= Space focusing
<Space focusing> in <time-of-flight mass spectrometry> reduces differences in arrival time caused by the initial positions of otherwise equivalent <ions>. An <ion> starting farther from the exit of an accelerating region spends longer there but can acquire greater <kinetic energy>, allowing it to catch up during the drift. Electrode geometry and <electric fields> are chosen so that these effects compensate at the detector.<Time-of-Flight Mass Spectrometer with Improved Resolution by W. C. Wiley and I. H. McLaren>[pp. 1151–1154; initial space distribution and ion-gun focusing analysis]{ref}
A simple idealized derivation makes the compensation explicit. Consider positive <ions> of <mass> $m$ and <electric charge> $q$, initially at rest, with distance $a$ remaining in a uniform accelerating <electric field> $E$. Let $L$ be the field-free distance beyond its exit. Their total flight time is
$$
t(a)=\sqrt{\frac{2ma}{qE}}+L\sqrt{\frac{m}{2qEa}}.
$$
The first term is acceleration time; the second is drift time. Setting the <derivative> $dt/da$ to zero at a reference position $a_0$ gives
$$
\left.\frac{dt}{da}\right|_{a_0}=0\quad\Longleftrightarrow\quad L=2a_0.
$$
For example, $a_0=1\,\mathrm{cm}$ gives a first-order space focus at $L=2\,\mathrm{cm}$ in this simplified single-field model. Nearby position errors then affect arrival time only at higher order in the <Taylor expansion>. Practical two-field sources allow the focus to be placed at more useful drift distances. Initial velocity spread is a separate contribution to peak width, motivating <time-lag focusing>.
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