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Neil Marks; DLS/CCLRC L ecture to Cockcroft Institute 2005/6. © N.Marks MMIV Resonances Neil Marks, DLS/CCLRC, Daresbury Laboratory, Warrington WA4 4AD, U.K. Tel: (44) (0)1925 603191 Fax: (44) (0)1925 603192

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Neil Marks; DLS/CCLRC L ecture to Cockcroft Institute 2005/6. © N.Marks MMIV Philosophy To present a short, qualitative overview of linear and non- linear resonances in circular accelerators, driven by harmonic field errors in lattice magnets as constructed, and stray fields as are present. This is preceded by a brief summary of betatron oscillations and the appropriate nomenclature.

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Neil Marks; DLS/CCLRC L ecture to Cockcroft Institute 2005/6. © N.Marks MMIV Betatron oscillations. the transverse focusing (in both planes) produces oscillations in those particles which are not on the closed equilibrium orbit appertaining to the particular particles momentum; that is to say that the oscillations are associated with the beam emitance, not its momentum distribution; the number of oscillations per single revolution is known as the tune of the accelerator; the tune is given the symbol Q in Europe and in USA; radial and vertical tunes are different: Q R and Q V ; values of tune vary widely between different accelerators; with weak-focusing accelerators Q 1; in strong focusing large accelerators Q>>1 (eg Q R = 28.xx in Diamond); that is to say that there are many 2 phase advances per revolution.

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Neil Marks; DLS/CCLRC L ecture to Cockcroft Institute 2005/6. © N.Marks MMIV The integer resonance Consider a magnetically perfect lattice with an exact integer Q R ; then introduce a small dipole error at one position: the deflection causes an increase in oscillation amplitude, which grows linearly per revolution; this would also occur for any error 2n away; vertical dipole error field. 2n

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Neil Marks; DLS/CCLRC L ecture to Cockcroft Institute 2005/6. © N.Marks MMIV Half integer resonance Now consider a lattice with a fractional part of Q R that is exactly a half integer: dipole error – less serious effect: quadrupole field error: the oscillations will build up on each revolution; also for a quadrupole field error 2n displaced. quad field error

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Neil Marks; DLS/CCLRC L ecture to Cockcroft Institute 2005/6. © N.Marks MMIV Higher order resonances/harmonics Likewise: sextupole errors will blow-up the beam with the fractional part of Q = 1/3 or 2/3; octupole errors will blow-up the beam with the fractional Q = 1/4, 3/4; etc. for higher orders, for both Q R and Q V. General equation for a resonance phenomena: n Q R + m Q V = p;n,m,p any integers; n + m is the order of the resonance; p is the periodicity of the error in the lattice; Note: n and m non-zero is a coupling resonance; n or m are -ve is a difference resonance.

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Neil Marks; DLS/CCLRC L ecture to Cockcroft Institute 2005/6. © N.Marks MMIV Resonance diagram In the region 10 to 10.5 :

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Neil Marks; DLS/CCLRC L ecture to Cockcroft Institute 2005/6. © N.Marks MMIV Resonances shown First order (integer):Q R = 10; Q V = 10; Second order (half integer):2 Q R = 21; Q R + Q V = 21; 2 Q V = 21; Q R – Q V = 0 Third order:3 Q R = 31; 2 Q R + Q V = 31; Q R + 2 Q V = 31; 3 Q V = 31; Forth order:4 Q R = 41; 3 Q R + Q V = 41; etc, plus some third order difference resonances.

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