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CIRCULAR ACCELERATOR IV

CIRCULAR ACCELERATOR IV. CHROMATICITY CHROMATIC ABERATION. FOCUSING STRENGTH OF LATTICE QUADRUPOLE. VARIES WITH MOMENTUM P ie B r. SINCE TUNE OF THE MACHINE Q VARIES WITH k. : CHROMATICITY. NATURAL CHROMATICITY. IN GENERAL IS ABOUT 1.3 TIMES OF Q.

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CIRCULAR ACCELERATOR IV

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  1. CIRCULAR ACCELERATOR IV

  2. CHROMATICITY CHROMATIC ABERATION FOCUSING STRENGTH OF LATTICE QUADRUPOLE VARIES WITH MOMENTUM P ie Br SINCE TUNE OF THE MACHINE Q VARIES WITH k : CHROMATICITY NATURAL CHROMATICITY

  3. IN GENERAL IS ABOUT 1.3 TIMES OF Q IN SMALL MACHIN LIKE SNS OR ISIS NATURAL CHROMATICITY IS GOOD FOR STABILIZING THE MACHINE HOWEVER IN LARGE MACHINE LIKE SPS FERMI’S MAIN INJECTOR NATURAL CHROMATICITY IS TO BIG TO CONTAINE THE TUNE SPREAD SEXTUPOLE HIGH DISPERSION AREA -DP +DP DIFFERENT FOCUSING FOR HIGH AND LOW MOMENTUM

  4. HIGH MOMENTUM HOR. FOCUSING VER. DEFOCUSING LOW MOMENTUM HOR. DEFOCUSING VER. FOCUSING REDUCEHORIZONTAL CHROMATICITY INCREASE VERTICAL CHROMATICITY NEED STRING OF SEXTUPOLES BOTH IN HORIZONTAL BETA MAX AND VERTICAL BETA MAX NUMARICALLY

  5. RESONANCES RESONANCES OCCURE WHEN INTEGER INTEGER ORDER OF RESONANCES THE RESONANCE IS DRIVEN BY STH FOURIER COMPONENT OF THE 2l POLE MAGNETIC FIELD ESPECIALLY IF WHERE p IS MACHINE PERIODICITY AND i IS AN INTEGER CALLED STRUCTURE RESONANCES THEREFORE ONE SEEKS HIGHER PERIODICITY THE AGS HAS p=12 BUT HAS PSEUDO PERODICITY OF 60

  6. 9 2/3 1/2 1/3 8 1/3 1/2 9 2/3 8

  7. DIPOLE ERROR SOMETIMES IT IS CONVENIENT TO WORK WITH NORMALIZED SPACE V V’ where AND IN THIS SYSTEM PARTICLES GOING AROUND THE CIRCLE AND ORIGINAL EQUATION OF MOTION CAN BE REPRESENTED REWRITING IN Y PLAIN WITH NORMALIZED TO NOW INTRODUCE SMALL DIPOLE KICK

  8. NOW TAKE THE POINT DIAMETRICALLY OPOSIT TO THE KICK TO BE SINCE ORBIT HAS TO CLOSE BY DEFINITION MATCHES THE CHANGE IN SLOPE AT THE KICK OR SOLUTION IS

  9. THUS IF Q IS INTEGER THE AMPLITUDE GOES INFINITY AND ANYWHER NEAR INTEGER AMPLTUDE GROWS THE SOURCE OF ERROR DIPOLE FIELD ERROR INCLUDING MAGNETIZATION REMENNANT FIELD DIPOLE ROLL QUADRUPOLE SURVEY ERROR STRAY FIELD LIKE EARTH FIELD ONE CAN SUM UP ALL THE ERRORS BUT MOST IMPORTANT IS THE FOURIER COMPONENT OF TERM NEAR Q LIKE 8TH AND 9TH HARMONIC FOR THE AGS

  10. SECOND ORDER RESONANCE SUPPOSE THERE IS QUADRUPOLE ERROR AT THE PHASE PROTON IS ON THE NORMALIZED PHASE SPACE OF CIRCLE OF RADIUS THE UNPERTURBED PROTON AT THE LOCATION IS ADD A DIVERGENCE KICK SLIGHT INCREASE IN AMPLITUDE. BUT MORE SERIOUS PROBLEM IS SMALL PHASE ADVANCE CHANGE IN Q

  11. FOR EACH TURN THE Q CHANGES BY THIS AMOUNT. IF Q IS CLOSE TO HALF INTEGERR, THE TUNE MAY REACH AND COINCIDE WITH HALF INTEGER. FROM THAT POINT ON THE TUNE IS LOCKED TO THE 2ND ORDER STOP BAND AND AMPLITUDE GROWS COHERENTLY AND REACH VACUUM CHAMBER WALL

  12. SEPTUM 1 3 FIXED POINT FIXED POINT 2

  13. THIRD ORDER RESONANCES A THIRD ORDER RESONANCE IS FIRST OF NON-LINEAR RESONANCE CAUSED BY SEXTUPOLE THEREFORE Q IS CLOSE TO 3RD INTEGER THEN 2ND TERM AVERAGES TO 0

  14. TUNE WILL OSCILLATE AROUND UNPERTURBED TUNE WHEN TUNE HIT A 3RD INTEGER IT LOCKS IN AND AMPLITUDE WILL GROW COHERENTLY THIS PHENOMENA IS USED FOR SLOW EXTRACTION FROM THE MACHINE

  15. 5 2 septum Fixed point Fixed point Fixed point 3 1 4 6

  16. .HOW TO CORRECT RESONANCES INTEGER POLE ERROR THE RESONANCE IS CAUSED BY FURTHER MORE IT IS DRIVEN BY Sth FOURIER COMPONENT OF THE MULTIPOLE IF ONE CAN CREATE THE FOURIER COMPONENT OF OPOSIT SIGN IN THE RING THE RESONACE IS ELIMINATED SINCE ONE DOES NOT KNOW NITHER AMPLITUDE NOR THE PHASE OF THE ERROR COMPONENT NEED TWO SETS THE MULTIPOLE TO CORRECT THE ERROR ACTUALLY CANCELTHE FOURIER COMPONENT REMEMBERING THE s IS EITHER EVEN OR ODD, NEXT EVEN OR ODD RESONANCE OF SAME ORDER IS UNIT TUNE AWAY FROM IT CREATE TWO PAIRS OF MULTIPOLES DIAMETRIALLY OPPOSITE LOCATION AND I/2p APART

  17. B A +/-A +; EVEN -: ODD +/-B

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