The power irradiated by the Earth through its surface is about 40 TW. A large fraction of this energy is due the decay of radioactive isotopes. The most important are ²³⁸U, ²³²Th and ⁴⁰K. In the ‘standard’ model built by the geologists, the Bulk Silicate Earth Model, radioactivity is assumed to contribute 50% of the total power,i.e. with 20 TW. The model contains important uncertainties and can be tested by detecting the electron antineutrinos produced in the decays. Their flux can be measured by observing the process ν̅_e + p → e⁺ + n in a medium containing free protons such as a liquid scintillator. The energy threshold is E_ν ≥ 1.8 MeV (see Problem 10.8). The maximum antineutrino energy is E_{ν, max} = 3.26 MeV for 238 U,E_{ν, max} = 2.25 MeV for ²³²Th and E_{ν, max} = 1.31 MeV for ⁴⁰K, so only U and Th produce antineutrinos above threshold. The total antineutrino flux at the surface dueto ²³⁸U and ²³²Th is expected to be Φ_ν = 3.5 x 10¹⁰ m^-2 s^-1. It is evaluated that the fraction P_ee  ≈  0.6 will reach our detector as ν̅_e due to the oscillations. Moreover,only the fraction f ≃ 0.05 of the flux is above detection threshold. Assume an average cross-section value σ ≃ 10^-47 m². We take as scintillator a blend of 20% PXE (C₁₆H₁₈) and 80% dodecane (C₁₆H₂₆) as proposed by the LENA proposal. Calculate the sensitive scintillator mass necessary to observe 1000 events.