A 201-MHz RF Cavity Design with Non-Stressed, Pre-Curved Be Windows for Muon Cooling Channels

Derun Li+, A. Ladran, J. Staples, S. Virostek, M. Zisman, Lawrence Berkeley National Laboratory, Berkeley, CA, USA
W. Lau, S. Yang, Dept. of Physics, Oxford University, Oxford, UK
R.A. Rimmer, Jefferson National Accelerator Facility, Newport News, VA, USA

We present a 201-MHz radiofrequency (RF) cavity design for muon (the second flavor of charged lepton, in order of increasing mass, with electric charge -1) cooling channels with non-stressed and pre-curved beryllium (Be) foils to terminate the beam apertures. The Be foils are necessary to improve the cavity shunt impedance and large beam apertures are needed to accommodate the large transverse-size muon beams. Be is a low Z material with good electrical and thermal properties. It presents an almost transparent window to muon beams, but terminates the RF cavity electro-magnetically. Previous designs used pre-stressed, flat Be foils in order to keep the cavity from detuning as a result of RF heating on the window surface. Be foils are expensive and difficult to make under the pre-stress condition needed to accommodate thermal expansion. An alternative design is to use non-stressed and pre-curved Be foils where the buckling direction is known and frequency shifts can be properly predicted. We will present mechanical simulations of such Be window designs.

INTRODUCTION

High-gradient RF cavities at 201 MHz are required for muon cooling channels in a neutrino factory or a muon collider, and also for a recently proposed international Muon Ionization Cooling Experiment (MICE) at Rutherford Appleton Laboratory (RAL). An accelerating gradient as high as 16 MV/m at a frequency of 201 MHz (or 1.07 Kilpatrick) is required. Eight 201-MHz cavities are needed for the MICE cooling channels, but the accelerating gradient for MICE will have to be limited to ~ 8 MV/m due to budget restrictions for RF power sources, not by cavity design. The cavity design supports a gradient of 16+ MV/m. In order to achieve such a high gradient for naturally large-dimension muon beams, the conventional open iris structures would inevitably introduce very high peak surface fields, which are a limiting factor of the accelerating gradient that can be achieved. A closed-cell (pillbox-like) cavity design was proposed and reported [1] where the beam apertures are electro-magnetically terminated by low Z and thin Be foils. To keep the cavity from detuning due to RF heating power, these Be foils are under tension that is introduced by a small CTE (coefficient of thermal expansion) difference between the thin Be foils and thick Be window frame during the brazing process. The pre-stressed windows should stay flat up to a certain temperature gradient limit where the pre-tension becomes zero. This temperature gradient limit determines how much heating power it can take for a given window thickness.

Manufacturing of the pre-stressed Be windows is expensive and predicting the temperature limit is difficult. Previous Be window designs for the 201-MHz cavity were scaled from the pre-stressed Be windows for an 805-MHz cavity. Recent experimental tests on the 805-MHz cavity found that the cavity frequency started to shift at a lower than predicted temperature gradient on the windows (the frequency shift was quite small and well within the klystron bandwidth). By taking advantage of the pillbox-like profile, non-stressed and pre-curved Be windows should result in smaller frequency shifts when both windows are installed and oriented in the same direction in a cavity.

THE 201-MHZ CAVITY

The cavity shape has a slightly re-entrant round profile with a large beam aperture of 21 cm in radius. The cavity profile has been updated recently, and a small 2° tilt angle has been added at a radius out of the iris region to avoid having a two-parallel-plane configuration in consideration of possible multipacting problem, as shown in Figure 1.

The cavity parameters are listed in Table 1 for Study-II and MICE muon cooling channels.

Table 1: 201-MHz cavity parameters

Name	Study-II	MICE
Length (cm)	43	43
Radius (cm)	61.2	61.2
Accelerating Gradient (MV/m)	16.2	8.0
Voltage on crest (MV)	5.76	2.84
Peak forward power* (MW)	4.63	1.0
Peak surface field (MV/m)	26.5	13.1

* Assumed 3 filling time and 85% of theoretical Q_o

NON-STRESSED AND PRE-CURVED BE WINDOWS

Experimental studies on pre-stressed, flat Be windows have shown that the windows start buckling at a limiting temperature gradient where the tension in the foil becomes zero. The measurements were conducted on 16 cm diameter windows in an 805-MHz cold-test cavity using a halogen lamp as a heating source. Both non-stressed aluminum (Al) and pre-stressed Be windows were measured. These measurement results are summarized in Figure 2. It is worth pointing out that it is difficult to predict the buckling direction of the pre-stressed flat windows once the pre-tension becomes zero.

