The Mechanical Analogue

The mechanical analogue of the Paul Trap: Trapping a ball in a time-varying quadrupole gravitational potential.

The shape of the quadrupole electric potential produced by a Paul trap can be quite hard to visualise, however, we can use a mechanical analogue to help us picture how the charged particles are confined in the trap.

The mechanical analogue consists of a ball trapped within a rotating hyperbolic saddle point surface, powered by a DC motor. Whilst it is not exactly equivalent to the electric quadrupole potential (it is actually a gravitational quadrupole potential), it gives us a good idea of how the particles are trapped.

The shape of the saddle point is given by where and are constants.

The shape of the saddle representing the gravitational quadrupole potential

As the saddle rotates, the ball experiences a changing potential, represented by the trap’s saddle shape, that alternates between being positive and negative, analogous to the time-varying electric quadrupole potential produced in a Paul trap [1].

Inside the ball’s stable rotation range it will remain trapped in the saddle for an extended period. This is because as the ball starts to roll down the negative potential it is subjected to a periodic force pushing it back up to the trap centre when it hits the positive potential (twice every revolution). As the ball rolls through the saddle’s centre each rotation it experiences a torque, causing it to spin like a spinning top at the centre, with a small oscillatory motion. The ball roughly follows simple harmonic motion if the oscillations are small enough, allowing its motion to be described by Mathieu’s linear equation [1].

The ball’s motion is stable at the centre of the saddle

However, if the rotation speed is too low the ball will simply roll out of the trap because the saddle is not rotating fast enough to ‘catch’ the ball with the positive potential and push it back to the trap centre. On the other hand, if the rotation speed is too high then the ball will fly out of the trap. This is because its motion within the trap rapidly becomes unstable as the greater impulses from the incoming positive potential displace the ball far beyond the saddle centre each time, causing it to struggle to keep contact with the surface as its inertia stops it from following the saddles contours as easily as for lower rotation speeds. Gravity helps the ball regain some contact with the saddle surface, but now it begins to bounce and is flung out of the trap. Here,  the ball’s motion obeys the nonlinear Mathieu equation [1].

The ball’s motion has now become unstable as it moves away from the trap centre

The ball looses contact with the saddle’s surface and is flung out of the trap

The stability and trapping time of the ball in the saddle depends on several factors, including:

• The saddle’s rotation speed
• The saddle’s curvature
• The ball’s curvature
• The ball’s moment of inertia

Ideally, both the ball and the saddle’s curvature should be of the same order of magnitude to limit the ball’s motion on the saddle’s surface before the restoring force can reach it.  The moment of inertia of a hollow ball () is greater than the moment of inertia of a solid ball () of the same curvature, so it requires more energy to roll, meaning we can use slower rotation speeds and the ball should remain trapped for longer. Heavier balls should be easier to trap, as the gravitational force holding them to the saddle’s surface is greater. In addition, saddles with a shallower curvature can trap the ball for longer periods. This is because the gradient of the potential is less severe, so the ball cannot fall away from the trap centre as easily, making the trap more stable.

This demonstration is an example of parametric excitation, where the system is affected by the periodically varying gravitational potential, causing parametric resonance to occur within a range about twice the natural frequency of the ball rolling along the saddle.  (Note – this is different to the system being driven by an external periodic force. Also, the natural frequency is the frequency at which the ball rolls along the saddle when it is stationary and subject to no external forces).

As mentioned earlier, this mechanical analogue is not exactly equivalent to the electric quadrupole potential in the Paul trap. The main difference is in how the potential varies in each case. For the mechanical analogue, the saddle rotated about the saddle’s centre, whereas, in the Paul trap the potential alternates above and below the trap centre (the middle of the ring electrode).  However, despite the difference, it is still a useful test to help us visualise how the particles are trapped.

This video explains how the mechanical analogue can be used to describe how charged particles are trapped in a Paul trap, using a variety of different balls and rotation speeds.

References:

[1] http://www.fas.harvard.edu/~scidemos/OscillationsWaves/SaddleShape/SaddleShape.html