MARKUS KLUTE: Welcome back to 8.701.
So in this section, we'll look at tracking detectors.
And before we look at tracking detector technologies,
we want to remind ourselves how we measure the momentum
of a charged particle.
And this measurement is possible because charged particles are
reflected in magnetic fields.
We have already seen that in a homogeneous magnetic field,
a particle follows a circle.
And so from the measurement of the radius
and the knowledge of the magnetic field,
we can infer the transverse momentum of the particle.
Typical particles also have a longitudinal momentum.
And so therefore, the trajectory is actually
of the form of a helix.
And so we then can get back to the total momentum
of the particle by knowing the angle or the component
of the longitudinal and transverse momentum
and then just calculate total momentum from there.
So we have seen that particles, when
they go through a piece of material, that the energy loss
through ionization or bremsstrahlung.
And so what we have to do now is put some material
in the way of the particles without really changing
its momentum or changing its trajectory.
So this is typically done with tracking detectors.
And so what you then measure is not just the radius directly,
but you measure individual points
that's part of the trajectory of the particles.
And so from the measurement of the points,
you can then reconstruct the trajectory of the particles,
and then therefore measure the curvature.
And that has been used to measure
the transverse momentum.
So if you look at this picture here,
the radius can be given as L squared over 8s plus s half.
s is typically called the sagitta of the trajectory.
If L is much, much larger than s, it simplifies to r--
about L squared over 8s.
And the uncertainty on the sagitta the uncertainty on s,
is limiting the uncertainty on the momentum measurement.
And this is described in a paper which
is very nice from the '60s.
And the formula is called the Gluckstern formula.
So what you see here is that the more measurements you actually
perform along the sagitta, the better your measurement
is performed.
And this goes with the square root of 1
over the number of measurements.
The total momentum uncertainty, or the relative momentum
uncertainty, sigma pt over pt, is
proportional to the uncertainty in the sagitta.
And that also is then proportional to the pt
of the measurement.
So if you want to improve your measurement of your momentum,
you want to decrease the uncertainty,
then you can do this by having a larger L--
and you see this goes with the square of the length--
you want to increase your magnetic field,
and you want to reduce the uncertainty on the second term.
Those are the elements you have in play in order
to improve the transverse momentum,
or the momentum measurement of your charged particles.
So this is a screenshot of this paper.
The measurement error is not the only error
on the transverse momentum, as we discussed previously
already.
Multiple scattering, so the scattering of the particle--
and it goes with momentum-- also reduces the uncertainty
in the momentum measurement.
And we can show that this component
on the relative momentum uncertainty is flat.
So what you find typically is that the transverse momentum
measurement is limited by, at high pt through the measurement
error, and at low pt through multiple scattering.
We have those two components entering the measurement
uncertainty.
So the actual detectors, I just give you
a couple of examples here.
The first one-- and the first one,
which actually really has allowed us to make measurements
with a lot--
devices with huge statistic experiments where
you have colliding beams, and you
look at many, many of those interactions
was a multi wire proportional chamber,
at a reasonable resolution.
Typically, we have hundreds or thousands
of wires with a spacing of up to 1 millimeter.
And this 1-millimeter spacing in a wire chamber
limits the spatial resolution.
If you don't have any knowledge of where the particle was
in that area, then the uncertainty of the spatial--
on just one hit, uncertainty is given by d,
the spacing between those two wires--
1 millimeter in this case--
divided by square root of 12.
So you find typically resolutions of 300 microns
per measurement.
But you may gain off the fact that you typically have
many, many, many measurements.
And as you just saw, this reduces the uncertainty
on the measurement.
You knowing that there was a hit on a wire
gives you two-dimension information.
If you then built the wire chamber such
that there's angles between individual wires,
you can use that in order to gain
three-dimensional information in the geometry of where
the particle went through.
Wire chambers can be operated in different modes.
And it depends on the level of voltage you apply on the wires.
So the higher the voltage--
the level of the voltage goes down here,
see a very limited voltage applied, then the ions--
you have ionization, and the ions
recombine without being collected.
In ionization mode, you collect the ions.
And there's typically a gain factor of 1.
You measure each ion, basically, separately.
You can increase the voltage and run the wire chamber
in proportional mode so you have a gas multiplication
factor so the ions are being accelerated,
and then they produce many more, more ions to be measured.
And you typically have gain factors of 10 to the fourth.
If you drive this further up, you
go into the limited proportional mode, and then in Geiger mode,
where basically you have an avalanche,
and so you have no information about the initial ionization
anymore.
And typically, the chamber breaks down
and needs to recover afterwards.
The signal formation is quite an interesting concept.
So you think that you do ionization,
the electrons are there, and then they're being collected.
Yes, that's true, and this happens very rapidly, typically
in the order of nanoseconds.
But the actual real signal comes from the ions themselves.
And so what happens here is you have your wire here,
you have your electrons.
The ions then themselves, they build this cloud of charges.
And this cloud of charges then gives you a signal on the wire.
And this happens typically via influence,
or you basically have a mirror charge on the conductor.
And that is really where the bulk of the information
comes from.
We just discussed that the spatial resolution is limited
by the spacing of the wires.
You can actually get better measurements
by using the timing information.
If you are able to measure the time profile of your signal,
that profile has information about the distance
between the ionization and the wire itself.
And that then helps you to improve the resolution further.
So I really don't do justice to this,
but nowadays, specifically in high dense--
particle-dense environments, you use solid-state tracking
detectors.
And they're used in many, many, many areas in particle physics.
The idea is to use mostly silicon,
but you can use other semiconductors as well.
You dope them, meaning that you create
more holes by using p-doping and n-type doping for more
electrons.
And then you bring n-doped and p-doped materials together.
And you-- you know, it's the same way as in a diode
in that then you apply a voltage in the opposite way
to create a very large depleted zone.
So in this depletion zone, there is no electrons
and no holes available.
But when a charged particle travels through,
like is shown in this picture here,
you create electron-hole pairs, and they then
can be used in amplification to create a signal.
That's the general concept in which solid states or silicon
detector [INAUDIBLE].
The nice thing about silicon detector
is that you can read them out with very
fine spatial resolution.
So from this 300 micrometers we just saw,
you go down to several micrometers
in spatial resolution when you measure particles
going through the detectors.