116
E. Metral et al.
Fig. 4.3 Horizontal (driving) impedance of a cylindrical one-meter long LHC collimator (even
if in reality the LHC collimators are composed of two parallel plates), with b = 2 mm and
ρ = 10 μm. The real part is in red while the imaginary part is in green (note that in the classical
thick-wall regime, the real and imaginary parts are equal). The dashed curves correspond to the
case with a copper coating of 5 μm [1]
through the magneto-resistance and the surface impedance can also increase due to
the anomalous skin effect ([44] and references therein).
In the case of a cavity, an equivalent RLC circuit can be used with R s the
longitudinal shunt impedance, C the capacity and L the inductance. In a real cavity,
these three parameters cannot be separated easily and some other related parameters
are used, which can be measured directly such as the resonance (angular) frequency
ω r = 1/
√
LC, the quality factor Q = R s
√
C/L = R s / (Lω r ) = R s Cω r and
the damping rate α = ω r /(2Q). When Q = 1, the resonator impedance is called
“broad-band”, and this model was extensively used in the past in many analytical
computations. The longitudinal and transverse impedances and wake functions
(with R ⊥ the transverse shunt impedance) are given by [45] (see Fig. 4.4):
Z
m (ω) =
R s
1+jQ
ω
ωr −
ωr
ω
, W
m (t) =
ω r R s
Q e −αt
cos (ω r t) −
α
ω r
sin (ω r t)
Z ⊥
m (ω) =
ω r
ω
R ⊥
1+jQ
ω
ωr −
ωr
ω
, W ⊥
m (t) =
ω 2
r R ⊥
Qω r
e −αt sin (ω r t) , ω r = ω r
1 −
1
4Q 2 .
(4.23)
Finally, all the transverse impedances (dipolar or driving and quadrupolar or
detuning) should be weighted by the betatron function at the location of the
impedances, as this is what matters for the effect on the beam, i.e. for the beam
dynamics. Furthermore, all the weighted impedances can be summed and lumped
at a single location around the ring (as the betatron phase advance does not play a
role [46]) such that the transverse impedance-induced instabilities can be studied
by considering only one interaction per turn, as it was confirmed in the past
by performing macro-particle tracking simulations and comparing the cases of
E. Metral et al.
Fig. 4.3 Horizontal (driving) impedance of a cylindrical one-meter long LHC collimator (even
if in reality the LHC collimators are composed of two parallel plates), with b = 2 mm and
ρ = 10 μm. The real part is in red while the imaginary part is in green (note that in the classical
thick-wall regime, the real and imaginary parts are equal). The dashed curves correspond to the
case with a copper coating of 5 μm [1]
through the magneto-resistance and the surface impedance can also increase due to
the anomalous skin effect ([44] and references therein).
In the case of a cavity, an equivalent RLC circuit can be used with R s the
longitudinal shunt impedance, C the capacity and L the inductance. In a real cavity,
these three parameters cannot be separated easily and some other related parameters
are used, which can be measured directly such as the resonance (angular) frequency
ω r = 1/
√
LC, the quality factor Q = R s
√
C/L = R s / (Lω r ) = R s Cω r and
the damping rate α = ω r /(2Q). When Q = 1, the resonator impedance is called
“broad-band”, and this model was extensively used in the past in many analytical
computations. The longitudinal and transverse impedances and wake functions
(with R ⊥ the transverse shunt impedance) are given by [45] (see Fig. 4.4):
Z
m (ω) =
R s
1+jQ
ω
ωr −
ωr
ω
, W
m (t) =
ω r R s
Q e −αt
cos (ω r t) −
α
ω r
sin (ω r t)
Z ⊥
m (ω) =
ω r
ω
R ⊥
1+jQ
ω
ωr −
ωr
ω
, W ⊥
m (t) =
ω 2
r R ⊥
Qω r
e −αt sin (ω r t) , ω r = ω r
1 −
1
4Q 2 .
(4.23)
Finally, all the transverse impedances (dipolar or driving and quadrupolar or
detuning) should be weighted by the betatron function at the location of the
impedances, as this is what matters for the effect on the beam, i.e. for the beam
dynamics. Furthermore, all the weighted impedances can be summed and lumped
at a single location around the ring (as the betatron phase advance does not play a
role [46]) such that the transverse impedance-induced instabilities can be studied
by considering only one interaction per turn, as it was confirmed in the past
by performing macro-particle tracking simulations and comparing the cases of