I wanted to offer you a paper published exactly 100 years ago, so I went searching for something interesting in the Physical Review issue of October 1920. And I found a paper by some Fernando Sanford, a teacher and science administrator who was among the founders of Stanford University, and a respectable chemical-physicist of his times. The title reads: “On the variation of the factor h in the equation 1/2mv2=hv ”. Already a bit shocking for us “moderns”: wait a minute… variation of the Planck’s… constant ??

The subject of the paper is a reinterpretation of H. Moseley’s experimental analysis of atomic spectra, and led me to dig out a number of interesting quirks. We are in the year 1920, and the atomic model proposed by Niels Bohr (and improved by Arnold Sommerfeld) is still the best offer on the market. However, the amount of things the new model explained is way smaller than the number of questions it left open; DeBroglie and Schroedinger are still 5-6 years down the road, and our colleagues from a century ago are not short of options, in a time rich and bubbling with ideas. This very creative professor Fernando Sanford is already in the last year of his career, during which he invented many curious devices, among which an “electric photography” procedure, one example of which is shown in the top picture. Still a few years before dying at age 95, in 1948, he was still publishing a comment in Science about the adjusted value of the solar constant. In the paper that I present you today, he tries to push forward the consequences of a previous work of his, On the nuclear charges of atoms, he had published 4 years before. So, I ended up studying two papers instead of one.

In that first 1916 paper, Sanford had used Moseley’s empirical equation to estimate the nuclear charges of the atoms. In 1916 the concept of atomic number was still under discussion, and most scientists kept using the atomic weight A as a – not very satisfactory – ordering parameter in Mendeleev’s table. However, five years before (and two years before Moseley’s experiments) the Dutch lawyer and amateur physicist Antonius van de Broeck already surmised that the atomic ordering parameter should coincide with the nuclear charge Z, rather than with “just about half” of the atomic mass; with a strike of visionary genius, he also suggested that the heavier the atom, the worse should be such an empirical correspondence (since, as we know, the number of neutrons grows with increasing atomic charge, therefore the ratio A/Z becomes progressively larger than 1)… all this written in one concise paragraph of less than 200 words appeared in February 1911’s Nature !

Our dear Fernando starts from Einstein’s equation for the photoelectric effect, published 13 years before, entirely neglecting both the relativistic and the (still very young) quantum details. It is  notable that he calls Planck’s constant h the “Einstein factor”, and links the classical kinetic energy of the orbiting electron (remember, we are still at stake with Bohr-Sommerfeld model) with the centripetal force from the nucleus, assumed as a central point charge Q. He thus obtains another equation that, like Moseley’s, relates the “atomic charge” Q to the inverse square root of the X-ray wavelength 𝜆 . Not surprisingly, he finds back the Rydberg relation (but without mentioning it), and henceforth recalculates the values of Q. However, while he states that “[Moseley] did not show what this unit of positive electricity is, nor how the nuclear charge may be computed”, his values of charge are all larger by a factor of about 5/3 than the atomic number Z, as we know it today.

This may seem surprising, since already the first of Moseley’s 1913 papers contained a detailed equivalence between the empirical proportionality constant (which he wrote as 𝜈0), the Rydberg constant R, and the full spelling of R into its fundamental constants, as explained just a few weeks before by Niels Bohr, in his earthshaking Phil. Mag. paper. However, because of this misunderstanding (that could be solved only if you replace the classical kinetic energy, , by Bohr’s energy formula for the atomic levels), Fernando’s values of Q become “almost exact” multiples of some charge value, about 1.86e. Given the finiteness of the charge, such a non-integer figure should have been regarded suspiciously. But instead, inspired by the observation that “the radioactive elements in their transformations give off 𝛼 particles and electrons, and form new elements” and that “the temptation is great to assume that all elements are built […] by the addition of 𝛼 particles”, Sanford then decides that the elementary charge in the nucleus must be exactly twice the charge e of the electron (however a little smaller, 1.86e, for some very complicate reason that he tries to spell out). That is, the Q of any nucleus is a multiple of an 𝛼 particle’s charge, 2e.

