Electron Spin

Electron spin is largely just another term for the magnetic field of an electron. There are many problems associated with it. Too many phenomena seem to violate it, others seem crucially dependent on it. This page is “wild science”, an attempt to think through some consequences of the physics involved.

The classic analysis on spin was the Stern-Gerlach experiment which showed in 1922 that a beam of silver atoms directed through an inhomogeneous magnetic field would be forced into just two beams. This can be explained by an intrinsic magnetic field that reacts to the external magnetic field gradient by taking one of just two absolute orientations - spin “up” and spin “down”. This is taken to prove both the intrinsic nature of the spin of the electron, and further that it can be in only one of two possible states. It is only fair to say that there were some problems with this interpretation regarding silver atoms, but the experiment has been accepted as valid.

Another totally independent observation was of the closely spaced splitting of the hydrogen spectral lines, called fine structure. Against these observations we have a range of devices that are crucially dependent on the electron having no intrinsic magnetic field. As is usual in conflicts of this sort, these belonged in a totally different field of knowledge - the field of electrical engineering - and practitioners in each field did not connect with those in the other.

The most important of these effects lies in the older cathode-ray style of television tube. Here electrons are “fired” from an electron gun, and spread out as they travel from their source, rather like a shotgun blast. They have to be focussed to a single point on the tube screen, so are focussed through a magnetic field acting as a lens, that both focusses them and performs the line- and frame-scan functions. Because of the natural gradient across the lens, if the electrons had an intrinsic magnetic field then one would expect a point focus on the screen to be surrounded by a ring halo of those electrons which had been in spin opposition to the focussed electrons - spin “up” rather than spin “down”. But this halo has never been observed, and this appears to prove that the electron has no intrinsic spin, in direct conflict with the Stern-Gerlach experiment - not that electrical engineers care, if the TV works that is enough for them.

Another observation lies in the measurements of the inertial mass of electrons. If there is an intrinsic magnetic field then one might expect it to orient randomly with respect to the direction of acceleration during such experiments, unless forced to orient itself via an external magnetic field. Since a moving magnetic field generates an electric field, and that takes energy, and since the electric field so generated has a strength dependent on its orientation with respect to the direction of motion, it follows that attempts to find the inertia of an electron would result in a spread of inertias rather than a single value if the electron had an intrinsic magnetic field.

Yet again, when an electron enters a cloud chamber it describes only one path. If the magnetic field really made a difference then the spin state should create two variations of the path.

So can we review the Stern-Gerlach experiment in terms of an extrinsic (externally -driven) magnetic field rather than in terms of an intrinsic field? The first thing to note is that there is a hidden assumption in the interpretation of the results, that the magnetic field is intrinsic - it certainly does not prove that the field is intrinsic, only that if it were, then the spin can take only one of two states. To understand why, let us go through it using the opposite premise.

Let us assume (this will be described in more detail later) that when an electron enters an external magnetic field it will start to spin, generating a magnetic field that always opposes that external field - that is, the electron has an externally driven or extrinsic magnetic field. Under this assumption if the electron passes through a strong magnetic field gradient, vertically oriented, then that half of the electrons that are in the upper part of the beam will start to spin to generate a field that opposes the local field, and therefore will be forced downwards. Those in the lower part of the beam will by a similar argument be forced upwards; there is no complete void in the centre, but for most typical field gradients the forces are greater the further from the centre of the beam, so there is a focussing effect on each of the two exit beams creating higher concentrations in each beam spot. From this we could say that the Stern-Gerlach experiment proves that electron spin is extrinsic, but in fact we cannot say this at all, since that is the underlying assumption in this interpretation of the experiment.

The fine structure in a hydrogen atom could also be caused by extrinsic fields rather than intrinsic ones.

