Solenoidal time-of-flight energy window ($K\in [K_0,K_0+q^2{B}^2{a}^2/(8m)]$)

A nonrelativistic particle accepted only at a fixed flight time through an ideal uniform-field tube has fixed parallel speed $v_{\parallel }=l/t_0$. On-axis entry and wall clearance bound the independent perpendicular speed by $|q|Ba/(2m)$, provided a full gyration occurs before exit. Adding parallel and perpendicular kinetic energies gives the displayed ideal energy window, where $K_0=m{v}_{\parallel }^2/2$. The upper endpoint is a grazing limit; strict wall exclusion makes it a supremum.