside. The applied voltage generates an electric field that causes the yanked-off electrons to accelerate towards the central wire. They hit the wire, and we measure their presence as current.

Naturally, though, the whole business is slightly more complicated than that; there's actually a variety of regions in the behavior of the detector based on the voltage you apply to the central wire.

The various regions in ionization detector behavior. Figure from W.R. Leo, chapter 6.

Each of these regions has something different going on physically.

• Recombination region: The electric field in the detector is low enough that ionized atoms are able to recombine (electrons find ions and re-merge) with some probability. As a result, not all the primary ionization electrons are captured, so as you increase the voltage on the wire, the number of ionization electrons you capture increases fairly dramatically.
• Ionization chamber region: All ionized atoms remain ionized and travel to the anode/cathode. As such, there's a plateau here: increasing or decreasing the voltage applied to the central wire doesn't change how many of the ions are caught.
• Proportionality region: The voltage in this range is high enough that it attracts those primary electrons very strongly - so strongly that they in turn ionize more atoms on their way to the central wire. This causes an avalanche, and in this region at least, the number of captured electrons is proportional to the number that were ionized in the first place, with a constant of proportionality dependent on the voltage.
• Region of limited proportionality: Increase the voltage higher still, and the cascades of electrons develop such high charge densities near the anode that they distort the electric field, which reduces the proportionality of the detector.
• Geiger-Muller counter: Beyond the region of limited proportionality is another plateau in the number of detected electrons. Physically, at this point, the cascades don't stop, and photons emitted by excited atoms ionize more and more atoms. These detectors end up with a self-sustaining chain reaction, so that even the slightest ionization in the gas results in exactly the same current as a massive, highly charged particle passing through. To avoid getting a constant current after just a single hit, these detectors have to have a quenching gas inside, in order to capture and disperse the energy from these emitted photons. 

If you're interested in learning more about ionization detectors (as well as a variety of other interesting detectors and techniques), I highly recommend W.R. Leo's Techniques for Nuclear and Particle Physics Experiments, which gives an excellent explanation of the theory behind many detector types.

Sonic booms

Many people have heard of the sonic boom, a loud noise that happens when something (an airplane, a bullet, the tip of a bull whip) starts moving faster than the speed of sound. What a lot of people don't realize, though, is that it's not just a single boom - it keeps going for as long as the object is moving faster than Mach 1 (1 times the speed of sound). In the case of an airplane, it will pass overhead, and the shock wave that is the sonic boom will travel outward to you, the observer, at the speed of sound. That is, slower than the airplane is moving. As a result, you end up hearing the sonic boom fairly substantially after the object actually passes over you. It expands in what looks like a cone shape, like this:

The red object, moving from left to right at Mach 1.5, leaves behind a sonic boom cone. When the edge of the cone reaches an observer on the ground, they hear the boom.

 

Where does this shock wave come from, anyway? An airplane in normal flight sends out vibrational waves (some in the form of sound, others at frequencies we can't hear) in all directions. These vibrations progress outwards at, you guessed it, the speed of sound.* But what happens when an object starts moving at the speed of sound? All the vibrations it's been beaming forward suddenly can't leave the vicinity of the object, so they just pile up in approximately one place, right along the object itself!

The red object, (still) moving from left to right, now at Mach 1, appears to create a massive shock wave, thanks to many many waves piling on top of each other at the object's location. This is the Prandtl-Glauert singularity.

This hypothesis, the Prandtl-Glauert singularity (which also suggests infinite pressure at Mach 1), turns out to be not entirely true, because other effects, like turbulence and viscosity, start to assert themselves, and compressible fluids under extreme conditions are hard to describe in a precise mathematical way. In other words, physics is awesome.

* This, by the way, is the origin of the Doppler effect: when an object is moving towards you, you hear a higher-pitched sound than when it passes you and starts moving away, because the sound waves are compressed in front. That compression lessens the time between peaks of the sound's oscillation, which manifests itself as a higher-pitch noise. Conversely, once the object is past you, the wave peaks grow further apart in time, so you hear it as a lower-pitch sound.

