Off-momentum DA of RCS3D_BM & QFF & CC

AP meeting / December 8, 2004

Alexander Molodozhentsev

Etienne Forest

KEK

Tune diagram

-QX+2Qy=6 “Sextupole”

-Qx+3Qy=12 “Octupole”

-Qx+4Qy=18 “Decapole”

Structure resonances:

“Bare” tunes:Qx = 6.68Qy = 6.27

-Qx+5Qy=24

RCS_3DBM

--- scanning regions

Chromatic tune_shift ~ 0.08

(dp/p= 0.01)

MADX:RCS Twiss parameters (dp=0.0)

MADX_Chromaticity: ~ -15 !!!

MAX_Dx ~ 10 m !!! WHY?Because MADX uses ‘time”as the independent parameterinstead of ‘path-length’…

Wkin = 181 MeV … = 0.54523

Off-momentum particle motion(lattice#1)

Observation point : SFX Dx ~ ???

-15cm 15 cm

5.5 cm

p/p = 0.01

PTC tracking:Use “path” instead of “time”…SAD, MAD8

Dx (SFX) ~ 5.5 m

PTC: Chromaticity (x/y) ~ -8

RCS_3DBM

Chromaticity definition

dsssbsp

qppQ xxx )()()(

2

1)//( 2

0

Sextupole field component

Path_length in ring

If the code uses the time instead of the path, the chromaticitywill be different in the case of the low kinetic energy: ds = (c) dt

Wkin = 181 MeV … = 0.54523

TIMEPATH

… then PATH ~ 0.545 TIME !!!

Dispersion function definition

dQsQ

ss H

Ls

sH

])()(cos[)(

)(

)sin(2

)()(

Periodic dispersion function can be written as[Mario Conte, p.85]

… also depends on choice “path” or “time”

… MAD dispersion (“path”) is about two time bigger than SAD dispersion for Wkin=181 MeV …

Off-momentum particle motion(lattice#1)

p/p = 0.01

Observation point : SFX Dx ~ 10 m

xp/p ~ 10 cm

-15cm 15 cm

TDR: Bore radius of SFX is (330/2) mm=165 mm

SFX: x ~ 13m, y ~ 6mxcenter->chamber ~ 16.5-10.0=6.5 cmx,MAX ~ (x)2 / x ~ (6.5e-2)2/13 ~ 325 .mm.mrad… Momentum Acceptance for p/p = 0.01

PTC tracking:Use “time” instead of “path” …MADX

Effects of the quadrupole fringe field

(2) De-tuning effect for the off-momentum particles … … the end-field of the quadrupole magnet gives the beam two kicks in the opposite directions, which cancel for the paraxial trajectories, but which gives a net focusing to trajectories passing though at an angle. It means that the fringe field of the quadrupole magnets, which are located in the non- zero dispersion sections, will provide the de-tuning of the off-momentum particles. This effect depends on the particle momentum and the betatron amplitude, the dispersion and the slope of the dispersion at the location of the quadrupole magnet.

(1) ‘Octupole-like’ effect … … at the leading order – excitation of the normal octupole resonances

(3) Sextupole feeding-down component … … for the off-momentum particles the pseudo-octupole nonlinearity of the quadrupole magnet will be “feeding-down” due to the closed orbit displacement in the quadrupole magnets in the non-zero dispersion regions (in the ARCs). In this case the quadrupole fringe field can contribute additional excitation to the normal sextupole resonances.

