Our lovely 2018 calendar reflects a summer holiday spent in Sicily in 2017. It almost goes without saying that the weather and food were both amazing – it is part of Italy, after all! But Palermo in particular had an edgier feel than most of mainland Italy, and a sense of modern day neglect that counterpoised the historical aspects of the city quite starkly.
The second week of our holiday was spent in Syracuse, with its amazing Greek and Roman history and architecture. Here, on the island of Ortigia, at the foot of Syracuse, we came across 5th century EBC Greek architecture that was still functioning today, in the form of the Duomo di Siracusa, built into the great Greek Temple of Athena. The temple was originally a Doric building with six columns o
n the short sides and 14 on the long sides. Today it has gained a Baroque frontage (built around 1740s), but the walls have been fitted very satisfactorily around the original columns. It was amazing to see and feel the continuity of use to this day.
Syracuse also has another claim that is still relevant today: it was the home of Archimedes (c. 287 – c. 212 BC), the Greek mathematician, physicist, engineer, inventor and astronomer. Ortigia has a statue celebrating him at the entrance to the island.
Archimedes is known primarily today for his ‘Eureka’ moment, but for modern purposes his definition of Pi was an invaluable tool for our technological age:
Mathematical Pi and Ion Beams
Mathematical Pi and ion beams have a very interesting and unusual relationship . We find Pi here, there and everywhere in the physics of beams but unlike most parameters, Pi is almost always helpful in improving beam performance. Kicking off with the basic derivation of the beam current from the beam energy, the Pi appears in the numerator of the beam perveance, so making the current bigger. Pi re-emerges again in the definition of the beam emittance as the phase space (radius and divergence angle) area of the beam divided by Pi which makes the divergence angle smaller by the square root of Pi. Helpful again!
That is not the end of the story of Pi. It resurfaces in the Wilson-Moliere equation for the mean square scattering angle when beam ions collide with gas molecules where it appears in the denominator (helpful yet again!) which in turn controls the growth of the aforementioned emittance. We have another run-in with Pi in the generalised beam perveance (not to be confused of course with the plain vanilla perveance mentioned above) where this time it is in the denominator but again it is helpful because the generalised perveance controls space charge forces involved with beam divergence and a small value of this kind of perveance is good news.
Pi also appears in the value of the beam space potential where it occurs again in the denominator which is helpful as we do not want a large space potential in a beam and a final appearance in the beam pinch force arising from the beam’s own magnetic field where it is also in the denominator (not quite so helpful this time) but in this instance the pinch force is extremely weak so it does not matter.
So mathematical Pi and beams have on the whole a very co-operative and harmonious relationship with Pi doing its best to keep the current up and divergence, scattering, emittance and space potential down. Just what we want!!
 Many of the equations described above are in “The Physics and Technology of Ion Sources” edited by I. G. Brown, John Wiley and Sons, 1989