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Resonances

Resonances. Neil Marks, DLS/CCLRC, Daresbury Laboratory, Warrington WA4 4AD, U.K. Tel: (44) (0)1925 603191 Fax: (44) (0)1925 603192. Philosophy.

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Resonances

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  1. Resonances Neil Marks, DLS/CCLRC, Daresbury Laboratory, Warrington WA4 4AD, U.K. Tel: (44) (0)1925 603191 Fax: (44) (0)1925 603192

  2. 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.

  3. 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 particle’s 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 ‘n’ in USA; • radial and vertical tunes are different: QR and QV; • values of tune vary widely between different accelerators; • with weak-focusing accelerators Q<1, with strong focusing Q>1; • in strong focusing large accelerators Q>>1 (eg QR = 28.xx in Diamond); • that is to say that there are many 2p phase advances per revolution.

  4. The integer resonance • Consider a magnetically perfect lattice with an exact integer QR; 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 2np away; 2np vertical dipole error field.

  5. Half integer resonance • Now consider a lattice with a fractional part of QR 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 2np displaced. quad field error

  6. 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 QR and QV. • General equation for a resonance phenomena: • n QR + m QV = 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.

  7. Resonance diagram • In the region 10 to 10.5 :

  8. Resonances shown • First order (integer): QR = 10; • QV = 10; • Second order (half integer): 2 QR = 21; • QR + QV = 21; • 2 QV = 21; • QR – QV = 0 • Third order: 3 QR = 31; • 2 QR + QV = 31; • QR + 2 QV = 31; • 3 QV = 31; • Forth order: 4 QR = 41; • 3 QR + QV = 41; • etc, plus some third order difference resonances.

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