RCS_3DBM

ON-momentum particles…

OFF-momentum particles…

Tune variation for off-momentum particles

-0.010 -0.005 0.000 0.005 0.0106.15

6.20

6.25

6.30

6.35

6.40

6.45

6.50

6.55

6.60

pp

Qx /

Qy

#3D_BM Q

x

Qy

# IDEAL Q

x

Qy

Without sextupole magnets for the chromaticity correction…

Linear chromaticity ~ (-8)

RCS_3DBM

Off-momentum DA (3D_BM)

6.3 6.4 6.5 6.6 6.7 6.83.0

3.5

4.0

4.5

5.0

5.5

6.0-Q

x+2Q

y=6

* Tunes include the chromatic tune shift

4Qx=

27

-Qx+

3Q

y=1

2

QV * ~ 6.222... 6.224

dp/p=0.005

Max

imum

X=

Y [c

m]

"Chromatic" horizontal tune *

3D_BMPlus: QFFHE

NO Sextupole CC

Observation point:BM1 entrance (#47)

RCS_3DBM

6.3 6.4 6.5 6.6 6.7 6.82.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

-Qx+

3Q

y=1

2

4Q

X=

27

-QX+

2Q

y=6

QV ~ 6.2620 ... 6.2630

BM_3D & QFF (1000 turns / entrance BM1)

Ma

xim

um

X=

Y [cm

]

Horizontal Tune

dp/p=0

6.3 6.4 6.5 6.6 6.7 6.83.0

3.5

4.0

4.5

5.0

5.5

6.0

Lattice tune

-Qx+

2Q

y=6

-Qx+

3Q

y=1

2

4Q

x=2

7

QV ~ 6.26

dp/p= 0.005

Ma

xim

um

X=

Y [cm

]

Horizontal tune

Off-momentum DA (3D_BM)

3D_BMPlus: QFFHE

NO Sextupole CC

Observation point:BM1 entrance (#47)

Resonance [-1,2]driving terms

Resonance [-1,2]driving terms for this case is 3 times smaller than for the corresponding tunes of the on-momentum particles

RCS_3DBM

Off-momentum DA (3D_BM)

6.3 6.4 6.5 6.6 6.7 6.83.0

3.5

4.0

4.5

5.0

5.5

6.0

* Tunes include the chromatic tune shift

QV

* ~ 6.183 ...6.188

-Qx+

2Q

y=6

-Qx+

3Q

y=1

2

4Q

x=2

7

dp/p=0.010

Ma

xim

um

X=

Y [

cm

]

'Chromatic' horizontal tune*

3D_BMPlus: QFFHE

NO Sextupole CC

Observation point:BM1 entrance (#47)

RCS_3DBM

Increasing the resonance4Qx=27 for the off-momentumparticles … by the feeding-downthe high-order field componentsof the 3D_BM …

Effect of the quadrupole fringe field for the off-momentum particles

6.3 6.4 6.5 6.6 6.7 6.83.0

3.5

4.0

4.5

5.0

5.5

6.0

4Q

x=2

7

-Qx+

2Q

y=6Ma

xim

um

X=

Y [

cm

]

'Chromatic' horizontal tune*

3DBM & QFF 3DBM

dp/p=0.01 3D_BMPlus: QFFHE

NO Sextupole CC

RCS_3DBM

(1)

(1) Contribution to the normal sextupole resonance by the sextupole ‘feeding-down’ of the QFF “pseudo-octupole”… some compensation effect (2)

(2) Increasing of the normal octupole resonance…

Off-momentum DA (3D_BM)

6.2 6.3 6.4 6.5 6.6 6.7 6.83.0

3.5

4.0

4.5

5.0

5.5

6.0

Lattice tune

-Qx+

3Q y=

12

-Qx+

2Q y=

6

4Q

x=2

7Ma

xim

um

X=

Y [cm

]

QV ~ 6.263 ... 6.265

dp/p=0.01

Horizontal tune

3D_BMPlus: QFFHE

NO Sextupole CC

Observation point:BM1 entrance (#47)

RCS_3DBM

Main limitation of the off-momentum DA iscaused by the normal octupole resonance.

