The Linearization of the Equations of Motion
89
Since the case of rotationally symmetric s-dependent potential is not covered by the assumption used for the linearized equations of motion (4.2) derived in the beginning of this chapter, we go back to the general equations of
motion (3.22) in Chapter 3. In this case, h = 0, and hence the set of equations
is given by
x
=
p 0
p s
a,
a
=
1 + η
1 + η 0
E x
χ e0
p 0
p s
−
B y
χ m0
+
B s
χ m0
p 0
p s
b,
y
=
p 0
p s
b,
b
=
1 + η
1 + η 0
E y
χ e0
p 0
p s
+
B x
χ m0
−
B s
χ m0
p 0
p s
a,
l
=
1 + η
1 + η 0
p 0
p s
− 1
κ
v 0
,
δ
= 0,
(4.8)
where
η = η 0 (1 + δ) −
ZeV
mc 2 ,
p 0
p s
=
η (2 + η)
η 0 (2 + η 0 )
− a
2
− b
2
−1/2
,
κ
v 0
= −
1 + η 0
2 + η 0
,
χ m0 =
p 0
Ze
,
χ e0 =
p 0 v 0
Ze
.
In the following subsections, we study the two important regularly employed
classes of lenses, the magnetic round lens and the electric round lens. As we
will see, different from the cases of the electric and magnetic quadrupoles,
their focusing properties arise from different mechanisms, and the magnetic
and electric round lenses require a quite different treatment.
4.4.1 The Electrostatic Round Lens
Electrostatic round lenses are arrangements of metallic electrodes with rotational symmetry that act as equipotential surfaces and thus determine the
on-axis potential. Electrostatic round lenses are frequently used in electron
microscopes and also in the shaping of low-energy beams near the source. For
a rotationally symmetric electrostatic lens, we have the electric field, linear in
x and y, given as
E x =
1
2
V
0 (s) x, E y =
1
2
V
0 (s) y, E s = −V
0 (s) ,
and the corresponding electrostatic potential
V = V 0 (s) −
1
4
V
0 (s)
x
2 + y
2
.
89
Since the case of rotationally symmetric s-dependent potential is not covered by the assumption used for the linearized equations of motion (4.2) derived in the beginning of this chapter, we go back to the general equations of
motion (3.22) in Chapter 3. In this case, h = 0, and hence the set of equations
is given by
x
=
p 0
p s
a,
a
=
1 + η
1 + η 0
E x
χ e0
p 0
p s
−
B y
χ m0
+
B s
χ m0
p 0
p s
b,
y
=
p 0
p s
b,
b
=
1 + η
1 + η 0
E y
χ e0
p 0
p s
+
B x
χ m0
−
B s
χ m0
p 0
p s
a,
l
=
1 + η
1 + η 0
p 0
p s
− 1
κ
v 0
,
δ
= 0,
(4.8)
where
η = η 0 (1 + δ) −
ZeV
mc 2 ,
p 0
p s
=
η (2 + η)
η 0 (2 + η 0 )
− a
2
− b
2
−1/2
,
κ
v 0
= −
1 + η 0
2 + η 0
,
χ m0 =
p 0
Ze
,
χ e0 =
p 0 v 0
Ze
.
In the following subsections, we study the two important regularly employed
classes of lenses, the magnetic round lens and the electric round lens. As we
will see, different from the cases of the electric and magnetic quadrupoles,
their focusing properties arise from different mechanisms, and the magnetic
and electric round lenses require a quite different treatment.
4.4.1 The Electrostatic Round Lens
Electrostatic round lenses are arrangements of metallic electrodes with rotational symmetry that act as equipotential surfaces and thus determine the
on-axis potential. Electrostatic round lenses are frequently used in electron
microscopes and also in the shaping of low-energy beams near the source. For
a rotationally symmetric electrostatic lens, we have the electric field, linear in
x and y, given as
E x =
1
2
V
0 (s) x, E y =
1
2
V
0 (s) y, E s = −V
0 (s) ,
and the corresponding electrostatic potential
V = V 0 (s) −
1
4
V
0 (s)
x
2 + y
2
.