= Mass spectrometry
{wiki}
<Mass spectrometry> separates and detects <ions> according to <mass-to-charge ratio>. It can distinguish <isotopes> which belong to the same <chemical element> but have different <atomic masses>. The instrument measures <ions>: interpreting the result as a neutral <atomic mass> requires accounting for <Electron> removal and its <ionization energy>. For a singly positive <ion> in its <ground state>, $m_+=m_{\rm a}-m_e+I/c^2$, where $m_e$ is the <Electron> <rest mass>, $I$ is the <ionization energy>, and $c$ is the <speed of light>.
For a simple illustrative analyzer, positive <ions> of <mass> $m$ and <electric charge> $q=ze$ gain <kinetic energy> $qV=mv^2/2$ across an <electric potential difference> $V$ before entering a transverse uniform <magnetic field> $B$. The <Lorentz force> supplies <centripetal acceleration>, so
$$
qvB=\frac{mv^2}{r},\qquad\frac{m}{q}=\frac{B^2r^2}{2V}.
$$
At fixed $V$ and $B$, the measured bending radius $r$ separates <mass-to-charge ratios>, not <masses> independently of <electric charge>. This ideal model assumes negligible initial <kinetic energy> and speeds small compared with the <speed of light>; the historical focusing <mass spectrograph> used the geometry described below.
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