An object traveling from left to right at Mach 0.7 emits sound (and other vibrations) in all directions. But it's catching up to the waves it sends forwards, and is running away from the waves it sends behind it. The Doppler effect is the result.

Optimization in c++

Recently, a question arose about the most efficient way to store certain data in a program. In essence, a unique identification is needed for a detector component, which can be broken down into properties A, B, C, and D. Many of these things will need to be initialized, stored, and sorted, and we wanted to know if a c++ struct (essentially identical to a c++ class, but conventionally used for storing related information together; at least in my experience, they tend to have far fewer methods and their fields are almost entirely public) or a sort of hashed integer (multiply property A by 10000, add B times 1000, etc.) would be more efficient.
A struct can have a manual comparison function, so it would be sortable, but we were interested in determining how efficient this sorting process is compared to sorting plain old integers, so I ran a quick little study to compare them.

The three candidate data structures are:

• An integer, which is defined as $(A \times 10000) + (B \times 1000) + (C \times 100) + D$. The sorting function is a simple comparison, but is provided manually in order to ensure similarity between the trials.
• A Specification struct, which contains fields for track number, view, plane, and wire, with a comparison function that accesses one or more of these fields in order, henceforth known as the naive struct.
• A Specification struct with the same fields as above, but with an additional integer uniqueID field, calculated after all setting is done according to the formula above. The sorting function compares only the uniqueID. This type is referred to as the smart struct.

What I investigated was the execution time to create and sort $10^7$ objects of these different data types, using c++'s clock type and clock() method to count CPU cycles spent in the program (so as to avoid influences from system interruptions).

I timed both creation and sorting the data structures with no optimization and then with gcc -O3 optimization. Results are summarized in the figures below, and discussion is below the relevant figure.

Unoptimized times to create and sort $10^7$ objects of the three types

Here, unsurprisingly, we see that the integer is faster in all respects than the structs. Since we have a total of $10^7$ objects, the total time difference boils down to around 150 nanoseconds per object of extra time. As we would expect, it takes a little longer to create the smart struct than the naive one, since we have to set up the uniqueID, but it sorts much faster. The interesting stuff starts to happen when the code is compiled with maximum (standards-compliant) optimization.

Optimized times to create and sort $10^7$ objects of the three types.
The purple/pink is an integer with its default sorting mechanism.

For one thing, it's obvious that optimization speeds up the code, especially the sorting, by a lot: it runs around a factor of four faster than in the unoptimized version. It also brings the performance of the three data structures much closer together, and for some strange reason, the smart struct appears to actually sort faster than the integers; what a mystery!
The last consideration is what I call the bare integers; that is, the same uniqueID as held by the regular integers, but with the default integer sorting mechanisms, rather than a custom-provided function. The running time difference between the bare and bloated integer types is on the order of 20 nanoseconds, which corresponds to around 40 assembly instructions; probably around the right number to correspond to a function call.

My conclusion from this study is that when compiled with optimization, structs and ints are roughly identical in terms of CPU time required to create and sort them. Since structs pretty dramatically increase readability of the code, I'm sticking with those instead of tiptoeing around a funky hashed integer.

Amor asteroids

The Amor asteroids are a class of asteroids that get very close to the Earth from the outside, usually without crossing the Earth's orbit. Some of these are classified as potential collision hazards, but most just keep their distance. I learned about them as a result of more playing with Mathematica, this time with its AstronomicalData. A particularly handy option for the package is the "Classes" argument (so AstronomicalData["Classes"]), which will return a list of possible classes, like InnerMainBeltAsteroid and DwarfSpheroidalGalaxy and so on. These classes can be used to get a list of list of astronomical objects that fall into the class.

In any case, here's a fun little diagram showing the orbits of various Amor asteroids. The thick black lines are Earth, Mars, and Jupiter's orbits, from smallest to largest. You can see that many of the Amor asteroids cross Mars's orbit, and a few even get as far away as Jupiter.

Orbits of the Amor asteroids. Black lines show
Earth, Mars, and Jupiter. Blue lines show the asteroid
orbits. The big yellow dot represents the Sun.

But overall, this isn't a terribly revealing diagram. A slightly more intriguing plot is shown below: eccentricity of the orbit as compared to the asteroid's semimajor axis.

