An **OmniTrak** feature that may not be familiar is the capability to include two-dimensional field solutions from **Estat** or **PerMag**. I recently had a consulting job where this was just the thing I needed, so I thought I would take this opportunity to review why the feature is valuable and how to apply it.

There are two application areas that come to mine where two-dimensional field solutions are useful. The first is when a three-dimensional beam propagates through a symmetric magnetic focusing device (like a solenoidal lens). If we want to include space-charge effects, the electric-field solution is certainly three-dimensional. On the other hand, the beam may have little or no effect on the strong magnetic field. In this case, it is much quicker and more accurate to construct a solution with **PerMag** rather than **Magnum**.

The second area can be illustrated with an application example. Figure 1 shows the cross section of a configuration proposed by Harald Enge[1] for an achromatic bending magnet for electrons and ions. The vertical field in the plane *z* = 0.0 varies as

*Bz(x,0,0) = A x^n*

Particles enter near the point (0.0, 0.0, 0.0) though a small slot in the left-hand pole at an angle ? with respect to the x-axis. The configuration has the property that particles follow a loop trajectory such that they leave the magnet at the same position with angle -?, independent of their kinetic energy. A value *n* = 0.8 corresponds to a 90° reflection (? = 45°). A value *n* = 1.0 gives a 81.4° deflection (? = 40.7°). A field with *n* = 1 is easily achieved — the configuration of Fig. 1 represents one-half of a magnetic quadrupole lens. In this case, the pole faces adjacent to the vacuum gap have a hyperbolic shape, *xz = K* [2].

The bending-field properties hold if the assembly of Fig. 1 has infinite length in *y* (out of the page). On the other hand, a practical system has finite length. One of the goals of calculations is to determine the effect fringing fields. In other words, adding length in y costs money — what is the shortest length consistent with accuracy? We would like to compare trajectories in a finite quadrupole to those in an infinite system. Years of experience have taught me that it is relatively easy to model real-world devices with three-dimensional codes, but extremely difficult and wasteful to approximate ideal devices like the infinite quadrupole. It’s usually much more effective to use a two-dimensional solution.

For this example, I generated the **PerMag** field solution of Fig. 1 for input to **OmniTrak**. The goal was to calculate three-dimensional electron trajectories with horizontal and vertical input displacements in the single-particle limit (*TRACK* mode). There are two points to note:

**OmniTrak**requires at least one three dimensional mesh to record trajectory vectors. Therefore, it is necessary to include a dummy electric field solution with no applied field. I defined a region covering the volume of anticipated orbits (0.0 ?*x*? 280.0, -100.0 ?*y*? 100.0, -20.0 ?*z*? 20.0) with grounded plates at the top and bottom boundaries in*z*(*i.e.*, zero electric field everywhere).- In planar calculations, the
**PerMag**convention is that the system has variations in*x-y*with infinite length in z. The field variations in the Enge paper have the same*x*axis but infinite length in*y*. Therefore we must rotate the two-dimensional field about the*x*axis on entry to**OmniTrak**.

Here is the complete **OmniTrak** input script to represent electron beams at 2.0, 4.0, 8.0, 16.0 and 20.0 MeV entering at ?= 40.7° with a horizontal with of +-3.2 mm:

FIELDS EField3D: DummyField.HOU BField2D: PMSolution.POU Rotate B 90.0 0.0 0.0 DUnit: 1.0000E+03 END PARTICLES TRACK PLIST * Mass Charge Ke x y z ux uy ux * ========================================================= 0.0 -1.0 2.0E6 0.001 -5.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 2.0E6 0.001 0.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 2.0E6 0.001 5.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 4.0E6 0.001 -5.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 4.0E6 0.001 0.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 4.0E6 0.001 5.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 8.0E6 0.001 -5.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 8.0E6 0.001 0.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 8.0E6 0.001 5.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 16.0E6 0.001 -5.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 16.0E6 0.001 0.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 16.0E6 0.001 5.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 20.0E6 0.001 -5.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 20.0E6 0.001 0.0 0.0 0.7501 0.6613 0.0000 0.0 -1.0 20.0E6 0.001 5.0 0.0 0.7501 0.6613 0.0000 END DT 1.0E-12 END DIAGNOSTICS PARTLIST END ENDFILE

Here, *DummyField.HOU* is the dummy electric field solution to define the three-dimensional mesh and *PMSolution.POU* is the field calculated by **PerMag**. The time step of 1.0^{-12} s corresponds to a vector length of 0.3 mm. For this choice, the particle list shows that energy is conserved to better than 0.5 parts in 10^6. Figure 2 shows the resulting trajectory projections in the plane *z* = 0.0. The results are not quite perfect because the calculated field for the geometry of Fig. 1 had *n* = 0.99.

**Footnotes**

[1] H.A. Enge, *Achromatic magnetic mirror for ion beams*, Rev. Sci. Instrum **35** (1963), 385.

[2] My book **Principles of Charged Particle Acceleration** discusses magnetic quadrupoles on page 96.

[3] Contact us : techinfo@fieldp.com.

[4] Field Precision home page: www.fieldp.com.