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.