4 Impedance and Collective Effects
111
of the centre of mass):
F x = Λ c x, F y = Λ c y, Λ c =
λe
2πε 0 γ 2 b 2 .
(4.6)
It can be seen that the transverse coherent space charge force of Eq. (4.6) is
similar to the transverse incoherent space charge force of Eq. (4.4, left): 2σ 2 has
been replaced by b 2 .
The same analysis can be performed in the case of two infinite (horizontal)
parallel plates spaced by 2h and the results are the following (assuming that the
transverse beam sizes are much smaller than h, assuming only the “ac” magnetic
part and keeping only the linear terms)
F x = Λ c
π 2
24
x −
π 2
24
x
, F y = Λ c
π 2
12
y +
π 2
24
y
.
(4.7)
Therefore, compared to the circular case, the coherent force is smaller by
π 2 /24 ≈ 0.4 in the horizontal plane and π 2 /12 ≈ 0.8 in the vertical one. Furthermore,
there is a second incoherent (or quadrupolar, as it is linear with the particle
position) term with opposite sign in both planes. The coefficients are linked to
the Laslett coefficients usually used in the literature [23], and they are the same
as the ones obtained by Yokoya [24] in the case of a resistive beam pipe under
some assumptions (see Sect. 4.2). General formulae exist for the “real” tune shifts
of coasting or bunched beams in pipes with different geometries, considering both
the “ac” and “dc” magnetic parts and can be found for instance in Refs. [6, 7].
4.2 Wake Fields and Impedances
E. Metral
If the wall of the beam pipe is perfectly conducting and smooth, as it was the
case in the previous section, a ring of negative charges is formed on the walls of
the beam pipe where the electric field ends, and these induced charges travel at the
same pace with the particles, creating the so-called “image” (or induced) current,
which leads to real tune shifts. However, if the wall of the beam pipe is not perfectly
conducting or contains discontinuities, the movement of the induced charges will
be slowed down, thus leaving electromagnetic fields (which are proportional to the
beam intensity) mainly behind: this is why these electromagnetic fields are called
wake fields. The latter will create complex tune shifts leading to instabilities (see
Sect. 4.3). What needs to be computed are the wake fields at the distance z = s − vt
behind the source particle (which is at position s source = vt; with this convention,
one has z < 0) and their effects on the test or witness particles that compose the
111
of the centre of mass):
F x = Λ c x, F y = Λ c y, Λ c =
λe
2πε 0 γ 2 b 2 .
(4.6)
It can be seen that the transverse coherent space charge force of Eq. (4.6) is
similar to the transverse incoherent space charge force of Eq. (4.4, left): 2σ 2 has
been replaced by b 2 .
The same analysis can be performed in the case of two infinite (horizontal)
parallel plates spaced by 2h and the results are the following (assuming that the
transverse beam sizes are much smaller than h, assuming only the “ac” magnetic
part and keeping only the linear terms)
F x = Λ c
π 2
24
x −
π 2
24
x
, F y = Λ c
π 2
12
y +
π 2
24
y
.
(4.7)
Therefore, compared to the circular case, the coherent force is smaller by
π 2 /24 ≈ 0.4 in the horizontal plane and π 2 /12 ≈ 0.8 in the vertical one. Furthermore,
there is a second incoherent (or quadrupolar, as it is linear with the particle
position) term with opposite sign in both planes. The coefficients are linked to
the Laslett coefficients usually used in the literature [23], and they are the same
as the ones obtained by Yokoya [24] in the case of a resistive beam pipe under
some assumptions (see Sect. 4.2). General formulae exist for the “real” tune shifts
of coasting or bunched beams in pipes with different geometries, considering both
the “ac” and “dc” magnetic parts and can be found for instance in Refs. [6, 7].
4.2 Wake Fields and Impedances
E. Metral
If the wall of the beam pipe is perfectly conducting and smooth, as it was the
case in the previous section, a ring of negative charges is formed on the walls of
the beam pipe where the electric field ends, and these induced charges travel at the
same pace with the particles, creating the so-called “image” (or induced) current,
which leads to real tune shifts. However, if the wall of the beam pipe is not perfectly
conducting or contains discontinuities, the movement of the induced charges will
be slowed down, thus leaving electromagnetic fields (which are proportional to the
beam intensity) mainly behind: this is why these electromagnetic fields are called
wake fields. The latter will create complex tune shifts leading to instabilities (see
Sect. 4.3). What needs to be computed are the wake fields at the distance z = s − vt
behind the source particle (which is at position s source = vt; with this convention,
one has z < 0) and their effects on the test or witness particles that compose the