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In this paper, the charge flipping method is proposed for ab initio structure determination using neutron diffraction data alone. For this purpose, a new variant of the dual-space iterative algorithm is introduced, which is called band flipping. Unlike the basic algorithm, it reverses the sign of scattering density only within a zero-centred band, develops large plateaus without forcing positivity, and often leads to Babinet solutions. Its phasing power was tested on two organic structures. These behave similarly when using X-ray diffraction data and the basic algorithm but, with neutron data and band flipping, their solution becomes orders-of-magnitude more difficult and strongly dependent on the hydrogen content. Surprisingly, when the constraint of positivity is added, convergence speeds up to the point where structure determination using neutron diffraction data is not more difficult than the X-ray case. However, by following the evolution of the R factor, such a solution can be easily missed, and band flipping must be used both as a probe of convergence and as a tool for developing negative densities. Apart from demonstrating the feasibility of charge flipping for ab initio neutron crystallography, the present study also leads to an important byproduct: the type of traps that occasionally block the iterative process are identified and a mathematical analysis of their origin is given.

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