1. Magnets for the FFAG ‘Pumplet’ Neil Marks ASTeC STFC Daresbury Laboratory FFAG Worksop, Grenoble, April 2007

2. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Basic Parameters (*) (*) Electron Model for a 3-10 GeV, NFFAG Proton Driver – G.H.Rees. Particles:electrons; Purpose:to model a 3 to 10 GeV proton driver; Number of cells:27; Energy range:3.0 MeV to 5.44 MeV; Gama range:6.87 to 11.66; Orbit circumference:23.778 m to 23.760 m; Betatron tune:8.30 (h) x 6.23 (v).

3. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Layout (*) -8.0° 7.5° 14.333° 7.5° -8.0° bd(-) BF(+) BD(+) BF(+) bd(-) 17.5 5.5 5.0 5.5 5.0 11.0 (cm) 5.0 5.5 5.0 5.5 17.5 Magnet bend angles and cell lengths for the 5.446 MeV, electron reference orbit. (*) Electron Model for a 3-10 GeV, NFFAG Proton Driver – G.H.Rees.

4. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Magnetic Field requirements. Fields are defined as a function of energy: T(MeV) −bd(gauss) BF(gauss) BD(gauss) 5.4463 502.61014 471.19703 450.25492 5.270 502.07071 448.66274 460.62734 5.100 500.66550 425.95657 470.66254 4.950 498.59188 404.89207 479.60411 4.800 495.64991 382.83608 488.61128 4.650 491.87125 359.83102 497.63974 4.500 487.29235 335.94282 506.63590 4.350 482.02073 311.29840 515.61337 4.200 475.93152 285.87404 524.32281 4.050 469.26794 259.89203 532.89601 3.900 462.04515 233.43397 541.20674 3.750 454.35053 206.63685 549.18928 3.600 446.28753 179.65751 556.77548 3.450 437.97707 152.67488 563.89485 3.300 429.50293 125.79372 570.51425 3.150 420.84710 99.000552 576.61410 3.000 411.92391 72.234761 582.16910

5. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Magnetic Field requirements. Position are defined as a function of energy: TMeV Kv(m-2) bd X(mm)  Kv(m-2) BF X(mm) Kv(m-2) BD X(mm) 5.446… 0.25 17.984403 76.346784 14.587926 60.835520 8.4826225 5.270 2.67 16.065935 78.355600 13.093898 60.188660 7.6037669 5.100 5.07 14.138198 80.235645 11.579084 59.408088 6.7170154 4.950 7.69 12.365652 82.175093 10.169445 58.542845 5.8925272 4.800 10.15 10.517979 83.865580 8.6837834 57.513771 5.0247381 4.650 12.45 8.5913580 85.314328 7.1189881 56.318225 4.1124628 4.500 14.57 6.5810794 86.512030 5.4695276 54.953655. 3.1544761 4.350 16.49 4.4828050 87.447889 3.7365221 53.416830. 2.1501785 4.200 18.20 2.2911316 88.114074 1.9145014 51.699838 1.0988121 4.050 19.67 0.0000000 88.487316 0.0000000 49.796092 0.0000000 3.900 20.88  2.3975051 88.544675  2.0106491 47.692107  1.1464574 3.750 21.79  4.9095333 88.245813  4.1212940 45.373991  2.3403381 3.600 22.36  7.5457418 87.539848  6.3360636 42.818705  3.5810667 3.450 22.54  10.317906 86.354699  8.6596111 39.992161  4.8677926 3.300 22.56  13.242444 84.684963  11.105317 36.706618  6.2099523 3.150 22.58  16.349105 82.420989  13.702231 32.732805  7.6314832 3.000 22.60  19.685474 79.272406  16.497373 27.898608  9.1732244 The two tables provide data on fields as a function of horizontal position (x):

6. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Determining Pole Shape. Magneto-static Maxwell’s equations: div B = 0; curl H = j; In the absence of currents j = 0 so B can be expressed as the gradient of a scalar potential: B = -  So B is perpendicular to lines of constant scalar potential; And B is perpendicular to surfaces of very high  (poles!) So ideal pole shapes are lines of constant . All we have to do is identify the lines of constant  !

7. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Procedure for 2D pole shape 1.Resolve B into the field components: dipole, quadrupole, sextupole, octupole, etc, (n = 1 to 4); 2. Obtain the coefficients for each term in the harmonic series; 3. Write down the expression for the total scalar potential summing the potential for each term in the harmonic series; 4. Resolve the family of curves in 2D for  = constant, choosing appropriate value of  to give required pole gap..

8. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Example: magnet bd- Curve of B(x) fitted up to 4 th order for series: B(x) = b 1 +b 2 x + b 3 x 2 +b 4 x 3 gives the coefficients: b 1 = 4.6930 E-02 T; b 2 = 2.9562 E-04 Tm -1 ; b 3 = -2.9366 E -06 Tm -2 ; b 4 = -1.6919 E -07 Tm -3. with = 3.668 E-05 T

9. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Graphic fit for bd-  = 3.668E-05

10. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Area of largest mismatch

11. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Significance of octupole term (b 4 )  = 5.125E- 04

12. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Expressions for scalar potential  General expression for  in polar coordinates (r,  ):  =  n =1  (J n r n cos n  +K n r n sin n  ) In 2D Cartesian coordinates (x,y) for harmonics 1 to 4 in upright (non-skew) magnets J n = 0 and  is given by: dipole:  = K 1 y; quadrupole:  = 2 K 2 xy; sextupole:  = K 3 ( 3 x 2 y – y 3 ); octupole:  = 4 K 4 (x 3 y – x y 3 ); where: K 1 = b 1 ; K 2 = b 2 /2 ; K 3 = b 3 /3 ; K 4 = b 4 /4. For the FFAG magnets, the potential is a sum of all above; pole shapes (  = constant) need numerical solutions.

13. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Resulting pole with vac vessel for bd-

14. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 2D ‘OPERA’ Model of bd- with flux lines

15. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 OPERA 2D prediction of B(x) for bd-  = 3.751 E-05

16. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Area of largest mismatch (predicted vs defined)

17. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Area of largest mismatch (predicted vs fitted curve)  = 1.81E-05

18. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Magnet BD+ curve fit  = 3.67 E-05 b 1 = 5.3289 E-02 T; b 2 = -7.6285 E-04 Tm - 1 ; b 3 = -2.4679 E -05 Tm - 2 ; b 4 = 2.6617 E -08 Tm - 3.

19. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Pole and vacuum vessel for BD+

20. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 OPERA 2D model for BD+ (with flux lines shown)

21. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 By(x) predicted by OPERA 2D.

22. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Defined data and prediction of OPERA 2D for BD+  = 3.668 E-05

23. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Curve fit for BF  = 2.217 E-05 b 1 = 0.02599 b 2 = 1.339 E-03 b 3 = 9.860 E-06 b 4 = -1.535 E-07

24. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Obtaining lines of equi-potential for the BF magnet. Considerable difficulty was encountered in finding the shape of the upper and lower poles in the dipole for the BF, until the extrapolation of the fitted curve was examined: Its a sextupole!

25. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Sextupole arrangement needed for BF

26. Magnets for the ‘Pumplet’ Scaling FFAG – N.Marks, FFAG Workshop, Genoble, April 2007 Problems associated with the sextupole. Four of the 6 poles normally found in a sextupole will be needed; this creates problems: magnet is much increased in size; cost rises; power consumption is strongly increased. Solutions are being considered.