Sunday, May 9, 2010

An interesting feature of the quasiclassical atom

As everyone who's taken a college level physics class should know, the problem of the classical atom in the days of Rutherford, with (for hydrogen, say) the electron orbiting the proton like the moon orbits the earth, is radiative decay. This system has a nonzero, nonstationary, electric dipole moment, and the radiation intensity due to this is proportional to the square of the magnitude of the electric dipole acceleration. This causes energy loss via radiation reaction, and the classical atom decays to a point in a small fraction of a second.

Something I've come to realize just recently however is that if the electron has an intrinsic magnetic moment (which of course it does, unbeknownst to Rutherford, or Bohr in 1913), this radiative decay is no longer so obviously necessarily the case. This is because a moving magnetic dipole necessarily acquires an electric dipole moment, as the cross product of the magnetic moment vector with its velocity, and it's easy to arrange things so that this opposes the electric dipole moment part that's due to the separation of charge. For example, if the electron is in a circular orbit and its intrinsic magnetic moment is oriented perpendicular to the orbital plane, the electric dipole moments from the two sources (separation of charge and moving magnetic dipole) are either parallel or antiparallel, depending on the orbital direction.

So then, for a perfectly circular orbit, since the magnitude of the motion-acquired electric dipole is proportional to the velocity, and the velocity increases as the orbital radius shrinks, whilst the dipole moment due to the separation of charge shrinks, there is a unique radius where the total electric dipole moment vanishes. If the quasiclassical atom were dynamically stable otherwise at this radius, it would not need to decay radiatively.

When this realization came to me the other day, the next thing I wanted to do was determine what this radius would be and whether it would turn out to be the Bohr radius. After all, it will certainly involve Planck's constant via the intrinsic magnetic moment being the Bohr magneton (which is e * hbar/ (m_e * c) in Gaussian units).

Turns out, the radius where the electric dipole moment vanishes is not the Bohr radius. The formula for it is easy to get, but it isn't much like the Bohr radius formula, although it also involves the charge and mass and h-bar but in the wrong powers and also has the speed of light which is absent in the Bohr radius formula. In magnitude however the radius is about a couple of orders of magnitude smaller than the Bohr radius. So it's between the atomic scale and the nuclear scale. Isn't it interesting that the scale where the two dipole moments can cancel is near the atomic scale? I think so. It's worth remembering here that in the classical atom without spin there is nothing obvious to determine scale at all.

I suppose, there's no reason for this particular radius to be dynamically preferrred, in the sense of it being a local energy minima. The dipole moment can momentarily vanish yet its acceleration not. So it has no direct particular significance. I think it's interesting though that acquired electric dipole moment is about the right magnitude so that if it jiggled around at the orbital frequency and just so, it might cancel the charge-separation dipole acceleration. After all, it's not the total electric dipole moment that has to vanish but rather its acceleration.

This is something I'm going to pursue further. I suspect pretty strongly that nobody has ever investigated it. When people originally tried to make the classical atom work, they had no knowledge of the electron spin. Then, within a year or so of the spin becoming known, both the Schroedinger and Heisenberg formulations of quantum theory were established and completely supplanted the classical atom idea (for good reason). A couple of years later the existence of the spin and the spin orbit coupling were even fully described by Dirac's relativistic quantum theory. Physicists have never looked back, but fortunately they have me willing to go back and make sure they didn't miss something important.

Here is how to get an equation of nonradiativity of electric dipole radiation for a classical atom where the particles are allowed to have intrinsic spin and associated intrinsic magnetic moment.

Let d represent the total electric dipole moment of the atom. Since the electric dipole radiation intensity is proportional to |d^2d/dt^2|^2, and assuming d is an ordinary vector, the nonradiativity requirement can be expressed simply as d^2d/dt^2 = 0.

For simplicity, I will treat hydrogen, but the approach generalizes to more complex atoms easily enough.

