Medical therapy and imaging fixed-field alternating-gradient accelerator with realistic magnets

NORMA is a design for a normal-conducting racetrack fixed-field alternating-gradient accelerator for protons from 50 to 350 MeV. In this article we show the development from an idealized lattice to a design implemented with field maps from rigorous two-dimensional (2D) and three-dimensional (3D) FEM magnet modeling. We show that whilst the fields from a 2D model may reproduce the idealized field to a close approximation, adjustments must be made to the lattice to account for differences brought about by the 3D model and fringe fields and full 3D models. Implementing these lattice corrections we recover the required properties of small tune shift with energy and a sufficiently large dynamic aperture. The main result is an iterative design method to produce the first realistic design for a proton therapy accelerator that can rapidly deliver protons for both treatment and for imaging at up to 350 MeV. The first iteration is performed explicitly and described in detail in the text.

Figure 1

NORMA cell, showing the midplane magnetic field strength in the three scaling FFAG magnets forming the FDF triplet, and the closed orbits for a range of energies.

Figure 2

NORMA round lattice, with minimum and maximum closed orbits shown in red.

Figure 3

Multipole Taylor expansion and fit to an example analytic scaling field; both with dipole, quadrupole and sextupole terms.

Figure 4

Dynamic aperture as a function of real space angle for the nominal design modeled with FFAG, DIPOLES and POLARMESH elements.

Figure 5

Lines of constant scalar potential for the D-magnet (blue). The good-field region required is represented by the rectangle (red) and the origin of the coordinate system is the center of the good-field region. The vectors (green) are proportional to the local gradient of the scalar potential.

Figure 6

A test of the pole optimization procedure performed on the 2D F magnet model. Distribution of the relative field error obtained from three different iterations. The inset shows the 2D flux density distribution as obtained with opera 2D illustrating the low-field and high-field regions of the magnets. The red color corresponds to high flux density.

Figure 7

Multipole components of the 2D field profiles compared to the nominal field. $r$ is the radial distance from the machine center.

Figure 8

Correction to the pole profile of the F magnet obtained after seven iterations.

Figure 9

3D model of the F magnet. (a) Side view without the clamping plates. (b) View from the top showing the pole edge profile without the clamping plates. (c) Half of the pole area with the clamping plates. Their role is to limit the extent of the fringe fields and to eliminate cross talk between adjacent magnets. The magnetizing coil is not shown.

Figure 10

Relative integrated field error for the F magnet for a 6°-sector model, 5°-sector model and a model with an optimized edge. Here $r$ is the radial coordinate of the auxiliary cylindrical coordinate system $(r,\phi ,y)$ introduced.

Figure 11

The pole edge profile $\phi =\phi (r)$ of the F magnet obtained from Eq. (8) is plotted in a Cartesian coordinate system $X=r\cos \phi (r)$ and $Z=r\sin \phi (r)$ and compared to a straight 6°-edge and a 5°-edge.

Figure 12

F and D magnet peak and integrated field for map compared to the nominal design. Dashed vertical lines show the good field region extents. The horizontal axis is the radial distance from the machine center.

Figure 13

Using the 2D opera model as input to the POLARMESH element.

Figure 14

Using the 2D opera model as input to the DIPOLES element.

Figure 15

Cell tune shift for 2D magnets modeled with POLARMESH and DIPOLES compared to the nominal design.

Figure 16

DA for 2D field profiles compared to nominal magnets.

Figure 17

DA for random quadrupole and sextupole error distributions around the 30 MeV orbit. Crosses show individual seeds and the blue bars show their mean and standard deviation. Dotted vertical lines represent the actual size of the errors in the 2D magnet model.

Figure 18

Effect on cell tune at 30 MeV as the fringe field extent length is varied.

Figure 19

Black diamonds show the DA at 30 MeV as the fringe length is adjusted. Recovery of DA by rematching tune to the original value is shown in blue squares.

Figure 20

Effect on cell tune of fringe field that varies with radius.

Figure 21

Rematching the cell tunes with a 6 cm fringe extent at 30 MeV. The star marker shows the working point shift due to changing the fringe extent. The dot shows the working point after rematching to the original tunes. The square and diamond markers show the highest DA point, and the region of highest DA.

Figure 22

Full midplane fields from 3D opera simulation.

Figure 23

Using the 3D opera model as input to the POLARMESH element.

Figure 24

Overlap of individual magnet fields and the full field found by linear summation, along the 50 MeV orbit.

Figure 25

Using the 3D opera model as input to the DIPOLES element.

Figure 26

Horizontal cell tune as a function of energy for 3D magnets and rematched fields.

Figure 27

Vertical cell tune as a function of energy for 3D magnets and rematched fields.

Figure 28

DA as a function of energy for 3D magnets and rematched fields.

Figure 29

Field profile for rematched F magnet.

Figure 30

Field profile for rematched D magnet.

Figure 31

Lines of constant scalar potential for the original and adjusted D magnet.