Eccentricity of the Amor asteroids as compared to their
semimajor axes. Looks interesting, right?

It's interesting to see that as the asteroid's semimajor axis increases, its eccentricity does too. There's not exactly a concrete reason that we would physically expect this result. So what causes this trend?

It turns out that it's actually all determined by our definition of an Amor asteroid: its perihelion (closest approach to the Sun) falls between 1.0 AU (astronomical units - the average distance from Earth to the Sun) and 1.3 AU. From the geometry of ellipses, the closest an orbit gets to one of its foci (which is the perihelion by definition), can be expressed in terms of its semimajor axis and eccentricity as $r_{min}=a(1-e)$ ($a$ is the semimajor axis, $e$ the eccentricity). Plotting this, we find that the eccentricities that we observed in such a promising-looking trend above are actually just adhering to the constraints imposed by the definition.

Perihelion of Amor asteroids compared to their
semimajor axes. Looks like that trend was just a
figment of our imaginations.

Another way of seeing this is to look at the possible values of the eccentricity for various semimajor axes:

Sure enough, all those eccentricities are based on
the definition of an Amor asteroid!

The upper bound there is $e=1-\frac{1}{a}$; that is, the highest possible eccentricity (for a given semimajor axis) the asteroid can have without crossing Earth's orbit. The lower bound is $e=1-\frac{1.3}{a}$, which is the lowest possible eccentricity it can have while still getting close enough (1.3 AU) to be considered an Amor asteroid. Mystery successfully solved.

Kirkwood gaps

Well, my recent post on resonance in beam dynamics got me thinking about one of my other favorite resonances: Kirkwood gaps. If you look at the semimajor axes (that's half the sum of closest and furthest distances from the Sun throughout the asteroid's orbit) of all the known asteroids and put them in a histogram of some sort, there's a very peculiar structure. There are noticeable gaps in the distribution, like there are some distances asteroids just don't want to live. Take a look:

What could cause this? The answer, it turns out, is our solar system's most massive planet, Jupiter, whose mass is within an order of magnitude of becoming an actual star. You see, the big gaps correspond to asteroid orbital periods that are rational number multiples of Jupiter's orbital period. Take the giant 3:1 gap around 2.5 AU. If some asteroid occupied that particular region, for every three of its turns around the Sun, Jupiter would go around once, which means that every three asteroid-years, Jupiter would be in the same relative position. This produces a resonance, and Jupiter, over the course of many, many asteroid-years, either slings the asteroid out of the solar system or adjusts its orbit so that its semimajor axis is no longer in that 'danger zone.' Similar stories apply to the three other prominent resonances.

These resonant regions are called Kirkwood gaps. It's just one of many really interesting phenomena resulting from resonances in physics.

Resonance in weak focusing

Since I'm sure you're not yet sick of weak focusing, I want to write a bit about the dangers of resonance in weak focusing systems. In the case I am most familiar with, a storage ring has a (mostly) uniform vertical magnetic field and some strong-focusing quadrupoles to ensure vertical stability. Muons are stored for many hundreds of turns around the ring, so it's critical that the orbits be stable in the 'long' term. The problem is that it's possible to have a localized disruption in the field at just one or two points along the ring. To ensure stability in the long run, the simple betatron oscillations that I mentioned have to be at a different point in their oscillation each time they hit that instability. This has to account for both vertical (frequency $f_y=\sqrt{n}f_C$) and horizontal (frequency $f_x=\sqrt{1-n}f_C$) oscillation frequencies (remember, $f_C$ is the cyclotron frequency, the rate at which the bunches move around the ring). Basically, for the beam to stay stored, any (integer-coefficient) linear combination of these frequencies must not be an integer multiple of the cyclotron frequency. If it is, then every few turns, the beam will be at the same phase of its oscillation at the location of the instability. This is called a resonance, and it can boost the beam out of its stable orbit. As a result, care must be taken in choosing the 'tune' (primarily the magnetic field index) of the storage ring.

