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Transcript of Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Analytic Approximations of Tune Shifts...
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
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
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
Betatron Oscillations
Forces acting on beam
vanishes at b
Nominal beam equilibrium position
Linearized forces :
,
Incoherent regime
(single particle dyn.)
Coherent regime
(whole beam dyn.)
Nominal tune, H-V simmetry no H-V coupling under guiding field (can be relaxed)
,
nominal orbit radius
angular circulation freq.
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
Betatron Oscillations, contd.
Normal-mode Lasletts:
(for the two incoherent normal modes one has always ).
U is a non-diagonal tensor,
Betatron oscillations ( )
=,
tune-shift tensor Laslett tensor
Square vs. Circular Liner
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
a
= /a (square), = /R (circle)
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
Twin-Beam Toy Model
R
b
a
High r yoke
Several possible regimes :
Incoherent-incoherent :Coherent-coherent :Mixed :
( ) ( ) ( ) ( ). ., , 1, 2i i i i
b eq eqr r r r i
( ) ( ) ( ). , 1, 2i i i
b eqr r r i
( ) ( ) ( ) ( ) ( ) ( ) ( ). . ., ,I i i i j j j
b eq eq b eqr r r r r r r
Different boundary conditions to be imposedon the electric () and magnetic (Az) potentialstatic (=) and dynamic () parts :
1
( ) ( )
0, n r npipe wall r R yoke r R yoke
A A
1
( ) ( )n r n
r R yoke r R yoke
A A
0pipe wall
A
, high-frequency, non penetrating modes (skin depth << pipe wall thickness)
, low-frequency, penetrating modes (skin depth >> pipe wall thickness)
…yielding different coherent and mixed dynamics.
[S. Petracca, Part. Accel., 62 (1999) 241]
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
R
b
a
High r yoke
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):
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
Twin-Beam Toy Model, contd.
R
b
a
(, ) = local (pipe) polar coordinates
R=b=2a r = 104
| | | |
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
Twin-Beam Toy Model – One Beam
R
b
a
(, ) = local (pipe) polar coordinates
R=b=2a r = 104
| | | |
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
Real-World Geometries: Numerics
[S. Petracca, et al. Part. Accel. 63 (1999) 37]
[S. Petracca, F. Ruggiero, et al., Proc. PAC-97, 1753]
Random Paths Computes the (complex) potential only on a circle, using stochastic calculus, and then uses Cauchy integral formula for computing the Lasletts without theneed of approximating derivatives with finite differences.
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.
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
MoMs : Lasletts vs. Round-Corner Radius
(fixed off-axis distance; several angular position)
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
MoMs : Lasletts vs. Round-Corner Radius
(fixed off-axis distance; several angular position)
MoMs : Hard-Cut Circle
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
Random Paths : Cold Bore HOM
Scaled 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
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
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
Beam Coupling Impedances from Reciprocity Theorem
Beam coupling impedances from potentials…
Reciprocity…
source (transverse) position
field (transverse) position
Potential – unperturbed pipe (known)Potential – perturbed pipe (unknown)
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
Impedance (Leontóvich) b.c.,non-perfectly conducting wall
Beam Coupling Impedances from Reciprocity Theorem, contd.
S
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
Beam Coupling Impedances from Reciprocity Theorem, contd.
[S. Petracca, Part. Accel., 50 (1995) 211.]
constitutive wall perturbation
geometric wall perturbation (E0z= 0 on S0)
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
Example: Rounded Square Liner
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
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 ?
Leontóvich B.C. and Beyond
Originally formulated for a planar surfacebounding some highly reflecting transversely homogeneous lossy half-space…
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
[T. Senior, J. Volakis, Approximate Boundary Conditions in EMtics, IEE Press, 1995][L.N. Trefethen, L. Halpern, Math. Comput. 47 (1986) 421]
…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)
-curved surfaces, provided e.g.,
local radius of curvature-inhomogeneous media, since
=0
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
reproduces Gluckstern’s 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 Kurennoy’s result for Z|| [S. Kurennoy, Part Accel. 50, 167, 1995], when used in our perturbative formula for Z||).
external hole polarizabilities
,
skin depth, cold bore
TEM field, cold bore
holes’ position
,
Axially aligned holes in a pipe surrounded by lossy co-axial shield (LHC cold bore)
linear hole density
n= surface density of holes
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 (1977)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 Bethe’s formula for polarizabilities beyond the underlying quasi-static (kD << 1) assumption (very short bunches)
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 GuidedWaves, IEEE-McGraw-Hill, 1998]
Example: LHC Parasitic Loss Budget
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
,(holes)(holes)
(ohmic(ohmic holes)holes)
(ohmic,(ohmic, holes)holes)
(ohmic)(ohmic) (ohmic)(ohmic)
surface fraction not covered by holes
perforatedCu strips
bunch length (gaussian)
Example: Parasitic Loss Budget, contd.
,
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
perforatedCu strips
(ohm
ic)
Example: Parasitic Loss Budget, contd.
,
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
perforatedCu strips
in good agreement with measurements (Caspers, Morvillo and Ruggiero)
"Beauty is truth, truth beauty,“ that is all Ye know on earth, and all ye need to know.
[John Keats, 1795-1821]
…What’s it all for ?
Thanks, Francesco.