Being old, I have studied these things in late 50's in Prague. At that time we students had no idea that computers were something a mere mortal could use for his/her work and it was automatically clear to us that the early researchers of the 20's had only the pen-and-paper technology to do their calculations, plus a lot of amazing enthusiasm to collect large amounts of spectroscopic data (a decent visible-light spectrometer was easy to build, even then). So we students instinctively knew that diagrams like the above were a result of a patient, manual, and 90% empirical work. From the theory one had an overall picture (the pattern of the orbitals and the selection rules for transitions), and from experiments one had a large bunch of transition wavelengths, clasified as strong, weak, and very weak. Since the transitions are many more than the expected energy levels, their set is heavily redundant and fitting it to the expected energy levels pattern is a pleasant crossword puzzle. The solution is generally unique (when not, one went back to the lab and hunted for a missed doubt-solving spectral line). The uniqueness of these solutions was what really made quantum physics win an otherwise widespread mistrust - its opponents simply had no idea how to even start looking for a solution, much less to find one.

Mathematically, this points to a problem of this type: Given a set S of real numbers, what is the minimal (smallest cardinality) set R of real numbers such that every element of S is a difference between two elements of R. I am not sure, but something tells me that this might be an NP problem :-) I am not even aware of any algorithm doing it.

Thank you for your input. As you stated, this is a mathematical problem of finding differences and series. It must be a very known approach of atomic spectra analysis. I also agree that Grotrian must have compiled tons of spectra to make such figures but a surprising observation is that one cannot find a protocol or an example of a simple element. Was the process so well known that nobody wrote it out in detail.

Please look at the following below links which may help you in your analysis:

https://www.utdallas.edu/~goeckner/plasma_tech_class/Glows.pdf

http://cdn.intechopen.com/pdfs/38537/InTech-Electronic_absorption_spectra_of_3d_transition_metal_complexes.pdf

https://www.reed.edu/physics/courses/Physics332.s17/pdf/N2%20Spectrum.pdf

Thanks

The analysis or level assignment is much easier now a days but one must appreciate the Germans who did this initial and very laborious work. Fist recording the emission spectra on the photographic films and the trying to fit the observed lines in to some systematic sequence of energy levels. It all started with the analysis of the simplest atom; hydrogen and particularly the absorption spectrum of hydrogen, the Balmer series. The book you mentioned was published in 1927 and the classical book on the subject is by White published in 1934. It is the contribution of Theoretical understanding and experimental evidence. On purely theoretical side, Hartree published a book in 1934 to calculate the atomic transitions. I enjoyed reading the book by Herzberg, published in 1942.

Aslam

Dear Dr. Aslam, I had copies of Herzberg and White's Atomic Spectra (really nice books) but as far as I remember they do not give any experimental example of spectral assignments. They do show Grotrian diagrams but never explain how they are constructed. It must be laborious work for sure and German scientists must be commended for this type of work. Are you aware of any practical examples / paper (pre-1930s) in which the spectral assignment is taught for simple atoms like lithium etc. I have asked physicists and others but none had a definite answer. For instance, if we have experimental spectrum of lithium in the UV-Vis range, how would be go about making a Grotrian diagram of lithium from scratch?

Dear Farooq,

Let me find some original paper about the way to start the analysis. I have seen a huge collection of atomic emission spectra of different elements that were taken by one of the leading German physicist at Bonn University. In fact, I had my office in that room where the slit of the classical spectrograph was fixed, a very touchy moments. I will try to find the old book by Bohr, published in 1922, there might be some old references. The way Balmer started the assignment of the Balmer series by some assumption, I think now a days it is possible to teach our students how to start the level assignments.

Regards,

Aslam

Dr. Sahab, Thanks. Please share if you happen to find a protocol. It is unique feeling to see the original apparatus and one can only admire the simplicity of those objects. You can find all the classical books including that of Bohr's on Internet Archive. The translation of his three great essays is here https://archive.org/details/ost-physics-theoryofspectraa00bohriala/page/n8. It is also searcheable This a consortium of many US and worldwide libraries to provide free access to everyone.

You can check the NIST atomic spectra Database (https://www.nist.gov/pml/atomic-spectra-database) with atomic and ionic energy levels.

I am not sure if they have a description on how the transitions were identified.

Condon and Shortleys classic book from 1935 (with later editions) or Cowan (1981) probably gives more information.

You might also check Russel and Edlen in History of Science journals, as they worked on this theoretically and experimentally, respectively.

Preston Guynn Yes, Internet Archive is indeed amazing service. I found all the classical books I could ever think of. If you are interested in other resources, there is a Hathi Trust. Sometimes university subscription is required.