Heisenberg antiferromagnetic (HAF) S = ½ chains are famous for their gapless magnetic excitation spectrum. Interestingly, in SrCuO2 and Sr2CuO3, which are some of the best realizations of such chains, a spin pseudogap was found to open upon introduction of impurities. While studying SrCuO2 with neutrons we came up with an explanation for the pseudogap based on theoretical treatment of broken chains which reproduced the data very well. Unfortunately, SrCuO2 has a complication of the chains being arranged in pairs via weak and frustrated ferromagnetic coupling and the role of such a double-chain structure is unclear. In order to avoid this ambiguity and to understand the origin of the spin pseudogap, we propose to study Sr2CuO3 which has well isolated single-chains. The key question of the proposed experiment is how the size of the pseudogap is related to the concentration of impurities.