Beryllium Windows Lorentz Force Detuning and Field Enhancement F. Marhauser, MuPlus, Inc. 2014-01
Be Windows • Beryllium windows (‘transparent’ for muons) close cavities such that neighboring cavities (closely packed) can be independently powered and adjusted in phase • Window R&D steered already in early 2000s by LBNL within the frame of the MICE project • First tests of thin (flat) windows with 805 MHz high power pillbox-like cavity • Space outside cavity was evacuated to prevent differential pressure on windows • Thinnest Be RF windows ever built and tested were flat with the following dimensions: thickness = 157 um, Radius = 8 cm, brazed between annular Be frames 2001 window still wavy Be foil (99% purity, IF-1) brazed between two annular Be frames (98.5% purity, PS-200) F. Marhauser , 01/2014, p.2
Be Windows • Pre-stress was required to keep the foils flat • Pre-stress was induced ‘naturally’ during the brazing process, but exact pre-stress unknown as difficult to measure and simulate • Pre-stress effected by CTE difference of ring and Be foil and braze temperature • Pre-stress can vary considerably with small changes in braze temperature 2002 11.5 ppm/K for 99% pure Be and noted deviations for ring with 98.5% Be purity F. Marhauser , 01/2014, p.3
MICE Windows • Progress for MICE cavity design required mechanically more robust RF windows pre-curved, pre-stressed Be windows conceived (designed by Oxford University) designed to remain rigid during thermal cycling • Dimensions for MICE: thickness = 380 um, Radius = 21 cm, brazed between annular copper frames • Be windows are TiN coated to mitigate multipacting • Brazing process still delicate (only one specific company in California has expertise so far to do it right) 2005 Prototype 201.25 MHz (MICE) cavity First successful Beryllium window manufactured for a prototype cavity, Be window is brazed between two annular copper frames F. Marhauser , 01/2014, p.4
805 MHz High Power Test Observations • Be has larger breakdown limit than copper, i.e. copper limits performance (LINAC04 paper) • Flat Windows: • Flat Be windows caused unpredictable pulse-to-pulse instabilities at high field levels • ∆ f = ± 30 kHz (no field level cited) • Instabilities are due to RF detuning and instantaneous RF heating and subsequent oscillation of the window • Curved Windows: • Flat RF windows were replaced by pre-curved (380 um thick), pre-stressed windows • Both RF windows were installed with curvature in same direction • RF heating relaxes window and detuning was predictable ( no pulse-to-pulse oscillations ) • The PAC07 paper states that the 380 thick pre-curved windows were thinner than the flat windows (but no thickness cited). Yet, the latter caused pulse-to-pulse instabilities • Eddy current braking (when external magnet is switched on) has been mentioned to eliminate pulse- to-pulse instabilities, not really explained in this of other papers, very questionable if this can help us • Even is so, would yet not allow to operate cavity without magnets and very thin windows conceived for HCC cavities (slides to come) F. Marhauser , 01/2014, p.5
201 MHz High Power Test Observations • Flat Windows: • 11 mm thick windows (not 11 cm as stated in PAC07 paper) were employed first to avoid potential deflections • not sure, why this thick • Curved Windows: • Flat RF windows were replaced by pre-curved (380 um thick), pre-stressed windows (frequency shift observed by geometry change was -379 kHz) • Cavity reached up to 18 MV/m without issues • No visible arc pits or damage in the interior F. Marhauser , 01/2014, p.6
HCC Cavity Be Window Design • Katsuya (December, 6 th , 2013): • 325 MHz (maybe re-entrant) cavities (segments 1-3), maybe 120 μ m thick Be windows • 650 MHz dielectric loaded cavities (segments 4-6), maybe 30 μ m thick Be windows • Cooling performance: simulation done with 0, 15, 30, 60 µm thin windows (both sides of cavities) • RF window thickness influences equilibrium emittance: 1) reference (0 thickness) equilibrium transverse emittance: 0.35 mm 2) 30 µm thick windows equilibrium transverse emittance: 0.60 mm, i.e. almost factor 2 larger F. Marhauser , 01/2014, p.7
Lorentz Force Detuning • Very questionable whether such thin windows are practicable and reasonable to use • Studied Lorentz Force Detuning (LFD) for a typical re-entrant cavity (here 650 MHz) • Did simulations studies with ACE3P/Tem3P on NERSC calculating LFD deflection/detuning as function of window thickness • Extremely dense mesh used for window to provide proper mechanical solution, i.e. several mesh cells per thickness ( down to 500 µm window thickness was feasible) • Cavity has 27.4 mm gap, no dielectric to ease calculations • Note: Tem3P requires single CPU run to export deformed result files (e.g.: ~4 days CPU time required for 500 µm case) cavity vacuum and mesh cavity wall and mesh F. Marhauser , 01/2014, p.8
