m.apollonio/j.cobbmice uk meeting- ral - (9/1/2007) 1 single particle amplitude m. apollonio –...
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m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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Single Particle Amplitude
M. Apollonio – University of Oxford
m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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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/j.cobb MICE UK meeting- RAL - (9/1/2007)
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)(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
m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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2
22
1
2
2
xxxxA
: optical Twiss parameters
1
1
B
xBxA T
1
1
B
xBxA T
m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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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
m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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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’
m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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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
m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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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 xBxA T
p14
... 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.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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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
m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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profile plot
reg-2 ~ centre of 1st tracker reg-92 ~ centre of 2nd tracker
cooling
m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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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
m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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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/
m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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PZ=200 MeV/c, abs=42 cm=0.6cm rad
=1.0cm rad
transmission through MICEstep VI
m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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MICE STEP VI ~90m of MICE Channel
RF ABS tracker
A (m rad) A (m rad)
=2.0cm rad
m.apollonio/j.cobb MICE UK meeting- RAL - (9/1/2007)
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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, allows 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