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CIRCULAR ACCELERATOR

CIRCULAR ACCELERATOR. INTRODUCTION. IN CONSTANT MAGNETIC FIELD, PARTICLE MAKE CIRCULAR MOTION WITH. AND THE FREQUENCY OF ROTATION. WHERE. CYCLOTRON FREQUENCY. BETATRON CYCLOTRON---SYNCHRO-CYCLOTRON SYNCLOTRON TRANSVERSE STABILITY UNIFORM MAGNETIC FIELD IS NOT CAPABLE OF CONTAIN

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CIRCULAR ACCELERATOR

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  1. CIRCULAR ACCELERATOR

  2. INTRODUCTION IN CONSTANT MAGNETIC FIELD, PARTICLE MAKE CIRCULAR MOTION WITH AND THE FREQUENCY OF ROTATION WHERE CYCLOTRON FREQUENCY

  3. BETATRON CYCLOTRON---SYNCHRO-CYCLOTRON SYNCLOTRON TRANSVERSE STABILITY UNIFORM MAGNETIC FIELD IS NOT CAPABLE OF CONTAIN PARTICLES TRANSVERSLY DIPOLE P DIPOLE + GRADIENT P NEED GRADIENT TO KEEP PROTONS IN PLACE

  4. THE GUIDE FIELD DECREASE SLIGHTLY ALL AROUND THE RING WITH STABILITY CONDITION n ; FIELD INDEX VERTICAL TUNE OF THE RING HORIZONTAL TUNE OF THE RING NEGATIVE n IS NEEDED FOR VERTICAL STABILITY BUT HORIZONTAL STABILITY PREFER A POSITIVE n THERE IS SMALL REGION OF NEGATIVE N HORIZONTAL MOTION IS STABLE THE CONDITION SEVERLY LIMIT THE ENERGY OF THIS KIND OF ACCELERATOR WIDELY CONSIDERED <10 GeV HILLS AND VALLY DESIGHN TO INCREASE TRANSVERSE FOCUSING FFAG

  5. STRONG FOCUSING LENSES---QUADRUPOLE HORIZONTALLY FOCUSING VERTICALLY DEFOCUSING VERY WEAK QUADRUPOLE SECTION IS KIND OF GRADIENT MAGNET A CYCLOTRON OR WEAK FOCUSING MACHINE USE USE STRONG QUADRUPOLE PLACED ALTERNATELY FOCUSING AND DEFOCUSING

  6. SINGLE PARTICLE DYNAMICS USE MINIMUM AMOUNT OF MATHEMATICS CURVILINEAR COORDINATE SYSTEM S ALONG THE MOTION OF EQUILIBRIUM ORBIT X HORIZONTAL Y VERTICAL y s x rO USE COORDINATE SYSTEM MOVING WITH THE REFERENCE PARTICLE

  7. EQUATION OF MOTION WITH THE LORENTZ FORCE ON THE PARTICLE x y where or The solution of the equation should be periodic motion. BETATRON MOTION Non-linear terms are ignored. Will be discussed when discussing resonances

  8. STABILITY OF BETATRON MOTION DP=0 THEN THE x,y MOTIONS ARE SAME AND IN SYNCHROTRON k IS PERIODIC—SUPER PERIODE SOLUTION OF THE EQUATION CAN BE WRITTEN Where C and S are sin like and cos like function and

  9. IF WRITE IN MATRIX FORM MATRIX M RELATES THE MOTION OF THE PARTICLE IN ONE POSITION TO THE OTHER M CAN BE LATTICE FUNCTION OR INSERTION ie k; CONSTANT BETWEEN S AND SO IF k IS PIECEWISE CONSTANT, MATRIX CAN BE WRITTEN WHEN k<0

  10. NOW FUNCTION C AND S IS BOUND EVERYWHERE AND PERIODIC OVER SUPERPERIODE, M CAN BE SIMPLIFIED FOR TIME BEING a b g ARE SOME CONSTANT THE EIGENVALUE OF THE MATRIX IS AND THE MATRIX ALSO CAN BE EXPRESSED IIS UNITMATRIX

  11. SINCE AND THEREFORE Ie M IS BOUND FOR ALL K

  12. AMPLITUDE OF BETATRON MOTION FLOQUET’S THEOREM SAYS EQUATION WITH HAS 2 PARTICULAR SOLUTION OF FORM ALSO FROM MATRIX EQUATION THEREFORE

  13. BY DIFFERENTIATING ALSO COMBINING WITH ORIGINAL EQUATION OF MOTION COMBINING TWO EQUATION Since first term is all real COMBINING WITH

  14. BY INTEGRATING where BETATRON OSCILLATION BEHAVES LIKE QUASI-HARMONIC LOCAL AMPLITUDE IS WAVE LENGTH l~b m(S) : PHASE ANGLE THE TUNE OF THE RING AND

  15. PHASE SPACE, ADMITTANCE, EMITTANCE TWO CONJUGATE VARIABLE u AND PU OR u’ u-u’ SPACE ---PHASE SPACE COURANT-SNYDER INVARIANT AND TWISS PARAMETERS

  16. NOW INTRODUCE WITHOUT A DERIVATION DISPERSION AND ITS DERIVATIVE COUSIN OF COURANT SNYDER INVARIANT AND IT IS PSEUDO-INVARIANT 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

  17. DISPERSIONS MOMENTUM SPREAD VARIABLE LET D(S,SO) BE THE SOLUTION THE TRANSFER MATRIX WOULD BE IF k AND r ARE PIECEWISE CONSTANT where

  18. IF kx<0 IF WE REPRESENT THE MATRIX AS SOLUTION OF THE DISPERSION EQUATION

  19. INTEGRAL REPRESENTATION OF THE SOLUTION IN X X’ SYSTEM THE DISPERSION AVERAGE IS CALLED MOMENTUM COMPACTIONFACTOR ONE OF A USEFUL NUMBER DENOTED a

  20. LINEAR LATTICE THIN LENS APPROXIMATION FODO CELL OF LENGTH L L/2 L/2

  21. COMPARE WITH

  22. BETA FUNCTION VS PHASE ADVANCE

  23. USEFUL MATRIX---TRANSFER MATRIX FOR TWISS PARAMETER TWO DIMENSIONAL TRANSFER MATRIX ELEMENT PHASE ADVANCE CAN BE CALCULATED

  24. ANOTHER USEFUL FORMULA WHEN ONE KNOWS TWISS PARAMETER AND PHASE ADVANCE 2 X 2 MATRIX ELEMENT IS BACK TO THIN LENS APPROXIMATION IF A BNDING MAGNET IS IN THE CELL

  25. AT THE FOCUSING AND DEFOCUSING QUADRUPLE THE h’ MUST BE 0 THEREFORE DISPERSION SUPPRESORS IT IS USEFUL SOME TIMES TO HAVE DISPERSION FREE STREIGHT SECTIONS f f f f f1 f1 f2 f2 CELL

  26. USING LAST 2 CELLS(4 DIPOLES) SUPPRESS THE DISPERSION. IF ONE HAS FOR m=p/2 f1=f2=f/2 FOR m=p/3 f1=0 f2=f MORE PRACTICAL WAY TO SUPPRESS DISPERSION SUPPOSE ONE HAS TWO BENDS h=0 h=0 p

  27. CHASMAN-GREEN DOUBLE BEND ACHROMAT USED IN MANY LIGHT SOURCE FOR 60OLATTICE p p ANY PAIR OF 6TH MAGNET CAN BE ELIMINATED FROM THIS STRING FOR 90O LATTICE p p p

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