Counting Cosmic Rays through the passage of matter By Edwin Antillon.

Energy loss of charged particles (collision loss) Inelastic collisions with atomic electrons of the material and elastic scattering due to nuclei. Collision with atomic electrons are far more probable and it’s the main component of energy loss by collisions. (radiation loss) Radiation loss by Bremsstrahlung radiation occurs from deceleration influenced by the presence of the electric field from the nucleus. Only for electrons is this radiation substantial since it is proportional to  m – 2. Cherenkov radiation can also be experienced by particles travelling faster than light on that same medium, however cherenkov radistion is small as compared to collision loss. (hard component) Muons mostly lose energy through collision loss by ionizing atoms on their path due to their large mass. (soft component) Electrons experience both radiation and collision loss.

If there are more than one scattering target, we need the number of scattering centers per unit volume. (N) Density(  )  At.Weight (A) = # moles / unit volume Then, N = N a  A Collision Energy Loss and Stopping Power

The energy dissipated depends on the product of the density x thickness = “Mass thickness”. Equal mass thickness has the same effect on same energy. Energy loss due to Cherenkov radiation is already included in the stopping power formula as seen in the logarithmic and  and  dependence. Bethe-Bloch Formula:

Stopping Momentum vs Mass Thickness For a charged particle of given energy, the minimum amount of mass thickness neccesary to stop the particle (Range) can be calculated. Stopping momentum as a function of Range are plotted on the right (due to Rossi) The lack of overlap for the two metals shown comes from the fact that the Z/A ratio for aluminum is about 17% bigger than Lead.

Flux vs. Energy Muons result from the interaction of primary Cosmic Rays with the atmosphere and their subsequent decay of produced particles. An integral distribution spectrum of muons as a function of energy at sea level is shown to the right (due to Sandstrom) Our estimation of the minimum energy obtained from a given mass thickness, provides the expected flux of particles.

Electron / Positron loss Electrons/Positrons lose energy additionally by Bremsstrahlung radiation. The critical energy corresponds to the energy where radiation and collision losses are equal. Ec=1600 m e c 2 / Z (60 MeV for Al) Electron above this energy can give rise to Electromagnetic showers.

Electromagnetic showers A high energetic photon produces pair production (e -+ pair) and this pair in turn radiates energetic photons via bremsstrahlung on average after one radiation length. (1 rad. length (t) =9cm for aluminum,.56 cm for Lead) So number of particles produced is N ~ 2 t and the initial energy decreases as ~ Eo/2 t Since the cascade stops at the critical energy Ec, then the max. number of particles produced N~ Eo / Ec.

A closer approach A closer fit on the number of particles produces is given by Leo [Eq ] From the flux rates below a fit was done to find the dependece on Energy. > 10 Mev 30/ (m^2 s sr) >100 Mev 6 / (m^2 s sr) >1000 Mev 0.2 / (m^2 s sr) Shower development has to be taken into account Where N is the number of shower particles at a depth t

Theory and Experiment. VariableMininumMaximun Mass thickness (R) Composite density ~ Aluminum density r dx = (2.75 g/cm 3 ) ( 45 cm) = 124 g/cm 2 r dx = (2.75 g/cm 3 ) ( 100 cm) = 275 g/cm 2 Radiation Length (Al) 1 Lrad = 8.9 cm Lrad = 5Lrad = 11 Mass thickness /m  c g cm -2 /10 8 eV260 g cm -2 /10 8 eV Momentum (p)3.1 m  c = 330 MeV /c5.6 m  c = 590 MeV /c Kinetic Energy (KE) m  c 2 = 160 MeV2. 20 m  c 2 = 230 MeV Integral muon flux= 122 / (m 2 s)= 105 / (m 2 s) Integral electron flux = 3 / (m 2 s)= 1 / (m 2 s) Total Flux= 126 / (m 2 s)= 118 / (m 2 s) Counts = total Flux * area (=3.92E.2 m 2) = 300 per min.= 280 per min. Observed rates= 270 counts per min= 270 counts per min.

PositionExpected counts/ m 2 s. Observed Roof180 +_ 5%148 +_12 4 th floor116 _ nd Floor90_+ 5%104 _+ 10 Basement  PositionMuonElectronTotalobserved Roof th Floor