# Presentation: 196434

## Correction of sextupole resonances

## w.p. (6.36,6.22) - N=0.6*10^14,Correction of 3Qx=19 at low intensity

**Total emittance pi mm mrad**

**% outside**

**blue**** – no errors**
**red**** – errors, no correction**
**green**** – correction of islands**

## w.p. (6.36,6.22) – N=2.0*10^14

**Blue – no sextupole errors,**
**Grey**** – sextupole errors, no correction**
**Green**** – sextupole errors, 3Qx=19 corrected**

## w.p. (6.36,6.22)

**3****rd**** order resonances**
**excited by sextupole**
**errors**

**For intensity >1*10^14**
**requires simultaneous correction**
**of ****3Qx=19**** and ****2Qy-Qx=6**** resonances**

## w.p. (6.36,6.22) – N=2.0*10^14, simultaneous correction of 3Qx=19 and 2Qy-Qx=6 resonances 

**Blue – no errors **
**Grey**** – errors, no correction**
**Green****- errors, correction of 3Qx=19**
**Pink**** – errors, ****simultaneous correction of 3Qx=19 and 2Qy-Qx=6**

## Towards VERY high intensity, w.p. and crossed resonances

**No that we found that we can have good**
**correction in the presence of space charge,**
**we want to see whether we can push intensity very high by considering the **
**w.p. far away from coherent structure resonance at integer.**
**We then take ****(6.4,6.3)**** and perform simultaneous correction of ****Qx+2Qy=19**** sum **
**resonance and ****3Qx=19**** with only sextupole errors being introduced.**

## w.p. (6.4,6.3) – towards very high intensity,Simultaneous correction of both sum coupling and 1D resonance

- blue

**Blue – no errors,**
**Red**** – errors, no correction, N=2*10^14**
**Green**** – errors, correction of Qx+2Qy=19 and 3Qx=19 resonances, N=2*10^14**
**Pink**** – errors, ****correction of Qx+2Qy=19 and 3Qx=19 resonances, N=3*10^14**
**Grey**** – errors, correction of Qx+2Qy=19 and 3Qx=19 resonances, N=4*10^14**

## Correction of sextupole resonances:w.p. (6.36,6.22) - Correction of 3Qx=19 and 2Qy-Qx=6

**Total emittance pi mm mrad**

**% outside**

**N=0.6*10^14**
**blue – no errors**
**red**** – errors, no correction**
**green**** – errors, correction of 3Qx=19**

**N=2*10^14**
**blue – no errors**
**grey**** – errors, no correction**
**green**** – errors, correction of 3Qx=19**
**pink**** – errors, ****simultaneous correction of 3Qx=19 and 2Qy-Qx=6 resonances**

- 3Qx

- 2Qy-Qx

- Qx

- Qy

## Correction of sextupole resonances:w.p. (6.4,6.3) - Correction of sum coupling resonance Qx+2Qy=19 and 3Qx=19 resonance

**N=0.6*10^14**
**blue- no errors**
**red**** – errors, no correction**
**pink ****– errors, correction of Qx+2Qy=19**

**N=2.0*10^14**
**blue- no errors**
**red****- errors, no correction**
**green****- errors, ****simultaneous corrections of **
**Qx+2Qy=19 and 3Qx=19 resonances**
**N=3.0*10^14**
**pink**** – errors, corrections of Qx+2Qy=19 and**
**3Qx=19 resonances**

**Total emittance pi mm mrad**

**% outside**

**% outside**

## Tentative conclusions (based on correction of sextupole resonances – March 2003)

**For the test  cases of local errors, we found out that detailed correction of resonance without space  charge gives good results for the case when such resonance are crossed at high intensity.**
**Strength of correctors used is well within the range.**
**These studies suggest important conclusions:**** **
**   ****For ****high intensity**** operation one can chose working point above the dangerous nonlinear resonance and ****cross these resonance with relatively low losses**** at high intensity, provided that one did careful correction of these resonance – it was demonstrated that it is possible to do so.**

## Correction of octupole resonances:w.p. (6.36,6.22), N=1*10^14, Correction of octupole resonance 2Qx+2Qy=25

**blue**** – no errors**
**red ****– b3=60 units, no correction**
**green**** – ****b3=60**** units ****plus correction of 2Qx+2Qy=25**** resonance**

## Correction of octupole resonances:w.p. (6.36,6.22)- N=1*10^14 correction of strong octupole resonance 2Qx+2Qy (b3=120 units)

**blue – no errors**
**gray**** – b3=120 units, no correction**
**pink**** – ****b3=120**** units ****plus correction of 2Qx+2Qy**** resonance**

## Correction of sextupole and octupole resonance simultaneously: w.p. (6.36,6.22) – N=2*10^14

**blue – no errors**
**red**** – b2=30 units, b3=60 units, no correction**
**green**** – ****simultaneous correction**** of**
** ****sextupole:  ****3Qx=19, 2Qy-Qx=6**** and octupole: ****2Qx+2Qy****  resonances**

## Correction of skew octupole resonances

**We do not have skew octupole**
**correctors.**
** ****However, these resonances**
**are there.**
**For completeness, we thus explore**
**their importance and **
**their correction.**

## Correction of skew octupole resonance:w.p. (6.36, 6.22) – Qx+3Qy=25 –sum resonances

**red**** –     a3 errors, no correction**
**green**** – a3 errors plus skew-octupole correction**

## Effective strength of resonances 

**Approximate strength**
**of resonances for the case**
**of lumped errors in units of Ks**
**of Qx+2Qy=19 resonance:**
**Sext.           (1,2)=19 –>  Ks**
**Skew sext. (2,1)=19  ->  Ks**
**Oct.            (0,4)=25  ->  0.3*Ks**
**Skew oct.  (1,3)=25  ->  0.7*Ks**
**Oct.            (2,2)=25  ->  0.9*Ks**
**Sext.          (3,0)=19 ->   0.6*Ks**
**Skew Oct.  (3,1)=25  ->  0.7*Ks**
**Sext.          (1,-2)=6  ->   Ks**
**Skew oct.  (3,-1)=12  -> 0.7*Ks**

## Error budget

**       ****In resonance correction studies we assumed lumped b2=30 units and b3, a3 = 60 units which corresponds to **
**5 times stronger random sextupole errors than measured**
**10 times stronger skew octupole errors than measured**
**      ****With the present error budget coming just from arc dipoles and arc quadrupoles the dominant contribution comes from normal sextupole and normal octupole resonances –  good correction.**
**       ****If the budget of skew octupole errors becomes factor of  5 stronger than the present - one will need skew octupole correctors.**