Material Properties (at r.t.) Units Copper (OFHC) Beryllium I Beryllium II Source NIST Wikipedia Brush Wellman IF1 Alloys, > 99.8% Be Mechanical Calculations Young’s modulus GPa 128.8 287 303 Poisson ratio 0.344 0.032 0.070 CTE 1/K 1.66e-5 1.13e-5 1.15e-5 Thermal calculations (usually non-linear function with temperature used) Thermal Conductivity W/(m·K) 390 200 216 RF Calculations Electrical Conductivity 1/( Ω ·m) 5.81e7 2.78e7 2.33e7 F. Marhauser , 01/2014, p.9
Window Deflection due to LFD (static) • Objective: Extrapolate to very thin windows as conceived for HCC cavities • RF windows (both sides) are flat and have a radius of 6 cm • Deflections/detuning is for field level within range conceived, here: E eff ( β = 1) = 20 MV/m Note: TEM3P does not consider β < 1 • Equates to ~24 MV/m on axis in this cavity Be II, note: No detuning due to RF heating considered yet F. Marhauser , 01/2014, p.10
LFD Coefficient (static) • Same case, LFD normalized to peak field on axis F. Marhauser , 01/2014, p.11
LFD Coefficient (static) • Be II (larger Poisson ratio) lowers LFD coefficient, i.e. improvement, but not dramatic • Expect similar ‘ballpark’ numbers F. Marhauser , 01/2014, p.12
Results for Flat Windows (Be I) Window Max. long. Frequency Detuning Comments on Feasibility thickness Deflection at 10/20/30 MV/m (µm) @ 24.1 MV/m (MHz) (mm) 30 355 277/1107/2490 impossible 60 46 36/144/324 impossible 120 6 5/19/42 very unlikely to work 380 (like MICE) 0.2 0.2/0.6/1.4 maybe possible at lower field levels (but expect significant emittance increase) 1000 0.012 0.01/0.04/0.09 likely to work (but expect significant emittance increase) F. Marhauser , 01/2014, p.13
Results for Flat Windows (Be I) • Won’t bet on Eddy current braking, also not when theoretically two neighboring cavities would cancel pressure since any power trip in one cavity could disrupt window • Curved window design is more robust, but seems to be not practical for very thin foils Window Max. long. Frequency Detuning Comments on Feasibility thickness Deflection at 10/20/30 MV/m (µm) @ 24.1 MV/m (MHz) (mm) 30 355 277/1107/2490 impossible 60 46 36/144/324 impossible 120 6 5/19/42 very unlikely to work 380 (like MICE) 0.2 0.2/0.6/1.4 maybe possible at lower field levels (but expect significant emittance increase) 1000 0.012 0.01/0.04/0.09 likely to work (but expect significant emittance increase) F. Marhauser , 01/2014, p.14
Thermal Calculation • Example: Be windows with 500 µm thickness (thermal: BNL notebook) • Copper walls are RRR=300 (NIST) • P avg ~ 260 Watt (E eff ( β = 1)= 20 MV/m) • Assumptions: GH 2 at 160 atm, T ref = 298 K • Cooling everywhere outside: Forced flow past a circular cylinder (R = 140.1 mm) equivalent heat transfer coefficient = 72 W/(m 2 ·K) based on Reynolds and Prandtl number, could be larger depending on average velocity of gas (which is an input parameter) • Cooling inside disregarded, but present, probably less than outside temperature rise within windows seems 0.5 mm windows to be small thanks to gas cooling h = 72 W/(m 2 ·K) temperature (K) F. Marhauser , 01/2014, p.15
Thermal Calculation 2002 0.5 mm windows h = 72 W/(m 2 ·K) temperature (K) F. Marhauser , 01/2014, p.16
Past Ideas on Window Design (2001) 1) Use stepped window to improve thermal capability over uniform foil 2) Use ribbed window to decrease muon scattering, while providing mechanical integrity e.g. twice the thickness at some radius e.g. ‘waffle’ pattern with reinforced ribs) F. Marhauser , 01/2014, p.17
Past Ideas on Window Design (2003) 3) Gridded windows with various concepts and arrays (touching, non-touching, merged) F. Marhauser , 01/2014, p.18
Past Ideas on Window Design (2004) 4) Gridded windows with various concepts and arrays (touching, merged) 2004 Thesis F. Marhauser , 01/2014, p.19
Thermal Calculation (2004) • Example: Be 4 x4 waffle window with 254 um thin tubes • Gas heat transfer assumption: 250 W/(m 2 ·K ) T ref = 0 deg. C F. Marhauser , 01/2014, p.20
Past Ideas on Window Design (2004) • Surface field enhancement on gridded window designs studied • Field enhancement (FE) defined as ratio of maximum surface field to on-axis electric field baseline diameter = 9.53 mm F. Marhauser , 01/2014, p.21
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