Redo problem 3 of homework 5 for extra credit. (Pair creation problem.)

Interactions of Particles with Bulk Matter

Process | Description |
---|---|

ionization | passing particle ionizes an atom, leaving behind an electron and positive ion. |

scintillation | passing particle excites (or ionizes) an atom which emits light (visible or u.v.) as it returns to an unexcited state |

showers (electromagnetic and hadronic) | cascade of secondary particles created when particles that interact electromagnetically and strongly pass through high A/Z material. |

Cerenkov radiation | emission of light (visible and u.v.) when the speed of a particle, βc, exceeds the speed (phase velocity) of light in a medium, c/n where n is the index of refraction (analogy of sonic boom for a vehicle exceeding the speed of sound) |

transition radiation | emission of light (visible and u.v.) when a particle passes from one medium to another of differing EM properties |

particle | lifetime |
---|---|

e^{+}, e^{-} | infinite |

p, pbar | infinite |

γ | infinite |

μ^{+}, μ^{-} | 2.2×10^{-6} s |

π^{+}, π^{-} | 2.6×10^{-8} s |

K^{+}, K^{-} | 1.2×10^{-8} s |

K^{0}_{S} | 0.89×10^{-10} s |

K^{0}_{L} | 5.2×10^{-8} s |

Λ, Λbar | 2.6×10^{-10} s |

These are the particles we are concerned with measuring.
Other particles that we may be interested in -- such as W^{±}, Z^{0}, t quark, D^{0}, B^{0} -- eventually decay into the particles above, sometimes after several intermediate stages.
We learn about the unstable particles by reconstructing them from their decay products.

Electromagnetic and hadronic showers.

We'd like to know whether a charged track is due to a e^{+}, μ^{+}, π^{+}, K^{+}, or p.
This is the task of particle identification.
There are several ways to differentiate particle types:

- differences in the way they interact -- electrons readily bremstrahl and compton scatter, pions, kaons, and protons interact strongly, muons do neither.
- differences in mass.
- sort out the possibilities by the kinematics of a decay.

The differences in the way particles interact show up in calorimetry. Let's consider how to "weigh" a particle to determine its mass. From the measurement of track curvature we determine p = βγm. If we can measure another quantity that depends differently on mass, then we should be able to determine the mass from these two measurements.

- β is the speed of the particle and is found by measuring the time-of-flight of the particle of some distance.

As a prelude to the calculations for a particle traversing bulk matter, let's consider the situation where a particle of mass m, four-momentum p_{i}, scatters from atomic electrons which we'll treat as being at rest.
After the collision, the particle has 4-momentum p_{f} and the electron is ejected from the atom with 4-momentum p_{e}.
We can solve this situation in general, but we are specifically interested in the case of maximum momentum transfer.
This occurs when the electron momentum is in the same direction as the momentum of the incoming particle.
In this situation, conservation of energy/momentum gives:

|**p**_{i}| = |**p**_{f}| + |**p**_{e}|

and
(p_{i}^{2} + m^{2})^{½} = (p_{f}^{2} + m^{2})^{½} + (p_{e}^{2} + m_{e}^{2})^{½}

Φ_{col}(E_{i},E_{f})dE_{f} = 2Cm_{e}c^{2}dE_{f}[E_{i}/{E_{f}(E_{i} - E_{f})} - 1/E_{i}]^{2}

or by factoring out the first term in the square brackets:

Φ_{col}(E_{i},E_{f})dE_{f} = 2C {m_{e}c^{2}E_{i}^{2} dE_{f}/((E_{i} - E_{f})^{2} E_{f}^{2})} [1 - E_{f}/E_{i} + (E_{f}/E_{i})^{2}]^{2}

Φ_{col}(E_{i},E_{f})dE_{f} = 2C (m_{e}c^{2}dE_{f}/E_{f}^{2}) [1 - E_{f}/E_{i} + (E_{f}/E_{i})^{2}]^{2}

Φ_{col}(E_{i},E_{f})dE_{f} = (2Cm_{e}c^{2}/β^{2}) (dE_{f}/E_{f}^{2}) [1 - β^{2} (E_{f}/E_{f max})]

Φ_{col}(E_{i},E_{f})dE_{f} = (2Cm_{e}c^{2}/β^{2}) (dE_{f}/E_{f}^{2}) [1 - β^{2} (E_{f}/E_{m}) + ½{E_{f}/(E_{i} + mc^{2})}^{2}]

Φ_{col}(E_{i},E_{f})dE_{f} = (2Cm_{e}c^{2}/β^{2}) (dE_{f}/E_{f}^{2}) [(1 - β^{2} (E_{f}/E_{m}))(1 + (1/3)(E_{f}/E_{c})) + (1/3)(E_{f}/(E_{i} + mc^{2}))^{2}(1 + ½(E_{f}/E_{c}))]

where EThe detection of particles relies on a transfer of energy from the incoming particle to the bulk medium of detection.

Copyright © Robert Harr 2003