A Velocity Filter for Electrons and Ions

If a charged particle, moving along the $x$-axis with a velocity $v$, encounters successively two identical alternating electric fields of frequency $\nu$, which are everywhere perpendicular to the $x$-axis, it will emerge from the last field undisplaced and traveling in the original direction under the following conditions: (1) Each field has two similar halves whose distance between centers is $a$; (2) The distance between the fields, center to center, is $D=\frac{\mathrm{sa}}{n}$, where $s$ and $n$ are odd integers; (3) The velocity of the particle is $v_0=\frac{2a\nu }{n}$. The distribution about the velocity $v_0$ is computed when particles enter and leave the system through slits of width $y_0$ on the $x$-axis. The results show that the emergent beam can be confined to a verv narrow velocity range.