The voltage that bends an electron like glass

One number for every point replaced the electric field’s three components by a single function, the potential, and found that nothing was lost. Nothing can be held still by a static field found what the potential cannot do: it obeys Laplace’s equation in empty space, so it has no minimum there, and a charge has always somewhere to fall. The potential is where the wanderers stop read the same equation as an average. And the field that keeps Greenwich time found the potential of the whole atmosphere set by the world’s thunderstorms.

Each of those treated the potential as something that pushes. This essay treats it as something that bends. A charged particle’s kinetic energy is fixed, wherever it is, by the potential it has fallen through, so a map of the potential is a map of the particle’s speed. A map of speed is what a refractive index is for light. The analogy is not loose: it is exact enough that the whole of geometrical optics — refraction, lenses, focal lengths, aberrations — carries over to electrons. Doing so was what made the electron microscope possible, and it also exposed a limit that no arrangement of electrodes could get round for sixty years.

Snell’s law with a square root #

Take an electron that started at rest where the potential was zero. Wherever the potential is $V$ it has kinetic energy $eV$, so its speed is proportional to $\sqrt{V}$ and so is its momentum. Let it cross a surface on which the potential jumps, held in place by a fine grid of wires at each potential.

At the surface, the force on the electron is perpendicular to the surface — that is what an equipotential means — so the component of its momentum along the surface does not change. The component across it grows, because the total has grown. If $\theta$ is the angle to the normal, the conserved quantity is $p \sin\theta$, and with $p \propto \sqrt{V}$,

$$\sqrt{V_1}\,\sin\theta_1 = \sqrt{V_2}\,\sin\theta_2.$$

That is Snell’s law, with $\sqrt{V}$ standing where the refractive index stands. From 100 volts to 400 the index doubles and the rays turn towards the normal, 45 degrees becoming 20.7. Going the other way, from 400 volts back into 100, an electron arriving at more than 30 degrees to the normal does not have enough momentum across the surface to climb the step, and it is turned back: total internal reflection, for a particle.

The direction of the bending is the historically interesting part. Newton explained refraction by supposing that light is a stream of corpuscles attracted towards glass, so that they speed up as they enter and their path bends towards the normal. The prediction that light should be faster in glass was the one the wave theory contradicted, and when Léon Foucault measured the speed of light in water in 1850 and found it slower, the corpuscles were finished. For electrons, Newton’s account is exactly right. They are attracted by the higher potential, they speed up, and they bend towards the normal on entering the region where they are fast. Light bends the same way for the opposite reason, because what a boundary conserves for a wave is the spacing of its crests along the surface, and that, as the law that only asks about one component found, is the tangential wavenumber.

The two pictures agree because they are two halves of one principle. Fermat’s principle makes light take the path of least time, which favours regions where light is fast; Maupertuis’s principle makes a particle at fixed energy take the path of least action, the integral of momentum along the path, which favours regions where the particle is slow. Hamilton noticed in the 1830s that the two have the same mathematics once the refractive index is identified with the momentum rather than with the inverse of the speed. Electron optics is that identification put to work, and a century later it found its wave: an electron’s de Broglie wavelength is $h/p$, so a region of high momentum is a region of short wavelength, which is what a high refractive index means for light.

A grid makes a sharp step. Without grids the potential changes smoothly, and the electron moves through what for light would be a graded-index medium, bending continuously rather than at a surface, as light does in the hot air of a mirage. The simplest case is a uniform field, in which the potential rises linearly and the index as its square root; the ray traced through that medium by Snell’s law applied layer by layer is a parabola, which is the path a thrown stone follows in a uniform gravitational field and a beam of electrons follows between the plates of an oscilloscope. Nothing new is being said about the electron’s motion. The same parabola is being reached from the other end, by optics instead of by Newton’s second law, and the agreement is the content of Hamilton’s analogy.

A lens made of three tubes #

A step between grids is a flat surface, and a flat surface does not focus. A curved equipotential does, and curved equipotentials are what any electrodes without grids make in the space between them. The simplest lens is three metal tubes on one axis, with the outer two at the potential the electrons arrive with and the middle one at a different potential. It is called an einzel lens, from the German for single, because the electrons leave it at the same speed they entered with, as light leaves a single glass lens.

The field between the tubes is computed from the potential along the axis, which for tubes separated by small gaps is well approximated by a smooth step in each gap. Off the axis the potential follows from Laplace’s equation, which fixes how fast it can change sideways given how it changes along the axis: near the axis, $V(r, z) \approx V(z) - \tfrac14 r^2 V''(z)$. The sideways push on an electron is therefore proportional to its distance from the axis and to the curvature of the axial potential. Where the potential along the axis curves upward, the push is towards the axis; where it curves downward, away.

