m.apollonioCM17 -CERN- (22/2 - 25/2 2007)1 Single Particle Amplitude M. Apollonio – University of Oxford

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)2 amplitude: is a single particle concept  Consider first a 2D case  field strength (1) (3) (2) c=cos(  )

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)3 x x’ area=  A A: for a linear system this is a constant of the motion (Liouville’s theorem)  : describes the optical properties of the channel x z envelope motion of a particle in the lattice

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)4  : optical Twiss parameters

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)5 if the beam is gaussian and matched there is a relation between V and B here B and  describe beam envelope properties. B can be inferred from V and A too A is still a single particle amplitude BUT describes a level of constant probability for a gaussian distributed beam V: covariance matrix of the beam

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)6 x emittance: RMS amplitude property of the beam it can be derived from the COVARIANCE MATRIX of the beam emittance/amplitude are normalized multipling by a factor p/mc optical parameters: from the covariance matrix OR from our knowledge of the magnetic field x’

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)7 from 2D to 4D  (x,x’,y,y’)   solenoidal field introduces couplings (assume  x =  y )

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)8 we can still think about single particle amplitude but we need to be a little more careful and take into account (x-y) correlations the definition of 4D A from a cov. mat. V is different w.r.t. the 2D case because of a (possible) non-zero canonical angular momentum

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)9 NORMALIZED amplitudes (x,x’,y,y’)  (x,p x,y,p y ) l = /2mc  N  T =  p V1+ l 2  T =  p V1+ l 2 the single particle amplitude is independent from the beam we can use this variable to characterize cooling and transmission through the channel

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)10 profile plot reg-2 ~ centre of 1 st tracker reg-92 ~ centre of 2 nd tracker cooling

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)11  =3.0 cm rad P Z =200 MeV/c,  abs =42 cm cooling N2/N1  =2.0 cm rad N2/N1

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)12  amplitude vs aperture p = 200 MeV/c R (cm)  (cm) A n max (cm) Absorber RF Tracker A n MAX = p/mc R 2 /   in a focus/unif. field the max allowed amplitude has a very simple expression  in a general case it is more complicated but still the same concept  we can study transmission as a function of amplitude A n MAX = p/mc R 2 / 

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)13 P Z =200 MeV/c,  abs =42 cm  =0.6cm rad  =1.0cm rad transmission through MICE step VI

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)14 MICE STEP VI ~90m of MICE Channel RF ABS tracker A (m rad)

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)15 A n MAX physical aperture R  we can define the max allowed amplitude at the end of the channel  useful for the acceleration stage in the NF

m.apollonioCM17 -CERN- (22/2 - 25/2 2007)16 conclusion  amplitude has been introduced as a single particle property  MICE is a capable of measuring single particle kinematic parameters which, combined with the optical functions, allow to define the amplitude of each muon  idependent from beam  useful to study the specific effects of scraping... TRANSMISSION ... and COOLING:  definable as an increase of the phase space density (rather than an emittance reduction)  useful to understand the fraction transmissable to the stage after the NF front-end