1. Main Injector Slow Extraction Studies f Y. Alexahin, E. Gianfelice-Wendt, J. Johnstone (APC) D. Morris, M.-J. Yang (AD) MI Slow Extraction Group Meeting, June 29, 2012

2. Issues  What are the optimal conditions for ½-integer resonance extraction?  The amplitude and phase of the quad harmonic?  Nonlinear tuneshift with amplitude?  Where we are:  Tune vs harmonic quad strength and amplitude of oscillations.  TBT optics measurements.  Nonlinear chromaticity  Optimum ES angle vs ES position  Plan of action 2 MI Slow Extraction Group Meeting, June 29, 2012

3. Theory Best orientation (  )? Optimum parameters? 3 r 2-turn move for  < 0: clockwise counter-clockwise a0a0  /2 c Model: MI Slow Extraction Group Meeting, June 29, 2012

4. Theory - 2 4 For this reason, and to maximize the step size, 0 <  <  (better <<  ) orientation for maximum 2-turn step in x and maximum  x and … With orientation like that (or on the right) Im a < 0 for x ES < 0 the extraction trajectory can be diverted from the axis, with the angle maximum tangential 2-turn step here r 2-turn move for  < 0: clockwise counter-clockwise a0a0  /2 c allowed ES position  +0.0011 for  x =31m,  x =1.72, x ES =-2cm MI Slow Extraction Group Meeting, June 29, 2012

5. Measurements 5 There was a few rounds of measurements: 03/21/12: TBT optics with harmonic quads: “nominal”; off; minimum Qx (QC206=+1.5A, QC328 = -2A); and QC206 = -3A Qx vs amplitude with harmonic quads for minimum Qx TBT optics with vertical kicks at QC206 = -3A (also gives Qx vs amplitude) 04/02/12: Qx vs amplitude with OF on and off (data not saved) 04/10/12: TBT optics and Qx vs amplitude at QC328 = -6.5A 04/25/12: TBT optics with individual harmonic quads offset by 12A from minimum Qx settings Extraction with QC328 = -6.5A attempted Also, Ming-Jen and Denton measured 2 nd order chromaticity with OF on and off MI Slow Extraction Group Meeting, June 29, 2012

6.  x - function @ HBPMs 6 Q206=-3A, Q328=-2A, Qx=0.4895 -  x @ HP520 goes down, not up! design Distance from HP624 (m) HP520measurement “Nominal” harm. quads, Qx=0.485627 Q206=+1.5A, Q328=-2A, Qx=0.484487 - John’s algorithm works! MI Slow Extraction Group Meeting, June 29, 2012

7. QC206 vs QC328 7 nominal Distance from HP624 (m) HP520 measurement  QC206=-1A,  QC328=-4.5A, Qx=0.4899 -  x @ HP520 goes up!  QC206=-4.5A,  QC328=0, Qx=0.4895 -  x @ HP520 goes down, not up! xx xx Distance from HP624 (m) MI Slow Extraction Group Meeting, June 29, 2012

8. Closer Look 8 MAD model Distance from HP624 (m) ES entrance  QC206=-1A,  QC328=-4.5A, Qx=0.4899 -  x @ ES larger while max  x is smaller  QC206=-4.5A,  QC328=0, Qx=0.4895 xx xx Distance from HP624 (m) measurement nominal MI Slow Extraction Group Meeting, June 29, 2012

9. Phase Space @ ES 9 Design optics used to compute variables @ES from HP520 & HP522 data QC206=-3A QC328=-6.5A The ellipse tilt with QC206 is wrong:  /2 = 108  > 90  With QC328 it is still wrong:  /2 = 145  Inversion of QC206 polarity will give  (-) =  (+) -  = 216  - 180  = 36  Still some contribution from QC328 may be helpful “minimum Qx”  /2 MI Slow Extraction Group Meeting, June 29, 2012

10. How it looks in ordinary coordinates 10 From Ming-Jen’s I90 simulations QC328=-6.5A MI Slow Extraction Group Meeting, June 29, 2012

