The field at which no electron reaches the anode

In 1921 Albert Hull, a physicist at the General Electric laboratory in Schenectady, was looking for a way to switch large currents without a control grid. His device was the simplest vacuum tube there is — a straight heated wire along the axis of a metal cylinder, the cylinder held positive, electrons boiling off the wire and crossing to the cylinder — with a coil wound round the outside to make a magnetic field along the axis. With no field, the current was whatever the voltage and the wire’s temperature allowed. As the field rose, the current did not change. At one field, sharply, it fell to nothing.

Hull called the tube a magnetron. Its descendant, with the cylinder cut into resonant cavities, is in every microwave oven, and it works by sitting just inside the boundary Hull measured. The boundary itself is one of the cleanest results in electromagnetism, because it can be found without solving for the motion — without even knowing the electric field between the wire and the cylinder, which in a working tube depends on the electrons themselves.

A force that only turns #

The electron leaving the cathode is pulled outwards by the electric field and turned sideways by the magnetic one. The force that does no work found that the magnetic part can never change the electron’s speed, only its direction: all the energy the electron has at any radius was supplied by the electric field, $eV(r)$, where $V(r)$ is the potential there. What the magnetic field changes is how that energy is shared between moving outwards and moving round.

In a weak field the share that goes into moving round is small, the paths are gently curved, and every electron reaches the anode. In a stronger field more of the energy goes into circling, and an electron arrives at the anode with less outward speed, at a glancing angle. In a strong enough field the circling takes all of it before the electron has got there, its outward motion stops, and it turns back towards the cathode.

The transition between those two cases is sharp, as the figure shows. At 0.98 of the cut-off field the electron grazes the anode and is collected; at 1.02 it turns back short of it and returns to the cathode, arc after arc, never collected. A few per cent of field separate a full current from none.

The quantity the field fixes #

The sharpness, and the simplicity of the answer, come from one conservation law. The diode is symmetric about its axis: its electric field points along radii and its magnetic field along the axis, and rotating the whole arrangement about the axis changes nothing. The conservation law a symmetry hands over found that every such symmetry comes with a conserved quantity, and for rotation the quantity is angular momentum — but in a magnetic field it is not the ordinary angular momentum. It is the canonical one, which includes a term from the vector potential: $mr^2\dot\theta - eA_\theta r$, the same correction the potentials that are not unique found the vector potential making to a charge’s momentum. For a uniform axial field $A_\theta = Br/2$, and for an electron that left the cathode surface at rest it gives, at every radius,

$$r^2\dot\theta = \frac{eB}{2m}\left(r^2 - a^2\right).$$

The angle is the coordinate the forces cannot see — the force a coordinate cannot see gave that name to any coordinate a system’s energy does not depend on, and found its momentum conserved — so the electron’s rate of circling at radius $r$ is fixed by the field and by $r$, and by nothing else. In particular it does not depend on the electric field.

That leaves only energy. At radius $r$ the electron’s kinetic energy is $eV(r)$, and its circling takes $\tfrac12 m r^2\dot\theta^2 = (e^2B^2/8m)(r - a^2/r)^2$ of it. Whatever is left over is outward motion. The electron reaches the anode at radius $b$ if, and only if, there is something left over there:

$$eV \ge \frac{e^2B^2}{8m}\left(b - \frac{a^2}{b}\right)^2,$$

and the cut-off voltage is

$$V_c = \frac{eB^2b^2}{8m}\left(1 - \frac{a^2}{b^2}\right)^2.$$

Why the voltage profile does not matter #

The striking thing about that derivation is what it never needed. Between cathode and anode, the potential $V(r)$ could be anything. In a vacuum diode it varies as the logarithm of the radius. In a working tube the cloud of electrons between the electrodes is a space charge that changes it — flattening it near the cathode, so that in the limit of large current it rises as the four-thirds power of the distance, the law the beam that stops pushing itself apart touched on for a beam’s own charge. None of that enters the cut-off, because the condition is only tested at the anode, where the electron’s energy is $eV$ by definition and its circling is fixed by the field.

Between the electrodes the profile matters a great deal. It decides how far a cut-off electron gets before turning back, how fast it goes, and the shape of its path. The figure follows the farthest radius an electron reaches against the field, for three profiles: the vacuum one, a uniform field, and a space-charge-limited one.

Above the cut-off the three curves disagree about everything — the space-charge profile, which leaves the electron with least energy far from the anode, turns it back closest to the cathode — and they meet at one point: the field at which the turning radius reaches the anode is the same for all three. Each curve was computed by stepping the full Lorentz force forward in time, not from the conservation law, and each keeps the canonical angular momentum constant to a part in a thousand along its path, which is the check that the integration and the law agree.

