m.apollonio CM17 -CERN- (22/2 - 25/2 2007) 1

Single Particle Amplitude

M. Apollonio – University of Oxford

m.apollonio CM17 -CERN- (22/2 - 25/2 2007) 2

amplitude: is a single particle concept

Consider first a 2D case

)(

02

z

xx

)(1)(

))(cos()()(

zz

zzAzx

field strength

222

2

22

4

22

ssccA

x

scAcxx

cAx

(1)

(3)

(2)

0442 222

c=cos()

m.apollonio CM17 -CERN- (22/2 - 25/2 2007) 3

)(zA

))(cos()( zzA

x

x’

area=

A

A: for a linear system this is a constant of the motion (Liouville’s theorem)

: describes the optical propertiesof the channel

x

z

envelope

motion of a particle in the lattice

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2

22

1

2

2

xxxxA

: optical Twiss parameters

1

1

B

xBxA T

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if the beam is gaussian and matched there is a relationbetween V and B

BVxxxx

xxxx

here B and describe beam envelope properties.B can be inferred from Vand A too ...

... A is still a single particle amplitude BUT describes a level of constant probability for a gaussian distributed beam

xVxA T 1

V: covariance matrix of the beam

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x

emittance: RMS amplitudeproperty of the beamit can be derived from the COVARIANCE MATRIX of the beam

emittance/amplitude are normalizedmultipling by a factor p/mc

optical parameters: from the covariance matrix OR from our knowledge of the magnetic field

x’

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from 2D to 4D

(x,x’,y,y’)

solenoidal field introduces couplings (assume x=y)

yy

yy

xx

xx

B

00

00

00

00

ppp

ppp

ppp

ppp

B

0

0

0

0

4

ppp

ppp

ppp

ppp

B

0

0

0

0

4

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we can still think about single particle amplitude but weneed to be a little more careful ...

p

ppp

pp

ppppp xyyxyyxxyxyxA

222

2222

1

2

22

xBxA Tp

14

... 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

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NORMALIZED amplitudes (x,x’,y,y’) (x,px,y,py)

xypyxpyx

z

pzp

np ypxpypxppp

pyxp

mA

221 2222

xyTyxTyx

z

TzT

nT ypxpypxppp

pyxp

mA

221 2222

l=<Lcan>/2mcN T=pV1+l2 T=pV1+l2

the single particle amplitude isindependent from the beam

we can use this variable to characterize cooling and transmissionthrough the channel

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profile plot

reg-2 ~ centre of 1st tracker reg-92 ~ centre of 2nd tracker

cooling

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=3.0 cm rad

PZ=200 MeV/c, abs=42 cm

coolingN

2/N

1

N2/

N1

=2.0 cm rad

N2/

N1

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amplitude vs aperture

p = 200

MeV/c R (cm) (cm) Anmax (cm)

Absorber 15 42 10.1

RF 21 110 7.6

Tracker 15 33 12.9 AnMAX = p/mc R2/

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

AnMAX = p/mc R2/

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PZ=200 MeV/c, abs=42 cm=0.6cm rad

=1.0cm rad

transmission through MICEstep VI

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MICE STEP VI ~90m of MICE Channel

RF ABS tracker

A (m rad) A (m rad)

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AnMAX

physical aperture R

we can define the max allowed amplitude at the end of the channel

useful for the acceleration stage in the NF

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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