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How did people determine the proton number for each element from experiment in each decade of 20th century?

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Moseley, the physicist who 'fixed' the Periodic Table at the start of the 20th Century, did it by measuring X-ray spectra. The energy of the $K_\alpha$ X-ray emission line is proportional to $(Z^2 - 1)$, where $Z$ is the atomic number.

The results of Moseley's experiment fitted his formula so perfectly that he was able to predict the existence of several as-yet-undiscovered elements by looking at the gaps in his graphs. He also re-ordered the controversial placement of nickel and cobalt. Sadly he was killed in World War One before he was able to become the great scientific figure he surely would have been.

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This is certainly true, but doesn't Rutherford scattering at large angles give you the nuclear charge too, and in a more direct manner? –  Ron Maimon Aug 3 '12 at 5:10
    
Yes, and I'm sure there are other ways I haven't thought of (I know, for example, the X-ray scattering form factor depends on Z). I only gave one example in my answer because it seems superfluous to give several, and historically this was an important one. Not sure what you mean by 'more direct' though. –  Benjamin Hodgson Aug 3 '12 at 8:56
    
More direct in that it doesn't involve an uncontrolled inner-shell screening approximation, replacing Z by Z-1. The Rutherford scattering is clean at large angles, because it's off the nucleus itself. The x-ray scattering might be confounded by electronic scattering. –  Ron Maimon Aug 3 '12 at 17:43
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