This independence is what makes Hull’s result useful. The space charge in a real tube is hard to calculate — it depends on the current, which depends on the space charge — and a prediction that needed it would be uncertain. The cut-off needs only the geometry, the voltage and the field.

Hull’s parabola #

Read as a curve in the plane of field and voltage, the cut-off is a parabola: doubling the voltage raises the cut-off field by $\sqrt2$. Hull measured it in exactly this form, sweeping the field at fixed voltage and recording where the current vanished, and the agreement with the formula was his evidence that the electrons left the cathode with negligible speed.

For flat, parallel electrodes the same argument gives $V_c = eB^2d^2/2m$, where $d$ is their separation, and the conserved quantity is the canonical momentum along the electrodes rather than round an axis. The cylinder needs a third more field than flat electrodes the same distance apart, because the circling speed the field forces on an electron arriving at a cylindrical anode is $(eB/2m)(b - a^2/b)$, smaller than the $eBd/m$ forced across a flat gap. The thinner the cathode relative to the anode, the closer the cylinder’s rule comes to $eB^2b^2/8m$, the cut-off for a cathode that is a mere wire, which is the case Hull built.

The window a magnetron works in #

Hull’s magnetron was a switch, and a crude one. It became a source of microwaves when the anode was cut into resonant cavities — the cavity magnetron of John Randall and Harry Boot at Birmingham in 1940, which made centimetre radar possible — and the way it works depends on the electrons not reaching the anode directly.

Below the cut-off parabola, electrons leaving the cathode turn back before the anode and form a cloud circling the cathode. In the crossed fields, radial electric and axial magnetic, the cloud drifts round the axis at the electric field divided by the magnetic — the drift the drift that does not care what the charge is found for any charge in crossed fields. The cavities, resonating, set up a wave of electric field travelling round the anode. If the electrons’ drift keeps pace with that wave, they bunch on it: electrons in the decelerating phase of the wave give it energy, lose speed, are pushed outwards and eventually reach the anode in spokes; electrons in the accelerating phase take energy, gain speed and are returned to the cathode early. The wave gains energy from the drifting electrons, and the electrons’ potential energy, carried out to the anode in the spokes, is what the oven’s food absorbs.

The condition that the drift can keep pace with the wave is the Buneman–Hartree condition, and it is a straight line, nearly, in the same plane as Hull’s parabola. It is tangent to the parabola, and the tube oscillates only in the sliver between them: below the parabola, so that electrons do not simply cross; above the line, so that they drift fast enough.

The dimensions in the figure are close to those of an oven magnetron: a cathode four millimetres across, an anode block of ten vanes, a permanent magnet giving about 0.17 tesla, and four kilovolts across the tube. At that field the window runs from 2.7 to 5.7 kilovolts. Below it the electrons drift too slowly to keep up with the wave, and the tube passes no current and makes no microwaves; above it they cross straight to the anode, the current jumps, and the tube heats without oscillating. The magnet in an oven is chosen so that the supply voltage sits in the window, and the oven’s power is varied by switching the magnetron on and off in cycles, because the window does not allow it to be turned down.

The two speeds can be checked against each other directly. In the oven-sized tube the average radial field is about four kilovolts across two millimetres, two million volts per metre, and in 0.17 tesla the drift is $E/B \approx 1.2\times10^7$ metres a second — four per cent of the speed of light. The cavities’ wave has ten vanes to go round, five full wavelengths in the mode where neighbours are in opposite phase, and at 2.45 gigahertz its crests travel round the anode at $\omega b/n = 2\pi \times 2.45\times10^9 \times 0.004/5$, which is also $1.2\times10^7$ metres a second. The electrons and the wave are matched by construction: the magnet, the voltage and the number of vanes are chosen together so that the drift equals the wave’s speed at the anode, and the Buneman–Hartree line is that equality written as a voltage.

Before the cavities: the split anode #

Microwaves came out of magnetrons before the cavity magnetron existed. In the 1920s Erich Habann in Germany and Kinjiro Okabe in Japan split the anode cylinder lengthwise into halves and connected the halves through a tuned circuit. Near the cut-off, electrons skimming the anode spend longer near whichever half is momentarily less positive and are collected by the other, so the tube behaves as a negative resistance between the halves, and the circuit oscillates. Okabe obtained wavelengths of a few centimetres this way in 1927. The power was small, because the electrons doing the work were the few near the cut-off and the anode was a single resonator; Randall and Boot’s insight was to make the anode block itself a ring of resonators, so that every electron in the circling cloud could interact with the wave. The window between the parabola and the line is what that ring made possible.

