= Magnesium isotope mass spectrum
For a modern chosen example, the <National Institute of Standards and Technology> lists neutral <magnesium> <atomic masses> $23.985041697\,\mathrm u$, $24.985836976\,\mathrm u$, and $25.982592968\,\mathrm u$, with reference <isotopic abundances> $0.7899$, $0.1000$, and $0.1101$. They describe the <isotopes> with <mass numbers> 24, 25, and 26, respectively.<Atomic Weights and Isotopic Compositions for Magnesium by National Institute of Standards and Technology>[NIST, Atomic Weights and Isotopic Compositions for Magnesium: rows 24–26, neutral relative atomic masses and reference compositions]{ref} These modern inputs are not a reconstruction of historical observations.
Use the <dimensionless> <mass-to-charge ratio> coordinate $x=(m/\mathrm u)/z$. For the illustration only, approximate <ion> <masses> by the neutral <atomic masses>. The singly charged positions are then near $23.98504$, $24.98584$, and $25.98259$; the doubly charged companions are near $11.99252$, $12.49292$, and $12.99130$. The neutral-to-ion correction already given above is essential at higher accuracy. A peak near $x=12$ alone need not identify a <chemical element>: another <ion> with a different <electric charge> can occupy the same <mass-to-charge ratio> interval.
\Image[/magnesium-charge-states-and-resolution.png]
{title=An original synthetic <magnesium> <mass spectrum>: charge states, finite <mass resolution>, and response bias}
{id=image-magnesium-charge-states-and-resolution}
{description=All curves are stipulated sums of unit-area <Gaussian functions>, not measured data. The first two panels use equal isotope responses and selected <full width at half maximum> values $0.12$ and $1.20$ in $x$. The last panel keeps the reference <isotopic abundances> fixed but multiplies the mass-25 response by two, then normalizes the areas. Neutral <atomic masses> approximate <ion> <masses>; charge-state yields are chosen independently, and no historical intensity or resolving-power claim is made.}
For the narrow singly charged peak at $x\approx24.98584$, the chosen <full width at half maximum> $\Delta x=0.12$ gives <mass resolving power> $R\approx208.2$; widening to $\Delta x=1.20$ gives $R\approx20.82$. Adjacent isotope spacings here are approximately one unit in $x$. Peaks are visibly separated in the first chosen profile and overlap strongly in the second. A small uncertainty in a fitted peak centre would not by itself make the overlapping components resolved.
Even integrated peak areas need a response model. For isotope fraction $f_i$, let $\eta_{i,z}$ collect the probability of <ionization> into <electric charge> state $z$, transmission through the analyzer, and the detection response. In a linear instrument with overall exposure factor $C$, the component area is
$$
A_{i,z}=C f_i\eta_{i,z}.
$$
Equal $\eta_{i,z}$ at a fixed $z$ makes normalized areas equal the fractions; otherwise it does not. In the chosen example, doubling only the mass-25 response changes the normalized areas to $0.71809$, $0.18182$, and $0.10009$, although the neutral sample is unchanged. Photographic darkening introduces another possible nonlinearity: neither line darkness nor peak height is a universal measure of neutral <isotopic abundance>. Calibration of position and calibration of intensity answer different questions. Matched standards, a known <electric charge> assignment, and a response correction make isotope analysis useful; a visually impressive set of peaks alone does not supply those controls.
Back to article page
