The scattering of slow neutrons by atoms is considered, assuming that, in addition to the ordinary nuclear forces, there is a magnetic interaction between the neutron and the atomic electrons. It is found that the neutrons scattered from an unpolarized beam will be partially polarized in virtue of this magnetic interaction. Since the scattered intensity depends not only upon the intensity, but also upon the spin density of the incident beam, the polarization thus produced can manifest itself by a second scattering. An expression is derived for the neutron intensity after double scattering from magnetized iron plates. Under optimum conditions, it is found that the scattered intensity with parallel orientation of magnetizations is 15 times that with antiparallel orientation. The partial polarization of the scattered neutrons indicates that the undeviated neutron beam will also have a nonvanishing spin density. Expressions are derived for the intensity and spin density of a neutron beam after traversing a certain thickness of magnetized iron. These results are used in the discussion of three types of experiments for producing and detecting a polarized beam of neutrons.