mode - CERN Accelerator School
mode - CERN Accelerator School
mode - CERN Accelerator School
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Transverse dynamics - Space<br />
charge<br />
→ Large numbers of particles per bunch ( ~10 10 ).<br />
→ CCoulomb l b repulsion l i between b t particles ti l ( (space charge) h ) plays l an iimportant t t role. l<br />
→ But space charge forces ~ 1/γ 2 disappear at relativistic velocity<br />
E<br />
B<br />
E<br />
r<br />
Force on a particle inside a long bunch<br />
with density n(r) traveling at velocity v:<br />
=<br />
e<br />
2πε r<br />
∫<br />
r<br />
0<br />
n(<br />
r)<br />
r<br />
dr<br />
F = e(<br />
Er<br />
− vBϕ<br />
) = eEr<br />
( 1−<br />
2<br />
2<br />
) = eE ( 1 β )<br />
2 r − =<br />
v<br />
c<br />
B<br />
ϕ<br />
eE<br />
γ<br />
r<br />
2<br />
=<br />
μ ev<br />
2π r<br />
∫<br />
r<br />
0<br />
n(<br />
r)<br />
r<br />
dr<br />
37
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