Question about an electron beam traveling through static B-fields

[mesa]

In the figures below is a sketch of an electron beam entering alternating static magnetic fields perpendicular to that of the motion of an electron beam. This beam is injected from a 45° angle below the x-axis. Figure 1 is a representation of this and is supposed to be an illustration of a basic undulator from a synchrotron.

It would seem that the electron beam would behave in the same manner as in a cyclotron as it goes through each of the alternating magnetic fields with each bending the beam 90°. The radius in each magnetic field is easily calculated and this information can be used to figure out the length of this arc along the x axis. From here we simply add in the spacing between the magnets and get the λ for our beam in an undulator.

There would be some 'fringe' effect on the outer edges of each magnet (please make a correction if that is the wrong term for this application) but due to symmetry the bending of the beam would cancel out and therefore can be represented as linearly alternating beams 45° off the x-axis for calculating the period of the beams wave like pattern.

Is this correct?

*edit, after some thinking there are several approaches to setting up an undulator where the 'arc' is less than 90 degrees of a circle so this formula only applies to those that have an arc specifically of this degree. The formula will require a re-work to accommodate for differing arcs and using a period based on the length along x based of a set of alternating magnets.

*edit, and finally the formula for cyclotron radius is mv/qB not qv/mB, formula dyslexia...

 

[Simon Bridge]

You can get all kinds of cute effects by carefully shaping electric and magnetic fields - yes.
The trick is getting the field ... oh, and finding an application.

 

[mesa]

You can get all kinds of cute effects by carefully shaping electric and magnetic fields - yes.
The trick is getting the field ... oh, and finding an application.

It is a very interesting device. So since I have you here, it seems all forces act radially inward on the x/y plane of the electron beam so the tangential velocity at any point is the same as the initial v_{e beam} as well correct?

Although we can get 'cute effects' trying to produce an accurate sketch of the B_{f ind} with an electron beam moving at near 'c' in a semi sinusoidal pattern in one of these devices is quite challenging (fun too!)

 

[Simon Bridge]

Try the same setup, but with the uniform B field bits as squares, and injecting the electron beam perpendicular to one side. Make a step-pattern of alternating direction fields to keep it going.

You can get an actual sinusiodal path by varying the field strength continuously.

The main advantage of the cyclotron setup is that you only need one magnet.

Alternating fields like that can get you edge effects - the magnets need to be close together - they will attract each others field lines. Alternatively, use smaller magnets placed farther apart.

There's all sort of stuff you can do.
Enjoy.

 

[mesa]

Try the same setup, but with the uniform B field bits as squares, and injecting the electron beam perpendicular to one side. Make a step-pattern of alternating direction fields to keep it going.

You can get an actual sinusiodal path by varying the field strength continuously.

So something like this:

So the field gets weaker by the edges at each transition of alternating magnets due to canceling of fields which are a gradient. There really wouldn't be a perfect arc but instead one that gradually tapers off as the field gets weaker as there is less acceleration radially inward because of this.

Doesn't this effect stretch our wave in the wrong direction if we are trying to mimic a sinusoidal path for the electron beam? or did I set this up incorrectly? or am I just looking at this wrong? :P

 

[Simon Bridge]

So something like this:

[pic]That's the one :)

So the field gets weaker by the edges at each transition of alternating magnets due to canceling of fields which are a gradient. There really wouldn't be a perfect arc but instead one that gradually tapers off as the field gets weaker as there is less acceleration radially inward because of this.

It's just the same as yours from post #1 ... each arc is a quarter circle.

Doesn't this effect stretch our wave in the wrong direction if we are trying to mimic a sinusoidal path for the electron beam? or did I set this up incorrectly? or am I just looking at this wrong? :P

If you are trying to mimic a sine wave - yes of course it does. It's not a sine wave at all, just a series of quarter circles.

You can get an exact sine wave by varying the magnetic field continuously.
You can vary it in strips to get a close approximation.
The B strength determines the curvature of the path - so find the curvature of the sine wave you want to mimic and set the B fields to that function.

 

[mesa]

[pic]That's the one :)

Very good. I would imagine setups for magnet placement are all over the map depending on application. Are there many designs incorporating elctromagnets?

It's just the same as yours from post #1 ... each arc is a quarter circle.

Let me make sure I understand this, there would be no reduction of the radially inward acceleration even with the opposing B_f of the magnet next to it?

If you are trying to mimic a sine wave - yes of course it does. It's not a sine wave at all, just a series of quarter circles.

You can get an exact sine wave by varying the magnetic field continuously.
You can vary it in strips to get a close approximation.
The B strength determines the curvature of the path - so find the curvature of the sine wave you want to mimic and set the B fields to that function.

Okay, I misread your earlier post, sorry about that.
These are interesting devices.