# How do you trap an electron?

I’ve always found the concept of two forces on an object cancelling themselves out strange. We say they cancel if the changes they exert completely offset each other, leaving the object unaffected. But is the object really unaffected? If the two forces act in absolute opposition and at the exact same time, the object may be unaffected. But practically speaking, this is seldom the case and the object experiences some net force, to which in may not respond in a meaningful timeframe or respond in an imperceptible or negligible way.

For example, imagine you are standing exactly still and two people standing on either side of you punch you hard on your upper arm, in an attempt to move you in the other direction. The two impulses may cancel each other out but you will still feel the pain in your arms. You might counterargue that this is true only because the human body has a considerable bulk, which means a force applied on one side of the body is transmitted through a series of media before it manifests on the other side, and that en route it loses some of its energy as the stress and strain through your muscles. This is true – but the concept of cancellation is actually imperfect even with microscopic objects.

Consider the case of the quadrupole trap – a device used to hold charged particles like electrons and ions in place, i.e. at a fixed point in three dimensions. This device was invented because it’s impossible to confine a charged particle in a static electric field. Imagine eight electrons are placed at the vertices of an imaginary cube, and a ninth electron is placed at the centre. You might reason that since like charges repel, the repulsive force exerted by the eight electrons should hold the ninth, central electron in place – but no. They won’t. The central electron will drift away if another force acts on it, instead of getting displaced by a little and then returning to its original position.

This is because of Earnshaw’s theorem. Thanks to Twitter user @catwbutter for explaining it to me thus:

You can understand the theorem as saying the following: In a configuration of n charges, you ask if one is in equilibrium. [Imagine the cubic prison of n = 8 electrons at the vertices and one at the centre – this one needs to be at equilibrium.] You displace it from its point a little bit. For there to be equilibrium, the force on it needs to point radially inward at the original point you displaced it from, regardless of where you displaced the charge to. This is only possible if there is a charge at the original point – but there isn’t in the setup.

Formally, Earnshaw’s theorem states that a collection of charged particles (of the same kind, i.e. only electrons or only protons or only ions, etc.) can’t maintain a stable and stationary equilibrium if the only thing maintaining that equilibrium is the electrostatic forces between them. In this case, the concept of ‘cancelling out’ becomes irrelevant because of the way the electric fields around the charged particles behave. One way to make it relevant is to use an exception to Earnshaw’s theorem: by using moving charges or time-varying forces.

Imagine you’re walking along a path when a cat appears in front of you and blocks the way. You step to the cat’s right but it moves and still blocks you. You step to the left and it moves again. You’re stepping right and left respectively because you see a gap there for you to go through, but every time you try, the cat moves quickly to block you. Scientists applied a similar kind of thinking with the quadrupole ion trap. They surround a clump of electrons, or any charged particles, with three objects. One is a hyperbolic cylinder (visualised below) called a ring electrode; it is capped at each end by two hyperbolic electrodes. The ring electrode needs to be exactly halfway between the capping electrodes. The electrons are injected into the centre.

Note that in the first image above, the ring electrode and the sides of the capping electrodes should ideally be inclined at an angle of a little over 53º relative to the z axis. But whatever the angle is, when an electric current is applied to the electrodes, the resulting electric field inside the trap will have four poles – thus the name ‘quadrupole’ – and the field along the poles (the hazy area in the image below) will be asymptotic to the electrodes.

This electric field has two important properties. The first is that it is inhomogenous: it is not uniform in different directions. Instead, it is weakest at the centre and becomes stronger as the field becomes squeezed between the electrodes. Second, the electric field is periodic, meaning that it constantly changes between two directions – thanks to the alternating current (AC) supplied to the electrodes. (Recall that AC periodically reverse its direction while DC doesn’t.)

The resulting periodic inhomogenous electric field exerts a unique influence on the electrons at the centre of the trap. If the field had been periodic homogenous instead and if something had knocked an electron away from the centre, the electron would have oscillated about its new point, moving back and forth. But because the field is inhomogenous, one half of the electron’s oscillation will be in an area where the field is stronger and the other half will be through an area where the field is weaker. And the stronger-field area will exert a stronger force on the electron than the force exerted by the weaker-field area. The result will be that the electron will experience a net force towards the weaker field area. This is called the ponderomotive force. And because the weakest field lies at the centre – where the electrons are originally confined – the apparatus will move any displaced electrons back there. Thus, it’s a trap.

When Wolfgang Paul, Helmut Steinwedel and others first developed the quadrupole ion trap in the latter half of the 20th century, they found that the motion of the charged particles within the trap could be modelled according to Mathieu’s equation. This is a differential equation that the French mathematician Émile Léonard Mathieu had uncovered in the 19th century itself, when he was studying the vibrating membranes of elliptical drums.

During the operation of the quadrupole ion trap, the charged particles experience ponderomotive forces in two directions in alternating fashion: a radial force exerted by the capping electrodes and an axial force exerted by the ring electrode (roughly, from the sides and from the top-bottom). The frequency of the AC applied to the electrodes has to be such that the forces switch sides faster than the electrons can escape. This is the cat analogy from earlier: the cat is the electric field configuration and you are the trapped particle.

With this device in mind, ask yourself: have the electrons been kept in place because counteracting forces have cancelled themselves out? No – that is a static picture that doesn’t allow for any deviations from the normal. If an electron does get displaced from the cubic prison described earlier, Earnshaw’s theorem ensures that it can just escape altogether.

The quadrupole ion trap represents a more dynamic picture. Here, electrons are either held in place or coaxed back into place by a series of forces interacting in a sophisticated way, sometimes in opposite directions but never quite simultaneously, such that particles can get displaced, but when they are, they are gently but surely restored to the desired state. In this picture, counteracting forces still leave behind a net force. In this picture, erring is not the end of the world.