How defocusing magnets help a beam stay in a ring

The next magnet meets a different particle position

A bending magnet guides charged particles around a curve. Focusing magnets control their departures from the centre line. A quadrupole has four magnetic poles around the pipe. It can pull a horizontally displaced particle inward, yet push a vertical displacement outward. Rotate the poles through ninety degrees and those roles exchange.

Imagine particles moving parallel to the axis, slightly to one side. The first magnet turns them inward. During the following drift they move closer to the axis. A second magnet defocuses them, but now they are closer to its centre and receive a smaller outward turn. The combined action can leave an inward tendency. The distance between the magnets helps determine the result.

In the other direction, the first magnet defocuses. The particles reach the next magnet farther from the axis, where the focusing action is stronger. Following both position and angle explains why the same sequence can confine motion in both directions. The intervening flight changes the strength of the second turn.

A focusing magnet, an open section, a defocusing magnet and another open section form a FODO cell. Bending magnets can occupy the open sections. Repeated cells make a ring in which particles oscillate across a reference path. Suitable settings keep these excursions bounded. An unsuitable sequence lets them grow until the particles reach the pipe wall.

The particles in a bunch occupy different positions and travel at slightly different angles. Their width shrinks and expands around the ring. Stable focusing keeps that envelope within the aperture. A narrow spot can be a crossing point of many trajectories, followed by renewed expansion downstream.

Sources: [1][6]

Repeated passages reveal an oscillation

Insulated metal electrodes inside the pipe pick up a signal as a bunch passes. The nearer electrode usually receives the larger signal. Taking the right-minus-left difference and dividing by their sum reduces sensitivity to changes in bunch intensity. A second opposing pair measures the vertical position.

Those signals need a position scale. A thin wire carrying a high-frequency current can stand in for a beam on a test bench. Moving it through independently known positions measures how electrode response changes with displacement. Electronic gains and pedestals need their own checks. A very weak bunch or a displacement outside the calibrated region can leave the instrument without a reliable position reading.

At one pickup, a list of positions from successive turns reveals transverse oscillation. The number of oscillations per revolution is called the tune. Changing the focusing changes this rhythm. An error that repeatedly nudges particles at nearly the same point in their oscillation can build a much larger excursion.

One observation per turn also hides information, much like a wheel under a flashing light. Different rhythms can leave identical samples, and complete extra oscillations between observations disappear. More positions around the ring, or a small known change in focusing followed by another observation, can help distinguish the possibilities. The extra observations add spatial clues to the turn sequence.

Sources: [2][5][3][4]

Probe a small change before steering the ring

Particles with different momenta bend by different amounts in the same field. Their average paths can therefore differ even with unchanged focusing. Dispersion describes how that average position changes with momentum. Comparing independently measured momenta shows which locations around the ring are most sensitive to that difference.

A displaced magnet can move the average orbit around much of the ring. A small steering magnet changes the particle direction; the effect travels downstream and returns through the same lattice. Several pickups respond together. Improving the nearest reading alone may worsen positions elsewhere.

An operator can first move one corrector slightly in each direction and record every pickup. Repeating the experiment with the other correctors builds a table of measured responses. Combinations of small adjustments can then reduce several orbit deviations together. Nearly identical steering effects, or effects buried in noise, call for more independent information.

After the adjustment, the pickups observe another passage. Those new signals show whether the average orbit improved. A position pickup measures the bunch centre; a separate profile instrument is needed to measure its width. Alternating focusing confines the motion, and small measured probes help an operator choose the next useful adjustment.

Sources: [1][3][2]