This choice can be illustrated with a complicated-looking plot. Below, the x-axis shows $\nu_x=\sqrt{1-n}$, and the y-axis shows $\nu_y=\sqrt{n}$. Integer combinations of $\nu_x$ and $\nu_y$ that yield integers (corresponding to frequencies that are integer multiples of the cyclotron frequency) are plotted as black lines. By the definition of $\nu_x$ and $\nu_y$, we see that $\nu_x^2+\nu_y^2=1$, which is shown on the plot as a red curve. Blue dots are along the intersection of this tune curve and the forbidden lines, and represent bad tunes, while the bright green dots show acceptable tunes, at least to the degree that we've plotted the resonance lines. Neah, eh?

Vertical focusing

So far, I've discussed what sort of a field will provide horizontal beam focusing. There are two problems left. For one thing, in my original discussion of weak focusing (link), I showed that the field index $n$ had to be less than one for horizontal weak focusing. Later on, the constraints had mysteriously tightened, and I stated that for weak focusing to occur, the field index had to satisfy $0\le n<1$. Secondly, we've seen that horizontal focusing can occur without vertical focusing in a uniform magnetic field, so we need a new constraint on the field index to ensure vertical focusing.  Well, in this case, two wrongs almost make a right, and the second of the above-mentioned problems explains the first. Here's how it works.

In order to stabilize the vertical structure of the beam, the magnetic field needs to provide a restoring force in the vertical direction, something along the lines of $F_z=-cz$. In order to produce that, the magnetic field needs a horizontal component: $B_x=-c'z$. Well, from this we know that $\frac{\partial B_x}{\partial z}=-c'$. One of Maxwell's famous equations tells us $\vec\nabla\times \vec{B}=0$, so we clearly see that
\[ \frac{\partial B_x}{\partial z} = \frac{\partial B_z}{\partial x}=\frac{\partial B_z}{\partial r}=-c' \]
Observe! We have shown that for vertical focusing, we need $\frac{\partial B_z}{\partial r}$ to be negative. From our definition of the field index ($n=\frac{-\rho}{B_0}\frac{\partial B_z}{\partial x}$), then, we see that for vertical focusing to occur, the field index must be positive. Voila!

Geometric focusing

A uniform magnetic field (field index 0, since the rate of change of the field is zero) provides a certain level of horizontal focus, in a phenomenon called geometric focusing.  From elementary E&M, we know that a particle in a uniform magnetic field with momentum perpendicular to the field lines will follow a perfectly circular path.  Let's examine the behavior of a nonconforming particle in the beam; call it Fred. If at some point it (he?) is in the ideal location, moving in the ideal direction, but has a lower momentum than a particle tracing out the ideal orbit (henceforth referred to as Ida), then the magnetic field will cause him to run around the ring in a smaller circle than Ida's trajectory. But after going all the way around, Fred ends up right back where he started, and while the beam may have defocused somewhat azimuthally (that is, the bunch is longer now, so it takes up a greater portion of the ring), it's once again focused horizontally. This is shown in the leftmost part of the figure below. Similarly, if Fred's momentum vector is pointed in a different direction than Ida's, he'll have a different trajectory, but that'll intersect Ida's twice, so we have geometric focusing.  Finally, just turning the previous example on its side, if Fred is slightly displaced relative to Ida, their trajectories once again meet twice in their trips around the ring. That's the premise of geometric focusing in accelerator physics; it's just a special case of horizontal weak focusing.

Some examples of geometric focusing. Black shows the ideal trajectory (Ida),
and red shows the trajectory of another particle (Fred) that differs slightly from
Ida in initial conditions (at the left of the image). In all cases, the trajectories
of Fred and Ida meet up at least once in each full revolution.

In (a), Fred starts in the same position as an ideal particle but with
lower magnitude momentum. In (b), he starts in the same position and with the same
magnitude momentum as an ideal particle, but pointed slightly outwards compared
to Ida's trajectory. And in (c), Fred has the same momentum as the ideal particle but
is slightly offset spatially.

There are a couple of problems with relying on geometric focusing, though.  For one thing, a very small deviation in the beginning can send Fred on a trajectory that is fairly far away from Ida's at certain points. In order for Fred to stay in the beam, he needs to not run into the walls, which creates a real headache for the accelerator designers. For another, a uniform magnetic field doesn't provide any vertical focusing effect; if Fred has even a tiny vertical component of his momentum, the magnetic field won't affect it, and Fred will end up moving higher and higher in the accelerator pipe until he runs into the material there and meets with an untimely end.

