francesco ruggiero memorial symposium cern, october 3 2007 analytic approximations of tune shifts...
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![Page 1: Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Analytic Approximations of Tune Shifts and Beam Coupling Impedances for the LHC Stefania Petracca](https://reader035.vdocument.in/reader035/viewer/2022081519/56649f3a5503460f94c5771f/html5/thumbnails/1.jpg)
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
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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 ?
![Page 3: Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Analytic Approximations of Tune Shifts and Beam Coupling Impedances for the LHC Stefania Petracca](https://reader035.vdocument.in/reader035/viewer/2022081519/56649f3a5503460f94c5771f/html5/thumbnails/3.jpg)
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.
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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
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Square vs. Circular Liner
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
a
= /a (square), = /R (circle)
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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]
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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):
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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
| | | |
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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
| | | |
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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.
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Francesco Ruggiero Memorial Symposium CERN, October 3 2007
MoMs : Lasletts vs. Round-Corner Radius
(fixed off-axis distance; several angular position)
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Francesco Ruggiero Memorial Symposium CERN, October 3 2007
MoMs : Lasletts vs. Round-Corner Radius
(fixed off-axis distance; several angular position)
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MoMs : Hard-Cut Circle
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
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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
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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 ?
![Page 16: Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Analytic Approximations of Tune Shifts and Beam Coupling Impedances for the LHC Stefania Petracca](https://reader035.vdocument.in/reader035/viewer/2022081519/56649f3a5503460f94c5771f/html5/thumbnails/16.jpg)
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)
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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)
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Francesco Ruggiero Memorial Symposium CERN, October 3 2007
Impedance (Leontóvich) b.c.,non-perfectly conducting wall
Beam Coupling Impedances from Reciprocity Theorem, contd.
S
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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)
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Francesco Ruggiero Memorial Symposium CERN, October 3 2007
Example: Rounded Square Liner
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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 ?
![Page 22: Francesco Ruggiero Memorial Symposium CERN, October 3 2007 Analytic Approximations of Tune Shifts and Beam Coupling Impedances for the LHC Stefania Petracca](https://reader035.vdocument.in/reader035/viewer/2022081519/56649f3a5503460f94c5771f/html5/thumbnails/22.jpg)
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
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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
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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]
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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)
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Example: Parasitic Loss Budget, contd.
,
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
perforatedCu strips
(ohm
ic)
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Example: Parasitic Loss Budget, contd.
,
Francesco Ruggiero Memorial Symposium CERN, October 3 2007
perforatedCu strips
in good agreement with measurements (Caspers, Morvillo and Ruggiero)
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"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 ?
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Thanks, Francesco.