Calculating trajectory by matrix methods

[Kolahal Bhattacharya]

For last few days,I was trying to calculate the trajectory of charged particles by matrix method.We have learned geometrical optics by matrix methods in our course.So,applying the same token,various interesting results might be obtained for a charged particle beam subjected to electromagnetic environment.
For freshers,it is enough to know that what we do is to essentially relate two points on a ray through a transfer matrix.Suppose,there is an EM lens that curves a beam.We take two points on the ray-one just before entering the beam,the other just after the beam has emerged.
Below I give a couple of cases from those I have got so far.Please check them.And contribute in this thread to enrich the discussion. I wish everyone to take part in this discusion...
The fundamental operations that a charged particle beam may be supposed to undergo are simple translation; deflection in electric field and deflection in magnetic field.
I am using spherical polar co-ordinates in the frame of right handed Cartesian axes.Taking an arbitrary ray (of charged particle) I orient my axes in such a way the ray lies in yz-plane of paper.Let it passes through the origin..The two points on it from which the projections are drawn onto z axis are P1 and P2.the lower is P1,say.P1P2 =D.The lengths of the projections are y1 and y2.From P1,I have drawn a line || to z axis.This will serve to determine the angles.Let the state of ray at P1 be (r1,θ1,Ф1) and that at P2 be (r2, θ2,Ф2).Then from geometry,
θ1= θ2
Ф1= Ф2
The r-components may not be written directly.So,I will use the y’s.
y2=y1+D sinθ1
we must assume always that these angles θ1 and θ2 are very very small,i.e.<< 1 radian.So sin θ1= θ1. Also,y1,y2<<D.Note that if these are not satisfied, the approximation regarding linearity will be violated and we may not work with paraxial approximation.So,you may construct now the transformation matrix: relating (the co-ordinates) P1 and P2.

3x1:(y2,θ2,Ф2)
3x1:(y1,θ1,Ф1)

3x3:

R1:D 0 1
R2:1 0 0
R3:0 1 0


There is no new physics hidden here.However,from the next example the charged particle behaviour is more prominent than this one.
Take a finite uniform electric field in x direction.Suppose the ray is in zx plane.It has a high z component of velocity.Its direction is sligthly changed as it emerges out of the field.
Also let the speed of the ray is v. As in the previous case, Ф is constant in the plane of consideration.
The state of ray at P1 (lower point) is (x1, θ1,Ф1) and that at P2 is (x2,θ2,Ф2).

Note that before entering the region,the v_z1=v cos θ1=v
And v_x1=v sin θ1=v (θ1)
After leaving the region,
v_z2=v
v_x2=v_x1-at
= v(θ1)-[eE/m](D/v)
Note that the approximate time to cross the region is precisely D/v.
You may even do better. θ2=v_x2/v_z2= θ1-(eE/m)(D/v2)
This much there is no problem.Now I am to relate x2 and x1.
Should I relate like this:
X2=x1-(1/2)(eE/m)(D/v)2?
Once this is clarified,the transformation matrix may be established.


There may be many more configurations.I have the treatment for a couple of them.I will send you those within a few days.we migh t encounter a cases where transcendental functions may come.I really do not know if a series expansion would do in those cases.ButI hope that will.I am not writing more as it is very cumbersome to borough constantly those thetas,phis etc.Ok.good luck.

 

[eljose79]

Thank you for sharing your thoughts and calculations on the trajectory of charged particles using the matrix method. It is an interesting and important topic in the field of electromagnetism and particle physics.

Firstly, I would like to clarify that the matrix method you are using is based on the paraxial approximation, which assumes that the angles of deflection are very small and the distances traveled are also small compared to the size of the system. This approximation is valid for most practical cases, but it may not hold true for extreme scenarios.

To answer your question, yes, you can relate x2 and x1 as X2=x1-(1/2)(eE/m)(D/v)2, which is based on the equations of motion for a charged particle in an electric field. This will give you the approximate trajectory of the particle after it has passed through the electric field.

However, as you mentioned, there may be cases where transcendental functions may arise, such as in the case of a non-uniform electric field. In those cases, a series expansion may not be sufficient and more advanced mathematical techniques may be required.

I look forward to reading about your results and further discussions on this topic. Thank you for initiating this discussion and inviting others to contribute. It is through such collaborations and discussions that we can enrich our understanding of complex physical phenomena.

 

[charng]

I find your approach to calculating the trajectory of charged particles by matrix methods to be very interesting and potentially useful. It is a creative way to apply the principles of geometrical optics to the behavior of charged particles in an electromagnetic environment. Your use of spherical polar coordinates and the paraxial approximation is a clever way to simplify the calculations and make them more manageable.

I also appreciate your willingness to share your findings and invite others to contribute to the discussion. This kind of collaboration and exchange of ideas is essential in scientific research.

I do have a few suggestions for your approach. In your first example, where you use the transfer matrix to relate two points on a ray, it would be helpful to include the equations for the transformation matrix. This would make it easier for others to follow and replicate your calculations.

In your second example, where you consider a charged particle in a uniform electric field, it would be beneficial to include the equations for the electric and magnetic fields, as well as the equations for the velocity and acceleration of the particle. This would provide a more complete understanding of the system and help others to verify your results.

Furthermore, in your final example, where you mention the possibility of transcendental functions arising, it would be helpful to discuss potential solutions or approaches to dealing with these functions. This would make your work more practical and applicable to a wider range of scenarios.

Overall, I am impressed by your approach and look forward to seeing more of your findings in the future. Keep up the good work and thank you for sharing your ideas with the scientific community.