1 Angular Momentum from diffuser Beam picks up kinetic angular momentum (L kin ) when it sits in a field –Canonical angular momentum (L can ) is conserved in the absence of material In material L kin is damped leading to non-conservation of L can –At higher fields we have more L kin and so the change in L can is greater This leads to a mismatch –Stronger Bz => bigger mismatch –Motivates pulling the diffuser out of the solenoid Two questions: –Does this seriously effect the amount of cooling? –Does this require serious amounts of reweighting to get a beam distribution with no mismatch? –Additionally consider an alternative matching condition

2 Change in Angular Momentum Kinetic angular momentum given by L kin = In a material change in angular momentum given by dL kin /L kin ~ dp z /p z –Thin foil approximation Monte Carlo (ICOOL) shows this is reasonably accurate –dL kin /L kin in black, dp z /p z in grey –The usual 4 T, 333 mm, 200 MeV, 6 pi beam

3  L kin from MICE diffuser Without knowing the precise beamline design we can make an estimate for the diffuser thicknesses –Assume beamline produces roughly 2  beam –Expect this to be good to ~10% –Gives lead thicknesses: –Gives dL kin /L kin : 2pi6pi10pi pi6pi10pi

4 Effect on beta function Introducing angular momentum will knock the beta function off –10,000 muons, no energy spread/absorbers/rf/windows/scifi in these plots –Black plot shows beam with “normal” beta function –Red plot shows beam with mismatch that would be induced by a material So take  L kin /L kin =0.1 but keep  (x),  (x’) the same –Blue plot shows a slightly different beam matched with the L can taken into account –Take out some of the transverse momentum spread

5 Define matched covariance matrix by –And use When I go through a diffuser  doesn’t change If I want a beam with  (x) constant I have to be careful to choose  (p x ) with this L dependence –L is basically the canonical angular momentum i.e. –L = 0 w/o diffuser, <~ 0.1 with diffuser (depending on thickness) “Rematching”

6 Angular Momentum Kinetic angular momentum varies wildly –The three plots that vary between +/ mm MeV/c are kinetic angular momentum Canonical angular momentum is really conserved very well Blue and red plots have non-zero canonical angular momentum –Again blue plot has been rematched to account for the L can Black plot is again for standard solution with L can = 0 L can L kin

7 Full Cooling Channel Go on to consider MICE VI with absorbers and RF –No SciFi/detectors, still 10,000 events Energy looks spot on Slight mismatch induced by the momentum change even in case of L can =0 –Black is L can = 0 –Red has L can ~0.1L kin in the tracker –Blue has L can ~0.1L kin but rematched

8 Effect on Cooling Slightly worse performance from the matched channel with angular momentum vs standard channel Slightly better performance from the unmatched channel with angular momentum –But higher initial emittance Regardless, the change in cooling performance from this effect (i.e.  /  ) ~ 5% This is well within specification (Perhaps beyond limits of statistics)

9 Phase Space Density Phase space density contours in x-py phase space –6  beams but density scales –Looks like any reduction in phase space density will be a tweak

10 Underdensity Due to L kin This is [n Lkin=0.1 (mu) - n design (mu)]/n design (mu) in x-py phase space –Left is for unmatched beam –Right is for rematched beam –Black contours are phase space density contours for the ideal beam –Only underdensities are shown Depletion in the fringes –Low statistics in this region ( mu total) –Compared to the gain in rate through quad aperture, this is not an issue

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12 Effect of Energy Spread on Cooling Beta function for several different beams –Black has 1 MeV energy spread –Red has 25 MeV energy spread –Blue has 25 MeV energy spread and tracking in ICOOL –Green has 25 MeV energy spread and RF is at 40 degrees Left hand plot has no RF/Absorbers, RH has full cooling

13 Transverse Emittance Non-linear effects dominate with a small NuFact energy spread –Typical NuFact dE ~ MeV –No cooling!? –Note blue & red have input beam but with different scaping (~1-5%) In ICOOL I killed particles at r>250 mm Did it properly in G4MICE 1 MeV 25 MeV/G4MICE 25 MeV/ICOOL25 MeV/RF 0 o 25 MeV/RF 40 o 1 MeV

14 Longitudinal Phase Space Longitudinal phase space at z = (centre of RF-8) –Note energy scale of RF bucket/Contours of Hamiltonian Running on-crestRunning at 40 0