254
An Introduction to Beam Physics
of freedom and the longitudinal map is
l f = l i + M l (δ i ) ,
δ f =
K 0
K 0 + qE 0 LT cos (φ 0 )
δ i +
qE 0 LT
K 0 + qE 0 LT cos (φ 0 )
·
cos
φ 0 +
ω
κ
l f
− cos (φ 0 )
.
Next, we keep only the linear terms in the map and solve for the oscillation
frequency. The linear map is
l f =l i + (l|δ)δ i ,
δ f =
K 0
K 0 + qE 0 LT cos (φ 0 )
δ i
+
qE 0 LT
K 0 + qE 0 LT cos (φ 0 )
ω
κ
(l i + (l|δ)δ i ) sin (φ 0 ) .
Writing in matrix form, we have
l f
δ f
=
1 (l|δ)
M δl M δδ
l i
δ i
,
where
M δl =
qE 0 LT
K 0 + qE 0 LT cos (φ 0 )
ω
κ
sin (φ 0 ) ,
M δδ =
1
K 0 + qE 0 LT cos (φ 0 )
K 0 + qE 0 LT
ω
κ
(l|δ) sin (φ 0 )
.
It is easy to verify that the determinant of the matrix is
K 0
K 0 + qE 0 LT cos (φ 0 )
,
which entails that the motion is non-symplectic when qE 0 LT cos(φ 0 ) = 0.
Furthermore, the longitudinal emittance of the beam decreases as the beam is
accelerated. It is obvious that the same is true for the transverse emittance,
which is called adiabatic damping. By redefining the relative energy deviation
as
δ =
ΔK
K 0
,
we obtain the new matrix
l f
δ f
=
1 (l|
δ)
M
δl M
δ δ
l i
δ i
,