That is the whole mechanism, and it shows why the lens converges. In each gap the axial potential first curves one way and then the other, so the electron is pushed towards the axis in one half and away from it in the other, and the two pushes would cancel if the electron took as long over each. It does not. In both gaps the inward push acts where the potential is lower and the electron slower, and a slower electron lingers longer and is deflected more. Each gap converges on balance, and the rays drawn come to a focus about four tube radii beyond the lens centre.

Converging whichever way it is charged #

A glass lens is converging or diverging according to its shape. An einzel lens is not a matter of choice: raise the middle tube, so the electrons accelerate through it, or lower it, so they decelerate, and the lens converges either way. The argument above shows why. Whichever way the axial potential bulges, each gap has an inward half and an outward half, and the inward half is always the one at lower potential, where the electron is slower. A lens with the same potential on both sides can only converge.

The two kinds differ in strength. A decelerating lens with its centre at three tenths of the outer potential focuses at six tube radii; an accelerating lens with the same difference of potential, its centre at 1.7 times the outer, focuses at twenty-four. In the decelerating lens the electron crawls through the middle and the fields have a long time to act. Pushed far enough, a decelerating lens brings the electrons nearly to rest and becomes a mirror, which electron optics also uses.

The same refraction runs every cathode-ray tube that ever made a picture: electrons accelerated from a hot filament through a sequence of apertures and tubes at increasing potentials, each pair a lens, brought to a spot a fraction of a millimetre across on the phosphor. Ion-beam machines, mass spectrometers and the electron guns of microscopes are built the same way, and the design of each is an exercise in geometrical optics with a refractive index of $\sqrt{V}$.

A prism as well as a lens #

Glass bends blue light more than red because its index depends on the wavelength. The index for an electron depends on its energy, so a beam whose electrons do not all have the same energy is dispersed, and the electron optician has both a nuisance and an instrument from the same fact.

The nuisance is chromatic aberration. An electron that left the filament with a few tenths of an electronvolt more than its neighbours has a slightly larger $\sqrt{V}$ everywhere in the lens, is bent slightly less, and comes to focus slightly further away. In an electron microscope at 200 kilovolts the spread of energies is about an electronvolt, a part in two hundred thousand, and it still blurs the image measurably, because the focal length of a strong lens changes in proportion to the fractional change of energy. Like spherical aberration, the chromatic aberration of a round electron lens has only one sign, for the same reason: there is no diverging round lens to pair it with.

The instrument is the electron spectrometer, which does on purpose what the lens does by accident. Two concentric hemispheres held at different potentials bend electrons round between them, and only those with the right energy follow the circle from the entrance slit to the exit; faster ones swing wide and slower ones fall in. The field between the hemispheres is the inverse-square field of a point charge at their centre, so the electrons move on Kepler ellipses, and ellipses of the same energy launched from one point at slightly different angles meet again after half a turn, which is what makes the analyser focus as well as disperse. Photoelectron spectroscopy, which reads the binding energies of the electrons in a surface’s atoms and so tells which elements are present and how they are bonded, is done with such hemispheres in almost every surface-science laboratory.

The aberration that has one sign #

The rays in the lens figure did not meet at one point. The ray entering at a tenth of the tube radius crossed the axis at 4.00 radii; the one at seven tenths, at 3.25. A lens that bends its outer rays more than its inner ones has spherical aberration, and it limits how small a spot the lens can make: the rays from different parts of the aperture arrive at different places.

Glass lenses have the same fault, and glass lens designers cure it by combining lenses: a converging element whose outer zones bend too much, paired with a diverging element whose outer zones bend too much in the opposite sense. The surface that images one point exactly is another cure, a single refracting surface of the right non-spherical shape. For round electron lenses neither cure is available. The figure shows three lenses, two accelerating and one decelerating, and in all three the crossing moves towards the lens as the entry height grows. No arrangement of round electrodes changes the sign.

Otto Scherzer proved it in 1936. For any electrostatic or magnetic lens that is rotationally symmetric, static and free of charge along its axis, the coefficient of spherical aberration is positive. The proof writes the aberration as an integral along the axis of terms in the axial potential and its derivatives, and uses Laplace’s equation to rearrange it into a sum of squares. It is the same equation that, read differently, says no static field can hold a charge still: Laplace’s equation is restrictive, and it removes a degree of freedom the optical designer would have needed. An electron lens cannot be diverging, and a lens that cannot diverge cannot be paired with one that corrects it.

From an analogy to a microscope #

The analogy was a mathematical curiosity for nearly a century after Hamilton. It became engineering in the 1920s, when two facts arrived within a few years of each other. Busch showed in 1926 that a short coil carrying a current acts on a beam of electrons exactly as a thin lens acts on light, with a focal length he could calculate from the coil’s field. And de Broglie’s proposal, confirmed by electron diffraction in 1927, gave the electron a wavelength thousands of times shorter than light’s. A lens and a short wavelength are the two ingredients of a microscope.

