# Einzel Lens Operating Principle

1. Nov 18, 2008

### craq

I'm trying to design an Einzel lens to focus an ion beam. The literature seems pretty consistent that the beam diverges in the first gap and converges in the second gap. (Gaps are between the three electrodes.) What I don't understand is that the potential is lowest at the centre of a cylindrical lens (highest closer to the central electrode). Why should the positive ions be attracted to a region of higher potential?

I have written a simulation which solves for voltage (the laplace equation by the finite difference method), finds the electric field (as the gradient of voltage) and then calculates the particle trajectory. The simulation results show the beam bending towards the axis at the first gap and hardly any change at the second one.

I'm obviously missing something here, can anyone point it out to me?

2. Nov 21, 2008

### craq

Ok, I figured it out... just needed to let the laplace solver run to a few (thousand) more iterations to get a better solution for the voltage profile. My initial conditions are V=0 and the centre of the cylinder stayed pretty close to 0, giving a radially inwards electric field everywhere. (Except very close to the axis, where it was very slightly outward, small enough that I dismissed it as a numerical error.)

3. Dec 8, 2008

### Ravaner

Hello. I've myself developped a software in order to determine trajectories of electrons and ions in every kind of lenses, so I'm interested by your own developpement ...

4. Dec 8, 2008

### Redbelly98

Staff Emeritus
If you (or your company, or your thesis professor) can afford it, there is commercially available software that does this:

http://www.simion.com/

5. Dec 8, 2008

### craq

Hi,
this was just for a one-off design, so I had a look at the Simion demo but decided to write it myself in Matlab. Oversimplifying for a second, its just solving the Laplace equation to give the potential distribution and then the Lorentz equation for the motion of the particles (later in the trajectory I have a confining magnetic field too). I ignore space charge effects, although I have been warned that at low velocities they can be dangerous.

I used a multi-grid Gauss seidel method to accelerate the convergence of the finite difference equation. These guys explain it really well:
grids.ucs.indiana.edu/ptliupages/jsucourse2005/jsuparallelpdesolution05.ppt
There were a couple of other problems I ran into, but nothing really tricky... And eventually I was able to show that an Einzel lens is unfortunately not going to work for us.

If you have any suggestions, or want to know anything specifically about my simulation I'd be glad to hear from you.

6. Dec 8, 2008

### Ravaner

Could you tell what application is yours. Mine is an homemade Time of flight Mass spectrometer. To return to your software development, after having solved Laplace equation, you directly use forces (Field) lines, in my software I apply Snell-Descartes law between 2 successive equipotential lines ...

7. Dec 8, 2008

### craq

I am building an ion beam, and the lens was intended to give real-time control of the beam diameter.

Snell-Descartes is an interesting idea, possibly it solves much faster, but I can't immediately see how to code it. My goal was to visualise particle trajectories, so I find the acceleration at each location and solve kinematic equations to give the velocity and position. The drawback here was that to get a good indication of the dependence on radial position and initial velocity meant I had to take a large number of test particles.