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This study was motivated by the recently proposed concept of remote detection of concealed radioactive materials by a focused terahertz (THz) radiation [V. L. Granatstein and G. S. Nusinovich, J. Appl. Phys. 108, 063304 (2010)]. According to this concept, a high-power THz radiation should be focused in a small spot where the field intensity exceeds the breakdown threshold. In the presence of free electrons in such a breakdown-prone volume, a THz discharge will occur there. However, this volume should be so small that in the absence of ionizing sources in its vicinity the probability to have there any free electrons is low. Then, the increased breakdown rate in a series of THz pulses would indicate the presence of hidden radioactive materials in the vicinity of the focused spot. For this concept, it is important to accurately determine the breakdown-prone volume created by a focused THz radiation. This problem is analyzed in this paper, first, for the case of a single wave beam and, then, for the case of crossing wave beams of different polarizations. The problem is studied first ignoring the diffraction spread of wave beams in the vicinity of the focal plane and, then, with the account for the diffraction spreading. Then, relations between the THz wave power, the range of such a system and the breakdown-prone volume are analyzed. Finally, the effect of the atmospheric turbulence on propagation and focusing of THz wave beams in air is considered.