So how might an electron’s extrinsic magnetic field occur? First consider a well-known phenomenon. Take a closed loop of copper wire, and push it between the poles of a magnet, normal to the magnetic field lines. A current will be induced in the loop, which creates a magnetic field that opposes that of the magnet. Hence you will experience a force that opposes the insertion of the loop into the magnet - it will take effort to push the coil into the magnetic field. Once you have pushed the coil into the magnet hold it steady - the resistance of the coil will cause the induced current to decay and the induced opposing magnetic field will disappear. If you now pull the loop out of the magnet a new current is induced which creates an attractive magnetic field that opposes your attempt to withdraw it from the magnet.

Next, what if the copper loop is replaced with a super-conducting loop. In this case the current will not decay after pushing the loop into the magnet, but continue to create an opposing magnetic field, causing the loop to continue to try and push its way out of the magnet. As it leaves the magnet the induced opposing field falls and the repulsion disappears. Hence the loop behaves like a spring - it bounces out of the magnet when you let it go. This is why a super-conducting ring will float in mid-air above a magnet - it acts as if it were supported by an invisible spring attached to the magnet.

All standard physics. But extend this to an electron surrounded by an electric field (note that this argument uses the Field Theory model rather than the “point charge” Quantum Mechanics model). Now as the electrons field is pushed into the magnet each electrical line of flux sees a transverse electric field induced by the magnets field, from the relative motion of the electron and the magnet. The same forces that cause an electron to follow a curved path in a magnetic field, will operate firstly on the first part of the electron’s field that enters the magnet - that means the leading part of the electron’s field will start to follow a curved path before the trailing part. Hence the electron will start to spin as well as to follow a curved path. This spin - through the rotating electric flux lines - will induce a magnetic field which opposes that of the magnet. There being no mechanism for loss, the electron will act as a superconductor. It will lose kinetic energy on entering the magnet (exchanging the energy for magnetic strain energy), but regain it on exit, at the same time losing its rotation.

Since the induced field is linearly proportional to the linear speed of the rotating electric flux lines, the electron’s rotation speed is linearly proportional to the magnet’s field strength. On this basis the gyroscopic moment is linearly proportional to the magnetic field. The torque involved between the electron’s induced dipole and the magnet, however, is proportional to the product of the magnetic field strengths of the electron and the magnet - since the induced field is linearly proportional to the magnet’s field, that means the torque is proportional to the square of the magnetic field. This in turn tells us that the precession frequency, being dependent on the Torque divided by the gyroscopic moment, is linearly dependent on the magnet’s field (in the classical model, the gyroscopic moment is constant, and the Torque is linearly dependent on the external magnet’s field strength, giving a precession rate that is also linearly dependent on the external magnet’s field). For a detailed explanation of this gyroscopic behaviour, click here.

The detailed mechanism depends on the fact that the magnet is moving with respect to the electron’s rest frame, inducing a transverse electric field that is normal to both the magnetci flux lines and the motion. This adds to the component of the flux line electric field vectors on one side of the electron, and opposes them on the other, causing the electron electric flux lines to want to move from the higher-energy side to the lower energy side; provided only part of the electron is submersed in the magnetic field the forces all round the electron will not be in balance and the electron will start to rotate. Once the electron is fully within the magnet the forces at the front and the rear of the electron will balance each other out and no more rotational acceleration will be imparted, leading to a constant rotation after this point.

Now although the precession frequency is linearly dependent on the magnetic field, as seen from measurement, the energy stored in the electron’s spin is proportional to the square of the induced field. If simple precession is involved in atomic energy level changes this result conflicts with current theory, which states that the energy should be linearly proportional to the external magnet’s field. However, if precession is not involved directly in atomic transitions it may not be an issue.

Problems with the extrinsic field idea...

The obvious one - that the energy stored in the electron’s spin is proportional to the square of the external field - is a radical departure from classical theory. However, although precession seems to be involved in magnetic resonance imaging, there is no current evidence that it is involved directly and simply in atomic transitions.

There are, however, a host of other experiments that would need to be used to test the idea - can all these experiments be reinterpreted successfully to allow for the electron to have an extrinsic magnetic field? A lot of work there for somebody if they have the interest and the time!