1. Simulations of the Rotating Positron Target in the Presence of OMD Field* S. Antipov+, W. Liu, W. Gai Argonne National Lab +also Illinois Institute of Technology *Collaboration with the ILC e+ Team

2. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 2 Problem formulation Conducting disk (target) rotates in a constant magnetic field of arbitrary distribution. Eddy currents are produced. Find their distribution, induced magnetic field etc depending on rotational frequency and geometry z

3. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 3 Equation for induced field To incorporate the fact, that disk is spinning we add an effective emf in Ohms law and get a non-standard equation: Velocity, B 0 -external, B-induced magnetic field

4. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 4 Set of equations for simulation Source for approximate models and simulations

5. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 5 Simulation of SLAC/LLNL experiment

6. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 6 Single pole disk Artificial subdomain to improve mesh quality The experiment geometry Courtesy SLAC/LLNL

7. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 7 External field, produced in simulation Measured field map, 1.14mm probe+/-0.36mm large error bars Thickness of the disk is 0.9 inch. It is hard to measure (and simulate) magnetic field near the face of magnet (divergence) – 20% error bars Input data B 0z -field Material: copper Diameter: 9 inch Thickness: 0.9 inch Magnet position: 0.47 inch off-center Distance between the disk and the magnet: 0.1, 0.05 and 0.01 inch

8. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 8 Arrows: Eddy currents Arrows: Force field y x Results: color – induced magnetic field (z)

9. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 9 Copper disk Parameter – distance between the magnet and the disk simulation exp

10. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 10 Higher frequencies With a single magnet the roll off is almost flat

11. ILC target geometry simulation

12. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 12 ILC target geometry 1m 6cm 1.4cm Typical simulation result: streamlines of solenoidal magnetic field (in simulation we consider returning flux – principal difference from the setup with magnet) Color: induced magnetic field (produced by eddy currents) at some frequency of rotation, ω. Solenoid positioned at 0.95m Outer domain Upper solenoid is not pictured Full domain

13. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 13 Results: simulation with the magnet

14. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 14 Results: simulation with the solenoid

15. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 15 Results: power vs RPMs

16. Parametric studies of the ring configuration of the ILC target

17. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 17 Simulation geometry 1m 3/6/12cm 1.4cm Typical simulation result: streamlines of solenoidal magnetic field (in simulation we consider returning flux – principal difference from the setup with magnet) Color: induced magnetic field (produced by eddy currents) at some frequency of rotation, ω. Solenoid positioned symmetrically over the ring Outer domain Upper solenoid is not pictured Full domain dr=1.5/3/6 cm

18. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 18 Color: external magnetic field. Max field inside the target – 5Tesla Simulation geometry Target ring magnet system artificial domain full domain

19. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 19 Low RPMNear roll off RPMHigh RPM Color – induced magnetic field, main component (into the screen) Disc rotates counter clockwise Results: induced field

20. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 20 Inside target m Tesla 100 980 1735 3367 (critical) rpms: Total z-field Frequency study of total field

21. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 21 Bx and By components, a deflection force on the beam. Color – B y Arrows {B x, B y } The level of transverse (to the beam) components is one order lower than the z- component Plotted 5mm from the surface of the target at 980rpms tesla

22. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 22 1.5cm3cm 278kW@980rpm488kW@980rpm 6cm300kW@980rpm657kW@980rpm 12cm325kW@980rpm770kW@980rpm Results for σ=3e6 @980rpm, 5Tesla Magnet aperture Ring width

23. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 23 1.5cm3cm 170kW@910rpm332kW@910rpm 6cm172kW@910rpm433kW@910rpm 12cm175kW@910rpm463kW@910rpm Results for σ=1.5e6 @910rpm, 5Tesla Magnet aperture Ring width

24. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 24 Results for σ=3e6, dr=1.5cm, 5Tesla

25. Transient Temperature Analysis of A Target Chamber Wanming Liu, Wei Gai HEP, ANL

26. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 Energy deposition (Calculated with EGS4) Beryllium window of 0.375mm thickness Undulator: K=1 u=1cm, 100m long with 150GeV 3nC drive electron beam  e-,e + and  ~0.32mJ per bunch deposited in upstream window ~8.4mJ per bunch deposited in downstream window e e+ For 100m long K=0.92, u=1.15cm undulator with 150GeV 3nC e- drive beam, ~0.265mJ per bunch deposited in upstream window ~5.9mJ per bunch deposited in downstream window.

27. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 27 Energy deposition distribution from EGS4 Simulation Upstream window Downstream window  r is ~3.5mm

28. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 28 How we solve transient response Under cylindrical coordinate system, the partial differential equation for this problem is given as: The Green’s function of (1) with an infinite body is given as (1) (2) Since the pulse length is very short, only 1ms, the effect of boundary around the window has nearly no effect to transient thermal response of window for a short time. Using the Green’s function for infinite body is accurate enough for our problem here. To simplify the calculation, we approximate the deposit energy profile using a Gaussian function as: where Q0 is the total deposit energy in window per bunch and d is the thickness of window. (3)

29. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 29 Since the bunch length is only ~1ps, we can simply replace it with a  (t). Then we have and now the solution becomes (4) (5) If one care about the long term response, we can simply replace the Green’s function in (5) with the Green’s function for a boundary value problem T=0 at r=b. A more accurate result could be obtained by using the energy distribution profile directly instead of fitting it into a Gaussian function.

30. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 30 Transient Response on downstream window. 100m undulator with K=1, u=1cm

31. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 31 Result for downstream windows, 100m undulator with K=0.92, u=1.15cm Melting point is 1560K for beryllium Downstream Windows melt in ~0.1ms. The smallest energy deposition among these windows is 0.265mJ for 0.375mm beryllium window, which is about 1/22 of the corresponding downstream window. The peak temperature rise on upstream window will be about 493K.