Weighing the electron with a magnet #

The cut-off contains the electron’s charge-to-mass ratio and nothing else about the electron, which makes it a measurement. Hold the voltage fixed, raise the field until the current vanishes, and $e/m = 8V/B_c^2b^2(1 - a^2/b^2)^2$. It is one of the ways students still measure $e/m$, with a commercial diode and a coil — often a pair of two coils one radius apart, whose flat field approximates the uniform one the derivation assumed — and its virtue is the one this essay has been about: the answer does not depend on the space charge in the tube, on the cathode’s temperature within reason, or on any detail of the electrons’ paths. Its weakness is the sharpness of the cut-off in practice. The current falls over a few per cent of the field rather than at a point, because the cathode emits electrons with a spread of speeds and the field is not quite uniform, and the measurement is as good as the judgement of where the fall happens.

A Trojan in a vacuum tube #

The cloud of electrons circling the cathode below the cut-off, held off the anode by the field and drifting round, has a counterpart far from any laboratory. The hilltop that holds the Trojans found asteroids held at a maximum of the effective potential by the Coriolis force — a velocity-dependent force that does no work, turning the asteroid’s outward slide into a slow circling round the hilltop, and stable only if the rotation is strong enough compared with the slope. Electrons in the magnetron sit at the same kind of arrangement: pulled outwards by the electric field, turned by a force that does no work, and held, provided the magnetic field is strong enough compared with the electric. Hull’s parabola is the same kind of statement as the condition that keeps the Trojans in place, and the same condition holds an ion in a Penning trap. In all three, a force that cannot change a particle’s energy changes where its energy can take it.

What the figures leave out #

The figures follow single electrons in static fields, starting from rest. Real electrons leave a hot cathode with thermal energies of a tenth of an electronvolt, which blurs the cut-off by a corresponding fraction of the voltage — negligible at kilovolts. The magnetic field is taken as uniform along the axis, which a short solenoid or a pair of magnets approximates only near the middle, and the ends of a real tube leak electrons along the axis. The cylinders are taken as perfect and coaxial; a cathode off-centre by a fraction of a millimetre changes the cut-off by a corresponding amount round its circumference and smears the current’s fall. And in the magnetron the window drawn is the simple one: the Buneman–Hartree line is computed for a single mode in the planar limit of the drift, and real tubes can lock into neighbouring modes whose lines sit elsewhere.

The electrons are also treated without relativity. At 4 kilovolts an electron’s energy is 0.8 per cent of its rest energy and the correction to the cut-off is of that order; in the relativistic magnetrons built for pulsed power, run at hundreds of kilovolts, the conserved angular momentum carries the Lorentz factor and the parabola bends towards higher fields, and the relativistic version of Hull’s condition is the one those designs use.

The domain of the cut-off itself is wider than any of this. It needs axial symmetry, a static uniform magnetic field, and electrons leaving the cathode at negligible speed. Within that, it holds for any radial electric field, including one shaped by the electrons themselves.

Still open: what the electron cloud does below the cut-off #

Hull’s analysis says no electron reaches the anode below the cut-off, and in an ideal tube none should. Real tubes pass a small current below the cut-off anyway, and the cloud of electrons circling the cathode is noisy, with fluctuations far larger than its thermal energy would give. The cause has been argued since the 1940s: instabilities of a rotating, sheared cloud of charge — the diocotron instability, the electron analogue of the instability that rolls up a shear layer in a fluid — break the symmetry on which the conservation law depends, and once the symmetry is broken, electrons can cross. How the instability saturates, how much current it lets through, and how it interacts with the cavities’ wave in an oscillating magnetron are questions that simulations of the full cloud address, and the answers matter for the noise that limits magnetrons in radar and in particle accelerators.

The cut-off itself is settled and exact. An electron leaving a cathode of radius $a$ in an axial field $B$ keeps $r^2\dot\theta = (eB/2m)(r^2 - a^2)$, so it reaches an anode of radius $b$ only if $V \ge (eB^2b^2/8m)(1 - a^2/b^2)^2$ — 142 mT at 4 kV for a 4 mm cathode in an 8 mm anode — whatever the potential does in between; a magnetron oscillates in the sliver between that parabola and the line where the electrons’ drift keeps pace with its cavities. A force that cannot do work cannot stop an electron. It can only see to it that the electron’s energy runs out sideways.