1. Example: Magnetic Quadrupole Lens

2. Introduction
This is an example model of a system of magnetic quadrupoles for use in a nuclear accelerator
The beam consists of Be5+ ions, traveling through three consecutive magnetic quadrupoles with the velocity 0.01c.
The quadrupole triplet acts as a lense, focusing the beam in the transversal plane.
Two quadrupoles, a ”doublet” can also focus an ion beam, but it acts as an astigmatic lens, i.e. with two separate focal planes, one per axis.

3. 3D Geometry
iron
magnets
N inward
S inward

4. 2D Geometry
iron M
magnet
Cross section of the 3D geometry

5. Results The magnetic field distribution.
The iron core enhances the field and makes it remain largely within the quadrupole. Without it, it wouldn’t be safe to keep sharp metal objects or wear pacemakers anywhere near!
The magnetic field distribution.

6. Results
The magnetic potential contour lines are also the streamlines of the magnetic flux density. In a perfect quadrupole, they are exactly hyperbolic.
The magnetic field distribution and approximately hyperbolic contour lines for the magnetic potential.

7. Particle Tracing The force on an ion is given by drift (1m) Q1 (1m)
The force expression is multiplied by -1 for L1<partt*vz<L1+L2, and 0 for partt*vz>L1+L2+L3 (no forces during drift). In 2D, only the transversal part of the force can be computed. This is fine, since the x and y velocities (and thereby the z component of the force) are negligible compared to the z velocity.
Q2 (2m)
Particle tracing in 2D is done with a logic expression reversing the polarity of the force when the ions are in Q2.

8. Results
Magnetic field and ion trajectories.

9. Results Ions starting from R = 3 cm all arrive within R = 1 cm. Q1 Q2
drift
1 cm
3 cm
The ions start out evenly distributed on a circle in the x-y plane (at z=0) of radius 3 cm. Q1 is focusing in y and defocusing in x. Q2 has the reverse polarity, and Q3 has the same polarity as Q1. The drift length beyond Q3 is about 2 m.
Ions starting from R = 3 cm all arrive within R = 1 cm.

10. Results 3D Magnetic field and ion trajectories.
The particle tracing shows better in the next slide.
Magnetic field and ion trajectories.

11. Results 3D
The force density is roughly proportional to the distance from the beam center. The dark blue parts (low force density) are at the ends of the quadrupoles (because the fields from two adjacent quadrupoles cancel each other) and beyond the triplet.
Ion trajectories only. Color scale shows local force on each ion.