Introduction to the orbit correction of electron storage rings. Theory, Practice and Reality Hiroshi Nishimura Advanced Light Source Lawrence Berkeley National Laboratory University of California September 7, 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 Closed Not closed

Eq. Of Closed Orbit

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

Local Bump 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 ALS has 96 BPMs and 94 Horizontal Correctors. The Smatrix for them is 96x94.

Role of Smatrix Smatrix Virtual RingReal Ring Operation Response Knobs Response

Inverting an Smatrix by SVD (1)

Inverting an Smatrix by SVD (2)

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 14Corrector –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 Field

Kick due to Earth Mag Field

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. 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 Motion H. Wiedemann, "Particle Accelerator Physics II", Springer, Eqs. (3.36) and (3.45) Therefore

Collective Effect COD, Smatrix,....  Single Particle Collective Single 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 –WinVC++, BC++, GCC (MingW) –LinuxGCC (3.3 or newer) Object-OrientedSimple and Flexible –Simple and Flexible –Modified version for neutral molecular accelerator! –Modified for UMER, too.