Alexander MolodozhentsevKEK

for MR-commissioning groupSeptember 20, 2005

for RCS-MR commissioning groupSeptember 27, 2005

Sextupole effect for MR- beam centroid beating

MR Technical Design

COD after correction should be less than 1 mm.

From the beam point of view it means that the deviation of the

beam centroid (center of “mass”) from the machine center (0,0,s)

should be less than 1 mm.

Quadrupole field

As ~ x·yAx = Ay =0

x/s ~ x

y/s ~ y

Single particle transverse kick from thin quadrupole

Multi-particle transverse kick from thin quadrupole

< x/s > ~ <x> = 0

< y/s > ~ < y > = 0 without COD

Vector potential for the quadrupole field

Average kick for the beam’s macroparticles is zero aroundthe central orbit…

Sextupole field (simplified model)

As = 1/6 B// (x3 – 3xy2)Ax = Ay =0

x/s ~ (a x2 – b y2)

y/s ~ xy

Single particle transverse kick from thin sextupole

Multi-particle transverse kick from thin sextupole

< x/s > ~ (a <x2> – b <y2>) ~ ( a x – b y )

< y/s > ~ < xy > = 0 without COD

Vector potential for the sextupole field

Average kick for the beam’s macroparticles is NOT zero aroundthe central orbit … shift of the beam centroid !!!

Sextupole effect on the beam centroid

yysxsxxsxsxsx

xxss

sx kx

;2/1;

2/3;

2;

2/1;

2

21

21

;0;0 4

1

4

1

2)sin(

)cos(

E.Forest book “Beam Dynamics: …” (p.285)

… sextupoles do move the average position of the beam.

The part which does not depend on amplitude is the regular 2 dispersion.It is a non-dynamical effect. It is the change of the fixed point as a functionof energy in a coasting beam normalization. The amplitude dependentterms are dynamical. A beam of finite size, on momentum (=0), appears shifted as if the fixed point moved.

This effect of the sextupole field nonlinearity is the ‘leading’ order effect,then it could be observed by using the second-order matrix formalism.

Sextupole effect on the beam centroid

Estimation of the effect for MR (3GeV_Beam) for a single sextupole

MR: <x> ~ 15 m, <x> ~ 1.5 m

= p/p = 0.004

x100% = y

100% = 54 mm.mrad … Uniform: x,RMS = y,RMS = 54/4 = 13.5 mm.mrad

“Dispersion” term … ~ 18 10-6

“Amplitude Dependent” term (X) … ~ 196 10-6

“AD” term is much bigger than “D” term.

Study approach …

• MAD second order transfer matrix between the ring elementsfor MR lattice for the sextupole magnets (OFF / ON).

• Teapot-type multi-particle tracker (ORBIT), based on the MADtransfer matrix.

• NO space charge effects.

• Observation of the first transverse moments (<X> & <Y>) aroundthe ring.

MR::

• Working point :: Qx = 22.428, Qy = 20.82• Transverse particle distribution ::

3GeV … 54 mm.mrad (Gaussian: 100% =8RMS / Parabolic: 6RMS / RCS_beam) 40 GeV … 6 mm.mrad

Test tracking #1

10’000 mp

S [m]

<X

> [

mm

]54 mm.mradp/p = 0NO CCSX

Test tracking #2

54 mm.mradp/p = 0NO CCSX

95’254 mp

S [m]

<X

> [

mm

]

Test tracking #3_1

54 mm.mradp/p = 0NO CCSX

250’000 mp

S [m]

<X

> [

mm

]

Test tracking #3_2

54 mm.mradp/p = 0NO CCSX

250’000 mp

S [m]

<Y

> [

mm

]

Some conclusion…from Test Tracking

Oscillation of <X> and <Y> around the ring for the case withoutCC_Sextupole_Magnets is caused by:

(1) Statistical effect (limited number of macro particles)

(1) Effect of the fringing field of the bending magnets (‘sextupole’- likeeffect … will be explained later.

Beam centroid motion around MR3 GeV

100% = 54 .mm.mrad (Gaussian-beam:: RMS = 54/8 .mm.mrad )

p/p = 0

95254 macro_particles

S [m]

<x>

[m

m]

Beam centroid motion around MR3 GeV

S [m]

<x>

[m

m]

100% = 54 .mm.mrad (Parabolic-beam:: RMS = 54/6 .mm.mrad )

p/p = 0

95254 macro_particles

Beam centroid motion around MR3 GeV

RCS-beam (v.050906) :: RMS x,y =7.190 / 8.325 .mm.mrad

p/p = 0

95254 macro_particles

S [m]

<x>

[m

m]

Turn_by_turnobservationpoint #OP

S [m]

<y>

[m

m]

Beam centroid motion around MR

y = 54 .mm.mrad (RCS-beam)

p/p = 0

50 turnsS [m]

<x>

[m

m]

Beam centroid motion around MR

x = 54 .mm.mrad (Parabolic-beam)

p/p = 0

S [m]

<y>

[m

m]

Beam centroid motion around MR

y = 54 .mm.mrad (RCS-beam)

-200 0 200 400 600 800 1000 1200 1400 1600-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

dp=0 dp=+-0.004

<x>

[m

m]

s [m]

Beam centroid motion around MR

x = 54 .mm.mrad (Parabolic-beam)

Turn_by_Turn changing of the beam centroid location

-2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

10

9

8 7

6

5

4

3

2

1

B

<Y

> [

mm

]

<X> [mm]

RCS_beam

Turn_by_turnobservationpoint #OP

Qx = 22.42

2Qx = 45

Turn numbers are presented on Fig.

Beam centroid motion around MR40GeV

x = 6 .mm.mrad (40GeV-beam)<

x> [

mm

]

S [m]

Beam centroid motion around MR40GeV

y = 6 .mm.mrad (40GeV-beam)<

y> [

mm

]

S [m]

Conclusions:

(2) At the injection energy of 3 GeV in the case of full linear chromaticity correction the maximum value of the beam centroid beating in the horizontal plane is about 2 mm (for the RCS_beam without any COD).

(3) The maximum shift the the beam centroid for MR has been observed at the center of a half of the MR_Arc.

(4) The contribution of the dispersion part into the beam centroid beating for MR is negligible in comparison with the amplitude dependent terms.

(1) Sextupole magnets, which are used for the chromaticity correction in MR, lead to the beam centroid beating around the ring in the horizontal plane. This is the ‘leading order’ effect of the sextupole field nonlinearity.

… to consider

… possibility to correct (reduce) the beam centroid shift

caused by the sextupole field nonlinearity for MR

Possible solutions:

… re-arrange the sextupole magnets (SDA)

… look at the effect of bump-orbit at the locations where

the beam centroid shift is maximum.

re-arrange the sextupole magnets (SDA) …

yxyxx 2/12/3Minimum …

SDA: x = 6.5 m, y = 19.9 m

SFA: x = 16.6 m, y = 7.78 m

Present SDA&SFA location

Move SDA to QFX … then … y(SDA) will be reduced to minimize contribution SDA to the shift of the beam centroid… CHECK !!!

Disadvantage … changing of the SDA location at the position with smaller y

will lead to increasing the SDA sextupole strength to keep the chromaticity after correction … CHECK !!! … increasing of the SDA strength could lead to reduction of DA … CHECK !!!

MR: Dispersion_SuperPeriod

SDA

MR: BetaX_ARC

SDASFA

x (SDA) = 6.5m

x (SFA) = 16.6 m

MR: BetaY_ARC

y (SDA) = 19.9m

y (SFA) = 7.78 m

Drift (SDA_QFX) ~ 2.3 m