 Aston velocity focusing

In Mass spectra and isotopes by Francis William Aston, printed pp. 10–12, Fig. 1, Francis William Aston describes successive electric and magnetic deflections in one plane, with opposite senses. This geometry focuses a range of ion speeds; it does not require the straight-through cancellation of a velocity selector.

The reason compensation is possible follows from the Lorentz force. In a short region of length ${\ell }_E$, a small electric deflection angle has magnitude ${\theta }_E\simeq qE{\ell }_E/(m{v}^2)$: the transverse acceleration acts for a time ${\ell }_E/v$, and the deflection angle is the acquired transverse velocity divided by $v$. A short magnetic region of length ${\ell }_B$ gives ${\theta }_B\simeq qB{\ell }_B/(mv)$, since the magnetic force already contains one factor of speed. Slower ions therefore receive a disproportionately larger electric deflection.

An original local approximation makes the focusing idea explicit. At fixed mass-to-charge ratio, replace each region by a small angular kick and include the drift distances in two positive constants $a$ and $b$. With $t=v/v_0$, write the detector coordinate asChoosing the geometry and fields so that $b=2a$ eliminates the first-order speed sensitivity. Indeed,A fractional speed error $\epsilon =t-1$ then contributes only at order ${\epsilon }^2$ in this approximation. This is a local thin-kick explanation, not Aston's exact pole geometry, focal-plane equation, or a prediction of its measured mass resolution. Finite slit widths, initial directions, and field boundaries still matter. Unlike a velocity selector, focusing can collect nearby speeds into a narrow image instead of discarding all but a narrow accepted range.