Ernst Ruska and Max Knoll built the first electron microscope in Berlin in 1931, with two magnetic lenses and a magnification of seventeen. By 1933 Ruska’s instrument exceeded the resolution of the best light microscope, and the first commercial microscopes followed before the end of the decade. Their designers were opticians in all but name, working with focal lengths, apertures, magnifications and aberration coefficients defined exactly as for glass, and it was in that setting, while the first instruments were still being improved, that Scherzer asked whether their aberrations could be corrected in the way a camera lens’s can.

Fifty times coarser than the wavelength #

The consequence was measured in atoms. The point of a microscope using electrons is their wavelength: an electron accelerated through 200 kilovolts has a de Broglie wavelength of 2.5 picometres, fifty times smaller than an atom, as everything has a wavelength would compute it. With light, wavelength is what limits resolution. With electrons it is not.

The spherical aberration blurs a point by an amount that grows as the cube of the aperture angle, and diffraction blurs it by an amount that shrinks as the aperture grows. Choosing the aperture to balance the two gives a resolution of about $0.61\,(C_s\lambda^3)^{1/4}$, which for a typical objective lens with an aberration coefficient of a millimetre is 0.22 nanometres at 200 kilovolts — about eighty-six times the wavelength. The aperture angle that achieves it is ten milliradians, half a degree, an aperture no light microscope would accept. For most of the twentieth century this was the limit of transmission electron microscopy: a resolution just good enough to see the columns of atoms in favourable crystals and not good enough to see light atoms at all.

Raising the voltage helps only as the fourth root of the cube of the wavelength, and voltage costs damage to the specimen; at a megavolt the gain is barely more than a factor of two. Scherzer himself listed in 1947 the ways out of his theorem: give up rotational symmetry, give up time-independence, or put charge on the axis. The first was made to work fifty years later, when Maximilian Haider, Harald Rose and Knut Urban, and independently Ondrej Krivanek, built correctors out of hexapole and quadrupole–octupole lenses, non-round elements whose combined effect on a round beam has a negative spherical aberration. Since about 2000 electron microscopes have reached resolutions below 0.05 nanometres, set by the next-order aberrations, the energy spread of the beam and the mechanical stability of the column rather than by Scherzer’s sign.

Where the refractive index stops being the square root of V #

Three assumptions are built into the figures. The electrons are slow enough for kinetic energy to be $\tfrac12 mv^2$; at 200 kilovolts they are at seven-tenths of the speed of light, and the index becomes $\sqrt{V(1 + eV/2mc^2)}$, which the wavelength curve includes and the lens traces do not. The lens is traced in the field of a smooth axial potential with the standard tanh form for each gap and its off-axis continuation to fourth order in the distance from the axis, accurate near the axis and increasingly approximate towards the tube wall; the rays drawn stop at seven tenths of the radius for that reason. And the beam is a set of independent electrons. Real beams carry enough charge that their own space charge diverges them, which is one of Scherzer’s ways round his theorem and in practice a nuisance rather than a cure.

What the figures cannot show is the magnetic half of electron optics. Hans Busch showed in 1926 that a short coil focuses electrons, and every high-resolution microscope’s objective is magnetic, because a magnetic lens can be made stronger without the electrodes sparking over. A magnetic lens also rotates the image and is described by a different index, involving the vector potential, but it obeys Scherzer’s theorem just the same.

Still open: how far an electron image can be pushed #

With spherical aberration corrected, the limits on an electron microscope are the chromatic aberration from the spread of electron energies, the higher-order geometric aberrations, the number of electrons a specimen can take before it is destroyed, and the stability of an instrument several metres tall to a few picometres. Chromatic correctors exist and are hard to build; monochromators that narrow the energy spread cost beam current. For radiation-sensitive specimens such as proteins, imaged frozen in ice, the count of electrons the sample survives sets the limit and optics no longer does, and how much further computational methods that combine thousands of noisy images can push that limit is an active question. Whether the resolution of a single image of a single unrepeated object can be pushed towards the wavelength itself is open.

The habit worth carrying away is to read a potential as a map of what a particle can do as well as what pushes it. An electron that has fallen through V carries momentum $\propto \sqrt{V}$, so every equipotential is a refracting surface obeying $\sqrt{V_1}\sin\theta_1 = \sqrt{V_2}\sin\theta_2$, and every arrangement of electrodes is a lens — always converging when the potential is the same on both sides, and always with spherical aberration of one sign, because the potential obeys Laplace’s equation. The same equation that leaves a charge nowhere to rest left electron microscopes eighty-six wavelengths short of their wavelength until someone stopped building them round.