1. V.Balbekov, 12/09/08 HCC simulation with wedge absorbers V. Balbekov, Fermilab Muon Collider Design Workshop December 8-12, 2008 JeffersonLab, Newport News, VA 0 Is this possible to improve HCC with help of wedge absorbers? Primary idea: low kappa channel with small transverse field, etc.

2. B. Palmer: Low κ helix cooling (08/26/08) V.Balbekov, 12/09/08 1

3. Outline 1.Stability region of “conventional” helical cooling channel (HCC) 2.Peculiarity of helical solenoid channel (HSC) 3.HCC and HSC with small pitch-factor and wedge absorbers 4.Wedge absorber as a tool for diminution of the HCC/HSC field and beam momentum 5.Cooling degeneration at excessive low momentum 6.HCC with small transverse field 2

4. V.Balbekov, 12/09/08, p.3 Characteristics of the HCC Most characteristics of the HCC: eigenmodes, eigentunes, beta-function, normalized dispersion, normalized transverse decrements can be found with help of the function F(X) which is normalized longitudinal magnetic field at r = XL/2 π, θ = 2πz/L: B z (X) = B sol F(X) = B sol [1+2F’∙I 1 (X)+F”∙I 2 (2X)+…] where z, r, θ are cylindrical coordinates, L – period length, B sol = B z (0) – solenoid field, I 1, I 2...- modified Bessel function, F’, F” … - coefficients. LB sol are required additionally to know reference momentum, slippage factor, and normalized longitudinal decrement. Absolute decrements are proportional to the ionization energy loss. For Helical Solenoid Channel (HSC, Kashikhin): the function F(X) depends on relative coil radius R coil /L and antisolenoid/solenoid field ratio B as /B hs. 3

5. V.Balbekov, 12/09/08 Example 2: HSC R coil /L = 0.2733, B as /B hs =-0.283 Example 1: HCC (Yonehara design) F’ = -0.2545, F” = 0.0777, F”’=0,… Normalization of the decrements: D t1 +D t2 +D l = 2β 2 !!! No stability at X < ~0.60 -- 0.65 !!! 4 In both cases: homogeneous H 2 absorber, LB sol = 6.97 T-m to get P refer = 250 MeV/c at X refer (= κ ) = 1 (unite length is L/2π)

6. Cooling simulation at ionization energy loss 14.7 MeV/m, RF 200 MHz, 29.4 MV/m Example 1: HCC V.Balbekov, 12/09/08 Example 2: HSC The results are very similar: Transverse emittances are about 2 mm ≈ L/500, longitudinal one ~3 mm ≈ L/300. Transmission 72% - 76 % at 200 MHz, but falls at higher frequency (reasonable requirement λ >~L ). 5

7. V.Balbekov, 12/09/08 Plot: HCC normalized cooling decrements in the at: W = 5 (circles), W = -5 (triangles down), and W = 5 (triangles up). Wedge strength: W = d(ln E ion )/dX. Decrements: orange – trans., magenta – long. 6 Effect of wedge absorber In considered cases, either one of transverse modes or longitudinal mode is unstable at X < ~0.6, if homogeneous absorber is used. Can a wedge absorber improve the situation? Wedge absorber does not improve the situation. Similar results are observed in HSC.

8. V.Balbekov, 12/09/08 7 Explanation Without transverse field and emittance exchange, only one transverse mode can be really cooled. Other one is neutral (no damping or rising), and longitudinal motion is unstable at E<~350 MeV. With additional dipole field and/or emittance exchange, sum of decrements of the neutral and unstable modes remains negative, if the reference radius is small enough. Small quadrupole etc. addition does not change the situation. Quad. dominated HCC has small acceptance. Therefore low kappa channel with any wedge is unsuitable for 6D cooling. However, one can use the wedge absorbers to reach a reasonable longitudinal decrement at pitch-factor ~1, using lower beam momentum and correspondingly lower magnetic field.

