# Why do different atomic isotopes have different atomic spectrums?

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?

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.

• 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? Oct 8 at 13:28
• That's right — though in a real spectrometer you'd mostly be comparing tritium to protium.
– rob
Oct 8 at 13:34
• 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.
– CWPP
Oct 8 at 14:23
• @CWPP Can you give a reference for such very large hyperfine effects? Oct 8 at 19:49
• How does the effect of the mass difference compare to that of the spin-spin interaction? Oct 9 at 2:12