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AP meeting / December 8, 2004. Off-momentum DA of RCS 3D_BM & QFF & CC. Alexander Molodozhentsev Etienne Forest KEK. RCS_3DBM. Tune diagram. Structure resonances:. -Q X +2Q y =6 “Sextupole”. Chromatic tune_shift ~  0.08 (dp/p=  0.01). -Q x +3Q y =12 “Octupole”.

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Off momentum da of rcs 3d bm qff cc

AP meeting / December 8, 2004

Off-momentum DA of RCS3D_BM & QFF & CC

Alexander Molodozhentsev

Etienne Forest

KEK


Tune diagram

RCS_3DBM

Tune diagram

Structure resonances:

-QX+2Qy=6 “Sextupole”

Chromatic

tune_shift ~  0.08

(dp/p=  0.01)

-Qx+3Qy=12 “Octupole”

-Qx+4Qy=18 “Decapole”

-Qx+5Qy=24

“Bare” tunes:

Qx = 6.68

Qy = 6.27

--- scanning regions


Madx rcs twiss parameters dp 0 0
MADX:RCS Twiss parameters (dp=0.0)

Wkin = 181 MeV

…  = 0.54523

WHY?

Because MADX uses ‘time”

as the independent parameter

instead of ‘path-length’…

MAX_Dx ~ 10 m !!!

MADX_Chromaticity: ~ -15 !!!


Off momentum particle motion lattice 1

RCS_3DBM

Off-momentum particle motion(lattice#1)

PTC tracking:

Use “path” instead of “time”…SAD, MAD8

Observation point : SFX Dx ~ ???

p/p = 0.01

Dx (SFX) ~ 5.5 m

PTC:

Chromaticity (x/y) ~ -8

5.5 cm

-15cm

15 cm


Chromaticity definition
Chromaticity definition

Sextupole field component

Path_length in ring

If the code uses the time instead of the path, the chromaticity

will be different in the case of the low kinetic energy: ds = (c) dt

Wkin = 181 MeV …  = 0.54523

… then PATH ~ 0.545 TIME !!!


Dispersion function definition
Dispersion function definition

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 11
Off-momentum particle motion(lattice#1)

PTC tracking:

Use “time” instead of “path” …MADX

Observation point : SFX Dx ~ 10 m

p/p = 0.01

TDR:

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

SFX: x ~ 13m, y ~ 6m

xcenter->chamber ~ 16.5-10.0=6.5 cm

x,MAX ~ (x)2 / x ~ (6.5e-2)2/13 ~ 325 .mm.mrad

… Momentum Acceptance for p/p = 0.01

xp/p ~ 10 cm

-15cm

15 cm


Effects of the quadrupole fringe field

RCS_3DBM

Effects of the quadrupole fringe field

ON-momentum particles…

(1) ‘Octupole-like’ effect …

… at the leading order – excitation of the normal octupole resonances

OFF-momentum particles…

(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.

(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.


Tune variation for off momentum particles

RCS_3DBM

Tune variation for off-momentum particles

Linear chromaticity ~ (-8)

Without sextupole magnets

for the chromaticity correction…


Off momentum da 3d bm

RCS_3DBM

Off-momentum DA (3D_BM)

3D_BM

Plus: QFFHE

NO Sextupole CC

Observation point:

BM1 entrance (#47)

dp/p=0


Off momentum da 3d bm1

RCS_3DBM

Off-momentum DA (3D_BM)

3D_BM

Plus: QFFHE

Resonance [-1,2]

driving terms

NO Sextupole CC

Observation point:

BM1 entrance (#47)

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)

3D_BM

Plus: QFFHE

NO Sextupole CC

Observation point:

BM1 entrance (#47)

Increasing the resonance

4Qx=27 for the off-momentum

particles … by the feeding-down

the high-order field components

of the 3D_BM …


Effect of the quadrupole fringe field for the off momentum particles

RCS_3DBM

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

3D_BM

Plus: QFFHE

NO Sextupole CC

  • Contribution to the normal

  • sextupole resonance by

  • the sextupole ‘feeding-down’

  • of the QFF “pseudo-octupole”…

  • some compensation effect

(1)

(2)

(2) Increasing of the normal

octupole resonance…


Off momentum da 3d bm2

RCS_3DBM

Off-momentum DA (3D_BM)

3D_BM

Plus: QFFHE

NO Sextupole CC

Observation point:

BM1 entrance (#47)

Main limitation of the off-momentum DA is

caused by the normal octupole resonance.




Synchrotron oscillation
… synchrotron oscillation …

Some speculation … look at summary012-file …


On momentum particle motion

RCS_3DBM

On-momentum particle motion

NO CC Sextupoles

Tune-scanning in the Qy direction

QX – fix

QY - vary

Main limitation of the

on-momentum DA is caused by the normal

sextupole resonance


Off momentum 3d bm plus chromatic sextupole magnets

RCS_3DBM

Off-momentum (3D_BM) plus chromatic sextupole magnets

Changing of the tune for

different (dp/p) before (1)

and after (2) the chromaticity

correction


Off momentum da after chromatic correction

RCS_3DBM

Off-momentum DA after chromatic correction

3D_BM

Plus: QFFHE

Sextupole CC

Observation point:

BM1 entrance (#47)

Twiss Parameters:

(βx)1/2 = 2.7848 m1/2, x = 0.2517

(βy)1/2 = 3.9030 m1/2, y = 0.1990

(X=Y)MIN = 2.75 cm

Ax = (Xmax)2x ~ 190  mm mrad

Ay = (Ymax)2y ~ 150  mm mrad


On momentum da after chromatic correction

RCS_3DBM

On-momentum DA after chromatic correction

… result was presented October 27, 2004.


To do

RCS_3DBM

To DO:

  • … correction of the normal sextupole resonance …

  • … off-momentum particle motion after the correction…

Presented for the AP-group

December 8, 2004


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