Big Numbers

Big Numbers
May 16, 2011

This article has some numbers in it. In principle, numbers are just language, like English or Japanese. Nevertheless, it is true that not everyone is comfortable or facile with numbers and may be turned off by too many of them. To those people, I apologize that this article pays less attention to maximizing the readership than some I do. But sometimes it’s just appropriate to indulge one’s self, so here goes.

When we discuss the performance of some piece of equipment, we usually use numbers to do the characterization. Now, good numbers can be small or big. Recently, our accelerators have been performing well, and I want to sing their praises. In this article, I want to talk about big numbers.

If I take a salami and cut off the end, I see a rough disk with some area. Let’s imagine that I want to throw a dart at the disk of the salami; the chances that I hit are dependent on the size of the salami, on the cross-sectional area of the salami and I can express that size as the area in, say, square centimeters. If the salami has a radius of 2 cm, its cross-section is 2*2*pi (3.1469) = 12.6 square centimeters. I can say that this is the interaction cross-section between my dart and the salami.

In exactly the same way, we express the interactions between sub-atomic particles, electrons, protons, neutrons and the like, using the term cross-section. Sometimes we use specially tailored units, such as a fermi squared, or a barn (10-24 cm2), or a picobarn (10-36 cm2), but often we just stay with the square centimeter. Now, if I throw some number of darts at some number of salamis, I will want to know how many times I get a hit, an interaction. Clearly, it depends on the cross-section, which we have discussed.

But it also depends on how many darts I throw, how many target salamis I have, and whether they are all tightly lined up. Put together, the factor, which tells me how many interactions I get, given the cross-section, is the luminosity. Its units are inverse area, and since we often ask how many interactions per second, we add that. As an example, with which I am familiar, when the top quark was found, the Tevatron Collider at Fermilab provided a luminosity of about 2 1031 cm-2. sec-1 at the beginning of a store. Now, 15 years later, the Tevatron luminosity is 3-4 1032 cm-2. sec-1. Ambitions for the Large Hadron Collider are to get to 1034 cm-2. sec-1. Similarly, when we discuss the Electron Ion Collider that we would like to build at Jefferson Lab, we would like to be in the 1034-35 cm-2. sec-1 range. In general, we would like the luminosity to be high.

A few weeks ago, we noted that Q-weak is a particularly sensitive experiment. When you do the arithmetic, you find that it is operating with luminosity in excess of 1039 cm-2. sec-1 - enormous!! Now, to achieve that, we do put in place a hydrogen target with a length of 35 cm, which certainly helps . With Avogadro’s number and the density of liquid hydrogen, we have about 1024 protons per cm2. We then take the beam of 160 microamperes, which is about 1015 electrons sec-1. That’s 1,000,000,000,000,000 electrons per second!!!

This was so far all about Q-weak in Hall C. Over the past weeks, high intensity, up to 120 microamperes, has also been regularly delivered to Hall A. Put them together, we have the CEBAF accelerator delivering 300 microamperes and a total of 1.6 1020 electrons per day, about 26 Coulombs. Of course that’s a little hypothetical, but the reality is that on May 7, 2011, CEBAF delivered a record charge of 18.8 Coulombs, more than 100,000,000,000,000,000,000 electrons, to the two halls.

But what about little sister, the FEL?

The aficionados of our FEL machine like to say that “despite the lower energy, the power of the accelerator is comparable to that of CEBAF.” What does that mean? The CEBAF machine with 6 GeV (Gigavolts) and 300 microamperes has a beam power of 1.8 Megawatts. The FEL in the past has operated with currents up to 9 milliamperes, with an energy of 100 MeV (Megavolts). Multiplying those together, we arrive at a power of 0.9 Megawatts. So, indeed, the FEL and CEBAF are comparable in power and, of course, because the energy is quite a lot lower, the FEL is quite a lot higher in current and thus electrons.

We talked above about the charge per day delivered by the CEBAF machine. All that charge comes from the cathode in the electron gun. In the FEL, one cathode yielded, over three years, a total of 7,000 Coulombs of charge. Around May 5, with 2.5 milliamps in the beam, 90 Coulombs were delivered in 10 hours and, overall, about 130 Coulombs have been delivered since the high intensity test run started on April 29.

So, big numbers indeed which directly reflect the capability of our two machines to impact science.

P.S. - For the accelerator junkies: the FEL delivers nearly a megawatt of beam power when the wall plug power available to the RF is closer to a half-Megawatt. That is because the machine is also an energy recovering linac (ERL) and the beam power is recovered as the beam is decelerated. Also, the CEBAF electrons sent to Q-weak were 85-89 percent polarized and without that added feature, and the ability to flip the spin at 960 times a second, no amount of beam would make the experiment a success.