Spin segregation of nonrelativistic charged particles by a helical magnetic field slab

We study a beam of nonrelativistic spin-$\frac{1}{2}$ charge carriers incident normally on a finite slab of helical magnetic field aligned along the beam axis. Starting from the Pauli--Schrödinger equation, we apply a unitary transformation expressed in terms of Pauli matrices to map the system to a spatially independent eigenvalue problem, from which the energy band structure is obtained analytically. Our results reveal two types of bandgap controlled by the field strength: a partial bandgap present for all field values, in which only one spin channel propagates, and a total bandgap that opens above a threshold field strength, in which both channels are evanescent. Using the eigenvectors and a transfer matrix approach, we solve the boundary value problem at the vacuum-magnetic interface and obtain spin-preserving and spin-flip transmittances and reflectances as functions of field strength and energy. The spin-filtering properties are discussed in terms of the alignment of the spin eigenstates with the effective field in spin space, with the Bloch sphere providing a geometric picture of the spin-channel selection mechanism.