Introduction to the orbit correction of electron storage rings.

Theory, Practice and Reality

Hiroshi NishimuraAdvanced Light Source

Lawrence Berkeley National Laboratory

University of California

September 7, 2005 @ UMER

UMER vs. ALS• UMER

– 10 KeV Electron Storage Ring with 36 Bend• C ~ 11 m• I = very high

• ALS – 1~2 GeV Electron Storage Ring with 36 Bend

• C ~ 200 m• I ~ 400 mA

Importance of a Good Orbit

• Injection

• Beam Quality– Beam Size– Intensity– Life– Stability

Closed Orbit)( inout rOneTurnr

inout rr

Closed

Not closed

Eq. Of Closed Orbit

skks

s kx

cossin2

Closed Orbit Distortion• Lattice Errors create COD.

– Magnet Misalignments• Transverse dK=dX*Kquad • Tilt -- Bend

– Field Error• Bend• Earth Field!

• Orbit Correction to cancel COD.

Orbit Correction• ~80’s

– Local • Local Bump

– Global• Most Effective Corrector• Harmonic

• 90’s ~

Smatrix + SVDIf BPMs and correctors are more or less uniformly distributed, this combination does almost all the orbit control jobs!

Local Bump

12

33

31

22

23

11

sinsinsin

kkk

1k2k

3k12 2331

How to apply this to the real ring?

How can you know these parameters?

Smatrix-based Correction

• Sensitivity (Response) Matrix.• Popular since early 90’s.• Model-independent.

– Compatible with theory (model Calibration).– Compatible with reality (measurement).

• Reproducibility and Linearity.• Combined with SVD for matrix inversion.

What is Smatrix?• Linear-relationship

kRx

94

1

1

94,962,961,96

94,22,21,2

94,12,11,1

96

3

2

1

:

..

:::

:::

..

..

:dk

dk

dk

RRR

RRR

RRR

dx

dx

dx

dx

ALS has 96 BPMs and 94 Horizontal Correctors.

The Smatrix for them is 96x94.

Role of Smatrix

Smatrix

Virtual Ring Real Ring

Operation

ResponseKnobsKnobs

Response

Inverting an Smatrix by SVD (1)

9494949696969496 VSUR t

EVV t

0..00

..00

:..::

0..0

0..0

94

2

1

S

S

S

SEUU t

0944321 SSSSS

USVR t11

Inverting an Smatrix by SVD (2)

U

S

S

S

VR t

0..00

..00

:..::

0..0

0..0

194

12

11

1

0944321 SSSSS

000....21 NSSS

SVD Inversion at ALS

Limit the number of “knobs”. (Do not to use inefficient knobs.)

• All Smatrix-based– Routine operations by SVD in Matlab– Special Machine Study by SVD in C++

Orbit Control at ALS

Orbit Control at UMER• BPM x 14 • Corrector

– Hor. x 36– Ver. x 18

• There are more correctors than BPMs!• Need for a linear model.

• If the nature is linear, Smatrix works.• SVD is not so crucial but still useful.

• The orbit can be set to zero at BPMs.

Earth Mag FieldEarth Mag Field (Measured)

-500-400-300-200-100

0100200300

0 60 120 180 240 300 360

T (deg)

Ma

g F

ield

(m

Ga

uss

)Br(mG)

Bv(mG)

Bs(mG)

Earth Mag Field replaced by ~ 500 Kicks

-500

-300

-100

100

300

0 60 120 180 240 300 360

T (deg)

Ma

g F

ield

(m

Ga

uss

)

Br(mG)

Bv(mG)

Bs(mG)

Kick due to Earth Mag Field

][3103,101

][4.3][10][1

104.3][][][10

32

4

4

mradL

mcmLFor

mTGaussB

mTBKeVT

COD Correction

To Model or Not

• Establish a linear model to fit the measured Smatrix.

• Use model to correct COD.

• Add more BPMs.

OR

Linear Kick Model

• Standard Matrix Formalism.

• Thin kicks for Earth Mag Field.

cossin

sin1

cos

KK

)( 01 ssK

xKspp

spxx

xx

x

Example

Earth Field as This Kicks

• Transverse– T=10 KeV, B=1 Gauss, L=1 cm 3 mrad– Add kicks every ~ 2 cm Total ~ 500 Kicks

• Longitudinal – Same as transverse kicks.– Vx, Vy << Vs=0.2xC.

Longitudinal MotionH. Wiedemann, "Particle Accelerator Physics II", Springer, Eqs. (3.36) and (3.45)

)()( sBcp

esS s

0)('2

1')(" ysSysSx

0)('2

1')(" xsSxsSy

))('2

1)(( ysSpsSLp yx

))('2

1)(( xsSpsSLp xy

Therefore

Collective Effect

• COD, Smatrix,. . . . Single Particle

CollectiveSingle Particle

Orbit Control

Simulation

• If you have a realistic code (PIC?), calculate the Smatrix and calibrate it.

• Apply orbit corrections!

My code is . . .• Goemon: C++ version of Tracy• It is a library.• Cross Platform

– Win VC++, BC++, GCC (MingW)– Linux GCC (3.3 or newer)

• Object-OrientedSimple and Flexible– Simple and Flexible– Modified version for neutral molecular

accelerator!– Modified for UMER, too.