1. Bogazici University, Department of Physics 1

2. 1. Emittance  Why we need to measure the emittance at the exit of the ion source ? 2. Quadrupole Variation Method  Our calculations and results 3. Forward Method  Forward Method using Quadrupole  Forward Method using Solenoid 2

3. A particle’s properties can be figured out via Lagrange Mechanics, however a system of particles had better be defined in Hamiltonian Mechanics which requires Phase Space. Phase Space includes position and (combination of) derivative of position (velocity, momentum). 3

4. In Accelerator Physics, we define 3 phase spaces for 3 dimensions (x,y,z). Again, phase space includes position(i.e. x) and its gradient (x’). where α,β,γ are twiss parameters which enables me to define the area uniquely. A= π. ϵ 4

5. Emittance with determined α,β,γ values gives information about the beam as a whole, and this information is used through the beam propagation simulations. This is why we had to figure out the emittance and twiss parameters. 5

6. This figure was drawn by a multiparticle simulation program PATH, and considered as the beam exiting the ion source. Due to radial symmetry at the exit of the ion source both xx’ and yy’ have the same emittance and twiss parameters. ϵ _ rms = 1.0000 π.mm.mrad β=0.2000 mm/π.mrad α=-2.0001 P.S: Beam pipe is taken with infinite radius such that all created particles can travel and are not annihilated. Also, no effect of space charge ! 6

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9. Transfer Matrices 9

10. We have 10 cm of quadrupole with effective length and 20 cm of drift space. Therefore our transfer matrix R(k) can be calculated, and notice that although focusing quadrupole stands before the drift, in transfer matrix they are in backwards sequence. 10

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12. So far we have seen that the emittance and twiss parameters are very close those ones simulated, but this is the case for 0 mA (No space charge effect). Normalized solutions are given for trials with equations more than the unknowns. 12 Denemeleremittance [pi*mm*mrad]Beta [pi/mm*mrad]alphaError in em[%]Error in b[%]Error in a[%] Gerçek Değerler 153,15 0,2000-2,0001 - - - 1.deneme (1,7,11) 155,950,1943-1,99881,82522,830,0650 2. Deneme (3, 7, 9) 156,300,1950-2,00552,05372,490,2700 3. Deneme (1, 3, 7, 9, 11) 157,300,1924-1,97752,70663,781,1299 4. Deneme (1, 3, 7) 155,980,1926-1,99071,84483,690,4700 5. Deneme (1, 2, 3, 5, 7) 155,840,1928-1,99271,75333,61250,3700 6. Deneme (bütün ölçümler) 157,160,1927-1,98072,61523,640,9700

13. In 2 mA space charge effect does not play such an important rule but in 10 mA, the beam emittance measurement blows up dramatically. What about beam pipe with physical length ? Trialsemittance [pi*mm*mrad]Beta [pi/mm*mrad]alphaError in em[%]Error in b[%]Error in a[%] Real Values 153,15 0,2000-2,0001 - - - 0 mA (ölçüm 3,7,9)156,300,1950-2,00552,053702,490,2699865 2 mA (ölçüm 3,7,9)156,30,19502-2,00522,053702,490,2549873 10 mA (ölçüm 3,7,9)170,560,17296-1,975811,3645513,521,2149393 13

14. Table is meant to give a taste about what is happening in the case of a 3 cm radius of beam pipe. Notice for small gradient x_rms values are close. Gradient (T/m)x_RMS [m] (no constraint)x RMS [m] (constraint)Transmission (%) 2,504,82E-034,84E-0342 3,456,37E-03 34 2,534,75E-034,78E-0342 2,964,67E-034,71E-0338 2,774,47E-034,51E-0340 1,897,78E-037,66E-0347 1,291,21E-021,10E-0251 1,041,42E-021,19E-0252 What is the effect of the beam pipe? 14

15.  In QVM, we calculate emittance via analytical way such that from x_rms and gradient values, we go backward by transfer matrices and find out the emittance and twiss parameters.  In FM, we have an inception beam, and we feed this emittance into simulations to get close to the x_rms and y_rms values using TRAVEL. 15

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17.  Now we insert our parameters to create a beam after analyses  We draw the beam with PLOTWIN 17

18. Beam ParametersReal BeamInception BeamForward Method alpha_x-2-2,1767-2,0401 beta_X0,20,19480,203 emittance_X pi.mm.mrad11,08511,007 alpha_y-2-2,1213-2,0201 beta_y0,20,20240,202 emittance_y11,03190,9748 error_alpha_x8,8352,005 error_beta_X2,61,5 error_emittance_X pi.mm.mrad8,510,7 error_alpha_y6,0651,005 error_beta_y1,21 error_emittance_y3,192,52 We used inception beam in simulations and the beam from FM is at good agreement with real beam. Notice that inception does not fit well with real beam. 18

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20. We have good agreement with real beam parameters by using solenoid, as well. Beam ParametersReal BeamInception BeamForward Method alpha_x-2-2,1767-2,0381 beta_X0,20,19480,2026 emittance_X pi.mm.mrad11,08511,0045 alpha_y-2-2,1213-2,0183 beta_y0,20,20240,2017 emittance_y11,03190,9752 error_alpha_x8,8351,905 error_beta_X2,61,3 error_emittance_X pi.mm.mrad8,510,45 error_alpha_y6,0650,915 error_beta_y1,20,85 error_emittance_y3,192,48 20

21.  Quadrupole Variation Method is useful and gives accurate results at 20 keV and 2 mA with shortcomings in beam transmission rate for 3 cm of radius beam pipe.  Forward Method are both applicable with Quadrupole and Solenoid magnets. Moreover, transmission rate does not follow since the particles do not hit solenoid surface. 21

22. Thanks for your kind attention … 22