Astroparticle Physics

7.3 Cosmic Rays Underground 153[ a −γN(> E,R) = Ab (ebR − 1)]. (7.12)For high energies (E µ > 1TeV, bE ≫ a) the exponentialdominates and one obtains( a) −γN(> E,R) = A e −γbR . (7.13)bFor inclined directions the absorbing ground layer increaseslike 1/ cos θ = sec θ (θ – zenith angle) for a flat overburden,so that for muons from inclined directions one obtainsa depth–intensity relation ofinclined muon directions( a) −γN(>E,R,θ)= A e−γbRsec θ . (7.14)bFor shallower depths (7.12), or also (7.9), however, leads toapowerlawN(> E,R) = A(aR) −γ . (7.15)The measured depth–intensity relation for vertical directionsis plotted in Fig. 7.19. From depths of 10 km water equivalent(≈ 4000 m rock) onwards muons induced by atmosphericneutrinos dominate the muon rate. Because of thelow interaction probability of neutrinos the neutrino-inducedmuon rate does not depend on the depth. At large depths(> 10 km w.e.) a neutrino telescope with a collection areaof 100 × 100 m 2 and a solid angle of π would still measurea background rate of 10 events per day.The zenith-angle distributions of atmospheric muonsfor depths of 1500 and 7000 meter water equivalent areshown in Fig. 7.20. For large zenith angles the flux decreasessteeply, because the thickness of the overburden increasesFig. 7.19Depth–intensity relation for muonsfrom vertical directions. Thegrey-hatched band at large depthsrepresents the flux ofneutrino-induced muons withenergies above 2 GeV (upper line:horizontal, lower line: verticalupward neutrino-inducedmuons) [2]Fig. 7.20Zenith-angle distribution ofatmospheric muons at depths of1500 and 7000 m w.e.