32. Modeling and Prototyping of Flux Concentrator and ILC AMD Design

33. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 33 Outline Circuit model of Flux concentrator Modeling of Brechna’s flux concentrator Prototype experiment ILC AMD design based on flux concentrator

34. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 34 Introduction of flux concentrator Cross-sectional and side view of a general flux concentrator. 1: primary winding, 2: core, 3: radial slot, 4: bore. Work as a pulsed transformer. The induced current generated by the primary coil tends to shift the primary coil flux into the smaller vacuum region inside the central bore and relieves the magnetic pressure on the primary coil. Simple transformer model

35. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 35 Circuit model of flux concentrator (1) Geometry division The flux concentrator modeling is started by dividing flux concentrator into thin disks along the longitudinal direction, and then each disk is subdivided into concentrating rings. These rings are interconnected with each other at the slot end. Each concentrating ring is modeled as a resistance and a inductance, and interconnection line along slot is modeled a resistance.

36. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 36 Circuit model of flux concentrator (2) Equivalent circuit

37. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 37 Modeling of Brechna’s flux concentrator(1) Geometry structure H. Brechna, D. A. Hill and B. M. Bally, “150 kOe Liquid Nitrogen Cooled Flux-Concentrator Magnet”, Rev. Sci. Instr., 36 1529, 1965. Primary coil This structure of flux concentrator is from Brechna’s paper. We will calculate its transient response and on-axis field profile using our equivalent circuit model.

38. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 38 Results from circuit model (Source R =0.12 Ω) Measurement results Modeling of Brechna’s flux concentrator(2) Comparison of transient response

39. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 39 Coil + flux concentrator Coil only Modeling of Brechna’s flux concentrator(3) Field profile along central axis Magnetic field is calculated at 20ms when a pulse is applied.

40. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 40 Prototype experiment (1) Geometric structure

41. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 41 Prototype experiment (2) Experimental setup Pulsed voltage is produced using a DC power supply and IGBT switch circuit. Pulsed magnetic field is measured using a magnetic sensor based on hall effect. Whole structure including coil and flux concentrator disk is placed in a cooler cooled by liquid nitrogen. Temperature is around 150ºK.

42. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 42 Two sets of data are measured, one works at room temperature (293ºK), and another is cooled by liquid nitrogen (around 150 ºK). After cooling by liquid nitrogen, current raise 21%, and magnetic field increase 45%. Prototype experiment (3) Measurement data

43. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 43 Magnetic field produced by coil can be calculated from measured current, and magnetic field from flux concentrator is obtained by reduction coil field from measured total B field. After cooling by liquid nitrogen, magnetic field produced by coil increase 21% at pulse end, magnetic field from flux concentrator increase 100%, and total magnetic field raise 45%. Working at room temperatureCooling by liquid nitrogen Prototype experiment (4) Comparison of magnetic fields

44. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 44 Prototype experiment (5) Comparing test data with modeling results Working at room temperature Cooling by liquid nitrogen Test Model

45. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 45 ILC AMD design (1) Geometric structure Target Flux concentrator Coils (DC)

46. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 46 Transient response at target exit Distribution of on-axis magnetic field (4ms after pulse is applied.) Target exit ILC AMD design (2) Transient response and field profile

47. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 47 ILC AMD design (3) Typical operating parameters Work modepulse Operation Temperature78ºK Pulse width5 ms Repetition rate5 Hz Number of turns of primary coil105 Peak power input to magnet5.1 MW Average power input113 KW Peak current7000 A Magnetic field at target exit5 Tesla Time constant of current in primary coil3 ms Wire size of primary coil0.475 × 0.381 cm² Work modeDC Operation Temperature293ºK Power input81 KW Current926 A Total Number of turns135 Wire size of coil 0.475 × 0.381 cm² Parameters of flux concentrator Parameters of DC coil

48. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 48 ILC AMD design (4) The effect of the fluctuation of B field in FC Time (ms)YieldPolarization 2.51.0959% 31.1359% 3.51.15859% 41.17459% 4.51.18759% 51.1959% 5.51.1259% 61.0559% As shown before, there is no flattop during the pulse duration(5ms) of the pulsed magnetic field. To investigate the effect of such fluctuating field, magnetic field distributions at different time near the peak of the pulse are applied in the end to end simulation. The results of positron yield and polarization are compared here in the following table. The field on target varies by about 8% during 4-5ms near the peak of pulse, but the yield varies less than 2% and the polarization barely changed.

49. The 2 nd ILC e+ Collaboration Meeting, IHEP, Beijing, Jan 31 – Feb 2, 2007 49 Summary We developed a circuit model based on frequency domain analysis to calculate transient response of a flux concentrator and its field profile. The circuit model was applied to calculate Brechna’s flux concentrator, and a good agreement is achieved. We designed a prototype flux concentrator with 50ms pulse width. Transient response of the flux concentrator were measured both at room temperature (293ºK) and at low temperature (around 150ºK), cooling by liquid nitrogen. The magnetic field increases 45% after cooling. The magnetic field from flux concentrator raise 100%. The circuit model gave a good prediction to the measured data. ILC AMD based on pulsed flux concentrator technique were designed using the equivalent circuit model. The designed AMD has a peak magnetic field at target exit equal to 5 Tesla. The peak power input to flux concentrator is about 5MW. The average power input to the entire AMD is around 200KW (flux concentrator + DC coil). There is no flattop during the pulse duration for the magnetic field. But simulation results show that during 4ms to 5ms, even though the magnetic field has a variation of about 8%, the positron yield fluctuates less than 2% and the polarization barely changes. According to the result, this design will work even though the time constant is relatively big.