= Time-of-flight mass spectrometry
<Time-of-flight mass spectrometry> separates <ions> by measuring how long they take to reach a detector after a timed acceleration pulse. The simplest instrument has an <ion> source, an accelerating <electric field>, and a drift tube kept under <vacuum>. For <ions> of equal positive <electric charge> accelerated through the same <electric potential difference>, smaller <mass> gives greater speed and earlier arrival.<Time-of-Flight Mass Spectrometer with Improved Resolution by W. C. Wiley and I. H. McLaren>[Review of Scientific Instruments 26, p. 1150, General Description]{ref}
Let $U$ be the accelerating <electric potential difference>, $q$ the positive <electric charge>, $m$ the <mass>, and $L$ the field-free drift length. Neglecting initial speed and the time spent accelerating, conservation of <energy> gives
$$
qU=\frac12 mv^2,\qquad t=\frac{L}{v}=L\sqrt{\frac{m}{2qU}},\qquad \frac{m}{q}=2U\left(\frac{t}{L}\right)^2.
$$
For an ideal $1\,\mathrm{m}$ drift at $1000\,\mathrm{V}$, a singly charged <ion> of <mass> $100\,\mathrm{u}$ arrives after approximately $22.8\,\mu\mathrm{s}$; $400\,\mathrm{u}$ takes $45.5\,\mu\mathrm{s}$. Here $\mathrm{u}$ is the <unified atomic mass unit>. Quadrupling <mass> doubles the ideal flight time. A doubly charged $200\,\mathrm{u}$ <ion> has the same <mass-to-charge ratio> as the singly charged $100\,\mathrm{u}$ <ion> and therefore the same ideal arrival time.
Real peaks broaden because <ions> begin at different positions and with different <kinetic energies>. In 1955, <W. C. Wiley> and <I. H. McLaren> analysed these effects and demonstrated an ion gun with two accelerating regions that reduced timing differences caused by initial position. Comparing calculated focusing conditions with measured spectra turned a timing idea into a practical improvement in <mass spectrometry>.<Time-of-Flight Mass Spectrometer with Improved Resolution by W. C. Wiley and I. H. McLaren>[pp. 1150–1157; resolution analysis, two-field ion gun and experimental spectra]{ref}
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