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## PowerPoint Slideshow about ' Off-momentum DA of RCS 3D_BM & QFF & CC' - lluvia

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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)

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)

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

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

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)

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

xp/p ~ 10 cm

-15cm

15 cm

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

Linear chromaticity ~ (-8)

Without sextupole magnets

for the chromaticity correction…

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

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

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_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 …

Some speculation … look at summary012-file …

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

Changing of the tune for

different (dp/p) before (1)

and after (2) the chromaticity

correction

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)2x ~ 190 mm mrad

Ay = (Ymax)2y ~ 150 mm mrad

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