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We know that different elements have different atomic spectrums as a result of the difference in charge and electron shielding that occurs when extra protons are added to a nucleus.

We also know that deuterium was discovered in 1931 by Harold Urey as a result of the different atomic spectra between hydrogen-1 and hydrogen-2. As there is no change in charge within the nucleus, why does the atomic spectrum change?

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When we solve the Schrödinger equation for the hydrogen atom, we find that the energy levels are

$$ E_n = -\frac{\alpha^2 m c^2}{2 n^2} = -\frac{\rm 13.6\,eV}{n^2} $$

where $\alpha \approx 1/137$ is the fine structure constant and $c$ is the speed of light. We usually approximate $m$ as the electron mass, but that's actually wrong. The correct mass parameter is the "reduced mass" $\mu$ of the electron-nucleus system, which obeys

$$ \frac 1\mu = \frac1{m_\text{e}} + \frac1{m_\text{n}} = \frac 1{m_\text{e}} \left( 1 + \frac{m_\text{e}}{m_\text{n}} \right) $$

The extra neutron in deuterium roughly doubles the nuclear mass, which changes $\mu$ (and therefore $E$) starting in its fourth or fifth significant figure.

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  • $\begingroup$ Does this mean that the relative change between the spectra of hydrogen-3 and hydrogen-2 is less than that of hydrogen-2 and hydrogen-1 because going from hydrogen-1 to hydrogen-2 roughly doubles the mass but going from hydrogen-2 to hydrogen-3 only increases the mass by ≈ 1.5x? $\endgroup$ Oct 8 at 13:28
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    $\begingroup$ That's right — though in a real spectrometer you'd mostly be comparing tritium to protium. $\endgroup$
    – rob
    Oct 8 at 13:34
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    $\begingroup$ There is also a contribution from the (small) change in the nuclear charge distribution. In hydrogen it is tiny compared to the mass effect, but at large Z this is the dominant effect. $\endgroup$
    – CWPP
    Oct 8 at 14:23
  • $\begingroup$ @CWPP Can you give a reference for such very large hyperfine effects? $\endgroup$
    – my2cts
    Oct 8 at 19:49
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    $\begingroup$ How does the effect of the mass difference compare to that of the spin-spin interaction? $\endgroup$
    – Sandejo
    Oct 9 at 2:12

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