How this instrument works
A hydrogen-like ion is any nucleus of charge Z stripped down to exactly one electron — plain hydrogen (Z = 1), singly-ionized helium He⁺ (Z = 2), doubly-ionized lithium Li²⁺ (Z = 3), all the way up to a bare uranium nucleus holding onto its very last electron. Because there is still only one charge pulling on one charge, Bohr's original balance of Coulomb attraction against centripetal force applies exactly as before; only the strength of the pull changes. Raise the nuclear charge by a factor of Z and the orbit radius at a given n shrinks by that same factor, while the potential energy holding the electron in grows by another factor of Z — multiply those two effects together and the bound-state energy scales as Z², not Z: E_n = −13.6·Z²⁄n² eV.
Spectroscopists lean on that scaling to read plasma conditions they cannot reach any other way. Stripping an iron atom down to its last electron takes several tens of millions of kelvin, so when an X-ray telescope catches the sharp emission line of hydrogen-like iron — near 6.97 keV, the ion's own Lyman-alpha — coming from a solar flare or the disk around a black hole, that single line certifies the plasma is that hot, because nothing cooler could produce it. At the opposite end of the periodic table, physicists at heavy-ion storage rings deliberately hold onto the single electron of hydrogen-like uranium, U⁹¹⁺: it is the cleanest one-electron system nature allows at high Z, and comparing its measured energy against this formula's prediction is how bound-state quantum electrodynamics gets tested under the strongest electric fields available in a laboratory.
The formula's simplicity has a price: it demands exactly one electron. Give the same nucleus a second one, as in ordinary neutral helium, and electron-electron repulsion breaks the clean Z² pattern immediately — that two-electron problem has no exact closed-form solution at all. Even for genuine one-electron ions the match is only non-relativistic; the electron in hydrogen-like uranium moves at a meaningful fraction of the speed of light, and relativistic corrections that scale as (Zα)² — utterly negligible for hydrogen, where Z = 1 — grow large enough by Z = 92 that measuring the gap between this Bohr prediction and the true energy is itself the physics experiment.
- Set Atomic number (protons) to the nuclear charge Z of the one-electron species — 1 for plain hydrogen, 2 for He⁺, 26 for hydrogen-like iron.
- Set Principal quantum number to the shell the lone electron occupies; 1 is the ground state, higher integers are excited states.
- Read Orbital energy, eV — the instrument evaluates E_n = −13.6·Z²⁄n² the moment either input changes.
- Compare two shells of the same ion by reading Orbital energy, eV twice at different n and subtracting to get a transition energy.
- Compare two different ions at the same shell by holding Principal quantum number fixed and changing Atomic number (protons) instead — that isolates the Z² scaling.
Worked example — singly-ionized helium's ground state
Set Atomic number (protons) to Z = 2 and Principal quantum number to n = 1 — singly-ionized helium, He⁺, the simplest hydrogen-like ion beyond hydrogen itself, with its lone remaining electron sitting in the ground state. The formula gives E₁ = −13.6 × 2² ⁄ 1² = −13.6 × 4 = −54.4 eV, so Orbital energy, eV reads −54.4. That is four times deeper than hydrogen's own −13.6 eV ground state — exactly what the Z² factor predicts — and the number matches He⁺'s measured spectral lines closely enough that it has served as a precision check on the Bohr picture since the 1920s.
Raise Atomic number (protons) to Z = 3 for doubly-ionized lithium, Li²⁺, and move Principal quantum number to n = 2, its first excited state: E₂ = −13.6 × 9 ⁄ 4 = −30.6 eV. That sits deeper than hydrogen's own n = 2 level of −3.4 eV — Li²⁺ pulls nine times harder — but shallower than Li²⁺'s own ground state, since climbing from n = 1 to n = 2 always loosens the grip, regardless of which nucleus is doing the pulling.
Questions
What exactly is a hydrogen-like atom?
Any nucleus with exactly one electron still attached to it: hydrogen itself (Z = 1), singly-ionized helium He⁺ (Z = 2), doubly-ionized lithium Li²⁺ (Z = 3), and onward up to a bare heavy nucleus holding its very last electron, such as hydrogen-like uranium U⁹¹⁺. Because only one charge is pulling on one charge, the same simple orbit-and-energy picture Bohr built for hydrogen applies unchanged — only the nuclear charge Z in the formula is different.
Why does the energy scale with Z squared rather than Z?
Two separate effects each contribute one power of Z. A stronger nuclear charge pulls the electron into a tighter orbit, shrinking the orbit radius by a factor of Z; that same stronger charge also deepens the potential well at any given radius by another factor of Z. Multiply a smaller radius by a larger charge and the two factors compound into Z² in the final energy — which is why He⁺'s ground state is four times deeper than hydrogen's, not merely twice.
Does the Z² formula work for neutral helium or other multi-electron atoms?
No. It requires exactly one electron orbiting the nucleus. Add a second electron, as in ordinary neutral helium, and that electron repels the first with its own Coulomb force, a term this formula has no room for — the clean closed-form solution disappears, and the true energy levels have to come from a numerical or perturbative calculation instead. The moment a species keeps even one electron more than the bare-nucleus-plus-one setup, this instrument no longer describes it.
Where does this formula actually get used?
In X-ray astronomy, the sharp Lyman-alpha line of hydrogen-like iron near 6.97 keV tells observers a plasma — a solar flare, or the disk around a black hole — has been heated past tens of millions of kelvin, since nothing cooler strips iron down to one electron. In atomic physics, storage-ring experiments hold onto the single electron of hydrogen-like uranium precisely because it is the cleanest one-electron system available at very high Z, ideal for testing quantum electrodynamics.
Why does the prediction drift for very heavy hydrogen-like ions?
Because the formula is non-relativistic, and it stops being a safe approximation once the electron moves fast enough for relativity to matter. That correction scales roughly as (Zα)², where α is the fine-structure constant — utterly negligible for hydrogen at Z = 1, but sizeable for a nucleus like uranium at Z = 92. Physicists exploit exactly that drift: measuring the gap between this formula's prediction and the real energy is how bound-state quantum electrodynamics gets tested in the lab.