# Cyclotron simulation

Nicholasfox shared this question 1 year ago

I'm trying to simulate a Cyclotron, as the attached mp4 video, but have no idea. Please help!

Some considerations,

1. time slider is the animation, so particle velocity changes can be shown.

2. Consider the effect of relativistic effects on the mass of the particle.

3. In CAS, I want to get the acceleration as a function of time, but failed.

Certainly an unexpected solution and without any calculations but assuming that the gap between the two electrodes is 0 (what is not the case).

The (constant) frequency is given by the speed of E. The visual speed increase is obtained by the (ggb) fact that the 11 semicircles (plus 1 segment = 12 elements) are passed each (element) in the same time (with longer way).

I mean: this corresponds exactly to the mode of action of a cyclotron. Of course without taking into account the voltage and the associated acceleration between the electrodes as well as the power of the magnets. This is implicitly given by the larger radius and speedE.

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and a snapshot

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Certainly an unexpected solution and without any calculations but assuming that the gap between the two electrodes is 0 (what is not the case).

The (constant) frequency is given by the speed of E. The visual speed increase is obtained by the (ggb) fact that the 11 semicircles (plus 1 segment = 12 elements) are passed each (element) in the same time (with longer way).

I mean: this corresponds exactly to the mode of action of a cyclotron. Of course without taking into account the voltage and the associated acceleration between the electrodes as well as the power of the magnets. This is implicitly given by the larger radius and speedE.

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Never thought it could be done this way. Thank you, very much.

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I still need time to understand your work.

By the way, relativistic effects cause the mass of the particle to become larger as the velocity increases, so the radius of motion grows less rapidly. Can you help to achieve this visual effect?

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quote: "By the way, relativistic effects...."

This question is beyond my knowledge.

But: after some research I found out that cyclotrons "get out of sync" at a particle velocity of 10% of the speed of light and the speed cannot accelerate further. They get out of sync because the particles are too slow (but somewhat heavier) and reach the gap between the electrodes after the polarity has change (too late).

I conclude from this (without mathematical proof) that the orbits (semicircle radii) do not become so much smaller that the speed loss would be compensated. Possibly the radii (due to the larger mass and the associated centrifugal force) are as large as they would be from a non-relavistic point of view.

However: with a constant clock rate, the particles arrive too late at the gap between the electrodes. Thus, they are slowed down instead of accelerated.

Assumption: At 10% of the speed of light, the way/time changes are so minor that they are no longer optically detectable (in a simulation).

Long speech short sense: if I could calculate it (and I can't) nothing (visible/recognizable) would change in the simulation.

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You're right, I made a mistake.

The reduction in the growth rate of the radius is not due to relativistic effects, It is supposed to be.

Without considering the real calculated value, I would like to qualitatively show this feature.

https://virtuelle-experimen...

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Based on your work, I modified it into another version.

Thanks.

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This looks good.

I am currently working on a slightly modified approach and still need some time.

This can be used to verify the two solutions.

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Attached my solution.

The two solutions agree (exactly enough).

In my solution the number of cycles are variable and I used your cleaned background image.

In addition, a checkbox lets you project your graphic semi-transparently over my solution.

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I interpreted your function for the radii as recursive and did not understand it either.

Therefore I have derived the function vRad_n as follows:

If we set all constant variables in the function for the velocity (v) equal to 1, the variable cycle remains.

This results in the function f(x)=sqrt(x) where x corresponds to the cycle. The radii are proportional to this.

The largest radius (maxRad) determines the factor by which f(x) must be multiplied to obtain the radii.

The first cycle results in an insignificant deviation from your solution, which I did not examine more.

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I think I found the difference. I treated it as the nth acceleration, you treated it as the nth lap.

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Yes, I see it.

3 possibilities: yours right, mine right, both wrong.

I can't decide.

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In either case, Its enough to show the visual effect. Thank you.

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I'm still working on this topic. After introducing the gap between the two electrodes, I was expecting to dynamically change PointE's speed by "speedTune" variable, But the results were not what I expected.

Can you help me? Thanks.

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What do you want to achieve and why (to show/demonstrate what).

How do you want to achieve it.

What (exactly) does not work where and why do you come to this judgment.

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By the way, I suspect that you may not have fully understood the handling of velocity in composed paths.

Note the question in the following attachment

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And...

If I assume that this formula is correct (personally I am not so sure, but my knowledge is too small), then I must accept that the speed (v) does not increase by the same amount with each pass (n) in the gap between the electrodes (not relativistic). This then has an effect on the increasingly reduced radii of the circulating routes, where each individual half-circle is passed through in the same time (cycle time).

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I wanted to get a visual effect of Point E accelerating smoothly.

I've checked your last attachment, point passes paths with different length costs same time, and that is just what I want to work out.

When point E crosses the gap, I tried to increase its speed by Multiply speedE by a factor(the length of the gap's previous semicircular arc divided by the gap's length), as follow,

b=floor(PathParameter(E) Length(l3))

If(Mod(b, 2) ≟ 1, speedE Element(l3, b) / vHigh, speedE)

But the result were not as smooth as expected，Its speed is fast and slow, not monotonically increasing.

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"...the speed (v) does not increase by the same amount with each pass (n) in the gap between the electrodes (not relativistic). This then has an effect on the increasingly reduced radii of the circulating routes, where each individual half-circle is passed through in the same time (cycle time)."

I totally agree.

In your first attachment, you used the characteristics of velocity in composed paths to achieve a acceleration effect, I think it was really a genius idea.

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I am a GGB beginner and I'm not very familiar with GGB. But I am very interested in GGB, Thank you very much for your many times of help.

Now I know "A speed of 1 means that the animation takes about 10 seconds to run once through the interval of the slider." So I have another idea, still to be verified. Unfortunately, I faced another difficult, How to get the index of the closest value(ceil or floor) to a specified value in a list, is there a command exist？

For example, list1={1,2,3,4,5}, how to get index=1(or 2) for a=1.2 (list1(1)<=a<=list1(2))

Thanks.

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The GGB parameter "speed" is unfortunate. Actually it should be "duration/time".

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Quote: How to get the index of the closest value(ceil or floor) to a specified value in a list, is there a command exist？

No, not exist.

But you can use for a point E on a path in a List with n Elements:

E_ndx = floor(PathParameter(E) * Length(<list>)) + 1

If you want to ask for even/odd then use:

IsEven = Mod(E_ndx,2)==0

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Thanks, I'll try it later.

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Hi, Rami, I think this version meets my needs, point E has a smooth acceleration. Thanks for all your help.

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If I have understood the cyclotron and your version 4 correctly, then the simulation basically does not correspond to reality.

The ion does NOT accelerate on its way through the semicircle but only at the transition between the two electrodes. If it would really accelerate in the semicircle, then the path would not be a semicircle (at constant magnetic field). Outside the gap (gap between the electrodes) there is no directional force (apart from the magnetic field) that could accelerate the velocity of particle in one direction or the other (an electrostatic field needs 2 poles to have an effect). The magnetic field does not accelerate the ion either, it forces the particle onto a circular path proportional to its velocity (partially comparable with gravity for a satellite)

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Yes, Acceleration exists only between the two electrodes.

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Sorry, I did not look closely and overlooked that in version 4 the point E does not skip the gap but passes through it. This is a very challenging task and solution to it. Congratulations.

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Limited by my English level, I may not have expressed my meaning accurately. In short, without your help I could not have completed this task. Thank you.