1. Trapped Modes in LHC Collimator (II) Liling Xiao Advanced Computations Department SLAC National Accelerator Laboratory

2. Using frequency domain code Omega3P to search for the trapped modes below 2GHz and provide HOM parameters for beam heating and coupled-bunch stability studies Beam Frequency Spectrum σ=7.6cm 900MHz Collimator beam pipe R=42mm, Fc(TE11)=2.1GHz, Fc(TM01)=2.73GHz

3. Simulation Model for LHC Collimator ¼ model for Omega3P runs Rectangular Vacuum Tank Round Vacuum Tank ¼ model for Omega3P runs y x z

4. Longitudinal Trapped Modes With magnetic boundary conditions on x and y symmetric planes, modes with Ez component on z-axis are determined. A quarter structure When the beam crosses the collimator, these modes will be excited that result in beam energy loss and collimator power dissipation. Beam heating due to these trapped modes will vary depending on the opening of the two jaws. When the two jaws are fully opened with gap=42mm, the beam heating is getting the largest. So we only calculated the trapped modes when the two jaws are fully opened. H H Ez H H

5. Form factor not included. Loss Factors Gap=42mm

6. Lower Longitudinal Trapped Modes Rectangular Tank F=82MHz F=334MHz F=196MHz F=481MHz F=630MHz F=779MHz F=936MHz B-Field E-Field

7. Lower Longitudinal Trapped Modes 92.8MHz 206MHz 337MHz 477MHz 621MHz 767MHz 912MHz B-Field E-Field Round Tank

8. Power Dissipation Vacuum tank is made of stainless steel, sigma=0.116e7s/m Two jaws are made of copper, sigma=5.8e7s/m For gap=42mm with rectrangular tank P dissipation =15 W For gap=42mm with round tank P dissipation =515 W In round tank, modes spread around the tank and have higher Q. In rectangular tank, modes localize between the jaw and the chamber wall and have lower Q.

9. Shunt Impedance R<10Kohm

10. How To Reduce the Q 1. Change the geometry 2. Put the lossy material along the jaws. What is the acceptable number for the power dissipation on the wall?

11. Transverse Trapped Modes With magnetic boundary condition on y-plane and electric boundary on x-plane, modes with strong Ey component between the two jaws are determined. When beam crosses the collimator, these modes will be excited and generate transverse kick in the y-direction as well as beam energy loss. Due to the small gap of the jaws, the Ey component is very strong over the full length of the collimator especially when the two jaws are fully inserted with gap=2mm. So we only calculated trapped modes when the two jaws are closed. H E H E Ey A quarter structure

12. Kick Factors of Trapped Modes Form factor not included

13. Loss Factors of Trapped Modes Rectangular TankRound Tank Form factor not included Loss factors of transverse modes depend on the beam offset.

14. Power Dissipation These modes can also cause beam energy loss that strongly depends on the beam offset. Round Vacuum Tank Rectangular Vacuum Tank 2.7W6W11WTransverse Modes (<2GHz) gap=2mm 15WLongitudinal Modes (<2GHz) gap=42mm 515WLongitudinal Modes (<2GHz) gap=42mm 6.7W15.2W26.6WTransverse Modes (<2GHz) gap=2mm 0.050mm Max. offset 0.075mm0.100mm Beam offset y0 P dissipation-wall