# Beam dynamics

1. Dec 12, 2014

### Alkass

Hello

I have this problem - From a generator, I get a compton scattering with the electrons theta and phi angles.
where I having the following equations for a particle

px = E_particle * sin (theta) * cos (phi);
py = E_particle * sin (theta) * sin (phi);
pz = E_particle * cos (theta);

where polar angle θ (theta), and azimuthal angle φ (phi). So, I am trying to build some kind of spherical-Cartesian transformation, as I need to have y'=dx/dy and x'=dz/dx in order to build the transfer matrix for beam dynamics (ie transport of the beam inside a magnet following thin lens approximation)

Would that be possible to have such transformation ?

Thanks

Alex

Last edited: Dec 12, 2014
2. Dec 12, 2014

### ofirg

Are you trying to express spherical coordinates in terms of the Cartesian coordinates? This is found in many sources.

For example http://mathworld.wolfram.com/SphericalCoordinates.html

3. Dec 12, 2014

### Alkass

Well, maybe I am missing something - The movement of the particles is along the z-axis (ie the reference obrit) and I need to calculate the y' = ds/dy and x'=ds/dx - So, the y' is actually the theta angle, but the phi angle accounts for the angle in the X-Y plane, while I would need the x' = the angle on the X-Z plane meaning I need the roll angle about the local s-axis...

4. Dec 12, 2014

### ofirg

This is entirely derivable in terms of the angles. You are trying to calculate the ratio between the change in y (or x ) to the change in z. Since you are keeping the direction constant, the angles don't change and the change in the total distance ($\Delta r$) cancels out. Use the relation between the coordinates given in the link to take the ratio and get the results.

5. Dec 12, 2014

### Alkass

I guess you mean relation (95) ? And what happens to the dr ? for me I am starting with some energy / px, py,pz instead...

6. Dec 12, 2014

### ofirg

The energy doesn't matter here. You have particles moving in some direction $(\theta,\phi)$ and want to calculate $\frac{dx}{dz},\frac{dy}{dz}$(or you can use s instead of z). Relation 95 is much more complicated than what you need since in your case $(\Delta \theta,\Delta \phi)$ are zero. From relations 4,5,6 you can get $(\Delta x, \Delta y, \Delta z$) for $(\Delta \theta = 0,\Delta \phi = 0)$. $\Delta r$ cancels out in the ratio and you get want your looking for. Notice that in the formulas in this link the roles of $\theta$ and $\phi$ are switched.

7. Dec 12, 2014

### Staff: Mentor

I would expect that to be y'=dy/ds and x'=dx/ds.
With s=z (to a very good approximation outside magnets if I remember correctly), this is y'=dy/ds=py/pz.
The energy cancels in this ratio.

8. Dec 12, 2014

### Alkass

So, then I guess I need to calculate the actual derivative *not* just the ratio, right ?

9. Dec 12, 2014

### Alkass

Yes, that is correct ie y'=dy/ds and x'=dx/ds as you care about the change of y/x on the direction of the movement ;-) So, when you say "outside of magnets" what do you mean ? and what about x' ? is there a similar approximation (and any reference would be great!)

Thanks bunch!

10. Dec 12, 2014

### ofirg

Since the ratio is constant it is the same.

11. Dec 12, 2014

### Staff: Mentor

(Dipole) magnets give a changing z-direction with s which is ugly, but should not matter as your compton scattering happens at a single point anyway.

Same as for y', of course.
Simple algebra, nothing you would find in a reference I guess. Books of beam dynamics should cover that somewhere.