I consider it an interesting accident of history that the first modern quantum theories, those of Heisenberg and Schroedinger, were developed just about simultaneously with the discovery of the so-called intrinsic spin of the electron. I wonder how different the development of theories of the atom would have been if the spin existence had been inferred sooner, so that it would have been known perhaps to such as Niels Bohr.
That an electron must possess an angular momentum separate from that associated with its orbital motion was inferred from atomic emission spectrum measurements performed by Zeeman, where the discharging gas was also subject to an externally-applied magnetic field. Although Zeeman published his observations in 1897, Goudsmit and Uhlenbeck published their conclusion that they implied an electron intrinsic spin and magnetic moment not until 1925, and it was in January 1926 that Schroedinger published his equation that is the foundation of wave mechanics, and its successful application to hydrogen-like atoms. Heisenberg's matrix mechanics formulation of quantum theory had already been published in 1925. Bohr's model had been published over a decade earlier, in 1913, and was followed in the next few years by Sommerfeld's generalization from circular to elliptical orbits. So, all the quantum theories prior to Dirac's relativistic theory (that found the first justification for the spin) were apparently developed with little or no knowledge of the electron spin, although both the Heisenberg and Schroedinger formulations were found to accommodate it readily enough, and strong contribution was made here by Pauli.
So then, Rutherford had no knowledge the electron was also a magnet when he developed the idea of an atom consisting of point-charge particles following trajectories determined by classical electrodynamics. Certainly he and others would have immediately recognized that the magnetic nature of the particles would influence the classical motion in interesting ways. Charged classical particles moving in magnetic fields feel forces accordingly, and further, when a magnetic dipole moves relative to a charged particle the charged particle will feel an electric force. This is due to the electric field induced by the time-varying magnetic field due to a moving magnet. The net result is that the dynamical picture of the point-charges-with-spin classical atom is vastly more complicated and interesting than the simple Rutherford charge-only model where the motion is essentially equivalent to Keplerian planetary orbital motion under gravitational force, apart from the small but important radiation reaction term that causes radiative decay of the classical Rutherfordian atom. In the absence of any working atomic model at the time, it's easy to imagine Rutherford and others such as Bohr setting to work to determine if this complex dynamics could lead to anything resembling the atom as understood from spectroscopic measurements and empirical models such as the Balmer and later Rydberg formulae.
An even greater interest would have been generated based on that the magnitude of the electron intrinsic angular momentum involves directly Planck's constant, h. Since 1888 and the publication of the Rydberg formula, it was known empirically that atomic spectra directly involve h. It was based on his knowledge of the Rydberg formula that Bohr proposed, with no particular justification other than that it worked, that atomic electron orbits could only be stable if the angular momentum were a whole-number multiple of h divided by two pi. In science new principles must usually be introduced only as the last resort, so prior to introducing this principle in such an ad hoc fashion, it's easy to imagine that Bohr would have worked to investigate what are the classical electrodynamics consequences of a spinning magnetic electron. Perhaps he would have noticed that his formula for the ground-state radius of the hydrogen atom can also be obtained by equating the mutual precession frequencies of the electron spin and orbital angular momenta. Then, attention would have been focused on why the latter is true, and whether perhaps atomic orbital quantization might merely be a classical-physics consequence of the existence of the intrinsic spin, rather than a new fundamental principle.
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9 hours ago