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.
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