Advances in Beam Dynamics for LHC: Analytic Approximations and Real-World Geometries

1. Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Analytic Approximations of Tune Shifts and Beam Coupling Impedances for the LHC Stefania Petracca University of Sannio at Benevento and INFN

2. Outlook - Laslett coefficients for Betatron Normal Modes Square vs Circular Pipes A Toy Twin-Liner Model Real-World Gometries: MoMs & Random Paths - Beam Coupling Impedances from Reciprocity Theorem Rounded corners and more - Leontòvich Boundary Conditions and Beyond Lossy Walls, Coated Walls, Pumping Holes, etc. …What’s it all for ? Francesco Ruggiero Memorial Symposium CERN, October 3 2007

3. vanishes at Incoherent regime (single particle dyn.) Coherent regime (whole beam dyn.) angular circulation freq. , nominal orbit radius Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Betatron Oscillations Forces acting on beam Nominal beam equilibrium position , Linearized forces : Nominal tune, H-V simmetry no H-V coupling under guiding field (can be relaxed)

4. tune-shift tensor Laslett tensor , Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Betatron Oscillations, contd. Betatron oscillations ( ) = U is a non-diagonal tensor, Normal-mode Lasletts: (for the two incoherent normal modes one has always ).

5.   a = /a (square), = /R (circle) Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Square vs. Circular Liner

6. R a b High r yoke  , high-frequency, non penetrating modes (skin depth << pipe wall thickness) , low-frequency, penetrating modes (skin depth >> pipe wall thickness)   Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Twin-Beam Toy Model Several possible regimes : Incoherent-incoherent : Coherent-coherent : Mixed : Different boundary conditions to be imposed on the electric () and magnetic (Az) potential static (=) and dynamic () parts : …yielding different coherent and mixed dynamics. [S. Petracca, Part. Accel., 62 (1999) 241]

7. R a b High r yoke Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Twin-Beam Toy Model, contd. The Al-frame holding the beam pipes in the LHC design will prevent the dynamic magnetic field from coupling the beams, even at the lowest frequency associated with collective beam oscillations. As a result, the two beams are dynamically uncoupled. Neglecting space-charge effects, all regimes (incoherent, coherent, & mixed) merge together in the limit as 1, yielding (both pipes):

8. R a b Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Twin-Beam Toy Model, contd. R=b=2a r = 104 | | | | (, ) = local (pipe) polar coordinates

9. R a b Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Twin-Beam Toy Model – One Beam R=b=2a r = 104 | | | | (, ) = local (pipe) polar coordinates

10. Random Paths Computes the (complex) potential only on a circle, using stochastic calculus, and then uses Cauchy integral formula for computing the Lasletts without the need of approximating derivatives with finite differences. [S. Petracca, F. Ruggiero, et al., Proc. PAC-97, 1753] Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Real-World Geometries: Numerics MoM : Rounded Corners & Stadium Shape Variations Based on efficient representation of the (exact) Green’s function for rectan- gular and circular domains, allowing to shrink the unknown charge density support (and related number of unknown charge expansion coeffs) to a min. [S. Petracca, et al. Part. Accel. 63 (1999) 37]

11. Francesco Ruggiero Memorial Symposium CERN, October 3 2007 MoMs : Lasletts vs. Round-Corner Radius (fixed off-axis distance; several angular position)

12. Francesco Ruggiero Memorial Symposium CERN, October 3 2007 MoMs : Lasletts vs. Round-Corner Radius (fixed off-axis distance; several angular position)

13. Francesco Ruggiero Memorial Symposium CERN, October 3 2007 MoMs : Hard-Cut Circle

14. Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Random Paths : Cold Bore HOM Scaled cutoff wavelength of 1st HOM in the coax guide between the beam screen and the cold bore vs. (scaled) corner radius for several beam screen (scaled) sizes

15. Outlook - Laslett coefficients for Betatron Normal Modes Square vs Circular Pipes A Toy Twin-Liner Model Real-World Gometries: MoMs & Random Paths - Beam Coupling Impedances from Reciprocity Theorem Rounded corners and more - Leontòvich Boundary Conditions and Beyond Lossy Walls, Coated Walls, Pumping Holes, etc. …What’s it all for ? Francesco Ruggiero Memorial Symposium CERN, October 3 2007

