Alexander Molodozhentsev

KEK

for MR-commissioning group

September 20, 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

Sextupole field (simplified model)

As = 1/6 B// (x3 – 3xy2)Ax = Ay =0

x/s ~ (a x2 – b y2)

y/s ~ y

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 > ~ < y > = 0 without COD

Vector potential for the sextupole field

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

x = y = 54 mm.mrad

“Dispersion” term … ~ 72 10-6

“Amplitude Dependent” term (X) … ~ 784 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 (Parabolic, max=6) 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

S [m]

<x>

[m

m]

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

p/p = 0

95254 macro_particles

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 (RCS-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 (RCS-beam)

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:

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

The maximum shift the the beam centroid for MR has been observed atthe center of a half of the MR_Arc.

The contribution of the dispersion part into the beam centroid beating for MRis negligible in comparison with the amplitude dependent terms.

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.

MR: Dispersion_SuperPeriod

MR: BetaXY_SuperPeriod