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Grotrian diagrams often taught in instrumental methods of chemical analysis to chemists. After a decade, I finally found the original book of 1927 by Grotrian "Graphische Darstellung der Spektren von Atomen und Ionen mit ein, zwei und dreivalenzelektronen" It turned out it is a classical German style handbook with these diagrams for each element. A question which always bothered me as a chemist, is how do we experimentally assign electronic transitions from an atomic spectra for complex atoms. What was the procedure, which seems to be established as early as 1920s? I searched so many texts but nobody talks about experimental procedure of making assignments. For example in this diagram for Cs? How do we experimentally determine or calculate that a spectral line of 894 nm is a transition from 1s to 2p in Cs atom or a 1002 nm is transition from 3d to 4f?

Thanks.

Cs spectrum

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$\let\l=\lambda$ AFAIK It was a long and complex undertaking. You can't look at a single atomic species by itself. The understanding of level assignments to series slowly grew starting from the simplest cases.

Of course the first was hydrogen, where - at least initially - no series are discernible but one - there is almost perfect degeneration on $l$. But that helped, as it allowed the first step, the one from wavelengths $\l$ (or wave numbers $k=1/\l$) to terms, later identified with energy levels. It was Ritz' combination principle: $$k = T_m - T_n.$$ Already at this introductory level a difficulty was to be overcome: the difference between emission and absorption spectra. The former are much richer. This was explained as a consequence of atoms in absorption initially being in ground state alone, whereas in emission photons are emitted by excited atoms jumping to a lower state, not necessarily the ground one. For hydrogen this is especially notable, as there are no absorption lines in the visible region - the Balmer series has $n=2$.

Then alkaline spectra were explored. In absorption spectra only lines $T_n - T_1$ are observable, where $T_1$ is what later would become the ground state energy (divided by $hc$). Note that here $n=1$ means ground state, but this is not the principal quantum number of hydrogen-like classification, which is 2 for Li, 3 for Na, etc.

The main difference between hydrogen and alkaline spectra is that in the latter $l$ degeneracy is broken. This required to separate terms in several series. Surely you know the origin of symbols S, P, D,... related to a different appearance of lines starting from different series towards the same final term. An instance is transitions S-P and D-P (in emission), In absorption only S-P transition is visible, giving rise to the principal series (therefrom P-terms).

But a second feature appears in alkaline spectra: the so-called "fine structure". It exists in hydrogen lines too, but is much less prominent, whereas the famous sodium doublet requires a modest resolving power to be seen. Its interpretation required the discovery of electron spin and of L-S coupling. The effect was a doubling of columns for all series, S excepted.

Well, this is a rough history, as I'm able to follow. Maybe a book could be of help: G. Herzberg, "Atomic Spectra and Atomic Structure". It's outdated as far as theory is concerned, but thanks to its publication date (first edition 1936) is nearer to the times when the facts were happening.

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