11. Harmonic Quad Strength 11 When QC206 is downshifted by 4.5A Qx=0.485  0.4885, so that  0 =0.015   =0.0115 that gives |G 2 |  0.0052 One more way is to use the transfer function – would I have known it – to calculate K1 QC206 for a given current. The last necessary parameter is To obtain the observed (by eye) beta-beat with MAD K1 QC206  0.0011m -2 is needed that gives |G 2 |  0.0055 Another way is to look at the tuneshift: MI Slow Extraction Group Meeting, June 29, 2012

12. Effect of Octupoles MAD8 with octupole components provided by D.Johnson seems to underestimate the effect, while I90 seems to overestimate it. 12 From 03/21/12 studies, harm. quads set for minimum Qx (QC206=+1.5A, QC328=-2A), Action variable determined from all HBPM data by a special algorithm Qx Jx (  m) casemeasuredMAD8 STATICI90 “minimum Qx” 1.68  10 3 1.03  10 3 - QC206=-3A 8.0  10 3 3.4  10 3 12.0  10 3 QC328=-6.5A 10.8  10 3 2.6  10 3 - Tuneshift with amplitude dQx/dCSI=1.68  10 3 [1/m] (Courant-Snyder Invariant = 2  J) MI Slow Extraction Group Meeting, June 29, 2012

13. Tune vs Δ p/p May 02, 2012 Harmonic quads? 04/28/12 studies by D.Morris & M.-J. Yang

14. 2 nd Order Chromaticity (/2) 13 OF (A)measuredMAD8 STATICI90 0326.9178.0 249.0  326.7*** +10454.2 (+127.3)311.9** (+133.9)455.8 (+129.1) –10198.6* (-128.3)44.2 (-133.8)200.9 (-125.8) *) using more points for  p/p>0 **) using transfer function 2.8 T-m/m^3/A (multiplied by 3!/LMRO) ***) increasing the built-in octupole component by 32.5% Correspondence between measured OF effect and I90 is amazing! MAD8 underestimates built-in octupoles by a factor of 1.84 and exaggerates OF (slightly) With this factor in the MAD8 detuning will be dQx/dCSI=1.90  10 3 [1/m] – not too far from the measured dQx/dCSI=1.68  10 3 [1/m]  G 4 =dQx/dJx/2=dQx/dCSI G 4 =1.68  10 3 [1/m] MI Slow Extraction Group Meeting, June 29, 2012

15. Step Size 14 - this is just a half of the available ES aperture (14mm). To increase  x max we can increase |G 2 | (and  0 /2) by up to 58% (or reduce G 4 by a factor of 4?)  /2 = 18  (the same |G 2 |) 2-turn step =- 6.5mm @ x ES =-1cm, x=-0.0003: x=-1cm  -1.2cm over ES With larger |G 2 | x ES can be increased At the stopband boundary (  0 =-2|G 2 |) and  =0 (left pic.) the step size in x reaches its maximum:  x (m) x x (m)  /2 = 108  2-turn step =- 4.5mm @ x ES =-2cm, x=+0.0007: x=-2cm  -1.5cm over ES  x (m) x (m) x MI Slow Extraction Group Meeting, June 29, 2012

16. Central Separatrix Area 15 x c =  1.7mm for  /2 = 108  and x c =  5.1mm for  /2 = 18  - no problem with x ES =-1cm Still,  can be further optimized by mixing QC206 and QC328 From Leo & John: To begin the squeeze of  95%N = 20  mm  mrad out of the central separatrix A c <  95%N /   |  0 | - 2|G 2 | < 0.0016 ( | a 0 /c| < 0.39 ) At this moment the separatrix extends to MI Slow Extraction Group Meeting, June 29, 2012

17. Plans  Analyze measurement data from 03/21 - 04/25  extract BPM calibration errors  find  -functions at important locations (harm. quads, septa)  fit the optics model to the data  Look for optimal harmonic quad configuration ( , |G 2 |)  16-mult for resonance excitation?  Extracted beam transport? (Larger step-size  larger emittance)  Simulate extraction with account of nonlinearities and space-charge 16 MI Slow Extraction Group Meeting, June 29, 2012