How did Antonius Van den Broek come up with his hypothesis that the atomic number matches the charge of the nucleus?

Question

Antonius van den Broek was the first person to suggest that the atomic number (which at that time only referred to an element's position in the periodic table and had no physical interpretation) of an element was exactly equal to the charge of its nucleus, then later Henry Moseley found evidence that the atomic number is a more fundamental quantity (Moseley's Law), thus making Van Den Broek's statement more likely to be true.

My question is how did Broek even come up with his hypothesis? He published his hypothesis in 1911. The idea of 'individual' protons of unit charge (hydrogen nuclei) making up the nucleus was discovered in 1919. in 1911, the model of the atom was essentially 'positive dense nucleus in center surrounded by electrons.'

According to rutherford's model (1911), most of the mass is concentrated in the center and is positively charged, so I think it makes sense for scientists at that time to believe that the charge of a nucleus increases with atomic weight, as heavier atoms would likely have heavier nuclei and therefore likely to have more charge. But how did Van Den Broek think that the value of the charge (in electron units) was EXACTLY equal to its atomic number (or roughly half its atomic weight, why not 1/4th, or twice, etc)

Were there experiments (in 1911 or prior) that showed that the charge of a nucleus is roughly half its atomic weight?

Answer 1

Van den Broek's train of thought is described in detail by Scerri in The Gulf between chemistry and philosophy of chemistry. His starting points in the 1907 article were Rutherford's remarks on the nature of $\alpha$ particles creatively combined with the old Prout's hypothesis of 1815 that all atoms are made from "protyles" (from Greek proto+hyle, prime matter), i.e. copies of hydrogen atom.

The $1/2$ came from the experiments of Rutherford and Barkla, who suggested the approximate proportional relationship between atomic charges and weights with this coefficient. But van den Broek made a further connection with the ordering of elements in the periodic table that his more eminent sources missed. Although the idea is already present in a 1911 paper, it is entangled there with his "cubic periodic table" and a modification of Prout's hypothesis, both of which he soon abandoned. The clearest formulation appears in a 1913 paper cited by Bohr:"The serial number of every element in the sequence ordered by increasing atomic weight equals half the atomic weight and therefore the intra-atomic charge".

Finally, after examining experimental data of Geiger and Marsden published the same year, van den Broek noticed that the ratio of $α$-particle scattering per atom to the square of the charge showed much better constancy than its ratio to the square of atomic weight. So he took the final step and dropped the shaky middle term of atomic weight:

"If now in these values the number M of the place each element occupies in Mendeléeff’s series is taken instead of A, the atomic weight, we get a real constant (18.7 +/− 0.3); the hypothesis proposed holds good for Mendeléeff’s series, but the nuclear charge is not equal to half the atomic weight".

Here are some relevant excerpts from Scerri's account:

"One of Rutherford’s three suggestions was that an α-particle might be half of a helium atom with a charge of twice the hydrogen atom. This idea appealed to van den Broek who coined the term alphon to describe such a particle. He then proposed that such an alphon particle might better serve the role that Prout had intended for the hydrogen atom in his famous hypothesis of 1815. According to van den Broek, the atoms of the elements should therefore consist of a series corresponding to the even whole numbers from 2 up to 240 such that there should be a total of 120 elements, each one made up of a whole number of alphons each with a weight of 2 units.

Van den Broek’s reason for this proposal rested on experiments by Rutherford and Charles Barkla who had independently concluded that the charge of any atom is approximately half of its atomic weight. Van den Broek’s article of 1907 does not yet show any direct signs of the concept of atomic number, unless one divides each of the atomic weights in van den Broek’s table by two to obtain a sequence of values from one to 120.

In 1911... van den Broek published a remarkably short and suggestive statement in Nature magazine... Stated otherwise, van den Broek is suggesting that since the charge on an atom is half of its atomic weight, and since the weights of successive elements differ by 2 units in a stepwise fashion, the charge on an atom defines its position in the periodic table. Neither Rutherford, Barkla, nor anybody else had considered the elements in the periodic table as a whole, and consequently, they had missed this key point, whereas Rutherford and Barkla recognized that $\textrm{charge}\approx A/2$ van den Broek went further in seeing that, $\textrm{charge}\approx A/2=\textrm{atomic number}$.

In 1913... the most significant development took place in another article that van den Broek placed in Nature magazine in which he abandoned the connection with atomic weight altogether. Van den Broek began with a set of experiments by Hans Geiger and Ernest Marsden, aimed at examining the ratio of scattering of α-particles per atom in several elements. According to Rutherford, this ratio needed to be constant, but this is not what Geiger and Marsden found when they divided the scattering by the atomic weight of each element. Nevertheless, Geiger and Marsden were not concerned about this discrepancy, believing that the error was small and could be ignored. Meanwhile, van den Broek got to work trying to make the ratio more constant. He achieved this by dividing the amount of scattering for each element by its charge rather than by its atomic weight."