= Isotope shift of a linear rotor
Changing an <isotope> changes a <mass> and therefore the <moment of inertia>. If the <bond length> is held fixed, $B\propto1/\mu$ and every corresponding transition shifts by the same fractional amount. This leading inverse-<reduced mass> scaling is also used in the more complete diatomic formulation.<Diatomic Spectral Database by National Institute of Standards and Technology/1Σ-Ground State Molecules>[§2a, equations (4)–(5): isotope dependence of rotational constants and reduced mass]{ref}
For a deliberately stipulated <carbon monoxide> example, take $m_{12}=12u$, $m_{13}=13u$, $m_{16}=16u$, $u=1.66053906660\times10^{-27}\,\mathrm{kg}$, $r=1.13\times10^{-10}\,\mathrm m$ for both isotopic forms, and $h=6.62607015\times10^{-34}\,\mathrm{J\,s}$. The integer <masses> are simplifying inputs, not accurate isotopic atomic <masses>; the shared <bond length> is an imposed model assumption. Calculation gives
$$
\begin{aligned}
\mu_{12,16}&=\frac{48}{7}u,& I_{12,16}&\approx1.45395\times10^{-46}\,\mathrm{kg\,m^2},& B_{12,16}&\approx57.7188\,\mathrm{GHz},\\
\mu_{13,16}&=\frac{208}{29}u,& I_{13,16}&\approx1.52080\times10^{-46}\,\mathrm{kg\,m^2},& B_{13,16}&\approx55.1817\,\mathrm{GHz}.
\end{aligned}
$$
In particular,
$$
\frac{B_{13,16}}{B_{12,16}}=\frac{\mu_{12,16}}{\mu_{13,16}}=\frac{87}{91}\approx0.956044.
$$
The heavier form's calculated $J=0\to1$ line is $110.363\,\mathrm{GHz}$ rather than $115.438\,\mathrm{GHz}$, a decrease of about $4.396\%$. These are model predictions from the stated inputs, not published measured <frequencies>. An <isotope> changes the spacing as well as the first line; the shift is not a fixed additive offset for the whole spectrum.
\Image[/molecular-rotation-isotope-and-distortion-spectra.png]
{title=Synthetic <rotational spectroscopy> comparison: a heavier <isotope> compresses the rigid spectrum, while <centrifugal distortion> bends the spacing pattern}
{id=image-molecular-rotation-isotope-and-distortion-spectra}
{description=Original equal-height sticks for $J=0\to1$ through $4\to5$, using only the stipulated <masses> and <bond length> above. The lower panel additionally stipulates $D/B=0.001$ for the lighter form and compares $2B(J+1)-4D(J+1)^3$ with dashed rigid positions. Neither heights, distortion constant nor line <frequencies> are measured data; no thermal-population or instrument model is included.}
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