## Question about calculating the electric field of an Octupole

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Justin Fozzard
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### Question about calculating the electric field of an Octupole

Hello,
I'm trying to calculate the electric field at any point within an octupole with a central electrode.

It is quite straightforward to model it in software like FEMM and also calculate the field of an octupole without a central electrode analytically, but I need equations for the field with a central electrode to then plot ion trajectories using Mathcad Prime.

Could you give me some hints how to calculate the field at any point given the electrode dimensions and voltages applied to them, please?

The outer electrodes have an alternating voltage applied and the central electrode is at a high dc potential with repect to them.

Here is the result of a typical simulation using FEMM 4.2

Electrode radii and applied potentials:
Bore 48mm
Shield 144mm 0V
Electrodes 18mm +/- 1000V
Central electrode 16mm +1000V
Thanks,
Justin Fozzard
Liam David
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### Re: Question about calculating the electric field of an Octupole

What method are you using to obtain the analytic octupole solution without the central electrode? A polar BVP on wedges, or a crap ton of image charges, comes to mind.
Justin Fozzard
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### Re: Question about calculating the electric field of an Octupole

Thanks for the reply, Liam,
I think one way of solving analytically for the electric field of the octupole of circular electrodes without the central electrode is to first find the field of a more easily analysed problem using Laplace's equation; there's a good example in Friedman, pp.12, 21 [1]. Here the fields for an arrangement of hyperbolic electrodes are analysed.

This solution can then be modified using the methods discussed in Baartman [2] & [3]and Rao [4] to provide an optimised solution for circular profile electrodes.

Gerlich [5] is also a very useful paper.

[1] Friedman 1981 Fundamentals of Ion Motion in Electric Radiofrequency Multipole Fields. [2} Baartman 1995 Electrostatic Multipole shapes
Baartman 1995 Electrostatic Multipole shapes.pdf
[3] Baartman 1998 Intrinsic Third Order Aberrations in Electrostatic and Magnetic Quadrupoles [4] Rao 2000 Electric hexapoles and octopoles with optimized circular section rods [5] Gerlich 1992 INHOMOGENEOUS RF FIELDS
Gerlich 1992 INHOMOGENEOUS RF FIELDS.pdf
Liam David
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### Re: Question about calculating the electric field of an Octupole

It looks like you're far ahead of me in analyzing the problem. I would have thought, given the symmetries, that an exact solution exists to multipoles with circular electrodes (as nasty as it might be with piecewise regions and such), but it seems not. I know that one can do a separation of variables BVP in bipolar coordinates, but I haven't found whether that's possible with a multipolar coordinate system. For what it's worth, I did a quick search and https://link.springer.com/book/10.1007/ ... 471-0495-7 has very brief discussion of multipolar coordinates (pg. 348). Unfortunately I don't think I'll be of much help. Best of luck!
Justin Fozzard
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### Re: Question about calculating the electric field of an Octupole

Thanks Liam,
I've had a quick look at page 348 online and my brain has broken. Looks like I will have to find the "Idiot's guide to multipolar coordinates" if I want to go further with that approach.

I have, however, found a possible way forward that may be easier.
There are some LUA commands within FEMM that I hope be able to use to export the electric field at each point for use in Mathcad as an array of vector components:
I have much to learn…
Frank Sanns
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### Re: Question about calculating the electric field of an Octupole

It is not a complete solution but a simple way to look at it is to take a slice of it. Technically it is just four points but repeated. Those four points are one constant voltage electrode with three other alternating voltage electrodes equidistant from it ( the radius of the circle). It will not be complete solution but it is a simple way to get an idea of what the ion paths will look like with the given configuration.
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Liam David
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### Re: Question about calculating the electric field of an Octupole

Justin, admittedly I know nothing about multipolar coordinates and I've never even worked with bipolar coordinates, much less done a BVP in them. It was just something I found when doing some research about your problem.

I've used FEMM's Matlab interface to extract the fields on a grid and compare them to values generated by my own code. Works pretty well by my experience, if a little slow for large grids.
Nicolas Krause
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### Re: Question about calculating the electric field of an Octupole

Hi Justin,

There's got to be some way of using the symmetry in your setup to find a nice solution. I think some kind of conformal mapping might do it, but that's a conjecture probably influenced by the fact I've spent a lot of time reviewing complex analysis in the past year. I might suggest you take a look at Electromagnetic Theory: Static Fields and Their Mapping by Ernst Weber. Chapter 7 discusses analytic solutions to two-dimensional fields.
Justin Fozzard
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### Re: Question about calculating the electric field of an Octupole

Thanks Nicolas, Frank and Liam, your comments are greatly appreciated.

You've given me some good clues for further research. The Weber book mentions fields around the electrodes in a triode, something I had forgotten about; I think I may have some textbooks and papers that explain the method to calculate these.
My main problem, though, is trying to remember stuff we did during my undergraduate physics course. I'm sure we did dozens of similar problems but it was over 35 years ago and my brain is now full of less useful stuff...
I will update the post when I find the solution.
Nicolas Krause
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### Re: Question about calculating the electric field of an Octupole

Justin,

I found this absolutely wonderful description of an old method of "calculating" electric potentials at the following link
Electric potential in a two-dimensional system of electrodes is analogous to physical height. An electode at a given potential is analogous to a level surface. Moreover, the height of a stretched sheet of thin rubber supported by "electrodes" of various heights is closely but not exactly analogous to the electric potential between actual electrodes of the same shapes.

In this rubber sheet analogy, the electric potential (height) and the electric field (slope) appear before ones very eyes. And, even better, by rolling ball bearings down the stretched rubber one can find very nearly, and certainly nearly enough, where electrons will go in the analogous electric fields. With this tool, it was easy to find a common shape for succeeding opposed and staggered electrodes at progressively higher potentials that would focus the secondary electrons from each electrode effectively on that of the next higher potential.

The electrode shapes that I worked out proved very effective in the photomultipliers that the tube shop on the second floor constructed for me. In 1941 I finally published a paper on this electrostatically focused photomultiplier with a colleague, R. C. Winans, who had made an improved structure. Electron multipliers of similar design are used today. A patent was applied for. Similar work had been done about the same time at RCA. I don't know who got the patent.
It's from the work of John R. Pierce, I believe the paper he is referencing is the following. In addition the electrolytic tank method he mentions is described in a paper here. I was absolutely unaware of these physical methods of determining electrostatic potentials, but was inspired to find them thanks to some comments Richard had written last year about using phosphors to determine electron paths in vacuum tubes.
Patrick Lindecker
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### Re: Question about calculating the electric field of an Octupole

Hello Justin and all,

An analytical solution seems extremely complicated (at least for me). If I had to calculate the electric induced potential at any point, I will solve it using a matrix solution and classical electrostatics formulas, decomposing the big problem in elementary problems.

I will first associate to each point of metal an electrical charge (unknown a priori) and a voltage (known apriori). Lets's say 1000 charges and 1000 voltages. Then, I will solve the system Vector(Voltages, known)=Matrix of the inverse of capacities (to calculate knowing the distance between the coordinates of a point of voltage and the coordinates of a point of charge)= Vector(charges).
With 1000 points, the vectors will be {1000,1] and the matrix [1000,1000]. To determine the charges, the system must be solved using any method (Cramer for example). Once the charges known, you can determine any induced potential at any point using the Coulomb formula. This method can be applied to any configuration, as it only needs to split your elements in discret points.

See https://elearning.univ-bejaia.dz/plugin ... ue%202.pdf page 33 (document in French).
The solution will not be exact but approximate as the number of points might be infinite for an exact solution.

Patrick Lindecker