Neil Marks; DLS/CCLRC L ecture to Cockcroft Institute 2005/6. © N.Marks MMIV Resonances Neil Marks, DLS/CCLRC, Daresbury Laboratory, Warrington WA4 4AD,

Neil Marks; DLS/CCLRC Lecture to Cockcroft Institute 2005/6. © N.Marks MMIV

Resonances

Neil Marks,DLS/CCLRC,

Daresbury Laboratory,Warrington WA4 4AD,

U.K.Tel: (44) (0)1925 603191Fax: (44) (0)1925 603192


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.


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 ‘’ 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 2 phase advances per revolution.


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 anincrease in oscillationamplitude, which growslinearly per revolution;

this would also occur forany error 2n away;

vertical dipole error field.

2n


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 buildup on each revolution;

also for a quadrupole fielderror 2n displaced.

quad field error


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.


Resonance diagram

In the region 10 to 10.5 :


Resonances shownFirst 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.

Documents

Neil Marks; DLS/CCLRC L ecture to Cockcroft Institute 2005/6. © N.Marks MMIV Resonances Neil Marks, DLS/CCLRC, Daresbury Laboratory, Warrington WA4 4AD,