In the OPERA experiment a beam is produced at CERN and aimed to the detector located at Gran Sasso at a distance L

In the OPERA experiment a ν_μ beam is produced at CERN and aimed to the detector located at Gran Sasso at a distance L = 730 km (CNGS = CERN Neutrinos to Gran Sasso). The experiment is designed to detect the appearance of tau neutrino by detecting the CC reaction ν_τ + n → τ^- + p. Neutrino energies are spread in a wide spectrum (see Problem 10.6), but we will assume all the neutrinos to have the same energy E_ν = 18 GeV.

a) Calculate the neutrino energy threshold.

b) Assuming a neutrino flux at Gran Sasso integrated over one year of running N_ν = 4.3 x 10⁸ m^-2, calculate the number of ν_μ CC interactions per year in a Pb target of mass M_t = 2000 t assuming the cross-section σCC = 10^-41 m². 

c) How many ν_τ CC interactions per year are expected in presence of oscillations with θ₂₃ = 45°, θ₁₃ = 0°, Δm² = 2500 m_eV²?

d) If it were θ₁₃ = 7°, how many ν_e CC interactions per year?