2.185),
(Φ
��
2
1 (Φ
IV + ρ
�� ) r
4
φ(r, z) = Φ − 4
1
+ ρ) r + 64
+ . . . .
(2.186)
The scalar kinetic momentum is (2.116, 2.186)
p = p − 4
1 p
−1 (Φ
�� + ρ) (1 + Φ) r
2

1 −1 (Φ
IV
4 −
1 −3 (Φ
��
4

+ p
+ ρ
�� ) (1 + Φ) r
p
+ ρ)
2 r + . . . .
64
32
(2.187)
In addition to the space charge, a beam in general has a global
(averaged) space current j. Maxwell’s equation is
v × B = v × (v × A) = j.
(2.188)
In the lab frame, the beam current is primarily along the beam
axis, with the transverse component relatively small. We therefore neglect the transverse components of j and A. It follows that
(2.188):
∂
2 A z 1 ∂A z
j z = −
−
.
(2.189)
∂r 2
r ∂ r
The solution for the axial component of the magnetic vector potential A z arising from the space current is
A z = −
1
4
j z r
2 = −
1
4
p ρ r
2 ,
(2.190)
1 + Φ
where the space current and space charge are related by
p
j z = ρ v z =
ρ.
(2.191)
1 + Φ
Using the earlier approach, we obtain the modified refractive index
(2.147) as follows:
√
m = p 1 + r � 2 + r 2 θ � 2 − rθ
� A θ − A z .
(2.192)
The expression of (2.159) with space charge terms added to (2.192)
gives the result
m = m 0 + m 2 + m 4 + . . .
67
2.5. Axial symmetry