1. marco apollonioAnalysis Meeting (9/12/2006)1 transmission vs amplitude with a finite size diffuser M. Apollonio – University of Oxford

2. marco apollonioAnalysis Meeting (9/12/2006)2 Amplitude – a single particle concept Amplitude – also known as ‘single particle emittance’ = SPE Focussing magnetic field  Particle (muon) performs oscillations about beam axis x’’ + k 2 (s) x = 0 (Hill’s eqn) k 2 (s) = focussing strength A = amplitude of betatron oscillations A is constant of motion in linear system Particle moves on ellipse of fixed area =  A in x, x’ space x x’ John’s talk at CM16

3. marco apollonioAnalysis Meeting (9/12/2006)3 Amplitude (single particle property) A =  x 2 + 2  xx’ +  x’ 2  are optical (Twiss) parameters Emittance (many particle description)   rms amplitude of beam Normalise by multiplying by p/mc Optical parameters from covariance matrix of a set of muons or from magnetic field At focus or in uniform field: A = x 2 /  +  x’ 2 A n = (p/mc) A = p x 2 /(  mc) + p t 2  / (p mc)  = p / (150 [MeV ] B [T] ) in uniform field x x’ x max = sqrt (  A) (ECALC9 does all this) MICE can measure single muon amplitudes

4. marco apollonioAnalysis Meeting (9/12/2006)4 Amplitude in 4D – a single particle concept trace space (x,x’,y,y’)  phase space (x,px,y,py) [coupled by the solenoidal field] Follwing Penn’s note we can write down the single particle amplitude (hold on tight): MICE can measure x,px,y,py while  are known from our set-up of the lattice (optics) So, a combination of tracker and knowledge of the optical system allow us to define Ap

5. marco apollonioAnalysis Meeting (9/12/2006)5 Scattering HEATS on average COOL by reducing p t  Increase central phase space density, i.e. increase density at low amplitudes Depending on thickness and material one mechanism dominates (or you can reach an equilibrium) We use this effect to inflate the emittance with a lead diffuser It can be shown that (1 st approx)  x ptpt

6. marco apollonioAnalysis Meeting (9/12/2006)6  (from beamline)  (from beamline)  (into MICE)  (into MICE) diffuser If you want  (in the channel) you must start somewhere (e.g. Q9) with  Which goes like WITHOUT diffuser

7. marco apollonioAnalysis Meeting (9/12/2006)7  I (re)studied the problem of diffuser position, which helped me understand it better (and appreciate all the effort done by Chris)  I chose to place the diffuser at Z=-6.010 m (best choice by Chris)  I have considered  =6mm rad, pz=200 MeV/c  I put TOF in the “game”  I evolve beta func with evbeta (2 nd order diff equation) imposing it to be =33.3 cm (and  =0) inside the solenoid  My solution gives me thickness_diff = 6.4 mm (instead of 7.6)  I generate a gaussian beam and “stick” it just before the diffuser:  It must have proper  =.1493,  =76.95 cm  correlations  I use this as an input beam for ICOOL

8. marco apollonioAnalysis Meeting (9/12/2006)8      const, 

9. marco apollonioAnalysis Meeting (9/12/2006)9 RF = OFF to study transmission

10. marco apollonioAnalysis Meeting (9/12/2006)10 NuNu NdNd T=N d /N u Scheme for transmission study: No selection on muons Ratio between downstream and upstream particles R=40 cm Z u ~-6.01 m Z d ~+5.4 m R=40 cm R=90 cm

11. marco apollonioAnalysis Meeting (9/12/2006)11  Does the finite size of the diffuser change the transmission of the channel?  Just consider the total crude number Nout/Nin  Define it as a T(Ap)  for all the muons  for Rm < Rd  for Rm > Rd  NB: Ap computed upstream (just after the diffuser)  match with particle downstream (if any) and assign the same amplitude  T(Ap) = Nout(Ap)/Nin(Ap)

12. marco apollonioAnalysis Meeting (9/12/2006)12  (from beamline)  (from beamline)  (into MICE)  matched to be 33.3 cm inside the solenoid  (into MICE): inflated the DIFFUSER “changes” the OPTICAL function!!! what happens when R D finite and a particle misses it? RD  (into MICE): NOT inflated  (into MICE)  unmatched !!!

13. marco apollonioAnalysis Meeting (9/12/2006)13 ...gut feeling, if a particle is not matched I have a greater chance to loose it in the subsequent channel  So the smaller the radius of the diffuser the smaller the overall T (true?)  Figures hardly prove that, indeed seem to go the other direction (but I want to have a better check) [Rd=7cm  98.9%, 10cm  98.45%, 40 cm  98.4% with 40K evt, sigma ~ 0.5%]  Define it as a T(Ap)  for all the muons  for R  < Rd (to be done)  for R  > Rd “  Three Ap distribution for Rd=4 cm, 7cm and 40 cm Bulk/depletion for RD=7cm Bulk/depletion for RD=4cm

14. marco apollonioAnalysis Meeting (9/12/2006)14 Partial inflation due to a finite radius RD = oo RD : finite RD(finite)/RD(oo) RD = 7cm = 8 cm = 9 cm Should be A~RD 2 /beta In this picture Ap is calculated with the proper  according to the radius of the muon

15. marco apollonioAnalysis Meeting (9/12/2006)15 CAVEAT !!!! take these considerations (and above all figures) as ***VERY*** preliminary many results are still not solid and need to be verified I must convince myself of what I have said (and put it in the right logical order probably) having said that … … I wish you a Merry Xmas and see you in 2007