2 - CERN Accelerator School
2 - CERN Accelerator School
2 - CERN Accelerator School
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ε<br />
after fil.<br />
rms<br />
=<br />
ε<br />
rms<br />
⎛<br />
⎜<br />
⎝<br />
l<br />
i<br />
Injection process - VIII<br />
Given the initial condition of the beam end of the transfer<br />
line<br />
x = A b / a sinψ<br />
, p = A a / b cosψ<br />
b / a<br />
2<br />
<<br />
2 2<br />
ri >= < Ai<br />
+ a / b<br />
⎞<br />
⎟<br />
⎠<br />
i<br />
By squaring and averaging over the particles’ particles distribution<br />
><br />
Massimo Giovannozzi CAS - 2-14 14 October 2005 21<br />
l<br />
i<br />
( b / a + a / b)<br />
The emittance after filamentation is given by<br />
ε<br />
F<br />
2<br />
after fil.<br />
rms<br />
=<br />
=<br />
ε<br />
rms<br />
1<br />
⎛<br />
⎜ β l<br />
2 ⎜ β m ⎝<br />
+<br />
F<br />
β m<br />
β<br />
l<br />
i<br />
⎛ α<br />
+<br />
⎜<br />
⎝ β<br />
m<br />
m<br />
αl<br />
⎞<br />
−<br />
⎟<br />
β l ⎠<br />
2<br />
β<br />
m<br />
⎞<br />
β<br />
⎟<br />
l ⎟<br />
⎠
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