Main Injector Slow Extraction Studies

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Y. Alexahin, E. Gianfelice-Wendt, J. Johnstone (APC)

D. Morris, M.-J. Yang (AD)

MI Slow Extraction Group Meeting, June 29, 2012

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

Theory

Best orientation ()?

Optimum parameters?

3

constrealG

iinGG

nQx

4

22 ),exp(||

,02/

)/(2/Im xxx xxa

observablesgn||4

0atdetuning

||||

||2

||2||

,/

02

220

20

20

2

2/

4

200

42

G

G

acr

eG

Ga

GGc

i

r

2-turn move for < 0: clockwise counter-clockwise

a0

/2

xxa 2/Re c

Model:

MI Slow Extraction Group Meeting, June 29, 2012

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) Ima < 0 for xES < 0 the extraction trajectory can be diverted from the axis, with the angle

maximum tangential2-turn step here

)/(2/Im xxx xxa

r

2-turn move for < 0: clockwise counter-clockwise

a0

/2

xxa 2/Re c

allowed ES position

axx xxESx Im/2/

+0.0011 for x=31m, x=1.72, xES=-2cm

MI Slow Extraction Group Meeting, June 29, 2012

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 2nd order chromaticity with OF on and off

MI Slow Extraction Group Meeting, June 29, 2012

x - function @ HBPMs 6

500 1000 1500 2000 2500 3000

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500 1000 1500 2000 2500 3000

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150

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

QC206 vs QC328 7

500 1000 1500 2000 2500 3000

25

50

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125

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500 1000 1500 2000 2500 3000

40

60

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100

120

140

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

Distance from HP624 (m)

MI Slow Extraction Group Meeting, June 29, 2012

Closer Look 8

2650 2700 2750 2800 2850 2900 2950 3000

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2650 2700 2750 2800 2850 2900 2950 3000

20

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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

Distance from HP624 (m)

measurement

nominal

MI Slow Extraction Group Meeting, June 29, 2012

Phase Space @ ES 9

-2 -1 1 2

-4

-2

2

4

-2 2 4

-3

-2

-1

1

2

3

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”

)(mmx

xx xx

)(mmx/2

xx xx

MI Slow Extraction Group Meeting, June 29, 2012

How it looks in ordinary coordinates 10

xx

From Ming-Jen’s I90 simulations

-2 2 4

-6

-4

-2

2

4

)(mmx

QC328=-6.5A

MI Slow Extraction Group Meeting, June 29, 2012

Harmonic Quad Strength 11

When QC206 is downshifted by 4.5A Qx=0.485 0.4885, so that 0=0.015 =0.0115that gives |G2|0.0052

One more way is to use the transfer function – would I have known it – to calculate K1QC206 for a given current.

The last necessary parameter is

x

x

dJ

dQG

2

14

To obtain the observed (by eye) beta-beat with MAD K1QC206 0.0011m-2 is needed that gives |G2|0.0055

Another way is to look at the tuneshift:

4||

2202

2

G

MI Slow Extraction Group Meeting, June 29, 2012

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

0.05 0.15 0.2 0.25 0.3

0.4846

0.4848

0.4852

0.4854

Qx

Jx (m)

case measured MAD8 STATIC I90

“minimum Qx” 1.68103 1.03103 -

QC206=-3A 8.0103 3.4103 12.0103

QC328=-6.5A 10.8103 2.6103 -

Tuneshift with amplitude dQx/dCSI=1.68103 [1/m] (Courant-Snyder Invariant = 2J)

MI Slow Extraction Group Meeting, June 29, 2012

Tune vs Δp/p

May 02, 2012Harmonic quads?

04/28/12 studies by D.Morris & M.-J. Yang

2nd Order Chromaticity (/2) 13

OF (A) measured MAD8 STATIC I90

0 326.9 178.0 249.0 326.7***

+10 454.2 (+127.3) 311.9** (+133.9) 455.8 (+129.1)

–10 198.6* (-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=1.90103 [1/m] – not too far from the measured dQx/dCSI=1.68103 [1/m] G4=dQx/dJx/2=dQx/dCSI

G4=1.68103 [1/m]

MI Slow Extraction Group Meeting, June 29, 2012

Step Size 14

- this is just a half of the available ES aperture (14mm).To increase xmax we can increase |G2| (and 0/2) by up to 58%(or reduce G4 by a factor of 4?)

/2 = 18 (the same |G2|)

2-turn step =- 6.5mm @ xES=-1cm,x=-0.0003: x=-1cm -1.2cm over ES

With larger |G2| xES can be increased

mm7)(/824

)168(24Re24Re22/1

42/3

2

342max

GG

rGrGaax

x

xxx

At the stopband boundary (0=-2|G2|) and =0 (left pic.) the step size in x reaches its maximum:

0.030 0.025 0.020 0.015 0.010 0.005

0.004

0.002

0.002

0.004

0.006

x (m)

x

x (m)

/2 = 108

2-turn step =- 4.5mm @ xES=-2cm,x=+0.0007: x=-2cm -1.5cm over ES

0.015 0.010 0.005 0.005 0.010

0.006

0.004

0.002

x (m)

x (m)x

MI Slow Extraction Group Meeting, June 29, 2012

Central Separatrix Area 15

xc =1.7mm for /2 = 108 and xc = 5.1mm for /2 = 18 - no problem with xES=-1cm

Still, can be further optimized by mixing QC206 and QC328

0Re2 ax xc

From Leo & John:

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||2||

10

11(

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3

22

...)5

11(

||3

||8)sin(2

2

202/1

24

2/320

2

03

02

G

G

GG

G

c

a

c

arAc

To begin the squeeze of 95%N = 20 mmmrad out of the central separatrix

Ac < 95%N / |0| - 2|G2| < 0.0016 ( |a0/c| < 0.39 )

At this moment the separatrix extends to

MI Slow Extraction Group Meeting, June 29, 2012

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 (, |G2|)

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