1. Modification of the vacuum chamber in ATLAS Effect on impedance and related quantities
E. Métral, N. Mounet and B. Salvant
with the help of R. Veness and M. A. Gallilee
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

2. Context and motivation
There is a proposition to reduce the diameter of the Beryllium vacuum chamber inside the ATLAS pixel detector:
Radius should change from 29 mm to 22.5 mm,
Need also to insert two new conical transitions (=tapering) from the existing 29 mm to the 22.5 mm radius, using the available 100 mm length.
Transition angle = arctan(6.5/100) ≈ 3.7 deg << 15 deg (value set by L. Vos in the past).
Issues to assess:
Longitudinal and transverse impedance for beam stability,
Longitudinal impedance for power loss.
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

3. Current situation 7.1 m, inner diameter 58 mm NEG coating (1 μm)
Beryllium (0.8 mm at -15°C)
Isolating aerogel (4 mm)
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

4. Current situation
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

5. Future situation inner diameter 58 mm 7.1 m, inner diameter 45 mm
100 mm transition
with the same layers of materials inside (NEG 1 μm – Be 0.8 mm – aerogel 4 mm)
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

6. Impedance contributions
Two different contributions are to be distinguished:
Resistive wall impedance, due to the resistivity of the beam pipe. Models assume a perfectly smooth cylindrical geometry.
Geometrical impedance, due to the conical transitions (tapering) in the modified situation. Models assume a perfectly conductive chamber.
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

7. Material parameters used for the resistive wall impedance
Beryllium:
Resistivity: Ω.m (from specifications)
Permittivity: 1
Aerogel:
Resistivity: Infinity (approximation)
Permittivity: 1.1 (from specifications)
NEG:
Resistivity: Ω.m (David Seebacher, F. Caspers, NEG properties in the microwave range, SPSU Meeting, 17th February, CERN)
Permittivity: 10
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

8. Resistive wall models Two models used here:
Exact multilayer analytical formula for an axisymmetric geometry (assumptions: linear materials, infinite length i.e. no side effects). Implemented in a Mathematica code (ReWall –
Classic thick wall formula (Chao’s book) giving a simple impedance formula for a resistive beam pipe, valid at “intermediate” frequencies:
Longitudinal impedance:
Transverse impedance:
with L the length of the element, b its radius, s the wall conductivity, m0 the vacuum permeability, Z0= m0 c the vacuum impedance (c=speed of light), and w the angular frequency.
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

9. Power loss due to the impedance
General expression for any impedance: loss is a function of the real part of the longitudinal impedance and of the bunch spectrum l(w) (=line density in frequency domain, here for a parabolic bunch):
with f0=w0 /2p the revolution frequency, M the number of bunches and e the electron charge.
For a classic resistive wall impedance and a parabolic line density, we can get an analytic formula:
for a bunch of total length 4sz=4stc (in meters), L= being the circumference of the LHC and Lelem the length of the element considered for the impedance (here 7.1 m).
G. Rumolo, USPAS 2009
Power loss per unit length :
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

10. Beam parameters (for power loss computation)
Energy
Bunch length (4 σt)
Number of bunches
Intensity per bunch
450 GeV
1.4 ns (during MD)
2808 *2 beams
p/b
3.5 TeV
0.8 ns (during MD)
2808*2 beams
7 TeV
1 ns (nominal)
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

11. Contribution of the aerogel to the power loss?
The skin depth of Beryllium at 20 kHz is 0.8 mm
 above 20 kHz, what is behind the 0.8 mm thick beam pipe contributes much less
Curve to integrate to get power loss
 below 20 kHz, the contribution to power loss is negligible
As a consequence, the material behind 0.8 mm beam pipe does not contribute (also checked by multilayer calculations)
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

12. Resistive wall contribution: power loss
Energy
Inner radius
Number of layers taken into account
Bunch length (4st)
Power loss (W/m) – exact
Power loss (W/m) – classic approx.
450 GeV
29 mm
1 (Be)
1.4 ns (MD)
0.62
22.5 mm
0.80 (+29%)
0.80
3.5 TeV
0.8 ns (MD)
1.44
1.86 (+29%)
1.86
7 TeV
1 ns (nominal)
1.03
1.33 (+29%)
1.33
2 (Be 0.8mm +air)
N/A
2 (NEG 1 μm + Be)
→ about 30% increase in the power loss, confirmed by classic approximation. Maximum at 3.5 TeV (~1.9 W/m).
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

13. Geometric impedance: ABCI electromagnetic simulations
Geometry simulated (perfect conductor):
1 m, inner diameter 58 mm
7.1 m, inner diameter 45 mm
100 mm transition
Other parameters for the simulation: 1 mm mesh size, 5 cm rms bunch length, Napoly integration with moving mesh and 4 m wake length.
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

14. ABCI cavity shape
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

15. Power loss ~ 10-7 W/m (analytic formula: exactly zero)
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011

16. Effect on beam stability: Longitudinal effective impedance
Sacherer longitudinal effective impedance (first headtail mode) for parabolic bunches:
Energy
Inner radius
Nb. of layers taken into account
Bunch length (4st)
(Z||/n)eff [W] resistive part (exact)
(Z||/n)eff [W] total (LHC ring)
450 GeV
29 mm
1 (Be)
1.4 ns (MD) j
j 0.09
22.5 mm j
7 TeV
1 ns (nominal) j
j 0.085 j
→ Longitudinal effective impedances for both current and future situations are estimated to be negligible with respect to the rest of the ring.

17. Effect on beam stability: Transverse effective impedance
Sacherer transverse effective impedance for water-bag bunches:
Energy
Inner radius
Nb. of layers taken into account
Bunch length (4st)
Im(Zteff ) [W/m] resistive part (exact)
Im(Zteff ) [W/m] geom. part (ABCI)
Im(Zteff ) [MW/m] total (LHC ring)
450 GeV
29 mm
1 (Be)
1.4 ns (MD)
169
~2.4
22.5 mm
361
282
7 TeV
1 ns (nominal)
141
~25
302
284
→ Transverse effective impedances for both current and future situations are estimated to be negligible with respect to the rest of the ring.

18. Conclusion
Power loss (from resistive-wall): at most 1.9 W/m (with bunches of 0.8 ns at 3.5 TeV), increased by 30% with respect to the previous configuration.
With our current understanding, there is a very little effect of the vacuum pipe modification on the beam stability (longitudinal and transverse).
E. Métral, N. Mounet & B. Salvant - BE/ABP/ICE - Effect on impedance of ATLAS pipe modifications - 21/01/2011