Transmission of ultra-cold neutrons through a weakly absorbing foil

Transmission, reflection and loss probability of ultra-cold neutrons (UCN) per bounce onto a weakly absorbing foil are studied in the framework of the Schrِdinger equation with a complex optical potential. Analytical expressions are given for UCN moving perpendicular to the foil and for an isotropic UCN gas. The formulas include the case, where the optical potentials on both sides of the foil are different. This situation is encountered in a new project of UCN generation through inelastic scattering of cold neutrons in a moderator of superfluid 4He. The source may be employed to fill a magnetic bottle for a neutron lifetime measurement using a new method also proposed here.