Quantum Diaries

Author’s note: I didn’t mean for this to end up so complicated that it had equations, figures, and footnotes, but that’s how it turned out. I do apologize for the inconvenience, and if it’s any compensation I can assure you that about half the footnotes are funny.

I’ve written before about how a pixel detector works, but at the time I left as a “topic for another day” the broader question of what a pixel detector is for.  I’m going to answer one part of that question today, and discuss the tracking system, of which a pixel detector is one possible component.¹ I’ll have to leave the question of the specific advantages of using pixels, as opposed to other tracking technologies, for another other day.

Regardless of the technology used, the basic idea of a tracker is to put together a bunch of stuff that measures the path a charged particle has taken.   The “stuff” could be silicon, in which electron-hole pairs are separated as the charged particle passes through, and can be used to produce a current, as I explained in my pixel detector entry.  It could also be gas, in which case electron-ion pairs are separated and produce a current in wires; this is the technology used in the ATLAS Transition Radiation Tracker.  If you want to “track” a baseball through the stands, the “stuff” is people: even if you can’t see the baseball in the crowd on other side of the stadium, you can see where it’s gone by who stands up or jumps down and starts grabbing under the seats.  An individual jumping person, or silicon pixel producing a current, is what we call a hit.

Our primary interest actually isn’t in how particles move through the detector, even though that’s what we directly measure.   So let me take a step back now and describe what we are measuring, first and foremost: momentum.

Momentum: What It’s Really All About

The best way I can think of to describe momentum in a few words is to quote Newton and call it the “quantity of motion.”² It reflects not just the speed and direction (i.e. velocity) of an object, but also the amount of stuff (i.e. mass) that makes up that object.  In ordinary life, if you double the mass then you double the momentum, and if you double the velocity you get double the momentum too; in other words:

• p = mv

where m is the mass, v is the velocity, and p is the momentum.³ Unfortunately, things get a little more complicated when the particle goes really fast, which they usually do in our detectors; then the equation doesn’t work anymore.  We’ll get to one that does in a minute.

Momentum intuitively seems the same as energy of motion, but technically the ideas aren’t exactly the same, and it just so happens that the difference is important to how the LHC detectors work.  One way to think of the energy of a particle is as follows: if you slammed the particle into a big block of metal and then extracted all the ensuing vibrations of the metal’s atoms⁴ and put them in a usable form, it’s the amount of mechanical work you could do.  In fact, that’s exactly what a detector’s calorimeter does, up to a point.  It’s made of big blocks of metal that absorb the particle’s energy, and then it samples that energy and turns it into an electrical current — not so we can do any kind of work with it, but just so we know how much energy there was in the first place.  So the calorimeter is the piece of ATLAS or CMS that measures the energy of particles and absorbs them; the tracker, by contrast, measures the momentum of particles and lets them pass through.   These two pieces of information are related by the following equation:

• E² = p²c² + m²c⁴

where p and m are still momentum and mass, E is the energy, and c is the speed of light.  The intuitive understanding of this equation is that the energy of a particle is partially due to its motion and partially due to the intrinsic energy of its mass.  The application to particle detectors is that if you know the mass of a particular particle, or if it’s going so fast that its energy and momentum are both huge so that the mass can be roughly ignored, then knowing the energy tells you the momentum and vice versa — and knowing at least one of the two is critical for analyzing where a particle might have come from and understanding the collision as a whole.  We have both kinds of systems because they have different strengths — for example, some kinds of particles don’t get absorbed by the calorimeter, and some kinds of particles (the uncharged ones) can’t be seen in the tracker — and together, they cover almost everything.

(By the way, the second equation is relativistic; that is, it’s compatible with Einstein’s Theory of Relativity.  That means it always works for any particle at any speed — it might assume that space is reasonably flat or that time really exists, but these are very reasonable assumptions for experimental physicists working on Earth.  For those who haven’t seen the equation before and enjoy algebra problems: what famous equation do you get if you take the special case of a particle that isn’t moving, i.e. with a momentum of zero?)

Particle Motion and Momentum

The next ingredient you need to understand what a tracker does is something I haven’t mentioned yet: the whole thing is enclosed in a huge solenoid magnet, which produces a more-or-less uniform magnetic field pointing along the direction of the LHC beam.  As a charged particle moves through a magnetic field, the force exerted on it by the field works at a right angle to both the direction of motion and the field — I tried to illustrate this in figure 1, where the magnetic field is pointing into your screen if you assume the particle is positively charged.⁵ This means that as the charged particle flies from the center of the detector, it curves (figure 2).  The amount it curves by is inversely proportional to the momentum, which means that higher-momentum particles curve less.  Along its path, it leaves hits in the detecting material, as I discussed above (red dots, figure 3).  Finally, in a process called track reconstruction, our software “connects the dots” and produces a track — which is just our name for “where we think the particle went” (figure 4).

You’ll notice that figure 2 looks a lot like figure 4, but the conceptual difference is a very important one.  The red line in figure 2 is the actual path followed by the particle, which we don’t see directly, while the black line in figure 4 is our track as determined by detector hits.  If we do our job right, the red line and black line should be almost exactly the same, but that job is complex indeed — literally thousands of person-years have been put into it, including two or three Seth-years⁶ spent on detector calibration and writing automated tools for making sure the tracking software works properly.

The detector is shown here with only three layers.  Although this would be enough to find a particle’s path in ideal circumstances, we actually have many more: this allows us to still make good measurements even when one layer somehow doesn’t see the particle, and to get a final result for the path that’s more accurate.  And don’t forget that there will actually be many particles passing through the detector at the same time — so we need lots of measurements to be sure that we’re seeing real tracks and not just a bunch of “dots” that happen to “line up”…!

More Than Just Momentum

If you measure the path of a particle, you can do more than just find its momentum; you can also see where it came from, or at least whether it could have come from the same place as another particle.  Pixel detectors excel at making accurate measurements to figure out this kind of thing, but as I said already, to do that subject justice will require another entry.

So there you have it.  In a very broad sense, that’s what I’m working toward when I talk about calibrating the pixel detector.  Tracking provides critical basic information about every charged particle that passes through our detector; combined with data from the calorimeter and the muon systems, this information is what will let ATLAS and CMS measure the properties of the new particles that we hope the LHC will produce.

¹ Both ATLAS and CMS have one, but many other detectors at colliders do not, because the technology is complex, relatively new, and expensive.
² See Corollary III here for what he says about it, if you like your science extra-opaque.
³ I’m really not sure why we always use p for momentum, although a good guess seems to be that it’s related to impetus or impulse.
⁴ A friend of mine, who has the mysterious superpower of understanding how bulk matter works rather than just mucking about with individual particles, looked at a draft of this and was very concerned that I’m implying that all the energy from such a happening would end up as atomic vibrations. So let the record show that this probably isn’t true. And now, if you’d be so kind, can we pretend it is true? It will make illustrating my point very much easier. Thanks!
⁵ The particle is definitely not actual size, and don’t ask me why it’s green.
⁶ A Seth-year doesn’t make nearly as big a contribution as a year of work by any of our real experts, but they do happen to be of particular interest to me.

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