Note that this idea of the nucleus as made up of a bunch of 𝛼 particles, was not just a stray bullet at that time, and the possibility of having electrons inside the nucleus had not yet been dispelled. Instead, the idea of a “nuclear electron” was still very popular (see e.g. Rutherford, Millikan, Sommerfeld, Aston…): conveniently enough, electrons living inside the nucleus allowed to neutralize the extra (almost) double charge Q. Given the new ordering based on the atomic number N (not yet our defined as our beloved Z), the only way to accommodate 2 units of positive charge for each unit increase of N, was to suppose some negative charge being added together (that is, two protons plus one electron make a unit charge and a double mass, the mass of the electron being negligible). Incidentally, this same idea led the great and much respected Rutherford to propose the existence of a “neutron” in the nucleus, which still in 1920 (in his Bakerian Lecture) he imagined as a particle doublet, made of a positive and a negative charge.

Now, in the second paper (that is, the one that started my interest in this silly question) professor Sanford goes several steps further. He now starts – more properly – from a relativistic assumption about the variation of mass with velocity (which he calls “Bucherer’s equation”, from Alfred Bucherer who firstly did experiments in 1908 to prove the relativistic increase of mass: now, he really hates to give credits to Einstein!), and observes that with this modification, a more correct dependence of Q on the velocity, and therefore on wavelength, is obtained. But, to his dismal, for increasingly heavier atoms the formula starts again deviating from the experimental results. Of course, we cannot blame Fernando and other scientists of his time, for being unaware of the notion of electron correlations and screening effects in many-electron atoms, which conjure to reduce electronic wavelengths. Therefore, it seems all in all rather reasonable that he wants to keep the (effective) charge of his (and Moseley’s) equation as invariable, and goes to look for the variation of something else. And he makes the wild guess that… Planck’s constant should not be constant indeed! To fit the data, h must vary with the logarithm of the mass… But while such a wrong attempt could be honest-minded, his final conclusion is ideologically biased : since h may be variable, there is no longer a need to assume a quantum of energy in X-ray emission, and in the atomic model at large… and this brings his prow to crash against the rocks. My feeling is that, like many other contemporaries, he didn’t want to bend to accept the “new” (actually, almost 15 years old already) energy quantisation ideas, and he tried to desperately prove that classical physics could explain as much as possible…

Last but not least, in the final part of his second paper Sanford even attempts at applying his same logarithmic formula for the function h(m) to explain the energies of the electrons from beta radioactivity, something that we know to be completely unrelated. He does this based on a seemingly curious analogy: since atomic X-ray transitions are linked (in his view) to Einstein’s photoelectron energies, also the beta electrons from radioactive elements should follow the same “kinetic” origin. In a Pindaric flight of mind, he thus retrieves the same logarithmic mass-dependence of h from fitting 𝛽-electron experimental energies of Radium-B and Radium-C; then (assuming beta decay to be of electromagnetic origin) he calculates a hypothetical electromagnetic wavelength corresponding to each beta electron energy; and surprisingly, several 𝛾-ray wavelengths measured from Ra-B and Ra-C correspond with these calculated values, within a few %.

However, we know that there is no such thing as discrete 𝛽-electron energies… but they should have known as well! In fact, already in 1910, Wilson demonstrated that electrons from radioactive elements have a continuous spectrum. And in 1913, Soddy and Fajans had proved the radioactive displacement law,[1] that clearly showed the existence of phenomena in the nucleus going beyond electromagnetic processes. But – again – this idea was strongly opposed by the community until much later: indeed, based on such a wrong belief in discrete beta-electron energies (which only originated from poorly made experiments) the most logical explanation of the 𝛾 and 𝛽 emission from radioactive elements, at that time, was some form of resonant energy conversion, transferred from the electromagnetic radiation to electrons and vice versa. Rutherford himself firmly believed it (Phil. Mag. 28 (1914) p.305) although he warned that any numerical correspondences between electron energies and gamma radiation wavelengths should have been seen as merely fortuitous.

