Analytic Approximations of Tune Shifts and Beam Coupling Impedances for the LHC

Stefania PetraccaUniversity of Sannio at Benevento and INFN

Francesco Ruggiero Memorial Symposium CERN, October 3 2007- 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 - Leontvich Boundary Conditions and Beyond Lossy Walls, Coated Walls, Pumping Holes, etc.Whats it all for ?

Betatron OscillationsForces acting on beamNominal beam equilibrium positionLinearized forces :,Nominal tune, H-V simmetry no H-V coupling under guiding field (can be relaxed)

Betatron Oscillations, contd.Normal-mode Lasletts:(for the two incoherent normal modes one has always ). U is a non-diagonal tensor,Betatron oscillations ( )=

Square vs. Circular Liner

Twin-Beam Toy ModelSeveral possible regimes :Incoherent-incoherent :Coherent-coherent :Mixed :

Different boundary conditions to be imposedon the electric () and magnetic (Az) potentialstatic (=) and dynamic () parts :yielding different coherent and mixed dynamics.[S. Petracca, Part. Accel., 62 (1999) 241]

Twin-Beam Toy Model, contd.The Al-frame holding the beam pipes in the LHC design will preventthe dynamic magnetic field from coupling the beams, even at thelowest 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):

Twin-Beam Toy Model, contd.(, ) = local (pipe) polar coordinates R=b=2a r = 104| || |

Twin-Beam Toy Model One Beam(, ) = local (pipe) polar coordinates R=b=2a r = 104| || |

Real-World Geometries: Numerics[S. Petracca, et al. Part. Accel. 63 (1999) 37]MoM : Rounded Corners & Stadium Shape VariationsBased on efficient representation of the (exact) Greens 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.

MoMs : Lasletts vs. Round-Corner Radius (fixed off-axis distance; several angular position)

MoMs : Lasletts vs. Round-Corner Radius (fixed off-axis distance; several angular position)

MoMs : Hard-Cut Circle

Random Paths : Cold Bore HOMScaled cutoff wavelength of 1st HOMin the coax guide between the beamscreen and the cold bore vs. (scaled) corner radius for several beam screen(scaled) sizes

Francesco Ruggiero Memorial Symposium CERN, October 3 2007- 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 - Leontvich Boundary Conditions and Beyond Lossy Walls, Coated Walls, Pumping Holes, etc.Whats it all for ?

Francesco Ruggiero Memorial Symposium CERN, October 3 2007LHC 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)

Beam Coupling Impedances from Reciprocity Theorem

Beam Coupling Impedances from Reciprocity Theorem, contd.[S. Petracca, Part. Accel., 50 (1995) 211.]

Example: Rounded Square Liner

Francesco Ruggiero Memorial Symposium CERN, October 3 2007- 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 - Leontvich Boundary Conditions and Beyond Lossy Walls, Coated Walls, Pumping Holes, etc.Whats it all for ?

Leontvich B.C. and BeyondOriginally formulated for a planar surfacebounding some highly reflecting transversely homogeneous lossy half-spacebut 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=0

Leontvich B.C. and Beyond : Perforated Walls

By solving a canonical b.v.p. we shew that perforated walls can be modeled using reproduces Glucksterns formula (circular pipe) [R. Gluckstern, J. Diamond, IEEE Trans. MTT-39, 274, 1991] when used in our perturbational formula for Z||)[S. Petracca, Phys. Rev. E60, 6041, 1999]Reproduces Kurennoys result for Z|| [S. Kurennoy, Part Accel. 50, 167, 1995], when used in our perturbative formula for Z||). external hole polarizabilities,Axially aligned holes in a pipe surrounded by lossy co-axial shield (LHC cold bore) linear hole density= surface density of holes

Leontvich 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]

Also, see [Van Bladel, Radio Sci. 14, 319, 1979] for corrections to Bethes formula for polarizabilities beyond the underlying quasi-static (kD

Example: LHC Parasitic Loss Budget(ohmic)surface fraction not covered by holesbunch length (gaussian)

Example: Parasitic Loss Budget, contd.,in good agreement with measurements (Caspers, Morvillo and Ruggiero)

Thanks, Francesco.

In the specific case of LHC, the thick aluminum frame holding the beam linerswill not allow the low frequency dynamic components of the magnetic field to penetrate across the liners wall, and the beams can thus be takenas being dynamically uncoupled. In addition if one neglects, as usual, the space charge forces, andassumes an ultrarelativistic limit, all regimes merge together.The Laslett coefficients for the two beams (carrying equal and opposite line charges) are the sameThe two coherent laslett coefficients differ only in sign, and their absolute valuecoincides with the incoherent one. We are accordingly left with a single and simple formuladescribing this problem. When one beam is off, the order of magnitude of the LaslettIs left unchanged but its angular dependence is markedly affected.When it comes to real-world geometries, for which the relevant potential boundary value problemCannot be solved in terms of elementary functions,One must resort to numerical methods.One must careful however, because the Laslett requires computing a second space- derivative of the potential, which therefore must be known pretty fine mesh for adequate accuracy.

With Francesco we developed two methods.

The first one was a version of Galerkin (or Moments) method, appropriate to modified rectangular pipes.

The second one was an elegant application of stochastic calculus, to solveThe potential equation using random As an example, you may calculate the cut-off wavelengthof the first higher-order mode in the coaxial region between abeam liner with rounded-square cross section, and a circular cold bore. The 1995 tentative design was indeed square, but with several featuresWhich made further investigation necessary, in particular as regards the beam couoling impedances Features include.

The beam coupling impedances can be derived from the electric Debye potential.This latter obeys the Lorentz reciprocity theorem yielding this well known Greens identityNow assume that you know the solution for the potential for some pipe geometry which we call the unperturbed caseWhile you are unable to solve the potential equation in the actual case, Which differs from the unperturbed one in terms of changes in the pipe geometry and boundary conditions.Using the reciprocity formula you obtain this identity which can be re-written in terms of impedancesUsing these formulas above, and a little more algebra---With a little algebra you end-up with these formulas

These formulas are nicely readable.Indeed these terms describe the effect of constitutive perturbations of the pipe-wall, modeled in terms of this wall impedance Z_wWhile this term, which is zero on the unperturbed pipe geometry, where the longitudinal component of the unperturbed field is null,Describes the effect of the geomegtrical changes in the pipe shape.

Of course, these formulas are of no use unless you make the crude approximation of letting the unperturbed field in place of the true one. This gives you a perturbational result,

Which turns out to be pretty accurate for practical purposes.

In some special cases, like the circular losy pipe it even gives back the exact result. ..When both approaches are applicable (Kurennoys requires a geometry for whichthe modal solutions are known; When both approaches are applicable (Kurennoys requires a geometry for whichthe modal solutions are known; Einstein handwriting, describing the expansion rate, density, size, and age of the universe on a single blackboard.