- #1
mesa
Gold Member
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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...
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...
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