Why do different atomic isotopes have different atomic spectrums

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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?

Best Answer

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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