The total electric dipole moment of the atom consists of a part due to separation of the point charges, and, since the electron is also intrinsically a magnetic dipole, a part v x m/c due to the translational motion of the electron, where "x" indicates the vector cross product. So the total electric dipole moment is, for hydrogen

d = -e r + v x m/c

where e is the fundamental charge, r is the displacement of the electron from the proton, v = dr/dt is the electron velocity, c is the speed of light (approximating that the proton is stationary), and m is the intrinsic magnetic moment of the electron (which is known to be the Bohr magneton). The condition d^2d/dt^2 = 0 is then easily found to be equivalent to

e c a = (da/dt) x m + v x (d^2m/dt^2) + 2a x (dm/dt)

where a = dv/dt = d^2r/dt^2

It's also possible to write an equation for magnetic dipole nonradiativity, and that one obtains something like quantization of the orbital angular momentum, if the nuclear magnetic moment is not neglected. However the composite nature of a nucleus even consisting of a single proton cannot be disregarded here, so I was forced to consider Positronium. In Positronium I get that the magnetic dipole radiation vanishes if the orbital angular momentum is (3/4)*h-bar, if it is assumed the electron spin component parallel to the orbital angular momentum is h-bar/2. It would be more interesting if the factor of 3/4 were not present, of course, but another problem is that the magnetic dipole radiation intensity is negligible compared to the electric dipole radiation intensity, in the atomic scale (although not at the nuclear scale, where the magnetic dipole radiation intensity becomes larger than the electric). So naturally it's more interesting if the electric dipole radiation could be shown to vanish for a nonzero atomic radius. I simply never until recently recognized it could even plausibly vanish, because I was not realizing the total electric dipole moment has the additional contribution due to the intrinsic magnetic moment. I knew about the acquired electric dipole moment of a moving magnetic dipole, but I failed to realize it must contribute to the radiation intensity in this way.

So now I am trying to determine if there are any interesting consequences of this equation, that might also correspond to known physics.


  1. I don't have enough background to understand much of what you're doing. But isn't it interesting that a positron-electron pair do indeed radiate until they disappear completely, while a proton-electron pair do not?

  2. J Thomas, thanks for being the first to comment here.

    Yes that's an interesting question and it has an answer that I think will be of interest.

    The gound state of positronium annihilates not because of radiative decay but rather simply because the wavefunctions of the electron and positron overlap sufficiently that they have a finite probability of being within the capture cross-section for electron-positron mutual annihilation within a fairly short time. This is the spherically-symmetrical or "s" state that has zero angular momentum. Quantum theory accurately predicts the decay time of positronium in this way.

    A similar situation exists in the case of the hydrogen atom. The ground state according to quantum theory is the s state, and this differs from the Bohr model which would have angular momentum of h-bar in its ground state. But, we know the modern quantum theory model is the right one, because of an analogous process in the hydrogen atom to the self-annihilation of positronium, the hyperfine splitting. The hyperfine splitting is a small spectral line splitting due to the interaction of the electron and proton magnetic moments. It is analogous to the spin-orbit coupling but much weaker due to the proton magnetic moment being about one two-thousandth of the magnetic moment of the electron (which is one Bohr magneton). This is simply due to the proton being more massive than the electron.

    The hyperfine splitting is measurable via emission spectra, and also correctly predicted by modern quantum theory based on the interaction strength and the overlapping of the electron and proton wavefunctions in the s state. It vanishes in the "l" states that have nonzero angular momentum. This is sufficient to prove the Bohr model ground state cannot be correct in having L = h-bar.

    This bothered me for a long time, as I was working on an assumption that the Bohr model is true in some sense, until one day I discovered the work of Manfred Bucher. Bucher shows that the generalized version of the Bohr model, the Bohr-Sommerfeld model, that allows for elliptical electron orbits and quantization of the orbital plane orientation as well, can be modified to be consistent with modern quantum theory in terms of both angular momentum and hyperfine coupling. The original Bohr-Sommerfeld model was successful in explaining the fine structure and other line series that the simple Bohr model could not, but it does not predict the hyperfine structure, and is incorrect on the angular momentum of the ground state. Bucher though modifies it so that the ground state instead of being an elliptical orbit is a degenerate motion that he calls the "Coulomb oscillator". In this state the electron is falling straight through the nucleus and has no angular momentum, and thus is in agreement with modern quantum theory. Also, remarkably, Bucher has calculated a value for the hyperfine splitting based on it and found it also in agreement with experiment. So, it seems possible that a quasiclassical atomic model such as that I've been pursuing is capable of agreeing with modern quantum theory to a higher degree than most people would realize.

    Here's a link to the Bucher paper on arxiv. (Several other Bucher papers incluing the hyperfine splitting one should be traceable from there):