We present a theoretical formalism for scattering of the twisted neutrons by nuclei in a kinematic regime where interference between Coulomb interaction and the strong interaction is essential. Twisted neutrons have definite quantized values of an angular momentum projection along the direction of propagation, and we show that it results in novel observable effects for the scattering cross section, spin asymmetries and polarization of the scattered neutrons. We demonstrate that additional capabilities provided by beams orbital angular momentum enable new techniques for measuring both real and imaginary parts of the scattering amplitude. Several possible observables are considered, for which the targets may be either well-localized with respect to the spatial beam profile, or the scattering occurs incoherently on nuclei in a bulk target. The developed approach can be applied to other nuclear reactions with strongly interacting twisted particles.
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