The logical fallacy that the many people like Fernando Sanford failed to appreciate is that photoelectron energies are linked to the absolute value of the atomic energy levels, while X-ray frequencies are linked to differences between energy levels. This should have been clear from Bohr’s trilogy of papers, which evidently were not yet well received by the community… In fact, Bohr’s formula was rather satisfactory for the frequencies of electronic transitions (of one-electron atoms), yet it is quite wrong about the photoelectron energies: eventually, Bohr’s formula sort of works just because of a cancellation of large errors.

Before leaving you to more constructive occupations, I wish to further notice that Moseley experiments, indeed, nailed down the values of (Z-q), rather than the absolute nuclear charge Z. The q is a constant, different for each electron shell in the atom, which is equal to about 1 for K-electrons, and to about 7.4 for L-electrons. In modern terms, q would be called a screening constant, something that practically nobody knows how to calculate exactly for a nuclear charge Z greater than 4. But then, how do we know for sure that the naked nucleus truly has a charge Z, and not (Z-q)? Of course, there is a vast amount of indirect proofs that should reassure us about that, but… has anybody measured this charge, on fully-ionized heavy atoms? Because, all we can actually measure in a mass spectrometer is the charge/mass ratio: but the absolute mass must be known from the outset if we want to get the absolute value of Z. And measuring atomic masses is another whole department, maybe good for another Sunday story!

Now, X-ray spectroscopy and Moseley-like equations seem like a great step to determine the nuclear charge, since they don’t require to know the mass… Or, do they? Actually yes, they do: what we call Rydberg’s constant is not a constant (again…). It changes for each chemical element, at least in principle. In practical calculations, this is hidden by the customary fact that for heavy elements we routinely use the “infinity” approximation of the Rydberg constant R, which only contains the electron rest mass. But the approximation of using M=infinity for the nuclear mass is a rather crude one, to the (hyper)fine eyes of the spectroscopists.

Eventually, the best way to make sure that the charge of the nucleus is indeed equal to Ze, is to use the theory of Quantum Electro Dynamics. Notably, the calculation of the g-factor for bound electrons is “the” most stringent test, as it depends on a slowly converging series of powers of Z. Today, extremely accurate experiments via the Penning trap apparatus can determine the value of g for one-electron heavy ions with extreme accuracy, to at least 10 significant digits or better.[2] Now, to get at this same level of precision with the theory, you must sum one-electron Feynman diagrams up to at least the fifth power of Z, and the agreement with the experiments is spectacular.

OK: but with just one electron around the nucleus there are no correlation effects… I can sense that you are starting to get tired of reading… but this is something so fundamental that one cannot just be happy with a rough estimate: we gotta beat it right! Well, it turns out that a group of scientists working between Darmstadt, Heidelberg and Mainz [3] has taken such a burden to compare the g-factor for silicon ions bearing one (Si13+) or three electrons (Si11+). Now, the theorist here must really get bald by picking his hair one by one, and sum up the humongously complicated two- and three-electron Feynman diagrams;[4] this is so difficult that it can be currently done only up to order Z2. As a result, the theoretical value is known only “just” up to 8 significant digits, however it is on top of the experimental result at better than 1 part in 108.  

So, I presume we can now be definitely sure that in each nucleus there should be Z, and only Z protons, each one carrying a charge +1e, not one more, not one less… Or, “rather” sure… at least, until we keep to energies of less than a few hundred GeV… but that is material for a next Sunday story.

([1]) Frederick Soddy and Kazimierz Fajans, independently and at about the same time in 1913, defined the rule governing the transmutation of elements during radioactive decay. For the beta decay, the transformation produced a new chemical element with nuclear charge +1 compared to the initial element, while for alpha decay the transformation resulted in a new chemical element of charge reduced by 2 units.

([2]) J. Verdu et al., Electronic g-Factor of Hydrogenlike Oxygen 16O7+, Phys. Rev. Lett. 92, 093002 (2004)

([3]) S. Sturm et al., g-Factor of Hydrogenlike 28Si13+Phys. Rev. Lett. 107, 023002 (2011); A. Wagner et al., g-Factor of Lithiumlike Silicon 28Si11+, Phys. Rev. Lett. 110, 033003 (2013)

([4]) V.M. Shabaev et al., Theory of bound electron g-Factor in highly charged ions, J. Phys. Chem. Ref. Data 44, 031205 (2015)

What is the real charge of an atom?

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