Physics 535 lecture notes: - 4 Sep 13th, 2007
Reading: Griffiths Chapter 2, Perkins Chapter 2 handed out in class.
Homework: Griffiths: 2.2, 2.4, 2.5, Perkins 2.4
1) Hints for homework.
Griffiths 2.4: A key elements in many high energy physics relativity problems is energy and momentum conservation. You should choose a frame, rest frame of a particle or lab frame as appropriate and then equate energy and momentum before and after the process. Also remember that m2 = E2 +p2 and since E = mc2 and p = mv then v = pc2/E = p/E.
In almost all cases you can avoid velocity transformations, which can make the problem quite complex.
Griffiths 2.5: How likely a process is will be determined by how many vertices there are in the process, what coupling constants are present at each vertex, and whether there are any additional multiplicative factors such as CKM matrix factors which make it less likely to change flavor between generations. The best way to approach a problem like this is to write down the starting and ending quarks on either end of your diagram and then try to figure out what goes on in-between. An interesting example related to the problem is D0 -> pi+pi compared to D0 -> K+K-
Perkins 2.4: Approach this problem using the same hints as above. Note that the solution they ask for makes the assumption that the mass of the incoming particles can be neglected. As an alternative to the crossing angle part of the problem you can try to solve for exact solution of the first part not neglecting the mass.
2) Units hbar=1 c=1?
c = d/t, for a particle traveling at c if c=1 then distance if distance is in cm then time is how long it takes the particle to travel 1cm
E=hbar 2pi/, if hbar=1 a photon with wavelength 2pi has an energy of 1.
The real convenience of this notation is that when dealing with mass, momentum and energy they all have the same values since c=1. Then it’s easy to see that given 2Tev of momentum you can convert that into 2TeV on energy or particle with 2TeV of mass.
For conversion back a useful number is that hbarc = 197 MeVfm
Thus a particle of rest mass m = 197MeV/c2 or mc2 = 197 MeV has a Compton wavelength of hbar/mc = hbarc/mc2 = 1fm
Also hc = 1240 eVnm
Thus for a photon with a wavelength of 1nm E = hc/ = =1240eV
3) Experimental particle physics.
We have new mysteries, but also better experiments to explore them.
In addition to cosmic ray experiments we can build accelerators where accelerate charged particles up to very high velocities and momentums. Then we annihilate electrons and positrons, protons and antiprotons or collide protons and protons to create other particles.
Remember that protons, neutrons, electrons, neutrinos, muons, pions and kaons are the only particles that have a long enough lifetime to be easily observed. These are the typical things we try to detect in our particle physics experiments. The whole experiments are built around the properties of these particles and how to measure the momentum or energy of the particles.
Particle detectors from inside to outside:
a) The tracker. A device immersed in a magnetic field, which will be used to measure the momentum of the particle by its curvature in the field. Particle going through a gas or a solid will ionize the material by exchanging photons, which leaves the detectable track. These ions are what are detected. Also the tracks can be traced back to find if they came from a common vertex and thus could be from the decay of a more massive particle.
There are two typical types of tracking detectors. Drift chambers: gas filled chambers with wires for detecting ions and silicon semiconductor sensors with metal strips for detecting ions.
What is the physics?
i) Radius of curvature of a charged particle track in a solenoidal magnetic field. r = 0.3Bp where p is the momentum in GeV/c, B is in Tesla and r is the radius of curvature in meters.
ii) Ionization charge drifting. The ionized particles drift toward wires that are held at high voltage. Measuring the drift time if the typical velocity is know can give a much more accurate measurement.
iii) The ionization process is a scattering interaction. Probability is proportional to 2 or e4. See Perkins page 39. dE/dx has an incident particle mass dependence at relativistic speeds that can be used to distinguish the particle type.
iv) Ionization is much larger in semiconductors, which makes them an excellent particle detector material since you can design a small volume detector that can precisely determine the position of particles.
b)Time of flight(TOF). Precise time measurements from the scintillation light of atoms excited by the passing particle. More massive particle move slower for the same momentum.
i) Momentums are measured in the tracker and velocity in the TOF and relativistic equations used to calculate the mass.
ii) The active material is usually a scintillating material, which is ionized by particles traversing the detector. The ionized electrons are recaptured by atoms emitting photons, which are detected by photomultipliers.
c) Cerenkov detectors. Identify particles by Cerenkov light.
i) When the particle velocity exceeds the speed of light in the medium v > c/n Cernkov light is emitter at an angle cos = c/vn. The particle velocity can be measured by measuring the angle or at least said to be above a certain value if the light is detected.
d) Electromagnetic calorimeter. Electrons and photons will interact electromagnetically in matter. This makes a cascade or shower of electrons and photons that can be measured to determine the total energy. muons and hadrons(pions kaons protons…)tend to go through these detectors without interaction.
i) Electrons will interact via brehmsstralung producing photons. Photons will pair product making electrons and positrons. These are 3 probability interactions proportional to e6. However, having a dense material with a high Z will make these interactions more likely. The secondary particles will continue to interact until all the electrons produce photons that are two low energy to ionize new electrons and these photons are measured by photomultipliers.
ii) Good materials for this type of detector are characterized by the radiation length, X0, the length of detector material electrons or photons will traverse before 1/e of the particles interact, and the Z of the material. You would like small X0 and large Z.
iii) Also once you are down to lower energy photons you would like to have a transparent material. Lead glass is a good example.
iv) These interactions are also characterized by the length, L, until the shower is done which is proportional to the energy of the incident particle.
e)Hadrons will interact strongly in dense matter such as iron and the energy of the resulting particles can be measured. Muons will go through without much interaction.
i) These interactions will happen for charged or neutral hadrons since they all have color charge.
ii) The scale of interaction length or absorption length, 0, is different since we are talking about the strong interaction. These lengths are actually larger since the hadrons have to get near within the typical strong force distance of the nucleus before there is a good probability of interaction.
iii) The interactions produce new charged and neutral hadrons with can undergo a shower of secondary interactions. The energy that is eventually detected is from ionization caused by the many charged hadrons produced in the shower. Also the lightest neutral meson, a pion, will decay electromagnetically to two photons which can be detected. A substantial portion of energy is lost to nuclear breakup which, can be detected if you material is U238 which will undergo fission when hit by fast neutrons and the photons from the fission process can be detected.
iv) Since you typically use a material like iron or Uranium which is not that transparent you need to alternate layers of metal and scintillator which are used with photomultipliers to collect the light.
f) Muon systems typically sit outside the large amount of metal of the hadronic calorimeter. They may consist of drift tubes, small version of the drift chamber to spot the charged muons. You can even put them in a field to measure momentums.
e)If you measure everything then any left over energy you expected is probably lost in neutrino form.