= Aston velocity focusing
{c}
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 $\theta_E\simeq qE\ell_E/(mv^2)$: 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 as
$$
X(t)=\frac{a}{t^2}-\frac{b}{t},\qquad
\left.\frac{dX}{dt}\right|_{t=1}=-2a+b.
$$
Choosing the geometry and fields so that $b=2a$ eliminates the first-order <speed> sensitivity. Indeed,
$$
X(t)-X(1)=a\frac{(t-1)^2}{t^2}.
$$
A fractional <speed> error $\varepsilon=t-1$ then contributes only at order $\varepsilon^2$ 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.
\Image[/aston-focusing-and-velocity-selector.png]
{title=Successive opposite deflections in an early <mass spectrograph>, compared with a chosen <velocity selector> and magnetic analyzer}
{id=image-aston-focusing-and-velocity-selector}
{description=Top: functional schematic of the sequence described by <Francis William Aston>; colored paths indicate one <mass-to-charge ratio> at nearby <speeds>. Paths use the local thin-kick illustration, and dimensions and plate orientation are not a reconstruction. Bottom: an independently chosen teaching geometry has overlapping selector fields followed by a separate semicircular magnetic path. Both diagrams show positive <ions>; the <Thomson charge-to-mass experiment> gives the corresponding negative-<electric charge> example.}
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