Quadrupoles

I mentioned last time that weak focusing is all well and good, but that in many cases, it just doesn't cut it. In such situations, experimentalists go for strong focusing, which involves electric and/or magnetic fields that are not radially symmetric, so that a particle traveling along its trajectory will see a different field as it goes along. In particular, quadrupole magnets or electrostatic quadrupoles can serve to focus a beam in one direction while defocusing it in the other. So for instance, one quadrupole magnet might focus the beam into a thin horizontal strip (vertical focusing, horizontal defocusing), and then another immediately afterwards could do the opposite. It turns out that such a setup can have a net focusing effect in both the horizontal and vertical orientations. This, like weak focusing, allows for thinner, higher-flux beams, critical for colliders and target experiments.

How do these things work? As I understand it, there's this mathematical approach to magnetic fields called the multipole expansion, in which a simple permanent magnet generates primarily a second-order (dipole) term. Other, higher-order terms, tend to be smaller than the low-order ones, especially at larger distances, so they can often be ignored. In a quadrupole, though, four magnetic dipoles (either permanent magnets or electromagnets) are positioned in such a way as to cancel the dipole moment, leaving only the quadrupole moment. This generates an interesting-looking magnetic field that is the source of the curvy bits in the Fermilab logo.

Logo courtesy of fnal.gov (upper left hand corner, when I
pulled it off). The curvy bits represent the quadrupoles in
the various particle accelerators on site, while the straight
lines represent the dipoles used to bend the beams.

Betatron Oscillations

This post ended up being a little longer and mathier than I'd expected, but I found out that weak focusing is really cool. Enjoy!

Theoretical physicists enjoy playing with perfect particles in a well-behaved world. Experimentalists would love it if that worked, but the real world is never so nice, so they have to deal with imperfectly calibrated beams. In particular, that means that if a particle deviates slightly from its ideal trajectory, there should be some mechanism in place to ensure that it stays close, rather than diverging away from the ideal beam location. The mechanisms that allow this to occur are called focusing, and they also serve to keep the beam narrow enough to allow precise knowledge of its structure and enhanced probabilities of interactions of opposing beams (like in the LHC, where protons are circling the ring in opposite directions and then collide head on).

For the following discussion of focusing techniques, I'll treat only circular beams/rings, as they're easiest to describe. This class includes colliders like the Tevatron and the LHC as well as, say, storage rings involved in intensity frontier experiments.

In a technique called weak focusing, a radially symmetric magnetic field is present in the region of the beam. The field gradient (both radially and vertically) means that when a particle isn't quite on the perfect trajectory, there's a restoring force. In the long run, this causes such particles to oscillate about the central orbit with a frequency determined by the magnetic field gradients in what's called betatron oscillation.

Let's take a quick look at how weak focusing gives horizontal beam stability. We'll take a beam that is ideally at radius $\rho$. Let's examine a single particle of charge $q$ that deviates slightly from this ideal radius, with a radius of $r$. For convenience in Taylor expansion, define $x=r-\rho$. In order for the beam to be stable, we want to have a restoring force; that is, there's more force on the particle if $r>\rho$ (or equivalently, if $x>0$), and less for $x<0$. The force is a result of the magnetic field at the location of the particle, and its magnitude is $F=qvB_z(r)$. Here $v$ is the velocity of the particle, and $B_z(r)$ is the magnitude of the vertical component of the magnetic field at radius $r$.  Since we're dealing with weak lensing, the magnetic field is radially symmetric, so we don't have to worry about its dependence on the azimuthal angle $\theta$.