16. LHC Pipe Progress Design (1995) Rounded corners (manufacturing limitations) Stainless steel (lossy) pipe (pure Cu would not sustain the quenching forces due to mag- netic field penetration (& para- sitic currents) in case of failure of the cryogenics) Copper-coated surface (uncoated SS would give exces- sive parasitic losses; coating restricted to flat faces, where fields and loss would be largest) Pumping holes (removal of gas desorbed tue to synchrotron radiation) Francesco Ruggiero Memorial Symposium CERN, October 3 2007

17. source (transverse) position Beam coupling impedances from potentials… field (transverse) position Reciprocity… Potential – unperturbed pipe (known) Potential – perturbed pipe (unknown) Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Beam Coupling Impedances from Reciprocity Theorem

18. Beam Coupling Impedances from Reciprocity Theorem, contd. S Impedance (Leontóvich) b.c., non-perfectly conducting wall Francesco Ruggiero Memorial Symposium CERN, October 3 2007

19. constitutive wall perturbation geometric wall perturbation (E0z= 0 on S0) Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Beam Coupling Impedances from Reciprocity Theorem, contd. [S. Petracca, Part. Accel., 50 (1995) 211.]

20. Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Example: Rounded Square Liner

21. Outlook - Laslett coefficients for Betatron Normal Modes Square vs Circular Pipes A Toy Twin-Liner Model Real-World Gometries: MoMs & Random Paths - Beam Coupling Impedances from Reciprocity Theorem Rounded corners and more - Leontòvich Boundary Conditions and Beyond Lossy Walls, Coated Walls, Pumping Holes, etc. …What’s it all for ? Francesco Ruggiero Memorial Symposium CERN, October 3 2007

22. local radius of curvature -curved surfaces, provided e.g., [T. Senior, J. Volakis, Approximate Boundary Conditions in EMtics, IEE Press, 1995] [L.N. Trefethen, L. Halpern, Math. Comput. 47 (1986) 421] Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Leontóvich B.C. and Beyond Originally formulated for a planar surface bounding some highly reflecting transversely homogeneous lossy half-space… =0 …but in fact much more versatile than this. Can be applied, e.g., to: -lossy stratified media (e.g., by repeated use of the TL impedance transport formula) -inhomogeneous media, since

23. skin depth, cold bore TEM field, cold bore , holes’ position Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Leontòvich B.C. and Beyond : Perforated Walls By solving a canonical b.v.p. we shew that perforated walls can be modeled using , = surface density of holes Reproduces Kurennoy’s result for Z|| [S. Kurennoy, Part Accel. 50, 167, 1995], when used in our perturbative formula for Z||). Axially aligned holes in a pipe surrounded by lossy co-axial shield (LHC cold bore) external hole polarizabilities linear hole density reproduces Gluckstern’s formula (circular pipe) [R. Gluckstern, J. Diamond, IEEE Trans. MTT-39, 274, 1991] when used in our perturbational formula forZ||) [S. Petracca, Phys. Rev. E60, 6041, 1999]

24. Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Leontòvich B.C. and Beyond : Perforated Walls, contd. • Hole polarizabilities available for a variety of hole-shapes - • see[F. de Meulenaere, J. Van Bladel, IEEE Trans., AP-25 • 198; also R. de Smedt, J. van Bladel, IEEE Trans., • AP-28 (1980) 703] Corrections for hole - hole couplings also worked out [S. Petracca, Phys. Rev. E60 (1999) 6030] in the quasi- static approximation[R.E. Collin, Field Theory of Guided Waves, IEEE-McGraw-Hill, 1998] Also, see [Van Bladel, Radio Sci. 14, 319, 1979]for corrections to Bethe’sformula for polarizabilities beyond the underlying quasi-static (kD << 1) assumption (very short bunches)

25. (ohmic holes) (ohmic, holes) , perforated Cu strips (holes) (ohmic) Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Example: LHC Parasitic Loss Budget (ohmic) surface fraction not covered by holes bunch length (gaussian)

26. Example: Parasitic Loss Budget, contd. (ohmic) , perforated Cu strips Francesco Ruggiero Memorial Symposium CERN, October 3 2007

27. perforated Cu strips Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Example: Parasitic Loss Budget, contd. , in good agreement with measurements (Caspers, Morvillo and Ruggiero)

28. …What’s it all for ? "Beauty is truth, truth beauty,“ that is all Ye know on earth, and all ye need to know. [John Keats, 1795-1821]

29. Thanks, Francesco.