Theory

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Theory:

When muons are created in the upper atmosphere they have a mean lifetime of only 2.2 x10-6 seconds.  Even though muons move at a velocity of .9994c they would never make it to the earthís surface if not for the effects of time dilation.  Since the particles are moving at a velocity so close to that of light their lifespan is, in the earthís frame of reference, stretched to 6.35 x10-5 sec.  This allows them to reach the surface of the earth where we will be able to make measurements of them.

When particles, such as a muon, travel through a material they lose energy because of interactions with the electrons of the material.  Interactions with the nucleus are negligible because of the relative size of the nucleus to that of the electron cloud.  This energy is lost when the muon ionizes the atoms.  The energy lost is described by the Bethe-Blocke equation; on which our experimentís analysis will be based:

                                                     (1)

This equation relates the term dE/dx, (which is the rate of energy loss of the particle with respect to the distance that it has traveled through the material), to the energy the particle had upon entering the material.  The energy term is found in the  and  terms, where                                       

                                                      and = .                                              (2)

Here v is the muons velocity, and c is the speed of light.  This equation takes into account the material in question with the materials atomic number and mass, Z and A respectively.  The term Tmax≠≠ is the maximum energy lost by the muon in any collision with the electron.  I is the mean excitation energy, which is the average energy a muon losses by an interaction with the electron.  The  term is a correction that is not applicable at the energy range we are measuring.  A table of the other constants of the material used in this equation can be found in note 1.

The Bethe-Bloch equation predicts the following relationship for -dE/dx of a muon passing through aluminum to its energy:

(a)

(b)

 

Figure 1:  (a) The Bethe-Bloch equation shown for a particle passing through aluminum.  We measured the stopping power of the muons over the energy range of 100-130 MeV.  (b) The Bethe-Bloch equation solved for numerous materials. 

 

Viewing the graph of the function is much more informative than the rather cumbersome equation.  It shows that slower moving particles, (i.e. particles with lower energy), lose energy more quickly as a function of distance.  This is due to the fact that slower moving particles when passing through a material have more time to interact with the electrons.  Thus, these particles lose more energy to ionization.  The graph of the Bethe-Bloch equation shows that the minimum value of -dE/dx, the MIP (minimum ionizing particles) point, is found at approximately the same momentum for various materials.  The rise in the stopping energy following the MIP point is due to relativistic effects. 

            We solved the Bethe-Bloch equation to find the muons range through aluminum as a function of incoming energy.  The energy term is found by writing v in terms of E:

                                                                ,                                                         (3)

where M is the rest mass of the muon.  Solving for v yields:

                                                          .                                                 (4)

Using these expressions and the expressions of  and  given in equation 2, dE/dx can be written in terms of E, and solved for range as a function of energy: 

                                                ,                                      (5)

here, Emin is the energy needed for a muon to penetrate a material, and dE/dx is determined from the Bethe-Bloch equation.  The Bethe-Bloch equation is too difficult to evaluate; the solution to this problem is to solve for the range numerically.  By using the trapezoid approximation we solved for the range in increments of 20 MeV, by evaluating dE/dx at each energy level.  This yields:

            Figure 2:  The range of a muon through aluminum as a function of its energy.

a particles range through a material increase with its incoming energy.