We know the centripetal force necessary to keep the ideal beam on a circular path is $F_c=\frac{mv^2}{r}$. Note that here, $m$ isn't the rest mass of the particle; it's the effective mass accounting for relativity, $m=\gamma m_0$, where as usual, $\gamma=\frac{1}{\sqrt{1-(v/c)^2}}$. Based on this observation, we define a restoring force 

\[ F_{rest} = \frac{mv^2}{r}-evB_z(r) \] 

Observe that since particles on the ideal orbit will happily orbit at radius $\rho$ until the end of time (or until they decay), the two terms are equal at $r=\rho$ ($x=0$), so we care only about the sign of the restoring force for small $x$ near zero. In particular, for beam stability, we want $F_rest$ and $x$ to have opposite signs.

Let's examine the second term first.  Taylor expanding the magnetic field about $x=0$ to first order in $x$, we see 

\[ B_z(x)\approx B_0+\frac{\partial B_z}{\partial x}x \]

where $B_0$ is the magnetic field strength at $x=0$, and the partial derivative is evaluated at $x=0$. By convention, we define the magnetic field gradient 

\[ n=\frac{-\rho}{B_0}\frac{\partial B_z}{\partial x}, \]

 which allows us to rewrite the magnetic field strength as 

\[ B_z(x)=B_0\left(1-\frac{x}{\rho}n\right). \]

Now let's look at the first term in the restoring force definition.  By the definition of $x$, we know that $r=\rho\left(1+\frac{x}{\rho}\right)$. The binomial approximation (for $x\ll1$, $(1+x)^n\approx 1+nx+\cdots$) allows us to write the first term as 

\[ \frac{mv^2}{r}=\frac{mv^2}{\rho\left(1+\frac{x}{\rho}\right)} \approx\frac{mv^2}{\rho}\left(1-\frac{x}{\rho}\right) \]

Based on the above approximations, the restoring force becomes 

\[ F_{rest}=\frac{mv^2}{\rho}\left(1-\frac{x}{\rho}\right) -qvB_0\left(1-\frac{x}{\rho}n\right) \]

Since the magnetic field at $r=\rho$ is exactly strong enough to keep the particles in the ideal circular orbit, we know that $\frac{mv^2}{\rho}=qvB_0$, which simplifies the above expression to 

\[ F_{rest}=qvB_0\left(1-\frac{x}{\rho}\right) -qvB_0\left(1-\frac{x}{\rho}n\right)=-qvB_0\,\frac{x}{\rho}\,(1-n). \]

As we saw, for the beam to be horizontally stable, we need $F_{rest}$ and $x$ to have opposite signs, so we find the weak focusing requirement on the field gradient: $n<1$.

One more quick(ish) note. By design, we have calculated this force only to first order in $x$, and that allows us to describe the motion as simply harmonic. Recall that if $F=-kx$, then the object's equation of motion is $\ddot{x}+\frac{k}{m}x=0$, so solutions have an angular frequency of $\sqrt{\frac{k}{m}}$. Based on this, we find the (angular) frequency of these betatron oscillations to be related to the cyclotron frequency $\omega_0$, which describes the frequency of the beam's rotation around the ring. By definition, $\omega_0=v/\rho$. The betatron oscillation frequency $\omega_{CBO}$ is 

\begin{align*} \omega_{CBO} &= \sqrt{\left(\frac{v}{\rho}\right)\left(\frac{Bq}{m}\right)(1-n)} \end{align*}

Recall that we have $\frac{mv^2}{\rho}=B_0 vq$, so the two terms in parentheses are actually equal. Furthermore, they are both equal to the cyclotron frequency, so we see 

\[ \omega_{CBO} = \omega_0\sqrt{1-n} \]

The critical thing to notice here is that because $0<n<1$, betatron oscillations must be lower in frequency than the cyclotron frequency; that is, it takes more than a full turn around the ring to complete a betatron oscillation. That means that these oscillations tend to have fairly large amplitudes.

Weak focusing is often convenient for its simplicity, but as we've seen, it also tends to result in fairly large-amplitude oscillations. This creates a headache for the beam pipe designers, since any portion of the beam that hits the pipe is quite abruptly no longer part of the beam. As a result, most modern experiments instead use strong focusing.

Strong focusing uses magnetic or electrostatic quadrupoles to provide alternating focusing in the horizontal and vertical directions. A single quadrupole focuses in one direction and defocuses in the other, so two quadrupoles in quick succession can provide a net focusing effect both horizontally and vertically. That's a topic for another day.