A non-stressed and pre-curved window design has been developed to replace the pre-stressed flat windows. The following criteria have been considered during the evolution of the window design:

Figure 3 shows the window profile evolutions during the design process. We started with a single bow (curvature) window and found the thermal stress was too high at the annular frame. Then, a new window design was developed having an intersection between a concave and convex shape in a region away from the edge of the frame, which makes the window more flexible and allowing for more free expansion. The thermal stress is thus further reduced.

The window thickness was varied from 0.125 mm to 0.5 mm to study its effect on thermal stress. Because windows thinner than 0.125 mm tend to be more expensive, we chose a window thickness of 0.38 mm as a baseline study parameter for a window with a 42 cm diameter.

Each window profile configuration has been modeled in 2-D and 3-D using FEA (finite element analysis) software from ALGOR. The following temperature distribution is then applied over the window for thermal and mechanical simulations [2].

where T_max is the temperature gradient limit and R is the window radius. This temperature distribution is a result of the magnetic field distribution in a pillbox cavity and the low radial thermal transfer of the thin material. The fact that the window center is the hottest spot is due to the limited thermal conduction within the thin Be foil. As one would expect, T_max is proportional to the total RF heating power over the window, and inversely proportional to window thickness. T_max=100 °C, which is higher than the temperature limit for an accelerating gradient of 16 MV/m, has been assumed and used for all of the simulations.

Figure 4 shows a 3-D FEA model of the non-stressed and pre-curved Be window with two curvatures, together with its mechanical resonant frequencies. In this example, the two lowest and one higher order mechanical resonant vibration distributions are shown.

Window displacements of the 42 cm diameter Be foils of different thickness were simulated after applying the above temperature distribution. Table 2 lists the maximum displacement at the window center from its natural position.

Table 2: Be window displacement versus thickness

Thickness (mm)	0.25	0.38	0.50
Displacement (mm)	2.42	2.25	2.13

The amount of displacement is considered to be acceptable as long as all of the windows are installed and oriented in the same direction, which also results in minimum frequency shift of the cavity.

Similarly, the corresponding stresses due to the thermal load (T_max=100 °C) are simulated and their results are listed in Table 3.

Table 3: Thermal stress versus window thickness

Thickness (mm)	0.25	0.38	0.50
Thermal Stress (MPa)	150	169	179

Considering that the temperature gradient for thicker windows is lower (scaled linearly with d, the window thickness), all of the above windows should work, and yet provide an adequate safety margin. The thermal stresses are much less than the Be stress limit of ~ 340 MPa.

A concept for fabrication of the pre-curved Be window is shown in Figure 5. A smaller window of 16 cm diameter using the same concept will be tested first. This test may be conducted in either room temperature or up to the recommended forming temperature of Be depending on the resulting window profile. We expect that the window may spring back slightly to its natural shape after the forming process. This will be quantified during the tests.

The fabrication concept of pre-curved windows will be realized in 16 cm diameter windows using materials having similar mechanical properties as Be. Time-domain simulations will be performed to study possible mechanical resonant excitations caused by an RF impulse and using the operational parameters for Study-II and MICE cooling channels.

CONCLUSION

A non-stressed, pre-curved Be window has been designed. FEA simulations show it has the desired mechanical flexibility and the thermal capability to handle the RF heating power while keeping the required cavity performance. Experimental tests and engineering efforts continue.

REFERENCES

[1] R. Rimmer, et al., "Closed-Cell 201.25 MHz RF Structures for a Muon Ionization Cooling Experiment", EPAC 2002, Paris, France

[2] D. Li, et al., "Temperature Distribution Calculations on Be windows in RF Cavities for Muon Colliders," International LINAC Conference 1998, Chicago, IL, USA

* Work supported by the US Department of Energy under contract No. DE-AC0376SF00098