4. Transversal beam optics

1. 4. Transversal beam optics 4. Introduction 4.1 Particle Motion, Phase space and Bush Theorem 4.2 Particle trajectories and transformation matrixes 4.3 Deflection in a Dipole and edge focussing 4.4 Electrostatic Einzellenses 4.5 Soleniods 4.6 Quadrupoles and alternating gradient focussing 4.7 Space charge and Plasma lenses

2. Introduction

3. Particle Motion, Phase space and Busch theorem

4. Particle trajectories and transformation matrixes

5. Particle trajectories and transformation matrixes

6. Dipoles for beam deflection Y Field limit for normal conducting magnets due to saturation Set up of a conventional C-type dipole magnet

7. Dipoles for beam deflection Beam deflection and weak focussing in the horizontal plane for a homogeneous dipole field dipole particle trajectory z

8. Edge focussing with dipoles Edge focussing in a homogeneous dipole field with tilted end planes z Shorter beam path in the horizontal plane reduces deflection, this can be interpreted as a defocussing of the beam, but in the horizontal plane the additional field components focus the beam.

9. Electrostatic Einzellenses Assuming that the volume of the lens is free of charges (Dirichlet boundary conditions) and starting and ending beam trajectories outside the lens at zero potential, it can be shown that the integral over all longitudinal and transversal electric field components equals zero. 100 12 kV 12 kV 0 kV 0 kV ] m m defocusing focusing defocusing [ r 0 0 kV 300 0 z [mm]

10. 100 100 ep = 0,0141 mmmrad 12 kV 100%,rms,n 12 kV 0 kV 0 kV ] 50 m ] m d [ a r r m 0 [ ´ x 0 0 kV 300 0 z [mm] -50 -100 40 10 -10 0 -5 5 100 ep = 0,0131 mmmrad x [mm] 100%,rms,n 8 mA 100 30 ep = 0,0146 mmmrad 50 100%,rms,n ] ] m d a 50 m r 20 0 m [ 2 mA ] [ d r ´ a x r 0 m [ 10 -50 ´ x 1nA -50 0 % 0 -100 10 -10 0 -5 5 50 0 250 100 200 150 300 x [mm] z [mm] -100 10 20 -20 -10 0 x [mm] 200 ep = 0,149 mmmrad 100%,rms,n 100 ] d a r 0 m [ ´ x -100 -200 10 20 -20 0 -10 x [mm] Electrostatic Einzellenses Influence of beam current and there from space charge forces on beam transport (He+, 14 keV) using einzellenses. Input emittance 5 G: 8.2*10 m/s -5 3 F;H:1.3*10 As/m Due to the deceleration of the particles the space charge forces are enhanced within the lens => space charge limit ! 0

11. Electrostatic Einzellenses Influence of the initial phase space distribution (equal diameters and emittances) on beam transport (He+, 14 keV, 0 mA) without space charge using einzellenses. The emittance growth is strongly coupled with the radial beam particle distribution within the lens. This is due to the non linear fields near the electrodes which causes non linear deformation of the emittance pattern (spherical aberations)

12. Electrostatic Einzellenses Influence of the initial phase space distribution on beam transport for space charge dominated beams (He+, 14 keV, 8 mA) using einzellenses. While in the first half of the transport the “ideal” beam with low emittance is nearly unchanged while the other beam doubles its emittance due to redistributions forced by self fields, in the second part, due to non linear external fields, the emittances grow drastically and the final emittances are nearly equal.

13. Solenoids

14. Soleniods

15. Quadrupoles and alternating gradient focussing

16. Comparison of Quadrupoles with solenoids and Einzellenses

17. Principle Gabors suggestion coils space charge cloud iron yoke ion beam cathode F anode r electron cloud z ion beam Isolator Coils Anode 10 cm ground potential Space charge lenses => strong focussing; for homogeneousr : F is linear with radius

18. Space charge lenses

19. Space charge lenses Calculation of space charge density distribution within the lens by numerical self consistent simulations assuming a thermalized particle cloud Electric fields allowed by theory of Gabor (left a, right blue), taking external fields into account (left b, right green) and calculated by simulation (left c, right red) Particle density allowed by theory of Gabor (right blue), taking external fields into account (right green) and calculated by simulation (left, right red)

20. Space charge lenses Comparison of phase space distributions between measurement and calculations of the beam transport using two space charge lenses to focus a space charge dominated ion beam (He+, 14 keV, 8 mA) into an RFQ

21. Plasma lenses, lithium lenses and horn Using a high current driven trough a plasma to produce a magnetic field to focus the particle beam (strong focussing, particle velocity transversal to magnetic field) Assuming a current of 500 kA and a radius of 10 mm the magnetic field strength is 10 T at the border of the discharge column.

22. Plasma lenses, lithium lenses and horn pinch hollow

23. Plasma lenses are favourable for the final focus of high energy particle beams like the HIIF, LC or the GSI HEDIM experiments, and to collect particles emitted with high transversal momentum from a target, like the anti proton production at CERN etc. Plasma lenses Pepper pot measurement of focussing a particle beam by the use of a plasma lens (GSI). The spot size is a function of the time varying current through the plasma lens