9. F’ = -0.2545 => -0.2535 F” = 0.0777 => 0.0780 W = 0 => 0.05 L = 1 m B sol = 6.97 => 5.24 T X refer (=κ) = 1 P refer = 250 => 189 MeV/c E’ refer = 14.7 MeV/m (wedge absorber – 5%) RF 200 MHz, 29.4 MV/m Example 1: HCC without/with wedge absorber V.Balbekov, 12/09/08 8 With wedgeW/o wedge

10. Antisol/Sol -0.2733=>-0.283 Rcoil/L = 0.2733 2πR helix /L = 1 W = 0 => 0.07 L = 1 m B sol = 6.98 => 5.24 T X refer (=κ) = 1 P refer = 250 => 189 MeV/c, E’ refer = 14.7 MeV/m (wedge absorber – 7%) RF 200 MHz, 29.4 MV/m V.Balbekov, 12/09/08 9 Example 2: HSC without/with wedge absorber W/o wedgeWith wedge

11. V.Balbekov, 12/09/08 10 RESULTS About 25% reduction of operational beam momentum and required magnetic field of the HCC is possible with wedge absorbers. Achievable beam emittance nearly does not change, and transmission a bit falls in the case (the fall is more in the helical solenoid). However, an attempt of more momentum/field reduction leads to a strong degradation resulting a huge particles loss. Important question: what is a reason of the degradation? The main reason is a deterioration of longitudinal phase space. Stronger dependence dE/dx on velocity at lower energy has but a minor effect.

12. Lower momentum HCC with wedge absorber V.Balbekov, 12/09/08 12 F’ = -0.196 F” = 0... W = 0.57 L = 1 m B sol = 5.23 T FX refer (=κ) = 0.62, P refer = 159 MeV/c, F = 50 MHz V’ = 29.4 MV/m E’ refer = 14.7 MeV/m Parameters Cooling simulation Longitudinal ph. space Parameters vs momentum Cooling is impossible at these parameters because of huge particles loss. The reason is “bad” longitudinal phase space because of small slippage factor. Betatron motion results fast phase oscillations and bring about an additional deterioration of the space. Blue – longitudinal phase trajectory at betatron oscillations

13. Cooling at artificial media: dE/dx does not depend on E HSC with artificial cooling media is considered. Longitudinal emittance is less than “normal” one because of more longitudinal decrement. Transmission slightly increases also. However, the transmission drops as before when beam momentum decreases from 250 MeV/c to 189 MeV/c: 76% / 62% “normal” 79% / 68% “artificial” V.Balbekov, 12/09/08 11 It means that dE/dx(E) dependence produces a minor effect.

14. Lower transverse field / pitch-factor V.Balbekov, 12/09/08 13 Old simulation L = 1.8 m B sol = 5 T X refer (κ) = 0.14 P refer = 240 MeV/c LiH, 260 mrad, 3.9 cm E’ = 7 MeV/m 201 MHz, 14 MV/m Parameters (V.B., MCnote 146, 2000) F’ =-0.06, F”...= 0, W = 1.9 No equilibrium transverse emittance Both simulation demonstrate that the cooling is unachievable in this regime.

15. Conclusion 1.HCC or HSC with small transverse field and small pitch-factor (kappa) are unsuitable for 6D cooling, both with and without wedge absorbers. 2.Wedge absorber can be used to decrease momentum and field of a “conventional” HCC or HSC. 25% reduction is achievable without unreasonable degradation of the cooling. The wedge absorbers should provide 5-7% of ionization energy loss, 95-93% -- high pressure hydrogen. 3.More momentum/field reduction leads to unacceptable particles loss. The main reason is a deterioration of longitudinal phase space because of an approach of transition energy to the working area. My attempts to remedy this flaw were unsuccessful. 4.After optimization, production LB sol can be decreased from ~7 T-m (no wedge) to ~5 T-m (with wedge). Achievable transverse emittance is ~L/500 in both cases. 5.RF separatrix should be long enough to neutralize dependence of time of flight on betatron oscillations. The wavelength λ ≈ L (or more) looks reasonably providing achievable longitudinal emittance ~L/300. V.Balbekov, 12/09/08 14