Off-momentum DA (3D_BM):dp/p= 0.005

6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.03.0

3.5

4.0

4.5

5.0

5.5

6.0

-QX

* +3Q

Y

* =12

HV-coupling

-QX

* +2Q

Y

* =6 H-unstable

4QX

* =27

HV-coupling

Qy

*=6.301...6.303

dp/p=-0.005

Ma

xim

um X

=Y

[cm

]

"Chromatic" horizontal tune *

6.3 6.4 6.5 6.6 6.7 6.83.0

3.5

4.0

4.5

5.0

5.5

6.0

-Qx* +

2Qy* =

6

* Tunes include the chromatic tune shift

4Qx* =

27

-Qx* +

3Qy* =

12

QV * ~ 6.222... 6.224

dp/p=0.005

Max

imum

X=

Y [c

m]

"Chromatic" horizontal tune *

Off-momentum DA (3D_BM):dp/p= 0.01

6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.03.0

3.5

4.0

4.5

5.0

5.5

6.0

4Q

X

* =2

7

-QX

* +2

QY

* =6

Overlapping ofresonances

HV-unstable

H-unstable

QV

* ~ 6.344 ...6.346

dp/p=-0.01

Max

imu

m X

=Y

[cm

]

'Chromatic' horizontal tune*

6.3 6.4 6.5 6.6 6.7 6.83.0

3.5

4.0

4.5

5.0

5.5

6.0

* Tunes include the chromatic tune shift

QV

* ~ 6.183 ...6.188

-Qx* +

2Qy* =

6

-Qx* +

3Qy* =

12

4Qx* =

27

dp/p=0.010

Max

imum

X=

Y [c

m]

'Chromatic' horizontal tune*

… synchrotron oscillation …

Some speculation … look at summary012-file …

On-momentum particle motion

6.0 6.1 6.2 6.3 6.4 6.53.0

3.5

4.0

4.5

5.0

5.5

6.0-Q

x+5

Qy=

24

-Qx+

2Q

y=6

-Qx+

3Q

y=1

2

-Qx+

4Q

y=1

8

3DBM & QFF / dp=0Q

X ~ 6.69

Ma

xim

um

X=

Y [cm

]

Vertical tune

QX – fixQY - vary

RCS_3DBM

Tune-scanning in the Qy direction

Main limitation of theon-momentum DA is caused by the normalsextupole resonance

NO CC Sextupoles

Off-momentum (3D_BM) plus chromatic sextupole magnets

-0.010 -0.005 0.000 0.005 0.0106.10

6.15

6.20

6.25

6.30

6.35

6.40

6.45

6.50

6.55

6.60

Qx /

Qy

pp

3D_BM:(1) NO_CC

B C

(2) YES_CC D E

Changing of the tune fordifferent (dp/p) before (1)and after (2) the chromaticitycorrection

RCS_3DBM

Off-momentum DA after chromatic correction

6.3 6.4 6.5 6.6 6.7 6.82.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

4Q

x=2

7

-Qx+

2Q

y=6

3D_BM & QFF & CCQ

V ~ 6.25...6.26

#47

dp/p=0.01

Ma

xim

um

X=

Y [

cm]

Horizontal tune

3D_BMPlus: QFFHE

Sextupole CC

Observation point:BM1 entrance (#47)

(X=Y)MIN = 2.75 cm

Twiss Parameters:(βx)1/2 = 2.7848 m1/2, x = 0.2517(βy)1/2 = 3.9030 m1/2, y = 0.1990

Ax = (Xmax)2x ~ 190 mm mrad

Ay = (Ymax)2y ~ 150 mm mrad

RCS_3DBM

On-momentum DA after chromatic correction

6.3 6.4 6.5 6.6 6.7 6.82.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

4Q

H=

27

-QH+

2Q V

=6

QV=6.2620...6.2630

Ma

xim

um

X=

Y [cm

]

Horizontal Tune

BM_3D&QFF BM_3D&QFF&CC

… result was presented October 27, 2004.

RCS_3DBM

To DO:

• … correction of the normal sextupole resonance …

• … off-momentum particle motion after the correction…

RCS_3DBM

Presented for the AP-groupDecember 8, 2004