A problem which has been around for some time is the full description of fringe fields as a bunch goes into or out of a multipole magnet. A complete description of fringe fields for dipoles has been around for quite some time. In the dipole case, the magnetic field is confined to one dimension (vertical for a horizontally bending magnet) when the bunch is inside the magnet. Then, in order for the field to be able to decay to zero in a continuous way whilst still satisfying Maxwell’s equations, a second, longitudinal, component needs to be introduced. After some algebra, this second component can be made to satisfy the required equations easily for any desired fall-off longitudinally. This is shown in Fig. 1 below for the simplest kind of Enge-type fall-off given by a single parameter only. Note that the two singularities appearing lie at either side of the dipole, in a region where the beam will never go. They could be thought of as the place where the current goes to the coils of the magnet. Analytically, the singularities are necessary given that any solution to Laplace’s equation, and hence the magneto-static Maxwell equations, that is bounded at infinity either has singularities or is constant.
Figure 1: Field components By and Bz in the fringe field region of a dipole magnet with the fall-off of the field, longitudinally (z), given by a single parameter Enge function (courtesy of PRST-AB).
The same is not true for multipoles where the field inside the magnet already requires the two transverse components of the magnetic field for a complete description. However, the same situation as in the dipole case still holds and a third, longitudinal component of the magnetic field, needs to be introduced to enable the two transverse fields to decay to zero whilst still satisfying all the magneto-static Maxwell equations. Until now, this was addressed by making an expansion about the origin and hence solving and satisfying Maxwell’s equations iteratively. This is a very good approximation if the bunch is centred about the origin throughout the accelerator under consideration. However, this procedure is far from exact and leads to considerable mistakes when applied to novel machines like EMMA where the beam goes considerably off-axis and, therefore, an expansion about the origin is unlikely to give sufficient accuracy. By solving the full three dimensional magneto-static Maxwell equations, it is possible to have a complete fall-off description for any kind of decay desired for all three required components of the magnetic field for any multipole. This problem was recently solved, for any multipole, by STFC ASTeC and Liverpool University at the Cockcroft Institute and a complete article was published in PRST-AB and is available for download here (http://dx.doi.org/10.1103/PhysRevSTAB.18.064001).
The expressions constitute the first time a closed form solution, for all field components, was given for fringe fields to all order of multipoles. The field components are shown in Fig. 2 as various snapshots from inside to outside a quadrupole, for illustration. In this case, a very simple one parameter Enge-type fall-off was again used, the details of which can be found in the paper, but here again, any desired fall-off may be used.
Figure 2: Quadrupole fringe field components Bx (left), By (center) and Bz (right) going from inside to outside the magnet longitudinally (courtesy of PRST-AB).
Once again, Fig. 2 shows the appearance of singularities and, again, these can be arranged to lie outside the region of interest. To better illustrate these fall-offs, three movies can be seen below showing the evolution of each of the components of the magnetic field as the bunch goes from inside the quadrupole to outside.
With the analytic expressions for quadrupole fringe fields, it is also possible to look at pole-faces for the magnets which quickens the initial design for them. Such an example is worked out in detail in the paper and results in the pole-face in Fig. 3 below for one of the HL-LHC inner triplet quadrupoles.
Figure 3 Surface of constant scalar potential in a representation of the HL-LHC inner triplet quadrupole (courtesy of PRST-AB)
It is also possible to include these closed expressions in such programs as MAD (Methodical Accelerator Design) or GPT (General Particle Tracer), and thereby include fringe fields right from the start of an accelerator design. The alternative involves waiting for a full magnetic field map to be available which requires precise modelling and therefore knowledge of the specifications required. Full expressions for quadrupole fringe fields have been implemented into GPT, together with the help of the authors of that code. The resulting smooth fields not only yielded more accurate results but also showed an impressive reduction in the CPU time required to do computations of more than two orders of magnitude with respect to the equivalent hard edged model and barely increasing at all when higher accuracies were desired. This reduction is shown in Fig. 4 below:
Figure 4: Increased accuracy vs. CPU time for hard edged (black) and fringe field models. (courtesy of S.B. van der Geer and M.J. de Loos (authors of GPT)).
Authors:
Bruno Muratori (ASTeC , STFC Daresbury Laboratory and Cockcroft Institute) (bruno.muratori@stfc.ac.uk),
James Jones (ASTeC , STFC Daresbury Laboratory and Cockcroft Institute) (james.jones@stfc.ac.uk),
Andy Wolski (Liverpool University and Cockcroft Institute) (a.wolski@liverpool.ac.uk).