# Why is the spectrum of helium different from that of hydrogen?

Jun 14, 2018

Well, is it not just a different atom, with more than one electron? The spectrum of helium must be more complex, because now angular momentum becomes a factor to which transitions are allowed; it must change by $1$ each time.

Examples:

$1 s \to 2 p$ ($\text{58.4 nm}$)
$2 s \to 3 p$ ($\text{501.6 nm}$)
$2 p \to 4 d$ ($\text{492.2 nm}$)
$2 p \to 4 s$ ($\text{504.8 nm}$)

The energy levels of the hydrogen atom are well-known:

${E}_{n} = - \text{13.6058 eV} \cdot {Z}^{2} / {n}^{2}$

where $Z = 1$ for hydrogen atom.

Those for helium have no straightforward formula, but are known experimentally.

Using Excel, and the energy levels of helium given numerically here (estimating the $4 s$ and $5 s$), I've superposed them next to those of hydrogen:

These energy level gaps are different, and since transitions between them lead to a spectrum, the spectrum is of course also different...

To be fair, I ignored the $2 p$, $3 p$, $3 d$, $4 p$, $4 d$, and $4 f$ energy levels, which ARE present AND split away from the $s$ levels in helium (but are degenerate in hydrogen), because they are too subtle on the above scale:

That occurs because having two electrons in helium introduces electron correlation, which splits levels of different angular momentum, because they no longer have spherical symmetry.

Beyond that difference, which is readily seen in multi-electron atoms having, e.g. orbital potential energies ${V}_{2 s} \ne {V}_{2 p}$, ${V}_{3 s} \ne {V}_{3 p} \ne {V}_{3 d}$, etc., we can see the following trends:

• The lowest energy levels become lower for heavier atoms, knowing that the energy levels depend directly on atomic number squared.
• The lowest energy levels become more spread out from the rest, in larger atoms, obviously because bigger atoms have a greater effective nuclear charge ${Z}_{e f f}$, which is most easily seen via the most significant attraction of the core energy levels ($n = 1$ in both atoms).

Because the energy level gaps widen, we expect to see shifts in electronic transitions towards lower wavelength for helium compared to hydrogen.

(Indeed, the $1 s \to 2 s$ transition is $\text{58.4 nm}$ for helium compared to $\text{121.5 nm}$ for hydrogen.)